Super Cartan Geometry and the Super Ashtekar Connection

Abstract

This work is devoted to the geometric approach to supergravity. More precisely, we interpret $\mathcal {N}=1$, $D=4$ supergravity as a super Cartan geometry which provides a link between supergravity and Yang–Mills theory. To this end, we first review important aspects of the theory of supermanifolds and we establish a link between various different approaches. We then introduce super Cartan geometries using the concept of so-called enriched categories. This, among other things, will enable us to implement anticommutative fermionic fields. We will then also show that non-extended $D=4$ supergravity naturally arises in this framework. Finally, using this gauge-theoretic interpretation as well as the chiral structure of the underlying supersymmetry algebra, we will derive graded analoga of Ashtekar’s self-dual variables and interpret them in terms of generalized super Cartan connections. This gives canonical chiral supergravity the structure of a Yang–Mills theory with gauge supergroup similar to the self-dual variables in ordinary first-order Einstein gravity. We then construct the parallel transport map corresponding to the super connection in mathematical rigorous way using again enriched categories. This provides the possibility of quantizing gravity and matter degrees of freedom in loop quantum gravity in a unified way. Kindly check and confirm inserted city and country name are correctly identified.City and country name are correct.

1 Introduction

Soon after the first discovery of supergravity by Freedman, Ferrara and van Nieuwenhuizen [1] in 1976, Ne’eman and Regge [79] studied a new geometric approach to supergravity based on the ideas of Cartan of a purely geometric interpretation of gravity.¹ In this theory, now commonly known as Cartan geometry, gravity arises by considering the underlying symmetry groups of flat Minkowski spacetime, i.e., a Klein geometry consisting of the isometry group given by the Poincaré group and the Lorentz group as stabilizer subgroup of a particular spacetime event. Gravity is then obtained by deforming this flat initial data in a particular way by studying a certain kind of connection forms, called Cartan connections, taking values in the Lie algebra of the isometry group of the flat model. This Cartan geometric approach gives gravity a very clear geometric interpretation and even allows the inclusion of matter fields via Kaluza–Klein reduction of higher-dimensional pure gravity theories leading for instance to Einstein–Yang–Mills theories. However, it still has some limitations, as, for instance, it does not include fermionic fields. This changes in case of supersymmetry, as graded Lie algebras, by definition, naturally include fermionic generators. It was then realized extending Cartan geometry to the super category that this in fact leads to supergravity. The fermion field, given by the superpartner of the graviton field, then arises from the odd components of a super Cartan connection taking values in the graded extension of the Poincaré algebra. Besides, this description also yields a geometric interpretation of supersymmetry transformations in terms of local gauge transformations along the odd directions of the underlying (graded) structure group.

These ideas were studied more systematically and developed even further by D’Auria and Fré et al. [2, 3] to include extended and higher-dimensional supergravity theories. Moreover, generalizing the Maurer–Cartan equations to include higher p-form gauge fields which naturally appear in higher dimensions, such as the supergravity C-field in the unique maximal $D=11$, $\mathcal {N}=1$ supergravity theory, then lead to the concept of free graded differential algebras (FDA). These types of algebras then turned out to have a rigorous geometric interpretation in higher category theory describing the higher gauge fields as components of a higher Cartan connection [4, 5].

In the canonical approach to (quantum) supergravity such as, most prominently, in the framework of loop quantum gravity (LQG) (see, e.g., [6, 7] and references therein), this naturally raises the question whether at least some remnant of the underlying geometrical structure of the full theory still can be seen in the resulting canonical theory. In fact, it was observed by Fülöp in the seminal paper [8] while studying a specific subfamily of the constraints of canonical $\mathcal {N}=1$, $D=4$ anti-de Sitter supergravity that the corresponding constraint algebra has the structure of a graded Lie algebra leading to some kind of a graded generalization of Ashtekar’s self-dual variables. Using this graded connection, Gambini et al. [9] as well as Ling and Smolin [10] then studied the possibility of generalizing the notion of spin network states to super spin networks.

In this paper, we will study this idea more systematically using the strong link between supergravity and Cartan geometry. We will therefore provide a mathematical rigorous account to the formulation of a super Cartan geometry. However, the problem of modeling anticommuting classical fermion fields, which is crucial in the context of supersymmetry, turns out be by far non-straightforward. This seems to be usually ignored in the physical literature. Nevertheless, motivated from algebraic geometry, this problem has a beautiful resolution using the concept of enriched categories first studied already by Schmitt [11]. We will apply these ideas to our situation and then see that the resulting picture resembles very closely the description of fermion fields in mathematical rigorous approaches to quantum field theory involving anticommuting classical fermion fields such as perturbative algebraic quantum field theory (pAQFT) [12, 13]. We will then see that non-extended $D=4$ supergravity arises naturally in this framework. For an interesting approach which is different from the present one, using the notion of ’integral forms’ see [14,15,16].

Based on this geometric formulation, we will be able to interpret the graded analog of Asthekar’s self-dual variables in terms of generalized super Cartan connections and give a conceptual explanation for the observation of Fülöp. As it turns out, this connection appears very naturally when studying the chiral structure of the underlying supersymmetry algebra and is rooted on the special property of the (bosonic) self-dual variables. Moreover, as we will see, this particular structure of the supersymmetry algebra even survives in case of extended supersymmetry which shows that the existence of the super Ashtekar connection is not just a mere coincidence.

Another advantage of deriving the super Ashtekar connection using the strong relation between supergravity and super Cartan connection is that this naturally leads to an interpretation of the canonical theory as a Yang–Mills theory with supergroup as a gauge group. This is in fact in complete analogy to the classical theory since the self-dual variables give first-order general relativity the structure of a $\textrm{SL}(2,\mathbb {C})$ Yang–Mills theory. We then use this connection in order to construct the corresponding parallel transport map using again the concept of enriched categories. These results are of course of independent interest and may even have applications in other areas of physics that involve super gauge fields.

As a final step, we will then use these holonomies in order to construct the state space of loop quantum supergravity based on the ideas of [9, 10]. Therefore, we will also derive the Haar measure of the underlying gauge supergroup. As it turns out, this gives the resulting Hilbert space a very intriguing structure which also uncovers some of the mathematical beauty underlying the standard quantization of fermion fields in LQG as proposed in [17, 18].

The structure of this paper is as follows: At the beginning, we will provide a detailed account of the concept of supermanifolds and establish a link between the various different approaches to this subject. We will then recall some basic super Lie groups which play a fundamental role in supersymmetry and supergravity. At the end of this section, we summarize the construction of the super holonomies as considered in detail in [19] and which is based on the concept of enriched categories in order to describe anticommuting classical fermion fields.

In Sect. 3, we describe the geometrical interpretation of gravity in terms of a Cartan geometry. We will then define super Cartan geometries in the framework of enriched categories in Sect. 4. Finally, this formulation will be used in order to derive $\mathcal {N}=1$, $D=4$ (anti-de Sitter) supergravity via the super MacDowell–Mansouri action.

With all these ingredients, the graded Ashtekar connection will be derived in Sect. 5 studying the chiral structure of the underlying (extended) super Poincaré/anti-de Sitter group. Using the relation to Yang–Mills theory, we will then use the super holonomies to construct the Hilbert space in the manifest approach to loop quantum supergravity. In particular, we will derive the invariant Haar measure of the gauge supergroup and describe the link to the standard quantization scheme of fermion fields in LQG. Finally, the generalization of the concept of invariant connections to the super category will be considered. This provides a mathematical solid basis to apply these results to symmetry reduced models in the framework of supersymmetric LQC which will be studied in more detail in [20].

2 Supersymmetry and Supergeometry

2.1 Three Roads Toward a Theory of Supermanifolds

In the literature, there exist various different approaches to formulate the notion of a supermanifold. Probably, the most popular one is the so-called algebro-geometric approach introduced by Berezin et al. [21, 22]. As can be already inferred from its name, this approach borrows techniques from algebraic geometry and starts formulating supermanifolds based on the observation that ordinary smooth manifolds can be equivalently be described in terms of the function sheaf defined on it. Albeit being very elegant, its definition is very abstract making it less accessible for physicists for concrete applications. Roughly speaking, this is due to the lack of points as points in this framework are implicitly encoded in the underlying structure sheaf.

Hence, another approach to supermanifolds, the so-called concrete approach, was initiated by DeWitt [23] and Rogers [24] defining them similar to ordinary smooth manifolds in terms of a topological space of points, i.e., a topological manifold that locally looks a flat superspace. However, as it turns out, this definition has various ambiguities in formulating the notion of a point yielding too many unphysical degrees of freedom.

It was then found by Molotkov [25] and further developed by Sachse [26] that both approaches are in fact two sides of the same coin. More precisely, as will be explained in more detail in what follows, it was shown that Rogers–DeWitt supermanifolds can be interpreted in terms of a functor constructed out of a algebro-geometric supermanifold. This functorial interpretation then resolved the ambiguities arising in the Rogers–De Witt approach and also opens the way for a generalization of the theory to infinite-dimensional supermanifolds.

In the following, we want to introduce the Berezin–Kostant–Leites approach to supermanifold theory and explain in some detail its relation to the Rogers–DeWitt approach using the functor of points which naturally leads to their functorial interpretation as observed by Molotkov and Sachse.

The Berezin–Kostant–Leites approach is based on the observation that ordinary smooth manifolds can equivalently be described in terms of locally ringed spaces. Therefore note that any smooth manifold canonically yields the locally ringed space $(M,C^{\infty }_M)$ which is locally isomorphic to some $(V,C^{\infty }_{\mathbb {R}^n}|_V)$ with $V\subseteq \mathbb {R}^n$ open.

In fact, it turns that all smooth manifold M can be described this way. That is, if $(M,\mathcal {O}_M)$ is a locally ringed space with $\mathcal {O}_M$ a sheaf on M such that $(M,\mathcal {O}_M)$ is locally isomorphic to some $(V,C^{\infty }_{\mathbb {R}^n})$ with $V\subseteq \mathbb {R}^n$ open. Then, M can be given the structure of smooth manifold in a unique way such that $\mathcal {O}_M\cong C^{\infty }_M$. Even more, it follows that both categories are equivalent.

Based on this idea, one defines supermanifolds as some sort of locally super ringed spaces generalizing appropriately the notion of a smooth function. Therefore, a so-called supersmooth function or superfield f on the superspace $\mathbb {R}^{m|n}=\mathbb {R}^m\oplus \mathbb {R}^n$ is defined as a function of the form.

$$\begin{aligned} f=\sum _{I\in M_n}{f_I\theta ^I} \end{aligned}$$

(1)

with $f_I$ ordinary smooth functions on $\mathbb {R}^m$ for any multi-index $I=(i_1,\ldots ,i_k)\in M_n$, $0\le |I|=k\le n$, where $\theta ^I:=\theta ^{i_1}\ldots \theta ^{i_k}$. In the following, we will follow very closely [27] for the definition of algebro-geometric supermanifolds and the construction of the functor of points (see also Appendix A for our choice of conventions in super linear algebra). Therefore, we will omit most of the proofs.

Definition 2.1

An algebro-geometric supermanifold of dimension (m, n) is a locally super ringed space $\mathcal {M}=(M,\mathcal {O}_{\mathcal {M}})$ that is locally isomorphic to the superspace $\mathbb {R}^{m|n}$. More precisely, $(M,\mathcal {O}_M)$ consists of a topological space M which is Hausdorff and second countable as well as a sheaf $\mathcal {O}_{\mathcal {M}}$ over M of super commutative rings called structure sheaf such that, for any $x\in M$, the stalk $\mathcal {O}_{M,x}$ is a local super ring. Moreover, for $x\in M$, there exists an open neighborhood $U\subset M$ of x as well as an isomorphism $\phi _U=(|\phi _U|,\phi _U^{\sharp })$ of ordinary locally ringed spaces²

$$\begin{aligned} \phi _U=(|\phi _U|,\phi _U^{\sharp }):\,(U,\mathcal {O}_{\mathcal {M}}|_U)\rightarrow \left( |\phi _U|(U),C^{\infty }_{\mathbb {R}^m}|_{|\phi _U|(U)}\otimes \bigwedge {[\theta ^1,\ldots ,\theta ^n]}\right) \nonumber \\ \end{aligned}$$

(2)

such that $\phi _U^{\sharp }:\,C^{\infty }_{\mathbb {R}^m}|_{|\phi _U|(U)}\otimes \bigwedge {[\theta ^1,\ldots ,\theta ^n]}\rightarrow \mathcal {O}_{\mathcal {M}}|_U$, in addition, is a (even) morphism of sheaves of super algebras. The tuple $(U,\phi _U)$ is called a local chart or superdomain of x. A family $\{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in \Lambda }$ of charts is called an atlas of $(M,\mathcal {O}_{\mathcal {M}})$ if $\bigcup _{\alpha \in \Lambda }{U_{\alpha }}=M$. A morphism of $f=(|f|,f^{\sharp }):\,(M,\mathcal {O}_{\mathcal {M}})\rightarrow (N,\mathcal {O}_{\mathcal {N}})$ of algebro-geometric supermanifolds is a morphism of the underlying ordinary locally ringed spaces such that $f^{\sharp }:\,\mathcal {O}_{\mathcal {N}}\rightarrow f_{*}\mathcal {O}_{\mathcal {M}}$ also is an (even) morphism of super algebras. Algebro-geometric supermanifolds together with morphisms between them form a category $\textbf{SMan}_{\textrm{Alg}}$ called the category of algebro-geometric supermanifolds.

Remark 2.2

Choosing a chart $(U,\phi _U)$ of an algebro-geometric supermanifold $(M,\mathcal {O}_{\mathcal {M}})$, this induces a local coordinates $(t^{\sharp i},\theta ^{\sharp j})$ on $\mathcal {M}|_U:=(U,\mathcal {O}_{\mathcal {M}}|_U)$ via $t^{\sharp i}:=\phi _U^{\sharp }(t^i)$ and $\theta ^{\sharp j}:=\phi _U^{\sharp }(\theta ^j)$ $\forall i=1,\ldots ,m$, $j=1,\ldots ,m$ where $\textrm{dim}(M,\mathcal {O}_{\mathcal {M}})=(m,n)$. Moreover, any function $f\in \mathcal {O}_{\mathcal {M}}|_U$ is of the form

$$\begin{aligned} f=\sum _{I\in M_n}{f_I\theta ^{\sharp I}} \end{aligned}$$

(3)

where, for $I=(i_1,\ldots ,i_k)\in M_n$, $\theta ^{\sharp I}:=\theta ^{\sharp i_1}\cdots \theta ^{\sharp i_n}$ and $f_I=\phi _U^{\sharp }(g_I)$ for some smooth function $g_I\in C^{\infty }(|\phi _U|(U))$.

Any supermanifold naturally contains an ordinary smooth manifold as a submanifold. Therefore, for any algebro-geometric supermanifold $\mathcal {M}\!=\!(M,\mathcal {O}_{\mathcal {M}})$ and $U\subset M$ open, consider the set $\mathcal {J}_{\mathcal {M}}(U):=\{f\in \mathcal {O}_{\mathcal {M}}(U)|\,f\text { is nilpotent}\}$. Then, it follows that $\mathcal {J}_{\mathcal {M}}(U)$ is an ideal in $\mathcal {O}_{\mathcal {M}}(U)$ yielding another sheaf $U\mapsto \mathcal {J}_{\mathcal {M}}(U)$. Hence, one can construct the quotient sheaf $\mathcal {O}_{\mathcal {M}}/\mathcal {J}_{\mathcal {M}}$ whose sections locally have the structure of an ordinary smooth functions. Hence, this yields a locally ringed space

$$\begin{aligned} M_0:=(M,\mathcal {O}_{\mathcal {M}}/\mathcal {J}_{\mathcal {M}}) \end{aligned}$$

(4)

which is a submanifold and has the structure of an ordinary smooth manifold. Before we continue, let us mention a central result in the theory of algebro-geometric supermanifolds as it will appear quite frequently in the discussion in what follows. It states that morphisms are uniquely characterized via the pullback of a basis of global sections.

Theorem 2.3

(Global Chart Theorem [27]). Let $\mathcal {M}$ be an algebro-geometric supermanifold and $\mathcal {U}^{m|n}=(U,C^{\infty }_U)\subseteq \mathbb {R}^{m|n}$ be a superdomain with $U\subseteq \mathbb {R}^{m}$ open. There is a bijectice correspondence between supermanifold morphisms $\psi :\,\mathcal {M}\rightarrow \mathcal {U}^{m|n}$ and tuples $(t^{\sharp i},\theta ^{\sharp j})$ of global sections of $\mathcal {O}(\mathcal {M})$ with $t^{\sharp i}$ even and $\theta ^{\sharp j}$ odd, $i=1,\ldots ,m$ and $j=1,\ldots ,n$, such that $(t^{\sharp i}(x),\theta ^{\sharp j}(x))\in U$ $\forall x\in M$.

Proof

For a section $f\in \mathcal {O}(\mathcal {M})$, the value $f(x)\equiv \textrm{ev}_x(f)$ of f at $x\in M$ is defined as the unique real number such that $f-f(x)$ is not invertible in any open neighborhood of x in M. It follows that, if $\phi :\,\mathcal {M}\rightarrow \mathcal {N}$ is a morphism of supermanifolds, then $\phi ^{\sharp }(g)(y)=g(|\phi |(y))$ for any $g\in \mathcal {O}(\mathcal {N})$ and $y\in N$. Hence, it is clear, by restricting the global sections $t^i$ and $\theta ^j$ of $\mathbb {R}^{m|n}$ to the superdomain $\mathcal {U}^{m|n}$, that their respective pullback $t^{\sharp i}:=\psi ^{\sharp }(t^i)$ and $\theta ^{\sharp j}:=\psi ^{\sharp }(\theta ^j)$ w.r.t. a morphism $\psi :\,\mathcal {M}\rightarrow \mathcal {U}^{m|n}$ indeed satisfy the properties as stated in the theorem. The inverse direction follows from the local triviality property of supermanifolds. $\square$

Example 2.4

(The split functor). Typical examples of a supermanifolds are obtained via their strong relationship to vector bundles. Let $V\rightarrow E{\mathop {\rightarrow }\limits ^{\pi }}M$ be a real vector bundle over an m-dimensional manifold with typical fiber given by a vector space V of dimensional n. This naturally yields a locally ringed space setting

$$\begin{aligned} \textbf{S}(E,M):=(M,\Gamma (\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}E^*)) \end{aligned}$$

(5)

where $\Gamma (\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}E^*)$ denotes the space of smooth sections of the exterior bundle $\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}E^*$. Since $\Gamma (\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}E^*)\cong \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}\Gamma (E)^*$ naturally carries a $\mathbb {Z}_2$-grading, it follows that it has the structure of a sheaf of local super rings, that is, $\textbf{S}(E,M)$ defines an algebro-geometric supermanifold of dimension (m, n) also called a split supermanifold. A morphism $(\phi ,f):\,(E,M)\rightarrow (F,N)$ between two vector bundles induces a morphism $\textbf{S}(\phi ,f):\,\textbf{S}(E,M)\rightarrow \textbf{S}(F,N)$ between the corresponding split supermanifolds. Hence, this yields a functor

$$\begin{aligned} \textbf{S}:\,\textbf{Vect}_{\mathbb {R}}\rightarrow \textbf{SMan}_{\textrm{Alg}} \end{aligned}$$

(6)

from the category of real vector bundles to the category of algebro-geometric supermanifolds which we call the split functor. In case that the vector bundle $(E,M)=(M\times V,M)$ is trivial, $\textbf{S}(E,M)$ will be called globally split and we also simply write $\textbf{S}(V,M)\equiv \textbf{S}(E,M)$. It is a general result due to Batchelor [28] that any algebro-geometric supermanifold is isomorphic to a split supermanifold of the form (5), i.e., (6) is surjective on objects. However, the split functor is not full, i.e., not every morphism $f:\,\textbf{S}(E,M)\rightarrow \textbf{S}(F,N)$ between split manifolds arises from a morphism between the respective vector bundles $(E,M),(F,N)\in \textbf{Ob}(\textbf{Vect}_{\mathbb {R}})$. Hence, the structure of morphisms between supermanifolds, in general, turns out to be much richer than for ordinary vector bundles. This is of utmost importance in modeling for instance supersymmetry transformations to be discussed in Sect. 4.

As a next step, we want to describe a relation between algebro-geometric and Rogers–DeWitt supermanifolds. A very elegant way in describing this relationship is given by the so-called functors of point approach. It is a general technique in algebraic geometry which can be used in order to give, a priori, very abstract objects a more concrete reinterpretation making proofs much easier in certain instances. As explained already in the introduction to this chapter, a general “issue” concerning algebro-geometric supermanifolds is the lack of points. In fact, in contrast to ordinary smooth manifolds, the points of the underlying topological space do not suffice to uniquely characterize the sections of the structure sheaf. As it turns out, this can be cured by studying the morphisms between them.

Definition 2.5

Let $\mathcal {M}$ be an algebro-geometric supermanifold. The functor of points of $\mathcal {M}$ is defined as the covariant functor $\mathcal {M}:\,\textbf{SMan}_{\textrm{Alg}}^{\textrm{op}}\rightarrow \textbf{Set}$ on the opposite category $\textbf{SMan}_{\textrm{Alg}}^{\textrm{op}}$ associated with $\mathcal {M}$³ which on objects $\mathcal {T}\in \textbf{Ob}(\textbf{SMan}_{\textrm{Alg}})$ is defined as

$$\begin{aligned} \mathcal {M}(\mathcal {T}):=\textrm{Hom}(\mathcal {T},\mathcal {M}) \end{aligned}$$

(7)

also called the $\mathcal {T}$-point of $\mathcal {M}$ and on morphisms $f\in \textrm{Hom}(\mathcal {T},\mathcal {S})$, $\mathcal {M}(f)\in \textrm{Hom}(\mathcal {M}(\mathcal {S}),\mathcal {M}(\mathcal {T}))$ is given by

$$\begin{aligned} \mathcal {M}(f):\,\mathcal {M}(\mathcal {S})\rightarrow \mathcal {M}(\mathcal {T}),\,g\mapsto g\circ f \end{aligned}$$

(8)

Hence, the functor of points of $\mathcal {M}$ coincides with the partial $\textrm{Hom}$-functor $h_\mathcal {M}\equiv \textrm{Hom}(\mathcal {M},\,\cdot \,)$ on $\textbf{SMan}_{\textrm{Alg}}^{\textrm{op}}$.

If $\phi :\,\mathcal {M}\rightarrow \mathcal {N}$ is a morphism between algebro-geometric supermanifolds, this yields a map $\phi _\mathcal {T}:\,\mathcal {M}(\mathcal {T})\rightarrow \mathcal {N}(\mathcal {T})$ between the associates $\mathcal {T}$-points setting $\phi _\mathcal {T}(f):=\phi \circ f$ $\forall f\in \mathcal {M}(\mathcal {T})=\textrm{Hom}(\mathcal {T},\mathcal {M})$. By definition, it then follows that for any morphism $f:\,\mathcal {S}\rightarrow \mathcal {T}$ one has

$$\begin{aligned} \mathcal {N}(f)\circ \phi _\mathcal {T}(g)=\phi _\mathcal {T}(g)\circ f=\phi \circ g\circ f=\phi _\mathcal {S}\circ \mathcal {M}(f)(g) \end{aligned}$$

(9)

$\forall g\in \mathcal {M}(\mathcal {T})$, that is, the following diagram is commutative

Hence, a morphism $\phi :\,\mathcal {M}\rightarrow \mathcal {N}$ induces a natural transformation between the associated functor of points. This poses the question whether all natural transformations arise in this way. This is an immediate consequence of the following well-known lemma.

Lemma 2.6

(Yoneda Lemma). Let $\mathcal {C}$ be a category and $F:\,\mathcal {C}\rightarrow \textbf{Set}$ be a functor. Then, for any object $X\in \textbf{Ob}(\mathcal {C})$, the assignment $\eta \mapsto \eta _X(\textrm{id}_X)$ yields a bijective correspondence between natural transformations $\eta :\,\textrm{Hom}(X,\,\cdot \,)\rightarrow F$ and the set $F(X)\in \textbf{Ob}(\textbf{Set})$.

Applied to our concrete situation, this implies that for the functor of points $h_\mathcal {M}:\,\textbf{SMan}_{\textrm{Alg}}^{\textrm{op}}\rightarrow \textbf{Set}$ and $h_\mathcal {N}:\,\textbf{SMan}_{\textrm{Alg}}^{\textrm{op}}\rightarrow \textbf{Set}$ associated with algebro-geometric supermanifolds $\mathcal {M}$ and $\mathcal {N}$, one has a bijective correspondence between natural transformations between $h_\mathcal {M}$ and $h_\mathcal {N}$ and elements in $\textrm{Hom}(\mathcal {N},\,\cdot \,)(\mathcal {M})=\textrm{Hom}(\mathcal {M},\mathcal {N})$. In particular, the supermanifolds $\mathcal {M}$ and $\mathcal {N}$ are isomorphic iff the associated functor of points are naturally isomorphic.

We next want to find an equivalent description of the $\mathcal {T}$-points of an algebro-geometric supermanifold $\mathcal {M}$ purely in terms of global sections of the structure sheaf $\mathcal {O}_{\mathcal {M}}$. Consider therefore the set $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}$ $(\mathcal {M})):=\textrm{Hom}_{\textbf{SAlg}}(\mathcal {O}(\mathcal {M}),\mathbb {R})$ called the real spectrum of $\mathcal {O}(\mathcal {M}):=\mathcal {O}_{\mathcal {M}}(M)$. Since a morphism $\phi :\,\mathcal {O}(\mathcal {M})\rightarrow \mathbb {R}$ in the real spectrum is always surjective, it follows that the kernel $\textrm{ker}(\phi )$ yields a maximal ideal in $\mathcal {O}(\mathcal {M})$, i.e., an element of the maximal spectrum

$$\begin{aligned} \textrm{MaxSpec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M})):=\{I\subset \mathcal {O}(\mathcal {M})|\,I\text { is a maximal ideal}\} \end{aligned}$$

(10)

In fact, it follows that all maximal ideals in $\mathcal {O}(\mathcal {M})$ are of this form. This is a direct consequence of the super version of the classical “Milnor’s exercise” [27].

Proposition 2.7

(Super Milnor’s exercise). For an algebro-geometric supermanifold $\mathcal {M}$ all the maximal ideals in $\mathcal {O}(\mathcal {M})$ are of the form $\mathfrak {J}_x:=\textrm{ker}(\textrm{ev}_x:\,\mathcal {O}(\mathcal {M})\rightarrow \mathbb {R})$ for some $x\in M$, where $\textrm{ev}_x\in \textrm{Hom}_{\textbf{SAlg}}(\mathcal {O}(\mathcal {M}),\mathbb {R})$ is the evaluation map at x mapping $f\in \mathcal {O}(\mathcal {M})$ to its real value $\textrm{ev}_x(f)\equiv f(x)$.

Proof

Let $I\subset \mathcal {O}_{\mathcal {M}}$ be a maximal ideal. On $M_0:=(M,\mathcal {O}_{\mathcal {M}}/\mathcal {J}_{\mathcal {M}})$ consider the subset $j^{\sharp }(I)\subseteq C^{\infty }(M_0)$ of $C^{\infty }(M_0)$, where $j^{\sharp }:\,\mathcal {O}_{\mathcal {M}}\rightarrow \mathcal {O}_{\mathcal {M}}/\mathcal {J}_{\mathcal {M}}\cong C^{\infty }_{M_0}$ is the pullback of the canonical embedding $j:\,M_0\hookrightarrow \mathcal {M}$. Since $j^{\sharp }$ is a surjective morphism of super rings and $1\notin I$, it follows that $j^{\sharp }(I)$ is a maximal ideal in $C^{\infty }(M_0)$. By the classical Milnor’s exercise, we thus have $j^{\sharp }(I):=\textrm{ker}(\textrm{ev}_x:\,C^{\infty }(M_0)\rightarrow \mathbb {R})$ for some $x\in M$. Hence, $I\subseteq \mathfrak {J}_x$ implying $I=\mathfrak {J}_x$ by maximality of I. $\square$

Hence, according to Proposition 2.7, we can identify the real spectrum with $\textrm{MaxSpec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ and even obtain a bijection $\Psi :\,M{\mathop {\longrightarrow }\limits ^{\sim }}\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ via

$$\begin{aligned}{} & {} M\ni x{\mathop {\mapsto }\limits ^{\sim }}(\textrm{ev}_x:\,\mathcal {O}(\mathcal {M})\rightarrow \mathbb {R})\in \textrm{Hom}(\mathcal {O}(\mathcal {M}),\mathbb {R}){\mathop {\mapsto }\limits ^{\sim }}\textrm{ker}(\textrm{ev}_x)\nonumber \\{} & {} \qquad \in \textrm{MaxSpec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M})) \end{aligned}$$

(11)

We want to define a topology on $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ such that $\Psi$ becomes a homeomorphism. Therefore, note that any section $f\in \mathcal {O}(\mathcal {M})$ canonically induces a morphsim $\phi _f:\,\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))\rightarrow \mathbb {R}$ setting

$$\begin{aligned} \phi _f(\textrm{ev}_x):=\textrm{ev}_x(f)=f(x) \end{aligned}$$

(12)

Hence, let us endow $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ with the Gelfand topology which is defined as the coarsest topology such that the maps $\phi _f$ for all $f\in \mathcal {O}(\mathcal {M})$ are continuous. A basis of this topology is generated by open subsets of the form

$$\begin{aligned} \phi _f^{-1}(B_{\epsilon }(x_0))=\{\textrm{ev}_{x}\in \textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))|\,|(\textrm{ev}_{x}-\textrm{ev}_{x_0})(f)|<\epsilon \} \end{aligned}$$

(13)

for some $f\in \mathcal {O}(\mathcal {M})$ and $B_{\epsilon }(x_0)\subset M$ an open ball of radius $\epsilon$ around $x_0\in M$. It then follows immediately that the map $\Psi :\,M{\mathop {\longrightarrow }\limits ^{\sim }}\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ is continuous w.r.t. this topology, since

$$\begin{aligned} \Psi ^{-1}(\phi _f^{-1}(B_{\epsilon }(x_0)))=|f|^{-1}(B_{\epsilon }(f(x_0))) \end{aligned}$$

(14)

is open in M as $|f|:\,M\rightarrow \mathbb {R}$ is continuous. In fact, $\Psi$ is even a homeomorphism. Therefore, consider a closed subset $X\subseteq M$ and let $\mathfrak {p}_X$ be the ideal in $\mathcal {O}(\mathcal {M})$ defined as the set of all sections $f\in \mathcal {O}(\mathcal {M})$ vanishing on X. Using a partition of unity argument, it can then be seen that

$$\begin{aligned} X=\{x\in M|\,f(x)=0,\,\forall f\in \mathfrak {p}_X\} \end{aligned}$$

(15)

and thus

$$\begin{aligned} \Psi (X)=\bigcap _{f\in \mathfrak {p}_X}\phi _f^{-1}(\{0\}) \end{aligned}$$

(16)

i.e., $\Psi (X)$ is closed in $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ proving that $\Psi$ is indeed a homeomorphism.

Theorem 2.8

Let $\mathcal {M}$ and $\mathcal {N}$ be algebro-geometric supermanifolds. Then, their exists a bijective correspondence between the set $\textrm{Hom}(\mathcal {M},\mathcal {N})$ of morphisms of algebro-geometric supermanifolds and the set $\textrm{Hom}_{\textbf{SAlg}}(\mathcal {O}(\mathcal {N}),\mathcal {O}(\mathcal {M}))$ of superalgebra morphisms between the superalgebras of global sections of the respective structure sheaves.

Proof

(Sketch of Proof). One direction is immediate, i.e., that the pullback of a supermanifold morphism $\phi :\,\mathcal {M}\rightarrow \mathcal {N}$ induces a morphism $\phi ^{\sharp }:\,\mathcal {O}(\mathcal {N})\rightarrow \mathcal {O}(\mathcal {M})$ of the respective structure sheaves. The proof of the inverse direction uses a standard tool in algebraic geometry called localization of rings. See [27] for more details. $\square$

Hence, according to this theorem, in the following, we will identify the $\mathcal {T}$-point $\mathcal {M}(\mathcal {T})$ of an algebro-geometric supermanifold $\mathcal {M}$ with $\textrm{Hom}(\mathcal {O}(\mathcal {M}),\mathcal {O}(\mathcal {T}))$. For instance, let us consider the $\mathcal {T}$-points $\mathbb {R}^{m|n}(\mathcal {T})=\textrm{Hom}(\mathcal {O}(\mathbb {R}^{m|n}),\mathcal {O}(\mathcal {T}))$ of the superspace $\mathbb {R}^{m|n}$. By the Global Chart Theorem 2.3, this set can be identified with

$$\begin{aligned} \mathbb {R}^{m|n}(\mathcal {T})&\cong \{(t^1,\ldots ,t^m,\theta ^1,\ldots ,\theta ^n)|\,t^i\in \mathcal {O}(\mathcal {T})_0,\,\theta ^j\in \mathcal {O}(\mathcal {T})_1\}\nonumber \\&=\mathcal {O}(\mathcal {T})_0^m\oplus \mathcal {O}(\mathcal {T})_1^n=(\mathcal {O}(\mathcal {T})\otimes \mathbb {R}^{m|n})_0 \end{aligned}$$

(17)

For $J(\mathcal {T}):=\{f\in \mathcal {O}(\mathcal {T})|\,\text {f is nilpotent}\}$ the ideal of nilpotent sections of $\mathcal {O}(\mathcal {T})$, this yields the canonical projection $\epsilon :\,\mathcal {O}(\mathcal {T})\rightarrow \mathcal {O}(\mathcal {T})/J(\mathcal {T})\cong C^{\infty }(T_0)$ with $T_0$ defined via (4) which can be extended to the body map

$$\begin{aligned} \epsilon _{m,n}:\,(\mathcal {O}(\mathcal {T})\otimes \mathbb {R}^{m|n})_0\rightarrow C^{\infty }(T_0)^m \end{aligned}$$

(18)

In the following, we want to restrict to a subclass of supermanifolds $\mathcal {T}\in \textbf{SMan}_{\textrm{Alg}}$ for which $C^{\infty }(T_0)\cong \mathbb {R}$, i.e., for which the underlying topological space $T=\{*\}$ just consists of a single point. Hence, it follows that $\mathcal {T}\cong (\{*\},\Lambda _N)=\mathbb {R}^{0|N}$ for some $N\in \mathbb {N}_0$.

Definition 2.9

An algebro-geometric supermanifold $\mathcal {T}$ is called a super point if the underlying topological space T only consists of a single point. The subclass of super points form a full subcategory $\textbf{SPoint}$ of $\textbf{SMan}_{\textrm{Alg}}$ called the category of superpoints.

Proposition 2.10

(see [25, 26]). Let $\textbf{Gr}$ be the category of (finite-dimensional) Grassmann algebras whose objects are given by equivalence classes of Grassmann algebras $\Lambda _N\in \textbf{Ob}(\textbf{Gr})$, $N\in \mathbb {N}_0$, and for $\Lambda _N,\Lambda _{N'}\in \textbf{Ob}(\textbf{Gr})$, $\textrm{Hom}_{\textbf{Gr}}$ $(\Lambda _N,\Lambda _{N'})$ is given by the set of superalgebra morphisms between Grassmann algebras. Then, the assignment

$$\begin{aligned} \textbf{Gr}^{\textrm{op}}\rightarrow \textbf{SPoint},\,\Lambda _N&\mapsto (\{*\},\Lambda _N)\nonumber \\ (\phi :\,\Lambda _N\rightarrow \Lambda _{N'})&\mapsto (\textrm{id}_{\{*\}},\phi ) \end{aligned}$$

(19)

yields an equivalence of categories. $\square$

In the following, we will therefore identify superpoints with finite-dimensi onal Grassmann algebras. From (17), it follows for $N\in \mathbb {N}_0$

$$\begin{aligned} \mathbb {R}^{m|n}(\Lambda _N)\cong (\Lambda _N\otimes \mathbb {R}^{m|n})_0=:\Lambda ^{m,n}_{N} \end{aligned}$$

(20)

with $\Lambda ^{m,n}_{N}$ the superdomain of dimension (m, n). We equip $\Lambda ^{m,n}_{N}$ with the coarsest topology such that the body map $\epsilon _{m,n}:\,\Lambda ^{m,n}_{N}\rightarrow \mathbb {R}^m$ is continuous, the so-called DeWitt topology. Hence, in this way, it follows that $\mathbb {R}^{m|n}(\Lambda _N)$ can be identified with a trivial supermanifold in the sense of Rogers–DeWitt.

2.2 Algebro-geometric and $H^{\infty }$ Supermanifolds: An Equivalence of Categories

With these preliminaries, in the following, we are ready to describe a concrete link between the algebro-geometric and Rogers–DeWitt approach using the functor of points technique. To this end, we first show that smooth functions on $\mathbb {R}^{m|n}(\Lambda _N)\cong \Lambda ^{m,n}_{N}$ can be described in terms of natural transformations between functor of points.

More precisely, by the Global Chart Theorem 2.3, a section $f\in \mathcal {O}(\mathbb {R}^{m|n})$ can be identified with a morphism $f:\,\mathbb {R}^{m|n}\rightarrow \mathbb {R}^{1|1}$. According to (9), this in turn induces a natural transformation $f_\mathcal {T}:\,\mathbb {R}^{m|n}(\mathcal {T})\rightarrow \mathbb {R}^{1|1}(\mathcal {T})$ between the respective functor of points. Sticking to Grassmann algebras, we want to find an explicit form of $f_{\Lambda _N}$. To this end, let $(x,\xi )\in \Lambda ^{m,n}_{N}$ which we can identify with a morphism $g:\,\mathbb {R}^{0|N}\rightarrow \mathbb {R}^{m|n}$ such that $g^{\sharp }(t^i)=x^i$ and $g^{\sharp }(\theta ^j)=\xi ^j$ $\forall i=1,\ldots ,m$, $j=1,\ldots ,n$. It then follows again from Theorem 2.3 that $f_{\Lambda _N}(x,\xi )$ can be identified with an element of $\Lambda _N^{1,1}\equiv \Lambda _N$ whose even and odd part is given by $g^{\sharp }(f^{\sharp }(t))$ and $g^{\sharp }(f^{\sharp }(\theta ))$, respectively, where t and $\theta$ denote the global sections of $\mathcal {O}(\mathbb {R}^{1|1})$. Thus, expanding $f=\sum _{\underline{I}}{f_{\underline{I}}\theta ^{\underline{I}}}$, this yields

$$\begin{aligned} f_{\Lambda _N}(x,\xi )&=g^{\sharp }(f^{\sharp }(t))+g^{\sharp }(f^{\sharp }(\theta ))=\sum _{\underline{I},\underline{J}}{\frac{1}{\underline{J}!}\partial _{\underline{J}}f_{\underline{I}}\big {|}_{|g|}s(g^{\sharp }(t))^{\underline{J}}(g^{\sharp }(\theta ))^{\underline{I}}}\nonumber \\&=\sum _{\underline{I},\underline{J}}{\frac{1}{\underline{J}!}\partial _{\underline{J}}f_{\underline{I}}(\epsilon _{m,n}(x))s(x)^{\underline{J}}\xi ^{\underline{I}}}=:\sum _{\underline{I}}{\textbf{G}(f_{\underline{I}})(x)\xi ^{\underline{I}}} \end{aligned}$$

(21)

where $s(x):=x-\epsilon _{m,n}(x)$ is the soul map and

$$\begin{aligned} \textbf{G}(f_{\underline{I}})(x):=\sum _{\underline{J}}{\frac{1}{\underline{J}!}\partial _{\underline{J}} f_{\underline{I}}(\epsilon _{m,n}(x))s(x)^{\underline{J}}} \end{aligned}$$

(22)

is called the Grassmann-analytic continuation of $f_{\underline{I}}$ or simply its $\textbf{G}$-extension. Functions of the form (21) are precisely supersmooth functions in the sense of Rogers–DeWitt! In the standard literature, they are also called of class $H^{\infty }$. As a result, $\Lambda ^{m,n}_{N}$ together with functions of the form (21) yields a Rogers–DeWitt or $H^{\infty }$ supermanifold. The assignment

$$\begin{aligned} \textrm{Hom}_{\textbf{SMan}_{\textrm{Alg}}}(\mathbb {R}^{m|n},\mathbb {R}^{1|1})&\rightarrow H^{\infty }(\Lambda ^{m,n}_{N})\nonumber \\ f&\mapsto f_{\Lambda _N} \end{aligned}$$

(23)

is clearly surjective but in general not injective unless $N\ge n$. We next want to extend these considerations from superspaces to arbitrary algebro-geometric supermanifolds. To this end, we make the following definition.

Definition 2.11

For $N\in \mathbb {N}$, the functor $\textbf{H}_N:\,\textbf{SMan}_{\textrm{Alg}}\rightarrow \textbf{Sets}$ is defined on objects $\mathcal {M}\in \textbf{Ob}(\textbf{SMan}_{\textrm{Alg}})$ via

$$\begin{aligned} \textbf{H}_N(\mathcal {M}):=\mathcal {M}(\Lambda _N)=\textrm{Hom}(\mathcal {O}(\mathcal {M}),\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}\mathbb {R}^N) \end{aligned}$$

(24)

and on morphisms $f:\,\mathcal {M}\rightarrow \mathcal {N}$ according to

$$\begin{aligned} \textbf{H}_N(f):\,\textrm{Hom}(\mathcal {O}(\mathcal {M}),\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}\mathbb {R}^N)\rightarrow \textrm{Hom}(\mathcal {O}(\mathcal {N}),\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}\mathbb {R}^N),\,\phi \mapsto \phi \circ f^{\sharp } \end{aligned}$$

(25)

The set $\mathcal {M}(\Lambda _N)$ contains the real spectrum $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))=\textrm{Hom}(\mathcal {O}(\mathcal {M}),$ $\mathbb {R})=\mathcal {M}(\mathbb {R})$ as a proper subset. According to Proposition 2.7 (see also (11)), this set can be identified with M and thus, in particular, naturally inherits a topology. Using this property, we again introduce the DeWitt topology on $\mathcal {M}(\Lambda _N)$ to be coarsest topology such that the projection⁴

$$\begin{aligned} \textbf{B}:\,\mathcal {M}(\Lambda _N)\rightarrow \textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))\cong M,\,\psi \mapsto \epsilon \circ \psi \end{aligned}$$

(26)

is continuous.

Proposition 2.12

Let $U\subseteq M$ be an open subset of the underlying topological space M of an algebro-geometric supermanifold $\mathcal {M}$. Let us identify U via $\Psi :\,M\rightarrow \textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M})),\,x\mapsto \textrm{ev}_x$ with an open subset in the real spectrum. Then, it follows that the open subsets $\textbf{B}^{-1}(U)$ in $\mathcal {M}(\Lambda _N)$ are given by

$$\begin{aligned} \textbf{B}^{-1}(U)=\{\psi :\,\mathcal {O}(\mathcal {M})\rightarrow \Lambda _N|\,\epsilon \circ \psi =\textrm{ev}_x\text { for some }x\in U\} \end{aligned}$$

(27)

In particular, one has $\textbf{B}^{-1}(U)=\mathcal {M}|_U(\Lambda _N)$ with $\mathcal {M}|_U:=(U,\mathcal {O}_{M}|_U)$.

Proof

The first assertion is immediate, since $\psi \in \textbf{B}^{-1}(U)$ if and only if $\epsilon \circ \psi \in \Psi (U)$, i.e., $\epsilon \circ \psi =\textrm{ev}_x$ for some $x\in U$.

To prove the last one, note that, by Theorem 2.8, one can identify a superalgebra morphism $\psi :\,\mathcal {O}(\mathcal {M})\rightarrow \Lambda _N$ with the pullback of a supermanifold morphism $\phi :=(|\phi |,\phi ^{\sharp }):\,\mathbb {R}^{0|N}=(\{*\},\Lambda _N)\rightarrow \mathcal {M}$. For any $f\in \mathcal {O}(\mathcal {M})$, $\epsilon (\phi ^{\sharp }(f))$ is defined as the unique real number such that $\phi ^{\sharp }(f)-\epsilon (\phi ^{\sharp }(f))$ is not invertible. This is precisely the definition of the value of a section of $\Lambda _N$ at $\{*\}$, i.e., $\epsilon (\phi ^{\sharp }(f))=\phi ^{\sharp }(f)(\{*\})=f(|\phi |(\{*\}))$. Since $\epsilon \circ \phi ^{\sharp }=\textrm{ev}_x$ for some $x\in M$, this yields $f(|\phi |(\{*\}))=\textrm{ev}_x(f)=f(x)$ for any $f\in \mathcal {O}(\mathcal {M})$ which implies $|\phi |(\{*\})=x$.

Note that $\phi ^{\sharp }$ is a morphism of sheaves and thus, in particular, commutes with restrictions. Hence, if, for $f\in \mathcal {O}(\mathcal {M})$, there exists an open neighborhood $x\in V$ such that $f|_V=0$, then $\phi ^{\sharp }(f)=0$. That is, $\phi ^{\sharp }$ is uniquely determined by the induced stalk morphism $\phi ^{\sharp }_x:\,\mathcal {O}_{\mathcal {M},x}\rightarrow \Lambda _N$. From this, it is immediate to see that any $\psi \in \mathcal {M}|_U(\Lambda _N)=\textrm{Hom}(\mathcal {O}_{M}(U),\Lambda _N)$ can trivially be extended to a morphism $\psi :\,\mathcal {O}(\mathcal {M})\rightarrow \Lambda _N$ satisfying $\epsilon \circ \psi =\textrm{ev}_x$ for some $x\in U$, i.e., $\psi \in \textbf{B}^{-1}(U)$. Conversely, it follows that any morphism in $\textbf{B}^{-1}(U)$ arises in this way. This proves the last assertion. $\square$

By the local property, for any $x\in M$, there exists an open subset $x\in U\subseteq M$ such that $\mathcal {M}|_U$ is isomorphic to a superdomain $\mathcal {U}^{m|n}$ which is a submanifold of the superspace $\mathbb {R}^{m|n}$. Applying the functor (24) and using Proposition 2.12, we thus obtain an isomorphism

$$\begin{aligned} \textbf{B}^{-1}(U)=\mathcal {M}|_{U}(\Lambda _N)\rightarrow \mathcal {U}^{m|n}(\Lambda _N)\subseteq \mathbb {R}^{m|n}(\Lambda _N) \end{aligned}$$

(28)

i.e., a local superchart of $\mathcal {M}(\Lambda _N)$. By (21), it follows immediately that the transition map between two local supercharts defines a $H^{\infty }$-smooth function. As a consequence, $\mathcal {M}(\Lambda _N)$ indeed carries the structure of a $H^{\infty }$ supermanifold. Hence, it follows that the $\Lambda$-points of an algebro-geometric supermanifold naturally define supermanifolds in the sense of Rogers–DeWitt (or more generally $\mathcal {A}$-manifolds in the sense of Tuynman [29]). Moreover, the corresponding topological space $\textbf{B}(\mathcal {M}(\Lambda _N))=\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ has the structure of an ordinary $C^{\infty }$ manifold.

Remark 2.13

Just for sake of completeness, note that each $\mathcal {M}\!\in \!\textbf{Ob}(\textbf{SMan}_{\textrm{Alg}})$ gives rise to the obvious functor

$$\begin{aligned} \mathcal {M}:\,\textbf{Gr}\rightarrow \textbf{Top} \end{aligned}$$

(29)

which maps Grassmann algebras $\Lambda$ to $\Lambda$-points $\mathcal {M}(\Lambda )$. This leads to the interpretation of a supermanifold in the sense of Molotkov–Sachse [25, 26, 30].

Similar to (23), for any $U\subseteq M$ open, one obtains a map

$$\begin{aligned} \mathcal {O}_{M}(U)\cong \textrm{Hom}(\mathcal {M}|_U,\mathbb {R}^{1|1})&\rightarrow H^{\infty }(\mathcal {M}|_U(\Lambda _N))=\textbf{B}_{*}H^{\infty }_{\mathcal {M}(\Lambda _N)}(U)\nonumber \\ f&\mapsto f_{\Lambda _N} \end{aligned}$$

(30)

which is generally surjective but injective iff $N\ge n$. In particular, one can show that it defines a morphism of sheaves, i.e., it commutes with restrictions.

Consider next a $H^{\infty }$ supermanifold $\mathcal {K}\in \textbf{Ob}(\textbf{SMan}_{H^{\infty }})$. To $\mathcal {K}$, one can associate the body $\textbf{B}(\mathcal {K})$ defined as the subset of $\mathcal {K}$ given by

$$\begin{aligned} \textbf{B}(\mathcal {K}):=\{x\in \mathcal {K}|\,f(x)\in \mathbb {R},\,\forall f\in H^{\infty }(\mathcal {K})\} \end{aligned}$$

(31)

which, by definition, has the structure of an ordinary smooth manifold. This can be extended to morphisms $f:\,\mathcal {K}\rightarrow \mathcal {L}$ between $H^{\infty }$ supermanifolds setting⁵$\textbf{B}(f):=f|_{\textbf{B}(\mathcal {K})}:\,\textbf{B}(\mathcal {K})\rightarrow \textbf{B}(\mathcal {L})$ yielding a functor $\textbf{B}:\,\textbf{SMan}_{H^{\infty }}\rightarrow \textbf{Man}$ called the body functor. In case $\mathcal {K}$ is given by a $\Lambda _N$-point $\mathcal {M}(\Lambda _N)$ of an algebro-geometric supermanifold $\mathcal {M}\in \textbf{Ob}(\textbf{SMan}_{\textrm{Alg}})$ with odd dimension bounded by N, one can identify $\textbf{B}(\mathcal {K})$ with the real spectrum $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$ justifying the notation.

To see this, note that, in this case, (30) implies that smooth functions on $\mathcal {K}$ are given by natural transformations $f_{\Lambda _N}$ induced by morphisms $f\in \textrm{Hom}(\mathcal {M},\mathbb {R}^{1|1})$. For $\phi \in \mathcal {K}=\textrm{Hom}(\mathcal {O}(\mathcal {M}),\Lambda _N)$, $f_{\Lambda _N}(\phi )$ can be identified with the element $\phi \circ f^{\sharp }\in \Lambda _N$. Hence, $\phi \in \textbf{B}(\mathcal {N})\Leftrightarrow f_{\Lambda _N}(\phi )\in \textrm{Hom}(\mathcal {O}(\mathbb {R}^{1|1}),\mathbb {R})\cong \mathbb {R}$ $\forall f\in \textrm{Hom}(\mathcal {M},\mathbb {R}^{1|1})$ if and only if $\phi \in \textrm{Hom}(\mathcal {O}(\mathcal {M}),\mathbb {R})$, that is, iff $\phi$ is contained in the real spectrum $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$.

To any $H^{\infty }$ supermanifold $\mathcal {K}$, one can associate the locally ringed space

$$\begin{aligned} \textbf{A}(\mathcal {K}):=(\textbf{B}(\mathcal {K}),\textbf{B}_{*}H^{\infty }_{\mathcal {K}}) \end{aligned}$$

(32)

which has the structure of an algebro-geometric supermanifold. A morphism $f:\,\mathcal {K}\rightarrow \mathcal {L}$ between $H^{\infty }$ supermanifolds $\mathcal {K}$ and $\mathcal {L}$ canonically induces a morphism

$$\begin{aligned} \textbf{A}(f)=(f|_{\textbf{B}(\mathcal {K})},f^*):\,\textbf{A}(\mathcal {K})\rightarrow \textbf{A}(\mathcal {L}) \end{aligned}$$

(33)

between the corresponding algebro-geometric supermanifolds, where $f^*$ denotes the ordinary pullback of smooth functions. Hence, this yields a functor

$$\begin{aligned} \textbf{A}:\,\textbf{SMan}_{H^{\infty }}\rightarrow \textbf{SMan}_{\textrm{Alg}} \end{aligned}$$

(34)

Let us restrict $\textbf{H}_N$ to the full subcategory $\textbf{SMan}_{\textrm{Alg},N}\subset \textbf{SMan}_{\textrm{Alg},N}$ of algebro-geometric supermanifolds with odd dimension bounded by N. Then, based on the previous observations, if follows $\textbf{A}(\textbf{H}_N(\mathcal {M}))\cong \mathcal {M}$ for any $\mathcal {M}\in \textbf{Ob}(\textbf{SMan}_{\textrm{Alg},N})$. In fact, we have the following.

Theorem 2.14

The functor $\textbf{A}\circ \textbf{H}_N:\,\textbf{SMan}_{\textrm{Alg},N}\rightarrow \textbf{SMan}_{\textrm{Alg},N}$ is naturally equivalent to the identity functor $\textrm{id}:\,\textbf{SMan}_{\textrm{Alg},N}\rightarrow \textbf{SMan}_{\textrm{Alg},N}$.

Proof

We have to show that the following diagrams are commutative

for any $\mathcal {M},\mathcal {N}\in \textbf{Ob}(\textbf{SMan}_{\textrm{Alg},N})$ and morphisms $f=(|f|,f^{\sharp }):\,\mathcal {M}\rightarrow \mathcal {N}$ where, in the diagram on the left, the lower arrow is given by the restriction of $\textbf{H}_N(f)$ to the real spectrum $\textrm{Spec}_{\mathbb {R}}(\mathcal {O}(\mathcal {M}))$.

That the left diagram commutes follows immediately, since, by Definition (25), we have

$$\begin{aligned} \textbf{H}_N(f)(\mathrm {ev_x})=\textrm{ev}_x\circ f^{\sharp }=\textrm{ev}_{|f|(x)} \end{aligned}$$

(35)

for any $x\in M$. To see the commutativity of the right diagram, note that, by identifying $g\in \mathcal {O}(\mathcal {N})$ with a morphism $g:\,\mathcal {N}\rightarrow \mathbb {R}^{1|1}$, the pullback $f^{\sharp }(g)$ is given by the morphism $g\circ f:\,\mathcal {M}\rightarrow \mathbb {R}^{1|1}$. Moreover, identifying $\phi \in \mathcal {M}(\Lambda _N)$ with a morphism $\phi :\,\mathbb {R}^{0|N}\rightarrow \mathcal {M}$, we have $\textbf{H}_N(f)(\phi )=f\circ \phi$. Thus, this yields

$$\begin{aligned} \textbf{H}_N(f)^*(g_{\Lambda _N})(\phi )=g_{\Lambda _N}(\textbf{H}_N(f)(\phi ))=g\circ f\circ \phi =f^{\sharp }(g)\circ \phi =(f^{\sharp }(g))_{\Lambda _N}(\phi ) \nonumber \\ \end{aligned}$$

(36)

for any $g\in \mathcal {O}(\mathcal {N})$ and $\phi \in \mathcal {M}(\Lambda _N)$. This proves the theorem. $\square$

Conversely, it is immediate to see that $\textbf{H}_N\circ \textbf{A}$ is naturally equivalent to the identity functor on the full subcategory $\textbf{SMan}_{H^{\infty },N}$ of $H^{\infty }$ supermanifolds with odd dimension bounded by N (see also [24, 31]). Thus, the functors $\textbf{H}_N:\,\textbf{SMan}_{\textrm{Alg},N}\rightarrow \textbf{SMan}_{H^{\infty },N}$ and $\textbf{A}:\,\textbf{SMan}_{H^{\infty },N}\rightarrow \textbf{SMan}_{\textrm{Alg},N}$ provide an equivalence of categories.

To summarize, any algebro-geometric supermanifold induces a functor of the form (29) assigning Grassmann algebras to the corresponding $H^{\infty }$ supermanifold. Moreover, in case that the number of odd generators of the Grassmann algebra is large enough, via (24), one even obtains an equivalence of categories which allows one to uniquely reconstruct the underlying algebro-geometric supermanifold. For this reason, many constructions on algebro-geometric supermanifolds can equivalently be performed on the corresponding $H^{\infty }$ supermanifolds (in fact, we will mainly do so in what follows as $H^{\infty }$ manifolds are often easier to handle for applications in physics).

However, the choice of a particular Grassmann algebra is completely arbitrary and therefore tends to introduce superfluous (physical) degrees of freedom. Consequently, any definition made on a $H^{\infty }$ supermanifold should not depend on a particular choice of a Grassmann algebra but, in the sense of Molotkov–Sachse, behave functorially under the change of Grassmann algebras. In the following, working with a particular $H^{\infty }$ supermanifold $\mathcal {M}$, we will only assume that the number of odd generators of the Grassmann algebra $\Lambda$ over which $\mathcal {M}$ is modeled is large enough, i.e., greater than the odd dimension of $\mathcal {M}$.⁶

2.3 The Super Poncaré and Anti-De Sitter Group

Let us recall the basic super Lie groups and their corresponding super Lie algebras which play a central role in supersymmetry and supergravity and which will provide the foundations for the derivation of the super Ashtekar connection for extended supergravity theories in four spacetime dimensions.

A $H^{\infty }$ super Lie group $\mathcal {G}$ is a group object in the category $\textbf{SMan}_{H^{\infty }}$ (see for instance [29, 32] and references therein). The super Lie module $\textrm{Lie}(\mathcal {G})$ is defined as the tangent space $T_e\mathcal {G}$ at the identity $e\in \mathcal {G}$ and turns out to have the structure of a super $\Lambda$-vector space $\textrm{Lie}(\mathcal {G})=:\mathfrak {g}\otimes \Lambda$ with $\mathfrak {g}$ a super Lie algebra over $\mathbb {R}$. In fact, $\mathfrak {g}$ can be identified with the super Lie algebra of the associated algebro-geometric super Lie group.

Definition 2.15

A (homogeneous) super bilinear form B of parity $|B|\in \mathbb {Z}_2$ on a super $\Lambda ^{\mathbb {C}}$-vector space $\mathcal {V}=V\otimes \Lambda ^{\mathbb {C}}$, $\Lambda ^{\mathbb {C}}:=\Lambda \otimes \mathbb {C}$, is a right-linear map $B:\,\mathcal {V}\times \mathcal {V}\rightarrow \Lambda ^{\mathbb {C}}$ which satisfies $B(\mathcal {V}_i,\mathcal {V}_j)\subseteq (\Lambda ^{\mathbb {C}})_{|B|+i+j}$ and is graded symmetric, i.e.,

$$\begin{aligned} B(v,w)=(-1)^{|v||w|}B(w,v),\quad \forall v,w\in \mathcal {V} \end{aligned}$$

(37)

Moreover, we require B to be smooth in the sense that $B(V,V)\subseteq \mathbb {C}$ [33]. Finally, the smooth super bilinear form B is called non-degenerate if for any $v\in V$ there exists $w\in V$ such that $B(v,w)\ne 0$.

Remark 2.16

From the smoothness requirement of a super bilinear form B, it follows immediately that $B|_{V\times V}$ defines a graded symmetric bilinear form on the super vector space V in the sense of [34], i.e., $B(V_i,V_j)=0$ unless $i+j=|B|$. Moreover, $B|_{V_0\times V_0}$ is symmetric and $B|_{V_1\times V_1}$ is antisymmetric. If B is furthermore even and non-degenerate then so is $B|_{V\times V}$ which implies that $V_1$ is necessarily even dimensional and one can always find a homogeneous basis $(e_i,f_j)$ of V (resp. $\mathcal {V}$) such that, w.r.t. this basis, B takes the form

$$\begin{aligned} \begin{pmatrix} \mathbbm {1}_m &{} 0\\ 0 &{} J_{2n} \end{pmatrix} \end{aligned}$$

(38)

where $\textrm{dim}\,V_0=m$ and $\textrm{dim}\,V_1=2n$ and $J_{2n}$ is the standard symplectic form on $\mathbb {C}^{2n}$

$$\begin{aligned} J_{2n}:=\begin{pmatrix} 0 &{} \mathbbm {1}_n\\ -\mathbbm {1}_n &{} 0 \end{pmatrix} \end{aligned}$$

(39)

We will call (39) the standard representation of B.

Definition 2.17

Let $(\mathcal {V},B(\,\cdot ,\,\cdot \,))$ be a super $\Lambda ^{\mathbb {C}}$-vector space equipped with an even non-degenerate super bilinear form $B:\,\mathcal {V}\times \mathcal {V}\rightarrow \Lambda ^{\mathbb {C}}$. The orthosymplectic super Lie group $\textrm{OSp}(\mathcal {V})$ is defined as the super Lie subgroup of the general linear group $\textrm{GL}(\mathcal {V})$ consisting of all those group elements that preserve B, i.e., $g\in \textrm{OSp}(\mathcal {V})$ if and only if

$$\begin{aligned} B(gv,gw)=B(v,w),\quad \forall v,w\in \mathcal {V} \end{aligned}$$

(40)

It follows from the “stabilizer theorem” (see, e.g., Prop. 5.13 in [29] or Prop. 8.4.7. in [27] in the pure algebraic setting) that $\textrm{OSp}(\mathcal {V})$ defines an embedded super Lie subgroup of the general linear group $\textrm{GL}(\mathcal {V})$ with super Lie algebra

$$\begin{aligned} \mathfrak {osp}(\mathcal {V}):=\{X\in \mathfrak {gl}(\mathcal {V})|\,B(Xv,w)+(-1)^{|X||v|}B(v,Xw)\,\forall v,w\in V\} \end{aligned}$$

(41)

If $(e_i,f_j)$ is homogeneous basis of $\mathcal {V}$ such that $\mathcal {V}\cong (\Lambda ^{\mathbb {C}})^{m|2n}$ and B acquires the standard representation (39), the orthosymplectic super Lie group is also simply denoted by $\textrm{OSp}(m|2n)$. Accordingly. the bosonic sub super Lie algebra takes the form $\mathfrak {osp}(m|2n)_0=\mathfrak {so}(m)\oplus \mathfrak {sp}(2n)$.

In order to find a graded generalization for the isometry group $\textrm{SO}(2,3)$ of anti-de Sitter spacetime $\textrm{AdS}_4$,⁷ we consider the following Lie algebra representation of $\mathfrak {so}(2,3)$.

Let $\gamma ^I$, $I=0,\ldots ,3$, be the gamma matrices satisfying the Clifford algebra relations in $D=4$

$$\begin{aligned} \{\gamma _I,\gamma _J\}=2\eta _{IJ} \end{aligned}$$

(43)

where $\eta _{IJ}=\textrm{diag}(-+++)$. Similar as in [35, 36], we define totally antisymmetric matrices $\Sigma ^{AB}$, $A,B=0,\ldots ,4$, via

$$\begin{aligned} \Sigma ^{IJ}:=\frac{1}{2}\gamma ^{IJ}:=\frac{1}{4}[\gamma ^I,\gamma ^J]\quad \text {as well as}\quad \Sigma ^{4I}:=-\gamma ^{I4}:=\frac{1}{2}\gamma ^I \end{aligned}$$

(44)

where indices are raised and lowered w.r.t. the metric $\eta _{AB}=\textrm{diag}(-+++-)$. These satisfy the following commutation relations

$$\begin{aligned}{}[\Sigma _{AB},\Sigma _{CD}]=\eta _{BC}\Sigma _{AD}-\eta _{AC}\Sigma _{BD}-\eta _{BD}\Sigma _{AC}+\eta _{AD}\Sigma _{BC} \end{aligned}$$

(45)

and thus indeed provide a representation of $\mathfrak {so}(2,3)$. In fact, choosing a real representation of the gamma matrices, it follows that the charge conjugation matrix C is of the form $C=-iJ_4$ and, by the symmetry properties of the gamma matrices,

$$\begin{aligned} (C\Sigma _{AB})^T=C\Sigma _{AB} \end{aligned}$$

(46)

Hence, the $\Sigma _{AB}$ generate $\mathfrak {sp}(4)$ the Lie algebra universal covering group $\textrm{Sp}(4,\mathbb {R})$ of $\textrm{SO}(2,3)$. Thus, a candidate for the graded extension of the anti-de Sitter group with $\mathcal {N}$-fermionic generators is given by the orthosymplectic Lie group $\textrm{OSp}(\mathcal {N}|4)$. We therefore choose $\mathcal {V}=(\Lambda ^{\mathbb {C}})^{\mathcal {N}|4}$ as super vector space equipped with the bilinear form

$$\begin{aligned} \Omega =\begin{pmatrix} \mathbbm {1} &{}\quad 0\\ 0 &{}\quad C \end{pmatrix} \end{aligned}$$

(47)

The algebra $\mathfrak {osp}(\mathcal {N}|4)$ is then generated by all $X\in \mathfrak {gl}(\mathcal {V})$ satisfying

$$\begin{aligned} X^{sT}\Omega +\Omega X=0 \end{aligned}$$

(48)

where $X^{sT}$ denotes the super transpose of X. Writing X in the block form

$$\begin{aligned} X=\begin{pmatrix} X_{11} &{}\quad X_{12}\\ X_{21} &{}\quad X_{22} \end{pmatrix} \end{aligned}$$

(49)

(48) is equivalent to

$$\begin{aligned} X_{11}^T=-X,\quad (CX_{22})^T=CX_{22},\quad X_{12}=-X_{21}^TC \end{aligned}$$

(50)

and thus, in particular, $X_{11}\in \mathfrak {so}(\mathcal {N})$ and $X_{22}\in \mathfrak {sp}(4)$. Thus, following [37], based on the above observation, we set

$$\begin{aligned} M_{AB}:=\begin{pmatrix} 0 &{}\quad 0\\ 0 &{}\quad \Sigma _{AB} \end{pmatrix}\quad \text {and}\quad T^{rs}:=\begin{pmatrix} A^{rs} &{}\quad 0\\ 0 &{}\quad 0 \end{pmatrix} \end{aligned}$$

(51)

as generators for the bosonic sub super Lie algebras $\mathfrak {sp}(4)$ and $\mathfrak {so}(\mathcal {N})$, respectively, where $(A^{rs})_{pq}:=2\delta _p^{[r}\delta _q^{s]}$, $p,q,r,s=1,\ldots ,\mathcal {N}$. For the fermionic generators, we set [37]

$$\begin{aligned} Q_{\alpha }^r:=\begin{pmatrix} 0 &{}\quad -\bar{e}_{\alpha }\otimes e_r\\ e_{\alpha }\otimes e_r^T &{}\quad 0 \end{pmatrix} \end{aligned}$$

(52)

where $(\bar{e}_{\alpha })_{\beta }=C_{\alpha \beta }$. It then follows by direct computation that

$$\begin{aligned}{}[M_{AB},Q_{\alpha }^r]=Q_{\beta }^r{(\Sigma _{AB})}{^{\beta }_{\alpha }}\quad \text {and}\quad [T^{pq},Q_{\alpha }^r]=\delta ^{qr}Q_{\alpha }^p-\delta ^{pr}Q_{\alpha }^q \end{aligned}$$

(53)

In order to compute the Lie bracket between two fermionic generators, one can use the Fierz identity $2(e_{\alpha }\bar{e}_{\beta }+e_{\beta }\bar{e}_{\alpha })=\gamma _I(C\gamma ^I)_{\alpha \beta }+\frac{1}{2}\gamma _{JI}(C\gamma ^{IJ})_{\alpha \beta }$ where the sum terminates after second order in the gamma matrices as the higher rank gamma matrices are antisymmetric with respect to the charge conjugation C. Thus, one finds

$$\begin{aligned}{}[Q_{\alpha }^r,Q_{\beta }^s]=\delta ^{rs}(C\Sigma ^{AB})M_{AB}-C_{\alpha \beta }T^{rs} \end{aligned}$$

(54)

Defining $P_I:=\Sigma _{4I}$ and reintroducing the cosmological constant by rescaling $P_I\rightarrow P_I/L$ and $Q_{\alpha }^r\rightarrow Q_{\alpha }^r/\sqrt{2L}$, one finally ends up with the following (graded) commutation relations

$$\begin{aligned}{}[M_{IJ},Q^r_{\alpha }]&=\frac{1}{2}Q_{\beta }^r{(\gamma _{IJ})}{^{\beta }_{\alpha }}\end{aligned}$$

(55)

$$\begin{aligned}&=-\frac{1}{2L}Q^r_{\beta }{(\gamma _I)}{^{\beta }_{\alpha }}\end{aligned}$$

(56)

$$\begin{aligned}&=\frac{1}{L^2}M_{IJ}\end{aligned}$$

(57)

$$\begin{aligned} =\delta ^{rs}\frac{1}{2}(C\gamma ^I)_{\alpha \beta }P_I+&\delta ^{rs}\frac{1}{4L}(C\gamma ^{IJ})_{\alpha \beta }M_{IJ}-\frac{1}{2L}C_{\alpha \beta }T^{rs} \end{aligned}$$

(58)

which is in the form we will use in what follows. Performing the Inönü–Wigner contraction, i.e., taking the limit $L\rightarrow \infty$, one obtains the super Poincaré Lie algebra.

Remark 2.18

The super Poincaré group can also be described as the isometry group $\textrm{ISO}(\mathbb {R}^{1,3|4})$ of super Minkowski spacetime $\mathbb {R}^{1,3|4\mathcal {N}}$. For instance, for $\mathcal {N}=1$, super Minkowksi spacetime is given by the split supermanifold $\textbf{S}(\Delta _{\mathbb {R}},\mathbb {R}^{1,3})$ associated with the trivial vector bundle $\mathbb {R}^{1,3}\times \Delta _{\mathbb {R}}$ with $\Delta _{\mathbb {R}}$ the four-dimensional real Majorana representation of $\textrm{Spin}^+(1,3)$. The super Poincaré group is then given by the semi-direct product⁸

$$\begin{aligned} \textrm{ISO}(\mathbb {R}^{1,3|4})=\mathcal {T}^{1,3|4}\rtimes _{\Phi }\textbf{S}(\textrm{Spin}^+(1,3)) \end{aligned}$$

(59)

where $\mathcal {T}^{1,3|4}\cong \textbf{S}(\Delta _{\mathbb {R}},\mathbb {R}^{1,3})$ is the super translation group (see, e.g., [32]) and $\Phi :\,\textbf{S}(\textrm{Spin}^+(1,3))\rightarrow \textrm{GL}(\mathcal {T}^{1,3|4})$ is the group action obtained by applying the split functor on the group representation

$$\begin{aligned} \textrm{Spin}^+(1,3)\ni g\mapsto \textrm{diag}(\lambda ^+(g),\kappa _{\mathbb {R}}(g))\in \textrm{GL}(\mathbb {R}^{1,3}\oplus \Pi \Delta _{\mathbb {R}}) \end{aligned}$$

(60)

of $\textrm{Spin}^+(1,3)$ on the super vector space $\mathbb {R}^{1,3}\oplus \Pi \Delta _{\mathbb {R}}$ with $\lambda ^+:\,\textrm{Spin}^+(1,3)\rightarrow \textrm{SO}^+(1,3)$ the universal covering map and $\Pi :\,\textbf{SVec}\rightarrow \textbf{SVec}$ the parity functor, i.e., $\Pi \Delta _{\mathbb {R}}$ is viewed as a purely odd super vector space.

2.4 Super Parallel Transport Map

One of the main issues while working in the standard category of supermanifolds, both in the $H^{\infty }$ or the algebraic category, is that when restricting on the body, the only nonvanishing field components turn out to be purely bosonic. Hence, there are no fermionic degrees of freedom on the body. This, however, seems to be incompatible with various constructions in physics. For instance, in the D’Auria-Fré formalism of supergravity [2, 3], one has the so-called rheonomy principle stating that physical fields are completely fixed by their pullback to the body manifold. Moreover, supersymmetry transformation depend on an anticommutative fermionic generator and, in particular, are nontrivial by restricting superfields to the body.

As we will see, all these issues can be remedied simultaneously by factorizing a given supermanifold $\mathcal {M}$ by an additional parameterizing supermanifold $\mathcal {S}$ and studying superfields on $\mathcal {S}\times \mathcal {M}$. Therefore, as will become clear in what follows, we are interested in a certain subclass of such superfields which depend as little as possible on this additional supermanifold making them covariant in a specific sense under a change of parameterization. This idea is based on a proposal formulated already by Schmitt [11] motivated by the functorial approach to supermanifold theory according to Molotkov [25] and Sachse [30] and which also recently found application in context of superconformal field theories on super Riemannian surfaces [38, 39] as well as the local approach to super QFTs [40].

The following presentation summarizes the main results obtained in [19]. We will therefore omit most of the proofs. We adopt the terminology of [40] introducing the notion of a relative supermanifold (see also [80]). However, unlike as in [40], in order to study fermionic fields, we will not restrict on superpoints as parameterizing supermanifolds. As we will see at the end of this section, the resulting picture resembles the construction of parallel transport map of super connections on super vector bundles as studied in the purely algebraic setting in [41, 42]. Moreover, the description of fermionic fields turns out to be quite similar to considerations in perturbative algebraic QFT (pAQFT) [12, 13].

Definition 2.19

Let $\mathcal {S}$ and $\mathcal {M}$ be $H^{\infty }$ supermanifolds. The pair $(\mathcal {S}\times \mathcal {M},\textrm{pr}_{\mathcal {S}})$ with $\textrm{pr}_{\mathcal {S}}:\,\mathcal {S}\times \mathcal {M}\rightarrow \mathcal {S}$ the projection onto the first factor is called a $\mathcal {S}$-relative supermanifold also denoted by $\mathcal {M}_{/\mathcal {S}}$.

A morphism $\phi :\,\mathcal {M}_{/\mathcal {S}}\rightarrow \mathcal {N}_{/\mathcal {S}}$ between $\mathcal {S}$-relative supermanifolds is a morphism $\phi :\,\mathcal {S}\times \mathcal {M}\rightarrow \mathcal {S}\times \mathcal {N}$ of $H^{\infty }$ supermanifolds preserving the projections, i.e., the following diagram is commutative

Hence, $\phi (s,p)=(s,\tilde{\phi }(s,p))$ $\forall (s,p)\in \mathcal {S}\times \mathcal {M}$ with $\tilde{\phi }:=\textrm{pr}_{\mathcal {N}}\circ \phi :\,\mathcal {S}\times \mathcal {M}\rightarrow \mathcal {N}$. This yields a category $\textbf{SMan}_{/\mathcal {S}}$ called the category of $\mathcal {S}$-relative supermanifolds.

The following proposition gives a different characterization of morphism between $\mathcal {S}$-relative supermanifolds.

Proposition 2.20

(After [40]). Let $\mathcal {M}_{/\mathcal {S}},\mathcal {N}_{/\mathcal {S}}\in \textbf{Ob}(\textbf{SMan}_{/\mathcal {S}})$ be $\mathcal {S}$-relative supermanifolds. Then, the map

$$\begin{aligned} \alpha _{\mathcal {S}}:\,\textrm{Hom}_{\textbf{SMan}_{/\mathcal {S}}}(\mathcal {M}_{/\mathcal {S}},\mathcal {N}_{/\mathcal {S}})&\rightarrow \textrm{Hom}_{\textbf{SMan}_{H^{\infty }}}(\mathcal {S}\times \mathcal {M},\mathcal {S}\times \mathcal {N})\nonumber \\ (\phi :\,\mathcal {S}\times \mathcal {M}\rightarrow \mathcal {S}\times \mathcal {N})&\mapsto (\textrm{pr}_{\mathcal {N}}\circ \phi :\,\mathcal {S}\times \mathcal {M}\rightarrow \mathcal {N}) \end{aligned}$$

(61)

is a bijection with the inverse given by

$$\begin{aligned} \alpha _{\mathcal {S}}^{-1}:\,\textrm{Hom}_{\textbf{SMan}_{H^{\infty }}}(\mathcal {S}\times \mathcal {M},\mathcal {N})&\rightarrow \textrm{Hom}_{\textbf{SMan}_{/\mathcal {S}}}(\mathcal {M}_{/\mathcal {S}},\mathcal {N}_{/\mathcal {S}})\nonumber \\ (\psi :\,\mathcal {S}\times \mathcal {M}\rightarrow \mathcal {N})&\mapsto ((\textrm{id}_{\mathcal {S}}\times \psi )\circ (d_{\mathcal {S}}\times \textrm{id}_{\mathcal {M}}):\nonumber \\ \mathcal {S}\times \mathcal {M}&\rightarrow \mathcal {S}\times \mathcal {N}) \end{aligned}$$

(62)

with $d_{\mathcal {S}}:\,\mathcal {S}\rightarrow \mathcal {S}\times \mathcal {S}$ the diagonal map. $\square$

Let $\lambda :\,\mathcal {S}\rightarrow \mathcal {S}'$ be a morphism between parameterizing supermanifolds, we will also call such a morphism a change of parameterization. Then, any smooth map $\phi :\,\mathcal {S'}\times \mathcal {M}\rightarrow \mathcal {N}$ can be pulled back via $\lambda$ to a morphism $\lambda ^*\phi :=\phi \circ (\lambda \times \textrm{id}_{\mathcal {M}}):\,\mathcal {S}\times \mathcal {M}\rightarrow \mathcal {N}$. Using 62, this yields the map [40]

$$\begin{aligned} \lambda ^{*}:\,\textrm{Hom}_{\textbf{SMan}_{/\mathcal {S}'}}(\mathcal {M}_{/\mathcal {S}'},\mathcal {N}_{/\mathcal {S}'})&\rightarrow \textrm{Hom}_{\textbf{SMan}_{/\mathcal {S}}}(\mathcal {M}_{/\mathcal {S}},\mathcal {N}_{/\mathcal {S}})\nonumber \\ \phi&\mapsto \alpha _{\mathcal {S}}^{-1}(\alpha _{\mathcal {S}'}(\phi )\circ (\lambda \times \textrm{id}_{\mathcal {M}})) \end{aligned}$$

(63)

Hence, for $\phi :\,\mathcal {M}_{/\mathcal {S}'}\rightarrow \mathcal {N}_{/\mathcal {S}'}$, $\lambda ^*(\phi )$ explicitly reads

$$\begin{aligned} \lambda ^*(\phi )(s,p)=(s,\textrm{pr}_{\mathcal {N}}\circ \phi (\lambda (s),p)) \end{aligned}$$

(64)

$\forall (s,p)\in \mathcal {S}\times \mathcal {M}$. The following proposition demonstrates that the set of morphisms between relative supermanifolds is functorial in the parameterizing supermanifold and thus indeed have the required properties under change of parameterization.

Proposition 2.21

(After [40]). The assignment

$$\begin{aligned} \textbf{SMan}\rightarrow \textbf{Set}:\,\textbf{Ob}(\textbf{SMan})\ni \mathcal {S}&\mapsto \textrm{Hom}_{\textbf{SMan}_{/\mathcal {S}}}(\mathcal {M}_{/\mathcal {S}},\mathcal {N}_{/\mathcal {S}})\in \textbf{Ob}(\textbf{Set})\\ (\lambda :\,\mathcal {S}\rightarrow \mathcal {S}')&\mapsto \lambda ^{*} \end{aligned}$$

defines a contravariant functor on the category $\textbf{SMan}$ of $H^{\infty }$ supermanifolds. Moreover, the map $\lambda ^{*}$ associated with the morphism $\lambda :\,\mathcal {S}\rightarrow \mathcal {S}'$ preserves compositions, i.e., $\lambda ^*(\phi \circ \psi )=\lambda ^*(\phi )\circ \lambda ^{*}(\psi )$ for any $\psi :\,\mathcal {M}\rightarrow \mathcal {N}$ and $\phi :\,\mathcal {N}\rightarrow \mathcal {L}$

Definition 2.22

• (i) Let $\mathcal {M}_{/\mathcal {S}}\in \textbf{Ob}(\textbf{SMan}_{/\mathcal {S}})$ be a $\mathcal {S}$-relative supermanifold. A vector field X on $\mathcal {S}\times \mathcal {M}$ is called $\mathcal {S}$-relative if

  $$\begin{aligned} X(f\otimes 1)=0\,\forall f\in H^{\infty }(\mathcal {S}) \end{aligned}$$

  (65)

  The $\mathcal {S}$-relative vector fields form a super $H^{\infty }(\mathcal {S}\times \mathcal {M})$-submodule $\Gamma (\mathcal {M}_{/\mathcal {S}})$ of $\Gamma (\mathcal {S}\times \mathcal {M})$ isomorphic to $H^{\infty }(\mathcal {S})\otimes \Gamma (\mathcal {M})$.

• (ii) A $\mathcal {S}$-relative 1-form $\omega$ on $\mathcal {M}_{/\mathcal {S}}$ is a left-linear morphism of super $H^{\infty }(\mathcal {S}\times \mathcal {M})$-modules $\omega \in \underline{\textrm{Hom}}_L(\Gamma (\mathcal {M}_{/\mathcal {S}}),H^{\infty }(\mathcal {S}\times \mathcal {M}))$. The set $\Omega ^1(\mathcal {M}_{/\mathcal {S}})$ of $\mathcal {S}$-relative 1-forms on $\mathcal {M}_{/\mathcal {S}}$ defines a super $H^{\infty }(\mathcal {S}\times \mathcal {M})$-submodule of $\Omega ^1(\mathcal {S}\times \mathcal {M})$ isomorphic to $\Omega ^1(\mathcal {M})\otimes H^{\infty }(\mathcal {S})$.

Let us consider a $H^{\infty }$ principal super fiber bundle $\mathcal {G}\rightarrow \mathcal {P}{\mathop {\rightarrow }\limits ^{\pi }}\mathcal {M}$ with $\mathcal {G}$-right action $\Phi :\,\mathcal {P}\times \mathcal {G}\rightarrow \mathcal {P}$ as well as a supermanifold $\mathcal {S}$. Taking products, this yields a fiber bundle

with projection $\pi _{\mathcal {S}}:=\textrm{id}_{\mathcal {S}}\times \pi$ and $\mathcal {G}$-right action $\Phi _{\mathcal {S}}:=\textrm{id}_{\mathcal {S}}\times \Phi :\,(\mathcal {S}\times \mathcal {P})\times \mathcal {G}\rightarrow \mathcal {S}\times \mathcal {P}$. Since, by definition, $\pi _{\mathcal {S}}$ and $\Phi _{\mathcal {S}}$ are morphisms of $\mathcal {S}$-relative supermanifolds, this yields a fiber bundle $\mathcal {G}\rightarrow \mathcal {P}_{/\mathcal {S}}{\mathop {\rightarrow }\limits ^{\pi _{\mathcal {S}}}}\mathcal {M}_{/\mathcal {S}}$ in the category $\textbf{SMan}_{/\mathcal {S}}$ of $\mathcal {S}$-relative supermanifolds which will be called a $\mathcal {S}$-relative principal super fiber bundle.

Definition 2.23

A $\mathcal {S}$-relative super connection 1-form $\mathcal {A}$ on the $\mathcal {S}$-relative principal super fiber bundle $\mathcal {G}\rightarrow \mathcal {P}_{/\mathcal {S}}{\mathop {\rightarrow }\limits ^{\pi _{\mathcal {S}}}}\mathcal {M}_{/\mathcal {S}}$ is an even $\textrm{Lie}(\mathcal {G})$-valued $\mathcal {S}$-relative 1-form $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g}):=\Omega ^1(\mathcal {P}_{/\mathcal {S}})\otimes \mathfrak {g}$ such that⁹

• (i) $\left\langle \widetilde{X}|\mathcal {A}\right\rangle =X$ $\forall X\in \mathfrak {g}$

• (ii) $(\Phi _{\mathcal {S}})_g^*\mathcal {A}=\textrm{Ad}_{g^{-1}}\circ \mathcal {A}$ $\forall g\in \textbf{B}(\mathcal {G})$ and $L_{\widetilde{X}}\mathcal {A}=-\textrm{ad}_{X}\circ \mathcal {A}$ $\forall X\in \mathfrak {g}$

where $\widetilde{X}:=(\mathbbm {1}\otimes X_e)\circ \Phi _{\mathcal {S}}^*\in \mathfrak {X}(\mathcal {P}_{/\mathcal {S}})$ is the fundamental vector field on $\mathcal {P}_{/\mathcal {S}}$ generated by $X\in \mathfrak {g}$.

Definition 2.24

Here and in the following, we define smooth (parameterized) paths on a $\mathcal {S}$-relative supermanifold $\mathcal {M}_{/\mathcal {S}}$ in terms of smooth maps $\gamma :\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {M}$ with $\mathcal {I}\subseteq \Lambda ^{1,0}$ also referred to as super interval.

Given such a smooth path as well as a $\mathcal {S}$-relative super connection 1-form $\mathcal {A}$ on $\mathcal {P}_{/\mathcal {S}}$, its corresponding horizontal lift on $\mathcal {P}_{/\mathcal {S}}$ is defined as a smooth path $\gamma ^{hor}:\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {P}$ such that $\pi \circ \gamma ^{hor}=\gamma$ and $\left\langle (\mathbbm {1}\otimes \partial _t)\alpha _{\mathcal {S}}^{-1}(\gamma ^{hor})(s,t)|\mathcal {A}\right\rangle =0$ $\forall (s,t)\in \mathcal {S}\times \mathcal {I}$.

The following theorem demonstrates that the horizontal lift of a smooth (parameterized) path on a $\mathcal {S}$-relative supermanifold always exists and is, in fact, even unique.

Proposition 2.25

Let $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})$ be a $\mathcal {S}$-relative super connection 1-form on the $\mathcal {S}$-relative principal super fiber bundle $\mathcal {G}\rightarrow \mathcal {P}_{/\mathcal {S}}{\mathop {\rightarrow }\limits ^{\pi _{\mathcal {S}}}}\mathcal {M}_{/\mathcal {S}}$ as well as $\gamma :\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {M}$ a smooth path. Let furthermore $f:\,\mathcal {S}\rightarrow \mathcal {P}$ be a smooth map. Then, there exists a unique horizontal lift $\gamma ^{hor}:\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {M}$ of $\gamma$ w.r.t. $\mathcal {A}$ such that $\gamma ^{hor}(\,\cdot ,0)=f$.

Proof

See [19]. $\square$

Remark 2.26

For a smooth map $f:\,\mathcal {S}\rightarrow \mathcal {M}$, one can consider the pullback super fiber bundle

$$\begin{aligned} f^*\mathcal {P}=\{(s,p)|\,f(s)=\pi (p)\}\subset \mathcal {S}\times \mathcal {P} \end{aligned}$$

(66)

over $\mathcal {S}$. A smooth section $\tilde{\phi }:\mathcal {S}\rightarrow f^*\mathcal {P}$ of the pullback bundle is then of the form $\tilde{\phi }(s)=(s,\phi (s))$ $\forall s\in \mathcal {S}$ with $\phi :\,\mathcal {S}\rightarrow \mathcal {P}$ a smooth map satisfying $\pi \circ \phi =f$. Hence, we can identify

$$\begin{aligned} \Gamma (f^*\mathcal {P})=\{\phi :\,\mathcal {S}\rightarrow \mathcal {P}|\,\pi \circ \phi =f\} \end{aligned}$$

(67)

Definition 2.27

Under the conditions of Proposition 2.25, the parallel transport in $\mathcal {P}_{/\mathcal {S}}$ along $\gamma$ w.r.t. the connection $\mathcal {A}$ is defined as

$$\begin{aligned} \mathscr {P}^{\mathcal {A}}_{\mathcal {S},\gamma }:\,\Gamma (\gamma _0^*\mathcal {P})&\rightarrow \Gamma (\gamma _1^*\mathcal {P})\nonumber \\ \phi&\mapsto \gamma ^{hor}_{\phi }(\,\cdot \,,1) \end{aligned}$$

(68)

where, for $\Gamma (\gamma _0^*\mathcal {P})\ni \phi :\mathcal {S}\rightarrow \mathcal {P}$, $\gamma ^{hor}_{\phi }$ is the unique horizontal lift of $\gamma$ with respect to $\mathcal {A}$ such that $\gamma ^{hor}_{\phi }(\,\cdot ,0)=\phi$.

Given a change of parameterization $\lambda :\,\mathcal {S}\rightarrow \mathcal {S}'$, due to Definition 2.22, this induces a map on relative smooth vector fields $\lambda ^*\equiv \lambda ^*\otimes \mathbbm {1}:\,\Gamma (\mathcal {M}_{/\mathcal {S}'})\rightarrow \Gamma (\mathcal {M}_{/\mathcal {S}})$. In a similar way, it follows that the change of parameterization induces a morphism on relative super connection 1-forms. The following proposition shows that the horizontal lift as defined in Definition 2.27 indeed has the right properties, that is, it transforms covariantly under change of parameterization.

Proposition 2.28

The parallel transport map enjoys the following properties:

• (i) $\mathscr {P}^{\mathcal {A}}_{\mathcal {S}}$ is functorial under compositions of paths, that is, for smooth paths $\gamma :\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {M}$ and $\delta :\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {M}$ on $\mathcal {M}_{/\mathcal {S}}$, one has

  $$\begin{aligned} \mathscr {P}^{\mathcal {A}}_{\mathcal {S},\gamma \circ \delta }=\mathscr {P}^{\mathcal {A}}_{\mathcal {S},\gamma }\circ \mathscr {P}^{\mathcal {A}}_{\mathcal {S},\delta } \end{aligned}$$

  (69)

• (ii) $\mathscr {P}^{\mathcal {A}}_{\mathcal {S},\gamma }$ is covariant under change of parameterization in the sense that if $\lambda :\,\mathcal {S}\rightarrow \mathcal {S}'$ is a morphism of supermanifolds, then the diagram

  

  (70)

  is commutative for any smooth path $\gamma :\,f\Rightarrow g$ on $\mathcal {M}_{/\mathcal {S}'}$.

Definition 2.29

A global gauge transformation f on the $\mathcal {S}$-relative principal super fiber bundle $\mathcal {G}\rightarrow \mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {M}_{/\mathcal {S}}$ is a morphism $f:\,\mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {P}_{/\mathcal {S}}$ of $\mathcal {S}$-relative supermanifolds which is fiber-preserving and $\mathcal {G}$-equivariant, i.e., $\pi _{\mathcal {S}}\circ f=\pi _{\mathcal {S}}$ and $f\circ \Phi _{\mathcal {S}}=\Phi _{\mathcal {S}}\circ (f\times \textrm{id})$. The set of global gauge transformations on $\mathcal {P}_{/\mathcal {S}}$ will be denoted by $\mathscr {G}(\mathcal {P}_{/\mathcal {S}})$.

Proposition 2.30

Their exists a bijective correspondence between the set $\mathscr {G}(\mathcal {P}_{/\mathcal {S}})$ of global gauge transformations on the $\mathcal {S}$-relative principal super fiber bundle $\mathcal {G}\rightarrow \mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {M}_{/\mathcal {S}}$ and the set

$$\begin{aligned} H^{\infty }(\mathcal {S}\times \mathcal {P},\mathcal {G})^{\mathcal {G}}:=\{\sigma :\,\mathcal {S}\times \mathcal {P}\rightarrow \mathcal {G}|\,\sigma \circ \Phi _{\mathcal {S}}=\alpha \circ (\sigma \times \textrm{id})\} \end{aligned}$$

(71)

via

$$\begin{aligned} H^{\infty }(\mathcal {S}\times \mathcal {P},\mathcal {G})^{\mathcal {G}}\ni \sigma \mapsto \Phi _{\mathcal {S}}\circ (\textrm{id}\times \sigma )\circ d_{\mathcal {S}\times \mathcal {P}}\in \mathscr {G}(\mathcal {P}_{/\mathcal {S}}) \end{aligned}$$

(72)

where $\alpha :\,\mathcal {G}\times \mathcal {G}\rightarrow \mathcal {G},\,(g,h)\mapsto h^{-1}gh$. In particular, global gauge transformations are super diffeomorphisms on $\mathcal {P}_{/\mathcal {S}}$ and $\mathscr {G}(\mathcal {P}_{/\mathcal {S}})$ forms an abstract group under composition of smooth maps.

Proposition 2.31

Let $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})$ be a $\mathcal {S}$-relative super connection 1-form and $f\in \mathscr {G}(\mathcal {P}_{/\mathcal {S}})$ a global gauge transformation on the $\mathcal {S}$-relative principal super fiber bundle $\mathcal {G}\rightarrow \mathcal {P}_{/\mathcal {S}}{\mathop {\rightarrow }\limits ^{\pi _{\mathcal {S}}}}\mathcal {M}_{/\mathcal {S}}$. Then,

• (i) $f^*\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})$ is a $\mathcal {S}$-relative connection 1-form and, in particular,

  $$\begin{aligned} f^*\mathcal {A}=\textrm{Ad}_{\sigma _f^{-1}}\circ \mathcal {A}+\sigma _f^*\theta _{\textrm{MC}} \end{aligned}$$

  (73)

• (ii) the diagram

  

  is commutative for any smooth path $\gamma :\,g\Rightarrow h$ on $\mathcal {M}_{/\mathcal {S}}$.

Example 2.32

We want to give an explicit local expression of the parallel transport map making it more accessible for applications in physics, in particular, in Sect. 5.3. Therefore, we assume that $\mathcal {G}$ is a super matrix Lie group, i.e., an embedded super Lie subgroup of the general linear group $\textrm{GL}(\mathcal {V})$ of a super $\Lambda$-vector space $\mathcal {V}=V\otimes \Lambda$. Let $\gamma :\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {M}$ be a smooth path which is contained within a local trivialization neighborhood of $\mathcal {P}_{/\mathcal {S}}$ and $\tilde{s}:\,U_{/\mathcal {S}}\rightarrow \mathcal {P}_{/\mathcal {S}}$ the corresponding smooth section. As shown in [19], the horizontal lift can then be written in the form $\gamma ^{hor}=(\tilde{s}\circ \gamma )\cdot g$ with $g:\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {G}$ satisfying the equation

$$\begin{aligned} (\mathbbm {1}\otimes \partial _t)g(s,t)=-\mathcal {A}^{\gamma }(s,t)\cdot g(s,t) \end{aligned}$$

(74)

with $\mathcal {A}^{\gamma }(s,t):=\left\langle (\mathbbm {1}\otimes \partial _t)\alpha _{\mathcal {S}}^{-1}(\gamma )(s,t)|\tilde{s}^*\mathcal {A}\right\rangle$. Furthermore, suppose that U defines a local coordinate neighborhood of $\mathcal {M}$. The 1-form $\tilde{s}^*\mathcal {A}$ on $\mathcal {S}\times U$ can then be expanded as

$$\begin{aligned} \tilde{s}^*\mathcal {A}=\textrm{d}x^{\mu }\mathcal {A}^{(\tilde{s})}_{\mu }+\textrm{d}\theta ^{\alpha }\mathcal {A}^{(\tilde{s})}_{\alpha } \end{aligned}$$

(75)

with smooth even and odd functions $\mathcal {A}^{(\tilde{s})}_{\mu }$ and $\mathcal {A}^{(\tilde{s})}_{\alpha }$ on $\mathcal {S}\times U$, respectively. This yields

$$\begin{aligned} \mathcal {A}^{\gamma }(s,t)=:\dot{x}^{\mu }\mathcal {A}^{(\tilde{s})}_{\mu }(s,t)+\dot{\theta }^{\alpha }\mathcal {A}^{(\tilde{s})}_{\alpha }(s,t) \end{aligned}$$

(76)

Hence, the solution of Eq. (74) with the initial condition $g(\,\cdot ,0)=\mathbbm {1}$ takes the form

$$\begin{aligned} g(s,t)=\mathcal {P}\exp \left( -\int _{0}^t{\textrm{d}t'\,\dot{x}^{\mu }\mathcal {A}^{(\tilde{s})}_{\mu }(s,t')+\dot{\theta }^{\alpha }\mathcal {A}^{(\tilde{s})}_{\alpha }(s,t')}\right) \end{aligned}$$

(77)

This is the most general local expression of the parallel transport map corresponding to a $\mathcal {S}$-relative super connection 1-form. This form is used for instance in [43] in the discussion about the relation between super twistor theory and $\mathcal {N}=4$ super Yang–Mills theory (see also [42]). Note that in case $\mathcal {S}=\{*\}$ is a single point, the odd coefficients in (77) become zero so that this expression just reduces the parallel transport map of an ordinary connection 1-form on a principal bundle.

Example 2.33

Finally, let us restrict to a subclass of smooth paths on $\mathcal {M}_{/\mathcal {S}}$ obtained via the lift of smooth paths $\gamma :\,\mathcal {I}\rightarrow \mathcal {M}$ on the bosonic sub supermanifold¹⁰$\mathcal {M}_0$ of $\mathcal {M}$. A $\mathcal {S}$-relative connection 1-form $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})$ induces via pullback along the inclusion $\iota :\,\mathcal {S}\times \mathcal {M}_0\hookrightarrow \mathcal {S}\times \mathcal {M}$ a $\mathcal {S}$-relative super connection 1-form $\iota ^*\mathcal {A}$ on the pullback bundle $\mathcal {G}\rightarrow \iota ^*\mathcal {P}_{/\mathcal {S}}\rightarrow (\mathcal {M}_0)_{/\mathcal {S}}$. Let

$$\begin{aligned} \iota ^*\mathcal {A}=\textrm{pr}_{\mathfrak {g}_0}\circ \iota ^*\mathcal {A}+\textrm{pr}_{\mathfrak {g}_1}\circ \iota ^*\mathcal {A}=:\omega +\psi \end{aligned}$$

(78)

be the decomposition of $\iota ^*\mathcal {A}$ according to the even and odd part of the super Lie algebra $\mathfrak {g}=\mathfrak {g}_0\oplus \mathfrak {g}_1$. It follows that $\omega$ can be reduced to a $\mathcal {S}$-relative super connection 1-form on the $\mathcal {S}$-relative principal super fiber bundle $\mathcal {G}_0\rightarrow (\mathcal {P}_0)_{/\mathcal {S}}\rightarrow (\mathcal {M}_0)_{/\mathcal {S}}$ which will be denoted by the same symbol. Hence, $\omega$ gives rise to a parallel transport map $\mathscr {P}^{\omega }_{\mathcal {S},\gamma }$ along $\alpha _{\mathcal {S}}(\textrm{id}\times \gamma ):\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {M}_{0}$.

Suppose that $\gamma$ is contained within a local trivialization neighborhood of $\mathcal {P}_0$ and let $\tilde{s}:\,U_{/\mathcal {S}}\rightarrow (\mathcal {P}_0)_{/\mathcal {S}}$ be the corresponding local section with $U\subset \mathcal {M}_0$ open. Let $g[\mathcal {A}]:\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {G}$ be the solution of the parallel transport Eq. (74), i.e.,

$$\begin{aligned} \partial _tg[\mathcal {A}](s,t)=-R_{g[\mathcal {A}](s,t)*}\mathcal {A}^{\gamma }(s,t) \end{aligned}$$

(79)

with the initial condition $g(\,\cdot ,0)=e$, where $\mathcal {A}^{\gamma }=:\omega ^{\gamma }+\psi ^{\gamma }$ Furthermore, let $g[\omega ]:\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {G}_0$ be the solution of the corresponding parallel transport equation of $\omega$. Set $g[\psi ]:=g[\omega ]^{-1}\cdot g[\mathcal {A}]:\,\mathcal {S}\times \mathcal {I}\rightarrow \mathcal {G}$. Using $\partial _t(g[\omega ]^{-1})=-L_{g[\omega ]^{-1*}}R_{g[\omega ]^{-1*}}(\partial _tg[\omega ])=L_{g[\omega ]^{-1*}}\omega ^{\gamma }$, it then follows

$$\begin{aligned} \partial _t g[\psi ]&=D\mu _{\mathcal {G}}(\partial _t(g[\omega ]^{-1}),\partial _tg[\mathcal {A}])=R_{g[\mathcal {A}]*}L_{g[\omega ]^{-1*}}\omega ^{\gamma }-L_{g[\omega ]^{-1*}}R_{g[\mathcal {A}]*}\mathcal {A}^{\gamma }\nonumber \\&=-R_{g[\mathcal {A}]*}L_{g[\omega ]^{-1*}}\psi ^{\gamma }=-R_{g[\psi ]*}R_{g[\omega ]*}L_{g[\omega ]^{-1*}}\psi ^{\gamma }\nonumber \\&=-R_{g[\psi ]*}\textrm{Ad}_{g[\omega ]^{-1}}(\psi ^{\gamma }) \end{aligned}$$

(80)

that is, $g[\psi ]$ is the solution of the equation

$$\begin{aligned} \partial _t g[\psi ]=-R_{g[\psi ]*}\textrm{Ad}_{g[\omega ]^{-1}}(\psi ^{\gamma }) \end{aligned}$$

(81)

For a super matrix Lie group $\mathcal {G}$, the solution of (81) can be explicitly written as

$$\begin{aligned} g[\psi ](s,t)=\mathcal {P}\textrm{exp}\left( -\int _0^t{\textrm{d}\tau \,(\textrm{Ad}_{g[\omega ]^{-1}}\psi ^{\gamma })(s,\tau )}\right) \end{aligned}$$

(82)

Hence, for instance, if $\gamma$ is closed loop on $\mathcal {M}_0$, in this gauge the Wilson loop takes the form

$$\begin{aligned} W_{\gamma }[\mathcal {A}]=\textrm{str}\left( g_{\gamma }[\omega ]\cdot \mathcal {P}\textrm{exp}\left( -\oint _{\gamma }{\textrm{Ad}_{g_{\gamma }[\omega ]^{-1}}\psi ^{(\tilde{s})}}\right) \right) :\,\mathcal {S}\rightarrow \mathcal {G} \end{aligned}$$

(83)

where $\psi ^{(\tilde{s})}:=\tilde{s}^*\psi$. It follows from 2.31 that $W_{\gamma }[\mathcal {A}]$ is invariant under local gauge transformations. In fact, $g_{\gamma }[\mathcal {A}]$ transforms as

$$\begin{aligned} g_{\gamma }[\mathcal {A}](s)\rightarrow \phi (s)\cdot g_{\gamma }[\mathcal {A}](s)\cdot \phi (s)^{-1},\quad \forall s\in \mathcal {S} \end{aligned}$$

(84)

for some smooth function $\phi :\,\mathcal {S}\rightarrow \mathcal {G}$. Hence, due to cyclicity of the supertrace, (83) is indeed invariant. Finally, by Proposition 2.28 (ii) the Wilson loop is also invariant under change of parameterizations. That is, if $\lambda :\,\mathcal {S}'\rightarrow \mathcal {S}$ is a supermanifold morphism, then

$$\begin{aligned} \lambda ^*W_{\gamma }[\mathcal {A}]=W_{\gamma }[\lambda ^*\mathcal {A}]:\,\mathcal {S}'\rightarrow \mathcal {G} \end{aligned}$$

(85)

Thus, due to these properties, $W_{\gamma }[\mathcal {A}]$ can be regarded as a fundamental physical quantity according to [11].

Remark 2.34

By definition, $g[\mathcal {A}]$ defines a smooth map $g[\mathcal {A}]:\,\mathcal {S}\rightarrow \mathcal {G}$ from the parameterizing supermanifold $\mathcal {S}$ to the gauge group $\mathcal {G}$. Since one has an equivalence of categories between algebro-geometric and $H^{\infty }$ supermanifolds, it follows that

$$\begin{aligned} H^{\infty }(\mathcal {S},\mathcal {G})\cong \textrm{Hom}_{\textbf{SMan}_{\textrm{Alg}}}(\textbf{A}(\mathcal {S}),\textbf{A}(\mathcal {G})) \end{aligned}$$

(86)

Thus, it follows that $g[\mathcal {A}]$ can be identified with a $\textbf{A}(\mathcal {S})$-point of $\textbf{A}(\mathcal {G})$ according to Definition 2.5. This coincides with the results of [41, 42] where the parallel transport of super connections on super vector bundles in the pure algebraic setting has been considered. It was found that the parallel transport map has the interpretation in terms of $\mathcal {T}$-points of a general linear group.

Remark 2.35

Let us finally comment on the choice of the parameterizing supermanifold $\mathcal {S}$. Working in the algebraic category, a typical choice for $\mathcal {S}$ is a superpoint $\mathcal {S}=(\{*\},\Lambda _N)$ with $\Lambda _N$ a Grassmann algebra with N fermionic generators. In this case, it follows $g[\mathcal {A}]\in \mathcal {G}(\Lambda _N)$, that is, $g[\mathcal {A}]$ is a $\Lambda _N$-point of $\mathcal {G}$. But this means, if N is large enough (i.e., larger than the odd dimension of $\mathcal {G}$), we again end up with a Rogers–DeWitt supermanifold $\mathcal {G}(\Lambda _N)$ and $g[\mathcal {A}]$ can be identified as elements of the group $\mathcal {G}(\Lambda _N)$. This, once more, reflects the strong link between these two approaches to supermanifold theory.

3 Gravity as Cartan Geometry

In this section, we want to review the interpretation of gravity as Cartan geometry as this will serve a starting point for a very elegant approach to supergravity as described in Sect. 4 and a derivation of a super analog of Asthekar’s connection in Sect. 5. A detailed account on the relation between Cartan geometry and general relativity can be found for instance in [44].

In his famous Erlangen program, F. Klein studied the idea of classifying the geometry of space via the underlying group of symmetries. For instance, one can consider Minkowski spacetime $\mathbb {M}=(\mathbb {R}^{1,3},\eta )$ and study the corresponding Lie group $\textrm{ISO}(\mathbb {R}^{1,3})$ of isometries which is isomorphic to the Poincaré group $\mathbb {R}^{1,3}\rtimes \textrm{SO}^+(1,3)$. If one then chooses a specific event $p\in \mathbb {M}$, one can consider the corresponding stabilizer subgroup $\textrm{SO}^+(1,3)$ which preserves that point. Since the isometry group acts transitively on $\mathbb {M}$, it follows that Minkowski spacetime can be described in terms of the coset space

$$\begin{aligned} \mathbb {M}\cong \textrm{ISO}(\mathbb {R}^{1,3})/\textrm{SO}^+(1,3) \end{aligned}$$

(87)

Hence, the collection of spacetime events can equivalently be described in terms of the underling symmetry groups. A similar kind of reasoning applies in case of the other maximally symmetric homogeneous spacetimes such as de Sitter of anti-de Sitter spacetime playing a central role in general relativity and cosmology. Hence, one makes the following definition.

Definition 3.1

A Klein geometry is a pair (G, H) consisting of a Lie Group G and an embedded Lie subgroup $H\hookrightarrow G$ such that G/H is connected.

Given a Klein geometry (G, H), the coset space G/H has the structure of principal H-bundle

Moreover, on G there exists a canonical $\mathfrak {g}$-valued 1-form $\theta _{\textrm{MC}}\in \Omega ^1(G,\mathfrak {g})$, called Maurer–Cartan form which, choosing a basis of left-invariant vector fields $X_i\in \mathfrak {g}$, $i=1,\ldots ,\textrm{dim}\,\mathfrak {g}$, is defined as

$$\begin{aligned} \theta _{\textrm{MC}}=X_i\otimes \omega ^i \end{aligned}$$

(88)

where $\omega ^i\in \Omega ^1(G)$ is the corresponding dual basis of left-invariant one-forms on G satisfying $\omega ^i(X_j)=\delta ^i_j$. It follows by definition that the Maurer–Cartan form is G-equivariant, i.e.¹¹

$$\begin{aligned} R^*_g\theta _{\textrm{MC}}=\textrm{Ad}_{g^{-1}}\circ \theta _{\textrm{MC}} \end{aligned}$$

(89)

$\forall g\in G$ with $R_g:\,G\rightarrow G$ denoting the right translation on G. By definition, $\theta _{\textrm{MC}}$ maps left-invariant vector fields to themselves, i.e., $\theta _{\textrm{MC}}(X)=X_e$ $\forall X\in \mathfrak {g}$ and, as a consequence, yields and isomorphism $\theta _{\textrm{MC}}:\,T_gG\rightarrow \mathfrak {g}$ of vector spaces at any $g\in G$. Moreover, it satisfies the Maurer–Cartan equation

$$\begin{aligned} \textrm{d}\theta _{\textrm{MC}}+[\theta _{\textrm{MC}}\wedge \theta _{\textrm{MC}}]=0 \end{aligned}$$

(90)

As seen above, standard examples of Klein geometries (G, H) arising in physics are given by the Minkowski spacetime $(\textrm{ISO}(\mathbb {R}^{1,3}),\textrm{SO}^+(1,3))$, de Sitter $(\textrm{SO}(1,pg{\!}4)$, $\textrm{SO}^+(1,3))$ and anti-de Sitter spacetime $(\textrm{SO}(2,3),\textrm{SO}^+(1,3))$. These have in common that the Lie algebra $\mathfrak {g}$ of G can be split into $\textrm{Ad}(H)$-invariant subspaces $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {g}/\mathfrak {h}$ with $\mathfrak {h}$ the Lie algebra of H. Moreover, on the moduli space $\mathfrak {g}/\mathfrak {h}$ there exists a canonical $\textrm{Ad}(H)$-invariant bilinear form. In this case, the Klein geometry is called metric and reductive. Hence, we see that flat spacetime can be equivalently be described in terms of Klein geometry. Based on this observation, Cartan formulated a theory now known as Cartan geometry which can be interpreted as a deformed Klein geometry such as gravity is a deformed version of flat Minkowski spacetime.

Definition 3.2

A metric reductive Cartan geometry $(\pi :\,P\rightarrow M,A)$ modeled on a metric reductive Klein geometry $(H,G;\eta )$ is a principal fiber bundle $H\rightarrow P\rightarrow M$ with structure group H together with a $\mathfrak {g}$-valued 1-form $A\in \Omega ^1(P,\mathfrak {g})$ on P called Cartan connection such that

• (i) $A_p(X_p)=X$ $\forall X\in \mathfrak {h}=T_e H$, $p\in P$

• (ii) $\Phi _h^*A=\textrm{Ad}_{h^{-1}}\circ A$ $\forall h\in H$

• (iii) the map $A_p:\,T_pP\rightarrow \mathfrak {g}$ defines an isomorphism of vector spaces for any $p\in P$.

where the last condition is also called the Cartan condition.

Given a metric reductive Cartan geometry $(\pi :\,P\rightarrow M,A)$, one can split the Cartan connection A by projecting it according to the decomposition $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {g}/\mathfrak {h}$ of the Lie algebra of G yielding

$$\begin{aligned} A=\textrm{pr}_{\mathfrak {h}}\circ A+\textrm{pr}_{\mathfrak {g}/\mathfrak {h}}\circ A=:\omega +\theta \end{aligned}$$

(91)

with 1-forms $\omega \in \Omega ^1(P,\mathfrak {h})$ and $\theta \in \Omega ^1(P,\mathfrak {g/\mathfrak {h}})$. Due to the conditions (i) and (ii) of the Cartan connections, it follows immediately that $\omega$ defines an ordinary principal connection 1-form in the sense of Ehresmann.

Let $\mathscr {H}:=\textrm{ker}(\omega )$ be the induced horizontal distribution of the tangent bundle TP. If $\tilde{X}\in \mathscr {V}$ is a vertical vector field with $X\in \mathfrak {h}$ one has $A(\tilde{X})=X=\omega (X)$ and thus $\theta (X)=0$. Hence, since $\mathfrak {g}/\mathfrak {h}$ defines a H-invariant subspace, together with property (ii), this immediately implies that the soldering form is horizontal of type $(H,\textrm{Ad})$, i.e., $\theta \in \Omega ^1_{hor}(P,\mathfrak {g}/\mathfrak {h})^{(H,\textrm{Ad})}$. In fact, $\theta$ even provides an identification of the principal bundle P as a H-reduction of the frame bundle $\mathscr {F}(M)$ explaining its name.

Therefore, note that $\theta _p^{-1}:=(\theta _p|_{\mathscr {H}_p})^{-1}$ for any $p\in P$ defines an isomorphism on $\mathfrak {g}/\mathfrak {h}$ and $\omega _p|_{\mathscr {H}_p}$ is an isomorphism onto $T_{\pi (p)}M$ so that $D_p\pi \circ \theta _p^{-1}:\,\mathfrak {g}/\mathfrak {h}{\mathop {\rightarrow }\limits ^{\sim }}T_{\pi (p)}M$ is a linear frame at $\pi (p)$. Hence, this yields a map

$$\begin{aligned} \iota :\,P\rightarrow \mathscr {F}(M),\,p\mapsto D_p\pi \circ \theta _p^{-1} \end{aligned}$$

(92)

By property (ii), we have $\Phi _h^*\theta _p(Y_p)=\theta _{ph}(D_p\Phi _{h}(Y_p))=\textrm{Ad}_{h^{-1}}(\theta _p(Y_p))$ $\forall Y_p\in T_pP$ and $h\in H$ and therefore

$$\begin{aligned} \theta ^{-1}_{ph}=D_p\Phi _{h}\circ \theta _p^{-1}\circ \textrm{Ad}_{g} \end{aligned}$$

(93)

from which we obtain

$$\begin{aligned} \iota (p\cdot h)=D_{ph}\pi \circ \theta _{ph}^{-1}=D_{ph}\pi \circ D_p\Phi _{h}\circ \theta _p^{-1}\circ \textrm{Ad}_h=\iota (p)\circ \textrm{Ad}_h \end{aligned}$$

(94)

$\forall p\in P,\,h\in H$. That is, $\iota :\,P\rightarrow \mathscr {F}(M)$ is H-equivariant and fiber-preserving so that P defines a H-reduction of the frame bundle w.r.t. the group morphism $\textrm{Ad}:\,H\rightarrow \textrm{GL}(\mathfrak {g}/\mathfrak {h})$.

Moreover, it follows that $\iota$ induces a an isomorphism (denoting by the same symbol)

$$\begin{aligned} \iota :\,P\times _{\textrm{Ad}}\mathfrak {g}/\mathfrak {h}&{\mathop {\longrightarrow }\limits ^{\sim }}TM\end{aligned}$$

(95)

$$\begin{aligned}&\longmapsto D_p\pi (\theta _p^{-1}(X)) \end{aligned}$$

(96)

between the associated vector bundle $P\times _{\textrm{Ad}}\mathfrak {g}/\mathfrak {h}$ and the tangent bundle of M.

To a Cartan connection A, one associates the Cartan curvature $F(A)\in \Omega ^2(P,\mathfrak {g})$ according to

$$\begin{aligned} F(A):=\textrm{d}A+\frac{1}{2}[A\wedge A] \end{aligned}$$

(97)

In case of a “flat” Klein geometry, the Cartan connection is given by the Maurer–Cartan form (88) which satisfies the structure Eq. (90), i.e., the Cartan curvature is identically zero. Thus F(A) indicates the deviation of a Cartan geometry from a flat Klein geometry. In fact, one can prove that a Cartan geometry is isomorphic to the homogeneous model $(G\rightarrow G/H,\theta _{\textrm{MC}})$ if and only if the corresponding Cartan curvature vanishes.

Decomposing F(A) according to the decomposition $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {g}/\mathfrak {h}$ of the Lie algebra, one obtains

$$\begin{aligned} F(A)=\textrm{pr}_{\mathfrak {h}}\circ F(A)+\textrm{pr}_{\mathfrak {g}/\mathfrak {h}}\circ F(A)=F(\omega )+\Theta ^{(\omega )}+\frac{1}{2}[\theta \wedge \theta ] \end{aligned}$$

(98)

where

$$\begin{aligned} F(\omega )=D^{(\omega )}\omega =\textrm{d}\omega +\frac{1}{2}[\omega \wedge \omega ] \end{aligned}$$

(99)

is the curvature of the connection 1-form $\omega$ and

$$\begin{aligned} \Theta ^{(\omega )}:=D^{(\omega )}\theta =\textrm{d}\theta +[\omega \wedge \theta ] \end{aligned}$$

(100)

is the corresponding torsion 2-form. To see that in fact encodes the torsion of the connection, note that $\omega$ indues a connection on the associated vector bundle $P\times _{\textrm{Ad}}\mathfrak {g}/\mathfrak {h}$ and thus, via (95), a connection $\nabla :\,\Gamma (TM)\rightarrow \Gamma (TM)$ on the tangent bundle. For vector fields $X,Y\in \Gamma (TM)$, it is given by

$$\begin{aligned} (\nabla _XY)_x=\iota ([p,X^{hor}\theta (Y^{hor})]) \end{aligned}$$

(101)

for any $x\in M$ and $p\in P$ with $\pi (p)=x$ where $X^{hor},Y^{hor}$ denote the horizontal lifts of X and Y, respectively.¹² Moreover, in general, given a representation $\rho :\,H\rightarrow \textrm{GL}(V)$ of H on a vector space V, there exists an isomorphism

$$\begin{aligned} \Omega ^{k}_{hor}(P,V)^{(H,\rho )}{\mathop {\rightarrow }\limits ^{\sim }}\Omega ^k(M,P\times _{\rho }V),\,\omega \mapsto \overline{\omega } \end{aligned}$$

(102)

between V-valued k-forms of type $(H,\rho )$ and k-forms with values in the associated bundle $P\times _{\rho }V$. Hence, we can associate with $\Theta ^{(\omega )}$ a 2-form $\overline{\Theta ^{(\omega )}}\in \Omega ^2(M,P\times _{\textrm{Ad}}\mathfrak {g}/\mathfrak {h})$, which, applying (95), yields another form $\iota \circ \overline{\Theta ^{(\omega )}}\in \Omega ^2(M)\otimes \Gamma (TM)$. For vector fields $X,Y\in \Gamma (TM)$, we then compute

$$\begin{aligned} \iota \circ \overline{\Theta ^{(\omega )}}(X_x,Y_x)&=\iota \circ [p,\textrm{d}\theta (X^{hor},Y^{hor})]\nonumber \\&=\iota \circ [p,X^{hor}\theta (Y^{hor})-Y^{hor}\theta (X^{hor})-\theta ([X^{hor},Y^{hor}])]\nonumber \\&=\nabla _XY-\nabla _YX-D_p\pi ([X,Y]^{hor})=T^{\nabla }(X_x,Y_x) \end{aligned}$$

(103)

for any $x\in M$ and $p\in P$ such that $\pi (p)=x$. Hence, $\Theta ^{(\omega )}$ indeed encodes the torsion of the associated affine connection $\nabla$ on the tangent bundle of M.

With all these observations, let us now make contact to general relativity. As seen already at the beginning, flat Minkowski spacetime can be described in terms of the metric Klein geometry $(\textrm{ISO}(\mathbb {R}^{1,3}),\textrm{SO}^+(1,3);\eta )$. Hence, we consider gravity as a metric reductive Cartan geometry $(P\rightarrow M,A;\eta )$ modeled on the metric reductive Klein geometry $(\textrm{ISO}(\mathbb {R}^{1,3}),\textrm{SO}^+(1,3);\eta )$ where $\textrm{SO}^+(1,3)\rightarrow P{\mathop {\rightarrow }\limits ^{\pi }} M$ is a principal bundle with structure group $\textrm{SO}^+(1,3)$ and $A\in \Omega ^1(P,\mathfrak {iso}(\mathbb {R}^{1,3}))$ is a Cartan connection.

By (92), we know that P defines a $\textrm{SO}^+(1,3)$-reduction of the frame bundle $\mathscr {F}(M)$ of M. As such, it induces a Lorentzian metric $g\in \Gamma (T^*M\otimes T^*M)$ on M which, for vector fields $X,Y\in \Gamma (TM)$, is defined as

$$\begin{aligned} g(X_x,Y_x):=\eta (\iota ^{-1}_p(X_x),\iota _p^{-1}(Y_x)) \end{aligned}$$

(104)

$\forall p\in P_x,\,x\in M$. Note that g is in fact well-defined, i.e., independent of the choice of $p\in P_x$, since $\iota$ is equivariant and $\eta$, by definition, is a bilinear form invariant under the Adjoint representation of $\textrm{SO}^+(1,3)$ on $\mathbb {R}^{1,3}$. Hence, M is in fact a Lorentzian manifold and P can be identified with the bundle $\mathscr {F}_{\textrm{SO}}(M)$ of Lorentz frames on M.

Let $e^I$ for $I=0,\ldots ,3$ be defined via $e=:e^IP_I$. Given a local section $s:\,M\supset U\rightarrow P$ of the bundle, the corresponding pullback then induces 1-forms (denoted by the same symbol for convenience) $e^I\equiv s^*e^I\in \Omega ^1(U)$ which satisfy

$$\begin{aligned} g_{\mu \nu }&=\eta (s^* e_{\mu },s^* e_{\nu })=e^I_{\mu }e^J_{\nu }\eta _{IJ} \end{aligned}$$

(105)

i.e., $(e^I)_I$ defines a local co-frame on M with the corresponding frame fields being given by $e_I:=s^*\iota (P_I)$. With these ingredients, we can define an action on M via

$$\begin{aligned} S(A)=\frac{1}{4\kappa }\int _{M}{s^*(F(\omega )^{IJ}\wedge e^K \wedge e^L)\epsilon _{IJKL}} \end{aligned}$$

(106)

where $\kappa =8\pi G$. This action precisely coincides with the first-order Palatini action of pure Einstein gravity. As we see, the whole theory including the underlying geometrical structure of the spacetime is completely encoded in the Cartan connection.

Following [44], there also exists another version of action (106) which depends on the Cartan connection in a more explicit way. This requires a nonvanishing cosmological constant which we take as negative for convenience (for a positive cosmological constant this in fact completely analogous (see, e.g., [44])). To this end, let us consider a Cartan geometry modeled on the Klein gravity $(\textrm{SO}(2,3),\textrm{SO}^+(1,3))$ corresponding to anti-de Sitter space. Since then $[P_I,P_J]=\frac{1}{L^2}M_{IJ}$ with L the anti-de Sitter radius, it follows that the Lorentzian part of the Cartan curvature acquires an additional contribution depending on the soldering form yielding

$$\begin{aligned} F(A)^{IJ}=F(\omega )^{IJ}+\frac{1}{L^2}e^I\wedge e^J \end{aligned}$$

(107)

One can then define the so-called MacDowell–Mansouri action as follows [44, 45]

$$\begin{aligned} S_{\text {MM}}(A)=\frac{L^2}{8\kappa }\int _{M}{s^*(F(A)^{IJ}\wedge F(A)^{KL})\epsilon _{IJKL}} \end{aligned}$$

(108)

which, in particular, solely depends on the curvature of the Cartan connection and thus has the structure of a Yang–Mills-type action. Expanding (108) using (107), it follows that the term quadratic in $F(\omega )$ is given by the well-known Gauss–Bonnet term an thus is purely topological. Hence, it follows that, up to boundary terms, (108) indeed leads back to first-order Einstein gravity with a nontrivial cosmological constant.

4 Super Cartan Geometry and Supergravity

As explained in the previous section, gravity has a very elegant geometric interpretation in terms of a Cartan geometry modeled on Klein geometry of flat Minkowski, de Sitter or anti-de Sitter spacetime. As it turns out, this description also carries over to the super category providing a geometrical foundation of supergravity. This is in fact the starting point of the D’Auria-Fré approach to supergravity [2, 3]. However, as already discussed in Sect. 2.4, in order to obtain nontrivial fermionic degrees of freedom as well as supersymmetry transformations on the body of a supermanifold, we will define the notion of super Cartan geometry using the concept of enriched categories. In [40], super Cartan structures on supermanifolds were introduced and also lifted trivially to Cartan structures in the relative category. However, to the best of the author’s knowledge, a precise definition of super Cartan geometry on (nontrivial) principal super fiber bundles in the framework of enriched categories seems not to exist so far in the mathematical literature (for a different approach toward a mathematical rigorous formulation of geometric supergravity using the notion of chiral triples see [46]). For a motivation of super Cartan geometry, let us consider first the ’flat’ case given by a super Klein geometry.¹³

Definition 4.1

A super Klein geometry is a pair $(\mathcal {G},\mathcal {H})$ consisting of a super Lie group $\mathcal {G}$ as well as an embedded super Lie subgroup $\mathcal {H}\hookrightarrow \mathcal {G}$ such that $\mathcal {G}/\mathcal {H}$ is connected.

Remark 4.2

Suppose one has given a pair $(\mathcal {G},\mathcal {H})$ of super Lie groups with $\mathcal {H}\hookrightarrow \mathcal {G}$ an embedded super Lie subgroup. By definition of the DeWitt topology, $\mathcal {G}/\mathcal {H}$ is connected iff $\textbf{B}(\mathcal {G}/\mathcal {H})\cong \textbf{B}(\mathcal {G})/\textbf{B}(\mathcal {H})$ is connected, that is, iff $(\textbf{B}(\mathcal {G}),\textbf{B}(\mathcal {H}))$ is a Klein geometry.

As shown in [33], as in the classical theory, a super Klein geometry $(\mathcal {G},\mathcal {H})$ canonically induces super fiber bundle with typical fiber $\mathcal {H}$ via

together with the natural $\mathcal {H}$-right action $\Phi :\,\mathcal {G}\times \mathcal {H}\rightarrow \mathcal {G}$ on $\mathcal {G}$. Hence $\mathcal {H}\rightarrow \mathcal {G}{\mathop {\rightarrow }\limits ^{\pi }}\mathcal {G}/\mathcal {H}$ has the structure of a principal $\mathcal {H}$-bundle. Let $(X_i)_i$ be a homogeneous basis of $\textrm{Lie}(\mathcal {G})\cong T_e\mathcal {G}=\mathfrak {g}\otimes \Lambda$ and $({}^i{\omega }{})_i$ the associated left-dual basis¹⁴ of left-invariant 1-forms ${}^i{\omega }{}\in \Omega ^1(\mathcal {G})$ on $\mathcal {G}$ satisfying $\left\langle X_{i}|{}^j{\omega }{}\right\rangle =\delta {_i^j}$, $\forall i,j=1,\ldots ,n$. On $\mathcal {G}$, one can then define the (super) Maurer–Cartan form $\theta _{\textrm{MC}}\in \Omega ^1(\mathcal {G},\mathfrak {g})$ via

$$\begin{aligned} \theta _{\textrm{MC}}:= {}^i\!{\omega }{}\otimes X_i \end{aligned}$$

(109)

yielding a smooth $\textrm{Lie}(\mathcal {G})$-valued left-invariant 1-form on $\mathcal {G}$. By definition, the fundamental vector fields on $\mathcal {G}$ correspond to the subspace of left-invariant vector fields $X\in \textrm{Lie}(\mathcal {H})$. Hence, it follows immediately from (109) that $\left\langle X|\theta _{\textrm{MC}}\right\rangle =X_e$ $\forall X\in \textrm{Lie}(\mathcal {H})$. Moreover, this also implies that the map $(\theta _{\textrm{MC}})_g:\,T_g\mathcal {G}\rightarrow T_e\mathcal {G}$ is an isomorphism of super $\Lambda$-modules for any $g\in \mathcal {G}$ (and even an isomorphism of super $\Lambda$-vector spaces if $g\in \textbf{B}(\mathcal {G})$). Finally, since the right action on $\mathcal {G}$ essentially coincides with the restriction of the group multiplication, it can be shown that [29]

$$\begin{aligned} R_{h}^{*}\theta _{\textrm{MC}}=\textrm{Ad}_{h^{-1}}\circ \theta _{\textrm{MC}} \end{aligned}$$

(110)

$\forall h\in \mathcal {H}$, where $R_{h}^{*}$ denotes the generalized pullback w.r.t. the right translation $R_h:=\Phi (\cdot ,h)$, on $\mathcal {G}$ w.r.t. $h\in \mathcal {H}$ (see [29] for more details). This motivates the following definition.

Definition 4.3

A super Cartan geometry $(\pi _{\mathcal {S}}:\,\mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {M}_{/\mathcal {S}},\mathcal {A})$ modeled on a super Klein geometry $(\mathcal {G},\mathcal {H})$ is a $\mathcal {S}$-relative principal super fiber bundle $\mathcal {H}\rightarrow \mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {M}_{/\mathcal {S}}$ with structure group $\mathcal {H}$ together with a smooth even $\textrm{Lie}(\mathcal {G})$-valued $\mathcal {S}$-relative 1-form $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})_0$ on $\mathcal {P}_{/\mathcal {S}}$ called super Cartan connection such that

• (i) $\left\langle \mathbbm {1}\otimes \widetilde{X}|\mathcal {A}\right\rangle =X$,   $\forall X\in \mathfrak {h}$

• (ii) $(\Phi _{\mathcal {S}})^*_{h}\mathcal {A}=\textrm{Ad}_{h^{-1}}\circ \mathcal {A}$,   $\forall h\in \mathcal {H}$

• (iii) choosing an embedding $\iota _{\mathcal {P}}:\,\mathcal {P}\hookrightarrow \mathcal {P}\times \mathcal {S}$, the pullback of $\mathcal {A}$ w.r.t. $\iota _{\mathcal {P}}$ yields an isomorphism $\iota _{\mathcal {P}}^*\mathcal {A}_p:\,T_p\mathcal {P}\rightarrow \textrm{Lie}(\mathcal {G})$ of free super $\Lambda$-modules for any $p\in \mathcal {P}$

where the last condition will be called the super Cartan condition. If (iii) is not satisfied, we call $\mathcal {A}$ a generalized super Cartan connection.

In the following, we will be interested on a particular subclass of super Cartan geometries. More precisely, we assume that the super Lie algebra $\mathfrak {g}$ of $\mathcal {G}$ admits a decomposition of the form $\mathfrak {g}=\mathfrak {g}/\mathfrak {h}\oplus \mathfrak {h}$ with $\mathfrak {h}$ the super Lie algebra of $\mathfrak {h}$ and $\mathfrak {g}/\mathfrak {h}$ a super vector space which is invariant w.r.t. the Adjoint action of $\mathcal {H}$. As in the classical theory, we will call such a super Cartan geometry reductive.

Thus, let $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})_0$ be a super Cartan connection corresponding to a reductive super Cartan geometry. Let us split this connection according to the decomposition $\mathfrak {g}=\mathfrak {g}/\mathfrak {h}\oplus \mathfrak {h}$ of the super Lie algebra of $\mathcal {G}$ yielding

$$\begin{aligned} \mathcal {A}=\textrm{pr}_{\mathfrak {g}/\mathfrak {h}}\circ \mathcal {A}+\textrm{pr}_{\mathfrak {h}}\circ \mathcal {A}=:E+\omega \end{aligned}$$

(111)

where E will be called the supervielbein. It then follows from condition (i) and (ii) above that $\omega$ defines $\mathcal {S}$-relative super connection 1-form according to Sect. 2.23. Moreover, the supervielbein is even and, since $\mathfrak {g}/\mathfrak {h}$ defines a super $\mathcal {H}$-module, horizontal of type $(\mathcal {H},\textrm{Ad})$, i.e., $E\in \Omega ^1_{hor}(\mathcal {P}_{/\mathcal {S}},\mathfrak {g}/\mathfrak {h})_0$.

Let $s\in \textbf{B}(\mathcal {S})$ and $\iota _{\mathcal {P}}:\,\mathcal {P}\rightarrow \mathcal {S}\times \mathcal {P}:\,p\mapsto (s,p)$ be a smooth embedding. This induces a smooth horizontal 1-form $\iota _{\mathcal {P}}^*E$ on $\mathcal {P}$ which, by condition (iii), is non-degenerate. Furthermore, it follows immediately that $\iota _{\mathcal {P}}^*\omega \in \Omega ^1(\mathcal {P},\mathfrak {h})_0$ defines an ordinary super connection 1-form on $\mathcal {P}$. Hence, as in Sect. 3, it follows that this induces a morphism between $\mathcal {P}$ and the frame bundle $\mathscr {F}(\mathcal {M})$ via

$$\begin{aligned} \mathcal {P}\rightarrow \mathscr {F}(\mathcal {M}),\,p\mapsto D_p\pi \circ E_p^{-1} \end{aligned}$$

(112)

where, for any $p\in \mathcal {P}$, $D_p\pi \circ E_p^{-1}:\,(\mathfrak {g}/\mathfrak {h})\otimes \Lambda {\mathop {\rightarrow }\limits ^{\sim }}T_{\pi (p)}M$ is an isomorphism of free super $\Lambda$-modules. Hence, $\mathcal {P}$ defines a $\mathcal {H}$-reduction of $\mathscr {F}(\mathcal {M})$. Moreover, it follows that $\mathcal {M}$ has the same dimension as the super vector space $\mathfrak {g}/\mathfrak {h}$.

We want to apply the above definition to supergravity. Hence, we consider super Cartan geometry $(\pi _{\mathcal {S}}:\,\mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {M}_{/\mathcal {S}},\mathcal {A})$ modeled over the Klein geometry¹⁵$(\textrm{ISO}(\mathbb {R}^{1,3|4}),\textrm{Spin}^+(1,3))$ corresponding to super Minkowski spacetime or $(\textrm{OSp}(1|4),\textrm{Spin}^+(1,3))$ for super AdS spacetime. In both cases the super Lie algebra $\mathfrak {g}$ can be decomposed as

$$\begin{aligned} \mathfrak {g}=\mathfrak {g}_0\oplus \mathfrak {g}_1\cong \mathfrak {spin}^+(1,3)\oplus \mathbb {R}^{1,3}\oplus \Delta _{\mathbb {R}}=:\mathfrak {spin}^+(1,3)\oplus \mathfrak {t} \end{aligned}$$

(113)

with $\Delta _{\mathbb {R}}$ the vector space of the four-dimensional real Majorana representation $\kappa _{\mathbb {R}}$ and the super vector space $\mathfrak {t}\equiv \mathfrak {t}^{1,3|4}$, which in case of vanishing cosmological constant, can be identified with the super Lie algebra of the super translation group $\mathcal {T}^{1,3|4}$ (see Remark 2.18). Hence, the supervielbein decomposes as

$$\begin{aligned} E=\textrm{pr}_{\mathfrak {t}}\circ \mathcal {A}=:\theta +\psi =:\theta ^IP_I+\psi ^{\alpha }Q_{\alpha } \end{aligned}$$

(114)

with $\psi \in \Omega ^1_{hor}(\mathcal {P}_{/\mathcal {S}},\Delta _{\mathbb {R}})_0$ and $\theta \in \Omega ^1_{hor}(\mathcal {P}_{/\mathcal {S}},\mathbb {R}^{1,3})_0$ defining $\mathcal {S}$-relative horizontal 1-forms of type $(\textrm{Spin}^+(1,3),\textrm{Ad})$ called the Rarita–Schwiger field and co-frame, respectively. Hence, the super Cartan connection takes the form

$$\begin{aligned} \mathcal {A}=\theta ^IP_I+\frac{1}{2}\omega ^{IJ}M_{IJ}+\psi ^{\alpha }Q_{\alpha } \end{aligned}$$

(115)

Due to (112), the supervielbein induces a $\textrm{Spin}^+(1,3)$-reduction $\mathcal {P}\rightarrow \mathscr {F}(\mathcal {M})$ of the frame bundle. Applying the body functor, this in turn induces a $\textrm{Spin}^+(1,3)$-reduction $P:=\textbf{B}(\mathcal {P})\rightarrow \mathscr {F}(M)$ of the frame bundle of the body $M:=\textbf{B}(\mathcal {M})$. That is, the body carries a spin structure.

Remark 4.4

Let $\iota :\,\mathcal {P}\rightarrow \mathcal {S}\times \mathcal {P}$ be an embedding and suppose $\mathcal {P}$ is trivial, i.e., $\mathcal {P}\cong \mathcal {M}\times \mathcal {G}$ with respect to a global trivialization $s:\,\mathcal {M}\rightarrow \mathcal {P}$. This in turn induces a (homogeneous) basis $(s_I,s_{\alpha })$ of global sections $s_I:=[s,P_I]$, $s_{\alpha }:=[s,Q_{\alpha }]$ of the associated super vector bundle $\mathcal {E}:=\mathcal {P}\times _{\textrm{Spin}^+(1,3)}(\mathfrak {t}\otimes \Lambda )$. This yields an isomorphism

$$\begin{aligned} \Omega ^1(\mathcal {M},\mathfrak {t}^*)&\rightarrow \Omega ^1(\mathcal {M},\mathcal {E})\cong \Omega ^1_{hor}(\mathcal {P},\mathfrak {t})^{(\textrm{Spin}^+(1,3),\textrm{Ad})}\nonumber \\ \omega&\mapsto \omega ^{I}s_{I}+\omega ^{\alpha }s_{\alpha } \end{aligned}$$

(116)

It thus follows from condition (iii) of a super Cartan connection that, via (116), the pullback $\iota ^*E\in \Omega ^1_{hor}(\mathcal {P},\mathfrak {t})$ induces a non-degenerate 1-form $\tilde{E}\in \Omega ^1(\mathcal {M},\mathfrak {t}^*)$. Consequently, the pair $(\mathcal {M},\tilde{E})$ defines super Cartan structure in the sense of [40]. Conversely, if $(\mathcal {M},\tilde{E})$ is a super Cartan structure with $\tilde{E}\in \Omega ^1(\mathcal {M},\mathfrak {t})$ being non-degenerate, one can use (116) to get a non-degenerate 1-form $E\in \Omega ^1_{hor}(\mathcal {P},\mathfrak {t})$ which can be lifted trivially to a $\mathcal {S}$-relative 1-form $\mathbbm {1}\otimes E\in \Omega ^1_{hor}(\mathcal {P}_{/\mathcal {S}},\mathfrak {t})$ satisfying condition (iii) above. Hence, definition (4.3) provides a generalization of super Cartan structures in the sense of [40] to a generalized notion of super Cartan connections on nontrivial $\mathcal {S}$-relative principal super fiber bundles.

The action of $\mathcal {N}=1$, $D=4$ supergravity can be obtained as a supersymmetric generalization of the MacDowell–Mansouri action (108) on AdS spacetime [45]. We therefore consider super Cartan geometry modeled on the super Klein geometry $(\textrm{OSp}(1|4),\textrm{Spin}^+(1,3))$. The Cartan curvature of $\mathcal {A}$ is given by

$$\begin{aligned} F(\mathcal {A})=\textrm{d}\mathcal {A}+\frac{1}{2}[\mathcal {A}\wedge \mathcal {A}]=\textrm{d}\mathcal {A}+\frac{1}{2}(-1)^{|T_{\underline{A}}||T_{\underline{B}}|}\mathcal {A}^{\underline{A}}\wedge \mathcal {A}^{\underline{B}}\otimes [T_{\underline{A}},T_{\underline{B}}]\nonumber \\ \end{aligned}$$

(117)

with respect to a homogeneous basis $(T_{\underline{A}})_{\underline{A}}$ of $\mathfrak {osp}(1|4)$, $\underline{A}\in (I,IJ,\alpha )$, where the minus sign in (117) appears due to the (anti)commutation of $T_{\underline{A}}$ and $\mathcal {A}^{\underline{B}}$. It then follows from $[M_{IJ},P_K]=\eta _{IK}P_J-\eta _{JK}P_I$ as well as (55)–(58) that the components of $F(\mathcal {A})$ in the translational part of the super Lie algebra, also called the supertorsion, take the form

$$\begin{aligned} F(\mathcal {A})^I&=\textrm{d}\theta ^I+{\omega }{^I_J}\wedge \theta ^J+\frac{1}{4}((-1)^{|Q_{\alpha }||Q_{\beta }|}\psi ^{\alpha }\wedge \psi ^{\beta }\otimes [Q_{\alpha },Q_{\beta }])^{I}\nonumber \\&=\Theta ^{(\omega ) I}-\frac{1}{4}\bar{\psi }\wedge \gamma ^I\psi \end{aligned}$$

(118)

since $(-1)^{|Q_{\alpha }||Q_{\beta }|}=-1$, with $\Theta ^{(\omega )}$ is the torsion 2-form associated with the spin connection $\omega$. For the spinorial components, we find

$$\begin{aligned} F(\mathcal {A})^{IJ}&=\textrm{d}\omega ^{IJ}+{\omega }{^I_K}\wedge \omega ^{KJ}+\frac{1}{2L^2}\theta ^I\wedge \theta ^J-\frac{1}{2}(\psi ^{\alpha }\wedge \psi ^{\beta }\otimes [Q_{\alpha },Q_{\beta }])^{IJ}\nonumber \\&=F(\omega )^{IJ}+\frac{1}{L^2}\theta ^I\wedge \theta ^J-\frac{1}{4L}\bar{\psi }\wedge \gamma ^{IJ}\psi \end{aligned}$$

(119)

with $F(\omega )$ the curvature of $\omega$. Finally, for the odd part, we obtain

$$\begin{aligned} \Sigma ^{\alpha }:=F(\mathcal {A})^{\alpha }&=\textrm{d}\psi ^{\alpha }+\frac{1}{4}\omega ^{IJ}{(\gamma _{IJ})}{^{\alpha }_{\beta }}\wedge \psi ^{\beta }-\frac{1}{2L}e^I\wedge \psi ^{\beta }{(\gamma _{I})}{^{\alpha }_{\beta }}\nonumber \\&=D^{(\omega )}\psi ^{\alpha }-\frac{1}{2L}e^I\wedge \psi ^{\beta }{(\gamma _{I})}{^{\alpha }_{\beta }} \end{aligned}$$

(120)

with $D^{(\omega )}\psi =\textrm{d}\psi +\frac{1}{4}\omega ^{IJ}\gamma _{IJ}\wedge \psi$ the exterior covariant derivative in the Majorana representation. Before we state the supergravity action, note that, by the rheonomy principle, the physical degrees of freedom are completely encoded on the body of the supermanifold. Hence, it suffices to consider local sections $\tilde{s}$ of the pullback bundle $\iota _{\mathcal {M}_0}^*\mathcal {P}_{/\mathcal {S}}$ on the bosonic sub supermanifold $\iota _{\mathcal {M}_0}:\,(\mathcal {M}_0)_{/\mathcal {S}}\hookrightarrow \mathcal {M}_{/\mathcal {S}}$. With respect to these type of localizations, the super MacDowell–Mansouri action for $\mathcal {N}=1$, $D=4$ supergravity then reads as follows [45, 47]

$$\begin{aligned} S(\mathcal {A})=\frac{L^2}{2\kappa }\int _{M}{\tilde{s}^*\left( \frac{1}{4}F(\mathcal {A})^{IJ}\wedge F(\mathcal {A})^{KL}\epsilon _{IJKL}+\frac{i}{L}\bar{\Sigma }\wedge \gamma _{*}\Sigma \right) } \end{aligned}$$

(121)

Since the trace over internal indices in (121) is manifestly $\textrm{Spin}^+(1,3)$-invariant, it is clear that the action invariant under local $\textrm{Spin}^+(1,3)$-gauge transformations and therefore does not depend on the choice of the local trivialization. To see that this in fact leads to $\mathcal {N}=1$ supergravity, let us further evaluate the individual terms in (121). From (119), we conclude

$$\begin{aligned} F(\mathcal {A})^{IJ}\wedge F(\mathcal {A})^{KL}\epsilon _{IJKL}=&\frac{2}{L^2}F(\omega )^{IJ}\wedge \theta ^K\wedge \theta ^L\epsilon _{IJKL}+F(\omega )^{IJ}\wedge F(\omega )^{KL}\epsilon _{IJKL}\nonumber \\&-\frac{1}{2L^3}\bar{\psi }\wedge \gamma ^{IJ}\psi \wedge \theta ^K\wedge \theta ^L\epsilon _{IJKL}\nonumber \\&-\frac{1}{2L}F(\omega )^{IJ}\wedge \bar{\psi }\wedge \gamma ^{KL}\psi \epsilon _{IJKL}\nonumber \\&+\frac{1}{L^4}\theta ^I\wedge \theta ^J\wedge \theta ^K\wedge \theta ^L\epsilon _{IJKL} \end{aligned}$$

(122)

For the second term, note that the conjugate $\bar{\Sigma }$ is given by

$$\begin{aligned} \bar{\Sigma }=\Sigma ^TC&=\textrm{d}\bar{\psi }+\frac{1}{4}\bar{\psi }\wedge \omega ^{IJ}\gamma _{IJ}-\frac{1}{2L}\bar{\psi }\gamma _I\wedge \theta ^I\nonumber \\&=D^{(\omega )}\bar{\psi }-\frac{1}{2L}\bar{\psi }\gamma _I\wedge \theta ^I \end{aligned}$$

(123)

Hence, this yields

$$\begin{aligned} \bar{\Sigma }\wedge \gamma _{*}\Sigma&=D^{(\omega )}\bar{\psi }\wedge \gamma _{*}D^{(\omega )}\psi +\frac{1}{2L}D^{(\omega )}\bar{\psi }\wedge \gamma _{*}\gamma _{I}\psi \wedge \theta ^I\nonumber \\&\quad +\frac{1}{2L}\bar{\psi }\wedge \gamma _{*}\gamma _ID^{(\omega )}\psi \wedge \theta ^I-\frac{1}{4L^2}\bar{\psi }\wedge \gamma _{IJ}\gamma _{*}\psi \wedge \theta ^I\wedge \theta ^J\nonumber \\&=D^{(\omega )}\bar{\psi }\wedge \gamma _{*}D^{(\omega )}\psi +\frac{1}{L}\bar{\psi }\wedge \gamma _{*}\gamma _ID^{(\omega )}\psi \wedge \theta ^I\nonumber \\&\quad -\frac{1}{8iL^2}\bar{\psi }\wedge \gamma ^{KL}\gamma _{*}\psi \wedge \theta ^I\wedge \theta ^J\epsilon _{IJKL} \end{aligned}$$

(124)

where in the last line we used that ${\epsilon }{_{IJ}^{KL}}\gamma _{KL}=2i\gamma _{IJ}\gamma _{*}$. If we insert (122) and (124) into (121), this then leads to

$$\begin{aligned} S(\mathcal {A})=&\frac{1}{2\kappa }\int _{M}\left( \frac{1}{2}F(\omega )^{IJ}\wedge e^K\wedge e^L\epsilon _{IJKL}-\frac{1}{4L}\bar{\psi }\wedge \gamma ^{IJ}\psi \wedge e^K\wedge e^L\epsilon _{IJKL}\right. \nonumber \\&\left. +\frac{1}{4L^2}e^I\wedge e^J\wedge e^K\wedge e^L\epsilon _{IJKL}+ i\bar{\psi }\wedge \gamma _{*}\gamma _ID^{(\omega )}\psi \wedge e^I\right) \nonumber \\&-\frac{L}{16\kappa }\int _M{\epsilon _{IJKL}F(\omega )^{IJ}\wedge \bar{\psi }\wedge \gamma ^{KL}\psi -8iD^{(\omega )}\bar{\psi }\wedge \gamma _{*}D^{(\omega )}\psi } \end{aligned}$$

(125)

where we set $\tilde{s}^*\theta ^I=:e^I$ and we have dropped a topological term proportional to the Gauss-Bonnet term. In fact, it turns out that (125) can be simplified even further. Therefore, recall that $\psi$, as odd part of the supervielbein E, defines a horizontal 1-form of type $(\textrm{Spin}^+(1,3),\kappa _{\mathbb {R}})$, i.e., $\psi \in \Omega ^1(P_{/\mathcal {S}},\Delta _{\mathbb {R}})^{(\textrm{Spin}^+(1,3),\kappa _{\mathbb {R}})}$. Hence, from the standard rules for the exterior covariant derivative, one obtains the Bianchi identity

$$\begin{aligned} D^{(\omega )}D^{(\omega )}\psi =\kappa _{\mathbb {R}*}(F(\omega ))\wedge \psi =\frac{1}{4}F(\omega )^{IJ}\gamma _{IJ}\wedge \psi \end{aligned}$$

(126)

One can thus equivalently write for the first term on the last line of (125)

$$\begin{aligned} \epsilon _{IJKL}F(\omega )^{IJ}\wedge \bar{\psi }\wedge \gamma ^{KL}\psi&=2i\bar{\psi }\wedge F(\omega )^{IJ}\gamma _{IJ}\gamma _{*}\wedge \psi \nonumber \\&=8i\bar{\psi }\wedge \gamma _{*}D^{(\omega )}D^{(\omega )}\psi \nonumber \\&=8iD^{(\omega )}\bar{\psi }\wedge \gamma _{*}D^{(\omega )}\psi -\textrm{d}(8i\bar{\psi }\wedge \gamma _{*}D^{(\omega )}\psi ) \end{aligned}$$

(127)

Thus dropping the boundary term, one finally ends up with

$$\begin{aligned} S(\mathcal {A})=&\frac{1}{2\kappa }\int _{M}\left( \frac{1}{2}F(\omega )^{IJ}\wedge e^K\wedge e^L\epsilon _{IJKL}+ i\bar{\psi }\wedge \gamma _{*}\gamma _ID^{(\omega )}\psi \wedge e^I\right. \nonumber \\&\left. -\frac{1}{4L}\bar{\psi }\wedge \gamma ^{IJ}\psi \wedge e^K\wedge e^L\epsilon _{IJKL}+\frac{1}{4L^2}e^I\wedge e^J\wedge e^K\wedge e^L\epsilon _{IJKL} \right) \end{aligned}$$

(128)

This is precisely the action of $\mathcal {N}=1$, $D=4$ anti-de Sitter supergravity as stated for instance in [48]. By definition, $S(\mathcal {A})\in H^{\infty }(\mathcal {S})_0$ describes an even functional on the parameterizing supermanifold $\mathcal {S}$. In particular, since we are working in the relative category, it transforms covariantly under change of parameterization. That is, given a change of parameterization $\lambda :\,\mathcal {S}'\rightarrow \mathcal {S}$, one has

$$\begin{aligned} \lambda ^*S(\mathcal {A})=S(\lambda ^*\mathcal {A})\in H^{\infty }(\mathcal {S}')_0 \end{aligned}$$

(129)

These are precisely the properties to be satisfied by a physical quantity according to [11] since physical degrees of freedom should not depend on the choice of a particular parameterizing supermanifold.

Remark 4.5

Due to (129), it is suggested in [11] to take $\mathcal {S}$ as the (infinite dimensional) configuration space as the action functional defined on any other finite-dimensional parameterizing supermanifold then may be obtained via pullback (i.e., change of parameterization). In this way, in particular, it follows that fermionic fields are described in terms of odd functionals on configuration space. This is in fact the interpretation of fermionic fields in pAQFT [12, 13].

Note that, for $\mathcal {S}\cong \{*\}$, action (121) reduces to the action (108) of pure Einstein gravity. One thus needs to ensure that $\mathcal {S}$ has nontrivial odd dimensions because otherwise all fermionic fields in $S(\mathcal {A})$ would simply drop off. But, as we have seen from the computations above, the anticommuting nature of the fermion fields has been crucial for the derivation of the action (128) and thus for the interpretation of supergravity in terms of super Cartan geometry. In fact, this is also important for the supersymmetry of the action (128).

As will become clear below, in the Cartan geometric picture, supersymmetry transformations have the interpretation in terms of (field dependent) gauge transformations. Therefore, we need to lift $\mathcal {A}$ to a principal connection. This can be done considering associated bundles.

Lemma 4.6

Let $\mathcal {H}\rightarrow \mathcal {P}\rightarrow \mathcal {M}$ be a principal super fiber bundle with structure group $\mathcal {H}$ and $\mathcal {H}$-right action $\Phi :\,\mathcal {P}\times \mathcal {H}\rightarrow \mathcal {H}$. Let $\lambda :\,\mathcal {H}\rightarrow \mathcal {G}$ be a morphism of super Lie groups and $\rho _{\lambda }:=\mu _{\mathcal {G}}\circ (\lambda \times \textrm{id}_{\mathcal {G}}):\,\mathcal {H}\times \mathcal {G}\rightarrow \mathcal {G}$ the induced $H^{\infty }$-smooth left action of $\mathcal {H}$ on $\mathcal {G}$. On $\mathcal {P}\times \mathcal {H}$ consider the map

$$\begin{aligned} \Phi ^{\times }:\,(\mathcal {P}\times \mathcal {H})\times \mathcal {G}\rightarrow \mathcal {P}\times \mathcal {G},\,((p,h),g)\mapsto (\Phi (p,g),\rho _{\lambda }(g^{-1},h)) \end{aligned}$$

(130)

Then, $\Phi ^{\times }$ defines an effective $H^{\infty }$-smooth $\mathcal {G}$-right action on $\mathcal {P}\times \mathcal {G}$. Let $\mathcal {E}:=\mathcal {P}\times _{\rho _{\lambda }}\mathcal {G}:=(\mathcal {P}\times \mathcal {H})/\mathcal {G}$ be the corresponding coset space and $\pi _{\mathcal {E}}:\,\mathcal {E}\rightarrow \mathcal {M}$ be defined as

$$\begin{aligned} \pi _{\mathcal {E}}:\,\mathcal {E}\rightarrow \mathcal {M},\,[p,g]\mapsto \pi _{\mathcal {P}}(p) \end{aligned}$$

(131)

Then, $\mathcal {E}$ can be equipped with the structure of a $H^{\infty }$ supermanifold such that $\pi _{\mathcal {E}}$ is a $H^{\infty }$-smooth surjective map and $(\mathcal {E},\pi _{\mathcal {E}},\mathcal {M},\mathcal {G})$ turns into a principal $\mathcal {G}$-bundle.

Furthermore, let $\iota :\,\mathcal {P}\rightarrow \mathcal {P}\times _{\mathcal {H}}\mathcal {G}$ be defined as $\iota (p):=[p,e]$ $\forall p\in \mathcal {P}$, then $\iota$ is smooth, fiber-preserving and $\mathcal {H}$-equivariant in the sense that $\iota \circ \Phi =\widetilde{\Phi }\circ (\iota \times \lambda )$. Moreover, if $\lambda :\,\mathcal {H}\hookrightarrow \mathcal {G}$ is an embedding, then $\iota$ is an embedding.

Proof

The proof is almost the same as in the classical theory (see [19, 29]). One only needs care about smoothness in the various constructions as smoothness is generally not preserved under partial evaluation of smooth functions. But, it turns out that everything works fine since $e\in \textbf{B}(\mathcal {G})$. $\square$

Remark 4.7

Given the associated bundle $\mathcal {P}\times _{\rho _{\lambda }}\mathcal {G}$ as constructed in Corollary 4.6, as in Sect. 2.4, we can lift it trivially to a $\mathcal {S}$-relative principal $\mathcal {G}$-bundle $(\mathcal {P}\times _{\rho _{\lambda }}\mathcal {G})_{/\mathcal {S}}\cong \mathcal {P}_{/\mathcal {S}}\times _{\rho _{\lambda }}\mathcal {G}$.

Proposition 4.8

Let $\mathcal {H}\rightarrow \mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {M}_{/\mathcal {S}}$ be a $\mathcal {S}$-relative principal super fiber bundle with structure group $\mathcal {H}$ as well as $(\mathcal {G},\mathcal {H})$ a super Klein geometry. Then, there is a bijective correspondence between generalized super Cartan connections in $\Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})_0$ and super connection 1-forms in $\Omega ^1(\mathcal {P}_{/\mathcal {S}}\times _{\mathcal {H}}\mathcal {G},\mathfrak {g})_0$ with $\mathcal {P}_{/\mathcal {S}}\times _{\mathcal {H}}\mathcal {G}$ the $\mathcal {G}$-extension of $\mathcal {P}_{/\mathcal {S}}$ as constructed in Remark 4.7.

Sketch of Proof.

One direction is immediate, i.e., given a $\mathcal {S}$-relative super connection 1-form $\mathcal {A}$ on $\mathcal {P}_{/\mathcal {S}}\times _{\mathcal {H}}\mathcal {G}$, the pullback $\hat{\iota }^*\mathcal {A}$ w.r.t. the embedding $\hat{\iota }:=\textrm{id}\times \iota :\,\mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {P}_{/\mathcal {S}}\times _{\mathcal {H}}\mathcal {G}$, with $\iota$ as defined in 4.6, yields a generalized super Cartan connection on $\mathcal {P}_{/\mathcal {S}}$ according to Definition 4.3.

Conversely, suppose $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})_0$ is a generalized super Cartan connection. Let $\hat{\pi }:\,\mathcal {P}_{/\mathcal {S}}\times \mathcal {G}\rightarrow \mathcal {P}_{/\mathcal {S}}\times _{\mathcal {H}}\mathcal {G}$ be the canonical projection. If $\hat{\Phi }_{\mathcal {S}}$ denotes the $\mathcal {G}$-right action on $\mathcal {P}_{/\mathcal {S}}\times _{\mathcal {H}}\mathcal {G}$, it follows that the fundamental vector fields are given by

$$\begin{aligned} \widetilde{Y}_{[p,g]}&=(\hat{\Phi }_{\mathcal {S}})_{[p,g]*}(Y_e)=D_{(p,g)}\hat{\pi }(0_{p},L_{g*}Y) \end{aligned}$$

(132)

for any $Y\in \textrm{Lie}(\mathcal {G})$ and $p\in \mathcal {P}_{/\mathcal {S}}$, $g\in \mathcal {G}$, where we used the generalized tangent map. Furthermore, for any $X_{p}\in T_{p}(\mathcal {P}_{/\mathcal {S}})$, one has

$$\begin{aligned} D_{(p,g)}\hat{\pi }(X_{p},0_g)=\hat{\iota }_{*}((\Phi _{\mathcal {S}})_{g*}X_{p}) \end{aligned}$$

(133)

$\forall (p,g)\in \mathcal {P}_{/\mathcal {S}}\times \mathcal {G}$. Hence, this yields

$$\begin{aligned} D_{(p,g)}\hat{\pi }(X_{p},Y_g)&=D_{(p,g)}\hat{\pi }(X_{p},0_g)+D_{(p,g)}\hat{\pi }(0_{p},Y_g)\nonumber \\&=\hat{\iota }_{*}((\Phi _{\mathcal {S}})_{g*}X_p)+D_{(p,g)}\hat{\pi }(0_p,L_{g*}\circ L_{g^{-1}*}(Y_g))\nonumber \\&=\iota _{*}((\Phi _{\mathcal {S}})_{g*}X_p)+\widetilde{\left\langle Y_g|\theta _{\textrm{MC}}\right\rangle }_{[p,g]} \end{aligned}$$

(134)

Therefore, if there exists a super connection 1-form $\hat{\iota }_{*}\mathcal {A}$ whose pullback under $\hat{\iota }$ is given by $\mathcal {A}$, then it necessarily has to be of the form

$$\begin{aligned} \left\langle D_{(p,g)}\hat{\pi }(X_p,Y_g)|\hat{\iota }_{*}\mathcal {A}_{[p,g]}\right\rangle =\textrm{Ad}_{g^{-1}}\left\langle X_p|\mathcal {A}_p\right\rangle +\left\langle Y_g|\theta _{\textrm{MC}}\right\rangle \end{aligned}$$

(135)

In particular, as $\hat{\pi }$ is a submersion, it is uniquely determined by (135). Using that the kernel of $\hat{\pi }_{*}$ is given by

$$\begin{aligned} \textrm{ker}\,D_{(p,g)}\hat{\pi }=\{((\mathbbm {1}\otimes Y)_{p},-R_{g*}Y)|Y\in \textrm{Lie}(\mathcal {H})\} \end{aligned}$$

(136)

it is easy to see that (135) is indeed well-defined. That $\hat{\iota }_{*}\mathcal {A}$ is $\mathcal {G}$-equivariant and maps fundamental vector fields to the corresponding generator follows from the respective properties of $\mathcal {A}$ and the Maurer–Cartan form. Finally, by construction, it is clear that $\hat{\iota }_{*}\mathcal {A}$ defines an even $\mathcal {S}$-relative 1-form. For more details see [19]. $\square$

By Proposition 4.8, we can thus lift the super Cartan connection $\mathcal {A}$ to principal connection

$$\begin{aligned} \widehat{\mathcal {A}}:=\iota _{*}\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}}\times _{\textrm{Spin}^+(1,3)}\textrm{OSp}(1|4),\mathfrak {osp}(1|4))_0 \end{aligned}$$

(137)

on the associated $\textrm{OSp}(1|4)$-bundle. Since, in this way, it defines a super connection 1-form à la Ehresmann, we can consider local gauge transformations generated by vectors $Q_{\alpha }$ in the odd part of the super Lie algebra.

Let $s:\,U_{/\mathcal {S}}\rightarrow \mathcal {P}_{/\mathcal {S}}$ be a local section which can be extended to a local section $\widehat{s}:=(\textrm{id}\times \iota )\circ s$ of the $\textrm{OSp}(1|4)$-bundle. We consider a local gauge transformation $g:\,\mathcal {M}_{/\mathcal {S}}\rightarrow \textrm{OSp}(1|4)$. If $\widehat{s}':=\widehat{s}\cdot g$, the connection $\widehat{\mathcal {A}}$ then transforms as

$$\begin{aligned} \widehat{\mathcal {A}}_{\widehat{s}'}=\textrm{Ad}_{g^{-1}}\circ \mathcal {A}_s+g^{-1}\textrm{d}g \end{aligned}$$

(138)

If we consider an infinitesimal local gauge transformation generated by a smooth map $\epsilon \in H^{\infty }(\mathcal {M}_{/\mathcal {S}},\Delta _{\mathbb {R}}\otimes \Lambda _1)$, this yields

$$\begin{aligned} \delta _{\epsilon }\mathcal {A}_s=\textrm{d}\epsilon +[\mathcal {A}_s\wedge \epsilon ]=D^{(\mathcal {A})}\epsilon \end{aligned}$$

(139)

Hence, using the super Lie algebra relations (55)–(58), it follows that the individual components of the super connection 1-form transform as

$$\begin{aligned} \delta _{\epsilon }e^I&=\frac{1}{2}\bar{\epsilon }\gamma ^I\psi \end{aligned}$$

(140)

$$\begin{aligned} \delta _{\epsilon }\psi ^{\alpha }&=D^{(\omega )}\epsilon ^{\alpha }-\frac{1}{2L}e^I{(\gamma _I)}{^{\alpha }_{\beta }}\epsilon ^{\beta } \end{aligned}$$

(141)

$$\begin{aligned} \delta _{\epsilon }\omega ^{IJ}&=\frac{1}{2L}\bar{\epsilon }\gamma ^{IJ}\psi \end{aligned}$$

(142)

The infinitesimal gauge transformations (140) and (141), when pulled back to the body manifold, are precisely the local supersymmetry transformations as stated in [48]. However, under the additional transformation (142), $S(\mathcal {A})$ is not invariant. This is only true in case of a vanishing cosmological constant, i.e., $L\rightarrow \infty$, where (142) simply drops off. Varying $S(\mathcal {A})$ w.r.t. $\omega$ yields the field equations of the spin connection which are equivalent to the supertorsion constraint

$$\begin{aligned} F(\mathcal {A})^I=0\quad \Leftrightarrow \quad \Theta ^{(\omega ) I}=\frac{1}{4}\bar{\psi }\wedge \gamma ^I\psi \end{aligned}$$

(143)

Thus, provided (143) holds, the supergravity action is indeed invariant under the full local gauge transformations generated by $\epsilon \in H^{\infty }((\mathcal {M}_0)_{/\mathcal {S}},\Delta _{\mathbb {R}}\otimes \Lambda _1)$ [49] (in fact, one can simply ignore the transformation of $\omega$ in this case). This is another nice feature of the super Cartan approach to supergravity as supersymmetry transformations have the geometrical interpretation in terms of gauge transformations which provides a link between supergravity and Yang–Mills theory [49]. This will also be crucial for the interpretation of the super Ashtekar connection in terms of a super gauge field in what follows.

Remark 4.9

Let us emphasize again that, technically, the existence of a nonvanishing $\epsilon$ relies crucially on the additional parameterizing supermanifold. Hence, working in the relative category resolves both nontrivial anticommuting fermionic fields and supersymmetry transformations on the body of a supermanifold.

5 The Super Ashtekar Connection

5.1 Derivation from the Full Theory

In the previous section, we have derived $\mathcal {N}=1$, $D=4$ anti-de Sitter supergravity considering it geometrically in terms of a super Cartan geometry. In particular, all the basic entities of the theory turn out to be completely encoded in the super Cartan connection

$$\begin{aligned} \mathcal {A}=e^IP_I+\frac{1}{2}\omega ^{IJ}M_{IJ}+\psi ^{\alpha }Q_{\alpha } \end{aligned}$$

(144)

taking values in the super Lie algebra $\mathfrak {osp}(1|4)$ corresponding to the underlying super Klein geometry.

In 1986 [50], Ashtekar introduced his self-dual variables which give ordinary gravity the structure of a $\textrm{SL}(2,\mathbb {C})$ Yang–Mills theory. This construction is based on a particular structure of the internal symmetry algebra. In fact, the complexification of the Lie algebra of the orthochronous Lorentz group $\textrm{SO}^+(1,3)$ has a decomposition of the form

$$\begin{aligned} \mathfrak {so}^+(1,3)_{\mathbb {C}}=\mathfrak {su}(2)_{\mathbb {C}}\oplus \mathfrak {su}(2)_{\mathbb {C}}\cong \textrm{sl}(2,\mathbb {C})\oplus \mathfrak {sl}(2,\mathbb {C}) \end{aligned}$$

(145)

and thus splits into two proper $\mathfrak {sl}(2,\mathbb {C})$ subalgebras (viewed as complex Lie algebras of complex $\textrm{SL}(2,\mathbb {C})$). This precisely corresponds to the decomposition of the spin connection $\omega$ into its self-dual $A^+$ and anti self-dual part $A^-$, respectively. In this sense, the self-dual variables can be regarded as chiral sub components of the 4D spin connection.

Hence, the natural question arises whether such a construction carries over to the super category. As we will see in what follows, this will be indeed the case, even for extended supersymmetry. Therefore, recall that the Ashtekar variables $A^{\pm }$ are defined as the (anti) self-dual part of the four-dimensional spin connection $\omega$ according to

$$\begin{aligned} A^{\pm }:=\frac{1}{2}\bigg {[}\frac{1}{2}\left( \omega ^{IJ}\mp \frac{i}{2}{\epsilon }{^{IJ}_{KL}}\omega ^{KL}\right) \bigg {]}M_{IJ} \end{aligned}$$

(146)

which takes values in the complexification $\mathfrak {spin}(1,3)_{\mathbb {C}}$ of the Lie algebra of the spin double cover $\textrm{Spin}^+(1,3)$ of the orthochronous Lorentz group generated by $M_{IJ}$. After some simple algebra, it follows that

$$\begin{aligned} A^{\pm }&=\frac{1}{2}\bigg {[}\frac{1}{2}\left( \omega ^{IJ}\mp \frac{i}{2}{\epsilon }{^{IJ}_{KL}}\omega ^{KL}\right) \bigg {]}M_{IJ}\nonumber \\&=\frac{1}{2}\left( \frac{1}{4}{\epsilon }{^i_{kl}}{\epsilon }{_i^{mn}}\omega ^{kl}M_{mn}\mp \frac{i}{2}{\epsilon }{^{0i}_{kl}}\omega ^{kl}M_{0i}\mp \frac{i}{2}{\epsilon }{_{0i}^{KL}}\omega ^{0i}M_{KL}+\omega ^{0i}M_{0i}\right) \nonumber \\&=\left( -\frac{1}{2}{\epsilon }{^i_{kl}}\omega ^{kl}\mp i\omega ^{0i}\right) \frac{1}{2}\left( -\frac{1}{2}{\epsilon }{_{i}^{kl}}M_{kl}\pm iM_{0i}\right) =:A^{\pm i}T_i^{\pm } \end{aligned}$$

(147)

where $A^{\pm i}:=\Gamma ^i\mp i K^i$, $i=1,\ldots ,3$, with $\Gamma ^i:=-\frac{1}{2}{\epsilon }{^i_{kl}}\omega ^{kl}$ the 3D spin connection and $K^i:=\omega ^{0i}$ the extrinsic curvature. Moreover, $T^{\pm }_i$ are given by

$$\begin{aligned} T_i^{\pm }=\frac{1}{2}(J_i\pm i\tilde{K}_i) \end{aligned}$$

(148)

with $J_i=-\frac{1}{2}{\epsilon }{_{i}^{jk}}M_{jk}$ and $\tilde{K}_i=M_{0i}$, the generators of local rotations and boosts, respectively. These satisfy the commutation relations

$$\begin{aligned}{}[T^{\pm }_i,T_j^{\pm }]={\epsilon }{_{ij}^k}T_k^{\pm } \end{aligned}$$

(149)

and therefore generate the chiral $\mathfrak {sl}(2,\mathbb {C})$ subalgebras of $\mathfrak {spin}^+(1,3)_{\mathbb {C}}$. Since ${\gamma }{_{0i}}=\frac{i}{2}{\epsilon }{_{ijk}}{\gamma }{^{jk}}\gamma _{*}$, one has the important identity

$$\begin{aligned} \frac{1}{4}\left( -{\epsilon }{_i^{jk}}\gamma _{jk}\pm i\gamma _{0i}\right) =\frac{\gamma _{*}\pm \mathbbm {1}}{2}\frac{i}{2}\gamma _{0i} \end{aligned}$$

(150)

Hence, using (150) it immediately follows that the exterior covariant derivative induced by $A^+$ (resp. $A^-$) acts on purely unprimed (resp. primed) spinor indices according to

$$\begin{aligned} D^{(A^+)}\psi ^A=\textrm{d}\psi ^A+{{A^{+}}}{^A_B}\wedge \psi ^B,\quad \text {and}\quad D^{(A^-)}\psi _{A'}=\textrm{d}\psi _{A'}+{{A^{-}}}{_{A'}^{B'}}\wedge \psi _{B'}\nonumber \\ \end{aligned}$$

(151)

respectively, where ${{A^{+}}}{^A_B}=A^{+i}{(\tau _i)}{^A_B}$ and ${{A^{-}}}{_{A'}^{B'}}=A^{-i}{(\tau _i)}{_{A'}^{B'}}$ (note that the second identity in (151) can be obtained taking the complex conjugate of the first one). Hence, focusing for the moment on the self-dual sector, let us consider the chiral subcomponents $Q^r_A$ of the Majorana charges $Q^r_{\alpha }$. From (55), it then follows, using again (150),

$$\begin{aligned}{}[T_k^+,Q^i_A]=Q^i_{B}{(\tau _k)}{^B_A} \end{aligned}$$

(152)

that is, the Weyl fermions $Q^r_A$ transform in the fundamental representation of $\mathfrak {sl}(2,\mathbb {C})$ as to be expected from (152). Next, let us consider the anti commutator relations between two Weyl fermions. In the Weyl representation, the charge conjugation matrix C admits a block diagonal form given by $C=\textrm{diag}(i\epsilon ,i\epsilon )$. From this, we immediately deduce

$$\begin{aligned} (C\gamma ^{IJ})_{AB}M_{IJ}&=\frac{i}{2}\left( \epsilon (\sigma ^I\bar{\sigma }^J-\sigma ^J\bar{\sigma }^I)\right) _{AB}M_{IJ}\nonumber \\&=2i(\epsilon \sigma ^i)_{AB}M_{0i}-{\epsilon }{^{ij}_k}(\epsilon \sigma ^k)_{AB}M_{ij}\nonumber \\&=2(\epsilon \sigma ^i)_{AB}\left( iM_{0i}-\frac{1}{2}{\epsilon }{_{i}^{jk}}M_{jk}\right) =2(\epsilon \sigma ^i)_{AB}T_i^+ \end{aligned}$$

(153)

Hence, using (58), it follows

$$\begin{aligned}{}[Q^i_{A},Q^j_{B}]=\delta ^{ij}\frac{1}{L}(\epsilon \sigma ^k)_{AB}T_k^+-\frac{i}{2L}\epsilon _{AB}T^{ij} \end{aligned}$$

(154)

The R-symmetry generators $T_{rs}$ do not mix the chiral components of the Majorana charges $Q^r_{\alpha }$. Thus, to summarize, we have found that $(T_i^+,T_{rs},Q_{A}^r)$ indeed form a proper chiral sub super Lie algebra of $\mathfrak {osp}(\mathcal {N}|4)_{\mathbb {C}}$ with the graded commutation relations

$$\begin{aligned}{}[T^+_i,T_j^+]&={\epsilon }{_{ij}^k}T_k^+ \end{aligned}$$

(155)

$$\begin{aligned}&=Q^r_{B}{(\tau _i)}{^B_A} \end{aligned}$$

(156)

$$\begin{aligned}&=\delta ^{rs}\frac{1}{L}(\epsilon \sigma ^i)_{AB}T_i^+-\frac{i}{2L}\epsilon _{AB}T^{rs} \end{aligned}$$

(157)

$$\begin{aligned}&=\delta ^{qr} Q_{A}^p-\delta ^{pr} Q_{A}^q \end{aligned}$$

(158)

which precisely coincide with graded commutation relations of the complex orthosymplectic Lie superalgebra $\mathfrak {osp}(\mathcal {N}|2)_{\mathbb {C}}$, the extended supersymmetric generalization of the isometry algebra of $D=2$ anti-de Sitter space [37]. Performing the Inönü–Wigner contraction, i.e., taking the limit $L\rightarrow \infty$, this yields the extended $D=2$ super Poincaré algebra. Similarly, considering the anti self-dual sector, one obtains a proper sub super Lie algebra generated by the anti chiral components $(T_i^{-},T_{rs},Q ^{A'})$ which again forms $\mathfrak {osp}(\mathcal {N}|2)_{\mathbb {C}}$.

For the construction of a super analog of Ashtekar’s self-dual variables, in what follows, let us restrict to the non-extended case $\mathcal {N}=1$ in which case the theory is described in terms of the super Cartan connection $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {osp}(1|4))$ (144). Based on the above observations, we introduce the following graded self-dual variables

$$\begin{aligned} \mathcal {A}^{+}:=A^{+ i}T_i^{+}+\psi ^AQ_A\quad \text {and}\quad \mathcal {A}^{-}:=A^{- i}T_i^{-}+\psi _{A'}Q^{A'} \end{aligned}$$

(159)

which define $\mathcal {S}$-relative 1-forms on the $\mathcal {S}$-relative principal super fiber bundle $\textrm{Spin}^+(1,3)\rightarrow \mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {M}_{/\mathcal {S}}$ with values in the chiral sub superalgebra $\mathfrak {osp}(1|2)_{\mathbb {C}}$ generated by $(T_i^+,Q_A)$ and $(T_i^-,Q^{A'})$, respectively. In case of a vanishing cosmological constant, this gives the $D=2$ super Poincaré algebra.

Remark 5.1

The $D=2$ super Poincaré algebra has an equivalent description in terms of a direct sum super Lie algebra $\textrm{sl}(2,\mathbb {C})\oplus \Pi \mathbb {C}^2$, where $\Pi \mathbb {C}^2$ is regarded as a purely odd super vector space. Given the fundamental representation $\rho :\,\mathfrak {sl}(2,\mathbb {C})\rightarrow \textrm{End}(\mathbb {C}^2)$ of $\mathfrak {sl}(2,\mathbb {C})$, the graded commutation relations are given by

$$\begin{aligned}{}[(x,v),(x',v')]:=([x,x'],\rho (x)(v)-\rho (x')(v')) \end{aligned}$$

(160)

$\forall (x,v),(x',v')\in \textrm{sl}(2,\mathbb {C})\oplus \Pi \mathbb {C}^2$. In the mathematical literature, such kind of superalgebras are usually called generalized Takiff Lie superalgebras [51].

For the rest of this section, let us focus on the chiral case (and the case of nonvanishing cosmological constant), the considerations for $\mathcal {A}^-$ are in fact completely analogous. Let us consider the complexification $\mathcal {P}^{\mathbb {C}}$ of $\mathcal {P}$ defined as the associated super $\textrm{Spin}^+(1,3)_{\mathbb {C}}$-bundle

$$\begin{aligned} \mathcal {P}^{\mathbb {C}}:=\mathcal {P}\times _{\textrm{Spin}^+(1,3)}\textrm{Spin}^+(1,3)_{\mathbb {C}} \end{aligned}$$

(161)

via the obvious mapping $\textrm{Spin}^+(1,3)\rightarrow \textrm{Spin}^+(1,3)_{\mathbb {C}}$. Due to (145), this bundle can be reduced to a super $\textrm{SL}(2,\mathbb {C})$-bundle $\mathcal {Q}\hookrightarrow \mathcal {P}^{\mathbb {C}}$. It then follows from the chiral nature of the self-dual Ashtekar connection $A^+$ as well as the chiral sub components $\psi ^A$ of the Rarita–Schwinger field that $\mathcal {A}^+$ induces a well-defined 1-form

$$\begin{aligned} \mathcal {A}^+\in \Omega ^1(\mathcal {Q}_{/\mathcal {S}},\mathfrak {osp}(1|2)_{\mathbb {C}})_0 \end{aligned}$$

(162)

which, by construction, satisfies the properties (i) and (ii) of definition 4.3. That is, $\mathcal {A}^+$ defines a generalized super Cartan connection on the $\mathcal {S}$-relative $\textrm{SL}(2,\mathbb {C})$-bundle $\mathcal {Q}_{/\mathcal {S}}$. This is precisely the super Ashtekar connection as first introduced in [8]. There, this connection arose by studying the constraint algebra of the canonical theory to be discussed in the following section. Here, we have derived it using the geometrical description of $\mathcal {N}=1$, $D=4$ supergravity in terms of super Cartan geometry and studying the chiral structure of the underlying supersymmetry algebra corresponding to the super Klein geometry. In particular, it follows that it has the interpretation in terms of a generalized super Cartan connection on the $\mathcal {S}$-relative $\textrm{SL}(2,\mathbb {C})$-bundle $\mathcal {Q}_{/\mathcal {S}}$.

Since the chiral structure of the supersymmetry algebra is even preserved in case of extended supersymmetry leading to the orthosymplectic Lie superalgebra $\mathfrak {osp}(\mathcal {N}|2)_{\mathbb {C}}$, this suggests that it might be possible to generalize this construction to extended $D=4$ supergravity theories including further chiral components of the supermultiplet in the definition of the super Ashtekar connection. In fact, graded Ashtekar variables for extended SUGRA theories have been studied in [52, 53] and in context of constrained super BF-theory in [54, 55]. Recently, in [56], these variables have been derived for pure $\mathcal {N}=2$, $D=4$ supergravity in the presence of boundaries in a purely geometric way following the Cartan geometric approach as studied in this present article.

Remark 5.2

Let us emphasize that, since the above construction relied crucially on the chiral description of the theory, this construction cannot be carried over to real Barbero-Immirzi parameters! In fact, real $\beta$ requires the consideration of both chiral components of the Majorana fermions $Q^r_{\alpha }$. But, the anti commutator between $Q^r_A$ and $Q^{r A'}$ is proportional to $P_I$ which is related to the soldering form $\theta$ corresponding to the dual electric field (see next section). Hence, this does not lead to a proper sub super Lie algebra and the super Ashtekar connection cannot be defined.

5.2 The Canonical Theory and the Constraint Algebra

As seen in he previous section, the chiral structure of the underlying supersymmetry algebra of $\mathcal {N}=1$, $D=4$ AdS supergravity enabled us to introduce a graded generalization of Ashtekar’s self-dual variables. So far, the discussion has been purely off-shell. As a next step, one thus needs to derive a chiral variant of the supergravity action (128) to specify the dynamics. We will therefore follow [57]. For a purely geometric discussion revealing the underlying $\textrm{OSp}(1|2)_{\mathbb {C}}$-gauge symmetry of the chiral theory in a manifest way, see [56]. Using the chiral variables $A^{+ i}$ and $\psi ^A$, it follows that the MacDowell–Mansouri action (121) (resp. (128)) can be split in its self-dual $S^+$ and anti self-dual part $\overline{S^+}$, respectively, i.e.,

$$\begin{aligned} S(\mathcal {A})=S^{+}+\overline{S^+} \end{aligned}$$

(163)

with

$$\begin{aligned} S^+&:=\frac{i}{2\kappa }\int _M{\bigg {(}{\Sigma }{^{AB}}\wedge {F(A^+)}{_{AB}}+\chi _A\wedge D^{(A^+)}\psi ^A}\nonumber \\&\quad +\frac{i}{2L}{\Sigma }{^{AB}}\wedge \psi _A\wedge \psi _B+\frac{i}{2L}\chi _A\wedge \chi ^A+\frac{i}{4 L^2}{\Sigma }{^{AB}}\wedge {\Sigma }{_{AB}}\bigg {)} \end{aligned}$$

(164)

where we set $\chi _A:=e_{AA'}\wedge \bar{\psi }^{A'}$ and $F(A^+)=\textrm{d}A^{+}+A^{+}\wedge A^+$ denotes the (self-dual) curvature of $A^+$. Moreover, note that the soldering form $e\in \Omega ^1_{hor}(\mathcal {P}_{/\mathcal {S}},\mathbb {R}^{1,3})_0$ induces a horizontal 2-form $\Sigma \in \Omega ^2(\mathcal {P}_{/\mathcal {S}},\mathfrak {spin}^+(1,3))_0$ which, in spinor indices, takes the form $\Sigma ^{AA'BB'}=e^{AA'}\wedge e^{BB'}$. Due to antisymmetry, it can be decomposed according to

$$\begin{aligned} \Sigma ^{AA'BB'}=\epsilon ^{AB}\Sigma ^{A'B'}+\epsilon ^{A'B'}\Sigma ^{AB} \end{aligned}$$

(165)

with $\Sigma ^{AB}$ and $\Sigma ^{A'B'}$ the self-dual anti self-dual part of $\Sigma ^{AA'BB'}$, respectively, given by

$$\begin{aligned} \Sigma ^{AB}:=\frac{1}{2}\epsilon _{A'B'}\Sigma ^{AA'BB'}\quad \text {and}\quad \Sigma ^{A'B'}:=\frac{1}{2}\epsilon _{AB}\Sigma ^{AA'BB'} \end{aligned}$$

(166)

As we see, action (164) only depends on the chiral degrees freedom $(A^+,\psi ^A)$ contained in the super Ashtekar connection as well as the soldering form e. The remaining field components are fixed via reality conditions (see discussion below). Provided these reality conditions are satisfied, it follows that the imaginary part of the action (164) becomes a boundary term [57]. Thus, in this way, one reobtains the field equations of ordinary real $\mathcal {N}=1$ supergravity.

The chiral action (164) can be written in a manifest invariant way using the super Ashtekar connection $\mathcal {A}^+$ as introduced in the previous section. To this end, let us define

$$\begin{aligned} \mathcal {E}:=\Sigma ^i T^+_i+L\chi ^A Q_A \end{aligned}$$

(167)

which we call the super electric field. It then follows that (164) can be written as

$$\begin{aligned} S^+=\frac{i}{2\kappa }\int _M{\textrm{str}\left( \mathcal {E}\wedge F(\mathcal {A}^+)+\frac{1}{4L^2}\mathcal {E}\wedge \mathcal {E}\right) } \end{aligned}$$

(168)

where $F(\mathcal {A}^+)$ is the curvature of $\mathcal {A}^+$ and “$\textrm{str}$” denotes the supertrace. Using the transformation relations (140)–(142) under local SUSY transformations, one finds that the super electric field transforms under the adjoint representation of the chiral subgroup $\textrm{OSp}(1|2)_{\mathbb {C}}$ of $\textrm{OSp}(1|4)$. Hence, the form (168) of the chiral action shows that $S^+$ is manifestly invariant under local $\textrm{OSp}(1|2)_{\mathbb {C}}$ gauge transformations.

As a next step, let us perform the canonical analysis of (164). The canonical decomposition of chiral $\mathcal {N}=1$ supergravity has been investigated in [8, 57] and, for arbitrary (real) Barbero-Immirzi parameters in [20, 58,59,60] and even higher spacetime dimensions in [61, 62]. Here, we want to follow a different strategy by using the manifest invariant form (168) of the chiral action.

Thus, let $M\cong \mathbb {R}\times \Sigma$ with $\Sigma$ a 3D Cauchy hypersurface. Furthermore, let $\partial _t$ denote the global timelike vector field which can be decomposed as

$$\begin{aligned} \partial _t=Nn+\vec {N} \end{aligned}$$

(169)

with $n:=e_0$ the unit normal vector field orthogonal to the tangent space $T\Sigma$ of $\Sigma$, the lapse function N and $\vec {N}$ the so-called shift vector field tangential to $\Sigma$. On TM, we define the projection $P^{\parallel }:\,TM\rightarrow TM$ via

$$\begin{aligned} P^{\parallel }(X):=X-\textrm{d}t(X)\partial _t \end{aligned}$$

(170)

$\forall X\in TM$. This induces the subspace

$$\begin{aligned} T^{\parallel }M:=P^{\parallel }(TM)=\left\{ X\in TM|\,\textrm{d}t(X)=0\right\} \end{aligned}$$

(171)

Moreover, the projection induces via pullback another projection $P_{\parallel }$ on co-vector fields according to

$$\begin{aligned} P_{\parallel }T:=T\circ P^{\parallel } \end{aligned}$$

(172)

where $P^{\parallel }$ on the right hand side acts on each slot. With these preparations, we are ready to perform the $3+1$-split of the action functional (168). To this end, let

$$\begin{aligned} \underset{{\leftarrow }}{\mathcal {A}}:=P_{\parallel }\mathcal {A} \end{aligned}$$

(173)

Furthermore, we set

$$\begin{aligned} \underset{{\leftarrow }}{\dot{\mathcal {A}}}:=L_{\partial _t}\underset{{\leftarrow }}{\mathcal {A}} \end{aligned}$$

(174)

Since $L_{\partial _t}\circ P_{\parallel }=P_{\parallel }\circ L_{\partial _t}$, this yields

$$\begin{aligned} \underset{{\leftarrow }}{\dot{\mathcal {A}}}=P_{\parallel }L_{\partial _t}\mathcal {A}=P_{\parallel }(\textrm{d}\mathcal {A}_0+i_{\partial _t}\textrm{d}\mathcal {A}) \end{aligned}$$

(175)

where $\mathcal {A}_0:=\mathcal {A}(\partial _t)$. Using the definition (170) and (172), one finds

$$\begin{aligned} F^+:=F(\mathcal {A}^+)=\underset{{\leftarrow }}{F}^+ +i_{\partial _t}F^+\wedge \textrm{d}t \end{aligned}$$

(176)

as well as

$$\begin{aligned} \mathcal {E}=\underset{{\leftarrow }}{\mathcal {E}}+i_{\partial _t}\mathcal {E}\wedge \textrm{d}t \end{aligned}$$

(177)

where, using (174), it follows

$$\begin{aligned} i_{\partial _t}F^+=\underset{{\leftarrow }}{\dot{\mathcal {A}}}-P_{\parallel }\textrm{d}\mathcal {A}^+_0+[\mathcal {A}^+_0\wedge \underset{{\leftarrow }}{\mathcal {A}}^+] \end{aligned}$$

(178)

Inserting all this into the covariant action functional (168), this yields

$$\begin{aligned} S(e,\mathcal {A}^+)=\frac{i}{\kappa }\int _{\mathbb {R}}{\textrm{d}t\int _{\Sigma }{\left\langle \mathcal {E}\wedge \dot{\mathcal {A}}^+\right\rangle +\left\langle \mathcal {A}_0^+,\textrm{d}_{\mathcal {A}^+}\mathcal {E}\right\rangle +\left\langle i_{\partial _t}\mathcal {E}\wedge \left( F^++\frac{1}{2L^2}\mathcal {E}\right) \right\rangle }}\nonumber \\ \end{aligned}$$

(179)

From (179), we can directly read off the canonically conjugate variables of the corresponding canonical theory. Choosing the real homogeneous basis $(T_{\underline{A}})_{\underline{A}}\equiv (T_i^+,Q_A)$ of $\textrm{OSp}(1|2)$, we can expand $\mathcal {A}^+=\mathcal {A}^{+\underline{A}}T_{\underline{A}}$. Moreover, we introduce the super electric field $\mathcal {E}^a_{\underline{A}}$ on $\Sigma$ defined as

$$\begin{aligned} \mathcal {E}^a_{\underline{A}}:=\frac{1}{2}\epsilon ^{abc}\mathscr {T}_{\underline{B}\underline{A}}\mathcal {E}^{\underline{B}}_{bc},\quad \text {with } \mathscr {T}_{\underline{A}\underline{B}}:=\left\langle T_{\underline{A}},T_{\underline{B}}\right\rangle \end{aligned}$$

(180)

Thus, according to (179), it follows that the pair $(\mathcal {A}_a^{+\underline{A}},\mathcal {E}^a_{\underline{A}})$ build up a graded symplectic phase space with graded Poisson relations

$$\begin{aligned} \{\mathcal {E}^a_{\underline{A}}(x),\mathcal {A}_b^{+\underline{B}}(y)\}=i\kappa \delta ^a_b\delta ^{\underline{B}}_{\underline{A}}\delta ^{(3)}(x,y) \end{aligned}$$

(181)

Moreover, from (179) we can immediately read off the super Gauss constraint

$$\begin{aligned} \mathscr {G}[\alpha ]:=\int _{\Sigma }{\left\langle \alpha ,\textrm{d}_{\mathcal {A}^+}\mathcal {E}\right\rangle } \end{aligned}$$

(182)

which, using the graded Poisson relations (181), satisfies

$$\begin{aligned} \{\mathscr {G}[\alpha ],\mathscr {G}[\alpha ']\}=\mathscr {G}([\alpha ,\alpha ']) \end{aligned}$$

(183)

Hence, the super Gauss constraint generates local $\textrm{OSp}(1|2)_{\mathbb {C}}$-gauge transformations. This was the starting point in [10, 63] for quantizing this theory studying super holonomies corresponding to $\mathcal {A}^+$ which we have constructed in mathematical rigorous way in Sect. 2.4. The construction of the state space of the quantum theory will be considered in the following section.

Remark 5.3

Actually, in context of Yang–Mills theory, we need principal connections. Therefore, as in the previous section, using Proposition 4.8, we simply lift $\mathcal {A}^+$ to the associated $\mathcal {S}$-relative $\textrm{OSp}(1|2)_{\mathbb {C}}$-bundle $\mathcal {Q}_{/\mathcal {S}}\times _{\textrm{SL}(2,\mathbb {C})}\textrm{OSp}(1|2)_{\mathbb {C}}$. With respect to this connection, we can define holonomies and construct the state space in the manifest approach to loop quantum supergravity (LQSG).

Finally, let us comment on the remaining constraints of the canonical classical theory. From (179), we can read off the constraint

$$\begin{aligned} \mathscr {H}[N,\vec {N},\eta ]=\int _{\Sigma }{ \left\langle i_{\partial _t}\mathcal {E}\wedge \left( F^++\frac{1}{2L^2}\mathcal {E}\right) \right\rangle } \end{aligned}$$

(184)

which depends on the lapse function, the shift vector field as well the spinorial smearing function $\eta _{A'}:=\bar{\psi }_{t A'}$. (184) contains all the remaining constraints, i.e., the vector $H_a$, the Hamiltonian H as well as the right SUSY constraint $S^{R A'}$, as usually considered in canonical supergravity. In fact, using that $\mathcal {E}^a=(E^a_i,-i\sqrt{\kappa }\pi ^a_A)$ with $E^a_i=\sqrt{q}e^a_i$ and

$$\begin{aligned} \pi ^a_A&={\epsilon }{^{abc}}\bar{\psi }_{b}^{A'}e_{c AA'} \end{aligned}$$

(185)

one finds that (184) can be expanded in the form

$$\begin{aligned} \mathscr {H}[N,\vec {N},\eta ]=\int _ {\Sigma }{\textrm{d}^3x\,\left( N^a H_a+NH+\eta _{A'}S^{R A'}\right) } \end{aligned}$$

(186)

with

$$\begin{aligned} H_a=\frac{i}{\kappa }E_i^{b}F(A^+)^i_{ab}-{\epsilon }{^{bcd}}e_{a AA'}\bar{\psi }_{b}^{A'}D^{(A^+)}_{c}\psi _{d}^A \end{aligned}$$

(187)

and

$$\begin{aligned} H=&-\frac{E_i^aE_j^b}{2\kappa \sqrt{q}}{\epsilon }{^{ij}_k}F(A^+)^k_{ab}-\epsilon ^{abc}\bar{\psi }_a^{A'}n_{AA'}D_b^{(A^+)}\psi _c^A\nonumber \\&+\frac{E^a_iE^b_j}{2L\sqrt{q}}{\mathring{\epsilon }}{^{ijk}}\left( \psi _{a A}n_{BA'}\sigma _k^{AA'}\psi _b^B-\bar{\psi }_a^{A'}n_{AA'}\sigma _k^{AB'}\bar{\psi }_{b B'}\right) +\frac{3}{\kappa L^2}\sqrt{q}\quad \end{aligned}$$

(188)

where $n^{AA'}$ is the spinor corresponding to the unit normal vector field $n^{\mu }$ orthogonal to the time slices $\Sigma _t$ in the 3+1-decomposition. Furthermore, for the right SUSY constraint, we find

$$\begin{aligned} S^{R A'}=\epsilon ^{ijk}\frac{E^b_jE^c_k}{2\sqrt{q}}\sigma _i^{AA'}\left( 2\epsilon _{AB}D_{[b}^{(A^+)}\psi _{c]}^B+\frac{1}{2L}\pi ^a_A\epsilon _{abc}\right) \end{aligned}$$

(189)

To conclude this section, let us finally comment on the reality conditions enforced on the self-dual Ashtekar connection in order to recover ordinary real supergravity. As discussed in detail in [20, 57], in the canonical theory, it follows that the reality conditions are equivalent to the requirement that the 3D spin connection part $\Gamma ^i:=-{\epsilon }{^i_{jk}}\omega ^{jk}$ of $A^+$ satisfies the torsion equation

$$\begin{aligned} D^{(\Gamma )}e^i\equiv \textrm{d}e^i+{\epsilon }{^{i}_{jk}}\Gamma ^j\wedge e^k=\Theta ^{(\Gamma ) i}=\frac{i\kappa }{2}\psi ^A\wedge \bar{\psi }^{A'}\sigma ^i_{AA'} \end{aligned}$$

(190)

This equation has the unique solution

$$\begin{aligned} \Gamma ^i\equiv \Gamma ^i(e)+C^i(e,\psi ,\bar{\psi }) \end{aligned}$$

(191)

with $\Gamma ^i(e)$ the torsion free metric connection

$$\begin{aligned} \Gamma ^i_a(e)=-\epsilon ^{ijk}e_j^b\left( \partial _{[a}e_{b]k}+\frac{1}{2}e^c_k e^l_a\partial _{[c}e_{b]l}\right) \end{aligned}$$

(192)

and $C^i$ the contorsion tensor given by

$$\begin{aligned} C^i_a=\frac{i\kappa }{4|e|}\epsilon ^{bcd}e_d^i\left( 2\psi ^A_{[a}\bar{\psi }_{b]}^{A'}e_{c AA'}-\psi ^A_{b}\bar{\psi }_{c}^{A'}e_{a AA'}\right) \end{aligned}$$

(193)

Thus, the reality conditions for the bosonic degrees of freedom are given by

$$\begin{aligned} A^{+ i}_a+(A^{+ i}_a)^*=2\Gamma ^i_a(e)+2C^i_a(e,\psi ,\bar{\psi }),\quad E^a_i=\Re (E^a_i) \end{aligned}$$

(194)

These ensure that, provided the initial conditions satisfy (194), the dynamical evolution remains in the real sector of the complex phase space, i.e., the phase space of ordinary real $\mathcal {N}=1$ supergravity.

5.3 Invariant Haar Measure on $\textrm{OSp}(1|2)_{\mathbb {C}}$ and the State Space of LQSG

In order to construct the (kinematical) Hilbert space of LQSG and to implement local $\textrm{OSp}(1|2)_{\mathbb {C}}$-gauge transformations, we want to derive the super analog of a Haar measure on the complex super Lie group $\textrm{OSp}(1|2)_{\mathbb {C}}$. In the literature, there exist various results in this direction in both the algebraic or concrete approach to supermanifold theory. In [64] for instance, invariant measures where constructed in the algebraic setting for various real super Lie groups including the super unitary groups $\textrm{U}(m|n)$ as well the real orthosymplectic groups $\textrm{OSp}(m|n)$. There, one uses the equivalent description of super Lie groups in terms of super Harish Chandra pairs. In fact, it turns out that super Lie groups $\mathcal {G}$ have a relatively simple structure that they are completely determined by the respective body $\textbf{B}(\mathcal {G})$ and the super Lie algebra $\mathfrak {g}$. However, in the algebraic setting, this correspondence remains rather implicit.

In the $H^{\infty }$ category (or more generally for $\mathcal {A}$-manifolds), in [29], a concrete algorithm was given constructing invariant Haar measures for arbitrary (real) super Lie groups. This is based on the existence of a concrete relation between a super Lie group $\mathcal {G}$ and the data $(\textbf{B}(\mathcal {G}),\mathfrak {g})$. More precisely, one has the following

Theorem 5.4

(Super Harish–Chandra pair (after [29, 33])). Let $\mathcal {G}$ be a $H^{\infty }$ super Lie group with body $G:=\textbf{B}(\mathcal {G})$. Then, $\mathcal {G}$ is globally split, that is, it is diffeomorphic to split supermanifold $\textbf{S}(\mathfrak {g}_1,G)$ associated with the trivial vector bundle $G\times \mathfrak {g}_1\rightarrow G$ via the canonical mapping

$$\begin{aligned} \Phi :\,\textbf{S}(\mathfrak {g}_1,G)&\rightarrow \mathcal {G}\nonumber \\ (g,X)&\mapsto g\cdot \exp (X) \end{aligned}$$

(195)

where $\textbf{S}(\mathfrak {g}_1,G)\cong \textbf{S}(G)\times (\mathfrak {g}_1\otimes \Lambda )_0$. In particular, there exists a unique super Lie group structure on $\textbf{S}(\mathfrak {g}_1,G)$ such that (195) turns into a morphism of super Lie groups. Hence, any $H^{\infty }$ super Lie group is uniquely determined via (195) by the data $(G,\mathfrak {g})$ called a super Harish–Chandra pair consisting of its body G as well as the super Lie algebra $\mathfrak {g}=\mathfrak {g}_0\oplus \mathfrak {g}_1$.

As it has been shown in [65] in the Molotkov–Sachse approach, such a correspondence via (195) even holds in case of infinite-dimensional Fréchet super Lie groups.

A $\mathbb {C}$-linear map $\int _{\mathcal {G}}:\,H_c^{\infty }(\mathcal {G},\mathbb {C})\rightarrow \mathbb {C}$ on $\mathcal {G}$ is called left-invariant integral on $\mathcal {G}$ if [66]

$$\begin{aligned} \mathbbm {1}\otimes \int _{\mathcal {G}}\circ \,\Theta _L^*=\mathbbm {1}\otimes \int _{\mathcal {G}} \end{aligned}$$

(196)

where $\Theta _L:\,\mathcal {G}\times \mathcal {G}\rightarrow \mathcal {G}\times \mathcal {G}$ is defined as $\Theta _L(g,h)=(g,\mu _{\mathcal {G}}(g,h))$. Similarly one can define right invariant integrals setting $\Theta _R:\,\mathcal {G}\times \mathcal {G}\rightarrow \mathcal {G}\times \mathcal {G},\,(g,h)\mapsto (\mu _{\mathcal {G}}(g,h),h)$. The reason for choosing $\Theta _L$ instead of just the group multiplication on $\mathcal {G}$ as usually done in the literature is that $\Theta _L$ is a proper map, i.e., the preimage of compact sets in $\mathcal {G}$ is compact in $\mathcal {G}\times \mathcal {G}$ which necessary for condition (196) to be well-defined (this is true for the group multiplication only in case $\mathcal {G}$ is compact).

Integrals on supermanifolds can be formulated in terms of Berezinian densities (see [67,68,69] for more details). A Berezinian density on a supermanifold $\mathcal {M}$ is defined as a smooth section $\Gamma _c(\textrm{Ber}(\mathcal {M}))$ with compact support of the Berezin line bundle,¹⁶$\textrm{Ber}(\mathcal {M}):=\mathscr {F}(\mathcal {M})\times _{\textrm{Ber}}\Lambda ^{\mathbb {C}}$ which is a bundle associated with the frame bundle $\mathscr {F}(\mathcal {M})$ via the one-dimensional dual representation $\textrm{GL}(m|n,\Lambda )\ni A\mapsto \textrm{Ber}(A)^{-1}\in \textrm{Aut}(\Lambda ^{\mathbb {C}})$ where $\textrm{dim}\,\mathcal {M}=(m,n)$.

For a super Lie group $\mathcal {G}$, a Berezinian density $\nu \in \Gamma _c(\textrm{Ber}(\mathcal {G}))$ then induces a left-invariant integral iff its trivial extension $\hat{\nu }$ on $\mathcal {G}\times \mathcal {G}$ satisfies

$$\begin{aligned} \Theta _L^*\hat{\nu }=\hat{\nu } \end{aligned}$$

(197)

Such a density will be called a left-invariant Haar measure on $\mathcal {G}$. To construct such an invariant measure note that, for any super Lie group $\mathcal {G}$, the tangent bundle $T\mathcal {G}$ is always trivializable with a global frame $\mathfrak {p}\in \Gamma (\mathscr {F}(\mathcal {G}))$ induced by a homogeneous basis $(e_i,f_j)$ of left-invariant vector fields $e_i,f_j\in \mathfrak {g}$ on $\mathcal {G}$, $i=1,\ldots ,m$ and $j=1,\ldots ,n$ with $\textrm{dim}\,\mathcal {G}=(m,n)$. In particular, this yields a global section $\nu _{\mathfrak {g}}:=[\mathfrak {p},1]\in \Gamma (\textrm{Ber}(\mathcal {G}))$ of the associated Berezin line bundle which, by construction, automatically defines a left-invariant Haar measure. With respect to local coordinates $(x,\xi )$, this density is of the form

$$\begin{aligned} \nu _{\mathfrak {g}}=[\mathfrak {p}=(\partial _{x^i},\partial _{\xi _j})\cdot X,1]=[(\partial _{x^i},\partial _{\xi _j}),\textrm{Ber}(X)^{-1}] \end{aligned}$$

(198)

where X denotes the matrix representation of the left-invariant vector fields w.r.t. the induced coordinate derivatives. To find an explicit expression for X, one can then use the equivalent description of $\mathcal {G}$ in terms of the corresponding super Harish–Chandra pair $(G,\mathfrak {g})$ via identification (195). This requires an intense use of the Baker–Campbell–Hausdorff formula and thus involves various powers of the (right) adjoint representation $\textrm{ad}_R:\,\textrm{Lie}(\mathcal {G})\rightarrow \textrm{End}_R(\textrm{Lie}(\mathcal {G})),\,X\mapsto [X,\cdot ]$. As shown in [33], the matrix representation then takes the form

$$\begin{aligned} X(x,\xi )=\begin{pmatrix} C(x) &{}\quad C(x)\cdot H(\xi )\\ A(\xi ) &{}\quad B(\xi ) \end{pmatrix} \end{aligned}$$

(199)

where C(x) as well as $H(\xi )$, $A(\xi )$ and $B(\xi )$ are submatrices depending purely on even and odd coordinates, respectively, and which are defined via

$$\begin{aligned}&\textrm{ad}_R(v)(e_i)=:f_j{A(\xi )}{^j_i},\quad b_{+}(\textrm{ad}_{R}(v))f_j=:f_k{B(\xi )}{^k_j}\nonumber \\&\qquad \text {and}\quad h(\textrm{ad}_{R}(v))f_j=e_i{H(\xi )}{^i_j} \end{aligned}$$

(200)

with $v:=f_j\xi ^j\in (\mathfrak {g}_1\otimes \Lambda )_0$ and real functions

$$\begin{aligned} b_{+}(t):= & {} \frac{t\cosh (t)}{\sinh (t)}=1+\frac{1}{3}t^2-\frac{1}{45}t^4+\ldots ,\nonumber \\ h(t):= & {} \frac{e^t-1}{e^t+1}=\frac{1}{2}t-\frac{1}{24}t^3+\ldots \end{aligned}$$

(201)

Moreover, C(x) is determined via the matrix representation of the even left-invariant vector fields $e_i$ via

$$\begin{aligned} e_i|_{(g,v)}=\partial _{x_j}{C}{^j_i}(x)+\partial _{\xi _k}{A}{^k_i}(\xi ) \end{aligned}$$

(202)

In particular, when restricting on the body, C can be identified with the matrix representation of the left-invariant vector fields on G. The left-invariant integral on $\textbf{S}(\mathfrak {g}_1,G)$ for smooth functions $f\in H^{\infty }_c(\textbf{S}(\mathfrak {g}_1,G))\cong C_c^{\infty }(G)\otimes \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}\mathfrak {g}_1^{*}$ then takes the form

$$\begin{aligned} \int _{\mathcal {G}}{f\nu _{\mathfrak {g}}}&=\int {\textrm{d}^nx\int _B{\textrm{d}^n\xi \,f(x,\xi )\textrm{Ber}(X)^{-1}(x,\xi )}}\nonumber \\&=\int {\textrm{d}^nx\,C(x)^{-1}\int _B{\textrm{d}^n\xi \,\frac{\det B(\xi )}{\det (\mathbbm {1}-H(\xi )B(\xi )^{-1}A(\xi ))}f(x,\xi )}}\nonumber \\&=:\int _{G}{\textrm{d}\mu _H(g)\int _B{\textrm{d}^n\xi \,\Delta (\xi )f(g,\xi )}} \end{aligned}$$

(203)

where $\mu _H$ is the induced left-invariant Haar measure on G and $\int _B$ denotes the usual Berezin integral on $\mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}\mathfrak {g}_1^{*}$. Hence, the derivation of the invariant integral on $\mathcal {G}$ boils down to the choice of an invariant Haar measure on the body G as well as the derivation of the density $\Delta (\xi )$ in the Berezin integral which, according to (200) and (201), only involves the computation of the matrix representation of the adjoint representation on $\mathfrak {g}$.

Let us apply this algorithm to compute the invariant Haar measure on the complex orthosymplectic group $\textrm{OSp}(1|2)_{\mathbb {C}}$ (in case of the real orthosymplectic group, this has been done explicitly already in [33]). Therefore, let us introduce a homogeneous basis $(e_1,e_2,e_3,f_1,f_2)$ on $\mathfrak {osp}(1|2)_{\mathbb {C}}$ defining

$$\begin{aligned}&e_1:=2iT_{3}^{+},\quad \quad e_{2}:=i(T_{1}^{+}+i T_{2}^{+}),\quad \quad e_{3}:=i(T_{1}^{+}-i T_{2}^{+}) \end{aligned}$$

(204)

$$\begin{aligned}&\quad f_1=\frac{1}{\sqrt{2}}(i-1)Q_{+},\quad \quad f_2:=\frac{1}{\sqrt{2}}(1-i)Q_{-} \end{aligned}$$

(205)

Setting $L=\frac{1}{2}$, it then follows from (155)–(158) that the commutators among the even generators of $\mathfrak {osp}(1|2)_{\mathbb {C}}$ satisfy

$$\begin{aligned}{}[e_1,e_2]=2e_2,\quad [e_1,e_3]=-2e_3,\quad [e_2,e_3]=e_1 \end{aligned}$$

(206)

which are the standard commutation relations of $\mathfrak {sl}(2,\mathbb {C})$. For the mixed commutators between even and odd generators, we find

$$\begin{aligned}&[e_1,f_1]=f_1,\quad \quad [e_2,f_1]=0,\quad \quad [e_3,f_1]=-f_2 \end{aligned}$$

(207)

$$\begin{aligned}&\quad [e_1,f_2]=-f_2,\quad \quad [e_2,f_2]=-f_1,\quad \quad [e_3,f_2]=0 \end{aligned}$$

(208)

and, finally, the anti commutators between odd generators yield

$$\begin{aligned}{}[f_1,f_1]&=-2e_2,\quad [f_2,f_2]=2e_3,\quad [f_1,f_2]=-e_1 \end{aligned}$$

(209)

These are precisely the graded commutation relations of real $\textrm{OSp}(1|2)$. Similar as in [33], using (195), we can then identify $\textrm{OSp}(1|2)_{\mathbb {C}}$ with the split super Lie group $\textbf{S}(\mathfrak {osp}(1|2)_{1},\textrm{SL}(2,\mathbb {C}))$ according to

$$\begin{aligned} \Phi (g,v)=g\exp (v)=g\cdot \begin{pmatrix} 1-\xi \eta &{}\quad \eta &{}\quad \xi \\ \xi &{}\quad 1+\frac{1}{2}\xi \eta &{}\quad 0\\ -\eta &{}\quad 0 &{}\quad 1+\frac{1}{2}\xi \eta \end{pmatrix} \end{aligned}$$

(210)

for $g\in \textrm{SL}(2,\mathbb {C})$ and $v:=\xi f_1+\eta f_2\in (\mathfrak {osp}(1|2)_{1}\otimes \Lambda ^{\mathbb {C}})_0$. Using the commutation relations above, it follows immediately that the matrix representation of $\textrm{ad}(v)$ is given by

$$\begin{aligned} \textrm{ad}(v)=\begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad -\eta &{}\quad \xi \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -2\xi &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 2\eta \\ \xi &{}\quad -\eta &{}\quad 0 &{}\quad 0 &{}\quad 0\\ -\eta &{}\quad 0 &{}\quad -\xi &{}\quad 0 &{}\quad 0 \end{pmatrix} \end{aligned}$$

(211)

Hence, it follows from (200) as well as (202), using $\textrm{ad}(v)^n=0$ for $n\ge 3$, [33]

$$\begin{aligned} \begin{pmatrix} \mathbbm {1} &{}\quad H\\ A &{}\quad B \end{pmatrix}=\begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{1}{2}\eta &{}\quad -\frac{1}{2}\xi \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad \xi &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad \eta \\ \xi &{}\quad -\eta &{}\quad 0 &{}\quad 1-\xi \eta &{}\quad 0\\ -\eta &{}\quad 0 &{}\quad -\xi &{}\quad 0 &{}\quad 1-\xi \eta \end{pmatrix} \end{aligned}$$

(212)

Actually, for the derivation of (203), it has been implicitly assumed that the super Lie group is real. Hence, we need to view $\textrm{OSp}(1|2)_{\mathbb {C}}$ as a real super Lie group. A homogeneous basis of the realification of $\mathfrak {g}:=\mathfrak {osp}(1|2)_{\mathbb {C}}$ (resp. $\textrm{Lie}(\mathcal {G}):=\mathfrak {g}\otimes \Lambda ^{\mathbb {C}}$) is then given by $(e_i,ie_i,f_j,if_j)$. Let $\mathcal {R}:\,\underline{\textrm{End}}_R(\textrm{Lie}(\mathcal {G}))\rightarrow \underline{\textrm{End}}_R(\textrm{Lie}(\mathcal {G})_{\mathbb {R}})$ be the morphism which identifies any $X\in \underline{\textrm{End}}_R(\textrm{Lie}(\mathcal {G}))$ with the corresponding real endomorphism $\mathcal {R}(A)$ on the realification $\textrm{Lie}(\mathcal {G})_{\mathbb {R}}$. For the density $\Delta \equiv \Delta (\xi ,\bar{\xi },\eta ,\bar{\eta })$ in the Berezin integral we then compute

$$\begin{aligned} \Delta =\frac{\det (\mathcal {R}(B))}{\det (\mathcal {R}(\mathbbm {1}-H\cdot B^{-1}\cdot A))}=(1+\xi \eta )(1+\bar{\xi }\bar{\eta }) \end{aligned}$$

(213)

Hence, going back to the original basis $(T_i^+,Q_A)$ of the super Lie algebra $\mathfrak {osp}(1|2)_{\mathbb {C}}$, we find the invariant integral for any smooth function $f\in V:=C_c^{\infty }(\textrm{SL}(2,\mathbb {C}),\mathbb {C})\otimes \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}[\theta ^A,\bar{\theta }^{A'}]$ is given by

$$\begin{aligned} \int _{\textrm{OSp}(1|2)_{\mathbb {C}}}{f\nu _{\mathfrak {g}}}=\int _{\textrm{SL}(2,\mathbb {C})}{\textrm{d}\mu _H(g)\int _B{\textrm{d}\mu (\theta ,\bar{\theta })\,f(g,\theta ,\bar{\theta })}} \end{aligned}$$

(214)

with $\textrm{d}\mu _H$ an invariant Haar measure on $\textrm{SL}(2,\mathbb {C})$ and

$$\begin{aligned} \textrm{d}\mu (\theta ,\bar{\theta }):=\textrm{d}\theta ^A\textrm{d}\bar{\theta }^{A'}\left( 1-\frac{i}{2}\theta ^A\theta _A\right) \left( 1+\frac{i}{2}\bar{\theta }^{A'}\bar{\theta }_{A'}\right) \end{aligned}$$

(215)

This Haar measure induces a super scalar product $\mathscr {S}:\,V\times V\rightarrow \mathbb {C}$, i.e., a graded symmetric and non-degenerate sesquilinear form on the super vector space V (cf. Definition 2.15) via

$$\begin{aligned} \mathscr {S}(f,g):=\int _{\textrm{OSp}(1|2)_{\mathbb {C}}}{\bar{f}g\,\nu _{\mathfrak {g}}} \end{aligned}$$

(216)

By construction, $\mathscr {S}$ is indefinite turning $(V,\mathscr {S})$ into an indefinite inner product space. In [33], it has been shown for arbitrary (real) super Lie groups that one can always find a linear map $J:\,V\rightarrow V$ such that $\left\langle \cdot ,\cdot \right\rangle :=\mathscr {S}(\cdot ,J\cdot )$ defines a positive definite scalar product on V. Hence, $(V,\mathscr {S},J)$ has the structure of a Krein space. Given such an endomorphism J, one can then use the induced scalar product to complete V to a Hilbert space

$$\begin{aligned} \mathcal {H}:=\overline{V}^{\Vert \cdot \Vert } \end{aligned}$$

(217)

However, the choice of such an endomorphism is, a priori, not unique and thus needs to be fixed by additional physical requirements. Therefore, as a next step, let us turn to the construction of the kinematical state space in the manifest approach to LQSG. However, due to the non-compactness of $\textrm{SL}(2,\mathbb {C})$, even in the purely bosonic theory, it is still unclear how the (kinematical) Hilbert space with self-dual variables can be defined consistently. Hence, the following considerations will only sketch some ideas following the standard procedure in LQG with real variables.

In the standard quantization scheme in loop quantum gravity, one considers graphs $\gamma \subset \Sigma$ that are embedded into spatial slices $\Sigma$ of the spacetime manifold $\mathcal {M}_0$ which can be regarded as a purely bosonic supermanifold. This is also reasonable in case of supersymmetry due to the rheonomy principle stating that physical degrees of freedom are completely encoded on the body [2, 3]. Hence, we consider graphs $\gamma$ that are embedded into spatial slices $\Sigma$ of the bosonic submanifold $\mathcal {M}_0\subset \mathcal {M}$ of the underlying supermanifold $\mathcal {M}$. Let us fix such a particular graph $\gamma$ consisting of n edges $e_i\in \gamma$, $i=1,\ldots ,n$, that intersect at certain vertices v and require that the graph is suitably refined such that the topology of $\Sigma$ can be resolved [70]. A cylindrical function $\Psi _{\gamma }[\mathcal {A}^{+}]$ on that graph will then generically be of the form

$$\begin{aligned} \Psi _{\gamma }:=f_{\gamma }(h_{e_1}[\mathcal {A}^{+}],\ldots ,h_{e_n}[\mathcal {A}^{+}]) \end{aligned}$$

(218)

for some smooth function $f_{\gamma }\in H^{\infty }_{(c)}(\textrm{OSp}(1|2)^{\times n}_\mathbb {C},\mathbb {C})\cong \bigotimes _{i=1}^n H^{\infty }_{(c)}(\textrm{OSp}(1|2)_{\mathbb {C}},\mathbb {C})$. In Sect. 2.4, choosing a particular gauge, it has been shown that the holonomies $h_{e}[\mathcal {A}^+]$ associated with the super Ashtekar connection are of the form¹⁷

$$\begin{aligned} h_{e}[\mathcal {A}^+]=h_e[A^+]\cdot \mathcal {P}\textrm{exp}\left( -\int _{e}{\textrm{Ad}_{h_{e}[A^+]^{-1}}\psi }\right) \end{aligned}$$

(219)

with $h_e[A^+]$ the induced holonomies of the (bosonic) self-dual Ashtekar connection $A^+$. According to (156), the adjoint representation of $\textrm{SL}(2,\mathbb {C})$ on the odd part of the Lie superalgebra $\mathfrak {osp}(1|2)_{\mathbb {C}}$ is given by the fundamental representation such that

$$\begin{aligned} \textrm{Ad}_{h_{e}[A^+]^{-1}}\psi =Q_B\,{(h_{e}[A^+]^{-1})}{^B_A}\psi ^A \end{aligned}$$

(220)

From (219) and (220) it follows that holonomies of $\mathcal {A}^+$ are holomorphic functions in the fermionic variables $\psi ^A$. Hence, in order to resolve the physical degrees of freedom, it is sufficient to restrict definition (218) of cylindrical functions to functions $f_{\gamma }$ of the form

$$\begin{aligned} f_{\gamma }\in V^{\otimes n}:=\bigotimes _{i=1}^n C^{\infty }_{(c)}(\textrm{SL}(2,\mathbb {C}),\mathbb {C})\otimes \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }}[\theta ^A] \end{aligned}$$

(221)

According to (218) and (221), we can identify the space of cylindrical functions w.r.t. $\gamma$ with $V^{\otimes n}$. The invariant Haar measure on $\textrm{OSp}(1|2)_{\mathbb {C}}$ as defined via (214) can be extended to invariant Haar measure on $\textrm{OSp}(1|2)_{\mathbb {C}}^{\times n}$ taking the tensor product (as may be checked by direct computation, this will automatically satisfy (197) (resp. (196))). This, in turn, induces a super scalar product $\mathscr {S}$ on $V^{\otimes n}$ which again has the structure of a Krein space. In order to complete to a Hilbert space and construct a positive definite inner product, one has to choose a particular endomorphism $J:V^{\otimes n}\rightarrow V^{\otimes n}$. Therefore, note that the fermionic component $\mathcal {E}_A$ of the super electric field, when implemented as an operator on the resulting Hilbert space, needs to satisfy the quantum version of the reality condition (185). Hence, $J:\,V^{\otimes n}\rightarrow V^{\otimes n}$ has to be chosen such that (185) can be implemented on the kinematical Hilbert space $\mathcal {H}_{\textrm{kin}}$ obtained via the completion of $(V^{\otimes n},\left\langle \cdot ,\cdot \right\rangle )$ w.r.t. to the positive definite scalar product $\left\langle \cdot ,\cdot \right\rangle :=\mathscr {S}(\cdot ,J\cdot )$. This is in fact in accordance with [71] where it is suggested that the inner product (for the bosonic degrees of freedom) has to be defined in such way that the reality conditions can be consistently implemented.

As discussed in detail in [33], there exists an endomorphism $J:\,V\rightarrow V$ such that, w.r.t. functions $f\in V$ written in the form

$$\begin{aligned} f=f_0+f_A\psi ^A+\frac{1}{2}f_2\psi ^A\psi _A \end{aligned}$$

(222)

with $f_0,f_A,f_2\in C_{(c)}^{\infty }(\textrm{SL}(2,\mathbb {C}),\mathbb {C})$, the induced scalar product is given by

$$\begin{aligned} \left\langle f,g\right\rangle =\left\langle \left\langle f_0,g_0\right\rangle \right\rangle +\left\langle \left\langle f_+,g_+\right\rangle \right\rangle +\left\langle \left\langle f_-,g_-\right\rangle \right\rangle +\left\langle \left\langle f_2,g_2\right\rangle \right\rangle \end{aligned}$$

(223)

with $\left\langle \left\langle \cdot ,\cdot \right\rangle \right\rangle$ the scalar product on $C^{\infty }_{(c)}(\textrm{SL}(2,\mathbb {C}))$ induced by the invariant Haar measure $\mu _H$ on $\textrm{SL}(2,\mathbb {C})$. This scalar product is then invariant under local $\textrm{SL}(2,\mathbb {C})$-gauge transformations.

As it is shown in [20], in the framework of symmetry reduced models, this in fact solves the reality conditions (185). This is also precisely the standard scalar product as used for instance in [18] in context of Dirac fermions or in [61] in context of the Rarita-Schwinger field in a complementary approach to LQSG with real variables.

Considering the holonomies (219), this may suggest to restrict $C^{\infty }_{(c)}(\textrm{SL}(2,\mathbb {C}))$ to holomorphic functions on $\textrm{SL}(2,\mathbb {C})$. However, due to Liouville’s theorem, if required to be nontrivial, general functions of this kind cannot be of compact support. This is of course problematic. Hence, either one does not restrict to holomorphic functions considering more general smooth functions on $\textrm{SL}(2,\mathbb {C})$ (regarded as a real manifold) or the measure on $\textrm{SL}(2,\mathbb {C})$ is changed appropriately. The last possibility has been studied in [19, 72] in the context of symmetry reduced models. There, it was shown that this indeed allows an exact implementation of the remaining reality conditions (194) where the measure turned out to be distributional. How this can be extended to the full theory, however, is still unclear and remains a task for future investigations.

Another proposal is to consider analytic continuations from real to imaginary Barbero–Immiri parameters (see for instance [70, 73,74,75,76] and references therein for recent advances in this direction). This is based on the fact that $\textrm{SL}(2,\mathbb {C})$ contains $\textrm{SU}(2)$ as a compact real form. Perhaps, these kind considerations can be generalized to the present situation in context of supersymmetry with gauge group $\textrm{OSp}(1|2)_{\mathbb {C}}$.

Nevertheless, in the present article, we have demonstrated that structure of the (kinematical) Hilbert space for the fermionic degrees of freedom as normally used in LQG has a very natural explanation in the framework of the manifest approach to LQSG. It would thus be very interesting to further analyze this structure and even generalize it including extended supersymmetries.

Remark 5.5

The Hilbert space of quantum chiral supergravity as outlined in this section has a very similar structure to standard LQG coupled to fermions (see, e.g., [17, 18]). However, there is a crucial difference as in ordinary LQG one works with half densitized fermionic variables in order to solve reality conditions. Furthermore, for the construction of the quantum theory, they are localized at spatial points. In contrast, in this manifest supersymmetric approach, the fermionic variables are smeared over one-dimensional edges similar to the bosonic degrees of freedom due to supersymmetry.

Nevertheless, let us emphasize that the structures of the resulting Hilbert spaces in both approaches are very similar. In fact, as explained in more detail in [66], the pre-Hilbert space in standard LQG coupled to Weyl fermions can be reinterpreted, for the choice of a specific graph $\gamma \subset \Sigma$, in terms of a superspace of supersmooth functions on the $\mathcal {S}$-point $\mathcal {M}(\mathcal {S})$ with $\mathcal {S}$ the parameterization supermanifold to account for anticommutativity and $\mathcal {M}$ a supermanifold of the form

$$\begin{aligned} G^{|E(\gamma )|}\times \prod _{i=1}^k\mathbb {C}^{0|2}_{x_i} \end{aligned}$$

(224)

with Lie group $G=\textrm{SU}(2)$. It turns out that the pre-Hilbert space indeed has the structure of a Krein space where the endomorphism J is fixed uniquely by the imposition of reality conditions. Hence, we see that similar structures appear in standard LQG as well as the manifest supersymmetric approach to quantum chiral supergravity. Moreover, in standard LQG, a consistent way has been found to account for the reality conditions which, as outlined here, can be reinterpreted in terms of choosing a particular Krein structure. Maybe, similar ideas can be transferred to this manifest supersymmetric approach in order to complete the construction of the quantum theory. This remains as a task for future research.

6 Symmetry Reduction in Supersymmetric Field Theories

Since chiral supergravity contains an enlarged gauge symmetry corresponding to a gauge supergroup, it seems suggestive to exploit this symmetry in order to construct symmetry reduced models by generalizing the notion of invariant connection 1-forms to the super category. The following discussion will provide a solid basis for the construction of (spatially) symmetry reduced models in the context of supersymmetric field theories. A respective discussion in the context of ordinary (bosonic) connection 1-forms defined on smooth principal fiber bundles can be found, e.g., in [77] (see also [78] for a nice introduction to this subject in the non-supersymmetric setting). The following results have been used in [20] to study minisuperspace models in the framework of loop quantum cosmology with local supersymmetry.

To this end, let us consider a general $H^{\infty }$ supermanifold $\mathcal {M}$ as well as a super Lie group $\mathcal {H}$ which, in the most situations of interest, will correspond to the super Lie group of isometries of a super Riemannian manifold $(\mathcal {M},g)$ (in fact, in most cases $\mathcal {M}$ will be a purely bosonic supermanifold corresponding to an ordinary smooth manifold). Suppose $\mathcal {H}$ acts from the left on $\mathcal {M}$, i.e., there exists a smooth map

$$\begin{aligned} f:\,\mathcal {H}\times \mathcal {M}\rightarrow \mathcal {M} \end{aligned}$$

(225)

such that

$$\begin{aligned} f\circ (\textrm{id}_{\mathcal {H}}\times f)=f\circ (\mu _{\mathcal {H}}\times \textrm{id})\quad \text {and}\quad f_e(x)=x\quad \forall x\in \mathcal {M} \end{aligned}$$

(226)

Furthermore, we assume that $\mathcal {H}$ acts transitively on $\mathcal {M}$. Hence, if $x\in \textbf{B}(\mathcal {M})$ is a body point and $\mathcal {H}_x$ is the stabilizer subgroup of $\mathcal {H}$, one can identify $\mathcal {M}\cong \mathcal {H}/\mathcal {H}_x$ which we want to do in what follows. The left action of $\mathcal {H}$ is then given by its standard action on the coset space $\mathcal {H}/\mathcal {H}_x$ which still will be denoted by f.

Let $\mathcal {G}\rightarrow \mathcal {P}{\mathop {\rightarrow }\limits ^{\pi }}\mathcal {H}/\mathcal {H}_x$ be a principal super fiber bundle over $\mathcal {H}/\mathcal {H}_x$ with structure group $\mathcal {G}$ and $\mathcal {G}$-right action $\Phi :\,\mathcal {P}\times \mathcal {G}\rightarrow \mathcal {P}$. We want to the ask the question about the existence of a $\mathcal {H}$-left action $\hat{f}:\,\mathcal {H}\times \mathcal {P}\rightarrow \mathcal {P}$ on $\mathcal {P}$ such that $\hat{f}$ is a $\mathcal {G}$-equivariant bundle automorphism on $\mathcal {P}$ projecting to the left multiplication of $\mathcal {H}$ on $\mathcal {H}/\mathcal {H}_x$, i.e.,

$$\begin{aligned} \hat{f}\circ (\textrm{id}_{\mathcal {H}}\times \Phi )=\Phi \circ (\hat{f}\times \textrm{id}_{\mathcal {G}})\text { and }\pi \circ \hat{f}=f\circ (\textrm{id}_{\mathcal {H}}\times \pi ) \end{aligned}$$

(227)

Therefore, applying the forgetful functor $\textbf{SMan}_{H^{\infty }}\rightarrow \textbf{Set}$, we consider the set of abstract group homomorphisms $\lambda :\,\mathcal {H}_x\rightarrow \mathcal {G}$. On this set, we introduce the equivalence relation

$$\begin{aligned} \lambda \sim \lambda ':\Leftrightarrow \exists g\in \mathcal {G}:\,\lambda '=\textrm{Ad}_{g}\circ \lambda \end{aligned}$$

(228)

which yields the set of conjugacy classes $\textrm{Conj}(\mathcal {H}_x\rightarrow \mathcal {G})$ of abstract group homomorphisms. An equivalence class $[\lambda ]\in \textrm{Conj}(\mathcal {H}_x\rightarrow \mathcal {G})$ will be called smoothly admissible, if it contains a $H^{\infty }$-smooth super Lie group homomorphism as a representative. The set of such smoothly admissible conjugacy classes yields a proper subset $\textrm{Conj}(\mathcal {H}_x\rightarrow \mathcal {G})_{\infty }\subset \textrm{Conj}(\mathcal {H}_x\rightarrow \mathcal {G})$.

Proposition 6.1

There exists a bijective correspondence between equivalence classes of principal $\mathcal {G}$-bundles over $\mathcal {H}/\mathcal {H}_x$ admitting an $\mathcal {H}$-left action which is $\mathcal {G}$-equivariant and projects to the standard left multiplication of $\mathcal {H}$ on the coset space $\mathcal {H}/\mathcal {H}_x$ and smoothly admissible conjugacy classes $[\lambda ]\in \textrm{Conj}(\mathcal {H}_x\rightarrow \mathcal {G})_{\infty }$ of group homomorphisms $\lambda :\,\mathcal {H}_x\rightarrow \mathcal {G}$.

Proof

Suppose $\lambda :\,\mathcal {H}_x\rightarrow \mathcal {G}$ is a smooth representative of a smoothly admissible conjugacy class of super Lie group homomorphisms. Consider then the associated principal supper fiber bundle $\mathcal {H}\times _{\lambda }\mathcal {G}$ with structure group $\mathcal {G}$. On $\mathcal {H}\times \mathcal {G}$, we define the smooth left action

$$\begin{aligned} \mathcal {H}\times (\mathcal {H}\times \mathcal {G})\rightarrow \mathcal {H}\times \mathcal {G},\quad (\phi ,(\psi ,g))\mapsto (\phi \circ \psi ,g) \end{aligned}$$

(229)

Since $(\phi \circ (\psi \circ \phi '),\lambda (\phi ')^{-1}(g))=((\phi \circ \psi )\circ \phi ',\lambda (\phi ')^{-1}(g))$ $\forall \phi ,\phi ',\psi \in \mathcal {H}$ and $g\in \mathcal {G}$, it follows that (229) is constant on $\mathcal {G}$-orbits so that (229) induces a well-defined smooth $\mathcal {H}$-left action on $\mathcal {H}\times _{\lambda }\mathcal {G}$ which is $\mathcal {G}$-equivariant and projects to the multiplication of $\mathcal {H}$ on $\mathcal {H}/\mathcal {H}_x$.

Conversely, let $\hat{f}:\,\mathcal {H}\times \mathcal {P}\rightarrow \mathcal {P}$ be a $\mathcal {H}$-left action on $\mathcal {P}$. Let $p\in \textbf{B}(\mathcal {P})$ be an element of the body. Since the $\mathcal {G}$-right action on $\mathcal {P}$ is transitive on each fiber and $\hat{f}$ is fiber-preserving, for any $\phi \in \mathcal {H}_x$, there exists a unique $\lambda (\phi )\in \mathcal {G}$ such that

$$\begin{aligned} \hat{f}_{\phi }(p)=\Phi _{\lambda (\phi )}(p) \end{aligned}$$

(230)

Moreover, since $p\in \textbf{B}(\mathcal {P})$, the map $\mathcal {H}\rightarrow \mathcal {P},\,\phi \mapsto f_{\phi }(p)$ is of class $H^{\infty }$ proving that the map $\lambda :\,\mathcal {H}_x\rightarrow \mathcal {G},\,\phi \mapsto \lambda (\phi )$ is smooth. By $\mathcal {G}$-equivariance (227), it follows for $\phi ,\psi \in \mathcal {H}_x$

$$\begin{aligned} \hat{f}_{\phi \circ \psi }(p)&=f_{\phi }(f_{\psi }(p))=f_{\phi }(\Phi _{\lambda (\psi )}(p))=\Phi _{\lambda (\psi )}(f_{\phi }(p))=\Phi _{\lambda (\phi )\circ \lambda (\psi )}(p)\nonumber \\&=\Phi _{\lambda (\phi \circ \psi )}(p) \end{aligned}$$

(231)

implying $\lambda (\phi \circ \psi )=\lambda (\phi )\circ \lambda (\psi )$, i.e., $\lambda$ is indeed a super Lie group homomorphism. If $p'\in \mathcal {P}$ is any other point, then, again by transitivity, there exists $g\in \mathcal {G}$ with $\Phi _{g}(p)=p'$. Hence,

$$\begin{aligned} \hat{f}_{\phi }(p')=\Phi _{g}(\hat{f}_{\phi }(p))=\Phi _{\textrm{Ad}_{g^{-1}}\lambda (\phi )}(p') \end{aligned}$$

(232)

with $\textrm{Ad}_{g^{-1}}\circ \lambda$ in the same equivalence class as $\lambda$. Finally, let $\mathcal {H}\times _{\lambda }\mathcal {G}$ be the associated principal $\mathcal {G}$-bundle with smooth $\mathcal {H}$-left action as constructed in the first part of this proof. For $p\in \textbf{B}(\mathcal {P})$ a body point, consider the map

$$\begin{aligned} \mathcal {H}\times _{\lambda }\mathcal {G}\rightarrow \mathcal {P},\quad [\phi ,g]\mapsto \Phi _{g}(f_{\phi }(p)) \end{aligned}$$

(233)

By (227), it follows immediately that (233) is well-defined and in fact yields an isomorphism of principal super fiber bundles. $\square$

Proposition (6.1) provides a complete classification of principal super fiber bundles admitting such a smooth left action by equivalence classes of smooth super Lie group morphisms $\lambda :\,\mathcal {H}_x\rightarrow \mathcal {G}$. We next want to study connections on $\mathcal {P}$ that are invariant under this left action. Since $\mathcal {M}$ is typically an ordinary smooth manifold and we would like to include fermionic degrees of freedom in our discussion, we go over to the category of relative supermanifolds. Hence, let us add a parameterizing supermanifold $\mathcal {S}$. We lift all objects and morphisms to the relative category in the obvious way. If $\mathcal {P}:=\mathcal {H}\times _{\lambda }\mathcal {G}$ is a principal super fiber bundle as in Proposition (6.1), it follows that $\mathcal {P}_{/\mathcal {S}}=\mathcal {H}_{/\mathcal {S}}\times _{\lambda }\mathcal {G}$. A $\mathcal {S}$-relative super connection 1-form $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})_0$ will be called $\mathcal {H}$-invariant, if

$$\begin{aligned} (\hat{f}_{\mathcal {S}})^*_{\phi }\mathcal {A}=\mathcal {A}\quad \forall \phi \in \mathcal {H} \end{aligned}$$

(234)

with $\hat{f}_{\mathcal {S}}:\,\mathcal {H}\times \mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {P}_{/\mathcal {S}}$ the lift of the left multiplication $f_{\mathcal {S}}:\,\mathcal {H}\times \mathcal {H}_{/\mathcal {S}}\rightarrow \mathcal {H}_{/\mathcal {S}}:\,(\phi ,(s,\psi ))\mapsto (s,\phi \circ \psi )$ to a smooth $\mathcal {H}$-left action on $\mathcal {P}_{/\mathcal {S}}$ defined via (cf. proof of Proposition 6.1)

$$\begin{aligned} \hat{f}_{\mathcal {S}}\circ (\textrm{id}_{\mathcal {H}}\times \hat{\pi })=\hat{\pi }\circ (f_{\mathcal {S}}\times \textrm{id}_{\mathcal {G}}) \end{aligned}$$

(235)

with $\hat{\pi }:\,\mathcal {H}_{/\mathcal {S}}\times \mathcal {G}\rightarrow \mathcal {H}_{/\mathcal {S}}\times _{\lambda }\mathcal {G}=\mathcal {P}_{/\mathcal {S}}$ the canonical projection.

Proposition 6.2

Let $\mathcal {P}:=\mathcal {H}\times _{\lambda }\mathcal {G}$ be the associated principal super fiber bundle induced by a smooth super Lie group homomorphism $\lambda :\,\mathcal {H}_x\rightarrow \mathcal {G}$.

The $\mathcal {H}$-invariant super connection 1-forms on $\mathcal {P}_{/\mathcal {S}}$ are in one-to-one correspondence to smooth maps $\Xi \in H^{\infty }(\mathcal {\mathcal {S}},\textrm{Hom}_L(\textrm{Lie}(\mathcal {H}),\textrm{Lie}(\mathcal {G})))$ from the parameterizing supermanifold $\mathcal {S}$ to even left linear super Lie algebra homomorphisms $\textrm{Hom}_L(\textrm{Lie}(\mathcal {H}),\textrm{Lie}(\mathcal {G}))$ satisfying

$$\begin{aligned} \Xi (s)\big {|}_{\textrm{Lie}(\mathcal {H}_x)}=\lambda _{*} \end{aligned}$$

(236)

and

$$\begin{aligned} \textrm{Ad}_{\phi ^{-1}}\diamond \Xi (s)=\Xi (s)\diamond \textrm{Ad}_{\lambda (\phi )^{-1}}\quad \text {on }\textrm{Lie}(\mathcal {H}) \end{aligned}$$

(237)

$\forall s\in \mathcal {S}$ and $\phi \in \mathcal {H}_x$.

Proof

In the following, let $\hat{\iota }:\,\mathcal {H}_{/\mathcal {S}}\rightarrow \mathcal {P}_{/\mathcal {S}}$ be the smooth map defined via $\hat{\iota }(s,\phi ):=[(s,\phi ),e]$ and $\Phi _{\mathcal {S}}$ denote the $\mathcal {G}$-right action on $\mathcal {P}_{/\mathcal {S}}$. Suppose $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})_0$ is a $\mathcal {H}$-invariant super connection 1-form. Consider then $\mathcal {A}_{\mathcal {H}}:=\hat{\iota }^*\mathcal {A}\in \Omega ^1(\mathcal {H}_{/\mathcal {S}},\mathfrak {g})_0$. Since $\hat{\iota }\circ f_{\mathcal {S}}=\hat{f}_{\mathcal {S}}\circ (\textrm{id}_{\mathcal {H}}\times \hat{\iota })$ by (235), it follows from the $\mathcal {H}$-invariance of $\mathcal {A}$ that

$$\begin{aligned} (f_{\mathcal {S}})_{\phi }^*\mathcal {A}_{\mathcal {H}}=\hat{\iota }^*((\hat{f}_{\mathcal {S}})^*_{\phi }\mathcal {A})=\hat{\iota }^*\mathcal {A}=\mathcal {A}_{\mathcal {H}} \end{aligned}$$

(238)

i.e., $\mathcal {A}_{\mathcal {H}}$ is left-invariant w.r.t. to the standard left multiplication on $\mathcal {H}$. As a consequence, $\mathcal {A}_{\mathcal {H}}$ is uniquely determined by its restriction $\mathcal {A}_{\mathcal {H}}|_{T_e\mathcal {H}}:\,\mathcal {S}\times T_{e}\mathcal {H}\rightarrow \textrm{Lie}(\mathcal {G})$. As this map is left-linear in the second argument it follows that it defines an even smooth map $\mathcal {A}_{\mathcal {H}}|_{T_e\mathcal {H}}:\,\mathcal {S}\rightarrow \textrm{Hom}_L(\textrm{Lie}(\mathcal {H}),\textrm{Lie}(\mathcal {G}))_0$. Moreover, since the Maurer–Cartan form $\theta ^{(\mathcal {H})}_{\textrm{MC}}|_{T_e\mathcal {H}}:\,T_{e}\mathcal {H}\rightarrow \textrm{Lie}(\mathcal {H})$ on $T_{e}\mathcal {H}$ is the identity, it follows that

$$\begin{aligned} \mathcal {A}_{\mathcal {H}}(s)=\theta ^{(\mathcal {H})}_{\textrm{MC}}\diamond \Xi (s)\quad \forall s\in \mathcal {S} \end{aligned}$$

(239)

on $T_{e}\mathcal {H}$ for some smooth map $\Xi \in H^{\infty }(\mathcal {S},\textrm{Hom}_L(\textrm{Lie}(\mathcal {H}),\textrm{Lie}(\mathcal {G})))$. By left-invariance, it follows that (239) indeed holds on all of $\mathcal {H}$.

Remains to proof that $\Xi$ satisfies the properties (236) and (237) of the proposition. To this end, for $X\in \textrm{Lie}(\mathcal {H}_x)$, we compute

$$\begin{aligned} \hat{\iota }_{*}(\mathbbm {1}\otimes X)_{p}&=D_{(p,e_{\mathcal {G}})}\hat{\pi }(0_s,X_{\phi },0_{e_{\mathcal {G}}})=D_{(p,e_{\mathcal {G}})}\hat{\pi }(0_{p},\lambda _{*}(X))\nonumber \\&=\widetilde{\lambda _{*}(X)}_{[p,e_{\mathcal {G}}]} \end{aligned}$$

(240)

$\forall p=(s,\phi )\in \mathcal {H}_{/\mathcal {S}}$, where in the second equality we used that the kernel of $\hat{\pi }_{*}$ is given by

$$\begin{aligned} \textrm{ker}\,D_{(p,g)}\hat{\pi }=\{(\mathbbm {1}\otimes Y_{p},-R_{g*}\lambda _{*}(X))|Y\in \textrm{Lie}(\mathcal {H}_x)\} \end{aligned}$$

(241)

Using (240), this yields

$$\begin{aligned} \lambda _{*}(X)&=\left\langle \widetilde{\lambda _{*}(X)}|\mathcal {A}\right\rangle =\left\langle (\mathbbm {1}\otimes X)_{p}|\mathcal {A}_{\mathcal {H}}\right\rangle \nonumber \\&=\left\langle \left\langle X_{\phi }|\theta _{\textrm{MC}}^{(\mathcal {H})}\right\rangle |\Xi (s)\right\rangle =\left\langle X|\Xi (s)\right\rangle \end{aligned}$$

(242)

$\forall X\in \textrm{Lie}(\mathcal {H}_x)$. Finally, since $\hat{\iota }\circ (\textrm{id}_{\mathcal {S}}\times R_{\phi })=(\Phi _{\mathcal {S}})_{\lambda (\phi )}\circ \hat{\iota }$ with $R_{\phi }$ the right translation on $\mathcal {H}$ w.r.t. $\phi \in \mathcal {H}_x$, it follows that

$$\begin{aligned} \left\langle X|\textrm{Ad}_{\phi ^{-1}}\diamond \Xi (s)\right\rangle&=\left\langle \textrm{Ad}_{\phi ^{-1}}\left\langle X|\theta _{\textrm{MC}}^{(\mathcal {H})}\right\rangle |\Xi (s)\right\rangle =\left\langle \left\langle R_{\phi *}X|\theta _{\textrm{MC}}^{(\mathcal {H})}\right\rangle |\Xi (s)\right\rangle \nonumber \\&=\left\langle R_{\phi *}X|\theta _{\textrm{MC}}^{(\mathcal {H})}\diamond \Xi (s)\right\rangle =\left\langle R_{\phi *}|\mathcal {A}_{\mathcal {H}}(s)\right\rangle =\textrm{Ad}_{\lambda (\phi )^{-1}}\left\langle X|\mathcal {A}_{\mathcal {H}}(s)\right\rangle \nonumber \\&=\left\langle X|\Xi (s)\diamond \textrm{Ad}_{\lambda (\phi )^{-1}}\right\rangle \end{aligned}$$

(243)

$\forall X\in \textrm{Lie}(\mathcal {H})$ as required.

Conversely, suppose one has given a smooth map $\Xi \in H^{\infty }(\mathcal {S},\textrm{Hom}_L(\textrm{Lie}(\mathcal {H}),\textrm{Lie}(\mathcal {G})))$ satisfying (236) and (237). We have to show that there indeed exists a unique super connection 1-form $\mathcal {A}\in \Omega ^1(\mathcal {P}_{/\mathcal {S}},\mathfrak {g})_0$ such that $\hat{\iota }^*\mathcal {A}(s)=\theta ^{(\mathcal {H})}_{\textrm{MC}}\diamond \Xi (s)$ for any $s\in \mathcal {S}$. This, in fact, follows along the lines of the proof of Proposition 4.8. As there, one can show that, if $\mathcal {A}$ exists, it necessarily has to be of the form

$$\begin{aligned} \left\langle D_{(p,g)}\hat{\pi }(X_p,Y_g)|\mathcal {A}_{[p,g]}\right\rangle =\textrm{Ad}_{g^{-1}}\left\langle X_p|\theta ^{(\mathcal {H})}_{\textrm{MC}}\diamond \Xi (s)\right\rangle +\left\langle Y_g|\theta ^{(\mathcal {G})}_{\textrm{MC}}\right\rangle \end{aligned}$$

(244)

Moreover, as $\hat{\pi }$ is a submersion, it is uniquely determined by (244). One then concludes that this indeed provides a well-defined super connection 1-form on $\mathcal {P}_{/\mathcal {S}}$. $\square$

Remark 6.3

Note that if $\lambda :\,\mathcal {H}_0\rightarrow \mathcal {G}$ is a group morphism corresponding to a left action of a bosonic super Lie group $\mathcal {H}_0$, i.e., a split super Lie group corresponding to an ordinary smooth symmetry group, then $\lambda$ only takes values in the bosonic super Lie subgroup $\mathcal {G}_0:=\textbf{S}(\textbf{B}(\mathcal {G}))$ of $\mathcal {G}$. Thus, it follows that condition (225) only encodes super connections that are invariant under purely bosonic gauge transformations.

This can be cured by considering a more general class of smooth $\mathcal {H}$-left actions $\hat{f}:\,\mathcal {H}\times \mathcal {P}_{/\mathcal {S}}\rightarrow \mathcal {P}_{/\mathcal {S}}$ on the $\mathcal {S}$-relative principal super fiber bundle $\mathcal {P}_{/\mathcal {S}}$ that are not merely trivial extensions of $\mathcal {H}$-left actions on $\mathcal {P}$ as considered above and which project to the left multiplication on $\mathcal {H}_{/\mathcal {S}}$, i.e., $\mathcal {H}\times \mathcal {H}_{/\mathcal {S}}\rightarrow \mathcal {H}_{/\mathcal {S}}:\,(\phi ,(s,\psi ))\mapsto (s,\phi \circ \psi )$. It then follows that a classification of these type of actions is given by smooth maps of the form $\lambda ':\,\mathcal {S}\times \mathcal {H}_x\rightarrow \mathcal {G}$ satisfying

$$\begin{aligned} \lambda '(s,\phi \circ \psi )=\lambda '(s,\phi )\circ \lambda '(s,\psi ) \end{aligned}$$

(245)

$\forall (s,\phi ),(s,\psi )\in \mathcal {S}\times \mathcal {H}_x$. Condition (234) for a $\mathcal {H}$-invariant super connection 1-form then again leads to (236) as well as (237), straightforwardly generalized to $\mathcal {S}$-parameterized group morphisms $\lambda ':\,\mathcal {S}\times \mathcal {H}_x\rightarrow \mathcal {G}$. In particular, since $\lambda '$ now explicitly depends on the parameterization, it follows, in case that the symmetry group $\mathcal {H}$ is purely bosonic, that $\lambda '$ can take values in the odd part of the gauge supergroup $\mathcal {G}$. Hence, in this way, one can model super connection 1-forms which are invariant under the spatial symmetry group up to super gauge transformations. In fact, as has been demonstrated in [20], this turned out to play important role in deriving symmetry reduced connections that contain nontrivial fermionic degrees of freedom.

7 Conclusions

In this paper, we have studied the Cartan geometric approach to supergravity as well as its application to loop quantum supergravity. To this end, we have provided a mathematical rigorous foundation for the formulation of super Cartan geometries. A crucial ingredient for supersymmetry is the anticommutative nature of fermionic fields. However, as we have seen, modeling anticommuting classical fermion fields turns out to be by far non-straightforward. A resolution is given considering enriched categories as initiated already in [11] based on standard techniques in algebraic geometry. This procedure requires the choice of an additional parameterizing supermanifold which encodes the fermionic degrees of freedom. Since the choice is arbitrary, one needs to ensure that physical quantities behave functorially under a change of parameterization. As we have seen, this property follows naturally, if one works in the category of relative supermanifolds. This also reflects the interpretation of supermanifolds in the sense of Molotkov–Sachse [25, 26] in terms of a functor $\textbf{Gr}\rightarrow \textbf{Top}$ assigning Grassmann algebras to Rogers—DeWitt supermanifolds.

Having formulated the notion of super Cartan geometries in the framework of enriched categories, we have then derived $\mathcal {N}=1$, $D=4$ supergravity via the super MacDowell–Mansouri action. Moreover it follows that, in this geometric framework, SUSY transformations have the interpretation in terms of (field dependent) super gauge transformations.

These results were then applied to the manifest approach toward loop quantum supergravity. To this end, studying the chiral structure of the underlying supersymmetry algebra (corresponding to the super Klein geometry) we have derived the graded analoga of Ashtekar’s self-dual variables. Moreover, we were able to interpret these variables in terms of generalized Cartan connections giving rise to super principal connections to the associated principal bundle. In this way, this provides a link between $\mathcal {N}=1$, $D=4$ supergravity and Yang–Mills theory with gauge group given the super anti-de Sitter or super Poincaré group in $D=2$ (where the latter can be obtained via a Inönü–Wigner contraction of the former) in case with or without a cosmological constant, respectively. This is in fact in complete analogy to the classical theory where the ordinary self-dual variables give ordinary gravity the structure of a $\textrm{SL}(2,\mathbb {C})$ Yang–Mills theory. Moreover, as it turns out, the possibility of defining the graded Ashtekar connection is based crucially on the particular properties of the (bosonic) self-dual variables making them in a sense unique. Furthermore, it follows that the chiral structure of the supersymmetry algebra is even preserved in case of extended supersymmetry.

This shows that the existence of the graded Ashtekar connection in the $\mathcal {N}=1$ case is not just mere coincidence and might also appear in context of extended SUGRA theories. In fact, chiral $\mathcal {N}=2$, $D=4$ AdS supergravity has been studied in [52, 53] and in context of constrained super BF-theory in [54, 55] using the graded Ashtekar connection associated with the superalgebra $\textrm{OSp}(2|2)_{\mathbb {C}}$. More recently, these variables have been derived in [56] for pure $\mathcal {N}=2$, $D=4$ AdS supergravity in the presence of boundaries in a purely geometric way following the Cartan geometric approach as studied in this present article. Thus, based on these observations, these graded variables seem to be right starting point for quantizing supergravity in the framework of loop quantum gravity. Moreover, this Cartan geometric approach may open the possibility for a systematic study of these type of variables including extended supergravity theories with $\mathcal {N}>2$ as well as matter coupled or even higher-dimensional SUGRA theories. Among other things, this may also lead to a very natural quantization of higher gauge fields in the framework of LQG. For an interesting treatment of higher gauge fields in a complementary approach that does not keep a part of the supersymmetry manifest but can handle higher-dimensional SUGRA theories, see [62].

By its very definition, the graded Ashtekar connection encodes both gravity and matter degrees of freedom. As such, it leads to a unified description of both. Since it provides a link between supergravity and Yang–Mills theory, it opens the possibility for a quantization of the theory using standard techniques of LQG following [9, 10]. Therefore, we have constructed the parallel transport map associated with the super connection 1-form in a mathematical rigorous way. In order to model anticommuting classical fermion fields, we again worked in the framework of enriched categories. It follows that the parallel transport map constructed this way indeed has the right properties, i.e., it behaves functorially under a change of parameterization. Moreover, Wilson-loop observables are indeed invariant under (parameterized) super gauge transformations.

Using the parallel transport map, we then studied the explicit construction of the state space of (manifest) loop quantum supergravity. To this end, we have, in particular, derived the invariant Haar measure of $\textrm{OSp}(1|2)_{\mathbb {C}}$. The resulting Hilbert space turned out to have a very intriguing structure, giving it the structure of a Krein space, i.e., an indefinite inner product space that can be completed to a Hilbert space with $\textrm{SL}(2,\mathbb {C})$-invariant scalar product. However, the quantization of fermions along edges is fundamentally different resulting in one-dimensional quantum excitations of the fermionic fields similar to gravity. It would be very interesting to see in which sense these different kinds of quantizations can be related.

The construction of the full state space of manifest LQSG, however, still remains incomplete, due to the non-compactness of the underlying gauge group as well as the implementation of the reality conditions. However, in [20] we made some progress in this direction studying certain symmetry reduced models where the reality conditions can be implemented exactly. We therefore have provided the required mathematical tools in the last section of this present article.

Finally, as we have outlined in Remark 5.5, this approach also uncovers some of the underlying mathematical structure behind the standard quantization scheme of fermions in LQG [17, 18]. Due to this observation, this approach may also give a concrete idea how matter fields could be quantized in the spin-foam approach to quantum gravity as suggested already in [63]. The Cartan geometric approach as studied in this article therefore seems to be the right starting point. References [79, 80, 81] are given in list but not cited in text. Please cite in text or delete from 1) References 27 and 81 are identical, but 27 was not cited correctly. Reference 27 has been corrected and 81 has been deleted. 2) References 79 and 80 are now cited in the text.

Appendices

A. Super Linear Algebra

This section is meant to fix some terminology of important aspects in super linear algebra used in the main text. Therefore, we will exclusively focus on $\mathbb {Z}_2$ grading as these are commonly used in physics in the context of (supersymmetric) field theories modeling commuting bosonic and anticommuting fermionic fields. A very deep and detailed account on this fascinating subject can be found, for instance, in the great reference of [27, 29].

Definition A.1

A $\mathbb {Z}_2$-graded or simply super vector space V is a vector space over a field $\mathbb {K}$ ($\mathbb {K}=\mathbb {R}$ or $\mathbb {C}$) of the form

$$\begin{aligned} V=V_0\oplus V_1 \end{aligned}$$

(246)

together with a map $|\cdot |:\,\bigcup _{i\in \mathbb {Z}_2}V_i\rightarrow \mathbb {Z}_2$ called parity map such that $|v|:=i$ $\forall v\in V_i$. Elements in $V_i$ are called homogeneous with parity $i\in \mathbb {Z}_2$. If the dimension of $V_0$ and $V_1$ are given by $\textrm{dim}\,V_0=m$ and $\textrm{dim}\,V_1=n$, respectively, then the dimension of V is denoted by $\textrm{dim}\,V=m|n$.

A morphism $\phi :\,V\rightarrow W$ between super vector spaces is a linear map between vector spaces preserving the parity, i.e., $\phi (V_i)\subseteq W_i$ for $i\in \mathbb {Z}_2$. The set of such super vector space morphisms is denoted by $\textrm{Hom}(V,W)$

Remark A.2

Instead of just looking at parity preserving morphisms between super vector spaces V and W, one can also consider all possible linear maps between the underlying vector spaces. This yields the internal $\underline{\textrm{Hom}}(V,W)$ which has the structure of a super vector space with the even and odd part $\underline{\textrm{Hom}}(V,W)_0$ and $\underline{\textrm{Hom}}(V,W)_1$ given by the parity preserving and parity reversing linear maps between V and W, respectively. Hence, $\underline{\textrm{Hom}}(V,W)_0$ coincides with $\textrm{Hom}(V,W)$ in Definition A.1.

Example A.3

A trivial but also very important example of a super vector space is given by $\mathbb {R}^{m|n}:=\mathbb {R}^m\oplus \mathbb {R}^n$ called the (m, n)-dimensional superspace. In fact, any super vector space $V=V_0\oplus V_1$ is isomorphic to a superspace. If $\textrm{dim}\,V=m|n$, let $\{e_i\}_{i=1,\ldots ,m+n}$ be a basis of V such that $\{e_i\}_{i=1,\ldots ,m}$ is a basis of $V_0$ and $\{e_j\}_{j=m,\ldots ,m+n}$ is a basis of $V_1$. Such a basis is called a homogeneous basis of V. Then, V is isomorphic, as a super vector space, to the superspace $\mathbb {R}^{m|n}$.

Definition A.4

A superalgebra A is a super vector space $A=A_0\oplus A_1$ together with a bilinear map $m:\,A\times A\rightarrow A$ such that

$$\begin{aligned} m(A_i,A_j)=:A_i\cdot A_j\subseteq A_{i+j}\quad \forall i,j\in \mathbb {Z}_2 \end{aligned}$$

(247)

The superalgebra A is called super commutative if

$$\begin{aligned} a\cdot b=(-1)^{|a||b|}b\cdot a \end{aligned}$$

(248)

for all homogeneous $a,b\in A$.

Definition A.5

Let A be a superalgebra. A super left A-module $\mathcal {V}$ is a super vector space which, in addition, has the structure of a left A-module such that

$$\begin{aligned} A_i\cdot \mathcal {V}_j\subseteq \mathcal {V}_{i+j}\quad \forall i,j\in \mathbb {Z}_2 \end{aligned}$$

(249)

A morphism $\phi :\,\mathcal {V}\rightarrow \mathcal {W}$ between super left A-modules is a map between the underlying super vector spaces such that $\phi (a\cdot v)=a\phi (v)$ $\forall a\in A$, $v\in \mathcal {V}$. Analogously, one defines a super right A-modules and morphisms between them.

Remark A.6

Given super left A-module $\mathcal {V}$ one can also turn it into a super right A-module setting

$$\begin{aligned} v\cdot a:=(-1)^{|v||a|}a\cdot v \end{aligned}$$

(250)

for homogeneous $a\in A$ and $v\in \mathcal {V}$. For this reason, in the following, we will simply say super A-module if we do not want to specify whether it should be regarded as a super left or right A-module. If $\mathcal {V}$ and $\mathcal {W}$ are super commutative super A-modules, their tensor product $\mathcal {V}\otimes \mathcal {W}$ is defined viewing $\mathcal {V}$ as a left and $\mathcal {W}$ as a right A-module.

Given a super left A-modules $\mathcal {V}$ and $\mathcal {W}$ we denote the set of left A-module morphisms $\phi :\,\mathcal {V}\rightarrow \mathcal {W}$ by $\textrm{Hom}_L(\mathcal {V},\mathcal {W})$ (and similarly $\textrm{Hom}_R(\mathcal {V},\mathcal {W})$ for right A-module morphisms). As in Remark A.2, instead of just looking at parity preserving morphisms, one can also consider all possible linear maps $\phi :\,\mathcal {V}\rightarrow \mathcal {W}$ between the underlying vector spaces satisfying

$$\begin{aligned} \phi (a\cdot v)=a\phi (v),\quad \forall v\in \mathcal {V},\,a\in A \end{aligned}$$

(251)

This again yields an internal $\underline{\textrm{Hom}}_L(\mathcal {V},\mathcal {W})$ which has the structure of a super right A-module with $\underline{\textrm{Hom}}_L(\mathcal {V},\mathcal {W})_0=\textrm{Hom}(\mathcal {V},\mathcal {W})$.

As usual, we denote the evaluation of a morphism $\phi \in \underline{\textrm{Hom}}_L(\mathcal {V},\mathcal {W})$ at $v\in \mathcal {V}$ via

$$\begin{aligned} \left\langle v|\phi \right\rangle \in \mathcal {W} \end{aligned}$$

(252)

This has the advantage that one does not need to care about signs due to super commutativity after right multiplication with elements $a\in A$, i.e., $\left\langle v|\phi a\right\rangle =\left\langle v|\phi \right\rangle a$ $\forall a\in A$, $v\in \mathcal {V}$.

Finally, let us define via $\phi \diamond \psi \in \underline{\textrm{Hom}}_L(\mathcal {V},\mathcal {W}')$ the composition of two left-linear morphisms $\phi :\,\underline{\textrm{Hom}}_L(\mathcal {V},\mathcal {W})$ and $\psi :\,\underline{\textrm{Hom}}_L(\mathcal {W},\mathcal {W}')$ given by

$$\begin{aligned} \left\langle v|\phi \diamond \psi \right\rangle :=\left\langle \left\langle v|\phi \right\rangle |\psi \right\rangle ,\quad \forall v\in \mathcal {V} \end{aligned}$$

(253)

which, by definition, is well-behaved under right multiplication.

Definition A.7

A super A-Lie module (or super Lie algebra or Lie superalgebra if $A=\mathbb {K}$) is a super A-module L such that bilinear map $m\equiv [\cdot ,\cdot ]:\,L\times L\rightarrow L$, also called the bracket, is graded skew-symmetric, i.e.,

$$\begin{aligned}{}[a,b]=-(-1)^{|a||b|}[b,a] \end{aligned}$$

(254)

and satisfies the graded Jacobi identity

$$\begin{aligned}{}[a,[b,c]]+(-1)^{|a|(|b|+|c|)}[b,[c,a]]+(-1)^{|c|(|a|+|b|)}[c,[a,b]]=0 \end{aligned}$$

(255)

for all homogeneous $a,b,c\in L$.

Example A.8

• (i) If V is a vector space (finite or infinite dimensional), then the exterior power $\bigwedge {V}:=\bigoplus _{k=0}^{\infty }{\bigwedge ^{k}V}$ also called Grassmann algebra naturally defines a superalgebra with even and odd part given by $(\bigwedge {V})_0$ $:=\bigoplus _{k=0}^{\infty }{\bigwedge ^{2k}V}$ and $(\bigwedge {V})_1:=\bigoplus _{k=0}^{\infty }{\bigwedge ^{2k+1}V}$, respectively. The Grassmann algebra is super commutative, associative and unital with unit $1\in \mathbb {R}=\bigwedge ^{0}V$.

• (ii) For A a superalgebra, the tensor product $A\otimes \mathbb {K}^{m|n}$ defines a super A-module with grading $(A\otimes \mathbb {K}^{m|n})_0=A_0\otimes \mathbb {K}^{m}\oplus A_1\otimes \mathbb {K}^{n}$ and $(A\otimes \mathbb {K}^{m|n})_1=A_0\otimes \mathbb {K}^{n}\oplus A_1\otimes \mathbb {K}^{m}$.

Definition A.9

A super A-module $\mathcal {V}$ is called free if it contains a homogeneous basis $\{e_i\}_{i=1,\ldots ,m+n}$, $\textrm{dim}\,\mathcal {V}=m|n$, such that any element $v\in \mathcal {V}$ can be written in the form $v=a^ie_i$ with coefficients $a^i\in A$ $\forall i=1,\ldots ,m+n$. Equivalently, $\mathcal {V}$ is a free super A-module iff it is isomorphic to $A\otimes \mathbb {K}^{m|n}$.

Definition A.10

• (i) Let $\mathcal {V}$ be a free super A-module. Two homogeneous bases $\{e_i\}_i$ and $\{f_j\}_j$ of $\mathcal {V}$ are called equivalent if they are related to each other by scalar coefficients, i.e., there exist real numbers ${a}{_i^j}\in \mathbb {K}$ such that $e_i={a}{_i^j}f_j$ $\forall i.j$ (for a proof that this indeed defines an equivalence relation see [29]).

• (ii) A free super A-module $\mathcal {V}$ together with a distinguished equivalence class of homogeneous bases of $\mathcal {V}$ is called a super A-vector space. A morphism $\phi :\,\mathcal {V}\rightarrow \mathcal {W}$ between super A-vector spaces is a morphism of super A-modules such that $\phi$ maps the equivalence class of bases of $\mathcal {V}$ to the real vector space spanned by the equivalence class of bases of $\mathcal {W}$.

Remark A.11

If $\mathcal {V}$ is a super A-vector space with equivalence class $[\{e_i\}_i]$ of homogeneous bases of $\mathcal {V}$, then $\mathcal {V}\cong A\otimes V$ with V the super vector space spanned by $\{e_i\}_i$ which, in particular, is independent on the choice of a representative of that equivalence class. Hence, the choice of such an equivalence class yields a well-defined super vector space V also called the body of $\mathcal {V}$. On the other hand, if $\mathcal {V}$ is a free super A-module, one can always choose a homogeneous basis $\{e_i\}_i$ of $\mathcal {V}$ such that $\mathcal {V}$ becomes a super A-vector space w.r.t. the equivalence class $[\{e_i\}_i]$. However, such a choice may not be canonical and various different bases exist which are not related by scalar coefficients.

B. Spinor Calculus

We summarize some important formulas concerning gamma matrices in Minko wksi spacetime and their chiral representation as we will need them in section and. The Clifford algebra $\textrm{Cl}(\mathbb {R}^{1,3},\eta )$ of four-dimensional Minkowski spacetime is generated by gamma matrices $\gamma _I$, $I\in \{0,\ldots ,3\}$, satisfying the Clifford algebra relations

$$\begin{aligned} \left\{ \gamma _{I},\gamma _{J}\right\} =2\eta _{IJ} \end{aligned}$$

(256)

where the Minkowski $\eta$ is chosen with signature $\eta =\textrm{diag}(-+++)$. In the chiral theory, we are working in the chiral or Weyl representation in which the gamma matrices take the form

$$\begin{aligned} \gamma _{I}=\begin{pmatrix} 0 &{}\quad \sigma _{I}\\ \bar{\sigma }_{I} &{}\quad 0 \end{pmatrix}\quad \text {and}\quad \gamma _{*}=\begin{pmatrix} \mathbbm {1} &{}\quad 0\\ 0&{}\quad -\mathbbm {1} \end{pmatrix} \end{aligned}$$

(257)

with $\gamma _*:=i\gamma _0\gamma _1\gamma _2\gamma _3$ the highest rank Clifford algebra element also commonly denoted by $\gamma _5$ in four spacetime dimensions. Here,

$$\begin{aligned} \sigma _{I}=(-\mathbbm {1},\sigma _i)\quad \text {and}\quad \bar{\sigma }_{I}=(\mathbbm {1},\sigma _i) \end{aligned}$$

(258)

with $\sigma _i$, $i=1,2,3$ the ordinary Pauli matrices. With respect to spinorial indices, (258) are written as $\sigma _I^{AA'}$ and $\bar{\sigma }_{I A'A}$, respectively. These can be used to map the internal indices I of the co-frame $e^I$ to spinorial indices setting

$$\begin{aligned} e_{\mu }^{AA'}=e^I_{\mu }\sigma _I^{AA'} \end{aligned}$$

(259)

Primed and unprimed Spinor indices are raised and lowered with respect to the complete antisymmetric symbols $\epsilon ^{A'B'}$ and $\epsilon ^{AB}$, respectively, with the convention

$$\begin{aligned} \psi _A=\psi ^B\epsilon _{BA}\quad \text {and}\quad \psi ^A=\epsilon ^{AB}\psi _B \end{aligned}$$

(260)

and analogously for primed indices. Due to $\epsilon \sigma _i\epsilon =\sigma _i^T$, one has the useful formula

$$\begin{aligned} \sigma _{I AA'}=\sigma _I^{BB'}\epsilon _{BA}\epsilon _{B'A'}=-\bar{\sigma }_{I A'A} \end{aligned}$$

(261)

Using (258) as well as (261), it is easy to see that

$$\begin{aligned} \sigma ^I_{AA'}\sigma _J^{AA'}=-2\delta _J^I \end{aligned}$$

(262)

Finally, in a $3+1$-decomposition $M\cong \mathbb {R}\times \Sigma$ of the four-dimensional spacetime M, one considers the spinor-valued one-forms $e_a^{AA'}$ which are related to the spatial metric q on $\Sigma$ according to

$$\begin{aligned} 2q_{ab}=-e_{a AA'}e_b^{AA'} \end{aligned}$$

(263)

with $a=1,2,3$. These, together with the future-directed unit normal vector field $n^{AA'}$ which is normal to the time slices $\Sigma _t$ and satisfies

$$\begin{aligned} n_{AA'}e_a^{AA'}=0\quad \text {and}\quad n_{AA'}n^{AA'}=2 \end{aligned}$$

(264)

form a basis of spinors with one primed and one unprimed index. Finally, let us mention the important identities

$$\begin{aligned} n_{AA'}n^{AB'}&=\delta _{A'}^{B'} \end{aligned}$$

(265)

$$\begin{aligned} n_{AA'}n^{BA'}&=\delta _{A}^{B} \end{aligned}$$

(266)

$$\begin{aligned} \sigma _{i AA'}\sigma ^{AB'}_j&=-\delta _{ij}\delta _{A'}^{B'}-i{\epsilon }{_{ij}^k}n_{AA'}\sigma ^{AB'}_k \end{aligned}$$

(267)

$$\begin{aligned} \sigma _{i AA'}\sigma ^{BA'}_j&=-\delta _{ij}\delta _{A}^{B}+i{\epsilon }{_{ij}^k}n_{AA'}\sigma ^{BA'}_k \end{aligned}$$

(268)