Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism

Abstract

This note describes a local Poisson structure (up to homotopy) associated with corners in four-dimensional gravity in the coframe (Palatini–Cartan) formalism. This is achieved through the use of the BFV formalism. The corner structure contains in particular an Atiyah algebroid that couples the internal symmetries to diffeomorphisms. The relation with BF theory is also described.

1 Introduction

The goal of this paper is to describe the Poisson structures (up to homotopy) that arise on two-dimensional corners of four-dimensional gravity in the coframe (Palatini–Cartan) formalism.

From a more general perspective, one expects quantum field theory on a cylinder to describe the quantum evolution of a system described by a Hilbert space attached to a boundary component. If the boundary has itself a boundary—a corner for space–time—the Hilbert space is expected to be a representation of some algebra associated with the corner. A standard example where this picture is considered is that of the vertex operator algebra arising from a punctured two-dimensional boundary.

At the classical level, one then expects a symplectic manifold to be associated with a boundary and a Poisson manifold to be associated with a corner. This picture is, however, problematic, since the constructions typically involve singular quotients.

A more suitable picture, which we use in this paper, is that of the Batalin–Fradkin–Vilkovisky (BFV) formalism [3,4,5], which replaces a (possibly singular) symplectic quotient by a cohomological resolution: namely, one extends the space of boundary fields to a superspace with additional structure (a symplectic structure—the BFV form—together with a Hamiltonian vector field that squares to zero—the BRST operator).

An added bonus of this formalism is that it naturally produces a structure on the corners [16, 17] which, upon choosing a “polarization” (i.e., a choice of a foliation by Lagrangian submanifolds) is associated with a Poisson structure (up to homotopy).

We recall this construction, together with background material, in the first part of Sect. 2, whereas in its second part we apply it to some instructive examples (Yang–Mills, Chern–Simons, and, notably, 4D BF theory).

In Sect. 4, we recall the BFV formulation of 4D Palatini–Cartan theory [13], and in Sect. 5.1, we apply the aforementioned construction for corners and observe that it is singular. Nonetheless, it is possible to study and describe, in Sect. 6, a naturally associated local Poisson algebra up to homotopy. This algebra is actually generated through Poisson brackets and a differential by the observables

$$\begin{aligned} J_\phi =\frac{1}{2}\int _\Gamma \phi ee, \end{aligned}$$

where $\Gamma$ is the two-dimensional corner, e is the coframe (tetrad) field (restricted to the corner), and $\phi$ is an $\mathfrak {so}(3,1)$-valued test function (Lie algebra pairing is tacitly understood in the notation).

These particular observables are reminiscent of the area observables considered in loop quantum gravity (see, e.g., [31] and references therein), where, however, $\Gamma$ is a closed surface inside the boundary instead of being a corner (and Ashtekar $\mathfrak {su}(2)$ variables are used instead). In particular, our observable has a similar form of the variable conjugate to the holonomy of the connection in loop quantum gravity. Namely, $J_\phi$ is the same as the variable (7.7) introduced in [32]. The corner structure leads to the Poisson bracket $\{J_{\phi _1},J_{\phi _2}\}_\text {corner}=J_{[\phi _1,\phi _2]}$, which is in line with the Poisson bracket of area observables, although we use here the Poisson bracket associated with the corner instead of that associated with the boundary¹ and, unlike in [15], no regularization is required in this context.

The above observables retain information of the internal $\mathfrak {so}(3,1)$ symmetry of Palatini–Cartan gravity. The other observables they generate, through the differential in the homotopy Poisson algebra, contain information about tangential and transversal vector fields encoding the diffeomorphism symmetry as well.

An interesting fact, which deserves further investigation, is that this corner structure actually turns out to be the corner structure for four-dimensional BF theory restricted to a submanifold of fields.

In order to understand better the algebra found, it is useful to consider some particular cases. In Sect. 7, we describe two possible restrictions of the general theory, called constrained and tangent theory, and produce a better description of a restricted version of the aforementioned local Poisson algebra up to homotopy. In the first (Sect. 7.1.2), we impose some ad hoc constraints that do not modify the classical structure of the theory, while in the second (Sect. 7.2.1)), we essentially freeze the generators of transversal diffeomorphisms. In the tangent theory, the associated Poisson manifold turns out to be a Poisson submanifold of the dual space of sections of an Atiyah algebroid associated with the corner (Sect. 7.2.4). We briefly discuss the quantization when the corner is a sphere and the fields are assumed to be constant—a situation that is relevant in the case of a punctured boundary (Sect. 7.2.5).

These results are of course expected to be related to the BMS group [8, 25, 30, 33, 38] at infinity, which has been extensively studied (see, e.g., [2, 23] and references therein). Furthermore, we expect some connections between the results of this paper and those derived recently in the context of the covariant phase space method (see, e.g., [19, 22, 28, 29] and references therein). This will be object of future work. The difference with our approach is that we assume the boundary of space–time to be a compact manifold with boundary. For a noncompact manifold, one should instead choose an appropriate compactification, related to the chosen asymptotic conditions for the fields. Given the differences in the formalism, the connection between these structures is not fully understood; hence, we plan to explore these topics in a forthcoming work. A similar Poisson structure on corners, in the case when a momentum map is available, can be found in [42].

Some of the results in this paper (in particular Sects. 5, 7.1 except Sect. 7.1.2, and the first part of 7.2 until Sect. 7.2.1) first appeared in the PhD thesis of the first author [12] available online but not published in a peer-reviewed journal.

2 Preliminaries and Relevant Constructions

In this section, we review how the BFV formalism is used to describe coisotropic reduction, which is relevant for the boundary structure of a field theory, how the $\hbox {BF}^2$V formalism is used to describe Poisson structures (possibly up to homotopy), which is relevant for the corner structure of a field theory, and how the two may be related.

Remark 1

We group here some references for this section, not to interrupt the flow of the following. For Poisson and symplectic structures, see, e.g., [6]. The notion of coisotrope was introduced in [41]. The notion of derived bracket was introduced in [27] and generalized in [39, 40]. The notion of $\hbox {BF}^m$V structures and their mutual relations, in particular arising from relaxed structures, was introduced in [16, 17], although not with this name; note that there is a parallel story developed in derived symplectic geometry, see [9, 10, 34] and references therein. The existence of BFV structures associated with coisotropic submanifolds is discussed in [21, 35,36,37].

2.1 Background Notions

We start recalling some important preliminaries.

2.1.1 Poisson and Symplectic Structures

Definition 2

A Poisson algebra is a pair $(A, \{\ ,\ \})$ where A is a commutative algebra (for our applications always over $\mathbb {R}$) and $\{\ ,\ \}$ is a bilinear, skew-symmetric operation on A which is a derivation w.r.t. each argument (Leibniz rule)—i.e., a biderivation—and satisfies the Jacobi identity. The operation $\{\ ,\ \}$ is called a Poisson bracket.

The simplest example of a Poisson algebra is any algebra with the zero Poisson bracket. Another interesting example is the symmetric algebra $S(\mathfrak {g})$ of a Lie algebra $\mathfrak {g}$, where the Lie bracket is extended by the Leibniz rule. Symplectic manifolds also produce Poisson algebras, as we recall below.

Definition 3

A Poisson manifold is a pair $(M,\{\ ,\ \})$ where M is a smooth manifold and $\{\ ,\ \}$ is a Poisson bracket on $C^\infty (M)$.

Again we have the simplest example of the zero Poisson bracket. The dual $\mathfrak {g}^*$ of a finite-dimensional Lie algebra $\mathfrak {g}$ is also an example, where the Poisson bracket on $S(\mathfrak {g})$, now viewed as the algebra of polynomial functions on $\mathfrak {g}^*$, is extended to the whole $C^\infty (\mathfrak {g}^*)$.

A biderivation $\{\ ,\ \}$ on a smooth manifold M is always determined by a bivector field $\pi$ via $\{f,g\}=-\pi (\textrm{d}f,\textrm{d}g)$. If we denote by $[\ ,\ ]$ the Schouten bracket of multivector fields, the Jacobi identity for the bracket is equivalent to the Maurer–Cartan equation $[\pi ,\pi ]=0$. In this case, $\pi$ is called a Poisson bivector field. Moreover, we can also write $\{f,g\}=[[\pi ,f],g]$, which is an example of derived bracket, on which we will elaborate below. In the trivial case, $\pi$ is the zero bivector field. In the case of the dual of a Lie algebra $\mathfrak {g}$, we have $\pi ^{ij}=-f^{ij}_k x^k$, where the $f^{ij}_k$s are the structure constant of $\mathfrak {g}$ in some basis and the $x^k$s are the coordinate on $\mathfrak {g}^*$ w.r.t. the same basis.

Definition 4

A symplectic manifold is a pair $(M,\varpi )$ where M is a smooth manifold and $\varpi$ is a closed nondegenerate two-form on M. If M is infinite dimensional, we require only weak nondegeneracy, namely, that at every point x

$$\begin{aligned} \varpi _x(v,w) = 0\ \forall v\in T_xM \implies w=0. \end{aligned}$$

This condition implies that a function f has at most one Hamiltonian vector field $X_f$: $\iota _{X_f}\varpi = \textrm{d}f$. We say that a function is Hamiltonian if it has a Hamiltonian vector field and we denote the space of such functions $C^\infty (M)_\text {hamiltonian}$. The Poisson bracket of two Hamiltonian functions f and g, with Hamiltonian vector fields denoted $X_f$ and $X_g$, respectively, is defined as

$$\begin{aligned} \{f,g\}:= X_f(g) = \iota _{X_f}\iota _{X_g}\varpi . \end{aligned}$$

It is a Poisson bracket on $C^\infty (M)_\text {hamiltonian}$. If M is finite dimensional, then $(M,\{\ ,\ \})$ is a Poisson manifold; the corresponding Poisson bivector field is the inverse of the symplectic structure.

Remark 5

The above can be generalized to the case when we drop the nondegeneracy condition. In this case, we say that a vector field X is in the kernel of $\varpi$ if $\iota _X\varpi =0$. We call a function f invariant if $X(f)=0$ for every X in the kernel of $\varpi$. We call, as before, f Hamiltonian if it possesses a Hamiltonian vector field $X_f$: $\iota _{X_f}\varpi = \textrm{d}f$. Note that in general the Hamiltonian vector field is no longer unique. A Hamiltonian function is automatically invariant. The action of a Hamiltonian function f on an invariant function g is defined as $\{f,g\}:= X_f(g)$, where it does not matter which Hamiltonian vector field we take and produces an invariant function. If also g is Hamiltonian, then the result is Hamiltonian as well, and $\{\ ,\ \}$ is a Poisson bracket on $C^\infty (M)_\text {hamiltonian}$.

2.1.2 Coisotropic Submanifolds and Reduction

Definition 6

A coisotrope in a Poisson algebra $(A, \{\ ,\ \})$ is an ideal I in the commutative algebra A which satisfies $\{I,I\}\subseteq I$: i.e., I is a Lie subalgebra of $(A, \{\ ,\ \})$.

Note that I naturally acts on the commutative algebra A/I via the bracket. We also have $(A/I)^I= N(I)/I$, where $N(I):=\{a\in A\ |\ \{a,I\}\subseteq I\}$ is the Lie normalizer of I in A. The latter description shows that ${\underline{A}}_I:=(A/I)^I=N(I)/I$ is a Poisson algebra, called the reduction of A w.r.t. to I.

Definition 7

A coisotropic submanifold of a Poisson manifold $(M,\{\ ,\ \})$ is a submanifold²C of M such that its vanishing ideal I is a coisotrope in $(C^\infty (M),\{\ ,\ \})$.

Remark 8

If C is the zero locus of constraints $\phi _i$, the latter condition is equivalent to having $\{\phi _i,\phi _j\}=f_{ij}^k \phi _k$, where summation over repeated indices is understood and the $f_{ij}^k$s are functions, called the structure functions. Constraints satisfying this condition are called first class in Dirac’s terminology.

If M is a finite-dimensional symplectic manifold, then this definition of coisotropic submanifold is equivalent to the geometric one that, for every $x\in C$, the subspace $T_xC$ be coisotropic, i.e., $(T_xC)^\perp \subseteq T_xC$,³ for every $x\in C$. The Hamiltonian vector fields of elements of the vanishing ideal span the involutive distribution $(TC)^\perp$.

Proposition 9

If the quotient space ${\underline{C}}$ has a smooth manifold structure for which the projection $\pi :C\rightarrow {\underline{C}}$ is a smooth submersion, then ${\underline{C}}$ is endowed with a unique symplectic structure ${\underline{\varpi }}$ such that $\pi ^*{\underline{\varpi }}=\iota ^*\varpi$, where $\iota :C\rightarrow M$ is the inclusion map. The pair $({\underline{C}},{\underline{\varpi }})$ is called the symplectic reduction of C. In this case, the resulting Poisson algebra $C^\infty ({\underline{C}})$ is the reduction ${\underline{A}}_I$ described above.

If M is an infinite-dimensional symplectic manifold, there are inequivalent ways of defining a coisotropic submanifold. In this paper, we will stick to the algebraic definition. More precisely, we assume that the vanishing ideal I is generated by its Hamiltonian part $I_\text {hamiltonian}:=I\cap C^\infty (M)_\text {hamiltonian}$ and that $I_\text {hamiltonian}$ is a coisotrope in $C^\infty (M)_\text {hamiltonian}$.

Remark 10

The importance of coisotropic submanifolds in field theory is related to the problem of finding the correct space of initial conditions for the Cauchy problem. Indeed, the coisotropic submanifold C arises as a submanifold of the space of boundary fields with the constraints determined by the Euler–Lagrange equations that do not involve transversal derivatives. In case, this construction arises from the Hamiltonian description associated with a Cauchy surface, the reduced phase space, i.e., the reduction ${\underline{C}}$ of C, is the correct space of initial conditions for the Cauchy problem.

2.1.3 The Graded Case: $\hbox {BF}^m$V Structures

All the above can be extended to the world of graded algebras and graded manifolds (supermanifolds with an additional $\mathbb {Z}$-grading on the local coordinates). Note that we assume both a grading and a parity, the latter being responsible for the sign rules. In all the examples in this paper, they are related, with the parity being the grading modulo two.

Definition 11

A graded Poisson algebra is a pair $(A, \{\ ,\ \})$ where A is a graded commutative algebra and $\{\ ,\ \}$ is a bilinear, graded skew-symmetric operation on A which is a graded derivation w.r.t. each argument (graded Leibniz rule) and satisfies the graded Jacobi identity.

It is important to notice that the grading of the bracket may be a shifted grading w.r.t. the original one.

An even bracket of degree 0—the straightforward generalization from the ungraded case—is also known as a BFV bracket. An odd bracket of degree $+1$ is also known as a BV bracket. We will call an odd bracket of degree $-1$ a $\hbox {BF}^2$V bracket.

Definition 12

An n-graded symplectic manifold is a pair $(M,\varpi )$ where M is a graded manifold and $\varpi$ is a closed nondegenerate two-form on M of homogenous degree n and parity $n\!\!\mod 2$. It defines a graded Poisson algebra structure on $C^\infty (M)_\text {hamiltonian}$ with bracket of degree $-n$.

An additional structure, important for the following, is that of cohomological vector field on a graded manifold M. This is an odd vector field Q of degree $+1$ satisfying $[Q,Q]=0$. Note that Q defines a differential on $C^\infty (M)$. For this reason, the pair (M, Q) is called a differential graded manifold (shortly, a dg manifold).

Definition 13

A dg manifold with a compatible symplectic structure, i.e., with $L_Q\varpi =0$, is called a differential graded symplectic manifold (shortly, a dg symplectic manifold).

We will always assume that Q is Hamiltonian, namely, that there is an $S\in C^\infty (M)_\text {hamiltonian}$ such that $\iota _Q\varpi = \textrm{d}S$ and $\{S,S\}=0$ (the master equation).⁴ If $\varpi$ has degree n, then S has degree $m=n+1$. In this case, we call the triple $(M,\varpi ,S)$ a $\hbox {BF}^m$V manifold.

Remark 14

BV manifolds arise in field theories as a generalization of the BRST formalism to discuss independence of gauge-fixing in the perturbative functional-integral quantization; we will not address this issue in this paper. BFV manifolds are used to give a cohomological description of reduced phase spaces. $\hbox {BF}^2$V manifolds describe Poisson structures (up to homotopy). We will recall these two constructions in Sects. 2.1.5 and 2.2, respectively.

2.1.4 Relaxed and Induced Structures

The above may be generalized by dropping the master equation, the condition that $\varpi$ is nondegenerate, and the strict relation among $(Q,\varpi ,S)$. Namely, we only assume that $\varpi$ is a closed two-form on M of homogenous degree $(m-1)$ and parity $(m-1)\!\!\mod 2$ and that Q is a cohomological vector field: we call this a relaxed $\hbox {BF}^m$V structure. We define $\widetilde{\alpha }:=\iota _Q\varpi -\textrm{d}S$ and $\widetilde{\varpi }=\textrm{d}\widetilde{\alpha }$. It turns out that Q and $\widetilde{\varpi }$ are compatible, i.e., $L_Q\widetilde{\varpi }=0$. We actually assume the slightly stronger condition $\iota _Q\widetilde{\varpi }=\textrm{d}\widetilde{S}$ for some function $\widetilde{S}$. One can also show the useful identity $\frac{1}{2}\iota _Q\iota _Q\varpi =\widetilde{S}$, called the modified master equation. We call the triple $(M,\widetilde{\varpi },\widetilde{S})$, or any of its partial reductions by an integrable subdistribution of the kernel of $\widetilde{\varpi }$, a pre–$\hbox {BF}^{m+1}$V manifold. If the whole reduction by the kernel is smooth, it is then a $\hbox {BF}^{m+1}$V manifold as defined above. In this case, we say that the relaxed $\hbox {BF}^m$V structure is 1-extendable.

Remark 15

In the case of field theory, we always assume locality. Namely, M is a space locally modeled on sections, the fields, of a vector bundle over some closed manifold $\Sigma$, and the structures $(Q,\varpi ,S)$ are integrals over $\Sigma$ of densities defined, at each point, in terms of jets of the fields. The relaxed structure typically arises when one extends the strict structure to a manifold with boundary,⁵ by taking the same triple $(Q,\varpi ,S)$. In this case, the “error term” $\widetilde{\alpha }$ arises by integration by parts and is concentrated on $\partial \Sigma$. Modding out by (part of) the kernel of $\widetilde{\varpi }$ then yields a (pre–)$\hbox {BF}^{m+1}$V structure depending on jets of the fields restricted to $\partial \Sigma$.

2.1.5 The BFV Formalism

If $(M,\varpi ,S)$ is a BFV manifold, then the zeroth cohomology group $H^0_Q(C^\infty (M)_\text {hamiltonian})$ is a Poisson algebra.⁶ Namely, if [f] and [g] are cohomology classes, we define $\{[f],[g]\}:=[\{f,g\}]$. This Poisson algebra is understood as the algebra of function of a would-be symplectic reduction.

This is justified by the BFV construction. Namely, one starts with a symplectic manifold $(M_0,\varpi _0)$ and a coisotropic submanifold C of $M_0$. One can then associate with it a BFV manifold $(M,\varpi ,S)$ that contains $(M_0,\varpi _0)$ as its degree zero part and such that C is recovered as the intersection of $M_0$ with the critical locus of S. (This construction works in general if $M_0$ is finite dimensional; in the infinite-dimensional case, it works at least when C is given by global constraints.) For example, if M is finite dimensional and C is locally defined by constraints $\phi _i$, then in local coordinates we have $S=c^i\phi _i + \cdots$, where the $c^i$s are the coordinates of degree $+1$ and the dots are in the ideal generated by the coordinates of degree $-1$. The dots here have to be added to ensure that the master equation is satisfied.

If C has a smooth reduction ${\underline{C}}$, then $H^0_Q(C^\infty (M)_\text {hamiltonian})$ is isomorphic, as a Poisson algebra, to $C^\infty ({\underline{C}})$. In general, one views $(M,\varpi ,S)$ as a good replacement (a cohomological resolution) for the reduction of C.

2.2 $P_\infty$ Structures from the $\hbox {BF}^2$V Formalism

In this case, $\varpi$ is an odd symplectic form of degree $+1$. We start with the finite-dimensional case. One then has that $(M,\varpi )$ is always symplectomorphic to a shifted cotangent bundle $T^*[1]N$, with canonical symplectic structure, for some graded manifold N (with this notation we mean that the fiber coordinates of $T^*N$ are assigned opposite parity and degree shifted by one w.r.t. the natural ones). We call this choice of N a polarization. Note that the Poisson algebra of functions on $T^*[1]N$ can be canonically identified with the algebra of multivector fields on N with the Schouten bracket. The function S, of degree $+2$, then corresponds to a linear combination $\pi =\pi _0+\pi _1+\pi _2+\cdots$, where $\pi _i$ is an i-vector field of degree $2-i$ on N. The master equation $\{S,S\}=0$ corresponds to the equations

$$\begin{aligned} {[}\pi _0,\pi _1]&=0,\\ {[}\pi _0,\pi _2]+\frac{1}{2}[\pi _1,\pi _1]&=0,\\ {[}\pi _0,\pi _3]+[\pi _1,\pi _2]&=0,\\ {[}\pi _0,\pi _4]+[\pi _1,\pi _3]+\frac{1}{2}[\pi _2,\pi _2]&=0,\\ \dots \end{aligned}$$

We start from the simpler case when N has only coordinates in degree zero (this is possible only if M has only coordinates in degree zero and one). In this case, $\pi =\pi _2$ and $[\pi _2,\pi _2]=0$, so $\pi$ is a Poisson structure on N. Algebraically, we can get the corresponding Poisson algebra as the algebra $C^\infty _0(T^*[1]N)$ of functions on $T^*[1]N$ of degree zero with Poisson bracket $\{f,g\}_2=[[\pi ,f],g]$.

In the general case, $\pi$ is called a $P_\infty$ structure on N (this stands for Poisson structure up to coherent homotopies). This structure is called curved if $\pi _0\not =0$. The $\pi _i$s, applied to the differentials of i functions on N, define multibrackets $\{\ \}_i$ on $C^\infty (N)$ which in turn define a (curved) $\hbox {L}_\infty$-algebra. Moreover, they are graded derivations w.r.t. each argument. The multibrackets may also be defined as derived brackets

$$\begin{aligned} \{f_1,\dots ,f_i\}_i = [[[[\cdots [\pi _i,f_1],f_2],\dots ],f_i] = P[[[[[\cdots [\pi ,f_1],f_2],\dots ],f_i], \end{aligned}$$

where P is the projection from multivector fields to functions. In particular, we have

$$\begin{aligned} \{\}_0&= \pi _0,\\ \{f\}_1&= \pi _1(f),\\ \{f,g\}_2&=[[\pi _2,f],g]. \end{aligned}$$

We will call these brackets, respectively, the nullary, unary and binary operations or, equivalently, the 0-bracket, 1-bracket and 2-bracket.

2.2.1 Generalizations

The above structure may be generalized as follows. Suppose we have a splitting $\mathfrak {a}=\mathfrak {p}\oplus \mathfrak {h}$ of an odd Poisson algebra $\mathfrak {a}$ (e.g., $C^\infty (M)$) into Poisson subalgebras with $\mathfrak {h}$ abelian (i.e., $\mathfrak {p}\cdot \mathfrak {p}\subseteq \mathfrak {p}$, $\mathfrak {h}\cdot \mathfrak {h}\subseteq \mathfrak {h}$, $\{\mathfrak {p},\mathfrak {p}\}\subseteq \mathfrak {p}$, $\{\mathfrak {h},\mathfrak {h}\}=0$). Let P be the projection $\mathfrak {a}\rightarrow \mathfrak {h}$. If $S\in \mathfrak {a}$ satisfies the master equation $\{S,S\}=0$, then the multibrackets

$$\begin{aligned} \{f_1,\dots ,f_i\}_i:= P\{\cdots \{S,f_1\},f_2\},\dots \},f_i\} \end{aligned}$$

make $\mathfrak {h}$ into a $P_\infty$ algebra. The previous case consisted in considering $\mathfrak {a}=C^\infty (T^*[1]N)$ and taking $\mathfrak {p}$ as the multivector fields on N of multivector degree larger than zero and $\mathfrak {h}$ as the functions on N; note that in this case $\mathfrak {h}$ is maximal as an abelian subalgebra. We call the more general choice of $(\mathfrak {p},\mathfrak {h})$ a weak polarization.

Remark 16

The algebraic construction makes sense also if $\varpi$ is degenerate. In this case, we consider a splitting, with the above properties, of the $-1$-Poisson algebra of hamiltonian functions: $C^\infty _\text {hamiltonian}(M)=\mathfrak {p}\oplus \mathfrak {h}$.

Remark 17

An important case is when $\varpi$ is degenerate, but its kernel has constant rank (i.e., the dimension of the kernel of $\varpi _x$ is the same for all $x\in M$). In this case, one calls it a presymplectic form. Note that the kernel is also involutive. If the quotient space of M by the kernel has a smooth structure, it is then symplectic, so it can be identified with some $T^*[1]N$. We can then take $\mathfrak {h}=p^* C^\infty (N)$, where p denotes the projection $M\rightarrow T^*[1]N$.

Remark 18

More generally, we can take the quotient of M by an involutive subdistribution of constant rank of the kernel of $\varpi$. If the quotient ${\underline{M}}$ has a smooth structure and p denotes the projection from M to ${\underline{M}}$, then we can take $\mathfrak {h}=p^* \mathfrak {h}'$, where $C^\infty _\text {hamiltonian}({\underline{M}})=\mathfrak {p}'\oplus \mathfrak {h}'$ is a splitting as above.

Let us now turn to the infinite-dimensional case. The first remark is that in this case M is symplectomorphic to a symplectic subbundle of $T^*[1]N$, for some infinite-dimensional graded manifold N. The only difference with the finite-dimensional case is that now not every function is Hamiltonian. We can anyway define the derived brackets, as above, on $C^\infty _\text {hamiltonian}(N):=C^\infty (N)\cap C^\infty _\text {hamiltonian}(M)$. The algebraic version for weak polarizations and its extension to the degenerate case works verbatim as above.

3 Corner Structures of Field Theories

In this section, we consider some illustrating examples of BFV and $\hbox {BF}^2$V structures in field theory (electromagnetism, Yang–Mills theory, Chern–Simons theory, BF theory). In particular, the example of BF theory is preliminary to our discussion of these structures in gravity.

Remark 19

From here on, we denote the differential on a space of fields by $\delta$, reserving the notation $\textrm{d}$ to the de Rham differentials on the underlying manifolds. Furthermore, we will denote with an apex $\partial$ all the quantities with fields defined on $\Sigma$ and with an apex $\partial \partial$ all the quantities with fields defined on $\partial \Sigma$. This notation is chosen in order to make contact with the one used in many previous articles. This is due to the fact that often the BFV theory can be induced from a BV theory when $\Sigma$ is considered as a boundary of a manifold M.

3.1 Electromagnetism

To warm up, we start with the simple example of electromagnetism in $d+1$ dimensions. In the Hamiltonian formalism, we then consider a d-dimensional Riemannian closed⁷ manifold $(\Sigma ,g)$, which for simplicity we assume to be oriented. The fields are the vector potential $\textbf{A}$ and the electric field $\textbf{E}$ with symplectic structure $\varpi ^{\partial }_0=\int _\Sigma \delta \textbf{A}\cdot \delta \textbf{E}\,\sqrt{\det g}$, where $\cdot$ denotes the inner product defined by the Riemannian metric g and $\sqrt{\det g}$ is the corresponding canonical density.

The constraints are given by the Gauss law ${\text {div}} \textbf{E}= 0$. To implement the BFV formalism, we then have to introduce a ghost $c\in C^\infty (\Sigma )[1]$ and its conjugate momentum $b\in \Omega ^d(\Sigma )[-1]$. We then have the BFV symplectic form

$$\begin{aligned} \varpi ^{\partial }=\int _\Sigma (\delta \textbf{A}\cdot \delta \textbf{E}\,\sqrt{\det g}+\delta b\,\delta c) \end{aligned}$$

and the BFV action

$$\begin{aligned} S^{\partial } = \int _\Sigma c\,{\text {div}} \textbf{E}\,\sqrt{\det g}. \end{aligned}$$

The variation of $S^{\partial }$ is

$$\begin{aligned} \delta S^{\partial } = \int _\Sigma (\delta c\,{\text {div}} \textbf{E}- c\,{\text {div}} \delta \textbf{E})\,\sqrt{\det g} = \int _\Sigma (\delta c\,{\text {div}} \textbf{E}+ {\text {grad}}c\cdot \delta \textbf{E}) \,\sqrt{\det g}, \end{aligned}$$

which shows that $S^{\partial }$ is Hamiltonian, $\iota _{Q^{\partial }}\varpi ^{\partial }=\delta S^{\partial }$, with $Q^{\partial }$ given by

$$\begin{aligned} Q^{\partial }\textbf{A}= {\text {grad}}c,\quad Q^{\partial }\textbf{E}\sqrt{\det g}=0,\quad Q^{\partial } b= {\text {div}} \textbf{E},\quad Q^{\partial } c=0. \end{aligned}$$

One can then see that the cohomology in degree zero consists of functionals of $\textbf{A}$ and $\textbf{E}$ that are gauge invariant modulo the ideal generated by ${\text {div}} \textbf{E}$. This is correctly the algebra of functions of the reduction of $C=\{(\textbf{A},\textbf{E})\ |\ {\text {div}} \textbf{E}= 0\}$.

If $\Sigma$ has a boundary, we instead get

$$\begin{aligned} \delta S^{\partial } = \int _\Sigma (\delta c\,{\text {div}} \textbf{E}+ {\text {grad}}c\cdot \textbf{E}) \,\sqrt{\det g} + \int _{\partial \Sigma } c\,\delta E_n \,\sqrt{\det g_{|_{\partial \Sigma }}}, \end{aligned}$$

where $E_n$ is the transversal component of $\textbf{E}$. This fits with the BFV-$\hbox {BF}^2$V prescription $\iota _{Q^{\partial }}\varpi ^{\partial }=\delta S^{\partial }+\widetilde{\alpha }^{\partial }$ with $\widetilde{\alpha }^{\partial }=\int _{\partial \Sigma } c\,\delta E_n \,\sqrt{\det g_{|_{\partial \Sigma }}}$. As $\widetilde{\varpi }^{\partial }=\delta \widetilde{\alpha }^{\partial }$ only depends on c and on $E_n$ on $\partial \Sigma$, we get the reduced space of fields $\mathcal {F}_{\partial \Sigma }=\{(c,E_n)\in C^\infty (\partial \Sigma )[1]\oplus C^\infty (\partial \Sigma )\}$ with $\hbox {BF}^2$V symplectic structure

$$\begin{aligned} \varpi ^{\partial \partial } = \int _{\partial \Sigma } \delta c\,\delta E_n \,\sqrt{\det g_{|_{\partial \Sigma }}}. \end{aligned}$$

As $Q^{\partial }$ is zero on the c and E coordinates, we get $Q^{\partial \partial }=0$ and $S^{\partial \partial }=0$. Therefore, we get a trivial structure.

We now make a change of coordinates that will make the other examples we want to describe easier to write. Namely, instead of the vector field $\textbf{A}$ we consider the corresponding 1-form A, via the metric g, and instead of the vector field $\textbf{E}$ we consider the $(d-1)$-form $B=\iota _{\textbf{E}}\sqrt{\det g}$. With these new notations, we get

$$\begin{aligned} \varpi ^{\partial }=\int _\Sigma (\delta B\,\delta A+\delta b\,\delta c), \end{aligned}$$

where we omitted the wedge product symbol from the notation, and

$$\begin{aligned} S^{\partial } = \int _\Sigma c\,\textrm{d}B. \end{aligned}$$

Note that any reference to the metric g has disappeared. Repeating the above computations, we now get

$$\begin{aligned} Q^{\partial }A = \textrm{d}c,\quad Q^{\partial }B=0,\quad Q^{\partial }b= \textrm{d}B,\quad Q^{\partial }c=0. \end{aligned}$$

If $\Sigma$ has a boundary, we get $\mathcal {F}_{\partial \Sigma }=\{(c,B)\in C^\infty (\partial \Sigma )[1]\oplus \Omega ^{d-1}(\partial \Sigma )\}$ with canonical symplectic structure $\varpi ^{\partial \partial }=\int _{\partial \Sigma }\delta c\,\delta B$ and with $Q^{\partial \partial }=0$ and $S^{\partial \partial }=0$. Note that, even though the corner operator $Q^{\partial \partial }$ is trivial, the corner theory still has a nontrivial symplectic space of fields.

3.2 Yang–Mills Theory

In the nonabelian case, the fields A, B, b, c are $\mathfrak {g}$-valued,⁸ where $\mathfrak {g}$ is a Lie algebra endowed with a nondegenerate, invariant inner product $\langle \ ,\ \rangle$. The Gauss law is $\textrm{d}_AB=0$, where $\textrm{d}_A$ denotes the covariant derivative. The BFV symplectic form now reads

$$\begin{aligned} \varpi ^{\partial }=\int _\Sigma (\langle \delta B,\,\delta A\rangle +\langle \delta b,\,\delta c\rangle ). \end{aligned}$$

As this notation is a bit heavy, we will omit the inner product $\langle \ ,\ \rangle$ throughout, so we simply write $\varpi ^{\partial }=\int _\Sigma (\delta B\,\delta A+\delta b\,\delta c)$ (one may think of the integral sign to contain the inner product as well, or one may think the inner product to be the Killing form and the integral to incorporate the trace sign). By the same convention, the BFV action reads

$$\begin{aligned} S^{\partial } = \int _\Sigma \left( c\,\textrm{d}_A B +\frac{1}{2} b[c,c]\right) , \end{aligned}$$

where the BRST term, linear in b, has now appeared. We can also easily calculate

$$\begin{aligned} Q^{\partial }A = \textrm{d}_A c,\quad Q^{\partial }B=[c,B],\quad Q^{\partial }b=\textrm{d}_AB+[c,b],\quad Q^{\partial }c=\frac{1}{2} [c,c]. \end{aligned}$$

If $\Sigma$ has a boundary, we get $\mathcal {F}_{\partial \Sigma }=\{(c,B)\in (C^\infty (\partial \Sigma )[1]\oplus \Omega ^{d-1}(\partial \Sigma ))\otimes \mathfrak {g}\}$ with canonical symplectic structure $\varpi ^{\partial \partial }=\int _{\partial \Sigma }\delta c\,\delta B$ and with $Q^{\partial \partial } B=[c,B]$ and $Q^{\partial \partial } c=\frac{1}{2} [c,c]$, which is the Hamiltonian vector field of

$$\begin{aligned} S^{\partial \partial } = \int _{\partial \Sigma } \frac{1}{2} B[c,c]. \end{aligned}$$

Now, the $\hbox {BF}^2$V structure is no longer trivial.

If we regard $\mathcal {F}_{\partial \Sigma }$ as $T^*[1](\Omega ^{d-1}(\partial \Sigma )\otimes \mathfrak {g})$, we then interpret $S^{\partial \partial }$ as the Poisson bivector field

$$\begin{aligned} \pi _2 = \int _{\partial \Sigma } \frac{1}{2} B\left[ \frac{\delta }{\delta B},\frac{\delta }{\delta B}\right] . \end{aligned}$$

As this is linear, it can actually be viewed (modulo subtleties due to dualization) as the Poisson structure on $\mathcal {G}^*$, where $\mathcal {G}$ is the Lie algebra $C^\infty (\partial \Sigma )\otimes \mathfrak {g}$ with pointwise Lie bracket induced by $\mathfrak {g}$. (We have identified $\mathfrak {g}^*$ with $\mathfrak {g}$ using the inner product, and we have regarded $\Omega ^{d-1}(\partial \Sigma )$ as the dual space of $C^\infty (\partial \Sigma )$.) For example, on linear functionals we have

$$\begin{aligned} \left\{ \int _{\partial \Sigma } fB,\int _{\partial \Sigma } gB\right\} _2=\int _{\partial \Sigma } [f,g]B. \end{aligned}$$

The other natural polarization consists in realizing $\mathcal {F}_{\partial \Sigma }$ as $T^*[1](C^\infty (\partial \Sigma )[1]\otimes \mathfrak {g})$. In this case, we interpret $S^{\partial \partial }$ as the cohomological vector field

$$\begin{aligned} \pi _1 = \int _{\partial \Sigma } \frac{1}{2} [c,c]\frac{\delta }{\delta c}, \end{aligned}$$

which gives $C^\infty (\partial \Sigma )[1]\otimes \mathfrak {g}$ the structure of a $P_\infty$-manifold. With the notations of the previous paragraph, this manifold is the same as $\mathcal {G}[1]$. Its algebra of functions is the exterior algebra $\Lambda \mathcal {G}^*$, regarded as a graded commutative algebra, and $\pi _1$ corresponds to the Chevalley–Eilenberg differential.

Remark 20

Note that for any $B_0\in \Omega ^{d-1}(\partial \Sigma )$ we can define a polarization choosing the $B_0$-section of $T^*[1](C^\infty (\partial \Sigma )[1]\otimes \mathfrak {g})$ instead of the zero section. In this case, in addition to $\pi _1$ as above, we also get a nontrivial $\pi _0=\int _{\partial \Sigma } \frac{1}{2} B_0[c,c]$, so we have a curved $P_\infty$ structure.

3.3 Chern–Simons Theory

In this case, $\Sigma$ is two-dimensional and the field is a $\mathfrak {g}$-connection one-form A, where $\mathfrak {g}$ again is a Lie algebra endowed with a nondegenerate, invariant inner product.⁹ The space of fields is endowed with the Atiyah–Bott symplectic form $\varpi ^{\partial }_0=\frac{1}{2}\int _\Sigma \delta A\,\delta A$ and the constraint is that the connection be flat. Therefore, we introduce the BFV structure

$$\begin{aligned} \varpi ^{\partial }= & {} \int _\Sigma \left( \frac{1}{2}\delta A\,\delta A+\delta b\,\delta c\right) , \\ S^{\partial }= & {} \int _\Sigma \left( c\,F_A + \frac{1}{2} b[c,c]\right) , \end{aligned}$$

where $F_A=\textrm{d}A+\frac{1}{2}[A,A]$ is the curvature of A. We now get

$$\begin{aligned} Q^{\partial }A = \textrm{d}_A c,\quad Q^{\partial }b=F_A+[c,b],\quad Q^{\partial }c=\frac{1}{2} [c,c]. \end{aligned}$$

If $\Sigma$ has a boundary, we get $\mathcal {F}_{\partial \Sigma }=\{(c,A)\in C^\infty (\partial \Sigma )[1]\otimes \mathfrak {g}\oplus \mathcal {A}(\partial \Sigma )\}$, where $\mathcal {A}$ denotes the space of connection one-forms, with canonical symplectic structure $\varpi ^{\partial \partial }=\int _{\partial \Sigma }\delta c\,\delta A$ and with $Q^{\partial \partial } A=\textrm{d}_A c$ and $Q^{\partial \partial } c=\frac{1}{2} [c,c]$, which is the Hamiltonian vector field of

$$\begin{aligned} S^{\partial \partial } = \int _{\partial \Sigma } \frac{1}{2} c\,\textrm{d}_Ac =\int _{\partial \Sigma } \left( \frac{1}{2} c\,\textrm{d}_{A_0}c+\frac{1}{2} c[a,c] \right) , \end{aligned}$$

where $A_0$ is a reference connection and $a=A-A_0$.¹⁰

If we regard $\mathcal {F}_{\partial \Sigma }$ as $T^*[1]\mathcal {A}(\partial \Sigma )$, we then interpret $S^{\partial \partial }$ as the Poisson bivector field

$$\begin{aligned} \pi _2 =\int _{\partial \Sigma } \left( \frac{1}{2} \frac{\delta }{\delta a}\textrm{d}_{A_0} \frac{\delta }{\delta a} +\frac{1}{2} a\left[ \frac{\delta }{\delta a},\frac{\delta }{\delta a}\right] \right) . \end{aligned}$$

In this case, we have an affine Poisson structure which can be viewed (modulo subtleties due to dualization) as the Poisson structure on $\mathcal {G}^*$ associated with the central extension of $\mathcal {G}=C^\infty (\partial \Sigma )\otimes \mathfrak {g}$ with pointwise Lie bracket induced by that on $\mathfrak {g}$ by the cocycle $c(f,g)=\int _{\partial \Sigma } f \textrm{d}_{A_0} g$. For example, on linear functionals we have

$$\begin{aligned} \left\{ \int _{\partial \Sigma } fa,\int _{\partial \Sigma } ga\right\} _2=\int _{\partial \Sigma } (f \textrm{d}_{A_0} g + [f,g]a). \end{aligned}$$

The other natural polarization consists in realizing $(\mathcal {F}_{\partial \Sigma })_{A_0}$ as $T^*[1](C^\infty (\partial \Sigma )[1]\otimes \mathfrak {g})$. In this case, we interpret $S^{\partial \partial }$ as the inhomogeneous multivector field $\pi =\pi _0+\pi _1$ with $\pi _0=\int _{\partial \Sigma } \frac{1}{2} c\,\textrm{d}_{A_0}c$ and

$$\begin{aligned} \pi _1 = \int _{\partial \Sigma } \frac{1}{2} [c,c]\frac{\delta }{\delta c}, \end{aligned}$$

which gives $C^\infty (\partial \Sigma )[1]\otimes \mathfrak {g}$ the structure of a curved $P_\infty$-manifold. Note that the curving $\pi _0$ is different from zero for every choice of $A_0$.

Remark 21

Chern–Simons theory is an example of an AKSZ theory [1]. In particular, this means that we can write the $\hbox {BF}^n$V structures in a compact way using superfields. For the cases at hand, we set $\widetilde{A} = c + A + b$ in the BFV case and $\widetilde{A} = c + A$ in the $\hbox {BF}^2$V case. The symplectic forms and actions on $\Sigma$ and on $\partial \Sigma$ now simply read $\frac{1}{2}\int _T \delta \widetilde{A}\delta \widetilde{A}$ and $\int _T\left( \frac{1}{2} \widetilde{A}\textrm{d}\widetilde{A}+\frac{1}{6}\widetilde{A}[\widetilde{A},\widetilde{A}]\right)$, by specializing T to $\Sigma$ or to $\partial \Sigma$, respectively.¹¹

3.4 BF Theory

In BF theory in $d+1$ dimensions, there are two fields: a $\mathfrak {g}$-connection A and a $\mathfrak {g}$-valued $(d-1)$-form B. Here, $\mathfrak {g}$ is, as before, a Lie algebra endowed with a nondegenerate, invariant inner product.¹² The symplectic form, for a d-manifold $\Sigma$, is $\varpi ^{\partial }_0=\int _\Sigma \delta B\,\delta A$ and the constraints are

$$\begin{aligned} \textrm{d}_AB=0\qquad \text {and}\qquad F_A+ \Lambda P(B) = 0, \end{aligned}$$

where $\Lambda$ is a constant and P an invariant polynomial of degree k such that $k(d-1)=2$.¹³ Note that P may be nontrivial only for $d=2,3$.

For $d=1$, for dimensional reasons the only nontrivial constraint is $\textrm{d}_AB=0$, so in this case the BFV structure is the same as in the case of Yang–Mills in $1+1$ dimensions.

For $d=2$, BF theory is actually a particular case of Chern–Simons theory with a Lie algebra structure, depending on $\Lambda$, on the vector space $\mathfrak {g}\oplus \mathfrak {g}$. If $\mathfrak {g}=\mathfrak {so}(1,2)$ (or $\mathfrak {so}(3)$) and B, viewed as a $3\times 3$ tensor field, is nondegenerate, it is $2+1$ (Euclidean) gravity with cosmological constant $\Lambda$ in the coframe formulation.

In the rest of the section, we focus on the case $d=3$, which, for $\mathfrak {g}=\mathfrak {so}(1,3)$ (or $\mathfrak {so}(4)$), is related to $3+1$ (Euclidean) gravity with cosmological constant $\Lambda$ in the coframe formulation. For definiteness, we write the constraints as

$$\begin{aligned} \textrm{d}_AB=0\qquad \text {and}\qquad F_A+ \Lambda B = 0. \end{aligned}$$

In the BFV formalism, we then need two kinds of ghosts to implement them. The beginning of the BFV action is

$$\begin{aligned} S^{\partial }=\int _\Sigma (c\,\textrm{d}_A B+\tau \, (F_A+ \Lambda B))+\cdots , \end{aligned}$$

with $c\in \Omega ^0(\Sigma )[1]\otimes \mathfrak {g}$ and $\tau \in \Omega ^1(\Sigma )[1]\otimes \mathfrak {g}$.

Note that the $\tau$-dependent Hamiltonian vector field acts on A as $\Lambda \tau$ and on B as $\textrm{d}_A\tau$. Therefore, if $\tau$ is of the form $\textrm{d}_A\phi$ for some 0-form $\phi$, it acts on A as a gauge transformation. Moreover, it acts on B as $[F_A,\phi ]$. If $F_A+\Lambda B=0$, which is what the constraint imposes, it acts also on B as a gauge transformation. This leads to a redundancy to the c-dependent Hamiltonian vector field. To avoid it, one has to mod out $\tau$ by such transformations. If the momentum for $\tau$ is denoted $B^+$, then we add the term $\int _\Sigma \phi \,\textrm{d}_AB^+$ to the BFV action, for its Hamiltonian vector field acts on $\tau$ precisely as $\textrm{d}_A\phi$. Note that $\phi$ is now considered as a new ghost (actually a ghost-for-ghost), which is assigned even parity and degree equal to two. It also comes with its own momentum.

As BF theory is an AKSZ theory, we will use the notation standard in that context. Namely, we group the fields into superfields,

$$\begin{aligned} \widetilde{A}&= c+A+B^++\tau ^+,\\ \widetilde{B}&= \phi + \tau + B + A^+, \end{aligned}$$

where the fields appear in decreasing order w.r.t. degree and in increasing order w.r.t. form degree. The BFV symplectic form is

$$\begin{aligned} \varpi ^{\partial }=\int _\Sigma \delta \widetilde{B}\,\delta \widetilde{A} =\int _\Sigma (\delta A^+\,\delta c + \delta B\,\delta A + \delta \tau \,\delta B^+ +\delta \phi \,\delta \tau ^+), \end{aligned}$$

from which it is clear that our notation for the momenta of c, $\tau$, and $\phi$ are $A^+$, $B^+$, and $\tau ^+$, respectively. The BFV action reads

$$\begin{aligned} S^{\partial }&=\int _\Sigma \left( \widetilde{B}F_{\widetilde{A}}+\frac{1}{2}\Lambda \widetilde{B}\widetilde{B}\right) \\&=\int _\Sigma \left( \frac{1}{2}A^+[c,c]+B\,\textrm{d}_Ac+\tau \,(F_A+[c,B^+])\right. \\&\quad \left. +\phi \,(\textrm{d}_AB^++[c,\tau ^+])+\Lambda \,(B\tau +A^+\phi ) \right) , \end{aligned}$$

from which we get

$$\begin{aligned} Q^{\partial } c&=\frac{1}{2}[c,c]+\Lambda \phi ,&\quad Q^{\partial } A&=\textrm{d}_Ac+\Lambda \tau ,\\ Q^{\partial } B^+&=F_A+\Lambda B+ [c,B^+],&\quad Q^{\partial } \tau ^+&=\textrm{d}_AB^++[c,\tau ^+]+\Lambda A^+, \end{aligned}$$

and

$$\begin{aligned} Q^{\partial }\phi&=[c,\phi ],&Q^{\partial }\tau&=\textrm{d}_A\phi +[c,\tau ],\\ Q^{\partial } B&=\textrm{d}_A\tau +[c,B]+[\phi ,B^+],&Q^{\partial } A^+&=\textrm{d}_AB+[c,A^+]+[B^+,\tau ]+[\tau ^+,\phi ]. \end{aligned}$$

If $\Sigma$ has a boundary, we get that the coordinates of $\mathcal {F}_{\partial \Sigma }$ can also be grouped into superfields

$$\begin{aligned} \widetilde{A}&= c+A+B^+,\\ \widetilde{B}&= \phi + \tau + B. \end{aligned}$$

The $\hbox {BF}^2$V symplectic form turns out to be

$$\begin{aligned} \varpi ^{\partial \partial }=\int _{\partial \Sigma }\delta \widetilde{B}\,\delta \widetilde{A} =\int _{\partial \Sigma }(\delta B\,\delta c + \delta \tau \,\delta A + \delta \phi \,\delta B^+ ). \end{aligned}$$

From

$$\begin{aligned} Q^{\partial \partial } c&=\frac{1}{2}[c,c]+\Lambda \phi ,&Q^{\partial \partial } A&=\textrm{d}_Ac+\Lambda \tau ,&Q^{\partial \partial } B^+&=F_A+\Lambda B+ [c,B^+],\\ Q^{\partial \partial } \phi&=[c,\phi ],&Q^{\partial \partial } \tau&=\textrm{d}_A\phi +[c,\tau ],&Q^{\partial \partial } B&=\textrm{d}_A\tau +[c,B]+[\phi ,B^+],\\ \end{aligned}$$

we get the $\hbox {BF}^2$V action

$$\begin{aligned}&S^{\partial \partial }=\int _{\partial \Sigma }\left( \widetilde{B}F_{\widetilde{A}}+\frac{1}{2}\Lambda \widetilde{B}\widetilde{B}\right) \\&\quad =\int _{\partial \Sigma }\left( \frac{1}{2}B[c,c]+\tau \,\textrm{d}_Ac+\phi \,(F_A+[c,B^+])+\Lambda \left( \frac{1}{2}\tau \tau +B\phi \right) \right) \\&\quad =\int _{\partial \Sigma }\left( \frac{1}{2}B[c,c]+\tau \,(\textrm{d}_{A_0}c+[a,c])+\phi \,\left( F_{A_0}+\textrm{d}_{A_0}a+\frac{1}{2}[a,a]+[c,B^+]\right) \right. \\&\qquad \left. +\Lambda \left( \frac{1}{2}\tau \tau +B\phi \right) \right) \end{aligned}$$

where $A_0$ is a reference connection and $a=A-A_0$.

Remark 22

Note that even in the abelian case the corner action is nontrivial.

One natural polarization consists in realizing $\mathcal {F}_{\partial \Sigma }$ as the shifted cotangent bundle of the space $\mathcal {N}$ with coordinates A, B, and $B^+$, by choosing $\{c=\phi =\tau =0\}$ as the reference Lagrangian submanifold. This corresponds to having $\pi =\pi _1+\pi _2$ with

$$\begin{aligned} \pi _1&= \int _{\partial \Sigma } (F_A+\Lambda B)\frac{\delta }{\delta B^+},\\ \pi _2&= \int _{\partial \Sigma }\left( \frac{1}{2}B\left[ \frac{\delta }{\delta B},\frac{\delta }{\delta B}\right] +\frac{\delta }{\delta a}\textrm{d}_{A_0}\frac{\delta }{\delta B} + a\left[ \frac{\delta }{\delta a},\frac{\delta }{\delta B}\right] \right. \\&\quad \left. +B^+\left[ \frac{\delta }{\delta B^+},\frac{\delta }{\delta B}\right] + \frac{1}{2}\Lambda \frac{\delta }{\delta a}\frac{\delta }{\delta a} \right) . \end{aligned}$$

In other words, we split functions on $\mathcal {F}_{\partial \Sigma }$ as $\mathfrak {p}\oplus \mathfrak {h}$ with $\mathfrak {p}$ the subalgebra of functions of positive degree and $\mathfrak {h}$ the subalgebra of functions of nonpositive degree, and the construction turns $\mathfrak {h}$ into a differential graded Poisson algebra. The degree zero part $\mathfrak {h}_0$, consisting of functions on $\mathcal {A}(\partial \Sigma )\oplus \Omega ^2(\partial \Sigma )\otimes \mathfrak {g}\ni (A,B)$, is a Poisson subalgebra. Actually, we may view the affine Poisson structure on $\mathcal {A}(\partial \Sigma )\oplus \Omega ^2(\partial \Sigma )\otimes \mathfrak {g}=(A_0+\Omega ^1(\partial \Sigma )\otimes \mathfrak {g})\oplus \Omega ^2(\partial \Sigma )$ as the one on the dual $\mathcal {G}^*$ associated with the central extension of $\mathcal {G}=(\Omega ^1(\partial \Sigma )\oplus \Omega ^0(\partial \Sigma ))\otimes \mathfrak {g}$ with pointwise Lie bracket induced by that on the semidirect sum $\mathfrak {g}\rtimes _{{\text {ad}}}\mathfrak {g}$ by the cocycle $c(\alpha \oplus f,\beta \oplus g)=\int _{\partial \Sigma }\left( \alpha \textrm{d}_{A_0}g- \beta \textrm{d}_{A_0} f+\Lambda \alpha \beta \right)$. For example, on linear functionals we have

$$\begin{aligned} \left\{ \int _{\partial \Sigma } \alpha a,\int _{\partial \Sigma } \beta a\right\} _2&=\Lambda \int _{\partial \Sigma } \alpha \beta ,\\ \left\{ \int _{\partial \Sigma } \alpha a,\int _{\partial \Sigma } fB\right\} _2&=\int _{\partial \Sigma } (\alpha \textrm{d}_{A_0} f + [\alpha ,f]a),\\ \left\{ \int _{\partial \Sigma } fB,\int _{\partial \Sigma } gB\right\} _2&=\int _{\partial \Sigma } [f,g]B. \end{aligned}$$

The degree-zero $\pi _1$-cohomology is the quotient of $\mathfrak {h}_0$ by the ideal generated by $F_A+\Lambda B$. Geometrically, this corresponds to restricting the above Poisson structure to the Poisson submanifold $\{(A,B)\ |\ F_A+\Lambda B=0\}$.

Another natural polarization consists in viewing $\mathcal {F}_{\partial \Sigma }$ as the shifted cotangent bundle of the space $\widetilde{\mathcal {A}}$ with coordinates c, A, and $B^+$, by choosing $\{\widetilde{B}=0\}$ as the reference Lagrangian submanifold. This corresponds to having $\pi =\pi _1+\pi _2$ with

$$\begin{aligned} \pi _1&= \int _{\partial \Sigma }\left( \frac{1}{2}[c,c]\frac{\delta }{\delta c} + \textrm{d}_Ac\frac{\delta }{\delta A}+(F_A+[c,B^+])\frac{\delta }{\delta B^+}\right) ,\\ \pi _2&=\Lambda \int _{\partial \Sigma }\left( \frac{1}{2}\frac{\delta }{\delta A}\frac{\delta }{\delta A}+\frac{\delta }{\delta c} \frac{\delta }{\delta B^+}\right) . \end{aligned}$$

In particular, on $C^\infty (\widetilde{\mathcal {A}})$ we have a differential defined by

$$\begin{aligned} \pi _1c=\frac{1}{2}[c,c],\quad \pi _1A=\textrm{d}_Ac,\quad \pi _1B^+=F_A+ [c,B^+]. \end{aligned}$$

If $\Lambda \not =0$, we also have a constant, nondegenerate Poisson bracket.

One last interesting polarization, which turns out to be important for the rest of this paper, consists instead in viewing $\mathcal {F}_{\partial \Sigma }$ as the shifted cotangent bundle of the space $\widetilde{\mathcal {B}}$ with coordinates $\phi$, $\tau$, and B, by choosing $\{\widetilde{A}=A_0\}$ as the reference Lagrangian submanifold. In this case, we have $\pi =\pi _0+\pi _1+\pi _2$ with

$$\begin{aligned} \pi _0&= \int _{\partial \Sigma }\left( \phi F_{A_0} +\Lambda \,\left( \frac{1}{2}\tau \tau +B\phi \right) \right) ,\\ \pi _1&= \int _{\partial \Sigma }\left( \textrm{d}_{A_0}\tau \frac{\delta }{\delta B}+\textrm{d}_{A_0}\phi \frac{\delta }{\delta \tau } \right) ,\\ \pi _2&= \int _{\partial \Sigma }\left( \frac{1}{2}B\left[ \frac{\delta }{\delta B},\frac{\delta }{\delta B}\right] +\tau \left[ \frac{\delta }{\delta \tau },\frac{\delta }{\delta B}\right] +\frac{1}{2}\phi \left[ \frac{\delta }{\delta \tau },\frac{\delta }{\delta \tau }\right] +\phi \left[ \frac{\delta }{\delta \phi },\frac{\delta }{\delta B}\right] \right) . \end{aligned}$$

This makes $C^\infty (\widetilde{\mathcal {B}})$ into a curved $P_\infty$ algebra. If $\Lambda =0$, it can be made flat by choosing the reference connection $A_0$ to be flat. It is useful, for further reference, to observe that there is a $P_\infty$ subalgebra generated by the following linear local observables:

$$\begin{aligned} J_\alpha = \int _{\partial \Sigma }\alpha B,\quad M_\beta = \int _{\partial \Sigma }\beta \tau ,\quad K_\gamma = \int _{\partial \Sigma }\gamma \phi , \end{aligned}$$

(1)

where $\alpha$, $\beta$, $\gamma$ are $\mathfrak {g}$-valued 0-, 1-, and $2-$forms, respectively. We have

$$\begin{aligned} \{\}_0= & {} \int _{\partial \Sigma }\left( \phi F_{A_0} +\Lambda \,\left( \frac{1}{2}\tau \tau +B\phi \right) \right) \\ \{J_\alpha \}_1= & {} M_{\textrm{d}_{A_0}\alpha },\quad \{M_\beta \}_1=K_{\textrm{d}_{A_0}\beta },\quad \{K_\gamma \}_1=0,\\ \{J_\alpha ,J_{\widetilde{\alpha }}\}_2= & {} J_{[\alpha ,\widetilde{\alpha }]},\quad \{J_\alpha ,M_\beta \}_2=M_{[\alpha ,\beta ]},\quad \{J_\alpha ,K_\gamma \}_2=K_{[\alpha ,\gamma ]},\\ \{M_\beta ,M_{\widetilde{\beta }}\}_2= & {} K_{[\beta ,\widetilde{\beta }]},\quad \{M_\beta ,K_\gamma \}_2=0,\quad \{K_\gamma ,K_{\widetilde{\gamma }}\}_2=0. \end{aligned}$$

Also note that $\{\{\}_0\}_1=0$, that $\{M_\beta ,\{\}_0\}_2=0=\{K_\gamma ,\{\}_0\}_2$, that $\{\{M_\beta \}_1\}_1=0=\{\{K_\gamma \}_1\}_1$, and that $\{\{J_\alpha \}_1\}_1=\{J_\alpha ,\{\}_0\}_2$. Observe that for $\Lambda = 0$ we can also write $\{\{J_\alpha \}_1\}_1=K_{[F_{A_0},\alpha ]}$. It is also instructive to compute the above expressions using the derived brackets corresponding to the splitting with $\mathfrak {h}=C^\infty (\mathcal {B})$ and $\mathfrak {p}$ the ideal in $C^\infty (\mathcal {F}_{\partial \Sigma })$ generated by $C^\infty (\mathcal {A}-A_0)$. In this case, the projection $P:C^\infty (\mathcal {F}_{\partial \Sigma })\rightarrow C^\infty (\mathcal {B})$ simply consists in setting A equal to $A_0$ and c and $B^+$ to zero. We see that $\{\}_0=PS^{\partial \partial }$. We can also, e.g., compute

$$\begin{aligned} \{J_\alpha \}_1=PQ^{\partial \partial } J_\alpha =P\int _{\partial \Sigma }\alpha (\textrm{d}_A\tau +[c,B]+[\phi ,B^+])= \int _{\partial \Sigma }\alpha \textrm{d}_{A_0}\tau =M_{\textrm{d}_{A_0}\alpha }. \end{aligned}$$

Similarly, we get

$$\begin{aligned} \{J_\alpha ,M_\beta \}_2= & {} P\{J_\alpha ,Q^{\partial \partial } M_\beta \} = P\left\{ \int _{\partial \Sigma }\alpha B,\int _{\partial \Sigma }\beta (\textrm{d}_A\phi +[c,\tau ])\right\} \\= & {} P\int _{\partial \Sigma }[\alpha ,\beta ]\tau =M_{[\alpha ,\beta ]}. \end{aligned}$$

Note that, when $\Lambda =0$, the above algebra closes also under the nullary operation, since we can write

$$\begin{aligned} \{\}_0 = K_{F_{A_0}}. \end{aligned}$$

Otherwise, we have to add more generators. First of all, we introduce

$$\begin{aligned} C_\mu = \int _{\partial \Sigma }\mu \,\left( \frac{1}{2}\tau \tau +B\phi \right) , \end{aligned}$$

where $\mu$ is a function, so that we have

$$\begin{aligned} \{\}_0 = K_{F_{A_0}}+C_\Lambda , \end{aligned}$$

where we view $\Lambda$ as a constant function. The algebra now closes as long as $C_\mu$ is defined for constant functions $\mu$ only.

It is, however, possible, and natural, to extend the algebra allowing for arbitrary functions $\mu$. In this case, we have to introduce

$$\begin{aligned} D_\nu&= \int _{\partial \Sigma }\nu \tau \phi ,\\ E_\rho&= \frac{1}{2}\int _{\partial \Sigma }\rho \phi ^2. \end{aligned}$$

It can be readily verified that the 2-brackets of C, D, and E among themselves or with J, M, and K all vanish. As for the unary operations, we have

$$\begin{aligned} \{C_\mu \}_1=D_{\textrm{d}\mu },\qquad \{D_\nu \}_1=E_{\textrm{d}\nu },\qquad \{E_\rho \}_1=0. \end{aligned}$$

4 Boundary Structure and BFV Data for Palatini–Cartan Theory

The starting point for the construction of the $\hbox {BF}^2$V structure is the BFV boundary structure. In the Palatini–Cartan formalism, this is described in [13].

We recall here the relevant quantities of this construction. We consider a four-dimensional closed, oriented¹⁴ smooth manifold M together with a reference Lorentzian structure so that we can reduce the frame bundle to an SO(3, 1)-principal bundle $P \rightarrow M$. We denote by $\mathcal {V}$ the associated vector bundle by the standard representation. Each fiber of $\mathcal {V}$ is isomorphic to a four-dimensional vector space V with a Lorentzian inner product $\eta$ on it. The inner product allows the identification $\mathfrak {so}(3,1) \cong \bigwedge ^2 {V}$. Let now $\Sigma =\partial M$ be the boundary of M and denote with $\mathcal {V}_\Sigma$ the restriction $\mathcal {V}|_{\Sigma }$. We define the following shorthand notation:

$$\begin{aligned} \Omega _{\partial }^{i,j}:= \Omega ^i\left( \Sigma , \textstyle {\bigwedge ^j} \mathcal {V}_{\Sigma }\right) . \end{aligned}$$

Remark 23

Throughout the article, we will refer to the local dimensions of the spaces $\Omega ^{i,j}$ as the number of degrees of freedom of the space. Note that this dimension is also the same as their rank as $C^\infty$ modules.

On $\Omega _{\partial }^{i,j}$, we also define the following maps

$$\begin{aligned} W_{\partial }^{(i,j)}:\Omega _{\partial }^{i,j}&\longrightarrow \Omega _{\partial }^{i,j}\\ X&\longmapsto X \wedge e|_{\Sigma }. \end{aligned}$$

Usually, we will omit writing the restriction of e to the manifold $\Sigma$. The properties of these maps are collected in Appendix A.

We assume $\mathcal {V}_\Sigma$ to be isomorphic to $T\Sigma \oplus \underline{\mathbb {R}}$, as is the case if we think of it as the restriction to the boundary of a vector bundle isomorphic to the tangent bundle of the bulk, and we take a nowhere vanishing section $\epsilon _n$ of the summand $\underline{\mathbb {R}}$. We then define the space $\Omega _{\epsilon _n}^1(\Sigma , \mathcal {V}_\Sigma )$ to consist of bundle maps $e:T\Sigma \rightarrow \mathcal {V}_\Sigma$ such that the three components of e together with $\epsilon _n$ form a basis. Equivalently, we may require $eee\epsilon _n$ to be different from zero everywhere.¹⁵

As a consequence of this, the field e together with $\epsilon _n$ defines an isomorphism $T\Sigma \oplus \underline{\mathbb {R}}\rightarrow \mathcal {V}_\Sigma$. Denoting by $f:\mathcal {V}_\Sigma \rightarrow T\Sigma \oplus \underline{\mathbb {R}}$ its inverse and by $\pi _{T\Sigma }$ the projection $T\Sigma \oplus \underline{\mathbb {R}}\rightarrow T\Sigma$, we have a map

$$\begin{aligned} \widehat{ \bullet }\ :\begin{array}{ccc} \Gamma (\mathcal {V}_\Sigma ) &{} \rightarrow &{}\mathfrak X(\Sigma ) \\ \nu &{}\mapsto &{} {\widehat{\nu }}:=\pi _{T\Sigma }(f(\nu )) \end{array} \end{aligned}$$

(2)

Note that the definition of the hat map really depends on the choice of $\epsilon _n$ and the field e, even though we hide it in the notation.

In local coordinates, the hat map has the following description. We denote by $e_a$, $a=1,2,3$, the three components of the $\mathcal {V}_\Sigma$-valued one-form e. Then, for a given $\nu \in \Gamma (\mathcal {V}_\Sigma )$, there are uniquely determined functions $\nu ^{(a)}$, $a=1,2,3$, and $\nu ^{(n)}$ such that

$$\begin{aligned} \nu = \nu ^{(a)}e_a + \nu ^{(n)}\epsilon _n. \end{aligned}$$

The induced hat vector field is then

$$\begin{aligned} \widehat{\nu }= \nu ^{(a)}\frac{\partial }{\partial x^a}. \end{aligned}$$

We also consider the space

$$\begin{aligned} T^* \left( \Omega _{\partial }^{0,2}[1]\oplus \mathfrak {X}[1](\Sigma ) \oplus C^\infty [1](\Sigma )\right) \end{aligned}$$

where the corresponding fields are denoted by $c \in \Omega _{\partial }^{0,2}[1]$, $\xi \in \mathfrak {X}[1](\Sigma )$, $\lambda \in \Omega ^{0,0}[1]$, $\gamma ^\dag \in \Omega _{\partial }^{3,2}[-1]$, and $y^\dag \in \Omega _{\partial }^{3,3}[-1]$.¹⁶ The space of boundary fields is the bundle

$$\begin{aligned} \mathcal {F}^{\partial } \longrightarrow \Omega _{\epsilon _n}^1(\Sigma , \mathcal {V}_\Sigma )\oplus T^* \left( \Omega _{\partial }^{0,2}[1]\oplus \mathfrak {X}[1](\Sigma ) \oplus C^\infty [1](\Sigma )\right) , \end{aligned}$$

with local trivialisation on an open $\mathcal {U}_{\Sigma } \subset \Omega _{\epsilon _n}^1(\Sigma , \mathcal {V}_\Sigma )\oplus T^* \Big (\Omega _{\partial }^{0,2}[1]\oplus \mathfrak {X}[1](\Sigma ) \oplus C^\infty [1](\Sigma )\Big )$ given by

$$\begin{aligned} \mathcal {F}^{\partial }\simeq \mathcal {U}_{\Sigma } \times \mathcal {A}^\text {red}(\Sigma ), \end{aligned}$$

where $\mathcal {A}^\text {red}(\Sigma )$ is the space of connections $\omega$ (on $P|_{\Sigma }$) such that

$$\begin{aligned} \epsilon _n \textrm{d}_{\omega }e + \iota _{\widehat{X}}\gamma ^{\dag }= e \sigma \end{aligned}$$

(3)

for some $\sigma \in \Omega _{\partial }^{1,1}$ and $X=[c, \epsilon _n]+ L_{\xi }^{\omega } \epsilon _n$. The constraint (3) is called structural constraint. The BFV action and symplectic form are, respectively:

$$\begin{aligned} S^{\partial }&= \int _{\Sigma } \Big (c e \textrm{d}_{\omega } e + \iota _{\xi } e e F_{\omega } +\lambda \epsilon _n e F_{\omega } + \frac{1}{3!}\lambda \epsilon _n \Lambda e^3 \nonumber \\&\quad +\frac{1}{2} [c,c] \gamma ^{\dag } - L^{\omega }_{\xi } c \gamma ^{\dag }+ \frac{1}{2} \iota _{\xi }\iota _{\xi } F_{\omega }\gamma ^{\dag }\nonumber \\&\quad +[c, \lambda \epsilon _n ]y^{\dag } - L^{\omega }_{\xi } (\lambda \epsilon _n)y^{\dag } - \frac{1}{2}\iota _{[\xi ,\xi ]}e y^{\dag }\Big ),\end{aligned}$$

(4)

$$\begin{aligned} \varpi ^{\partial }&= \int _{\Sigma } \left( e \delta e \delta \omega + \delta c\delta \gamma ^{\dag } - \delta \omega \delta (\iota _\xi \gamma ^{\dag }) + \delta \lambda \epsilon _n \delta y^\dag +\iota _{\delta \xi } \delta (e y^\dag )\right) . \end{aligned}$$

(5)

Remark 24

For simplicity, we consider in this paper only the case of dimension $N=4$. However, some of the considerations of this article can be extended to the higher-dimensional cases. This can be done in the same way in which we can extend the boundary results on the boundary from the case $N=4$ to a generic $N\ge 4$ (see [13]). Furthermore, in this and the following sections, we assume that the cosmological constant vanishes: $\Lambda =0$. In Sect. 8, we will discuss the small corrections that have to be implemented when the cosmological constant is nonzero.

The boundary structure is completed by the cohomological vector field $Q^{\partial }$ defined as the hamiltonian vector field of $S^{\partial }$ with $\partial \Sigma = \emptyset$. Its expression (in components) reads:

$$\begin{aligned} Q^{\partial } e&= [c,e] + L_\xi ^\omega e + \textrm{d}_{\omega }(\lambda \epsilon _n) + \lambda \sigma , \end{aligned}$$

(6a)

$$\begin{aligned} Q^{\partial } \omega&= \textrm{d}_\omega c - \iota _\xi F_\omega + \lambda (W_{\partial }^{(1,2)})^{-1}(\epsilon _n F_{\omega }+\iota _{\widehat{X}}y^\dag )+ \frac{1}{2}\lambda \epsilon _n \Lambda e + \mathbb {V}_{\omega }, \end{aligned}$$

(6b)

$$\begin{aligned} Q^{\partial } c&= \frac{1}{2}[c,c] + \frac{1}{2}\iota _\xi \iota _\xi F_\omega + \lambda \iota _\xi (W_{\partial }^{(1,2)})^{-1}(\epsilon _n F_{\omega }+X^{(a)}y_a^\dag )+ \iota _{\xi }\mathbb {V}_{\omega }, \end{aligned}$$

(6c)

$$\begin{aligned} Q^{\partial } \lambda&= [c, \lambda \epsilon _n ]^{(n)} + (L_\xi ^\omega \lambda \epsilon _n)^{(n)}, \end{aligned}$$

(6d)

$$\begin{aligned} Q^{\partial } \xi&= \lambda \widehat{X} + \frac{1}{2}[\xi , \xi ], \end{aligned}$$

(6e)

$$\begin{aligned} Q^{\partial } \gamma ^{\dag }&= e d_\omega e +[c,\gamma ^{\dag }] + L_\xi ^\omega \gamma ^{\dag } + [\lambda \epsilon _n, y^\dag ], \end{aligned}$$

(6f)

$$\begin{aligned} e_a Q^{\partial } y^\dag&= e_a[c,y^\dag ] + e_a L_\xi ^\omega y^\dag + e_a e F_\omega + (\gamma _a^{\dag } \lambda (W_{\partial }^{(1,2)})^{-1}(\epsilon _n F_{\omega }+\iota _{\widehat{X}}y^\dag )\nonumber \\&\qquad + \lambda \sigma _a y^\dag + \mathbb {V}_{\omega }\gamma _a^{\dag }, \end{aligned}$$

(6g)

$$\begin{aligned} \epsilon _n Q^{\partial } y^\dag&= \epsilon _n[c,y^\dag ] + \epsilon _n L_\xi ^\omega y^\dag + \epsilon _n e F_\omega + \frac{1}{3!}\Lambda \epsilon _n e^3, \end{aligned}$$

(6h)

where $X=[c, \epsilon _n ] + L_\xi ^\omega (\epsilon _n)$ and $e \mathbb {V}_{\omega }=0$.

Remark 25

The map $W_\partial ^{(1,2)}$ is surjective but not injective (see Appendix A for more details), so we can choose a preimage defined up to terms in the kernel of $W_\partial ^{(1,2)}$, denoted here by $\mathbb {V}_{\omega }$. This is fixed by requiring that the action of the vector field $Q^{\partial }$ preserves the structural constraint (3), for some choice of the action of $Q^{\partial }$ on $\sigma$; i.e., we require ( [13]) that

$$\begin{aligned} Q^{\partial }(\epsilon _n d_{\omega }e + \iota _{\widehat{X}}\gamma ^{\dag })= Q^{\partial }e \sigma + e Q^{\partial } \sigma . \end{aligned}$$

This way we get an inverse $(W_\partial ^{(1,2)})^{-1}$. Comparing this expression with the corresponding one of the three-dimensional theory [14], we also note that the terms containing the inverse of the function $W_{\partial }^{(1,2)}$ and the auxiliary field $\sigma$ constitute exactly the difference between the two expressions.

5 Corner Structure of Palatini–Cartan Formalism

5.1 Corner Induced Structure

From a boundary BFV action, we can now induce a corner structure following the procedure recalled in Sect. 2.1.4. From now on, we assume that the manifold $\Sigma$ has a nonempty boundary $\partial \Sigma = \Gamma$.¹⁷ In this and in the following sections, we describe the relaxed $\hbox {BF}^2$V structure on the corner. In particular, we have the following result:

Proposition 26

The BFV theory $\mathfrak {F}^{(1)}_{PC}=(\mathcal {F}^{\partial }_{PC}, S^{\partial }_{PC}, \varpi ^{\partial }_{PC}, Q^{\partial }_{PC})$ is not 1-extendable.

We will then construct some associated $P_\infty$ algebras and will highlight a relation with BF theory (Sect. 6). We will also describe particular cases where we freeze some of the fields or do some partial reductions (Sect. 7).

Remark 27

Note that the four-dimensional case differs from the three-dimensional case. In this last, it has been proven in [14] that it is possible to extend the BFV theory to a $\hbox {BF}^2$V theory on the corner. The difference between the three- and four-dimensional cases is to be accounted to mathematical properties of this particular formulation of the BV theories. At the moment, neither do we have a physical interpretation of Proposition 26, whose content might even just be a mathematical artifact, nor do we have a meaningful statement about such an interpretation; hence, we postpone these to future work.

Before proving Proposition 26, let us introduce some further piece of notation, similarly to what we have done for the boundary structure. Let M be a smooth manifold of dimension 4 with corners and let us denote by $\Sigma = \partial M$ its three-dimensional boundary and by $\Gamma = \partial \partial M$ its 2-dimensional corner. Furthermore, we will use the notation $\mathcal {V}_{\Gamma }$ for the restriction of $\mathcal {V}_\Sigma$ to $\Gamma$. We define

$$\begin{aligned} \Omega _{\partial \partial }^{i,j}:= \Omega ^i\left( \Gamma , \textstyle {\bigwedge ^j} \mathcal {V}_{\Gamma }\right) . \end{aligned}$$

On $\Omega _{\partial \partial }^{i,j}$, we define the following map:

$$\begin{aligned} W_{\partial \partial }^{ (i,j)}:\Omega _{\partial \partial }^{i,j}\longrightarrow & {} \Omega _{\partial \partial }^{i,j}\\ X\longmapsto & {} X \wedge e|_{\Gamma } . \end{aligned}$$

Remark 28

As before, we will omit writing the restriction of e to the corner $\Gamma$.

The properties of these maps are collected in Appendix A. Furthermore, we recall that the restriction to $\Gamma$ of a vector field $\nu \in \mathfrak {X}(\Sigma )$ contracted through the interior product with a one form $\beta \in \Omega ^1(\Sigma )$ reads

$$\begin{aligned} \iota _{\nu }\beta = \iota _{\nu |_{\Gamma }}\beta |_{\Gamma }+ \beta _m \nu ^m, \end{aligned}$$

where the index m denotes the components transversal to the corner. For simplicity, we will omit the restrictions to $\Gamma$.

Proof of Proposition 26

From the variation of the boundary action, using the formula

$$\begin{aligned} \delta S^{\partial } = \iota _{Q^{\partial }} \varpi ^{\partial } + \widetilde{\alpha }^{\partial }, \end{aligned}$$

we can deduce the pre-corner (or pre-codimension-2) one form

$$\begin{aligned} \widetilde{\alpha }^{\partial }&= \int _{\Gamma } ( c e \delta e - \iota _{\xi } e e \delta \omega - e_m \xi ^m e \delta \omega \\&\quad -\lambda \epsilon _n e \delta \omega - \delta c \gamma _m^{\dag }\xi ^m- \delta \omega \iota _{\xi }\gamma _m^{\dag }\xi ^m\nonumber \\&\quad - \delta (\lambda \epsilon _n) \iota _{\xi }y^{\dag }- \delta (\lambda \epsilon _n) y_m^{\dag } \xi ^m - \iota _{\delta \xi } e y_m^{\dag } \xi ^m + e_m \delta \xi ^m y_m^{\dag } \xi ^m ). \end{aligned}$$

Taking its variation, we obtain the pre-corner two-form:

$$\begin{aligned} \widetilde{\varpi }^{\partial }= \delta \widetilde{\alpha }^{\partial }&= \int _{\Gamma } ( \delta c e \delta e - \iota _{\delta \xi } e e \delta \omega - \iota _{\xi }( e \delta e) \delta \omega \nonumber \\&\quad -\delta e_m \xi ^m e \delta \omega + e_m \delta \xi ^m e \delta \omega - e_m \xi ^m \delta e \delta \omega \nonumber \\&\quad -\delta \lambda \epsilon _n e \delta \omega -\lambda \epsilon _n \delta e \delta \omega - \delta c \gamma _m^{\dag }\delta \xi ^m- \delta c \delta \gamma _m^{\dag }\xi ^m- \delta \omega \delta (\iota _{\xi }\gamma _m^{\dag }\xi ^m)\nonumber \\&\quad + \delta (\lambda \epsilon _n) \delta y_m^{\dag } \xi ^m+ \delta (\lambda \epsilon _n) y_m^{\dag } \delta \xi ^m \nonumber \\&\quad + \iota _{\delta \xi } \delta e y_m^{\dag } \xi ^m +\iota _{\delta \xi } e \delta y_m^{\dag } \xi ^m- \iota _{\delta \xi } e y_m^{\dag } \delta \xi ^m \nonumber \\&\quad + \delta e_m \delta \xi ^m y_m^{\dag } \xi ^m- e_m \delta \xi ^m \delta y_m^{\dag } \xi ^m + e_m \delta \xi ^m y_m^{\dag } \delta \xi ^m). \end{aligned}$$

(7)

In order to proceed, we have to check if this two-form is pre-symplectic, i.e., if the kernel of the corresponding map

$$\begin{aligned} \widetilde{\varpi }^{\partial \sharp }: T \widetilde{\mathcal {F}}^{\partial }&\rightarrow T^* \widetilde{\mathcal {F}}^{\partial }\\ X&\mapsto \widetilde{\varpi }^{\partial \sharp }(X)=\widetilde{\varpi }^{\partial }(X, \cdot ) \end{aligned}$$

is regular. The equations defining the kernel are:

$$\begin{aligned} \delta c:&\quad e X_e +X_{\gamma _m^{\dag }} \xi ^m- \gamma _m^{\dag } X_{\xi ^m}=0 , \end{aligned}$$

(8a)

$$\begin{aligned} \delta e:&\quad e X_c - e \iota _\xi X_{\omega } - \lambda \epsilon _n X_{\omega }- e_m \xi ^m X_{\omega } -\iota _{X_{\xi }}y_m^\dag \xi ^m=0, \end{aligned}$$

(8b)

$$\begin{aligned} \delta \xi :&\quad e_{\bullet } e X_{\omega }-X_{\omega }c_{m\bullet }^{\dag } \xi ^m+ (X_e)_\bullet y_m^{\dag } \xi ^m + e_{\bullet }X_{y_m^{\dag }} \xi ^m- e_{\bullet }y_m^{\dag }X_{\xi ^m}=0, \end{aligned}$$

(8c)

$$\begin{aligned} \delta \omega :&\quad - \iota _{X_\xi } e e - \iota _{\xi }( e X_e) -X_{e_m} \xi ^m e + e_m X_{\xi ^m} e - e_m \xi ^m X_e \nonumber \\&\qquad -X_\lambda \epsilon _n e -\lambda \epsilon _n X_e - X_{(\iota _{\xi }\gamma _m^{\dag }\xi ^m)}=0, \end{aligned}$$

(8d)

$$\begin{aligned} \delta e_m:&\quad - \xi ^m e X_\omega +X_{\xi ^m} y_m^{\dag } \xi ^m=0, \end{aligned}$$

(8e)

$$\begin{aligned} \delta \xi ^m:&\quad e_m e X_\omega - X_c \gamma _m^{\dag }- X_\omega \iota _{\xi }\gamma _m^{\dag }+ X_\lambda \epsilon _n y_m^{\dag } - \iota _{X_\xi } e y_m^{\dag } + X_{e_m} y_m^{\dag } \xi ^m \nonumber \\&\qquad - e_m X_{ y_m^{\dag }} \xi ^m + 2 e_m y_m^{\dag } X_{\xi ^m}=0, \end{aligned}$$

(8f)

$$\begin{aligned} \delta \lambda :&\quad -\epsilon _n e X_\omega + \epsilon _n X_{y_m^{\dag }} \xi ^m+ \epsilon _n y_m^{\dag } X_{\delta \xi ^m}=0, \end{aligned}$$

(8g)

$$\begin{aligned} \delta \gamma _m^{\dag }:&\quad - X_c \xi ^m+ \iota _{\xi }X_\omega \xi ^m=0, \end{aligned}$$

(8h)

$$\begin{aligned} \delta y_m^{\dag }:&\quad +X_\lambda \epsilon _n \xi ^m +\iota _{X_\xi } e \xi ^m - e_m X_{\xi ^m} \xi ^m =0. \end{aligned}$$

(8i)

Let us consider (8a) and (8b). They can be solved only if the functions $W_{\partial \partial }^{(1,1)}$ and $W_{\partial \partial }^{(0,2)}$ are invertible. However, from Lemma 54 in Appendix A we gather that both $W_{\partial \partial }^{(1,1)}$ and $W_{\partial \partial }^{(0,2)}$ are neither injective nor surjective. In particular, $\dim {{\,\textrm{Im}\,}}W_{\partial \partial }^{(1,1)}= \dim {{\,\textrm{Im}\,}}W_{\partial \partial }^{(0,2)}=5$, while the respective codomains $\Omega _{\partial \partial }^{1,1}$ and $\Omega _{\partial \partial }^{0,2}$ have dimension 6 and 8, respectively. Hence, we deduce that these two equations are singular and so is the kernel of $\widetilde{\varpi }^{\partial \sharp }$.

Therefore, it is not possible to perform a symplectic reduction, and the BFV data do not induce a 1-extended BFV theory. $\square$

5.2 Pre-corner theory

The failure of the standard procedure does not allow us to construct a $\hbox {BF}^2$V theory. It is, however, still possible to analyze the pre-corner structure. To complete the picture, along the pre-corner two form (7) we have to find the pre-corner action $\widetilde{S}^{\partial }$ and an expression for a Hamiltonian vector field. Even if the two-form is degenerate, we can still get a pair $\widetilde{Q}^{\partial }$ and $\widetilde{S}^{\partial }$ satisfying $\iota _{\widetilde{Q}^{\partial }} \widetilde{\varpi }^{\partial }= \delta \widetilde{S}^{\partial }$, out of the boundary data.

Before proceeding, let us recall the spaces on which the pre-corner fields are defined. In degree $-1$, we have $\gamma _m^\dag \in \Omega ^{2}(\Gamma , \wedge ^2\mathcal {V}))[-1]$ and $y_m^\dag \in \Omega ^{2}(\Gamma , \wedge ^4\mathcal {V}))[-1]$. In degree 1, we have the ghosts parametrizing the gauge symmetries, $c \in \Omega ^{0}(\Gamma , \wedge ^2\mathcal {V}_\Gamma ))[1]$, and the ones parametrizing the diffeomorphisms: respectively, $\xi \in \mathfrak {X}[1](\Gamma )$ tangential to $\Gamma$, $\xi ^m\in \Omega ^{0}(\Gamma )[1]$ transversal to $\Gamma$ into $\Sigma$, and $\lambda \in \Omega ^{0}(\Gamma )[1]$ transversal also to $\Sigma$. In degree zero, we first have the tangent part $e \in \Omega _\text {nd}^1(\Gamma , \mathcal {V}_\Gamma )$ of the coframe restricted to the corner and its transversal part $e_m\in \Omega ^0(\Gamma , \mathcal {V}_\Gamma )$, together with a fixed nowhere vanishing field $\epsilon _n \in \Omega ^0(\Gamma , \mathcal {V}_\Gamma )$ with the requirement that $eee_m\epsilon _n$ is nowhere 0.¹⁸ Furthermore, we also have a connection $\omega \in \mathcal {A}^\text {red}(\Gamma )$ where $\mathcal {A}^\text {red}(\Gamma )$ is the space of connections (on $P|_{\Gamma }$) such that the following equations are satisfied:

$$\begin{aligned} \epsilon _n \textrm{d}_{\omega }e +&\gamma _m^{\dag }\widehat{Z}^m= e \sigma ,\\ e_m \sigma&\in {{\,\textrm{Im}\,}}W_{\partial \partial }^{(0,1)}, \end{aligned}$$

where $Z=[c, \epsilon _n]+ L_{\xi }^{\omega } \epsilon _n+ \textrm{d}_{\omega _m} \epsilon _n \xi ^m$.

Remark 29

These last equations are a consequence of the fact that the starting data on the boundary were constrained by (3); hence, this constraint will also descend to the pre-corner. However, it will split into two separate equations:

$$\begin{aligned} \epsilon _n \textrm{d}_{\omega }e +&\gamma _m^{\dag }\widehat{Z}^m= e \sigma , \\ \epsilon _n \textrm{d}_{\omega _m}e + \epsilon _n \textrm{d}_{\omega }e_m +&\iota _{\widehat{Z}}\gamma _{m}^{\dag }= e_m \sigma + e \sigma _m. \end{aligned}$$

The second equation is dynamical but still gives some information about $\sigma$ and $\sigma _m$. In particular, we can rewrite it as

$$\begin{aligned} e_m \sigma \in {{\,\textrm{Im}\,}}W_{\partial \partial }^{(0,1)}. \end{aligned}$$

An interpretation of these constraints is given in Appendix C.

Remark 30

The map $\widehat{\bullet }$ has been defined in (2) for fields on the boundary $\Sigma$. However, when we have combinations of the type $\iota _{\widehat{X}}\alpha$ for some form $\alpha$ on the boundary and some section X of $\mathcal {V}_{\Sigma }$, we can pull them back to the corner and get $\iota _{\widehat{X}}\alpha + \alpha _m\widehat{X}^m$.

Let us now compute the pre-corner action. Since we have the boundary cohomological vector field, we can let $\partial \Sigma = \Gamma \ne \emptyset$ and, using the modified master equation $\iota _{Q^{\partial }} \iota _{Q^{\partial }} \varpi ^{\partial } = 2 \widetilde{S}^{\partial }$, find an expression for the pre-corner action. After a long but straightforward computation, we get

$$\begin{aligned} \widetilde{S}^{\partial }&= \int _{\Gamma } \Big (\frac{1}{4}[c,c] ee + \frac{1}{2}\iota _\xi (ee) \textrm{d}_\omega c + e e_m \xi ^m \textrm{d}_\omega c + \lambda \epsilon _n e \textrm{d}_\omega c \nonumber \\&\quad + \frac{1}{4}\iota _\xi \iota _\xi (ee) F_\omega + \iota _\xi e e_m \xi ^m F_\omega + \iota _\xi e \epsilon _n \lambda F_\omega + e_m \xi ^m \epsilon _n \lambda F_\omega \nonumber \\&\quad + \frac{1}{2}[c,c] \gamma _m^{\dag } \xi ^m + L_\xi ^\omega c \gamma _m^{\dag } \xi ^m + \frac{1}{2}\iota _\xi \iota _\xi F_\omega \gamma _m^{\dag } \xi ^m \nonumber \\&\quad + \frac{1}{2}\iota _{[\xi ,\xi ]}e y_m^\dag \xi ^m + L_\xi ^\omega (\lambda \epsilon _n) y_m^\dag \xi ^m + L_\xi ^\omega (e_m \xi ^m) y_m^\dag \xi ^m + [c, \lambda \epsilon _n]y_m^\dag \xi ^m\Big ). \end{aligned}$$

(9)

The last bit of information that is missing is a pre-corner cohomological vector field. This can be obtained by pushing forward the one on the boundary to the corner. We collect some technical lemmata that are useful for this computation in Appendix D.

Remark 31

Due to the degeneracy of the pre-corner two form, a Hamiltonian vector field defined through $\iota _{\widetilde{Q}^{\partial }} \widetilde{\varpi }^{\partial }= \delta \widetilde{S}^{\partial }$ is not unique and might differ from the projection of $Q^\partial$ by an element in the kernel of $\widetilde{\varpi }^\partial$.

Collecting all the above information, we get the following expression for the pre-corner cohomological vector field $\widetilde{Q}^{\partial }$:

$$\begin{aligned} \widetilde{Q}^{\partial } e&= [c,e] + L_\xi ^\omega e + \xi ^m \textrm{d}_{\omega _m} e +e_m \textrm{d}\xi ^m + \textrm{d}_{\omega }(\lambda \epsilon _n) + \lambda \sigma , \\ \widetilde{Q}^{\partial } e_m&= [c,e_m] + L_\xi ^\omega e_m + \iota _{\partial _m \xi } e + \textrm{d}_{\omega _m}(e_m \xi ^m) + \textrm{d}_{\omega _m}(\lambda \epsilon _n)+ \lambda \sigma _m, \\ \widetilde{Q}^{\partial } \omega&= \textrm{d}_\omega c - \iota _\xi F_\omega -F_{\omega _m} \xi ^m+ \lambda \mu + \frac{1}{2}\lambda \epsilon _n \Lambda e,\\ \widetilde{Q}^{\partial } \omega _m&= \textrm{d}_{\omega _m} c - \iota _\xi F_{\omega _m} + \lambda \mu _m+ \frac{1}{2}\lambda \epsilon _n \Lambda e_m,\\ \widetilde{Q}^{\partial } c&= \frac{1}{2}[c,c] + \frac{1}{2}\iota _\xi \iota _\xi F_\omega +\iota _\xi F_{\omega _m}\xi ^m +\lambda \iota _\xi \mu + \lambda \mu _m \xi ^m, \\ \widetilde{Q}^{\partial } \lambda&= Y^{(n)}, \\ \widetilde{Q}^{\partial } \xi&=\widehat{Y}+ \frac{1}{2}[\xi , \xi ],\\ \widetilde{Q}^{\partial } \xi ^m&=\widehat{Y}^{m}+ \frac{1}{2}[\xi , \xi ]^m,\\ \widetilde{Q}^{\partial } \gamma ^{\dag }&= e_m \textrm{d}_\omega e +e \textrm{d}_{\omega _m} e +e \textrm{d}_\omega e_m +[c,\gamma _m^{\dag }] + L_\xi ^\omega \gamma _m^{\dag } + \textrm{d}_{\omega _m} (\gamma _m^{\dag } \xi ^m) + [\lambda \epsilon _n, y_m^\dag ],\\ \widetilde{Q}^{\partial } y^\dag&= [c,y_m^\dag ] + L_\xi ^\omega y_m^\dag +\textrm{d}_{\omega _m} (y^\dag _m \xi ^m)+ e_m F_\omega + e F_{\omega _m}+ \frac{1}{2} \Lambda e_m e^2 \nonumber \\ {}&{\qquad + \lambda (\sigma _m y_m^\dag )^{(m)} + \lambda (\mu _m \gamma _m^{\dag })^{(m)}+ \lambda (\sigma _a y_m^\dag )^{(a)} + \lambda (\mu \gamma _{am}^{\dag })^{(a)}}, \end{aligned}$$

where

$$\begin{aligned} Y&= [c, \lambda \epsilon _n] + L_\xi ^\omega (\lambda \epsilon _n) + \xi ^m \textrm{d}_{\omega _m}(\lambda \epsilon _n), \\ \mu&=(W_{\partial \partial }^{(1,2)})^{-1}(\epsilon _n F_{\omega }+y_m^\dag \widehat{Y}^{m}), \\ \mu _m&= (W_{\partial \partial }^{(0,2)})^{-1}(e_m \mu + \epsilon _n F_{\omega _m}+\iota _{\widehat{Y}}y_{m}^\dag ),{} & {} \end{aligned}$$

and $e_a Z_{a}^{(a)}= Z_{a}$, $e_m Z_{m}^{(m)}= Z_{m}$. The data just collected do not form a $\hbox {BF}^2$V structure on the corner, since the closed two-form (7) is degenerate. Nonetheless, using the procedure described in Sect. 2.2.1, it is possible to extract information from this structure.

6 $P_{\infty }$ Structure of General Pre-Corner Theory

As explained in Sect. 2.2, $\hbox {BF}^2$V theories define a $P_\infty$ structure once a polarization is chosen on the space of corner fields. Furthermore (see Remark 16), this construction can be generalized to the cases when the two-form is degenerate, which is precisely the case at hand. In this section, we analyze these structures. In order to have a better understanding of the results that we find, we will afterward consider two simplified theories in Sect. 7, for which the structure will be more readable.

Since the two-form is not symplectic, we consider the construction explained in Remarks 5 and 16. Following the notation introduced in Sect. 2.2, we consider a splitting of the Hamiltonian functionals and define $\mathfrak {h}$ to be a subalgebra of functionals in the variables $e, \xi , \lambda , \xi ^m$ and $\gamma _m^\dag \xi ^m$. The projection to it is just obtained by setting $\omega =\omega _0$, a fixed background connection, and by putting to zero all the other fields.¹⁹ In particular, we consider the following Hamiltonian functionals and prove that they form a $P_\infty$ subalgebra of $\mathfrak {h}$:

$$\begin{aligned} J_{\varphi }&= \int _{\Gamma } \varphi \left( \frac{1}{2}ee+ \gamma _m^\dag \xi ^m\right) ,\\ M_{Y}&= \int _{\Gamma } Y \left( \iota _{\xi }\left( \frac{1}{2}ee+ \gamma _m^\dag \xi ^m\right) + \alpha e\right) ,\\ K_{Z}&= \int _{\Gamma } Z \left( \frac{1}{2}\iota _{\xi }\iota _{\xi }\left( \frac{1}{2}ee + \gamma _m^\dag \xi ^m\right) + \iota _{\xi }e \alpha +\frac{1}{2}\alpha ^2\right) , \end{aligned}$$

where $\alpha = \epsilon _n \lambda +e_m \xi ^m$. These functionals are Hamiltonian because it is possible to construct the corresponding Hamiltonian vector fields, which read

$$\begin{aligned} \mathbb {J}_{\varphi }&= \int _{\Gamma }\varphi \frac{\delta }{\delta c}, \\ \mathbb {M}_Y&= \int _{\Gamma } Y \frac{\delta }{\delta \omega },\\ \mathbb {K}_Z&= \int _{\Gamma } \Bigg (\left( -\iota _\xi Z + (W_{\partial \partial }^{(2,3)})^{-1}(\epsilon _n \lambda Z)\right) \frac{\delta }{\delta \omega }\\&\quad +\left( -\frac{1}{2}\iota _\xi \iota _\xi Z+ \iota _\xi (W_{\partial \partial }^{(2,3)})^{-1}(\epsilon _n \lambda Z)\right. \\&\quad \left. -(W_{\partial \partial }^{(2,3)})^{-1}(e_m \xi ^m (W_{\partial \partial }^{(2,3)})^{-1}(\epsilon _n \lambda Z)) \right) \frac{\delta }{\delta c}\\&\quad +\left( e_m Z+ \gamma _m^\dag (W_{\partial \partial }^{(2,3)})^{-1}(e_m (W_{\partial \partial }^{(2,3)})^{-1}(\epsilon _n \lambda Z))^{(m)}\right. \\&\quad \left. + (W_{\partial \partial }^{(2,3)})^{-1}(\epsilon _n \lambda Z)\gamma _{am}^\dag )^{(a)} \right) \frac{\delta }{\delta y_m^\dag }\Bigg ). \end{aligned}$$

We can then prove that they form a subalgebra by computing the various brackets. After a long but straightforward computation, we get the following result:

$$\begin{aligned}&\{\}_0=\int _{\Gamma }\left( \frac{1}{2}\iota _{\xi }\iota _{\xi }\left( \frac{1}{2}ee + \gamma _m^\dag \xi ^m\right) + \iota _{\xi }e \alpha +\frac{1}{2}\alpha ^2\right) F_{\omega _0}, \\&\{J_{\varphi }\}_1= M_{\textrm{d}_{\omega _0}\varphi }, \quad \{M_{Y}\}_1 = K_{\textrm{d}_{\omega _0}Y}, \quad \{K_{Z}\}_1 = 0, \\&\{J_{\varphi }, J_{\varphi '}\}_2 = J_{[\varphi , \varphi ']}, \quad \{J_{\varphi }, M_Y\}_2 = M_{[\varphi , Y]}, \quad \{M_Y, K_Z\}_2 = 0,\\&\{M_Y, M_{Y'}\}_2 = K_{[Y, Y']}, \quad \{J_{\varphi }, K_Z\}_2 = K_{[\varphi ,Z]}, \quad \{K_Z, K_{Z'}\}_2 = 0. \end{aligned}$$

Note that the nullary operation is here obtained by the nonvanishing part of the projection of the action to $\mathfrak {h}$. We can write

$$\begin{aligned} \{\}_0= K_{F_{\omega _0}}, \end{aligned}$$

so the algebra generated by J, M, and K closes also under the nullary operation. We also explicitly note that this structure is identical to the tangent theory and that of BF theory in (1).

Remark 32

As before, the similarity between the structure of the subalgebra of observables and that of BF theory is connected to the possibility of obtaining the constrained theory as BF theory for the Lie algebra $\mathfrak {so(3,1)}$, restricted to the submanifold of fields parametrized by

$$\begin{aligned} c&=c, \quad A = \omega , \quad B^\dag = 0,\\ \phi&= \frac{1}{4} \iota _\xi \iota _\xi (ee)+ \frac{1}{2}\iota _\xi \iota _\xi \gamma _m^\dag \xi ^m + \iota _\xi e \alpha + \frac{1}{2}\alpha ^2, \quad \tau = \frac{1}{2}\iota _\xi (ee)+ \iota _\xi \gamma _m^\dag \xi ^m + e \alpha , \\&\quad B = \frac{1}{2} ee + \gamma _m^\dag \xi ^m. \end{aligned}$$

7 Simplified Theories

The expressions of the pre-corner data without reduction are rather complicated, and the information contained in them is well hidden. For this reason, it is useful to consider some simplified cases in which the Poisson structure is more manifest. In this section, we propose two different simplified theory in which the physical content is more explicit. In the first, we impose some constraints on the boundary data, which do not change the on-shell boundary structure (i.e., we consider a smaller BFV theory still describing the same reduced phase space of the original one). In the second, we assume some ghost fields to vanish, thus not considering some symmetries (the diffeomorphisms normal to the boundary and to the corner).

7.1 Constrained Theory

This approach is based first on considering the BFV theory on a cylindrical boundary manifold (i.e., assuming $\Sigma =\Gamma \times I$, where I is an interval, and then, focusing on one of the two boundary components $\Gamma$). Next, we impose some further constraints, on the line of (3), to get a theory that is on-shell equivalent to the original one but better treatable with the $\hbox {BF}^2$V machinery.

Remark 33

This approach is based on the fact that the failure of the two-form (7) to have a regular kernel has similar causes to the same failure of the pre-boundary two-form [18]. As discussed in [11], it is anyway possible to overcome the problem by constructing a BV theory on the bulk with some additional constraints. Indeed, using the constraints suggested by the AKSZ construction, it is possible to construct a BV theory that induces a BFV theory on the boundary.

We now want to mimic this behavior in order to get a BFV theory that induces a $\hbox {BF}^2$V theory on the corner. Since we do not have at hand a corner theory, we cannot use any suggestion from the AKSZ construction and we can only try to guess the correct constraints.

Assume that the manifold $\Sigma$ has the form of a cylinder, $\Sigma =\Gamma \times I$, and call $x^m$ the coordinate along I. Then, a possible choice is given by the following constraints:

$$\begin{aligned}&\gamma _m^\dag = e K, \end{aligned}$$

(10a)

$$\begin{aligned}&e_m \textrm{d}_{\omega } e + e_m \textrm{d}\xi ^m K + \textrm{d}_{\omega }(\lambda \epsilon _n) K + \lambda \sigma K + [\lambda \epsilon _n, y_m^\dag ]= eL, \end{aligned}$$

(10b)

$$\begin{aligned}&\epsilon _n K =0, \end{aligned}$$

(10c)

$$\begin{aligned}&\epsilon _n L + \epsilon _n \textrm{d}_{\omega _m}e+ \epsilon _n \textrm{d}_{\omega } e_m + [c, \epsilon _n] K + L_{\xi }^{\omega }\epsilon _n K +\textrm{d}_{\omega _m} \epsilon _n \xi ^m K=0. \end{aligned}$$

(10d)

Remark 34

As we will see later on, these constraints are sufficient to get a simplified version of the pre-corner structure, but they still do not grant the possibility of doing a proper symplectic reduction.

Remark 35

Note that these constraints do not modify the boundary theory, in the sense that the constraints do not modify the classical critical locus of the unconstrained theory described in Sect. 4. Indeed, (10a) and (10c) are constraints on an anti-field and have no meaning in the classical interpretation. On the other hand, (10b) and (10d) encode part of the Euler–Lagrange equations on the boundary. To see this, we can rewrite the equation $e d_{\omega }e=0$ on the cylindrical boundary manifold $\Sigma =\Gamma \times I$ and get the equation

$$\begin{aligned} e_m d_{\omega }e + e (d_{\omega })_m e + e d_{\omega }e_m=0. \end{aligned}$$

Since $W_{\partial \partial }^{1,1}$ is neither injective nor surjective, besides the dynamical equation describing $\partial _m e$ we get also

$$\begin{aligned} e_m d_{\omega }e = e L' \end{aligned}$$

for some $L'$. This last equation, modulo anti-fields (which can be ignored at the classical level), is the same as (10b). Then, (10d) is added to guarantee the invariance under the action of $Q^{\partial }$, as proved in Lemma 37 below.

These constraints are fixing some components of the pre-corner fields $\omega$ and $\gamma _m^\dag$. Namely, we fix three components of $\omega$ in the kernel of $W_{\partial \partial }^{(1,2)}$ and four components of $\gamma _m^\dag$. More details can be found in C with the relevant proofs.

Remark 36

These additional constraints on the boundary simplify the expression of the structural constraints (3). Dividing them into tangential and transversal to the corner, we obtain

$$\begin{aligned} \epsilon _n \textrm{d}_{\omega }e + \widehat{Y}^{m}e K&= e \sigma , \\ \epsilon _n \textrm{d}_{\omega _m}e + \epsilon _n \textrm{d}_{\omega }e_m +\iota _{\widehat{Y}}(e K )&= e_m \sigma + e \sigma _m, \end{aligned}$$

where $Y=[c, \epsilon _n]+ L_{\xi }^{\omega } \epsilon _n + \textrm{d}_{\omega _m}(\epsilon _n) \xi ^m.$

Furthermore, it is worth noting that since $W_{\partial \partial }^{(1,1)}$ is surjective, we can write $y_m^\dag = e x_m^\dag$ for some $x_m^\dag$. Moreover, since $W_{\partial \partial }^{(1,1)}$ is not injective, we can also ask that $\epsilon _n x_m^\dag = e A$ for some A. Indeed, this condition fixes only some components of $x_m^\dag$ in the kernel of $W_{\partial \partial }^{(1,1)}$.

Lemma 37

The set of constraints (10) is conserved under the action of $Q^{\partial }$, i.e., it is possible to define $Q^{\partial }K$ and $Q^{\partial }L$ so that

$$\begin{aligned}&Q^{\partial } \gamma _m^\dag = Q^{\partial }e K + e Q^{\partial }K, \\&\epsilon _n Q^{\partial }K =0, \\&Q^{\partial }(e_m \textrm{d}_{\omega } e + e_m \textrm{d}\xi ^m K + \textrm{d}_{\omega }(\lambda \epsilon _n) K + \lambda \sigma K + [\lambda \epsilon _n, y_m^\dag ])= Q^{\partial }eL + e Q^{\partial }L,\\&\epsilon _n Q^{\partial } L + Q^{\partial }(\epsilon _n \textrm{d}_{\omega _m}e+ \epsilon _n \textrm{d}_{\omega } e_m + [c, \epsilon _n] K + L_{\xi }^{\omega }\epsilon _n K +\textrm{d}_{\omega _m} \epsilon _n \xi ^m K)=0. \end{aligned}$$

Proof

We use the expressions of the components of $Q^{\partial }$ recalled in (6). We start from (10a). After a short computation, it is possible to see that $Q^{\partial } \gamma _m^\dag = Q^{\partial }e K + e Q^{\partial }K$ is satisfied modulo a term proportional to (10b) by choosing

$$\begin{aligned} Q^{\partial }K = \textrm{d}_{\omega _m} e + \textrm{d}_{\omega }e_m + L_{\xi }^{\omega } K + [c,K] + \textrm{d}_{\omega _m} ( K \xi ^m) +L + \mathbb {K}, \end{aligned}$$

where $\mathbb {K} \in \textrm{Ker}{(W_{\partial \partial }^{(1,1)})}$ is not fixed by this equation. We use this freedom to choose a $Q^{\partial }K$ such that (10b) is invariant as well. Indeed, it is a long but straightforward computation to show that (10b) is invariant and the correct choice for $Q^{\partial }K$ is with $\mathbb {K}=0$ and

$$\begin{aligned} Q^{\partial }L =&L_{\xi }^{\omega } L + [c,L] + \textrm{d}_{\omega _m} (L\xi ^m) + \textrm{d}_{\omega }(\lambda \sigma _m)\\&+[(\mathbb {V}_{\omega })_m,e]+[\mathbb {V}_{\omega },e_m] + \iota _{\partial _m \xi } \textrm{d}_{\omega } e +[\lambda \epsilon _n, (F_{\omega })_m]\\ {}&+ \textrm{d}_{\omega _m}(\lambda \widehat{Y}^{m}K)+ \lambda \iota _{\widehat{Y}}(\textrm{d}_{\omega } K)+\iota _{\partial _m \xi } K \textrm{d}\xi ^m \\&+ [((W_{\partial \partial }^{(1,2)})^{-1}(\lambda \epsilon _n F_{\omega }))_m,e] +\mathbb {L}\\&+ [(W_{\partial \partial }^{(1,2)})^{-1}(\lambda \epsilon _n F_{\omega }),e_m]+ [(W_{\partial \partial }^{(1,2)})^{-1}(\lambda \widehat{Y}^{m}y_m^\dag ),e_m]\\&+[((W_{\partial \partial }^{(0,2)})^{-1}(\lambda \iota _{\widehat{Y}}y^\dag ))_m,e]\\&+\textrm{d}_{\omega }(\lambda \iota _{\widehat{Y}}K), \end{aligned}$$

where $\mathbb {L} \in \textrm{Ker}{(W_{\partial \partial }^{(1,1)})}$ is not fixed by this equation. Lastly, (10c) is invariant thanks to (10d), which in turn is invariant by choosing $\epsilon _n\mathbb {L}=0$. $\square$

From the previous lemma, we deduce that the constraints (10) define a submanifold of $\mathcal {F}^{\partial }$ compatible with $Q^{\partial }$. As a consequence, they define a pre-BFV theory.

7.1.1 Corner Theory

Starting from this new constrained BFV theory, it is possible to build a partial symplectic reduction on the new pre-corner two-form and to write the pre-corner symplectic form and the pre-corner action in more readable variables. First, we fix a section $\epsilon _m$ of $\mathcal {V}_{\Gamma }$ that is linearly independent from $\epsilon _n$, and we only allow fields e that form a basis together with $\epsilon _m$ and $\epsilon _n$. In other words, we have that the combination $ee\epsilon _m\epsilon _n \ne 0$ everywhere. Next, we consider the map

$$\begin{aligned} \widetilde{e}&= e + K \xi ^m,\\ \widetilde{\omega }&= \omega + x_m^\dag \xi ^m,\\ \widetilde{c}&= c + \iota _{\xi } x_m^\dag \xi ^m +W^{-1}(\lambda \epsilon _n x_m^\dag \xi ^m),\\ \epsilon _{m}&= k^m e_m+ k^a e_a + k^n \epsilon _n,\\ \widetilde{\xi }^{m}&= \frac{1}{k^m}\xi ^m,\\ \widetilde{\xi }^{a}&= \xi ^a + \frac{k^a}{k^m}\xi ^m,\\ \widetilde{\lambda }&= \lambda + \frac{k^n}{k^m}\xi ^m, \end{aligned}$$

where $k_a,k_n, k_m$ are functions, with $k_m\ne 0$, chosen so that $\widetilde{Q}^\partial \epsilon _m=0$. The target space is then defined as the direct sum

$$\begin{aligned} \underbrace{\Omega _{\partial \partial \text {nd}}^{1,1}}_{\widetilde{e}} \oplus \underbrace{\mathcal {A}_{\text {red}}^{\partial \partial }}_{\widetilde{\omega }}\oplus \underbrace{\Omega _{\partial \partial }^{0,2}[1]}_{\widetilde{c}}\oplus \underbrace{\mathfrak {X}[1](\Gamma )}_{\widetilde{\xi }^{}}\oplus \underbrace{\Omega _{\partial \partial }^{0,0}[1]}_{\widetilde{\xi }^{m}}\oplus \underbrace{\Omega _{\partial \partial }^{0,0}[1]}_{\widetilde{\lambda }}, \end{aligned}$$

where the fields must satisfy

$$\begin{aligned}&\widetilde{\xi }^{m} \epsilon _m \textrm{d}_{\widetilde{\omega }}\widetilde{e}+ \widetilde{\lambda }\epsilon _n \textrm{d}_{\widetilde{\omega }}\widetilde{e}= \widetilde{e}(\widetilde{\lambda }\widetilde{\sigma }+ \widetilde{\xi }^{m}\widetilde{L}),\\&\widetilde{\xi }^{m} \epsilon _n \textrm{d}_{\widetilde{\omega }}\widetilde{e}= \widetilde{e}\widetilde{\sigma }\widetilde{\xi }^{m},\\&\widetilde{\xi }^{m} \epsilon _m \widetilde{\sigma } + \widetilde{e}\widetilde{\sigma }_m \widetilde{\xi }^{m} + \widetilde{L}\epsilon _n\widetilde{\xi }^{m}=0, \end{aligned}$$

for some $\widetilde{\sigma }\in \Omega _{\partial \partial }^{1,1}$, $\widetilde{\sigma }_m\in \Omega _{\partial \partial }^{0,1}$ and $\widetilde{L}\in \Omega _{\partial \partial }^{1,1}$.

With these variables, the pre-corner two-form and the pre-corner action are, respectively,

$$\begin{aligned} \widetilde{\varpi }^{\partial \partial }&= \int _{\Gamma } \left( \delta \widetilde{c}\widetilde{e}\delta \widetilde{e}+ \delta (\iota _{\widetilde{\xi }^{}}\widetilde{e}\widetilde{e})\delta \widetilde{\omega }+ \delta (\epsilon _m\widetilde{\xi }^{m}\widetilde{e})\delta \widetilde{\omega }+ \delta (\widetilde{\lambda } \epsilon _n \widetilde{e})\delta \widetilde{\omega }\right) , \end{aligned}$$

(11)

$$\begin{aligned} \widetilde{S}^{\partial \partial }&= \int _{\Gamma }\Big (\frac{1}{4}[\widetilde{c},\widetilde{c}]\widetilde{e}\widetilde{e}+ \iota _{\widetilde{\xi }^{}}\widetilde{e}\widetilde{e}\textrm{d}_{\widetilde{\omega }} \widetilde{c}+ \epsilon _m\widetilde{\xi }^{m}\widetilde{e}\textrm{d}_{\widetilde{\omega }} \widetilde{c}+ \widetilde{\lambda } \epsilon _n \widetilde{e}\textrm{d}_{\widetilde{\omega }} \widetilde{c}\nonumber \\&\quad + \frac{1}{4}\iota _{\widetilde{\xi }^{}}\iota _{\widetilde{\xi }^{}}(\widetilde{e}\widetilde{e}) F_{\widetilde{\omega }}+ \iota _{\widetilde{\xi }^{}}\widetilde{e}\epsilon _m\widetilde{\xi }^{m}F_{\widetilde{\omega }}+ \iota _{\widetilde{\xi }^{}}\widetilde{e}\widetilde{\lambda } \epsilon _n F_{\widetilde{\omega }}+ \epsilon _m\widetilde{\xi }^{m} \widetilde{\lambda } \epsilon _n F_{\widetilde{\omega }}\Big ). \end{aligned}$$

(12)

It is also possible to give an explicit expression of the cohomological vector field $\widetilde{Q}^{\partial \partial }$. This can be either be computed as the Hamiltonian vector field of the action $\widetilde{S}^{\partial \partial }$ or pushed forward from the boundary vector field ${Q}^{\partial }$. Both these methods lead to the following expression:

$$\begin{aligned} \widetilde{Q}^{\partial \partial } \widetilde{e}&= [\widetilde{c}, \widetilde{e}] + L_{\widetilde{\xi }^{}}^{\widetilde{\omega }} \widetilde{e}+ \textrm{d}_{\widetilde{\omega }} (\epsilon _m \widetilde{\xi }^{m}+ \widetilde{\lambda } \epsilon _n) + \widetilde{\lambda } \widetilde{\sigma } + \widetilde{L}\widetilde{\xi }^{m}, \\ \widetilde{Q}^{\partial \partial } \widetilde{\xi }^{m}&= X_m^{[m]} + X_n^{[m]} + \widetilde{\lambda }\widetilde{\sigma }_m^{[m]} \widetilde{\xi }^{m},\\ \widetilde{Q}^{\partial \partial } \widetilde{\xi }^{a}&= X_m^{[a]} + X_n^{[a]} + \widetilde{\lambda }\widetilde{\sigma }_m^{[a]} \widetilde{\xi }^{m}+ \frac{1}{2}[\widetilde{\xi }^{}, \widetilde{\xi }^{}]^a,\\ \widetilde{Q}^{\partial \partial } \widetilde{\lambda }&= X_m^{[n]} + X_n^{[n]} + \widetilde{\lambda }\widetilde{\sigma }_m^{[n]} \widetilde{\xi }^{m},\\ \widetilde{Q}^{\partial \partial } \widetilde{\omega }&= \textrm{d}_{\widetilde{\omega }} \widetilde{c}- \iota _{\widetilde{\xi }^{}}F_{\widetilde{\omega }} + (W_{\partial \partial }^{(1,2)})^{-1}((\epsilon _m \widetilde{\xi }^{m}F_{\widetilde{\omega }}+\epsilon _n \widetilde{\lambda }F_{\widetilde{\omega }}) + \mathbb {V}_{\widetilde{\omega }},\\ \widetilde{Q}^{\partial \partial } \widetilde{c}&= \frac{1}{2} [\widetilde{c},\widetilde{c}] + \frac{1}{2} \iota _{\widetilde{\xi }^{}}\iota _{\widetilde{\xi }^{}}F_{\widetilde{\omega }} + \iota _{\widetilde{\xi }^{}}(W_{\partial \partial }^{(1,2)})^{-1}(\epsilon _m \widetilde{\xi }^{m}F_{\widetilde{\omega }}+\epsilon _n \widetilde{\lambda }F_{\widetilde{\omega }}) + \iota _{\widetilde{\xi }^{}}\mathbb {V}_{\widetilde{\omega }}\\&\quad + (W_{\partial \partial }^{(0,2)})^{-1}(\epsilon _m \widetilde{\xi }^{m}\mathbb {V}_{\widetilde{\omega }}+\epsilon _n \widetilde{\lambda }\mathbb {V}_{\widetilde{\omega }})+ (W_{\partial \partial }^{(0,2)})^{-1}\\&\quad ((\epsilon _m \widetilde{\xi }^{m}+\epsilon _n \widetilde{\lambda })(W_{\partial \partial }^{(1,2)})^{-1}(\epsilon _m \widetilde{\xi }^{m}F_{\widetilde{\omega }}+\epsilon _n \widetilde{\lambda }F_{\widetilde{\omega }})), \end{aligned}$$

where $X_m= [\widetilde{c}, \epsilon _m \widetilde{\xi }^{m}]+ L_{\widetilde{\xi }^{}}^{\widetilde{\omega }}(\epsilon _m \widetilde{\xi }^{m})$, $X_n= [\widetilde{c}, \epsilon _n \widetilde{\lambda }]+ L_{\widetilde{\xi }^{}}^{\widetilde{\omega }}(\epsilon _n \widetilde{\lambda })$, $\widetilde{\sigma }= \sigma + X^{(m)}K+[\epsilon _n, x^{\dag }_m\xi ^{m}]+[A\xi ^m, \widetilde{e}]$, $\widetilde{L}= Lk^m + k^n \widetilde{\sigma }+k^a (d_{\widetilde{\omega }}\widetilde{e})_a$, and $\widetilde{\sigma }_m=k^m \sigma _m +k^m X^{(a)}K_a + k^a \sigma _a + k^a X^m K_a$. The square brackets denote the components with respect to the basis $\{\widetilde{e}, \epsilon _m, \epsilon _n\}$, e.g., $X_m= X_m^{[a]}\widetilde{e}_a + X_m^{[m]}\epsilon _m+ X_m^{[n]}\epsilon _n$.²⁰ Since the two form (11) is still degenerate (see below), the Hamiltonian vector field $\widetilde{Q}^{\partial \partial }$ is not unique, as it can be seen by the presence of inverses of maps $(W_{\partial \partial }^{(1,2)})$ which are not injective.

The two-form (11) is not symplectic. The equations defining its kernel are the following:

$$\begin{aligned} \delta \widetilde{c}:&\quad \widetilde{e}X_{\widetilde{e}} =0, \\ \delta \widetilde{e}:&\quad \widetilde{e}X_{\widetilde{c}} - \widetilde{e}\iota _{\widetilde{\xi }^{}} X_{\widetilde{\omega }} - \widetilde{\lambda } \epsilon _n X_{\widetilde{\omega }}- \epsilon _m \widetilde{\xi }^{m} X_{\widetilde{\omega }}=0,\\ \delta \widetilde{\xi }^{}:&\quad \widetilde{e}_{\bullet } \widetilde{e}X_{\widetilde{\omega }}=0,\\ \delta \widetilde{\omega }:&\quad - \iota _{X_{\widetilde{\xi }^{}}} \widetilde{e}\widetilde{e}- \iota _{\widetilde{\xi }^{}}( \widetilde{e}X_{\widetilde{e}}) + \epsilon _m X_{\widetilde{\xi }^{m}} \widetilde{e}- \epsilon _m \widetilde{\xi }^{m} X_{\widetilde{e}} \nonumber \\ {}&\qquad -X_{\widetilde{\lambda }} \epsilon _n \widetilde{e}-\widetilde{\lambda } \epsilon _n X_{\widetilde{e}} =0,\\ \delta \widetilde{\xi }^{m}:&\quad \epsilon _m \widetilde{e}X_{\widetilde{\omega }}=0,\\ \delta \widetilde{\lambda }:&\quad -\epsilon _n \widetilde{e}X_{\widetilde{\omega }}=0. \end{aligned}$$

We can simplify this system by noting that the third and the last two equations together form the equation $\widetilde{e}X_{\widetilde{\omega }}=0.$ Hence, it can be rewritten as

$$\begin{aligned}&\widetilde{e}X_{\widetilde{e}} =0, \widetilde{e}(X_{\widetilde{c}} - \iota _{\widetilde{\xi }^{}} X_{\widetilde{\omega }}) - (\widetilde{\lambda } \epsilon _n + \epsilon _m \widetilde{\xi }^{m}) X_{\widetilde{\omega }}=0,\\&\widetilde{e}X_{\widetilde{\omega }}=0, \widetilde{e}( -\iota _{X_{\widetilde{\xi }^{}}} \widetilde{e}+ \epsilon _m X_{\widetilde{\xi }^{m}} -X_{\widetilde{\lambda }} \epsilon _n) - (\epsilon _m \widetilde{\xi }^{m} -\widetilde{\lambda } \epsilon _n ) X_{\widetilde{e}} =0. \end{aligned}$$

This system is still singular since the map $W_{\partial \partial }^{(0,2)}$ appearing in the second equation is neither injective nor surjective, and the map $W_{\partial \partial }^{(0,1)}$ appearing in the fourth is injective but not surjective. However, it is worth noting that with the extra requests $(\widetilde{\lambda } \epsilon _n + \epsilon _m \widetilde{\xi }^{m}) X_{\widetilde{\omega }}=0$ and $(\epsilon _m \widetilde{\xi }^{m} -\widetilde{\lambda } \epsilon _n ) X_{\widetilde{e}} =0$ we get $X_{\widetilde{e}} =0$, $X_{\widetilde{\omega }}=0$ from the first and the third equation, while the second identifies equivalence classes of [c] and the fourth can be solved yielding $X_{\widetilde{\xi }^{}}$, $X_{\widetilde{\xi }^{m}}$ and $X_{\lambda }$.

7.1.2 $P_\infty$ Structure

Let us now analyze the $P_\infty$ structure of this constrained theory. Since the two-form is not symplectic, as in the general case we have to consider the construction explained in Sect. 2.2.1. In order to keep the notation light, in this section we drop the tildes on the fields since no confusion can arise. The splitting that we consider here follows the one of the general theory described in Sect. 6. Indeed, we define $\mathfrak {h}$ to be a subalgebra of functionals in the variables $e, \xi , \lambda$ and $\xi ^m$. As before, the projection to it is obtained by fixing $\omega$ to a background connection $\omega _0$ and by setting to zero all the other fields. The Hamiltonian functionals that we consider are again derived from the general case (we also use the same notation) and are the following:

$$\begin{aligned} J_{\varphi }&= \int _{\Gamma } \frac{1}{2}\varphi ee,\\ M_{Y}&= \int _{\Gamma } Y (\iota _{\xi } e + \alpha )e,\\ K_{Z}&= \int _{\Gamma } Z \left( \iota _{\xi }e\left( \frac{1}{2}\iota _{\xi } e + \alpha \right) +\frac{1}{2}\alpha ^2\right) , \end{aligned}$$

where $\alpha = \epsilon _n \lambda + \epsilon _m \xi ^m$.²¹ These functionals are Hamiltonian because it is possible to construct the corresponding Hamiltonian vector fields, which read

$$\begin{aligned} \mathbb {J}_{\varphi }&= \int _{\Gamma }\varphi \frac{\delta }{\delta c}, \\ \mathbb {M}_Y&= \int _{\Gamma } Y \frac{\delta }{\delta \omega },\\ \mathbb {K}_Z&= \int _{\Gamma }\Bigg ( \left( -\iota _\xi Z + (W_{\partial \partial }^{(2,3)})^{-1}(\alpha Z)\right) \frac{\delta }{\delta \omega }\\&\quad +\left( -\frac{1}{2}\iota _\xi \iota _\xi Z+ \iota _\xi (W_{\partial \partial }^{(2,3)})^{-1}(\alpha Z)-(W_{\partial \partial }^{(2,3)})^{-1}(\alpha (W_{\partial \partial }^{(2,3)})^{-1}(\alpha Z)) \right) \frac{\delta }{\delta c}\Bigg ). \end{aligned}$$

These functionals form a $P_\infty$ subalgebra of $\mathfrak {h}$ and the corresponding brackets read exactly as in the general case.

Remark 38

As in the general case, there is a similarity between the structure of the subalgebra of observables and that of BF theory.

7.2 Tangent Theory

Let us now consider an even simpler case where we assume $\xi ^m=0$ and $\lambda =0$ on the corner.²² As we will see, these two conditions are sufficient in order to get a regular kernel, so we can perform a symplectic reduction and get a proper $\hbox {BF}^2$V theory.

Remark 39

Note that assuming either only $\xi ^m=0$ or only $\lambda =0$ is not sufficient to get a regular kernel. For example, considering the first case, we get that the pre-corner two-form becomes

$$\begin{aligned} \widetilde{\varpi }_{part}^{\partial }= \int _{\Gamma }&(\delta c e \delta e - \iota _{\delta \xi } e e \delta \omega - \iota _{\xi }( e \delta e) \delta \omega -\delta \lambda \epsilon _n e \delta \omega -\lambda \epsilon _n \delta e \delta \omega ) \end{aligned}$$

on the space $\widetilde{\mathcal {F}}^{\partial }_{\text {part}}$ (given by the restriction to the corner of the fields appearing above). The equations defining the kernel of the corresponding application $(\widetilde{\varpi }_{part}^{\partial })^{\sharp }$ are

$$\begin{aligned} \delta c:&\quad e X_e =0, \end{aligned}$$

(13a)

$$\begin{aligned} \delta e:&\quad e X_c -e \iota _\xi X_{\omega }-\lambda \epsilon _n X_{\omega }=0, \end{aligned}$$

(13b)

$$\begin{aligned} \delta \xi :&\quad e_{\bullet } e X_{\omega }=0, \end{aligned}$$

(13c)

$$\begin{aligned} \delta \omega :&\quad - \iota _{X_\xi } e e - \iota _{\xi }( e X_e)-X_\lambda \epsilon _n e -\lambda \epsilon _n X_e =0, \end{aligned}$$

(13d)

$$\begin{aligned} \delta \lambda :&\quad -\epsilon _n e X_\omega =0. \end{aligned}$$

(13e)

This system is still singular. Indeed, the third element of the second equation might not be proportional to e and the map $W_{\partial \partial }^{(0,2)}$ is not surjective.

Let us now consider, as announced, the case $\xi ^m=0$ and $\lambda =0$; i.e., we retain only the tangential vector fields. The pre-corner two-form now reads

$$\begin{aligned} \widetilde{\varpi }_{\text {part}}^{\partial }= \int _{\Gamma }&\left( \delta c e \delta e - \iota _{\delta \xi } e e \delta \omega - \iota _{\xi }( e \delta e) \delta \omega \right) . \end{aligned}$$

The only remaining fields are those displayed in this formula. Note that, in particular, the transversal component $e_m$ of the coframe has disappeared. The only remaining, open, condition is that $e \in \Omega ^1(\Gamma , \mathcal {V}_\Gamma )$ should satisfy

$$\begin{aligned} ee\epsilon _m\epsilon _n\not =0, \end{aligned}$$

(14)

where $\epsilon _m$ and $\epsilon _n$ are fixed linearly independent sections of $\mathcal {V}_\Gamma$.²³ The equations defining the kernel of the corresponding application $(\widetilde{\varpi }_{part}^{\partial })^{\sharp }$ are

$$\begin{aligned} \delta c:&\quad e X_e =0, \\ \delta e:&\quad e X_c -e \iota _\xi X_{\omega }=0, \\ \delta \xi :&\quad e_{\bullet } e X_{\omega }=0,\\ \delta \omega :&\quad - \iota _{X_\xi } e e - \iota _{\xi }( e X_e)=0. \end{aligned}$$

This system is not singular. Let us then define the following theory:

Definition 40

We call BF-like corner theory the $\hbox {BF}^2$V theory on the space of fields

$$\begin{aligned} \check{\mathcal {F}}^{\partial \partial }= T^*[1]\left( \Omega _{\partial \partial }^{2,2}\oplus (\Omega _{\partial \partial }^{2,4} \otimes \Omega ^1(\Gamma ) )\right) \end{aligned}$$

with symplectic form

$$\begin{aligned} \check{\varpi }^{\partial \partial }= \int _{\Gamma }&\left( \delta \widetilde{c}\delta \widetilde{E} - \iota _{\delta \widetilde{\xi }} \delta \widetilde{P}\right) \end{aligned}$$

and action

$$\begin{aligned} \check{S}^{\partial \partial } = \int _{\Gamma }&\left( \frac{1}{2}[\widetilde{c},\widetilde{c}] \widetilde{E} + \iota _{\widetilde{\xi }} \widetilde{E} \textrm{d}_{\omega _0} \widetilde{c}- \frac{1}{2}\iota _{[\widetilde{\xi }, \widetilde{\xi }]} \widetilde{P} + \frac{1}{2}\widetilde{E} \iota _{\widetilde{\xi }}\iota _{\widetilde{\xi }}F_{\omega _0}\right) , \end{aligned}$$

where $\omega _0$ is a reference connection.

Remark 41

It is a straightforward check that this is actually a $\hbox {BF}^2$V theory, i.e., that the action $\widetilde{S}^{\partial }$ satisfies the classical master equation.

Furthermore, we can define a map $\widetilde{\pi _\text {red}}:\widetilde{\mathcal {F}}^{\partial }\rightarrow \check{\mathcal {F}}^{\partial \partial }$:

$$\begin{aligned} \widetilde{\pi _\text {red}}:= {\left\{ \begin{array}{ll} \widetilde{E}= \frac{1}{2} e e &{}\\ \widetilde{c}= c + \iota _{\xi }(\omega -\omega _0) &{}\\ \widetilde{\xi }^i= \xi ^i &{}\\ \widetilde{P}_i = \frac{1}{2} e e (\omega _i-\omega _{0i}) &{} \end{array}\right. } \end{aligned}$$

Notice that here we are assuming to work around a connection $\omega _0$. It is a short computation to show that this map is compatible with the two-forms (respectively, the pre-corner form $\widetilde{\varpi }_{part}^{\partial }$ on $\widetilde{\mathcal {F}^{\partial }}$ and $\check{\varpi }^{\partial \partial }$ on $\check{\mathcal {F}}^{\partial \partial }$).

Define now the submanifold $\mathcal {E} \subset \check{\mathcal {F}}^{\partial \partial }$ such that $(E,P,c,\xi )\in \mathcal {E}$ if E is of the form $\frac{1}{2}ee$ for some e satisfying $ee\epsilon _m\epsilon _n\not =0$, with $\epsilon _m$ and $\epsilon _n$ fixed linearly independent sections of $\mathcal {V}_\Gamma$ as above.²⁴ These conditions may be translated to requiring that the Pfaffian of E vanishes and $E\epsilon _m\epsilon _n\ne 0$. In these cases, we drop the tilde. As a consequence of the first statement of Proposition 62, which we prove in Appendix B, $\mathcal {E}$ coincides with the image of $\widetilde{\pi _\text {red}}$.

Let now $p': \Omega _{\partial \partial }^{0,2} \rightarrow \Omega _{\partial \partial }^{0,2}$ be a projection to the complement of the kernel of the map $W_{\partial \partial }^{(0,2)}: \Omega _{\partial \partial }^{0,2} \rightarrow \Omega _{\partial \partial }^{1,3}$. Then, the characteristic distribution of $\mathcal {E}$ is given by the vector fields $X_{p'c}$. Hence, we have the following

Proposition 42

The $\hbox {BF}^2$V space of fields $\mathcal {F}^{\partial \partial }$ is symplectomorphic to the symplectic reduction of $\widetilde{\mathcal {F}}^{\partial }_{\text {part}}$.

We can express the symplectic form on the space of corner fields as

$$\begin{aligned} \varpi ^{\partial \partial }= \int _{\Gamma }&\left( \delta [c] \delta E - \iota _{\delta \xi } \delta P\right) , \end{aligned}$$

where E is a pure tensor as above and [c] denotes the equivalence class of elements $c \in \Omega _{\partial \partial }^{0,2}[1]$ under the equivalence relation $c+ d \sim c$ for $d \in \Omega _{\partial \partial }^{0,2}[1]$ such that $ed=0$.

From the expression of the pre-corner action in this particular case,

$$\begin{aligned} \widetilde{S}^{\partial } = \int _{\Gamma }&\left( \frac{1}{4}[c,c] ee + \frac{1}{2}\iota _\xi (ee) \textrm{d}_\omega c + \frac{1}{4}\iota _\xi \iota _\xi (ee) F_\omega \right) , \end{aligned}$$

we can deduce the corresponding action on the corner:

$$\begin{aligned} S^{\partial \partial }_{\omega _0} = \int _{\Gamma }&\left( \frac{1}{2}[[c],[c]] E + \iota _{{\xi }} (E) \textrm{d}_{\omega _0} [c] - \frac{1}{2}\iota _{[{\xi }, {\xi }]} {P} + \frac{1}{2}E \iota _{{\xi }}\iota _{{\xi }}F_{\omega _0}\right) . \end{aligned}$$

This expression is invariant under the quotient map above: $\frac{1}{2}[c,c] ee = [ce,c]e - [e,c] e c= [ce,ce]$, $\iota _\xi (ee) d c= -d\iota _\xi e e c= L_{\xi }(ee) c = 2 (L_{\xi }e) ec$.

Remark 43

The open condition $E\epsilon _m\epsilon _n\ne 0$ may possibly be dropped to get an extended version of the tangent corner theory (this is analogous to the observation that in $2+1$ PC gravity one may extend the theory dropping the condition that the coframe be nondegenerate). One might want, however, to retain the weaker open condition $E\not =0$ to ensure that the closed condition ${\text {Pf}}(E)=0$ still defines a submanifold.

Remark 44

The map $\pi _\text {red}$ is not strictly speaking the reduction with respect to the kernel of the pre-corner two-form but does satisfy the BV-BFV axioms.

7.2.1 $P_\infty$ Structure

We start our analysis of the $P_\infty$ structure of the tangent theory. Since it is a proper $\hbox {BF}^2$V theory, we can apply the results Sect. 2.2.

We first study the structure of the BF-like corner theory as in Definition 40, and then, we give an implicit description of the corner Poisson structure of gravity by means of a quotient with respect to a suitable ideal. Note that in this section we will drop the tilde on the fields, since no confusion can arise.

The case at hand is similar to that of BF theory. The first step is to choose a polarization and reinterpret the space of fields as a cotangent bundle. We will consider two interesting polarizations.

7.2.2 The First Polarization

Here, we choose the space of fields as the cotangent bundle of the space $\mathcal {N}$ with coordinates E and $\xi$ and choose $\{P=c=0\}$²⁵ as the Lagrangian submanifold. From the action, we get $\pi = \pi _0+ \pi _1 + \pi _2$ with

$$\begin{aligned} \pi _0&= \int _\Gamma \frac{1}{2} E \iota _{\xi } \iota _{\xi } F_{\omega _0},\\ \pi _1&= \int _\Gamma \left( \iota _{\xi } E \textrm{d}_{\omega _0} \frac{\delta }{\delta E} - \frac{1}{2}\iota _{[\xi ,\xi ]}\frac{\delta }{\delta \xi }\right) ,\\ \pi _2&= \int _\Gamma \frac{1}{2}\left[ \frac{\delta }{\delta E},\frac{\delta }{\delta E}\right] E. \end{aligned}$$

These equip $C^{\infty }(\mathcal {N})$ with the structure of a curved $P_\infty$ algebra. Note that this polarization roughly corresponds to the choice of subalgebra $\mathfrak {h}$ that we have made for the general and constrained theory in Sects. 6 and 7.1.2. Indeed, we consider a subalgebra of linear functionals of the form²⁶:

$$\begin{aligned} J_{\varphi }&= \int _{\Gamma } \varphi E,\\ M_{Y}&= \int _{\Gamma } Y \iota _{\xi } E,\\ K_{Z}&= \int _{\Gamma } \frac{1}{2}Z \iota _{\xi }\iota _{\xi } E. \end{aligned}$$

The derived brackets are as follows

$$\begin{aligned}&\{\}_0= \int _\Gamma \frac{1}{2} E \iota _{\xi } \iota _{\xi } F_{\omega _0},{} & {} {} & {} \\&\{J_{\varphi }\}_1= M_{\textrm{d}_{\omega _0}\varphi },{} & {} \{M_{Y}\}_1 = K_{\textrm{d}_{\omega _0}Y},{} & {} \{K_{Z}\}_1 = 0, \\&\{J_{\varphi }, J_{\varphi '}\}_2 = J_{[\varphi , \varphi ']},{} & {} \{J_{\varphi }, M_Y\}_2 = M_{[\varphi , Y]},{} & {} \{J_{\varphi }, K_Z\}_2 = K_{[\varphi ,Z]},\\&\{M_Y, M_{Y'}\}_2 = K_{[Y, Y']},{} & {} \{M_Y, K_Z\}_2 = 0,{} & {} \{K_Z, K_{Z'}\}_2 = 0. \end{aligned}$$

Observe the similarity with (1) in BF theory. Also, note that we can write

$$\begin{aligned} \{\}_0= K_{F_{\omega _0}}, \end{aligned}$$

so the algebra generated by J, M, and K closes also under the nullary operation.

Remark 45

The striking similarity between the structure of the subalgebra of observable proposed in the present section and that of BF theory is not accidental. In fact, the tangent theory (before the reduction) can be obtained as BF theory, for the Lie algebra $\mathfrak {so(3,1)}$, restricted to the submanifold of fields parametrized by

$$\begin{aligned} c&=c,&A&= \omega ,&B^\dag&= 0,\\ \phi&= \frac{1}{4} \iota _\xi \iota _\xi (ee),&\tau&=\frac{1}{2} \iota _\xi (ee),&B&=\frac{1}{2} ee. \end{aligned}$$

We now want to describe the $P_\infty$ structure of the real theory describing gravity. Hence, we have to consider the structure described above and assume that the Pfaffian of E vanishes. Instead of describing it directly, we can describe the subalgebra as the quotient of this $P_\infty$ algebra by the ideal generated by the following additional linear functionals:

$$\begin{aligned} P_{\mu }&= \int _{\Gamma } \mu \mathcal {P}_E,\\ Q_{\nu }&= \int _{\Gamma } \nu \iota _{\xi } \mathcal {P}_E,\\ R_{\sigma }&= \int _{\Gamma } \frac{1}{2} \sigma \iota _{\xi }\iota _{\xi } \mathcal {P}_E, \end{aligned}$$

where $\mathcal {P}_E= \sqrt{{\text {Pf}}(E)}$ is the square root of the Pfaffian of E.²⁷ It is worth noting that $\mathcal {P}_E$ is invariant under the action of the gauge transformations. Now, we have to compute the brackets of these new linear functionals to show that they form an ideal of the $P_\infty$ algebra generated by J, M, K, P, Q and R. Let us start from the 1-brackets. They read

$$\begin{aligned}&\{P_{\mu }\}_1= Q_{\textrm{d}_{\omega _0}\mu },{} & {} \{Q_{\nu }\}_1 = R_{\textrm{d}_{\omega _0}\nu },{} & {} \{R_{\sigma }\}_1 = 0. \end{aligned}$$

On the other hand, all the 2-brackets containing P, Q or R vanish.

Hence, we can describe the $P_\infty$ algebra of such linear functionals on the space of corner fields in the tangent theory as the quotient of the $P_\infty$ algebra generated by J, M, K, P, Q and R by the $P_\infty$ ideal generated by P, Q and R.

7.2.3 The Second Polarization

We can now consider another polarization: we choose the space of fields as the cotangent bundle of the space $\mathcal {N}$ with coordinates E and P and choose $\{\xi =c=0\}$ as the Lagrangian submanifold. From the action, we get $\pi = \pi _2$ with

$$\begin{aligned} \pi _2&= \int _\Gamma \left( \frac{1}{2}\left[ \frac{\delta }{\delta E},\frac{\delta }{\delta E}\right] E + \iota _{\frac{\delta }{\delta P}} (E) \textrm{d}_{\omega _0} \frac{\delta }{\delta E} - \frac{1}{2}\iota _{[\frac{\delta }{\delta P}, \frac{\delta }{\delta P}]} {P} + \frac{1}{2}E \iota _{\frac{\delta }{\delta P}}\iota _{\frac{\delta }{\delta P}}F_{\omega _0}\right) , \end{aligned}$$

which equips $C^{\infty }(\mathcal {N})$ with the structure of a Poisson algebra. As before, we can consider a subalgebra of linear functionals. Let

$$\begin{aligned} F_{X}= \int _{\Gamma } \iota _X P \quad \text {and} \quad J_{\varphi }&= \int _{\Gamma } \varphi E. \end{aligned}$$

Their binary operations are as follows:

$$\begin{aligned} \{J_{\varphi }, J_{\varphi '}\}_2 = J_{[\varphi , \varphi ']},{} & {} \{J_{\varphi }, F_X\}_2 = J_{\iota _X \textrm{d}_{\omega _0} \varphi },{} & {} \{F_X, F_{X'}\}_2 = F_{[X, X']}+ J_{\iota _X\iota _{X'}F_{\omega _0}}.\nonumber \\ \end{aligned}$$

(15)

As before, in order to get the structure on the gravity theory, we have to consider the ideal generated by the functional $P_{\mu }= \int _{\Gamma } \mu \mathcal {P}_E$. The only nonzero bracket is the one with $F_X$:

$$\begin{aligned} \{P_{\mu }, F_X\}_2= P_{\iota _X \textrm{d}_{\omega _0}\mu }. \end{aligned}$$

It is worth noting that, with this polarization, the structure of linear functionals corresponds to that of (a subalgebra of) an Atiyah algebroid.

Remark 46

An Atiyah algebroid is a way of describing a combination of a Lie algebra of internal symmetries with the Lie algebra of vector fields (the infinitesimal version of a combination of a Lie group of symmetries with the group of diffeomorphisms). From this point of view, it is not unexpected that an Atiyah algebroid should appear as part of the corner structure (and in fact it also appears in some work completed after ours [20]). What is interesting is the explicit way in which it emerges from our construction.

The goal of next section is to show this relation.

7.2.4 Atiyah Algebroids

Let us begin with some definitions.

Definition 47

Let M be a manifold. A Lie algebroid over M is a triple $(A, [\cdot , \cdot ], \rho )$ where $A \rightarrow M$ is a vector bundle over M, $[\cdot , \cdot ]:\Gamma (A) \times \Gamma (A) \rightarrow \Gamma (A)$ an $\mathbb {R}$-Lie bracket, and $\rho :A \rightarrow TM$ a morphism of vector bundles, called the anchor, such that

$$\begin{aligned}{}[X, g Y]= \rho (X) g \cdot Y + g[X, Y] \qquad \forall X,Y \in \Gamma (A), \; g \in C^{\infty }(M). \end{aligned}$$

The Atiyah algebroid is a particular example of a Lie Algebroid.

Definition 48

Let G be a Lie Group and $P \rightarrow M$ a G-principal bundle over M. The Atiyah algebroid is a Lie Algebroid with $A = TP/G$, the Lie bracket on sections that inherited from the tangent Lie algebroid of P, and the anchor induced by the quotient by G of the differential map $\textrm{d}\pi :TP \rightarrow TM$.

The Atiyah algebroid may be written in terms of the short exact sequence

$$\begin{aligned} 0 \rightarrow {\text {ad}} P \rightarrow A \rightarrow TM \rightarrow 0. \end{aligned}$$

The algebroid that we will construct out of the corner data will be of type $A= F \oplus TM$, corresponding to a splitting of the exact sequence. By well-known results, this corresponds to a map $\tau :TM \rightarrow A$ such that $\pi \circ \tau = \text {id}_{TM}$. Out of this map, we can construct an isomorphism between A and $F \oplus TM$ as follows:

$$\begin{aligned} \chi : F \oplus TM&\rightarrow A\\ ( a, X )&\mapsto \iota (a)+ \tau (X). \end{aligned}$$

This map is injective. Indeed, let $\chi (a,X)=0$, then $\pi (\chi (a,X))=X=0$. As a consequence, $\iota (a)=0$ implying $a=0$.

Using this isomorphism, we can induce an algebroid structure on $F \oplus TM$. After a short computation, we find the following structure:

$$\begin{aligned} {[}(a,X),(b,Y)]&=([a,b]+ \iota ^{-1}([\iota (a),\tau (Y)]+[\tau (X),\iota (b)]\\&\quad +[\tau (X),\tau (Y)]-\tau [X,Y]),[X,Y]) \end{aligned}$$

We can now introduce the map $\nabla ^\tau$

$$\begin{aligned} \nabla ^\tau :\Gamma (TM) \times \Gamma (F)&\rightarrow \Gamma (F)\\ (X,a)&\mapsto \nabla ^\tau _X(a) = \iota ^{-1}([\iota (a),\tau (X)]) \end{aligned}$$

Lemma 49

The map $\nabla ^\tau$ has the following properties:

• (1) $\nabla ^\tau$ is a connection for F.

• (2) The curvature of $\nabla ^\tau$ is given by

  $$\begin{aligned} R^\tau (X,Y)= \iota ^{-1}([\tau (X),\tau (Y)]-\tau [X,Y]). \end{aligned}$$

Proof

Easy computation. $\square$

Let us now denote by $\omega _0$ the connection one-form corresponding to the connection $\nabla ^\tau$. Then, we can rewrite the brackets on $F \oplus TM$ as

$$\begin{aligned} {[}(a,X),(b,Y)]&=([a,b] - \iota _X \textrm{d}_{\omega _0} (b) + \iota _Y \textrm{d}_{\omega _0} (a) \nonumber \\&\quad + \iota _X \iota _Y F_{\omega _0} ,[X,Y]). \end{aligned}$$

(16)

The Lie algebroid structure on A allows us to define a Poisson bracket on $\Gamma (A^*)$. We write this down for linear functionals. Namely, we define $U_\beta =\int _M\Phi \beta$, with $\Phi \in \Gamma (A^*)$ and $\beta \in \Gamma (A)$. We then define

$$\begin{aligned} \left\{ \int _{M}\Phi \beta _1,\int _{M}\Phi \beta _2\right\} = \int _{M}\Phi [\beta _1,\beta _2]. \end{aligned}$$

Let us now write $\Phi =\mathcal {F}+\mathcal {Q}$ with $\mathcal {F}\in \Gamma (\bigwedge ^{\text {top}} T^*M, F^*)$ and $\mathcal {Q}\in \Gamma (\bigwedge ^{\text {top}} T^*M, T^*M)$. Then, using (16) we get

$$\begin{aligned} \left\{ \int _{M}(\mathcal {F}a + \mathcal {Q}X) ,\int _{M}(\mathcal {F}b + \mathcal {Q}Y)\right\}&= \int _{M}\!\!(\mathcal {F}([a,b] - \iota _X \textrm{d}_{\omega _0} (b) + \iota _Y \textrm{d}_{\omega _0} (a) + \iota _X \iota _Y F_{\omega _0}) \nonumber \\&\quad + \mathcal {Q}[X,Y]). \end{aligned}$$

(17)

Theorem 50

The $\hbox {BF}^2$V structure of the tangent theory on a corner $\Gamma$ induces an Atiyah algebroid structure on ${\text {ad}} P \oplus T \Gamma$.

Proof

Let us define $B={\text {ad}}P \oplus T \Gamma$. Then, the space of corner fields is $\mathcal {F}^{\partial \partial }= T^*[1]\Gamma (B)^*$. As explained in the previous section, we can equip this space with a Poisson structure. Comparing (17) with (15), it is easy to see that on linear functionals these brackets coincide with the identification $E= \mathcal {E}$ and $P=\mathcal {Q}$. Hence, dualizing, the induced structure is the one of an Atiyah algebroid. $\square$

Remark 51

This construction does not depend on the final quotient. Hence, the symplectic space of corner fields identifies a Poisson subalgebra and consequently a subalgebroid.

7.2.5 Quantization

In the relatively simple tangent case, we may also describe the quantization of the corner structure for a very important particular situation that arises when we consider a point defect on a spacelike boundary $\Sigma$. By quantization, we mean here the deformation quantization of a Poisson manifold [7] (solved in the finite-dimensional case by Kontsevich [26]).²⁸ We take $\Gamma$ to be an infinitesimal sphere surrounding this point. On $\Gamma$, we only consider uniform fields (this is our formalization of its being infinitesimal). For $\xi$, which is a vector field, this implies $\xi =0$. Similarly, we get $P=0$. In the resulting theory, there are then no $\xi$ nor P. On the other hand, c and E are ${\text {SO}}(3)$-equivariant. Since the $\hbox {BF}^2$V action and 2-form are defined in terms of an invariant pairing, what matters are only the values of c and E at some point. We denote the first as $c\in \Lambda ^2V$ and the second as $E=\textbf{A}\,\text {vol}$, with $\textbf{A}\in \Lambda ^2V$ and $\text {vol}$ the standard, normalized volume form on the sphere $\Gamma$ evaluated at the chosen point. We then have the symplectic form

$$\begin{aligned} \varpi ^{\partial \partial }_q= \delta c\, \delta \textbf{A} \end{aligned}$$

and the $\hbox {BF}^2$V action

$$\begin{aligned} S^{\partial \partial }_q= \frac{1}{2} [c,c]\,\textbf{A}. \end{aligned}$$

(Note that both expressions take values in $\Lambda ^4V$ which we tacitly identify with $\mathbb {R}$.) Next, we will have to impose that E is a pure tensor satisfying $E\epsilon _m\epsilon _n\not =0$ for some fixed linearly independent sections $\epsilon _m$ and $\epsilon _n$ in V. This corresponds to imposing ${\text {Pf}}(\textbf{A})=0$ and $\textbf{A}\epsilon _m\epsilon _n\not =0$, and to reduce c accordingly. Note that the second condition on $\textbf{A}$ is an open condition, which, in particular, entails $\textbf{A}\not =0$.

We first analyze the theory without the conditions on $\textbf{A}$. In the polarization $c=0$, the above data yield as Poisson manifold the dual of the Lie algebra $\mathfrak {g}=\mathfrak {so}(3,1)\simeq \Lambda ^2V$. Its quantization may be identified with the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$ [24]. The module structures for $\Sigma$ minus the defect that we get from the quantization of the corner then correspond to representations of $U(\mathfrak {g})$, but this is the same as Lie algebra representations of $\mathfrak {g}$ or group representations of its simply connected Lie group $G=\textrm{SL}(2,\mathbb {C})$.

The conditions on $\textbf{A}$ select a five-dimensional Poisson submanifold of $\mathfrak {g}^*$. Since ${\text {Pf}}(\textbf{A})$ is quadratic in $\textbf{A}$ and invariant, it is a quadratic Casimir. If we ignore the open condition $\textbf{A}\epsilon _m\epsilon _n\not =0$, the quantization then simply amounts to considering representations of G in which this Casimir is represented as zero. Explicitly we write

$$\begin{aligned} \textbf{A}=\begin{pmatrix} 0 &{} A^{01} &{} A^{02} &{} A^{03}\\ -A^{01} &{} 0 &{} A^{12} &{} A^{13}\\ -A^{02} &{} -A^{12} &{} 0 &{} A^{23}\\ -A^{03} &{} -A^{13} &{} -A^{23} &{} 0 \end{pmatrix}=: \begin{pmatrix} 0 &{} M^{1} &{} M^{2} &{} M^{3}\\ -M^{1} &{} 0 &{} J^{3} &{} -J^{2}\\ -M^{2} &{} -J^{3} &{} 0 &{} J^{1}\\ -M^{3} &{} J^{2} &{} -J^{1} &{} 0 \end{pmatrix}. \end{aligned}$$

We then have

$$\begin{aligned} {\text {Pf}}(\textbf{A}) = A^{01}A^{23}-A^{02}A^{13}+A^{03}A^{12} = \textbf{M}\cdot \textbf{J} = \frac{\textbf{J}_+^2-\textbf{J}_-^2}{4}, \end{aligned}$$

with $\textbf{J}_\pm =\textbf{J}\pm \textbf{M}$. Note that $\textbf{J}_\pm ^2$ are the two standard $\mathfrak {su}(2)$ quadratic Casimirs of the two summands of $\mathfrak {sl}(2,\mathbb {C})=\mathfrak {su}(2)\oplus \mathfrak {su}(2)$. The condition ${\text {Pf}}(\textbf{A})=0$, i.e., $\textbf{J}_+^2=\textbf{J}_-^2$, therefore implies that we only have representations of $\textrm{SO}(3,1)^+$ with highest weight of the form (m, m) (here, 2m is a nonnegative integer).

The open condition $\textbf{A}\epsilon _m\epsilon _n\not =0$ is more difficult to understand algebraically. The induced open condition $\textbf{A}\not =0$ instead corresponds to $\textbf{J}_+^2\not =0$ and $\textbf{J}_-^2\not =0$, which would suggest that we have to exclude the case $m=0$. On the other hand, it might make sense to retain also this possibility in the quantization (essentially working with the extended theory of Remark 43).

To summarize the results of this section, we see that, in the case of small m, the point defect then corresponds to a scalar ($m=0$), a vector ($m=\frac{1}{2}$), and a traceless symmetric tensor ($m=2$).

8 Cosmological Term

In the previous sections, we have always assumed the vanishing of the cosmological constant. We now drop this assumption and add the following term to the boundary BFV action:

$$\begin{aligned} S^{\partial }_{\text {cosm}}= \int _{\Sigma }\frac{1}{6}\Lambda \lambda \epsilon _n e^3. \end{aligned}$$

Since it does not contain any derivatives, this additional term does not change the pre-corner two-form (7) and hence, the extendability of the BFV theory to a BFV-$\hbox {BF}^2$V theory. The only change in the pre-corner structure is an additional term in the pre-corner action (9) of the form

$$\begin{aligned} \widetilde{S}^{\partial }_{\text {cosm}}= \int _{\Gamma }\frac{1}{2}\Lambda \lambda \epsilon _n \xi ^m e_m e^2 . \end{aligned}$$

Since this term contains $\xi ^m$, the tangent case is unmodified and carries no information about the cosmological constant.

However, the action of the constrained case (12) gets a contribution of the form

$$\begin{aligned} \widetilde{S}^{\partial }_{\text {cosm}}= \int _{\Gamma }\frac{1}{2}\Lambda {\widetilde{\lambda }} \epsilon _n \widetilde{\xi }^{m} \epsilon _m \widetilde{e}^2 . \end{aligned}$$

In the constrained case and in the pre-corner case, there are some differences when the cosmological constant is present, similarly to what happens in BF theory. Indeed, even though the unary operation $\{\ \}_1$ and the binary operation $\{\,\ \}_2$ do not change, we have

$$\begin{aligned} \{\}_0&=\int _{\Gamma }\left( \iota _{\widetilde{\xi }^{}}\widetilde{e}\left( \frac{1}{2}\iota _{\widetilde{\xi }^{}} \widetilde{e}+ \alpha \right) +\frac{1}{2}\alpha ^2\right) F_{\omega _0} + \int _{\Gamma }\frac{1}{2}\Lambda \lambda \epsilon _n \xi ^m \epsilon _m \widetilde{e}^2,\\ \{\}_0&=\int _{\Gamma }\left( \iota _{\xi }e\left( \frac{1}{2}\iota _{\xi } e + \alpha \right) +\frac{1}{2}\alpha ^2\right) F_{\omega _0} + \int _{\Gamma }\frac{1}{2}\Lambda \lambda \epsilon _n \xi ^m e_m e^2, \end{aligned}$$

for the constrained and the pre-corner theories, where $\alpha$ is as defined in Sects. 7.1.2 and 6, respectively. As a result, the algebra generated by J, M, and K no longer closes under the nullary operation. To remedy for this, we can add a functional $C_{\beta }$ to the $P_\infty$ subalgebra to parametrize this new term as follows²⁹:

$$\begin{aligned} C_{\beta }&= \int _{\Gamma } \frac{1}{2}\beta ee\alpha ^2. \end{aligned}$$

We now have

$$\begin{aligned} \{\}_0= K_{F_{\omega _0}}+C_\Lambda . \end{aligned}$$

In order to get a closed set under the bracket operations, we also add the following two additional functionals:

$$\begin{aligned} D_{\gamma }&= \int _{\Gamma } \frac{1}{2}\gamma \iota _{\xi }(ee)\alpha ^2,\\ E_{\rho }&= \int _{\Gamma } \frac{1}{4}\rho \iota _{\xi }\iota _{\xi }(ee)\alpha ^2. \end{aligned}$$

The brackets of these functionals with themselves and with $J_{\varphi }$, $M_y$, $K_Z$ are all zero except for

$$\begin{aligned} \{C_\beta \}_1=D_{\textrm{d}\beta } \qquad \{D_\gamma \}_1=E_{\textrm{d}\gamma }. \end{aligned}$$

Appendices

Appendix A: Notation and Property of Maps

The goal of this appendix is to recall and collect in one place the relevant quantities and maps, to establish the conventions, and to summarize the technical results needed in the article.

Let us first recall some useful shorthand notation introduced in the previous sections. Let M be a smooth manifold of dimension 4 with corners, and let us denote by $\Sigma = \partial M$ its 3-dimensional boundary and by $\Gamma = \partial \partial M$ its 2-dimensional corner. Furthermore, we will use the notation $\mathcal {V}_{\Sigma }$ for the restriction of $\mathcal {V}$ to $\Sigma$ and $\mathcal {V}_{\Gamma }$ for the restriction of $\mathcal {V}$ to $\Gamma$. We define

$$\begin{aligned} \Omega _{\partial }^{i,j}:= \Omega ^i\left( \Sigma , \textstyle {\bigwedge ^j} \mathcal {V}_{\Sigma }\right) , \qquad \Omega _{\partial \partial }^{i,j}:= \Omega ^i\left( \Gamma , \textstyle {\bigwedge ^j} \mathcal {V}_{\Gamma }\right) . \end{aligned}$$

On $\Omega _{\partial }^{i,j}$ and $\Omega _{\partial \partial }^{i,j}$, we define the following maps:

$$\begin{aligned} W_{\partial }^{(i,j)}:\Omega _{\partial }^{i,j}&\longrightarrow \Omega _{\partial }^{i,j}\\ X&\longmapsto X \wedge e|_{\Sigma } , \\ W_{\partial \partial }^{ (i,j)}:\Omega _{\partial \partial }^{i,j}&\longrightarrow \Omega _{\partial \partial }^{i,j}\\ X&\longmapsto X \wedge e|_{\Gamma } . \end{aligned}$$

Remark 52

Usually, we will omit writing the restriction of e to the corresponding manifold $\Sigma$ or $\Gamma$.

The properties of these maps are collected in the following lemmata, where we condensate all the information in two tables, one for the boundary maps and one for the corner maps. We organize the $\Omega _{\bullet }^{ij}$ spaces in an array and connect them with arrows corresponding to the maps $W_{\bullet }^{\bullet (i,j)}$: a hooked arrow denotes an injective map, while a two-headed arrow denotes a surjective map. The proofs of these properties are similar to those proved in [13] and are left to the reader.

On the boundary, the index i runs only between 1 and 3.

Lemma 53

The maps $W_{\partial }^{(i,j)}$ on the boundary fields have the properties described in the following table:

Lemma 54

The maps $W_{\partial \partial }^{(i,j)}$ on the corner fields have the properties described in the following table:

The coframe e viewed as an isomorphism $e:TM \rightarrow \mathcal {V}$ defines, given a set of coordinates on M, a preferred basis on $\mathcal {V}$. If we denote by $\partial _i$, the vector field in TU, where U is a coordinate neighborhood in M, corresponding to the coordinate $x^i$, we get a basis on $\mathcal {V}|_{U}$ by $e_i:= e (\partial _i)$. On the boundary, since $T\Sigma$ has one dimension less than $\mathcal {V}_{\Sigma }$, we can complement the linear independent set $(e_i)$ with another independent vector that we will call $\epsilon _n$. On the corner $\Gamma$, the tangent space loses one further dimension; hence, we will have to introduce one more additional independent vector that will be denoted by $\epsilon _m$. Fixed a coordinate system on M (or $\Sigma$ or $\Gamma$), we call this basis the standard basis and, unless otherwise stated, the components of the fields will always be taken with respect to this basis.

Appendix B: Pfaffian and Pure Tensors

In this appendix, we discuss the relation between having ${\text {Pf}}(E)=0$ for an element $E \in \Omega _{\partial \partial }^{2,2}$ and requiring that E can be expressed as a pure tensor, i.e., that $E=\frac{1}{2}ee$ for some $e \in \Omega _{\partial \partial }^{1,1}$. We start with the local analysis. Let

$$\begin{aligned} \phi :\begin{array}{ccc} V\times V &{}\rightarrow &{}\Lambda ^2V\\ (e_1,e_2)&{}\mapsto &{}e_1e_2 \end{array} \end{aligned}$$

where V is a four-dimensional vector space and, as usual, we omitted the wedge multiplication symbol on the right hand side. We then have the following two lemmata.

Lemma 55

$e_1,\, e_2\ \text {linearly independent} \iff \phi (e_1,e_2)\not =0.$

Proof

If $e_1$ and $e_2$ are linearly independent, then we can complete them to a basis $\{e_1,e_2,e_3,e_4\}$, and we clearly have that $\phi (e_1,e_2)e_3e_4=e_1e_2e_3e_4\not =0$ as an element of $\Lambda ^4V$, so $\phi (e_1,e_2)\not =0$. If, on the other hand, $e_1$ and $e_2$ are linearly dependent, then we have $e_1=\alpha e_2$ or $e_2=\alpha e_1$, for some scalar $\alpha$, so $e_1e_2=0$. $\square$

Lemma 56

${\text {Pf}}(\phi (e_1,e_2))=0$ for all $e_1,e_2$.

Proof

For $E=(E^{ab})$ in some basis, we have

$$\begin{aligned} {\text {Pf}}(E)=\frac{1}{8}\epsilon _{abcd}E^{ab}E^{cd}. \end{aligned}$$

Therefore, if $E^{ab}=e_1^ae_2^b-e_2^ae_1^b$, we clearly have ${\text {Pf}}(E)=\frac{1}{2}\epsilon _{abcd}e_1^ae_2^be_1^ce_2^d=0$. $\square$

A further interesting remark is that, for $E=e_1e_2$, we have $Ee_1=Ee_2=0$. This can also be written in terms of matrix multiplication if we introduce , i.e., . Now, we have . For further reference, we also introduce the linear map .

Let us finally introduce

$$\begin{aligned} W:=\{(e_1,e_2)\in V\times V \ |\ e_1,\, e_2\ \text {linearly independent} \} \end{aligned}$$

and

$$\begin{aligned} B:=\{E\in \Lambda ^2 V\setminus \{0\}\ |\ {\text {Pf}}(E)=0\}. \end{aligned}$$

For every $E\in B$, we define as above and the corresponding linear map $\psi _E:V\rightarrow V^*$.

Lemma 57

The kernel of $\psi _E$ is two-dimensional.

Proof

Since the matrix representing E or $*E$ is skew-symmetric, its eigenvalues are either equal to zero or they come in pairs of conjugate nonzero imaginary numbers. Since $E\not =0$, they cannot all vanish. On the other hand, the condition ${\text {Pf}}(E)=0$ implies that E and $*E$ are singular; therefore, at least one eigenvalue must vanish. It then follows that exactly two eigenvalues vanish, whereas the other two are conjugate nonzero imaginary numbers. $\square$

Let $S_E:={\text {ker}}\psi _E$.

Lemma 58

Let $(e_1,e_2)$ be a basis of $S_E$. Then, there is a uniquely determined nonzero scalar $\lambda$ such that $E=\lambda e_1e_2$.

Proof

Let $E':=e_1e_2$. Then $S_{E'}=S_E$. Let us complete $(e_1,e_2)$ to a basis $(e_1,e_2,e_3,e_4)$ of V. In this basis, we then have and for every a. By skew-symmetry, we then have that the only nonzero entries of and are the 34 and the 43 ones, one opposite to the other. There is then a uniquely determined nonzero scalar $\lambda$ such that $E_{34}=\lambda E'_{34}$. $\square$

Collecting all the above, we then have the

Proposition 59

$\phi (W)=B$.

Proof

For every $E\in B$, we can choose a basis $(e_1,e_2)$ of $S_E$ and we then have $E=\lambda e_1e_2$. But then $(\lambda e_1,e_2)\in W$ and $E=\phi (\lambda e_1,e_2)$. $\square$

The map $\phi$ is clearly not injective. We can, however, relate this to a distribution that is the same as the one that we get from the kernel of the two-form in the tangent corner structure, see (13a). Namely, let $K\subset TW$ be the regular involutive distribution spanned by vector fields $X=(X_1,X_2)$ satisfying $e_1X_2+X_1e_2=0$ (wedge product symbols omitted). It is clear that $\phi$ is constant along K. Let ${\underline{\phi }}$ be the induced map $W/K\rightarrow B$.

Proposition 60

${\underline{\phi }}$ is a diffeomorphism.

Proof

We have already seen that every $E\in B$ is of the form $E=\phi (e_1,e_2)$ with $(e_1,e_2)$ of $S_E$ a basis of $S_E$. Choose an inner product on $S_E$ and a reference vector $v\not =0$. By moving along K (with $X_1=0$ and $X_2=e_1$), we can always arrange $e_1$ and $e_2$ to be orthogonal. By further moving along K (with $X_1=e_1$ and $X_2=-e_2$), we can arrange $e_1$ and $e_2$ to have the same length.

Now, suppose that $E=\phi (e_1,e_2)=\phi (e'_1,e'_2)$. By the above discussion, we may assume that $e_1$, $e_2$, $e'_1$, and $e'_2$ have the same length, that $e_1$ is orthogonal to $e_2$, that $e'_1$ is orthogonal to $e'_2$, and that the two pairs have the same orientation on $S_E$. We can now rotate the vectors $e_1$ and $e_2$ (by choosing $X_1=e_2$ and $X_2=-e_1$) to send $e_1$ to $e'_1$. This automatically sends $e_2$ to $e'_2$. $\square$

To get in touch with the corner structure, we need one more piece of information to implement condition (14); namely, the datum of two linearly independent vectors $\epsilon _m$ and $\epsilon _n$ in V. We then define

$$\begin{aligned} W':=\{(e_1,e_2)\in V\times V \ |\ (e_1,\, e_2,\,\epsilon _m,\,\epsilon _n)\ \text {linearly independent} \}\subset W \end{aligned}$$

and

$$\begin{aligned} B':=\{E\in \Lambda ^2V\ |\ E\epsilon _m\epsilon _n\not =0\text { and } {\text {Pf}}(E)=0\}\subset B. \end{aligned}$$

Note that $W'$ is an open subset of W and $B'$ is an open subset of B. It is immediately clear that $\phi (W')\subseteq B'$. On the other hand, if $E\in B'\subset B$, we can write $E=e_1e_2$. The condition $E\epsilon _m\epsilon _n\not =0$ implies that $e_1,\, e_2,\,\epsilon _m,\,\epsilon _n$ are linearly independent, so $(e_1,e_2)\in W'$. Moreover, the K-leaf of $(e_1,e_2)\in W'$ is contained in $W'$, as it has image a fixed $E\in B'$. Therefore, we have the following

Proposition 61

$\phi (W')=B'$, and ${\underline{\phi }}:W'/K\rightarrow B'$ is a diffeomorphism.

We finally move to the setting of the corner structure. The data are the following: a two-manifold $\Gamma$, a rank-four vector bundle $\mathcal {V}_\Gamma$ over $\Gamma$, which is assumed to be isomorphic to $T\Gamma \oplus {\underline{\mathbb {R}}}^2$, and two linearly independent sections $\epsilon _m,\epsilon _n$ of the ${\underline{\mathbb {R}}}^2$ summand of $\mathcal {V}_\Gamma$. We consider the map

$$\begin{aligned} \phi :\Omega _{\partial \partial }^{1,1}:=\Gamma (T^*\Gamma \otimes \mathcal {V}_\Gamma )&\rightarrow \Gamma (\Lambda ^2T^*\Gamma \otimes \Lambda ^2\mathcal {V}_\Gamma )=:\Omega _{\partial \partial }^{2,2}\\ e&\mapsto \frac{1}{2}ee \end{aligned}$$

In local coordinates, we write $e=e_1\textrm{d}x^1+e_2\textrm{d}x^2$, so $E=\phi (e)=-e_1e_2\textrm{d}x^1\textrm{d}x^2$, which is the same map $\phi$ (up to the density $-\textrm{d}x^1\textrm{d}x^2$) that we considered in the first part of this section when we restrict ourselves to a fiber of $\mathcal {V}_\Gamma$.

We then define

$$\begin{aligned} \mathcal {W}':=\{e\in \Omega _{\partial \partial }^{1,1} \ |\ ee\epsilon _m\epsilon _n\not =0 \} \end{aligned}$$

and

$$\begin{aligned} \mathcal {B}':=\{E\in \Omega _{\partial \partial }^{2,2}\ |\ E\epsilon _m\epsilon _n\not =0\text { and } {\text {Pf}}(E)=0\}. \end{aligned}$$

Proposition 62

$\phi (\mathcal {W}')=\mathcal {B}'$, and ${\underline{\phi }}:\mathcal {W}'/\mathcal {K}\rightarrow \mathcal {B}'$ is an isomorphism of fiber bundles where $\mathcal {K}$ is a distribution fiberwise defined as K.

Proof

Fiberwise we follow the proofs of the first part of this appendix. The only problem is to prove that globally we can write $E\in \mathcal {B}'$ as $\frac{1}{2}ee$. The point is that the condition $E\epsilon _m\epsilon _n\not =0$ implies that the distribution of two-planes $S_E$ is transversal to the distribution $S_{\epsilon _m\epsilon _n}$, i.e., the ${\underline{\mathbb {R}}}^2$ summand of V. This means that for a given isomorphism $e^{0}$ of $T\Gamma$ with a complement of the ${\underline{\mathbb {R}}}^2$ summand (chosen in such a way that $e^0e^0\epsilon _m\epsilon _n$ defines the same orientation as $E\epsilon _m\epsilon _n$), we have $E=\frac{1}{2}ee$ with e of the form $fe^{0}+\alpha \epsilon _m +\beta \epsilon _n$, with $\alpha ,\beta$ 1-forms on $\Gamma$ and f a nowhere vanishing function. $\square$

Appendix C: Analysis of the Constraints

In this appendix, we analyze the constraints (10) and show which fields are they fixing. Let us start with some preliminary results. Consider $W_{\partial \partial }^{(1,2)}:\Omega _{\partial \partial }^{1,2} \longrightarrow \Omega _{ \partial \partial }^{2,3}$. The dimensions of domain and codomain are $\dim \Omega _{ \partial \partial }^{1,2} = 12$ and $\dim \Omega _{ \partial \partial }^{2,3} = 4$. The kernel of $W_{\partial \partial }^{(1,2)}$ is defined by

$$\begin{aligned} X_{\mu _1}^{ab} e_a e_b e_{\mu _2} \cdots e_{\mu _{2}} \textrm{d}x^{\mu _1}\textrm{d}x^{\mu _2}\cdots \textrm{d}x^{\mu _{2}}=0, \end{aligned}$$

where we used $e_a$ as a basis for $\mathcal {V}_\Gamma$.³⁰ Since $\textrm{d}x^{1}\textrm{d}x^{2}$ is a basis for $\Omega ^{2}(\Gamma )$, we obtain one equation of the form

$$\begin{aligned} X_{1}^{ab} e_a e_b e_{2} - X_{2}^{ab} e_a e_b e_{1} =0. \end{aligned}$$

Recall now that $e_a e_b e_{\mu }$ for $\mu =1,2$ is a basis of $\wedge ^{3}\mathcal {V}_\Gamma$. Hence, we obtain the following equations:

$$\begin{aligned}&X_1^{13}+X_2^{23}=0,&\qquad X_1^{14}+X_2^{24}=0,\\&X_1^{34}=0,&X_2^{34}=0. \end{aligned}$$

Hence, the map $W_{\partial \partial }^{(1,2)}$ is surjective but not injective. In particular, $\dim \text {Ker}W_{\partial \partial }^{(1,2)}=8$ and the kernel is generated by the following components:

$$\begin{aligned}&X_1^{13}-X_2^{23},&X_1^{14}-X_2^{24},{} & {} X_1^{12},{} & {} X_2^{12},\\&X_1^{23},&X_2^{13},{} & {} X_1^{24},{} & {} X_2^{14}. \end{aligned}$$

Consider now $\psi _e:\Omega _{\partial \partial }^{1,2} \rightarrow \Omega _{\partial \partial }^{2,1}$, $\psi _e(v):=[v,e]$. The components of $\psi _e$ are defined by³¹

$$\begin{aligned} {[}v,e]_{\mu _1\mu _2}^a= v_{\mu _1}^{ab} g^{\partial \partial }_{b \mu _2} - v_{\mu _2}^{ab}g^{\partial \partial }_{b\mu _1}=0. \end{aligned}$$

Using now normal geodesic coordinates, we can diagonalize $g^{\partial \partial }$ with eigenvalues on the diagonal $\alpha _{\mu } \in \{1,-1,0\}$:

$$\begin{aligned} {[}v,e]_{\mu _1\mu _2}^a= v_{\mu _1}^{a\mu _2} \alpha _{ \mu _2} - v_{\mu _2}^{a\mu _1} \alpha _{\mu _1}. \end{aligned}$$

Let us now assume that $g^{\partial \partial }$ is nondegenerate and in particular space-like ($\alpha _{\mu }=1$). Then, the components of $\psi _e$ are defined by

$$\begin{aligned} {[}v,e]_{12}^1= v_{1}^{12},{} & {} [v,e]_{12}^3=v_{1}^{32} - v_{2}^{31}, \\ {[}v,e]_{12}^2= v_{2}^{12},{} & {} [v,e]_{12}^4=v_{1}^{42} - v_{2}^{41}. \end{aligned}$$

We can now analyze part of the constraints (10). At the beginning, we just consider the classical part of them (i.e., we assume $c=\xi =\xi ^m=\lambda =0$). The results will then straightforwardly generalize to the complete case.

Lemma 63

The constraints

$$\begin{aligned} \epsilon _n \textrm{d}_{\omega }e =e\sigma ,{} & {} \epsilon _n \textrm{d}_{\omega _m}e+\epsilon _n \textrm{d}_{\omega }e_m=e\sigma _m + e_m \sigma , \\ e_m \textrm{d}_{\omega }e=eL,{} & {} \epsilon _n L +e_m \sigma + e \sigma _m=0, \end{aligned}$$

fix four components of $\omega$.

Proof

Let us start with the restriction of the boundary constraint to the corner: $\epsilon _n \textrm{d}_{\omega }e=\epsilon _n \textrm{d}e + \epsilon _n [\omega , e] =e\sigma$. Let us denote $Y=\textrm{d}e$. Then, using the results of the previous lemmata, we get that this equation translates into the following equations for components of the fields:

$$\begin{aligned} \omega _1^{32}-\omega _2^{31}= Y_{12}^{3},{} & {} \sigma _{2}^{4}=\omega _1^{12} + Y_{12}^1,{} & {} \sigma _{1}^{4}=-\omega _2^{12} + Y_{12}^2,\\ \sigma _{1}^{3}=0,{} & {} \sigma _{2}^{3}=0,{} & {} \sigma _{1}^{1}+\sigma _{2}^{2}=0. \end{aligned}$$

The part transversal to the corner of the boundary structural constraint is $\epsilon _n \textrm{d}_{\omega _m}e+\epsilon _n \textrm{d}_{\omega }e_m=e\sigma _m + e_m \sigma$. On the corner, it is a dynamical equation but also introduces some relations between the components of $\sigma$ and $\sigma _m$. These are

$$\begin{aligned} \sigma _{m}^{2}=0,{} & {} \sigma _{m}^{1}=0,{} & {} \sigma _{1}^{2}=0,\\ \sigma _{2}^{1}=0,{} & {} \sigma _{m}^{3}+\sigma _{1}^{1}=0,{} & {} \sigma _{m}^{3}+\sigma _{2}^{2}=0. \end{aligned}$$

In a similar way, we get the following equations for the components from the equation $e_m \textrm{d}_{\omega }e=e_m \textrm{d}e + e_m [\omega , e] =eL$:

$$\begin{aligned} \omega _1^{24}-\omega _2^{14}= Y_{12}^{4},{} & {} L_{2}^{3}=\omega _1^{12} + Y_{12}^1,{} & {} L_{1}^{3}=-\omega _2^{12} + Y_{12}^2,\\ L_{1}^{4}=0,{} & {} L_{2}^{4}=0,{} & {} L_{1}^{1}+L_{2}^{2}=0. \end{aligned}$$

Lastly, we consider the constraint $\epsilon _n L +e_m \sigma + e \sigma _m=0.$ In components, we obtain some equations proportional to the previous ones and the following:

$$\begin{aligned} \sigma _{1}^{4}+L_1^3=0,{} & {} \sigma _{2}^{4}+L_2^3=0,{} & {} L_{1}^{2}=0,\\ L_{2}^{1}=0,{} & {} \sigma _{m}^{4}-L_{1}^{1}=0,{} & {} \sigma _{m}^{4}-L_{2}^{2}=0. \end{aligned}$$

Collecting all the information, we get the following equations for the components of $\omega :$

$$\begin{aligned} \omega _1^{32}-\omega _2^{31}= Y_{12}^{3}{} & {} \omega _1^{24}-\omega _2^{14}= Y_{12}^{4}{} & {} \omega _1^{12} + Y_{12}^1=0{} & {} \omega _2^{12} + Y_{12}^2=0. \end{aligned}$$

$\square$

To generalize this result to the case where also the ghosts are present, it is sufficient to modify the definitions of $\sigma , \sigma _m, L$, and Y. The components fixed will not change, but they will be fixed to a different combination of the other fields.

Let us now consider the two constraints $\gamma _m^{\dag }= eK$ and $\epsilon _n K=0.$

Lemma 64

The constraints (10a) and (10c) fix four components of the field $\gamma _m^{\dag }$.

Proof

In components, (10a) corresponds to the following relations:

$$\begin{aligned} (\gamma _m^{\dag })_{12}^{12}=K_1^1+ K_2^2,{} & {} (\gamma _m^{\dag })_{12}^{13}=K_2^3,{} & {} (\gamma _m^{\dag })_{12}^{14}= K_2^4,\\ (\gamma _m^{\dag })_{12}^{23}= -K_1^3,{} & {} (\gamma _m^{\dag })_{12}^{24}=-K_1^4,{} & {} (\gamma _m^{\dag })_{12}^{34}=0. \end{aligned}$$

On the other hand, (10c) correspond to the following relations:

$$\begin{aligned} K_1^1=0,{} & {} K_1^3=0,{} & {} K_1^2=0,{} & {} K_2^1=0,{} & {} K_2^3=0,{} & {} K_2^2=0. \end{aligned}$$

Hence, combining the two sets of equations, we get four equations for the components of $\gamma _m^{\dag }$:

$$\begin{aligned} (\gamma _m^{\dag })_{12}^{12}=0,{} & {} (\gamma _m^{\dag })_{12}^{13}=0,{} & {} (\gamma _m^{\dag })_{12}^{23}=0,{} & {} (\gamma _m^{\dag })_{12}^{34}=0. \end{aligned}$$

$\square$

Appendix D: Results About the Push-Forward of Hamiltonian Vector Fields

In this appendix, we present some technical results that are useful to push-forward the Hamiltonian vector field $Q^{\partial }$ from the boundary to the corner. Since the expression (6) of $Q^{\partial }$ contains nonexplicit terms involving the function $(W_{\partial }^{(i,j)})^{-1}$, we must find a way to invert it.

Lemma 65

Let $\widetilde{\gamma }\in \Omega _{\partial }^{i,j}$ and $\widetilde{X} \in \Omega _{\partial }^{i+1,j+1}$ be such that $\widetilde{\gamma }= (W_{\partial }^{(i,j)})^{-1}(\widetilde{X})$. If we let $\widetilde{e}=e|_{\Gamma } + e_m \textrm{d}x^m$, $\widetilde{\gamma }= \gamma |_{\Gamma } + \gamma _m \textrm{d}x^m$, and $\widetilde{X}= X|_{\Gamma }+ X_m \textrm{d}x^m$, then we have

$$\begin{aligned} \gamma |_{\Gamma }&= (W_{\partial \partial }^{(i,j)})^{-1}(\pi _I (X|_{\Gamma })),\\ \gamma _m&= (W_{\partial \partial }^{,(i-1,j)})^{-1}(\pi _I(-e_m (W_{\partial \partial }^{(i,j)})^{-1}(\pi _I (X|_{\Gamma })) + X_m)). \end{aligned}$$

Proof

Omitting the restriction to the corner, we have that

$$\begin{aligned} \widetilde{e} \widetilde{\gamma }= (e + e_m \textrm{d}x^m)(\gamma + \gamma _m \textrm{d}x^m) = X+ X_m \textrm{d}x^m = \widetilde{X}. \end{aligned}$$

This equation splits into two subequations, containing $\textrm{d}x^m$ or not:

$$\begin{aligned} e \gamma = X, \qquad \qquad e \gamma _m + e_m \gamma = X_m. \end{aligned}$$

From the first, we deduce $\gamma = (W_{\partial \partial }^{(i,j)})^{-1}(\pi _I (X))$, while from the second we find

$$\begin{aligned} \gamma _m = (W_{\partial \partial }^{(i-1,j)})^{-1}(\pi _I(-e_m (W_{\partial \partial }^{(i,j)})^{-1}(\pi _I (X)) + X_m)), \end{aligned}$$

where $\pi _I$ stands for the projection to the image of the map $W_{\partial \partial }^{(i,j)}$. $\square$

Remark 66

One has to be careful here because the map $W_{\partial \partial }^{(i,j)}$ can be noninvertible. Hence, technically here we are finding the values of $\gamma$ and $\gamma _m$ up to terms in the kernel of the map $W_{\partial \partial }^{(i,j)}$, and we need to keep using the projection $\pi _I$ at all times.

As an example, we consider $Q^\partial \omega$: it contains a term of the form $\lambda (W_{\partial }^{(1,2)})^{-1}(\epsilon _n F_\omega )$. Here, $X= \epsilon _n F_\omega$. Hence, we have

$$\begin{aligned} \widetilde{Q}^{\partial \partial } \omega&= \dots + (W_{\partial \partial }^{( 1,2)})^{-1}(\epsilon _n F_\omega ), \\ \widetilde{Q}^{\partial \partial } \omega _m&= \dots + (W_{\partial \partial }^{( 0,2)})^{-1}(\pi _I(-e_m (W_{\partial \partial }^{( 1,2)})^{-1}(\epsilon _n F_\omega ) + \epsilon _n F_{\omega _m})) + K, \end{aligned}$$

where $e K=0$. Notice that since $W_{\partial \partial }^{(1,2)}$ is surjective on $\Omega _{\partial \partial }^{1,2}$, we do not need the projection on $\epsilon _n F_\omega$, while, since the map $W_{\partial \partial }^{( 0,2)}$ is neither surjective nor injective on $\Omega _{\partial \partial }^{0,2}$, we need the projection $\pi _I$ on the second expression and we still miss something in the kernel of $W_{\partial \partial }^{( 0,2)}$, denoted by K.

A similar procedure is needed also for $Q^{\partial }{y^{\dag }}$. On the boundary, we have

$$\begin{aligned} \widetilde{e}_i \widetilde{Q^{\partial }{y^{\dag }}}= \lambda \widetilde{\sigma _i} \widetilde{y}^{\dag } + \widetilde{\mu }\widetilde{\gamma _i}^\dag \end{aligned}$$

for $i = a, m$. Hence, since $y_m^\dag$ is a top form on the boundary, we get

$$\begin{aligned} e_m Q^{\partial }{y^{\dag }_m} \textrm{d}x^m&= \lambda \sigma _m y^\dag _m \textrm{d}x^m + \mu _m \textrm{d}x^m \gamma _m^{\dag }, \\ e_a Q^{\partial }{y^{\dag }_m} \textrm{d}x^m&= \lambda \sigma _a y^\dag _m \textrm{d}x^m + \mu \gamma _{am}^{\dag } \textrm{d}x^m, \end{aligned}$$

from which we can easily deduce the expression of $\widetilde{Q}^{\partial }$ on the pre-corner.