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Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces

Abstract

We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables constructed in Mnev (Lett Math Phys 105:1735–1783, 2015) using the Batalin–Vilkovisky structure of AKSZ theories to a formal global version with methods of formal geometry. We will consider the case where the AKSZ theory is “split” which will give an explicit construction for formal vector fields on base and fiber within the formal global action. Moreover, we consider the example of formal global generalized Wilson surface observables whose expectation values are invariants of higher-dimensional knots by using BF field theory. These constructions give rise to interesting global gauge conditions such as the differential quantum master equation and further extensions.

Introduction

Observables play a fundamental role in theoretical and mathematical physics. They are used in several constructions, e.g., deformation quantization and factorization algebras. In [46], a method for constructing observables in the setting of AKSZ theories was introduced, where several examples, including Wilson-loop-type observables for different theories, have been addressed.

These constructions were given using the approach of supersymmetric field theory and methods of functional integrals. In particular, the focus lies within a special formalism dealing with gauge theories which is called the Batalin–Vilkovisky (BV) formalism. This formalism was developed by Batalin and Vilkovisky in a series of papers [5, 7, 8] during the 1970s and 1980s in order to deal with the functional integral quantization approach where the Lagrangian is invariant under certain symmetries and the integral is ill-defined. They have shown (later also formulated in a more mathematical language by Schwarz) that these issues can be resolved by replacing the ill-defined integral by a well-defined (after some regularization is also introduced) one without changing the final value. The mathematical structures of this powerful formalism have been studied since then by many different people.

AKSZ theories [1] (named after Alexandrov, Kontsevich, Schwarz and Zaboronsky) are a particular type of field theories where the space of fields is given by a mapping space between manifolds. It can be shown that these theories, regarded in a special setting, will give rise to field theories as formulated in the BV setting. Many interesting theories are in fact of AKSZ-type, e.g., Chern–Simons theory [3, 4, 18, 22, 58], the Poisson sigma model [16, 35, 51], Rozansky–Witten theory [50], the Courant sigma model [49], BF theory [20, 47], Witten’s A- and B-twisted sigma models [59] and 2D Yang–Mills theory [36].

The globalization idea originates from a field-theoretic approach to globalization of Kontsevich’s star product [39] in deformation quantization. The associated field theory is given by the Poisson sigma model. The Poisson sigma model is a two-dimensional bosonic string theory with target a Poisson manifold which was first considered by Ikeda [35] and Schaller–Strobl [51] by the attempt of studying 2D gravity theories and combine them to a common form with Yang–Mills theories. Using the Poisson sigma model on the disk, Cattaneo and Felder have proven that Kontsevich’s star product is exactly given by the perturbative expansion of its functional integral quantization [15]. Regarding the fact that the Poisson sigma model is a gauge theory, it is interesting to note that it is a fundamental non-trivial theory where the BRST gauge formalism [9,10,11, 56] does not work if the Poisson structure is not linear. In fact, to treat the Poisson sigma model and its quantization, one has to use the BV formalism. However, the field-theoretic construction of Kontsevich’s star product was only considered locally since Kontsevich’s formula was only given for the local picture on the upper half-plane. Later on, using techniques of formal geometry, developed by, e.g., Gelfand–Fuks [32, 33], Gelfand–Kazhdan [34] or Bott [13], it was possible to construct a globalization, similar to the approach of Fedosov for symplectic manifolds which only covers the case of constant (symplectic) Poisson structures [30].

In [12, 17], this approach was first extended to the field theoretic BV construction of the Poisson sigma model for closed source manifolds. In recent work [25] this construction was extended to the case of source manifolds with boundary. There one has to extend the BV formalism to the BV-BFV formalism which couples the boundary BFV theory to the bulk BV theory such that everything is consistent in the cohomological formalism. Here BFV stands for Batalin–Fradkin–Vilkovisky which formulated a Hamiltonian version of the BV construction in [6, 31]. The bulk-boundary coupling (the BV-BFV formalism) was first introduced classically in [19, 20] and extended to the quantum version in [21]. The globalization construction for the Poisson sigma model on manifolds with boundary was more generally extended in [24] to a special class of AKSZ theories which are called “split” where the case of the Poisson sigma model is an example.

The aim of this paper is to extend the constructions of [46] to a formal global construction. In fact, we will construct formal global observables by using the notion of a Hamiltonian Q-bundle [41] together with notions of formal geometry, and we will study the formal global extension of Wilson loop type observables for the Poisson sigma model.

Additionally, we discuss the formal global extension of Wilson surface observables which have been studied in [27] by using the AKSZ formulation of BF theories. We will show that these constructions lead to interesting gauge conditions such as the differential quantum master equation (and further extensions).

These constructions are expected to extend to manifolds with boundary by using the BV-BFV formalism as the globalization constructions have been studied for nonlinear split AKSZ theories on manifolds with boundary [24].

The Batalin–Vilkovisky (BV) Formalism

In this section, we will recall some aspects of the Batalin–Vilkovisky formalism as in [21, 48]. An introductory reference for learning about the formalism is [23], which also covers the most important concepts of supergeometry and the case of manifolds with boundary (BV-BFV).

Classical BV Picture

Let us start with the classical setting of the BV formalism.

Definition 2.1

(BV manifold). A BV manifold is a triple

$$ ({\mathcal {F}},{\mathcal {S}},\omega ) $$

such that \({\mathcal {F}}\) is a \({\mathbb {Z}}\)-graded supermanifoldFootnote 1, \({\mathcal {S}}\) is an even function on \({\mathcal {F}}\) of degree 0 and \(\omega \) is an odd symplectic form on \({\mathcal {F}}\) of degree \(-1\). Moreover, we want that \({\mathcal {S}}\) satisfies the Classical Master Equation (CME)

$$\begin{aligned} \boxed { \{{\mathcal {S}},{\mathcal {S}}\}_\omega =0,} \end{aligned}$$
(1)

where \(\{,\}_{\omega }\) denotes the odd Poisson bracket induced by the odd symplectic form \(\omega \). This odd Poisson bracket is also called BV bracket and, according to Batalin and Vilkovisky, is often denoted by round brackets \((,)\). We will call \({\mathcal {F}}\) the BV space of fieldsFootnote 2, \({\mathcal {S}}\) the BV action (sometimes also called the master action) and \(\omega \) the BV symplectic form.

Remark 2.2

In physics, the \({\mathbb {Z}}\)-grading is called the ghost number. We will denote the ghost number by \({\mathrm {gh}}\) and the form degree by \(\deg \).

Remark 2.3

The data of a BV manifold induces a symplectic cohomological vector field Q of degree \(+1\) which is given by the Hamiltonian vector field of \({\mathcal {S}}\), i.e.,

$$\begin{aligned} \iota _Q\omega =\delta {\mathcal {S}}, \end{aligned}$$
(2)

wher \(\delta \) denotes the de Rham differential on \({\mathcal {F}}\). The cohomological property means that \([Q,Q]=0\) and the symplectic property means \(L_Q\omega =0\), where L denotes the Lie derivative. Moreover, note that by definition

$$ Q=\{{\mathcal {S}},\}_\omega . $$

Definition 2.4

(Exact BV manifold). A BV manifold is called exact if \(\omega =\delta \alpha \) for some primitive 1-form \(\alpha \).

In what will follow, we will mostly consider exact BV manifolds. According to the use of sigma models we want to consider space-time manifolds as the source manifolds for our theory. Moreover, in this paper, we will restrict ourself to topological theories.

Definition 2.5

(BV theory). A BV theory is an assignment of a manifold \(\Sigma \) to a BV manifold

$$\begin{aligned} \Sigma \mapsto ({\mathcal {F}}_\Sigma ,{\mathcal {S}}_\Sigma ,\omega _\Sigma ,Q_\Sigma ). \end{aligned}$$
(3)

Quantum BV Picture

We continue with the quantum setting of the BV formalism.

Definition 2.6

(Quantum BV manifold). A quantum BV manifold is a quadruple \(({\mathcal {F}},\omega ,\mu ,{\mathcal {S}})\) such that \({\mathcal {F}}\) is a \({\mathbb {Z}}\)-graded supermanifold, \(\omega \) a symplectic form on \({\mathcal {F}}\) of degree \(-1\), \(\mu \) a volume elementFootnote 3 of \({\mathcal {F}}\) which is compatible with \(\omega \) in the sense that the associated BV Laplacian

$$\begin{aligned} \Delta :f\mapsto \frac{1}{2}{{\,\mathrm{div}\,}}_{\mu }\{f,\}_\omega \end{aligned}$$
(4)

satisfies

$$\begin{aligned} \Delta ^2=0, \end{aligned}$$
(5)

and \({\mathcal {S}}\) is a degree 0 function on \({\mathcal {F}}\) such that it satisfies the QME (8).

Remark 2.7

The BV Laplacian satisfies a generalized BV Leibniz rule. For two functions fg on \({\mathcal {F}}\), we have

$$\begin{aligned} \Delta (fg)=\Delta (f)g\pm f\Delta (g)\pm \{f,g\}_\omega . \end{aligned}$$

see also [38, 53] for a mathematical exposure to the origin of the BV Laplacian.

Moreover, define \(\delta _{{\mathrm {BV}}}\) to be the degree \(+1\) operator given by

$$\begin{aligned} \delta _{{\mathrm {BV}}}:=Q-{\mathrm {i}}\hbar \Delta \end{aligned}$$
(6)

which satisfies

$$\begin{aligned} \delta _{{\mathrm {BV}}}^2=0. \end{aligned}$$
(7)

The following theorem is one of the main statements in the formalism developed by Batalin and Vilkovisky. In its present form, it was stated by Schwarz on general manifolds [52].

Theorem 2.8

(Batalin–Vilkovisky). For any half-density f on \({\mathcal {F}}\), we have:

  1. (1)

    If \(f=\Delta g\), then

    $$ \int _{{\mathcal {L}}}f=0, $$

    for a Lagrangian submanifold \({\mathcal {L}}\subset {\mathcal {F}}\),

  2. (2)

    If \(\Delta f=0\), then

    $$ \frac{{\mathrm {d}}}{{\mathrm {d}}t}\int _{{\mathcal {L}}_t}f=0, $$

    for any continuous family \(({\mathcal {L}}_t)\) of Lagrangian submanifolds of \({\mathcal {F}}\).

Remark 2.9

The choice of Lagrangian submanifold is in fact equivalent to fixing a gauge. The second part of Theorem 2.8 tells us that if we have an integral over a Lagrangian submanifold which is ill-defined, but on the other hand \(\Delta f=0\), then we can deform the Lagrangian submanifold \({\mathcal {L}}\) continuously to a Lagrangian submanifold \({\mathcal {L}}'\) (choosing a different gauge) where the integral is well-defined. In application to quantum field theory, we have \(f={\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}}\). Hence, for gauge independence, we need to impose

$$\begin{aligned} \boxed { \Delta {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}}=0\Longleftrightarrow \{{\mathcal {S}},{\mathcal {S}}\}_\omega -2{\mathrm {i}}\hbar \Delta {\mathcal {S}}=0.} \end{aligned}$$
(8)

The condition (8) is called the Quantum Master Equation (QME). If we let \({\mathcal {S}}\) depend on \(\hbar \), we can see that in order zero we get the CME \(\{{\mathcal {S}},{\mathcal {S}}\}_\omega =0\). One can then solve (8) order by order.

\(L_\infty \)-Structure

Recall that a Q-manifold with trivial body induces an \(L_\infty \)-algebra structure (see, e.g., [45]). More generally, a Q-manifold with non-trivial body induces an \(L_\infty \)-algebroid structure. Similarly, a BV manifold endowed with its Q-structure induces an \(L_\infty \)-algebra structure on \({\mathcal {F}}\) [55]. This \(L_\infty \)-algebra encodes all the relevant classical information of the field theory. Hence, at the classical level, Lagrangian field theories can be equivalently described in terms of the underlying (cyclicFootnote 4) \(L_\infty \)-algebra structure. Moreover, equivalent theories induce quasi-isomorphic \(L_\infty \)-algebras. The unary operation \(\ell _1\) is in fact encoded in the linear part of the action \(Q=\{{\mathcal {S}},\}_\omega \) on the field corresponding to the image of \(\ell _1\). The higher brackets then make the linearized expressions covariant and to allow for higher interaction terms. The operator \(\delta _{\mathrm {BV}}\) in fact induces a quantum \(L_\infty \)-algebra (or loop homotopy algebra) on the same graded space. In particular, by a direct application of the homological perturbation lemma, one can prove a similar decomposition theorem and compute its minimal model as for the classical case, which leads directly to a homotopy between a quantum \(L_\infty \)-algebra and its minimal model in which the non-triviality of the action is fully absorbed in the higher brackets. Moreover, the homotopy Maurer–Cartan theoryFootnote 5 implies that for an arbitrary \(L_\infty \)-algebra the BV complex of fields, ghosts and anti fields is just the \(L_\infty \)-algebra itself. See, e.g., [37, 54, 55] for a more detailed discussion of \(L_\infty \)-structures for BV field theories.

AKSZ Theories

Preliminaries

In [1], Alexandrov, Kontsevich, Schwarz, and Zaboronsky have proposed a class of local field theories which are compatibel with the Batalin–Vilkovisky gauge formalism construction, in the sense that the constructed local actions are solutions to the Classical Master Equation. Hence, these theories give a subclass of BV theories. In this section we want to recall the most important notions of AKSZ sigma models. We start with defining the ingredients.

Definition 3.1

(Differential graded symplectic manifold). A differential graded symplectic manifold of degree k is a triple

$$ ({\mathcal {M}},\Theta _{\mathcal {M}},\omega _{\mathcal {M}}={\mathrm {d}}_{{\mathcal {M}}}\alpha _{\mathcal {M}}) $$

such that \({\mathcal {M}}\) is a \({\mathbb {Z}}\)-graded manifold, \(\Theta _{\mathcal {M}}\in C^\infty ({\mathcal {M}})\) is a function on \({\mathcal {M}}\) of degree \(k+1\), and \(\omega _{\mathcal {M}}\in \Omega ^2({\mathcal {M}})\) is an exact symplectic form of degree k with primitive 1-form \(\alpha _{\mathcal {M}}\in \Omega ^1({\mathcal {M}})\), such that

$$\begin{aligned} \left\{ \Theta _{\mathcal {M}},\Theta _{\mathcal {M}}\right\} _{\omega _{\mathcal {M}}}=0, \end{aligned}$$
(9)

where \(\{,\}_{\omega _{\mathcal {M}}}\) is the odd Poisson bracket induced by \(\omega _{\mathcal {M}}\). We have denoted by \({\mathrm {d}}_{{\mathcal {M}}}\) the de Rham differential on \({\mathcal {M}}\).

Remark 3.2

We denote by \(Q_{\mathcal {M}}\in {\mathfrak {X}}({\mathcal {M}})\) the Hamiltonian vector field of \(\Theta _{\mathcal {M}}\), defined by the equation

$$ \iota _{Q_{\mathcal {M}}}\omega _{\mathcal {M}}={\mathrm {d}}_{{\mathcal {M}}}\Theta _{\mathcal {M}}$$

with the properties \([Q_{\mathcal {M}},Q_{\mathcal {M}}]=0\) (cohomological) and \(L_{Q_{\mathcal {M}}}\omega _{\mathcal {M}}=0\) (symplectic). Note that \(Q_{\mathcal {M}}\) is of degree \(+1\). A quadruple \(({\mathcal {M}},Q_{\mathcal {M}},\Theta _{\mathcal {M}},\omega _{\mathcal {M}}={\mathrm {d}}_{{\mathcal {M}}}\alpha _{\mathcal {M}})\) as in Definition 3.1 is also called a Hamiltonian Q-manifold.

AKSZ Sigma Models

Let \(\Sigma _d\) be a d-dimensional compact, oriented manifold (possibly with boundary) and consider its shifted tangent bundle \(T[1]\Sigma _d\). Moreover, fix a Hamiltonian Q-manifold

$$ ({\mathcal {M}},Q_{{\mathcal {M}}},\Theta _{{\mathcal {M}}},\omega _{{\mathcal {M}}}={\mathrm {d}}_{{\mathcal {M}}}\alpha _{{\mathcal {M}}}) $$

of degree \(d-1\) for \(d\ge 0\). We can consider the mapping space of graded manifolds from \(T[1]\Sigma _d\) to \({\mathcal {M}}\) to be our space of fields:

$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}:={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,{\mathcal {M}}), \end{aligned}$$
(10)

where \({{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}\) denotes the mapping space between graded manifolds.Footnote 6 We would like to endow \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\) with a Q-manifold structure. This can be done by considering the lifts of the de Rham differential \({\mathrm {d}}_{\Sigma _d}\) on \(\Sigma _d\) and the cohomological vector field \(Q_{\mathcal {M}}\) on the target \({\mathcal {M}}\) to the mapping space. Hence, we get a cohomological vector field

$$\begin{aligned} Q_{\Sigma _d}:={\widehat{{\mathrm {d}}}}_{\Sigma _d}+{\widehat{Q}}_{{\mathcal {M}}}\in {\mathfrak {X}}\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) , \end{aligned}$$
(11)

where \({\widehat{{\mathrm {d}}}}_{\Sigma _d}\) and \({\widehat{Q}}_{{\mathcal {M}}}\) denote the corresponding lifts to the mapping space. Note that we can regard \({\mathrm {d}}_{\Sigma _d}\) as a cohomological vector field on \(T[1]\Sigma _d\). Consider the following push-pull diagram

$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\xleftarrow {{\mathrm {p}}} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\times T[1]\Sigma _d\xrightarrow {{\mathrm {ev}}} {\mathcal {M}}, \end{aligned}$$
(12)

where \({\mathrm {p}}\) denotes the projection onto \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\) and \({\mathrm {ev}}\) is the evaluation map. We can construct a transgression map

$$\begin{aligned} {\mathscr {T}}_{\Sigma _d}:={\mathrm {p}}_*{\mathrm {ev}}^*:\Omega ^\bullet ({\mathcal {M}})\rightarrow \Omega ^\bullet \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) . \end{aligned}$$
(13)

Note that the map \({\mathrm {p}}_*\) is given by fiber integration on \(T[1]\Sigma _d\). Now we can endow the space of fields \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\) with a symplectic structure \(\omega _{\Sigma _d}\) by setting

$$\begin{aligned} \omega _{\Sigma _d}:=(-1)^{d}{\mathscr {T}}_{\Sigma _d}(\omega _{{\mathcal {M}}})\in \Omega ^2\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) . \end{aligned}$$
(14)

Moreover, we will get a solution \({\mathcal {S}}_{\Sigma _d}\) to the CME, the BV action functional, by

$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}:=\underbrace{\iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _d}}{\mathscr {T}}_{\Sigma _d}(\alpha _{{\mathcal {M}}})}_{=:{\mathcal {S}}^{\mathrm {kin}}_{\Sigma _d}} +\underbrace{{\mathscr {T}}_{\Sigma _d}(\Theta _{{\mathcal {M}}})}_{=:{\mathcal {S}}^{\mathrm {target}}_{\Sigma _d}}\in C^\infty \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) . \end{aligned}$$
(15)

Indeed, one can check that

$$\begin{aligned} \left\{ {\mathcal {S}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d}\right\} _{\omega _{\Sigma _d}}=0. \end{aligned}$$
(16)

Note that the symplectic form \(\omega _{\Sigma _d}\) is of degree \((d-1)-d=-1\) as expected. Moreover, the action \({\mathcal {S}}_{\Sigma _d}\) is of degree 0. Thus this setting does indeed induce a BV manifold \(\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d},\omega _{\Sigma _d}\right) \). Consider local coordinates \((x^\mu )\) on \({\mathcal {M}}\) and let \((u^i)\) be local coordinates on \(\Sigma _d\) for \(1\le i\le d\). Denote the odd fiber coordinates of degree \(+1\) on \(T[1]\Sigma _d\) by \(\theta ^i={\mathrm {d}}_{\Sigma _d} u^i\). Then, for a field \({\mathcal {A}}\in {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\), we have the local expression

$$\begin{aligned} {\mathcal {A}}^\mu (u,\theta )= & {} \sum _{\ell =0}^d\,\,\underbrace{\sum _{1\le i_1<\cdots <i_\ell \le d}{\mathcal {A}}^\mu _{i_1\ldots i_\ell }(u)\theta ^{i_1}\wedge \cdots \wedge \theta ^{i_\ell }}_{{\mathcal {A}}^\mu _{(\ell )}(u,\theta )} \nonumber \\\in & {} \bigoplus _{\ell =0}^d C^\infty (\Sigma _d)\otimes \bigwedge ^\ell T^*\Sigma _d. \end{aligned}$$
(17)

The functions \({\mathcal {A}}^\mu _{i_1\ldots i_\ell }\in C^\infty (\Sigma _d)\) are of degree \(\deg (x^\mu )-\ell \) on \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\). The local expression of the symplectic form \(\omega _{{\mathcal {M}}}\) and its primitive 1-form \(\alpha _{{\mathcal {M}}}\) on \({\mathcal {M}}\) are given by

$$\begin{aligned} \alpha _{{\mathcal {M}}}&=\alpha _{\mu }(x){\mathrm {d}}_{{\mathcal {M}}}x^\mu \in \Omega ^1({\mathcal {M}}), \end{aligned}$$
(18)
$$\begin{aligned} \omega _{{\mathcal {M}}}&=\frac{1}{2}\omega _{\mu _1\mu _2}(x){\mathrm {d}}_{{\mathcal {M}}}x^{\mu _1}\wedge {\mathrm {d}}_{{\mathcal {M}}}x^{\mu _2}\in \Omega ^2({\mathcal {M}}). \end{aligned}$$
(19)

Locally, using the expressions above, we get the following expression for the BV symplectic form, its primitive 1-form and the BV action functional:

$$\begin{aligned} \alpha _{\Sigma _d}&=\int _{\Sigma _d}\alpha _\mu ({\mathcal {A}})\delta {\mathcal {A}}^\mu \in \Omega ^1\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) ,\end{aligned}$$
(20)
$$\begin{aligned} \omega _{\Sigma _d}&=(-1)^d\frac{1}{2}\int _{\Sigma _d}\omega _{\mu _1\mu _2}({\mathcal {A}})\delta {\mathcal {A}}^{\mu _1}\wedge \delta {\mathcal {A}}^{\mu _2}\in \Omega ^2\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) , \end{aligned}$$
(21)
$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}&=\int _{\Sigma _d}\alpha _\mu ({\mathcal {A}}){\mathrm {d}}_{\Sigma _d}{\mathcal {A}}^\mu +\int _{\Sigma _d}\Theta _{{\mathcal {M}}}({\mathcal {A}})\in C^\infty \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) . \end{aligned}$$
(22)

Note that we have denoted by \(\delta \) the de Rham differential on \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\). If we consider Darboux coordinates on \({\mathcal {M}}\), we get that

$$\begin{aligned}\omega _{\mathcal {M}}=\frac{1}{2}\omega _{\mu _1\mu _2}{\mathrm {d}}_{\mathcal {M}}x^{\mu _1}\wedge {\mathrm {d}}_{{\mathcal {M}}} x^{\mu _2}, \end{aligned}$$

where the \(\omega _{\mu _1\mu _2}\) are constant implying that \(\alpha _{\mathcal {M}}=\frac{1}{2}x^{\mu _1}\omega _{\mu _1\mu _2}{\mathrm {d}}_{\mathcal {M}}x^{\mu _2}\). Hence we get the BV symplectic form

$$\begin{aligned} \omega _{\Sigma _d}= & {} \frac{1}{2}\int _{T[1]\Sigma _d}\mu _{\Sigma _d}\left( \omega _{\mu _1\mu _2}\delta {\mathcal {A}}^{\mu _1} \wedge \delta {\mathcal {A}}^{\mu _2}\right) \nonumber \\= & {} \frac{1}{2}\int _{\Sigma _d}\left( \omega _{\mu _1\mu _2}\delta {\mathcal {A}}^{\mu _1}\wedge \delta {\mathcal {A}}^{\mu _2}\right) ^{\mathrm {top}} \end{aligned}$$
(23)

and the master action

$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}=\int _{T[1]\Sigma _d}\mu _{\Sigma _d}\left( \frac{1}{2}{\mathcal {A}}^\mu \omega _{\mu _1\mu _2}{\varvec{D}}_{\Sigma _d}{\mathcal {A}}^{\mu _2}\right) +(-1)^d\int _{T[1]\Sigma _d}\mu _{\Sigma _d}{\mathcal {A}}^*\Theta _{\mathcal {M}}, \end{aligned}$$
(24)

where \(\mu _{\Sigma _d}\) is a canonical measure on \(T[1]\Sigma _d\) and \({\varvec{D}}_{\Sigma _d}=\theta ^j\frac{\partial }{\partial u_j}\) the superdifferential on \(T[1]\Sigma _d\).

Hamiltonian Q-Bundles

We want to construct a combination of the notion of Q-manifolds and the concept of Hamiltonian vector fields together with the notion of vector bundles, where we want to extend most of our constructions on the fiber (see also [41]). We will see that the fiber will represent the target of an AKSZ theory for an embedded source manifold when lifted to an AKSZ-BV theory. We will call the fiber theory auxiliary. In this section we will give the main definitions as in [46]. Let us start with the definition of the trivial case.

Definition 4.1

(Trivial Q-bundle). Let \({\mathcal {N}}\) be a graded manifold and \(({\mathcal {M}},Q_{\mathcal {M}})\) a graded Q-manifold. A trivial Q-bundle is a trivial bundle

$$\begin{aligned} \pi :{\mathcal {E}}:={\mathcal {M}}\times {\mathcal {N}}\rightarrow {\mathcal {M}}\end{aligned}$$
(25)

such that \({\mathrm {d}}\pi (Q_{\mathcal {E}})=Q_{\mathcal {M}}\), where \(Q_{\mathcal {E}}\) denotes the Q-structure on the total space \({\mathcal {E}}\).

Remark 4.2

Note that this implies that

$$\begin{aligned} Q_{\mathcal {E}}=Q_{\mathcal {M}}+{\mathcal {V}}, \end{aligned}$$

where \({\mathcal {V}}\in \ker {\mathrm {d}}\pi \cong C^\infty ({\mathcal {M}}){\widehat{\otimes }}{\mathfrak {X}}({\mathcal {N}})\) denotes the vertical part of \(Q_{\mathcal {E}}\). The fact that \([Q_{\mathcal {E}},Q_{\mathcal {E}}]=0\) can be translated to

$$\begin{aligned} \underbrace{[Q_{\mathcal {M}},Q_{\mathcal {M}}]}_{=0}+[Q_{\mathcal {M}},{\mathcal {V}}]+\frac{1}{2}[{\mathcal {V}},{\mathcal {V}}]=0. \end{aligned}$$
(26)

Definition 4.3

(Trivial Hamiltonian Q-bundle). A trivial Hamiltonian Q-bundle of degree \(n\in {\mathbb {Z}}\) is a trivial Q-bundle

$$\begin{aligned} \pi :{\mathcal {E}}:={\mathcal {M}}\times {\mathcal {N}}\rightarrow {\mathcal {M}}\end{aligned}$$

as in Definition 4.1 with \(Q_{\mathcal {E}}=Q_{\mathcal {M}}+{\mathcal {V}}\) such that the fiber \({\mathcal {N}}\) is endowed with an exact symplectic structure \(\omega _{\mathcal {N}}={\mathrm {d}}_{\mathcal {N}}\alpha _{\mathcal {N}}\in \Omega ^2({\mathcal {N}})\) of degree n with \(\alpha _{\mathcal {N}}\in \Omega ^1({\mathcal {N}})\) and a Hamiltonian function \(\Theta _{\mathcal {E}}\in C^\infty ({\mathcal {E}})\) of degree \(n+1\) satisfying

$$\begin{aligned}&{\mathcal {V}}=\{\Theta _{\mathcal {E}},\}_{\omega _{\mathcal {N}}} \end{aligned}$$
(27)
$$\begin{aligned}&Q_{\mathcal {M}}(\Theta _{\mathcal {E}})+\frac{1}{2}\{\Theta _{\mathcal {E}},\Theta _{\mathcal {E}}\}_{\omega _{\mathcal {N}}}=0. \end{aligned}$$
(28)

We can now give the definition of a general Hamiltonian Q-bundle.

Definition 4.4

(Hamiltonian Q-bundle) A Hamiltonian Q-bundle is a Q-bundle \(\pi :{\mathcal {E}}\rightarrow {\mathcal {M}}\) where the total space \({\mathcal {E}}\) is endowed with a degree n exact pre-symplectic form \(\omega _{\mathcal {E}}={\mathrm {d}}_{\mathcal {E}}\alpha _{\mathcal {E}}\) such that \(\ker \omega _{\mathcal {E}}\subset T{\mathcal {E}}\) is transversal to the vertical distribution \(T^{\mathrm {vert}}{\mathcal {E}}\) and hence \(\ker \omega _{\mathcal {E}}\) defines a flat Ehresmann connection \(\nabla _{\omega _{\mathcal {E}}}\). Moreover, there is a Hamiltonian function \(\Theta _{\mathcal {E}}\in C^\infty ({\mathcal {E}})\) with

$$ \iota _{Q_{\mathcal {E}}}\omega _{\mathcal {E}}={\mathrm {d}}^{\mathrm {vert}}_{\mathcal {E}}\Theta _{\mathcal {E}}, $$

where \({\mathrm {d}}_{{\mathcal {E}}}^{\mathrm {vert}}\) denotes the vertical part of the de Rham differential on \({\mathcal {E}}\) as a pullback by the natural inclusion \(T^{\mathrm {vert}}{\mathcal {E}}\hookrightarrow T{\mathcal {E}}\). Finally, we also want that

$$\begin{aligned} \left( Q^{\mathrm {hor}}_{\mathcal {E}}+\frac{1}{2}Q^{\mathrm {vert}}_{\mathcal {E}}\right) (\Theta _{\mathcal {E}})=0, \end{aligned}$$
(29)

where we split \(Q_{\mathcal {E}}=Q^{\mathrm {hor}}_{\mathcal {E}}+Q^{\mathrm {vert}}_{\mathcal {E}}\) into its horizontal and vertical parts by using the Ehresmann connection \(\nabla _{\omega _{\mathcal {E}}}\) defined by \(\omega _{\mathcal {E}}\).

Observables in the BV Formalism

We want to define certain classes of observables arising within the BV construction which are compatible with the structure of an underlying Q-bundle. We will start with the classical setting.

Observables for Classical BV Manifolds

Definition 5.1

(BV classical observable) A classical observable for a BV manifold \(({\mathcal {F}},{\mathcal {S}},Q,\omega )\) is defined as a function \({\mathcal {O}}\in C^\infty ({\mathcal {F}})\) of degree 0 such that

$$\begin{aligned} Q({\mathcal {O}})=0. \end{aligned}$$
(30)

Definition 5.2

(Equivalence of BV classical observables). Two BV classical observables \({\mathcal {O}}\) and \({\widetilde{{\mathcal {O}}}}\) are said to be equivalent if

$$\begin{aligned} {\widetilde{{\mathcal {O}}}}-{\mathcal {O}}=Q(\Psi ),\quad \Psi \in C^\infty ({\mathcal {F}}), \end{aligned}$$
(31)

or equivalently, \({\mathcal {O}}\) and \({\widetilde{{\mathcal {O}}}}\) have the same Q-cohomology class.

Definition 5.3

(BV classical pre-observable). For a classical BV theory

$$ ({\mathcal {F}},{\mathcal {S}},Q,\omega ) $$

we define a pre-observable to be a Hamiltonian Q-bundle over \({\mathcal {F}}\) of degree \(-1\). We denote the fiber by \({\mathcal {F}}^{{\mathrm {aux}}}\) and call them the space of auxiliary fields, which itself is endowed with a symplectic structure \(\omega ^{{\mathrm {aux}}}\) of degree \(-1\) and an action functional \({\mathcal {S}}^{\mathrm {aux}}\in C^\infty ({\mathcal {F}}\times {\mathcal {F}}^{\mathrm {aux}})\) of degree 0 such that

$$\begin{aligned} Q({\mathcal {S}}^{\mathrm {aux}})+\frac{1}{2}\{{\mathcal {S}}^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}}\}_{\omega ^{\mathrm {aux}}}=0. \end{aligned}$$
(32)

Using the notions of quantum BV manifolds as in Definition 2.6, we can define a fiber auxiliary version which is compatible with the Hamiltonian Q-bundle construction as in Definition 4.4.

Definition 5.4

(BV semi-quantum pre-observable). For a classical BV theory \(({\mathcal {F}},{\mathcal {S}},Q,\omega )\) we define a BV semi-quantum pre-observable to be a quadruple

$$ ({\mathcal {F}}^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}}) $$

such that \(\mu ^{\mathrm {aux}}\) is a volume form on \({\mathcal {F}}^{\mathrm {aux}}\) compatible with \(\omega ^{\mathrm {aux}}\), i.e., the associated BV Laplacian on \(C^\infty ({\mathcal {F}}^{\mathrm {aux}})\) given by

$$\begin{aligned} \Delta ^{\mathrm {aux}}:f\mapsto \frac{1}{2}{\mathrm {div}}_{\mu ^{\mathrm {aux}}}\{f,\}_{\omega ^{\mathrm {aux}}} \end{aligned}$$
(33)

satisfies \((\Delta ^{\mathrm {aux}})^2=0\). Moreover, the action functional \({\mathcal {S}}^{\mathrm {aux}}\) satisfies

$$\begin{aligned} Q({\mathcal {S}}^{\mathrm {aux}})+\frac{1}{2}\{{\mathcal {S}}^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}}\}_{\omega ^{\mathrm {aux}}}-{\mathrm {i}}\hbar \Delta ^{\mathrm {aux}}{\mathcal {S}}^{\mathrm {aux}}=0, \end{aligned}$$
(34)

which is equivalent to

$$\begin{aligned} \delta _{\mathrm {BV}}^{\mathrm {aux}}{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathrm {aux}}}:=(Q-{\mathrm {i}}\hbar \Delta ^{\mathrm {aux}}){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathrm {aux}}}=0. \end{aligned}$$
(35)

Remark 5.5

The name “semi-quantum” is chosen since it is not a quantum observable yet, but rather the theory whose functional integral quantization will lead to a quantum observable in the sense that it is closed with respect to the infinitesimal symmetries.

We also want to extend the notion of equivalent pre-observables to the case of semi-quantum pre-observables.

Definition 5.6

(Equivalent BV semi-quantum pre-observables). Two BV semi-quantum pre-observables

$$\begin{aligned} ({\mathcal {F}}^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}})\quad \text {and}\quad ({\mathcal {F}}^{\mathrm {aux}},{\widetilde{{\mathcal {S}}}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}}) \end{aligned}$$

are said to be equivalent if there exists a function \(f^{\mathrm {aux}}\in C^\infty ({\mathcal {F}}\times {\mathcal {F}}^{\mathrm {aux}})\) such that

$$\begin{aligned} {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\widetilde{{\mathcal {S}}}}^{\mathrm {aux}}}-{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathrm {aux}}}=(Q-{\mathrm {i}}\hbar \Delta ^{\mathrm {aux}})\left( {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathrm {aux}}}f^{\mathrm {aux}}\right) . \end{aligned}$$
(36)

Proposition 5.7

([27, 46]). Let \(({\mathcal {F}}^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}})\) be a BV semi-quantum pre-observable. Define

$$\begin{aligned} {\mathcal {O}}_{\mathcal {L}}:=\int _{{\mathcal {L}}\subset {\mathcal {F}}^{\mathrm {aux}}}{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathrm {aux}}}\sqrt{\mu ^{\mathrm {aux}}}\vert _{\mathcal {L}}\in C^\infty ({\mathcal {F}}), \end{aligned}$$
(37)

where \({\mathcal {L}}\subset {\mathcal {F}}^{\mathrm {aux}}\) is a Lagrangian submanifold. Then \({\mathcal {O}}_{\mathcal {L}}\) is an observable, i.e., \(Q({\mathcal {O}}_{\mathcal {L}})=0\). Moreover, if for two Lagrangian submanifolds \({\mathcal {L}}\) and \({\widetilde{{\mathcal {L}}}}\) there exists a homotopy between them, then the observables \({\mathcal {O}}_{\mathcal {L}}\) and \({\mathcal {O}}_{{\widetilde{{\mathcal {L}}}}}\) are equivalent. Also for two equivalent BV semi-quantum pre-observables \({\mathcal {S}}^{\mathrm {aux}}\) and \({\widetilde{{\mathcal {S}}}}^{\mathrm {aux}}\), the corresponding observables \({\mathcal {O}}_{\mathcal {L}}\) and \({\widetilde{{\mathcal {O}}}}_{\mathcal {L}}\) are equivalent.

Definition 5.8

(Good auxiliary splitting). We say that a semi-quantum pre-observable \(({\mathcal {F}}^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}})\) has a good splitting if there is a decomposition

$$ {\mathcal {F}}^{\mathrm {aux}}={\mathsf {F}}^{\mathrm {aux}}\times {\mathscr {F}}^{\mathrm {aux}} $$

such that

$$\begin{aligned} \omega ^{\mathrm {aux}}&=\omega _1^{\mathrm {aux}}+\omega _2^{\mathrm {aux}}, \end{aligned}$$
(38)
$$\begin{aligned} \mu ^{\mathrm {aux}}&=\mu ^{\mathrm {aux}}_1\otimes \mu ^{\mathrm {aux}}_2, \end{aligned}$$
(39)

where \(\omega _1^{\mathrm {aux}}\) is a symplectic form on \({\mathsf {F}}^{\mathrm {aux}}\), \(\omega _2^{\mathrm {aux}}\) is a symplectic form on \({\mathscr {F}}^{\mathrm {aux}}\), \(\mu ^{\mathrm {aux}}_1\) is a volume form on \({\mathsf {F}}^{\mathrm {aux}}\) and \(\mu ^{\mathrm {aux}}_2\) is a volume form on \({\mathscr {F}}^{\mathrm {aux}}\).

Remark 5.9

This is in fact the trivial case. The general version, called hedgehog, is discussed in [21].

Remark 5.10

We split the auxiliary fields into high energy modes \({\mathscr {F}}^{\mathrm {aux}}\) and low energy modes \({\mathsf {F}}^{\mathrm {aux}}\). This splitting can be done by using Hodge decomposition of differential forms into exact, coexact and harmonic forms (see Appendix A of [21]). Note that, in addition, we might also have background fieldsFootnote 7. If \(\Sigma _d\) would have boundary, one can in general split the space of fields into three parts, the low energy fields, the high energy fields and the boundary fields. The boundary fields are generally given by techniques of symplectic reduction as the leaves of a chosen polarization on the boundary. This is the content of the BV-BFV formalism [20, 21, 23].

Proposition 5.11

([46]). Let \(({\mathcal {F}}^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}})\) be a semi-quantum pre-observable with a good splitting. Define \({\mathsf {S}}^{\mathrm {aux}}\in C^\infty ({\mathcal {F}}\times {\mathsf {F}}^{\mathrm {aux}})\) by

$$\begin{aligned} {\mathsf {S}}^{\mathrm {aux}}=-{\mathrm {i}}\hbar \log \int _{{\mathscr {L}}\subset {\mathscr {F}}^{\mathrm {aux}}}{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathrm {aux}}}\sqrt{\mu _2^{\mathrm {aux}}}\big |_{{\mathscr {L}}}, \end{aligned}$$
(40)

where \({\mathscr {L}}\) is a Lagrangian submanifold of \({\mathscr {F}}^{\mathrm {aux}}\). Then \(({\mathsf {F}}^{\mathrm {aux}},{\mathsf {S}}^{\mathrm {aux}},\omega ^{\mathrm {aux}}_1,\mu ^{\mathrm {aux}}_1)\) defines a semi-quantum pre-observable for the same BV theory. Moreover, the observable for the BV theory induced by \({\mathsf {S}}^{\mathrm {aux}}\) using Equation (37) with a Lagrangian submanifold \({\mathsf {L}}\subset {\mathsf {F}}^{\mathrm {aux}}\) is equivalent to the one induced by \({\mathcal {S}}^{\mathrm {aux}}\) using the Lagrangian submanifold \({\mathcal {L}}\subset {\mathcal {F}}^{\mathrm {aux}}\), if there exists a homotopy between \({\mathcal {L}}\) and \({\mathsf {L}}\times {\mathscr {L}}\) in \({\mathcal {F}}^{\mathrm {aux}}\).

Remark 5.12

Note that Equation (40) means that \({\mathsf {S}}^{\mathrm {aux}}\) is the low energy effective action (zero modes).

Observables for Quantum BV Manifolds

Definition 5.13

(BV quantum observable). A BV quantum observable for a quantum BV manifold is a function \({\mathcal {O}}\) on \({\mathcal {F}}\) of degree 0 such that

$$\begin{aligned} \delta _{{\mathrm {BV}}}{\mathcal {O}}=0\Longleftrightarrow \Delta \left( {\mathcal {O}}{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}}\right) =0. \end{aligned}$$
(41)

Definition 5.14

(Equivalent BV quantum observables). Two BV quantum observables \({\mathcal {O}}\) and \({\widetilde{{\mathcal {O}}}}\) are said to be equivalent if

$$\begin{aligned} {\widetilde{{\mathcal {O}}}}-{\mathcal {O}}=\delta _{\mathrm {BV}}\Psi ,\qquad \Psi \in C^\infty ({\mathcal {F}}), \end{aligned}$$
(42)

or equivalently, \({\mathcal {O}}\) and \({\widetilde{{\mathcal {O}}}}\) have the same \(\delta _{\mathrm {BV}}\)-cohomology class.

Definition 5.15

(BV quantum pre-observable). A BV quantum pre-observable for a BV manifold is a BV semi-quantum pre-observable

$$ ({\mathcal {F}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}}) $$

where \({\mathcal {S}}+ {\mathcal {S}}^{\mathrm {aux}}\) satisfies the QME

$$\begin{aligned} (\Delta +\Delta ^{\mathrm {aux}}){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }({\mathcal {S}}+{\mathcal {S}}^{\mathrm {aux}})}=0. \end{aligned}$$
(43)

Proposition 5.16

([46]). Let \(({\mathcal {F}}^{\mathrm {aux}},\omega ^{\mathrm {aux}},\mu ^{\mathrm {aux}},{\mathcal {S}}^{\mathrm {aux}})\) be a BV quantum pre-observable. Define

$$\begin{aligned} {\mathcal {O}}_{{\mathcal {L}}}:=\int _{{\mathcal {L}}\subset {\mathcal {F}}^{\mathrm {aux}}}{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathrm {aux}}}\sqrt{\mu ^{\mathrm {aux}}}\vert _{\mathcal {L}}\in C^\infty ({\mathcal {F}}), \end{aligned}$$
(44)

where \({\mathcal {L}}\subset {\mathcal {F}}^{\mathrm {aux}}\) is a Lagrangian submanifold. Then \({\mathcal {O}}_{\mathcal {L}}\) is an observable, i.e., \(\delta _{\mathrm {BV}}{\mathcal {O}}_{\mathcal {L}}=0\). Moreover, if for two Lagrangian submanifolds \({\mathcal {L}}\) and \({\widetilde{{\mathcal {L}}}}\) there exists a homotopy between them, then the observables \({\mathcal {O}}_{\mathcal {L}}\) and \({\mathcal {O}}_{{\widetilde{{\mathcal {L}}}}}\) are equivalent.

Formal Global Split AKSZ Sigma Models

The formal global construction for ASKZ sigma models is given by using methods of formal geometry (see [13, 34] for the formal geometry part, and [24] for a detailed discussion of the formal global split AKSZ construction and its quantization) where one constructs a BV action that depends on a choice of classical background by adding an additional term to the AKSZ-BV action. This construction leads to modifications in the usual BV gauge-fixing condition if we apply the BV construction to this new formal global action. The globalization arises in an equivalent way as for the constructions involving the underlying curvedFootnote 8\(L_\infty \)-structure for the space of fields (see, e.g., [42] for an exposition on curved \(\infty \)-structures and [29] for the field-theoretic concept).

In this section we want to recall some notions of formal geometry and describe the extension of AKSZ sigma models to a formal global version.

Notions of Formal Geometry

Let us introduce the main players.

Definition 6.1

(Generalized exponential map). Let M be a manifold and let \(U\subset TM\) be an open neighborhood of the zero section of the tangent bundle. A generalized exponential map is a map \(\phi :U\rightarrow M\) such that \(\phi :(x,p)\mapsto \phi _x(p)\) with \(\phi _x(0)=x\) and \({\mathrm {d}}\phi _x(0)={\mathrm {id}}_{T_xM}\). Locally, we have

$$\begin{aligned} \phi ^{i}_x(p)=x^{i}+p^{i}+\frac{1}{2}\phi ^{i}_{x,jk}p^jp^k+\frac{1}{3!}\phi ^{i}_{x,jk\ell }p^jp^kp^\ell +\cdots \end{aligned}$$
(45)

where \((x^{i})\) are coordinates on the base and \((p^{i})\) are coordinates on the fiber.

Definition 6.2

(Formal exponential map). A formal exponential map is an equivalence class of generalized exponential maps, where we identify two generalized exponential maps if their jets agree to all orders.

One can define a flat connection D on \({\widehat{{\text {Sym}}}}(T^*M)\), where \({\widehat{{\text {Sym}}}}\) denotes the completed symmetric algebra. Such a flat connection D is called classical Grothendieck connection [17] and it is locally given by \(D={\mathrm {d}}_M+R\), where

$$ R\in \Omega ^1\left( M,{\mathrm {Der}}\left( {\widehat{{\text {Sym}}}}(T^*M)\right) \right) $$

is a 1-form with values in derivations of the completed symmetric algebra of the cotangent bundle. Here R acts on sections \(\sigma \in \Gamma \left( {\widehat{{\text {Sym}}}}(T^*M)\right) \) by Lie derivative, that is \(R(\sigma )=L_R\sigma \). Note that we have denoted by \({\mathrm {d}}_M\) the de Rham differential on M. In local coordinates we have \(R=R_\ell {\mathrm {d}}_M x^{\ell }\), where \(R_\ell =R_\ell ^j(x,p)\frac{\partial }{\partial p^j}\) and

$$\begin{aligned} R_\ell ^j(x,p)=-\frac{\partial \phi ^k}{\partial x^\ell }\left( \left( \frac{\partial \phi }{\partial p}\right) ^{-1}\right) ^j_k=-\delta _\ell ^j+O(p). \end{aligned}$$
(46)

Hence, for \(\sigma \in \Gamma \left( {\widehat{{\text {Sym}}}}(T^*M)\right) \) we have

$$\begin{aligned} R(\sigma ):=L_R(\sigma )=R_\ell (\sigma ){\mathrm {d}}_M x^\ell =-\frac{\partial \sigma }{\partial p^j}\frac{\partial \phi ^k}{\partial x^\ell }\left( \left( \frac{\partial \phi }{\partial p}\right) ^{-1}\right) ^j_k{\mathrm {d}}_M x^\ell . \end{aligned}$$
(47)

Note that we can extend the connection D to the complex

$$ \Gamma \left( \bigwedge ^\bullet T^*M\otimes {\widehat{{\text {Sym}}}}(T^*M)\right) $$

of \({\widehat{{\text {Sym}}}}(T^*M)\)-valued differential formsFootnote 9. The following proposition tells us that the D-closed sections are exactly given by smooth functions.

Proposition 6.3

A section \(\sigma \in \Gamma \left( {\widehat{{\text {Sym}}}}(T^*M)\right) \) is D-closed if and only if \(\sigma ={\mathsf {T}}\phi ^*f\) for some \(f\in C^\infty (M)\), where \({\mathsf {T}}\) denotes the Taylor expansion around the fiber coordinates at zero. Moreover, the D-cohomology

$$ H^\bullet _D\left( {\widehat{{\text {Sym}}}}(T^*M)\right) $$

is concentrated in degree 0 and

$$\begin{aligned} H^0_D\left( {\widehat{{\text {Sym}}}}(T^*M)\right) ={\mathsf {T}}\phi ^*C^\infty (M)\cong C^\infty (M). \end{aligned}$$
(48)

Remark 6.4

Note that we use any representative of \(\phi \) to define the pullback.

Proof of Proposition 6.3

If we use (45) and (47), We can see that \(R=\delta +R'\) where \(\delta ={\mathrm {d}}x^{i}\frac{\partial }{\partial p^{i}}\) and \(R'\) is a 1-form with values in vector fields vanishing at \(p=0\). Then we have \(D=\delta +D'\) with

$$\begin{aligned} D'={\mathrm {d}}x^{i}\frac{\partial }{\partial x^{i}}+R'. \end{aligned}$$
(49)

One should note that \(\delta \) is itself a differential and that it decreases the polynomial degree in p, whereas \(D'\) does not decrease the degree. We can show that the cohomology of \(\delta \) consists of 0-forms which are constant in p. To show this, let

$$ \delta ^*=p^{i}\iota _{\frac{\partial }{\partial x^{i}}} $$

and note that

$$\begin{aligned} (\delta \delta ^*+\delta ^*\delta )\sigma =k\sigma , \end{aligned}$$
(50)

where \(\sigma \) is an r-form of degree s in p such that \(r+s=k\). By cohomological perturbation theory the cohomology of D is isomorphic to the cohomology of \(\delta \). \(\square \)

Note that in local coordinates we get for \(f\in C^\infty (M)\)

$$\begin{aligned} \textsf {T}\phi _x^*f=f(x)+p^{i}\partial _if(x)+\frac{1}{2}p^jp^k(\partial _j\partial _kf(x)+\phi ^{i}_{x,jk}\partial _if(x))+\cdots \end{aligned}$$
(51)

An interesting question is how the Grothendieck connection depends on the choice of formal exponential map. Let \(I\subset {\mathbb {R}}\) be an open interval and let \(\phi \) be a family of formal exponential maps depending on a parameter \(t\in I\). This family may be associated to a family of formal exponential maps \(\psi \) on \(M\times I\) by

$$\begin{aligned} \psi (x,t,p,\tau )=(\phi _{x,t}(p),t+\tau ), \end{aligned}$$
(52)

where \(\tau \) denotes the tangent coordinate to t. The associated connection \({\widetilde{R}}\) is defined by

$$\begin{aligned}&{\widetilde{R}}\left( {\widetilde{\sigma }}\right) =-({\mathrm {d}}_p{\widetilde{\sigma }},{\mathrm {d}}_\tau {\widetilde{\sigma }})\circ \begin{pmatrix} ({\mathrm {d}}_p\phi )^{-1}&{}0\\ 0&{}1 \end{pmatrix}\circ \begin{pmatrix} {\mathrm {d}}_x\phi &{}{\dot{\phi }}\\ 0&{}1 \end{pmatrix}, \nonumber \\&{\widetilde{\sigma }}\in \Gamma \left( {\widehat{{\text {Sym}}}}(T^*(M\times I)\right) . \end{aligned}$$
(53)

Thus we can write \({\widetilde{R}}=R+C{\mathrm {d}}t+T\) with R defined as before with the difference that it now depends on t, C is given by

$$\begin{aligned} C({\widetilde{\sigma }})=-{\mathrm {d}}_p{\widetilde{\sigma }}\circ ({\mathrm {d}}_p\phi )^{-1}\circ {\dot{\phi }}, \end{aligned}$$
(54)

and \(T=-{\mathrm {d}}t\frac{\partial }{\partial \tau }\). Note that \({\mathrm {d}}_xT=0\), \({\mathrm {d}}_tT=0\) and \([T,R]=0\), \([T,C]=0\). Thus, using the Maurer–Cartan equation for \({\widetilde{R}}\) and for R, we get

$$\begin{aligned} {\dot{R}}={\mathrm {d}}_xC+[R,C], \end{aligned}$$
(55)

which shows that under a change of formal exponential map, R changes by a gauge transformation with generator C. Moreover, if \(\sigma =\textsf {T}\phi _x^*f\) for some \(f\in C^\infty (M\times I)\), we get

$$\begin{aligned} {\dot{\sigma }}=-L_C\sigma . \end{aligned}$$
(56)

This can be thought of as an associated gauge transformation for sections.

Formal Global AKSZ Sigma Models

Let \(\Sigma _d\) be a closed, oriented, compact d-manifold and consider a Hamiltonian Q-manifold

$$ ({\mathcal {M}},\omega _{\mathcal {M}}={\mathrm {d}}_{\mathcal {M}}\alpha _{\mathcal {M}},\Theta _{\mathcal {M}},Q_{\mathcal {M}}) $$

of degree \(d-1\). As described in Sect. 3.2, we can consider its induced AKSZ theory with the space of fields

$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,{\mathcal {M}}). \end{aligned}$$
(57)

Consider now a formal exponential map \(\phi :T{\mathcal {M}}\rightarrow {\mathcal {M}}\). Then we can lift the space of fields by \(\phi \). For \(x\in {\mathcal {M}}\) we denote the lifted space of fields by

$$\begin{aligned} {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d}:={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,T_x{\mathcal {M}})\cong \Omega ^\bullet (\Sigma _d)\otimes T_x{\mathcal {M}}. \end{aligned}$$
(58)

Note that we have used the fact that

$$\begin{aligned} C^\infty (T[1]\Sigma _d)\cong \Omega ^\bullet (\Sigma _d). \end{aligned}$$
(59)

This construction gives us a linear space for the target and thus we can identify the fields with differential forms on \(\Sigma _d\) with values in the vector space \(T_x{\mathcal {M}}\) for \(x\in {\mathcal {M}}\). Consider the map

$$\begin{aligned} {\widetilde{\phi }}_x:{\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\rightarrow {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}, \end{aligned}$$
(60)

which is given by composition with \(\phi ^{-1}_x\), i.e., \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}=\phi ^{-1}_x\circ {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\). We can lift the BV symplectic 2-form \(\omega _{\Sigma _d}\), the primitive 1-form \(\alpha _{\Sigma _d}\) and the BV action \({\mathcal {S}}_{\Sigma _d}\) to the lifted space of fields. We will denote the lifts by

$$\begin{aligned} {\widehat{\alpha }}_{\Sigma _d,x}&={\widetilde{\phi }}^*_x\alpha _{\Sigma _d}\in \Omega ^1\left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\right) , \end{aligned}$$
(61)
$$\begin{aligned} {\widehat{\omega }}_{\Sigma _d,x}&={\widetilde{\phi }}^*_x\omega _{\Sigma _d}\in \Omega ^2\left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\right) , \end{aligned}$$
(62)
$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{{\mathrm {AKSZ}}}_{\Sigma _d,x}&=\iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _d}}{\widetilde{\phi }}^*_x{\mathscr {T}}_{\Sigma _d}(\alpha _{\mathcal {M}}) +{\mathsf {T}}{\widetilde{\phi }}^*_x{\mathscr {T}}_{\Sigma _d}(\Theta _{\mathcal {M}})\in C^\infty \left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\right) . \end{aligned}$$
(63)

Note that we can regard a constant map \(x:T[1]\Sigma _d\rightarrow {\mathcal {M}}\) in \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\) as an element of \({\mathcal {M}}\), hence there is a natural inclusion \({\mathcal {M}}\hookrightarrow {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\). For a constant field x and \({\mathcal {A}}\in {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\) We can construct a 1-form

$$\begin{aligned} R_{\Sigma _d}=(R_{\Sigma _d})_\mu (x,{\mathcal {A}}){\mathrm {d}}_{\mathcal {M}}x^\mu \end{aligned}$$
(64)

on \({\mathcal {M}}\) with values in differential operators on \({\mathcal {F}}_{\Sigma _d}^{\mathcal {M}}\). Moreover, we can lift this 1-form to \({\widehat{{\mathcal {F}}}}_{\Sigma _d}^{\mathcal {M}}\) and we denote the lift by \({\widehat{R}}_{\Sigma _d}\). Locally, we write

$$\begin{aligned} {\widehat{R}}_{\Sigma _d}=\left( {\widehat{R}}_{\Sigma _d}\right) _\mu \left( x,{\widehat{{\mathcal {A}}}}\right) {\mathrm {d}}_{\mathcal {M}}x^\mu . \end{aligned}$$
(65)

It is important to recall that classical solutions for AKSZ sigma models, i.e., solutions of \(\delta {\mathcal {S}}_{\Sigma _d}=0\), are given by differential graded maps

$$ (T[1]\Sigma _d,{\mathrm {d}}_{\Sigma _d})\rightarrow ({\mathcal {M}},Q_{\mathcal {M}}). $$

Hence we can consider the moduli space of classical solutions \({\mathrm {M}}_{{\mathrm {cl}}}\) for AKSZ theories which is given by constant maps \(x:T[1]\Sigma _d\rightarrow {\mathcal {M}}\) and thus we get an isomorphism \({\mathrm {M}}_{{\mathrm {cl}}}\cong {\mathcal {M}}\). We will refer to this constant solutions as being background fields. Choosing a background field \(x\in {\mathcal {M}}\), we can define a formal global AKSZ action.

Definition 6.5

(Formal global AKSZ action). The formal global AKSZ action is given by

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}=\iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _d}}{\widetilde{\phi }}_x^*{\mathscr {T}}_{\Sigma _d}(\alpha _{\mathcal {M}})+\textsf {T}{\widetilde{\phi }}^*_x{\mathscr {T}}_{\Sigma _d}(\Theta _{\mathcal {M}})+{\widehat{{\mathcal {S}}}}_{\Sigma _d,R,x}, \end{aligned}$$
(66)

where \({\widehat{{\mathcal {S}}}}_{\Sigma _d,R,x}\) is constructed locally such that

$$\begin{aligned} {\widehat{{\mathcal {S}}}}_{\Sigma _d,R,x}\left( {\widehat{{\mathcal {A}}}}\right) =\int _{\Sigma _d}\left( {\widehat{R}}_{\Sigma _d}\right) _\mu \left( x,{\widehat{{\mathcal {A}}}}\right) {\mathrm {d}}_{\mathcal {M}}x^\mu . \end{aligned}$$
(67)

Hence locally we get we get

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}= & {} \int _{\Sigma _d}{\widehat{\alpha }}_{\mu }\left( {\widehat{{\mathcal {A}}}}\right) {\mathrm {d}}_{\Sigma _d}{\widehat{{\mathcal {A}}}}^\mu +\int _{\Sigma _d}{\widehat{\Theta }}_{{\mathcal {M}},x}\left( {\widehat{{\mathcal {A}}}}\right) \nonumber \\&+\int _{\Sigma _d}\left( {\widehat{R}}_{\Sigma _d}\right) _\mu \left( x,{\widehat{{\mathcal {A}}}}\right) {\mathrm {d}}_{\mathcal {M}}x^\mu , \end{aligned}$$
(68)

where \({\widehat{\alpha }}_\mu \) are the coefficients of \({\widehat{\alpha }}_{\Sigma _d,x}:={\widetilde{\phi }}^*_x\alpha _{\Sigma _d}\) and \({\widehat{\Theta }}_{{\mathcal {M}},x}:={\mathsf {T}}{\widetilde{\phi }}_x^*\Theta _{{\mathcal {M}}}\).

Remark 6.6

This construction has to be understood in a formal way. The geometric meaning and the relation to a global construction is clear when using the relation of \(R_{\Sigma _d}\) to the Grothendieck connection D. This can be done if we start with a theory called split which we will introduce now.

Formal Global Split AKSZ Sigma Models

AKSZ theories can generally be more difficult to work with depending on the target differential graded symplectic manifold \({\mathcal {M}}\). Recall that, using the isomorphism (59), if the target is linear, we have an isomorphism

$$\begin{aligned} {{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,{\mathcal {M}})\cong \Omega ^\bullet (\Sigma _d)\otimes {\mathcal {M}}. \end{aligned}$$
(69)

Moreover, we can split the space of fields by considering \({\mathcal {M}}\) to be the shifted cotangent bundle of a linear space. At first, however, we only want \({\mathcal {M}}\) to be the shifted cotangent bundle of any graded manifold M. This leads to the following definition of AKSZ theories.

Definition 6.7

(Linear split AKSZ sigma model). We call a d-dimensional AKSZ sigma model linear split if the target is of the form

$$ {\mathcal {M}}=V\oplus V^* $$

for some vector space V.

Definition 6.8

(Split AKSZ sigma model). We call a d-dimensional AKSZ sigma model split if the target is of the form

$$ {\mathcal {M}}=T^*[d-1]M $$

for some graded manifold M.

This space can be lifted to a formal construction using methods of formal geometry as in Sect. 6.1 to the shifted cotangent bundle of the tangent space of M at some constant background in M. Consider a d-dimensional split AKSZ sigma model with space of fields given by

$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,T^*[d-1]M), \end{aligned}$$
(70)

for some graded manifold M, with its corresponding AKSZ-BV theory

$$ \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d},\omega _{\Sigma _d}\right) . $$

Note that, similarly as for general AKSZ theories, one type of classical solutions to the Euler–Lagrange equations for split AKSZ theories are given by fields of the form (x, 0) where \(x:\Sigma _d\rightarrow M\) is a constant background field. Note that the classical space of fields \(F_{\Sigma _d}\) is given by vector bundle maps \(T\Sigma _d\rightarrow T^*M\), i.e.,

$$\begin{aligned} F_{\Sigma _d}={{\,\mathrm{Map}\,}}_{{\mathrm {VecBun}}}(T\Sigma _d,T^*M). \end{aligned}$$

Then the BV space of fields is given by (70). Thus, for the classical space of fields \(F_{\Sigma _d}\), we have a moduli space of classical solutions

$$\begin{aligned} {\mathrm {M}}_{{\mathrm {cl}}}=\left\{ (A,B)\in {{\,\mathrm{Map}\,}}(T\Sigma _d,T^*M)\mid A=x=const,B=0\right\} \cong M. \end{aligned}$$
(71)

Moreover, for a chosen formal exponential map \(\phi :TM\rightarrow M\) and a constant background field \(x:\Sigma _d\rightarrow M\) regarded as an element of the moduli space of classical solutions \({\mathrm {M}}_{{\mathrm {cl}}}\), one can consider the lifted space of fields

$$\begin{aligned} {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,T^*[d-1]T_xM), \end{aligned}$$
(72)

which gives a linearization (or also coordinatization) of the space of fields in the target as we have seen before. Let \(({\varvec{A}},{\varvec{B}})\in {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\), where \({\varvec{A}}:T[1]\Sigma _d\rightarrow M\) denotes the base superfield and \({\varvec{B}}\in \Gamma (\Sigma _d,T^*\Sigma _d\otimes {\varvec{A}}^*T^*[d-1]M)\) the fiber superfield. Consider the corresponding lifts by \(\phi \) where the superfields are given by

$$\begin{aligned} {\widehat{{\varvec{A}}}}:=\phi _x^{-1}({\varvec{A}}),\qquad {\widehat{\varvec{B}}}:=({\mathrm {d}}\phi _x)^{*}{\varvec{B}} \end{aligned}$$
(73)

The BV action functional \({\mathcal {S}}_{\Sigma _d}\) then lifts to a formal global action.

Definition 6.9

(Formal global split AKSZ action). The formal global action for the split AKSZ sigma model is given by

$$\begin{aligned} \boxed { {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}:=\int _{\Sigma _d}{\widehat{{\varvec{B}}}}_\ell \wedge {\mathrm {d}}_{\Sigma _d} {\widehat{\varvec{A}}}^\ell +\int _{\Sigma _d}{\widehat{\Theta }}_{{\mathcal {M}},x}\left( {\widehat{\varvec{A}}},{\widehat{{\varvec{B}}}}\right) +\int _{\Sigma _d}R_\ell ^j\left( x,{\widehat{{\varvec{A}}}}\right) {\widehat{{\varvec{B}}}}_j\wedge {\mathrm {d}}_{M} x^\ell . }\nonumber \\ \end{aligned}$$
(74)

Remark 6.10

Note that in this case we get a lift of R as defined in Sect. 6.1 to the space of fields which splits into base and fiber fields by

$$\begin{aligned} {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\cong \Omega ^\bullet (\Sigma _d)\otimes T_xM\oplus \Omega ^\bullet (\Sigma _d)\otimes T^*_xM[d-1]. \end{aligned}$$
(75)

Hence the induced 1-form \({\widehat{R}}_{\Sigma _d}\) is indeed given by

$$\begin{aligned} {\widehat{R}}_{\Sigma _d}=R_\ell ^j\left( x,{\widehat{{\varvec{A}}}}\right) {\widehat{{\varvec{B}}}}_j\wedge {\mathrm {d}}_M x^\ell , \end{aligned}$$
(76)

where \(R_\ell ^j\) are the components of \(R\in \Omega ^1\left( M,{{\mathrm {Der}}}\left( {\widehat{{\text {Sym}}}}(T^*M)\right) \right) \).

The Q-structure is given by the Hamiltonian vector field of \({\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}\). Indeed, let \({\widehat{R}}_{\Sigma _d}\) denote the lift of the vector field \(R_{\Sigma _d}\) to \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\) and let

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _d,x}&:=\int _{\Sigma _d}{\widehat{{\varvec{B}}}}_\ell \wedge {\mathrm {d}}_{\Sigma _d}{\widehat{{\varvec{A}}}}^\ell +\int _{\Sigma _d}{\widehat{\Theta }}_{{\mathcal {M}},x}\left( {\widehat{{\varvec{A}}}},{\widehat{{\varvec{B}}}}\right) \end{aligned}$$
(77)
$$\begin{aligned} {\widehat{{\mathcal {S}}}}_{\Sigma _d,R,x}&:=\int _{\Sigma _d}R_\ell ^j\left( x,{\widehat{{\varvec{A}}}}\right) {\widehat{{\varvec{B}}}}_j\wedge {\mathrm {d}}_M x^\ell , \end{aligned}$$
(78)

such that

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}={\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _d,x}+{\widehat{{\mathcal {S}}}}_{\Sigma _d,R,x}. \end{aligned}$$
(79)

Denote by \({\widehat{\omega }}_{\Sigma _d,x}={\widetilde{\phi }}_x^*\omega _{\Sigma _d}\) the lift of the symplectic form on \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\) to a symplectic form on \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\). Then we can define a cohomological vector field \({\widehat{Q}}_{\Sigma _d,x}\) on \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\) by

$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}={\widehat{Q}}^{\mathrm {AKSZ}}_{\Sigma _d,x}+{\widehat{R}}_{\Sigma _d}, \end{aligned}$$
(80)

where \({\widehat{Q}}^{\mathrm {AKSZ}}_{\Sigma _d,x}\) is the Hamiltonian vector field of

$$\begin{aligned} {\mathrm {Back}}_{{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _d}}:x\mapsto {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _d,x}, \end{aligned}$$
(81)

and hence we have

$$\begin{aligned} \iota _{{\widehat{Q}}_{\Sigma _d,x}}{\widehat{\omega }}_{\Sigma _d,x}=\delta {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}. \end{aligned}$$
(82)

This is in fact true if the source manifold is closed, i.e., \(\partial \Sigma _d=\varnothing \). We have denoted the map by “Back” to indicate the variation of the “background”.

Proposition 6.11

If \(\partial \Sigma _d=\varnothing \), then

$$\begin{aligned} {\mathrm {d}}_x{\mathrm {Back}}_{{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _d}}=\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _d,R,x},{\mathrm {Back}}_{{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _d}}\right\} _{{\widehat{\omega }}_{\Sigma _d,x}}, \end{aligned}$$
(83)

where \({\mathrm {d}}_M\) denotes the de Rham differential on the moduli space space of classical solutions \({\mathrm {M}}_{{\mathrm {cl}}}\cong M\).

Using the formal global action, we get the following Proposition (see also Proposition 8.4 for the quantum version)

Proposition 6.12

(dCME). The differential Classical Master Equation for the formal global split AKSZ action holds:

$$\begin{aligned} \boxed { {\mathrm {d}}_x{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x},{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x}\right\} _{{\widehat{\omega }}_{\Sigma _d,x}}=0.} \end{aligned}$$
(84)

Definition 6.13

(Formal global split AKSZ sigma model). The formal global split AKSZ sigma model is given by the AKSZ-BV theory for the quadruple

$$\begin{aligned} \left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x},{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x},{\widehat{\omega }}_{\Sigma _d,x},{\widehat{Q}}_{\Sigma _d,x}\right) . \end{aligned}$$
(85)

Remark 6.14

Note that the CME has to be replaced by the dCME as in (84) in the formal global setting.

Pre-observables for AKSZ Theories

AKSZ Pre-observables

Let \(\Sigma _d\) be a closed and oriented source d-manifold and for some differential graded symplectic manifold \(({\mathcal {N}},\omega _{\mathcal {N}}={\mathrm {d}}_{\mathcal {N}}\alpha _{\mathcal {N}})\) let

$$\begin{aligned} \pi :{\mathcal {E}}={\mathcal {M}}\times {\mathcal {N}}\rightarrow {\mathcal {M}}\end{aligned}$$

be a trivial Hamiltonian Q-bundle of degree n over some Hamiltonian Q-manifold \(({\mathcal {M}},\omega _{\mathcal {M}}={\mathrm {d}}_{\mathcal {M}}\alpha ,Q_{\mathcal {M}},\Theta _{\mathcal {M}})\) of degree \(d-1\). Denote by \(\Theta _{\mathcal {E}}\in C^\infty ({\mathcal {E}})\) the Hamiltonian on the total space \({\mathcal {E}}\) and by \({\mathcal {V}}_{\mathcal {E}}\in \ker {\mathrm {d}}\pi \) the vertical part of \(Q_{\mathcal {E}}\), such that

$$\begin{aligned}Q_{\mathcal {E}}=Q_{\mathcal {M}}+{\mathcal {V}}_{\mathcal {E}}. \end{aligned}$$

Consider the corresponding AKSZ-BV theory with BV manifold given by

$$ \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d},\omega _{\Sigma _d},Q_{\Sigma _d}\right) $$

as it was constructed in Sect. 3. Let \(i:\Sigma _k\hookrightarrow \Sigma _d\) be the embedding of a closed oriented submanifold of dimension \(k\le d\) and let the auxiliary space of fields be given by

$$\begin{aligned} {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}:={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _k,{\mathcal {N}}). \end{aligned}$$
(86)

Moreover, consider the transgression maps

$$\begin{aligned} {\mathscr {T}}_{\Sigma _k}&:\Omega ^\bullet ({\mathcal {N}})\rightarrow \Omega ^\bullet \left( {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}\right) , \end{aligned}$$
(87)
$$\begin{aligned} {\mathscr {T}}^{{\mathcal {E}}}_{\Sigma _k}&:\Omega ^\bullet ({\mathcal {E}})\rightarrow \Omega ^\bullet \left( {{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _k,{\mathcal {E}})\right) \end{aligned}$$
(88)

corresponding to the fiber \({\mathcal {N}}\) and the total space \({\mathcal {E}}\). DefineFootnote 10

$$ p:=i^*:\underbrace{{{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,{\mathcal {M}})}_{=:{\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}}\rightarrow \underbrace{{{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _k,{\mathcal {M}})}_{=:{\mathcal {F}}^{\mathcal {M}}_{\Sigma _k}}. $$

Furthermore, let \({\widehat{{\mathrm {d}}}}_{\Sigma _k}\in {\mathfrak {X}}\left( {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}\right) \subset {\mathfrak {X}}^{\mathrm {vert}}\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\times {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}\right) \), where \({\mathfrak {X}}^{\mathrm {vert}}\) denotes the space of vertical vector fields, and let

$$\begin{aligned} {\widehat{{\mathcal {V}}}}_{{\mathcal {E}}}\in {\mathfrak {X}}^{\mathrm {vert}}\left( {{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _k,{\mathcal {E}})\right) \end{aligned}$$

be the lift of \({\mathcal {V}}_{{\mathcal {E}}}\in {\mathfrak {X}}^{\mathrm {vert}}({\mathcal {E}})\) such that \(p^*{\widehat{{\mathcal {V}}}}_{{\mathcal {E}}}\in {\mathfrak {X}}^{\mathrm {vert}}\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\times {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}\right) \), where

$$ p^*:C^\infty \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _k}\right) \rightarrow C^\infty \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) . $$

Proposition 7.1

([46]). Consider the data given by

$$\begin{aligned} {\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}&=\iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _k}}{\mathscr {T}}_{\Sigma _k}(\alpha _{\mathcal {N}})+p^*{\mathscr {T}}^{{\mathcal {E}}}_{\Sigma _k}(\Theta _{\mathcal {E}}), \end{aligned}$$
(89)
$$\begin{aligned} \omega ^{\mathcal {N}}_{\Sigma _k}&=(-1)^k{\mathscr {T}}_{\Sigma _k}(\omega _{\mathcal {N}}), \end{aligned}$$
(90)
$$\begin{aligned} {\mathcal {V}}^{\mathcal {E}}_{\Sigma _k}&={\widehat{{\mathrm {d}}}}_{\Sigma _k}+p^*{\widehat{{\mathcal {V}}}}_{{\mathcal {E}}}. \end{aligned}$$
(91)

Then the quadruple

$$\begin{aligned} \left( {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k},{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k},\omega ^{\mathcal {N}}_{\Sigma _k},{\mathcal {V}}^{\mathcal {E}}_{\Sigma _k}\right) \end{aligned}$$
(92)

defines a pre-observable for the AKSZ-BV theory as in (85), that is we have

$$\begin{aligned} Q_{\Sigma _d}\left( {\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}\right) +\frac{1}{2}\left\{ {\mathcal {S}}^{\mathcal {N}}_{\Sigma _k},{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}\right\} _{\omega ^{\mathcal {N}}_{\Sigma _k}}=0. \end{aligned}$$
(93)

Remark 7.2

This pre-observable is invariant under reparamterizations of \(\Sigma _k\) and under diffeomorphism of the ambient manifold \(\Sigma _d\). In fact, for \(({\mathcal {A}},{\mathcal {B}})\in {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\times {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}\), \(\varphi _d\in {\mathrm {Diff}}(\Sigma _d)\) and \(\varphi _k\in {\mathrm {Diff}}(\Sigma _k)\), one can immediately show that

$$\begin{aligned} {\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};\varphi _d\circ i\circ \varphi _k)={\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}\left( \varphi _d^*{\mathcal {A}},(\varphi _k)^{-1}{\mathcal {B}};i\right) \end{aligned}$$
(94)

Formal Global AKSZ Pre-observables

We want to extend the constructions above to a formal global lift by using methods of formal geometry as in Sect. 6. It turns out that the formal global lift of the pre-observable constructed in the previous section is not automatically a pre-observable. In particular, it is spoilt by an obstruction which can be phrased as an equation that has to be satisfied. Hence we get the following theorem.

Theorem 7.3

Let \(\left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d},\omega _{\Sigma _d},Q_{\Sigma _d}\right) \) be the AKSZ-BV theory constructed as before and let \(i:\Sigma _k\hookrightarrow \Sigma _d\) be a submanifold of \(\Sigma _d\). Moreover, consider constant background fields \(x\in {\mathcal {M}}\) and \(y\in {\mathcal {N}}\). Then its formal global AKSZ construction

$$\begin{aligned} \left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x},{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _d,x},{\widehat{\omega }}_{\Sigma _d,x},{\widehat{Q}}_{\Sigma _d,x}\right) , \end{aligned}$$
(95)

constructed by using a formal exponential map \(T{\mathcal {M}}\rightarrow {\mathcal {M}}\), together with the formal global fiber

$$\begin{aligned} \left( {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y},{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}={\mathrm {d}}_{\mathcal {N}}{\widehat{\alpha }}^{\mathcal {N}}_{\Sigma _k,y}, {\widehat{Q}}^{\mathcal {N}}_{\Sigma _k,y}\right) , \end{aligned}$$
(96)

constructed using a formal exponential map \(T{\mathcal {N}}\rightarrow {\mathcal {N}}\), defines a pre-observable if and only if

$$\begin{aligned} {\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}_{\Sigma _k,y}}=0. \end{aligned}$$
(97)

Remark 7.4

Moreover, for an exponential map \(\phi :T{\mathcal {N}}\rightarrow {\mathcal {N}}\), we set

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}=\underbrace{{\widetilde{\phi }}^*_y\iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _k}}{\mathscr {T}}_{\Sigma _k}(\alpha _{\mathcal {N}})}_{=:{\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y}}+\underbrace{{\mathsf {T}}{\widetilde{\phi }}_y^*p^*{\mathscr {T}}_{\Sigma _k}^{\mathcal {E}}(\Theta _{\mathcal {E}})}_{=:{\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}}, \end{aligned}$$
(98)

and thus we have a decomposition, similarly as in (79), of the formal global action as

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}={\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}+{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}. \end{aligned}$$
(99)

The following Lemma is going to be useful for the proof of Theorem 7.3.

Lemma 7.5

Let \(\Sigma \) be a compact, connected manifold and let \({\mathcal {M}}\) be a differential graded symplectic manifold. Moreover, let \(X\in {\mathfrak {X}}(T[1]\Sigma )\), \(Y\in {\mathfrak {X}}({\mathcal {M}})\), \(\Xi \in \Omega ^\bullet ({\mathcal {M}})\) and denote the lifts of X and Y to the mapping space by \({\widehat{X}}\) and \({\widehat{Y}}\) respectively. Then

$$\begin{aligned} L_{{\widehat{X}}}{\mathscr {T}}_\Sigma (\Xi )&=0, \end{aligned}$$
(100)
$$\begin{aligned} L_{{\widehat{Y}}}{\mathscr {T}}_\Sigma (\Xi )&=(-1)^{{{\,\mathrm{gh}\,}}({\widehat{Y}})\dim \Sigma }{\mathscr {T}}_{\Sigma }(L_Y\Xi ). \end{aligned}$$
(101)

Proof of Theorem 7.3

First we note that the lift

$$\begin{aligned} {\widehat{{\mathcal {V}}}}^{\mathcal {E}}_{\Sigma _k,y}={\widehat{{\mathrm {d}}}}_{\Sigma _k}+{\widetilde{\phi }}^*_yp^*{\widehat{{\mathcal {V}}}}_{\mathcal {E}}\end{aligned}$$
(102)

of \({\mathcal {V}}_{\Sigma _k}^{\mathcal {E}}\) to \({\mathfrak {X}}^{\mathrm {vert}}\left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\times {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\right) \), the space of vertical vector fields on the lifted mapping spaces, is the Hamiltonian vector field for \({\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\), i.e., we have

$$\begin{aligned} {\widehat{{\mathcal {V}}}}^{\mathcal {E}}_{\Sigma _k,y}=\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y},\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}. \end{aligned}$$
(103)

Indeed, we have

$$\begin{aligned} \iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _k}}{\widehat{\omega }}_{\Sigma _k,y}&=\iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _k}}(-1)^k{\widetilde{\phi }}^*_y{\mathscr {T}}_{\Sigma _k}(\omega _{\mathcal {N}})={\widetilde{\phi }}_y^*\iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _k}}\delta {\mathscr {T}}_{\Sigma _k}(\alpha _{\mathcal {N}}) \nonumber \\&={\widetilde{\phi }}^*_y\underbrace{L_{{\widehat{{\mathrm {d}}}}_{\Sigma _k}}{\mathscr {T}}_{\Sigma _k}(\alpha _{\mathcal {N}})}_{=0}+{\widetilde{\phi }}^*_y\delta \iota _{{\widehat{{\mathrm {d}}}}_{\Sigma _k}}{\mathscr {T}}_{\Sigma _k}(\alpha _{\mathcal {N}})=\delta {\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y} \nonumber \\&=\delta ^{\mathrm {vert}}{\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y}, \end{aligned}$$
(104)

where we have used Cartan’s magic formula \(L={\mathrm {d}}\iota +\iota {\mathrm {d}}\), Lemma 7.5 and the fact that \({\widetilde{\phi }}_y^*{\widehat{{\mathrm {d}}}}_{\Sigma _k}={\widehat{{\mathrm {d}}}}_{\Sigma _k}\). We have denoted by \(\delta ^{\mathrm {vert}}\) the vertical part of the de Rham differential \(\delta \) on the lifted total mapping space \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\times {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\), i.e., in the fiber direction \({\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\). The last equality holds since \({\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y}\) is constant in the \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\) direction. Similarly, we have

$$\begin{aligned} \iota _{{\widetilde{\phi }}^*_yp^*{\widehat{{\mathcal {V}}}}_{\mathcal {E}}}{\widehat{\omega }}_{\Sigma _k,y}&={\mathsf {T}}{\widetilde{\phi }}^*_yp^*\iota _{{\widehat{{\mathcal {V}}}}_{\mathcal {E}}}(-1)^k{\mathscr {T}}^{\mathcal {E}}_{\Sigma _k}(\omega _{\mathcal {N}}) \nonumber \\&=(-1)^k{\mathsf {T}}{\widetilde{\phi }}^*_yp^*{\mathscr {T}}^{\mathcal {E}}_{\Sigma _k}(\underbrace{\iota _{{\mathcal {V}}_{\mathcal {E}}}\omega _{\mathcal {N}}}_{=\delta ^{\mathrm {vert}}\Theta _{\mathcal {E}}}) \nonumber \\&=\delta ^{\mathrm {vert}}{\mathsf {T}}{\widetilde{\phi }}^*_yp^*{\mathscr {T}}^{\mathcal {E}}_{\Sigma _k}(\Theta _{\mathcal {E}})=\delta ^{\mathrm {vert}}{\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}. \end{aligned}$$
(105)

Moreover, we have

$$\begin{aligned}&{\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\right) +\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}} \nonumber \\&\quad ={\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right) +{\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) +\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}} \nonumber \\&\qquad +\underbrace{\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}}_{={\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}. \end{aligned}$$
(106)

The first two terms of the left hand side of (106) are given by

$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right)&={\widehat{Q}}^{\mathrm {AKSZ}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right) +{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right) \nonumber \\&={\widehat{{\mathrm {d}}}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right) +{\widetilde{\phi }}_y^*{\widehat{Q}}_{\mathcal {M}}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right) +{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right) , \end{aligned}$$
(107)
$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right)&={\widehat{Q}}^{\mathrm {AKSZ}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) +{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) \nonumber \\&={\widehat{{\mathrm {d}}}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) +{\widetilde{\phi }}_y^*{\widehat{Q}}_{\mathcal {M}}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) +{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) . \end{aligned}$$
(108)

Since \({\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y}\) and \({\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\) are constant in direction of \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\), we get

$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right)&=\underbrace{{\widehat{{\mathrm {d}}}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}\right) }_{=0}+{\widetilde{\phi }}_y^*{\widehat{Q}}_{\mathcal {M}}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}\right) +{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}\right) , \end{aligned}$$
(109)
$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right)&=0. \end{aligned}$$
(110)

Using (107), (108), (109) and (110) we get

$$\begin{aligned}&{\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\right) +\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}} \nonumber \\&\quad ={\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+{\widetilde{\phi }}_y^*{\widehat{Q}}_{{\mathcal {M}}}\left( {\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}\right) +{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) \nonumber \\&\qquad +\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}} \nonumber \\&\quad ={\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) \nonumber \\&\qquad +\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {kin}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+\underbrace{{\widehat{{\mathrm {d}}}}_{\Sigma _k}{\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}}_{=0}+{\widetilde{\phi }}^*_y\left( {\widehat{Q}}_{\mathcal {M}}+\frac{1}{2}p^*{\widehat{{\mathcal {V}}}}_{\mathcal {E}}\right) \left( {\widehat{{\mathcal {S}}}}^{\mathrm {target}}_{\Sigma _k,y}\right) \nonumber \\&\quad ={\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) \nonumber \\&\qquad +\underbrace{(-1)^k{\widetilde{\phi }}^*_yp^*{\mathscr {T}}_{\Sigma _k}^{\mathcal {E}}\left( Q_{\mathcal {M}}(\Theta _{\mathcal {E}})+\frac{1}{2}{\mathcal {V}}_{\mathcal {E}}(\Theta _{\mathcal {E}})\right) }_{=(-1)^k{\widetilde{\phi }}^*_yp^*{\mathscr {T}}_{\Sigma _k}^{\mathcal {E}}\left( Q_{\mathcal {M}}(\Theta _{\mathcal {E}})+\frac{1}{2}\{\Theta _{\mathcal {E}},\Theta _{\mathcal {E}}\}_{\omega _{\mathcal {N}}}\right) =0 \quad ({\text {by definition of Hamiltonian}} \, Q{\text {-bundle}})} \nonumber \\&\quad ={\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) \end{aligned}$$
(111)

Note that \({\widehat{R}}_{\Sigma _d}\) is a vector field on the lifted space \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\) which implies that \({\widehat{R}}_{\Sigma _d}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) =0\) because \({\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\in C^\infty \left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\times {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\right) \) is constant in the direction of \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\) and the claim follows. \(\square \)

Corollary 7.6

An equivalent condition for the formal global AKSZ-BV theory as in Theorem 7.3 to be a pre-observable is given by

$$\begin{aligned} {\widehat{{\mathcal {V}}}}^{\mathcal {E}}_{\Sigma _k,y}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) ={\widehat{{\mathcal {S}}}}_{\Sigma _k,{\mathrm {d}}_yR,y}. \end{aligned}$$
(112)

Proof

Note that we have

$$\begin{aligned}&{\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}} \nonumber \\&\quad =\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _kx}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}^{\mathcal {N}}_{\Sigma _k,y}} \nonumber \\&\quad ={\widehat{{\mathcal {V}}}}^{\mathcal {E}}_{\Sigma _k,y}\left( {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right) +\underbrace{\frac{1}{2}{\widehat{{\mathcal {S}}}}_{\Sigma _k,[R,R],x}}_{={\widehat{{\mathcal {S}}}}_{\Sigma _k,{\mathrm {d}}_yR,y}}, \end{aligned}$$
(113)

where we have used that \({\widehat{{\mathcal {V}}}}^{\mathcal {E}}_{\Sigma _k,y}\) is the Hamiltonian vector field of \({\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\) and the fact that [12]

$$\begin{aligned} \left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\right\} _{{\widehat{\omega }}_{\Sigma _k,y}^{\mathcal {N}}}={\widehat{{\mathcal {S}}}}_{\Sigma _k,[R,R],y}. \end{aligned}$$
(114)

the last equality (under the braces) follows from the fact that D is a flat connection on \({\widehat{{\text {Sym}}}}(T^*{\mathcal {N}})\) which can be translated into

$$\begin{aligned} {\mathrm {d}}_yR+\frac{1}{2}[R,R]=0. \end{aligned}$$
(115)

Moreover, it is easy to see that \({\widehat{{\mathcal {S}}}}_{\Sigma _k,\ell R,y}=\ell {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}\) for any \(\ell \in {\mathbb {R}}\). \(\square \)

Formal Global Auxiliary Construction in Coordinates

We want to describe the auxiliary theory as well as its formal global extension in terms of coordinates. The description follows similarly from the description of the ambient theory as in Sect. 3.2 and its formal global extension as in Sect. 6.2. Let \((v^j)\) be even local coordinates on \(\Sigma _k\) and consider the corresponding odd local coordinates \(\xi ^j={\mathrm {d}}_{\Sigma _k} v^j\) for \(1\le j\le k\). Then we can construct superfield coordinates

$$\begin{aligned} {\mathcal {B}}^\nu (v,\xi )= & {} \sum _{\ell =1}^k\,\,\underbrace{\sum _{1\le j_1<\cdots <j_\ell \le k}{\mathcal {B}}^\nu _{j_1\ldots j_\ell }(v)\xi ^{j_1}\wedge \cdots \wedge \xi ^{j_\ell }}_{={\mathcal {B}}^\nu _{(\ell )}(v,\xi )} \nonumber \\\in & {} \bigoplus _{\ell =0}^k C^\infty (\Sigma _k)\otimes \bigwedge ^\ell T^*\Sigma _k. \end{aligned}$$
(116)

associated to local homogeneous coordinates \((y^\nu )\) of \({\mathcal {N}}\). Note that locally we have

$$\begin{aligned} \alpha _{\mathcal {N}}&=\alpha _\nu ^{\mathcal {N}}(y){\mathrm {d}}_{\mathcal {N}}y^\nu \in \Omega ^1({\mathcal {N}}), \end{aligned}$$
(117)
$$\begin{aligned} \omega _{\mathcal {N}}&=\frac{1}{2}\omega ^{\mathcal {N}}_{\nu _1\nu _2}(y){\mathrm {d}}_{\mathcal {N}}y^{\nu _1}\wedge {\mathrm {d}}_{\mathcal {N}}y^{\nu _2}\in \Omega ^2({\mathcal {N}}). \end{aligned}$$
(118)

Hence we get

$$\begin{aligned} \alpha _{\Sigma _k}^{\mathcal {N}}&=\int _{\Sigma _k}\alpha ^{\mathcal {N}}_\nu ({\mathcal {B}})\delta {\mathcal {B}}^\nu \in \Omega ^1\left( {\mathcal {F}}_{\Sigma _k}^{\mathcal {N}}\right) , \end{aligned}$$
(119)
$$\begin{aligned} \omega _{\Sigma _k}^{\mathcal {N}}&=(-1)^k\frac{1}{2}\int _{\Sigma _k}\omega ^{\mathcal {N}}_{\nu _1\nu _2}({\mathcal {B}})\delta {\mathcal {B}}^{\nu _1}\wedge \delta {\mathcal {B}}^{\nu _2}\in \Omega ^2\left( {\mathcal {F}}_{\Sigma _k}^{\mathcal {N}}\right) , \end{aligned}$$
(120)

and thus we get an action for the auxiliary fields as

$$\begin{aligned} {\mathcal {S}}_{\Sigma _k}^{\mathcal {N}}({\mathcal {A}},{\mathcal {B}};i)= & {} \int _{\Sigma _k}\alpha ^{\mathcal {N}}_\nu ({\mathcal {B}}){\mathrm {d}}_{\Sigma _k}{\mathcal {B}}^\nu \nonumber \\&+\int _{\Sigma _k}\Theta _{\mathcal {E}}(i^*{\mathcal {A}},{\mathcal {B}})\in C^\infty \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\times {\mathcal {F}}_{\Sigma _k}^{\mathcal {N}}\right) , \end{aligned}$$
(121)

These expressions can be lifted to the formal global construction. Indeed, consider a formal exponential map \(\phi :T{\mathcal {N}}\rightarrow {\mathcal {N}}\). Let \({\widehat{{\mathcal {A}}}}=\phi _x^{-1}({\mathcal {A}})\) be the lift of \({\mathcal {A}}\) to \({\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\) and \({\widehat{{\mathcal {B}}}}=\phi _y^{-1}({\mathcal {B}})\) be the lift of \({\mathcal {B}}\) to \({\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\) for \(x\in {\mathcal {M}}\) and \(y\in {\mathcal {N}}\). Then we get

$$\begin{aligned} {\widehat{\alpha }}_{\Sigma _k,y}^{\mathcal {N}}&=\int _{\Sigma _k}{\widehat{\alpha }}_\nu \left( {\widehat{{\mathcal {B}}}}\right) \delta {\widehat{{\mathcal {B}}}}^\nu \in \Omega ^1\left( {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\right) , \end{aligned}$$
(122)
$$\begin{aligned} {\widehat{\omega }}_{\Sigma _k,y}^{\mathcal {N}}&=(-1)^k\frac{1}{2}\int _{\Sigma _k}{\widehat{\omega }}_{\nu _1\nu _2}\left( {\widehat{{\mathcal {B}}}}\right) \delta {\widehat{{\mathcal {B}}}}^{\nu _1}\wedge \delta {\widehat{{\mathcal {B}}}}^{\nu _2}\in \Omega ^2\left( {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\right) , \end{aligned}$$
(123)

where \({\widehat{\alpha }}^{\mathcal {N}}_{\nu }\) and \({\widehat{\omega }}^{\mathcal {N}}_{\nu _1\nu _2}\) are the coefficients of \({\widehat{\alpha }}_{\mathcal {N}}\in \Omega ^1(T{\mathcal {N}})\) and \({\widehat{\omega }}_{\mathcal {N}}\in \Omega ^2(T{\mathcal {N}})\) respectively. If we set \({\widehat{\Theta }}_{{\mathcal {E}},y}:={\mathsf {T}}{\widetilde{\phi }}^*_y\Theta _{\mathcal {E}}\), the auxiliary formal global AKSZ action is then given by

$$\begin{aligned} {\widehat{{\mathcal {S}}}}_{\Sigma _k,y}^{\mathrm {global}}\left( {\widehat{{\mathcal {A}}}},{\widehat{{\mathcal {B}}}};i\right)&=\underbrace{\int _{\Sigma _k}{\widehat{\alpha }}_\nu ^{\mathcal {N}}\left( {\widehat{{\mathcal {B}}}}\right) {\mathrm {d}}_{\Sigma _k}{\widehat{{\mathcal {B}}}}^\nu +\int _{\Sigma _k}{\widehat{\Theta }}_{{\mathcal {E}},y}\left( i^*{\widehat{{\mathcal {A}}}},{\widehat{{\mathcal {B}}}}\right) }_{={\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}} \nonumber \\&\quad +\underbrace{\int _{\Sigma _k}({\widehat{R}}_{\Sigma _k})_{\nu }\left( y,{\widehat{{\mathcal {B}}}}\right) {\mathrm {d}}_{\mathcal {N}}y^{\nu }}_{={\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}}. \end{aligned}$$
(124)

Formal Global Split Auxiliary Construction in Coordinates

If we consider a split AKSZ model with target \({\mathcal {M}}=T^*[d-1]M\), for some graded manifold M, for the ambient theory associated to \(\Sigma _d\), we can consider a split construction for the auxiliary theory associated to the embedding \(i:\Sigma _k\hookrightarrow \Sigma _d\). We set \({\mathcal {N}}=T^*[k-1]N\) for some graded manifold N. The description is analogously given by the one of the ambient theory as in Sect. 6.3. Hence we have

$$\begin{aligned} {\mathcal {F}}_{\Sigma _k}^{\mathcal {N}}={{\,\mathrm{Map}\,}}(T[1]\Sigma _k,T^*[k-1]N), \end{aligned}$$
(125)

and choosing a formal exponential map \(\phi :TN\rightarrow N\) together with \(y\in N\) we get

$$\begin{aligned} {\begin{matrix} {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}&{}={{\,\mathrm{Map}\,}}(T[1]\Sigma _k,T^*[k-1]T_yN)\\ &{}\cong \Omega ^\bullet (\Sigma _k)\otimes T_yN\oplus \Omega ^\bullet (\Sigma _k)\otimes T^*_yN[k-1]. \end{matrix}} \end{aligned}$$
(126)

Then we can write \({\widehat{{\mathcal {A}}}}=\left( {\widehat{{\varvec{A}}}}, {\widehat{\varvec{B}}}\right) \in {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\) and \({\widehat{{\mathcal {B}}}}=\left( {\widehat{{\varvec{\alpha }}}},{\widehat{{\varvec{\beta }}}}\right) \in {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}\), thus we have an auxiliary formal global split AKSZ action given by

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\left( {\widehat{{\varvec{A}}}},{\widehat{{\varvec{B}}}},{\widehat{{\varvec{\alpha }}}},{\widehat{{\varvec{\beta }}}};i\right)&=\int _{\Sigma _k}{\widehat{{\varvec{\beta }}}}_\ell \wedge {\mathrm {d}}_{\Sigma _k}{\widehat{{\varvec{\alpha }}}}^\ell +\int _{\Sigma _k}{\widehat{\Theta }}_{{\mathcal {E}},y}\left( i^*{\widehat{{\varvec{A}}}},i^*{\widehat{{\varvec{B}}}},{\widehat{{\varvec{\alpha }}}},{\widehat{{\varvec{\beta }}}}\right) \nonumber \\&\quad +\int _{\Sigma _k}R_\ell ^j\left( y,{\widehat{{\varvec{\alpha }}}}\right) {\widehat{{\varvec{\beta }}}}_j\wedge {\mathrm {d}}_{N} y^\ell , \end{aligned}$$
(127)

where \(R\in \Omega ^1\left( N,{\mathrm {Der}}\left( {\widehat{{\text {Sym}}}}(T^*N)\right) \right) \).

From Pre-observables to Observables

AKSZ-Observables

We want to construct the observables for the AKSZ theories out of pre-obsrvables by integrating out means of auxiliary fields similarly as in Proposition 5.7. For a submanifold \(i:\Sigma _k\hookrightarrow \Sigma _d\) we set

$$\begin{aligned} {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)=\int _{{\mathcal {L}}\subset {\mathcal {F}}_{\Sigma _k}}{\mathscr {D}}[{\mathcal {B}}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)}\in C^\infty \left( {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\right) . \end{aligned}$$
(128)

There are several things to note. First, \({\mathcal {O}}_{\Sigma _k}\) depends only on the fields in \({\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\) via the pullback of \(i:\Sigma _k\hookrightarrow \Sigma _d\), hence \(Q_{\Sigma _d}({\mathcal {O}}_{\Sigma _k})=0\) which is consistent with the definition of an observable. Moreover, the \(Q_{\Sigma _k}\)-cohomology class of \({\mathcal {O}}_{\Sigma _k}\) does not depend on deformations of the Lagrangian submanifold \({\mathcal {L}}\subset {\mathcal {F}}_{\Sigma _k}\) and is invariant under isotopies of \(\Sigma _k\). We get the following Proposition.

Proposition 8.1

Let \({\mathrm {Diff}}_0(\Sigma _k)\subset {\mathrm {Diff}}(\Sigma _k)\) be diffeomorphisms on \(\Sigma _k\) which are connected to the identity. Then for \(\varphi _k\in {\mathrm {Diff}}(\Sigma _k)\) we have

$$\begin{aligned} {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i\circ \varphi _k)={\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)+ Q_{\Sigma _k}{\text {-exact}}. \end{aligned}$$
(129)

Proof

Indeed, we have

$$\begin{aligned} {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i\circ \varphi _k)&=\int _{{\mathcal {L}}\subset {\mathcal {F}}_{\Sigma _k}}{\mathscr {D}}[{\mathcal {B}}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i\circ \varphi _k)}\nonumber \\&=\int _{{\mathcal {L}}}{\mathscr {D}}[{\mathcal {B}}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}\left( {\mathcal {A}},(\varphi _k^{-1})^*{\mathcal {B}};i\right) } \nonumber \\&=\int _{(\varphi _k^{-1})^*{\mathcal {L}}}(\varphi _k^{-1})^*{\mathscr {D}}[{\mathcal {B}}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)}\nonumber \\&=\int _{(\varphi _k^{-1})^*{\mathcal {L}}}{\mathscr {D}}[{\mathcal {B}}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)} \nonumber \\&=\int _{{\mathcal {L}}}{\mathscr {D}}[{\mathcal {B}}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)}+ Q_{\Sigma _k}{\text {-exact}}\nonumber \\&={\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)+ Q_{\Sigma _k}{\text {-exact}}, \end{aligned}$$
(130)

where we think of \(\int _{\mathcal {L}}{\mathscr {D}}[{\mathcal {B}}]\) to be in fact given by \(\int _{\mathcal {L}}\sqrt{\mu }\vert _{\mathcal {L}}\), with \(\mu \) being the functional integral measure on \({\mathcal {F}}_{\Sigma _k}^{\mathcal {N}}\). Moreover, we have used the isotopy property of \(\varphi _k\) to make sure that \({\mathcal {L}}\) and \((\varphi _k^{-1})^*{\mathcal {L}}\) are indeed homotopic. \(\square \)

There is a similar invariance result for diffeomorphisms of the ambient manifold \(\Sigma _d\) which is the content of the following Proposition.

Proposition 8.2

For a diffeomorphism \(\varphi _d\in {\mathrm {Diff}}(\Sigma _d)\) we get

$$\begin{aligned} {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};\varphi _d\circ i)={\mathcal {O}}_{\Sigma _k}(\varphi _d^*{\mathcal {A}},{\mathcal {B}};i). \end{aligned}$$
(131)

This is indeed true since \({\mathcal {O}}_{\Sigma _k}\) only depends of the ambient field \({\mathcal {A}}\) via the pullback by i.

Another important property is that the correlator of an observable should be invariant under ambient isotopies.

Proposition 8.3

For \(\varphi _d\in {\mathrm {Diff}}_0(\Sigma _d)\) we have

$$\begin{aligned} \langle {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};\varphi _d\circ i)\rangle =\langle {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)\rangle . \end{aligned}$$
(132)

Proof

Indeed, we have

$$\begin{aligned} \langle {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};\varphi _d\circ i)\rangle&=\int _{{\mathcal {L}}\subset {\mathcal {F}}_{\Sigma _k}}{\mathscr {D}}[{\mathcal {A}}]{\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};\varphi _d\circ i){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}_{\Sigma _d}({\mathcal {A}})} \nonumber \\&=\int _{{\mathcal {L}}}{\mathscr {D}}[{\mathcal {A}}]{\mathcal {O}}_{\Sigma _k}(\varphi _d^*{\mathcal {A}},{\mathcal {B}};i){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}_{\Sigma _d}({\mathcal {A}})}\nonumber \\&=\int _{\varphi _d^*{\mathcal {L}}}(\varphi _d^*)_*{\mathscr {D}}[{\mathcal {A}}]{\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}_{\Sigma _d}((\varphi ^{-1})^*{\mathcal {A}})} \nonumber \\&=\int _{\varphi _d^*{\mathcal {L}}}{\mathscr {D}}[{\mathcal {A}}]{\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}_{\Sigma _d}({\mathcal {A}})}\nonumber \\&=\int _{{\mathcal {L}}}{\mathscr {D}}[{\mathcal {A}}]{\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}_{\Sigma _d}({\mathcal {A}})} \nonumber \\&=\langle {\mathcal {O}}_{\Sigma _k}({\mathcal {A}},{\mathcal {B}};i)\rangle , \end{aligned}$$
(133)

where we have used Proposition 8.2 and that the AKSZ action \({\mathcal {S}}_{\Sigma _d}\) (see Remark 7.2) and the functional integral measure \({\mathscr {D}}[{\mathcal {A}}]\) are invariant under diffeomorphisms for our theory is topological. \(\square \)

Formal Global AKSZ-Observables

The construction above can be extended to a formal global one if we start with a formal global pre-observable. Then we have

$$\begin{aligned} {\widehat{{\mathcal {O}}}}_{\Sigma _k,y}\left( {\widehat{{\mathcal {A}}}},{\widehat{{\mathcal {B}}}};i\right) =\int _{{\widehat{{\mathcal {L}}}}\subset {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}}{\mathscr {D}}\left[ {\widehat{{\mathcal {B}}}}\right] {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\left( {\widehat{{\mathcal {A}}}},{\widehat{{\mathcal {B}}}};i\right) }\in C^\infty \left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d}\right) . \end{aligned}$$
(134)

If we start with a split AKSZ theory we get

$$\begin{aligned}&{\widehat{{\mathcal {O}}}}_{\Sigma _k,y}\left( {\widehat{{\varvec{A}}}},{\widehat{{\varvec{B}}}};i\right) \nonumber \\&\quad =\int _{{\widehat{{\mathcal {L}}}}\subset {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}}{\mathscr {D}}\left[ {\widehat{{\varvec{\alpha }}}}\right] {\mathscr {D}}\left[ {\widehat{{\varvec{\beta }}}}\right] {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\left( {\widehat{{\varvec{A}}}},{\widehat{{\varvec{B}}}},{\widehat{{\varvec{\alpha }}}},{\widehat{{\varvec{\beta }}}};i\right) }\in C^\infty \left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d}\right) . \end{aligned}$$
(135)

We have the following proposition (quantum version of (84)).

Proposition 8.4

(dQME). The differential quantum master equation (dQME) for the formal global split AKSZ-observable holds:

$$\begin{aligned} \boxed { {\mathrm {d}}_y{\widehat{{\mathcal {O}}}}_{\Sigma _k,y}-(-1)^d {\mathrm {i}}\hbar \Delta {\widehat{{\mathcal {O}}}}_{\Sigma _k,y}=0.} \end{aligned}$$
(136)

Proof

Note that we have

$$\begin{aligned} {\mathrm {d}}_y{\widehat{{\mathcal {O}}}}_{\Sigma _k,y}= & {} -\frac{{\mathrm {i}}}{\hbar }\int _{{\widehat{{\mathcal {L}}}}\subset {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}}{\mathscr {D}}\left[ {\widehat{{\varvec{\alpha }}}}\right] {\mathscr {D}}\left[ {\widehat{{\varvec{\beta }}}}\right] \nonumber \\&\quad {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\left( {\widehat{{\varvec{A}}}},{\widehat{{\varvec{B}}}},{\widehat{{\varvec{\alpha }}}},{\widehat{{\varvec{\beta }}}};i\right) }\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right\} _{\omega ^{\mathcal {N}}_{\Sigma _k,y}}, \end{aligned}$$
(137)

which we can write as

$$\begin{aligned}&-\frac{{\mathrm {i}}}{\hbar }\int _{{\widehat{{\mathcal {L}}}}\subset {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}}{\mathscr {D}}\left[ {\widehat{{\varvec{\alpha }}}}\right] {\mathscr {D}}\left[ {\widehat{{\varvec{\beta }}}}\right] {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\left( {\widehat{{\varvec{A}}}},{\widehat{{\varvec{B}}}},{\widehat{{\varvec{\alpha }}}},{\widehat{{\varvec{\beta }}}};i\right) }\left\{ {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y},{\widehat{{\mathcal {S}}}}^{\mathrm {AKSZ}}_{\Sigma _k,y}\right\} _{\omega ^{\mathcal {N}}_{\Sigma _k,y}} \nonumber \\&\quad =-\Delta \int _{{\widehat{{\mathcal {L}}}}\subset {\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}}{\mathscr {D}}\left[ {\widehat{{\varvec{\alpha }}}}\right] {\mathscr {D}}\left[ {\widehat{{\varvec{\beta }}}}\right] {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\left( {\widehat{{\varvec{A}}}},{\widehat{{\varvec{B}}}},{\widehat{{\varvec{\alpha }}}},{\widehat{{\varvec{\beta }}}};i\right) }{\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y} \end{aligned}$$
(138)

if we assume that \(\Delta {\widehat{{\mathcal {S}}}}_{\Sigma _k,R,y}=0\), which is true, e.g., if the Euler characteristic of \(\Sigma _k\) is zero or if \({{\,\mathrm{div}\,}}_{{\mathsf {T}}\phi ^*\mu }R=0\), where \(\mu \) is some volume form on \({\mathcal {N}}\). Note that \({\mathrm {d}}_y{\mathsf {T}}\phi ^*\mu =-L_R{\mathsf {T}}\phi ^*\mu \) which means that \({{\,\mathrm{div}\,}}_{{\mathsf {T}}\phi ^*\mu }R=0\) if and only if \({\mathrm {d}}_y{\mathsf {T}}\phi ^*\mu =0\). For any volume element \(\mu \) it is always possible to find a formal exponential map \(\phi \) such that the latter condition is satisfied. Note that this is then also translated into the differential quantum master equation

$$\begin{aligned} {\mathrm {d}}_y{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}+\frac{1}{2}\left\{ {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y},{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\right\} _{{\widehat{\omega }}_{\Sigma _k,y}}-{\mathrm {i}}\hbar \Delta {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}=0, \end{aligned}$$
(139)

and by the assumption \(\Delta {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}=0\), we obtain the differential CME as in (84). Hence the claim follows. \(\square \)

Remark 8.5

One can check that \({\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {O}}}}_{\Sigma _k,y}\right) =0\) and that Proposition 8.1 and 8.2 also hold for the formal global extension if we indeed start with a formal global pre-observable, i.e., that the assumption of Theorem 7.3 is satisfied.

Remark 8.6

The dQME as in (136) can be thought of as a descent equation for different form degrees. In fact we have

$$\begin{aligned} {\widehat{\delta }}_{{\mathrm {BV}}}{\widehat{{\mathcal {O}}}}_{\Sigma _k,y}=(-1)^d{\mathrm {d}}_y{\widehat{{\mathcal {O}}}}_{\Sigma _k,y} \end{aligned}$$
(140)

since \({\widehat{{\mathcal {O}}}}_{\Sigma _k,y}\) is a formal global observable. We have set \({\widehat{\delta }}_{{\mathrm {BV}}}={\widehat{Q}}_{\Sigma _d,x}-{\mathrm {i}}\hbar \Delta \).

Remark 8.7

Note that if \({\mathcal {N}}\) is a point, we have \({\mathcal {V}}_{\mathcal {E}}=0\), \(\omega ^{\mathcal {N}}_{\Sigma _k}=0\) and \(\Theta _{\mathcal {E}}\in C^\infty ({\mathcal {M}})\). The associated pre-observable is then given by

$$\begin{aligned} {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}={\mathrm {pt}},\quad {\mathcal {V}}^{\mathcal {E}}_{\Sigma _k}=0,\quad \omega _{\Sigma _k}^{\mathcal {N}}=0,\quad {\mathcal {S}}^{\mathcal {N}}_{\Sigma _k}({\mathcal {A}})=\int _{\Sigma _k}\Theta _{\mathcal {E}}(i^*{\mathcal {A}}). \end{aligned}$$
(141)

Hence, since there are no auxiliary fields \({\mathcal {B}}\), the constructed observable is given by

$$\begin{aligned} {\mathcal {O}}_{\Sigma _k}({\mathcal {A}};i)={\text {e}}^{\frac{{\mathrm {i}}}{\hbar }\int _{\Sigma _k}\Theta _{\mathcal {E}}(i^*{\mathcal {A}})}. \end{aligned}$$
(142)

This can be easily lifted to a formal global pre-observable by

$$\begin{aligned}&{\widehat{{\mathcal {F}}}}^{\mathcal {N}}_{\Sigma _k,y}={\mathrm {pt}},\quad {\widehat{{\mathcal {V}}}}^{\mathcal {E}}_{\Sigma _k,y}=0,\quad {\widehat{\omega }}_{\Sigma _k,y}^{\mathcal {N}}=0, \nonumber \\&{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _k,y}\left( {\widehat{{\mathcal {A}}}}\right) = \int _{\Sigma _k}{\widehat{\Theta }}_{{\mathcal {E}},y}\left( i^*{\widehat{{\mathcal {A}}}}\right) , \end{aligned}$$
(143)

Thus, we get a formal global observable by

$$\begin{aligned} {\widehat{{\mathcal {O}}}}_{\Sigma _k,y}\left( {\widehat{{\mathcal {A}}}};i\right) ={\text {e}}^{\frac{{\mathrm {i}}}{\hbar }\int _{\Sigma _k}{\widehat{\Theta }}_{{\mathcal {E}},y}\left( i^*{\widehat{{\mathcal {A}}}}\right) }. \end{aligned}$$
(144)

Loop Observables

Let us consider the case where \(S^1\) is embedded into \(\Sigma _d\), i.e., \(i:\Sigma _1:=S^1\hookrightarrow \Sigma _d\) and assume that \({\mathcal {N}}\) is given by an ordinary symplectic manifold with symplectic structure \(\omega _{\mathcal {N}}={\mathrm {d}}_{\mathcal {N}}\alpha _{\mathcal {N}}\), which means that \({\mathcal {N}}\) is concentrated in degree zero. Let \(\sigma \) denote the coordinate on \(\Sigma _1\). Then we can write the auxiliary field as

$$\begin{aligned} {\mathcal {B}}^\nu (\sigma ,{\mathrm {d}}_{\Sigma _1}\sigma )={\mathcal {B}}^\nu _{(0)}(\sigma )+{\mathcal {B}}^\nu _{(1)}{\mathrm {d}}_{\Sigma _1}\sigma , \end{aligned}$$
(145)

and hence we get a pre-observable by

$$\begin{aligned} {\mathcal {F}}^{\mathcal {N}}_{\Sigma _1}&={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _1,{\mathcal {N}}), \end{aligned}$$
(146)
$$\begin{aligned} \omega ^{\mathcal {N}}_{\Sigma _1}&=-\oint _{\Sigma _1}\omega ^{\mathcal {N}}_{\nu _1\nu _2}\left( {\mathcal {B}}_{(0)}\right) \delta {\mathcal {B}}_{(0)}^{\nu _1}\wedge \delta {\mathcal {B}}_{(1)}^{\nu _2} \nonumber \\&\quad +\oint _{\Sigma _1}\frac{1}{2}{\mathcal {B}}^{\nu _3}_{(1)}\partial _{\nu _3}\omega _{\nu _1\nu _2}^{\mathcal {N}}\left( {\mathcal {B}}_{(0)}\right) \delta {\mathcal {B}}_{(0)}^{\nu _1}\wedge \delta {\mathcal {B}}_{(0)}^{\nu _2}, \end{aligned}$$
(147)
$$\begin{aligned} {\mathcal {S}}^{\mathcal {N}}_{\Sigma _1}&=\oint _{\Sigma _1}\alpha ^{\mathcal {N}}_\nu \left( {\mathcal {B}}_{(0)}\right) {\mathrm {d}}_{\Sigma _1}{\mathcal {B}}^\nu _{(0)}+\oint _{\Sigma _1}\Theta _{{\mathcal {E}}}(i^*{\mathcal {A}},{\mathcal {B}}), \end{aligned}$$
(148)

where \(\omega ^{\mathcal {N}}_{\nu _1\nu _2}\) are the coefficients of \(\omega _{\mathcal {N}}\) and \(\alpha ^{\mathcal {N}}_\nu \) are the coefficients of \(\alpha _{\mathcal {N}}\). Note that in this setting we have

$$\begin{aligned} {\mathcal {F}}^{\mathcal {N}}_{\Sigma _1}=\left\{ \left( {\mathcal {B}}_{(0)},{\mathcal {B}}_{(1)}\right) \,\Big |\, {\mathcal {B}}_{(0)}:\Sigma _1\rightarrow {\mathcal {N}},\,\, {\mathcal {B}}_{(1)}\in \Gamma \left( \Sigma _1,T^*\Sigma _1\otimes {\mathcal {B}}_{(0)}^*T^*{\mathcal {N}}\right) [-1]\right\} .\nonumber \\ \end{aligned}$$
(149)

Hence we can construct the observable as

$$\begin{aligned} {\mathcal {O}}_{\Sigma _1}({\mathcal {A}};i)=\int _{{\mathcal {L}}}{\mathscr {D}}\left[ {\mathcal {B}}_{(0)}\right] {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }\oint _{\Sigma _1}\alpha _\nu ^{\mathcal {N}}({\mathcal {B}}_{(0)}){\mathrm {d}}_{\Sigma _1}{\mathcal {B}}_{(0)}+\frac{{\mathrm {i}}}{\hbar }\oint _{\Sigma _1}\Theta _{{\mathcal {E}}}\left( i^*{\mathcal {A}},{\mathcal {B}}_{(0)}\right) }, \end{aligned}$$
(150)

where we have chosen the natural Lagrangian submanifold

$$\begin{aligned} {\mathcal {L}}={{\,\mathrm{Map}\,}}_{{\mathrm {Mnf}}}(\Sigma _1,{\mathcal {N}})\subset {\mathcal {F}}^{\mathcal {N}}_{\Sigma _1}, \end{aligned}$$
(151)

which is obtained by setting all odd variables \({\mathcal {B}}_{(1)}\) to zero.

Remark 8.8

(Bohr–Sommerfeld). If \({\mathcal {N}}\) is a differential graded symplectic manifold of degree different from zero, we know that the symplectic form \(\omega _{\mathcal {N}}\) is always exact since we can write it as

$$\begin{aligned} \omega _{\mathcal {N}}={\mathrm {d}}_{\mathcal {N}}(\iota _E\omega _{\mathcal {N}}), \end{aligned}$$

(see [49]) where E is the Euler vector field. For the degree zero case, the symplectic form does not automatically have a primitive 1-form and hence one can not immediately define \({\mathcal {S}}^{\mathrm {kin}}_{\Sigma _1}\). However, one can also assume that \(\omega _{\mathcal {N}}\) satisfies the Bohr–Sommerfeld condition, which says that

$$\begin{aligned} \frac{\omega _{\mathcal {N}}}{2\pi }\in H^2({\mathcal {N}},{\mathbb {Z}}). \end{aligned}$$

Then the primitive 1-form can be understood as a Hermitian line bundle over \({\mathcal {N}}\) endowed with a U(1)-connection \(\nabla _{\mathcal {N}}\) such that its curvature is given by \((\nabla _{\mathcal {N}})^2=\omega _{\mathcal {N}}\). Thus we can define

$$\begin{aligned} {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\mathcal {S}}_{\Sigma _1}^{\mathrm {kin}}({\mathcal {B}})} \end{aligned}$$

to be given by the holonomy of \(\left( {\mathcal {B}}_{(0)}\right) ^*\nabla _{\mathcal {N}}\) around \(\Sigma _1\). Using Stokes’ theorem we get

$$\begin{aligned} {\mathcal {S}}_{\Sigma _1}^{{\mathrm {kin}}}({\mathcal {B}})=\int _{{\mathbb {D}}}\left( {\mathcal {B}}_{(0)}^{\mathrm {ext}}\right) ^*\omega _{{\mathcal {N}}}, \end{aligned}$$
(152)

where \({\mathbb {D}}\) is a disk with \(\partial {\mathbb {D}}=\Sigma _1\) and \({\mathcal {B}}_{(0)}^{\mathrm {ext}}\) is any extension of \({\mathcal {B}}_{(0)}\) to \({\mathbb {D}}\).

Remark 8.9

This construction can be obviously extended to the formal global case. The case of a dimension 1 submanifold gives the same auxiliary theory as for the case when our theory is split.

Formal Global Loop Observables

The following proposition is an extension of Proposition 5 in [46] to the formal global case.

Proposition 8.10

Let \(({\mathcal {N}},\omega _{\mathcal {N}})\) be a symplectic manifold and assume that it can be geometrically quantized to a complex vector space \({\mathcal {H}}\), the state space, and that the Hamiltonian \(\Theta _{\mathcal {E}}\in C^\infty ({\mathcal {E}})\) can be quantized to an operator valued function \({\varvec{\Theta }}_{\mathcal {E}}\in C^\infty ({\mathcal {M}})\otimes {\mathrm {End}}({\mathcal {H}})\). Moreover, for a formal exponential map \(\phi :T{\mathcal {M}}\rightarrow {\mathcal {M}}\), let \({\widehat{{\varvec{\Theta }}}}_{{\mathcal {E}},x}:={\mathsf {T}}{\widetilde{\phi }}^*_x{\varvec{\Theta }}_{\mathcal {E}}\) and assume that

$$\begin{aligned} {\widehat{Q}}_{{\mathcal {M}}}\left( {\widehat{{\varvec{\Theta }}}}_{{\mathcal {E}},x}\right) +{\widehat{R}}_{\Sigma _d}\left( {\widehat{{\varvec{\Theta }}}}_{{\mathcal {E}},x}\right) +{\mathrm {i}}\hbar \left( {\widehat{{\varvec{\Theta }}}}_{{\mathcal {E}},x}\right) ^2=0 \end{aligned}$$
(153)

for \(x\in {\mathcal {M}}\). Then for \(\Sigma _1:=S^1\) we get that

$$\begin{aligned} {\widehat{{\mathcal {O}}}}_{\Sigma _1,x}={{\,\mathrm{Tr}\,}}_{\mathcal {H}}{\mathcal {P}}\exp \left( \frac{{\mathrm {i}}}{\hbar }\oint _{\Sigma _1}{\widehat{{\varvec{\Theta }}}}_{{\mathcal {E}},x}\left( i^*{\widehat{{\mathcal {A}}}}\right) \right) \end{aligned}$$
(154)

is a formal global observable, where we have denoted by \({{\,\mathrm{Tr}\,}}_{\mathcal {H}}\) the trace map on \({\mathcal {H}}\) and \({\mathcal {P}}\exp \) denotes the path-ordered exponential.

Remark 8.11

Note that (153) is the formal global quantum version of (28).

Proof of Proposition 8.10

Let \(\gamma :\Sigma _1:=[0,1]\rightarrow \Sigma _{d}\) be a path in \(\Sigma _d\) which is parametrized by \(t\in [0,1]\). Denote by

$$ {\widehat{{\varvec{\psi }}}}:={\widehat{{\varvec{\Theta }}}}_{{\mathcal {E}},x}\left( \gamma ^*{\widehat{{\mathcal {A}}}}\right) \in \Omega ^\bullet ([0,1])\otimes C^\infty \left( {\widehat{{\mathcal {F}}}}^{\mathcal {M}}_{\Sigma _d,x}\right) \otimes {{\,\mathrm{End}\,}}({\mathcal {H}}). $$

Moreover denote by \({\widehat{{\varvec{\psi }}}}_{(0)}(t)\) and \({\widehat{{\varvec{\psi }}}}_{(1)}(t,{\mathrm {d}}t)\) the 0- and 1-form part of \({\widehat{{\varvec{\psi }}}}\). Then, for the 1-form part, we get

$$\begin{aligned} {\widehat{W}}_{\Sigma _1,x}^\gamma&={\mathcal {P}}\exp \left( \frac{{\mathrm {i}}}{\hbar }\int _0^1{\widehat{{\varvec{\psi }}}}_{(1)}\right) \nonumber \\&=\lim _{N\rightarrow \infty }\overleftarrow{\prod _{0\le r\le N}}\left( {\mathrm {id}}_{\mathcal {H}}+\frac{{\mathrm {i}}}{\hbar }\iota _{\frac{1}{N}\frac{\partial }{\partial t}}{\widehat{{\varvec{\psi }}}}_{(1)}\left( \frac{r}{N},{\mathrm {d}}t\right) \right) \in C^\infty \left( {\widehat{{\mathcal {F}}}}_{\Sigma _d,x}\right) \otimes {{\,\mathrm{End}\,}}({\mathcal {H}}) \end{aligned}$$
(155)

Then we get

$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{W}}^\gamma _{\Sigma _1,x}\right)&=-{\mathrm {i}}\hbar \int _0^1{\mathcal {P}}\exp \left( \frac{{\mathrm {i}}}{\hbar }\int _t^1{\widehat{{\varvec{\psi }}}}_{(1)}\right) {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\varvec{\psi }}}}(t,{\mathrm {d}}t)\right) {\mathcal {P}}\exp \left( \frac{{\mathrm {i}}}{\hbar }\int _0^t{\varvec{\psi }}_{(1)}\right) \nonumber \\&=-{\mathrm {i}}\hbar \int _0^1{\mathcal {P}}\exp \left( \frac{{\mathrm {i}}}{\hbar }\int _t^1{\widehat{{\varvec{\psi }}}}_{(1)}\right) \nonumber \\&\quad \left( {\mathrm {d}}t\frac{\partial }{\partial t}{\widehat{{\varvec{\psi }}}}_{(0)}(t)-{\mathrm {i}}\hbar \left[ {\widehat{{\varvec{\psi }}}}_{(0)}(t),{\widehat{{\varvec{\psi }}}}_{(1)}(t,{\mathrm {d}}t)\right] \right) {\mathcal {P}}\exp \left( \frac{{\mathrm {i}}}{\hbar }\int _0^t{\widehat{{\varvec{\psi }}}}_{(1)}\right) \nonumber \\&=-{\mathrm {i}}\hbar \lim _{N\rightarrow \infty }\sum _{\ell =0}^{N-1}\overleftarrow{\prod _{\ell<r<N}}\left( {\mathrm {id}}_{\mathcal {H}}+\frac{{\mathrm {i}}}{\hbar }\iota _{\frac{1}{N}\frac{\partial }{\partial t}}{\widehat{{\varvec{\psi }}}}_{(1)}\left( \frac{r}{N},{\mathrm {d}}t\right) \right) \nonumber \\&\quad \times \left( {\widehat{{\varvec{\psi }}}}_{(0)}\left( \frac{\ell +1}{N}\right) \left( {\mathrm {id}}_{\mathcal {H}}+\frac{{\mathrm {i}}}{\hbar }\iota _{\frac{1}{N}\frac{\partial }{\partial t}}{\widehat{{\varvec{\psi }}}}_{(1)}\left( \frac{\ell }{N},{\mathrm {d}}t\right) \right) \right. \nonumber \\&\qquad \left. -\left( {\mathrm {id}}_{\mathcal {H}}+\frac{{\mathrm {i}}}{\hbar }\iota _{\frac{1}{N}\frac{\partial }{\partial t}}{\widehat{{\varvec{\psi }}}}_{(1)}\left( \frac{\ell }{N},{\mathrm {d}}t\right) \right) {\widehat{{\varvec{\psi }}}}_{(0)}\left( \frac{\ell }{N}\right) \right) \nonumber \\&\quad \times \overleftarrow{\prod _{0\le r<\ell }}\left( {\mathrm {id}}_{\mathcal {H}}+\frac{{\mathrm {i}}}{\hbar }\iota _{\frac{1}{N}\frac{\partial }{\partial t}}{\widehat{{\varvec{\psi }}}}_{(1)}\left( \frac{r}{N},{\mathrm {d}}t\right) \right) \nonumber \\&=-{\mathrm {i}}\hbar \left( {\widehat{{\varvec{\psi }}}}_{(0)}(1){\widehat{W}}_{\Sigma _1,x}^\gamma -{\widehat{W}}_{\Sigma _1,x}^\gamma {\widehat{{\varvec{\psi }}}}_{(0)}(0)\right) . \end{aligned}$$
(156)

We have used (154), which gives us

$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\varvec{\psi }}}}\right) ={\mathrm {d}}_{\Sigma _1}{\widehat{{\varvec{\psi }}}}-{\mathrm {i}}\hbar \left[ {\widehat{{\varvec{\psi }}}},{\widehat{{\varvec{\psi }}}}\right] , \end{aligned}$$
(157)

where \([,]\) denotes the commutator of operators. Now if \(\Sigma _1:=S^1\) we have \(\gamma (0)=\gamma (1)\), and thus we get

$$\begin{aligned} {\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{{\mathcal {O}}}}_{\Sigma _1,x}\right)= & {} {{\,\mathrm{Tr}\,}}_{\mathcal {H}}{\widehat{Q}}_{\Sigma _d,x}\left( {\widehat{W}}_{\Sigma _1,x}^\gamma \right) \nonumber \\= & {} -{\mathrm {i}}\hbar {{\,\mathrm{Tr}\,}}_{\mathcal {H}}\left[ {\widehat{{\varvec{\Theta }}}}_{{\mathcal {E}},x}\left( {\widehat{{\mathcal {A}}}}_{(0)}(\gamma (0))\right) ,{\widehat{W}}_{\Sigma _1,x}^\gamma \right] =0, \end{aligned}$$
(158)

where \({\widehat{{\mathcal {A}}}}_{(0)}\) denotes the degree zero component of \({\widehat{{\mathcal {A}}}}\). \(\square \)

Remark 8.12

The construction in Proposition 8.10 does not require \(\omega _{\mathcal {N}}\) to be exact. It is in fact enough to require that \(\omega _{\mathcal {N}}\) satisfies the Bohr–Sommerfeld condition as discussed in Remark 8.8. This is necessary for the assumption that \({\mathcal {N}}\) can be geometrically quantized.

Formal Global Loop Observables for the Poisson Sigma Model

The Poisson sigma model is an example of a 2-dimensional AKSZ theory which is split as in Definition 6.8. Let M be a Poisson manifold with Poisson bivector \(\pi \in \Gamma \left( \bigwedge ^2TM\right) \). Moreover, consider a 2-dimensional source \(\Sigma _2\). Let (xp) be base and fiber coordinates on \(T^*[1]M\). Then we can define a differential graded symplectic manifold as the target of the AKSZ theory by the data

$$\begin{aligned} {\mathcal {M}}&=T^*[1]M, \end{aligned}$$
(159)
$$\begin{aligned} Q_{\mathcal {M}}&=\left\langle \pi (x),p\frac{\partial }{\partial x}\right\rangle +\frac{1}{2}\left\langle \frac{\partial }{\partial x}\pi (x),(p\wedge p)\otimes \frac{\partial }{\partial p}\right\rangle , \end{aligned}$$
(160)
$$\begin{aligned} \omega _{\mathcal {M}}&=\langle \delta p,\delta x\rangle , \end{aligned}$$
(161)
$$\begin{aligned} \alpha _{\mathcal {M}}&=\langle p,\delta x\rangle , \end{aligned}$$
(162)
$$\begin{aligned} \Theta _{\mathcal {M}}&=\frac{1}{2}\langle \pi (x),p\wedge p\rangle . \end{aligned}$$
(163)

The corresponding 2-dimensional AKSZ-BV theory is given by the data

$$\begin{aligned} {\mathcal {F}}_{\Sigma _2}&={{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _2,T^*[1]M) \nonumber \\&\cong \Omega ^\bullet (\Sigma _2)\otimes T_xM\oplus \Omega ^\bullet (\Sigma _2)\otimes T_x^*M[1]\ni ({\varvec{X}},{\varvec{\eta }}), \end{aligned}$$
(164)
$$\begin{aligned} \omega _{\Sigma _2}&=\int _{\Sigma _2}\langle \delta {\varvec{\eta }}, \delta {\varvec{X}}\rangle , \end{aligned}$$
(165)
$$\begin{aligned} {\mathcal {S}}_{\Sigma _2}&=\int _{\Sigma _2}\langle {\varvec{\eta }},{\mathrm {d}}_{\Sigma _2}{\varvec{X}}\rangle +\frac{1}{2}\int _{\Sigma _2}\langle \pi ({\varvec{X}}),{\varvec{\eta }}\wedge {\varvec{\eta }}\rangle . \end{aligned}$$
(166)

Choosing a formal exponential map \(\phi :TM\rightarrow M\) together with a background field \(x:T[1]\Sigma _2\rightarrow M\), the formal global action for the Poisson sigma model is given by

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _2,x}\left( {\widehat{{\varvec{X}}}},{\widehat{{\varvec{\eta }}}}\right)&=\int _{\Sigma _2}{\widehat{{\varvec{\eta }}}}_\ell \wedge {\mathrm {d}}_{\Sigma _2}{\widehat{{\varvec{X}}}}^\ell +\frac{1}{2}\int _{\Sigma _2}\left( \textsf {T}{\widetilde{\phi }}_x^*\pi \right) ^{ij}\left( {\widehat{{\varvec{X}}}}\right) {\widehat{{\varvec{\eta }}}}_i\wedge {\widehat{{\varvec{\eta }}}}_j \nonumber \\&\quad +\int _{\Sigma _2}R_{\ell }^j\left( x,{\widehat{{\varvec{X}}}}\right) {\widehat{{\varvec{\eta }}}}_j\wedge {\mathrm {d}}_Mx^\ell . \end{aligned}$$
(167)

We want to construct a formal global Wilson loop like observables using the Poisson sigma model toegtehr with an auxiliary theory for an embedding \(i:\Sigma _1:=S^1\hookrightarrow \Sigma _2\). Consider an exact symplectic manifold \(({\mathcal {N}},\omega _{\mathcal {N}}={\mathrm {d}}_{\mathcal {N}}\alpha _{\mathcal {N}})\). We can construct a vertical vector field \({\mathcal {V}}\) on the trivial bundle \({\mathcal {N}}\times M\rightarrow {\mathcal {N}}\) which can be viewed as a map \({\mathcal {N}}\rightarrow {\mathfrak {X}}(M)\) with the property

$$\begin{aligned} \frac{1}{2}\{{\mathcal {V}},{\mathcal {V}}\}_{\omega _{\mathcal {N}}}+[\pi ,{\mathcal {V}}]_{{\mathrm {SN}}}+R\wedge {\mathcal {V}}=0, \end{aligned}$$

where \([,]_{{\mathrm {SN}}}\) denotes the Schouten–Nijenhuis bracket defined on polyvector fields on M. We have a degree 0 Hamiltonian Q-bundle structure on

$$ T^*[1]M\times {\mathcal {N}}\rightarrow T^*[1]M $$

with fiber \({\mathcal {N}}\) endowed with the structure

$$\begin{aligned} {\mathcal {V}}_{\mathcal {E}}&=\langle p,\{{\mathcal {V}},\}_{\omega _{\mathcal {N}}}\rangle , \end{aligned}$$
(168)
$$\begin{aligned} \Theta _{\mathcal {E}}&=\langle p,{\mathcal {V}}\rangle , \end{aligned}$$
(169)

where \({\mathcal {E}}=T^*[1]M\times {\mathcal {N}}\). If we use the notation of Sect. 8.3, we can associate a pre-observable to the Poisson sigma model given by the data (146) and (147) together with the auxiliary action

$$\begin{aligned} {\mathcal {S}}^{\mathcal {N}}_{\Sigma _1}({\varvec{X}},{\varvec{\eta }},{\mathcal {B}};i)=\oint _{\Sigma _1}\alpha ^{\mathcal {N}}_\nu ({\mathcal {B}}){\mathrm {d}}_{\Sigma _1}{\mathcal {B}}^\nu +\oint _{\Sigma _1}\langle i^*{\varvec{\eta }},{\mathcal {V}}(i^*{\varvec{X}},{\mathcal {B}})\rangle . \end{aligned}$$
(170)

Choosing a formal exponential map \(\phi :T{\mathcal {N}}\rightarrow {\mathcal {N}}\) together with local coordinates we can lift this to a formal global auxiliary action

$$\begin{aligned} {\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _1,y}\left( {\widehat{{\varvec{X}}}},{\widehat{{\varvec{\eta }}}},{\widehat{{\mathcal {B}}}};i\right)&=\oint _{\Sigma _1}\alpha ^{\mathcal {N}}_\nu \left( {\widehat{{\mathcal {B}}}}\right) {\mathrm {d}}_{\Sigma _1}{\widehat{{\mathcal {B}}}}^\nu +\oint _{\Sigma _1}\textsf {T}{\widetilde{\phi }}_y^*\langle i^*{\varvec{\eta }},{\mathcal {V}}(i^*{\varvec{X}},{\mathcal {B}})\rangle \nonumber \\&\quad +\oint _{\Sigma _1}({\widehat{R}}_{\Sigma _1})_\nu \left( y,{\widehat{{\mathcal {B}}}}\right) {\mathrm {d}}_{\mathcal {N}}y^\nu . \end{aligned}$$
(171)

The corresponding auxiliary formal global observable is given by

$$\begin{aligned} {\widehat{{\mathcal {O}}}}_{\Sigma _1,y}\left( {\widehat{{\varvec{X}}}},{\widehat{{\varvec{\eta }}}};i\right) =\int _{{\widehat{{\mathcal {L}}}}} {\mathscr {D}}\left[ {\widehat{{\mathcal {B}}}}_{(0)}\right] {\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\widehat{{\mathcal {S}}}}^{\mathrm {global}}_{\Sigma _1,y}\left( {\widehat{{\varvec{X}}}},{\widehat{{\varvec{\eta }}}},{\widehat{{\mathcal {B}}}}_{(0)};i\right) }, \end{aligned}$$
(172)

where we use the gauge-fixing Lagrangian

$$\begin{aligned} {\widehat{{\mathcal {L}}}}={{\,\mathrm{Map}\,}}_{{\mathrm {Mnf}}}(\Sigma _1,T_y{\mathcal {N}})\subset {{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _1,T_y{\mathcal {N}})\cong \Omega ^\bullet (\Sigma _1)\otimes T_y{\mathcal {N}}. \end{aligned}$$
(173)

If we assume that \(({\mathcal {N}},\omega _{\mathcal {N}})\) can be geometrically quantized to a space of states \({\mathcal {H}}\) and \({\mathcal {V}}\) is quantized to an operator-valued vector field \({\varvec{{\mathcal {V}}}}\in {{\,\mathrm{End}\,}}({\mathcal {H}})\otimes {\mathfrak {X}}(M)\) such that \([\pi ,{\varvec{{\mathcal {V}}}}]_{{\mathrm {SN}}}+R\wedge {\varvec{{\mathcal {V}}}}+{\mathrm {i}}\hbar {\varvec{{\mathcal {V}}}}\wedge {\varvec{{\mathcal {V}}}}=0\), then we get that

$$\begin{aligned} {\widehat{{\mathcal {O}}}}_{\Sigma _1,x}\left( {\widehat{{\varvec{X}}}},{\widehat{{\varvec{\eta }}}};i\right) ={{\,\mathrm{Tr}\,}}_{\mathcal {H}}{\mathcal {P}}\exp \left( \frac{{\mathrm {i}}}{\hbar }\oint _{\Sigma _1} \widehat{\langle i^*{\varvec{\eta }},{\varvec{{\mathcal {V}}}}(i^* {\varvec{X}})\rangle }\right) , \end{aligned}$$
(174)

which is, by Proposition 8.10, indeed a formal global observable. Here we have chosen an exponential map for the base of the target of the Poisson sigma model \(TM\rightarrow M\) with background field \(x\in M\).

Wilson Surfaces and Their Formal Global Extension

BF Theory and Wilson Surfaces

Let G be a Lie group and denote by \({\mathfrak {g}}\) its Lie algebra. Moreover, consider a principal G-bundle P over some d-manifold \(\Sigma _d\) and construct the adjoint bundle of P, denoted by \({\mathrm {ad}}P\), given as the frame bundle \(P\times ^{\mathrm {Ad}}{\mathfrak {g}}\) with respect to the adjoint representation \({\mathrm {Ad}}:G\rightarrow {\mathrm {Aut}}({\mathfrak {g}})\) and let \({\mathrm {ad}}^*P\) denote its coadjoint bundle. Let \({\mathscr {A}}\) be the affine space of connection 1-forms on P and \({\mathscr {G}}\) the group of gauge transformations. For a connection \(A\in {\mathscr {A}}\), let \({\mathrm {d}}_A\) be the covariant derivative on \(\Omega ^\bullet (\Sigma _d,{\mathrm {ad}}P)\) and \(\Omega ^\bullet (\Sigma _d,{\mathrm {ad}}^*P)\). Let \(A\in {\mathscr {A}}\) and \(B\in \Omega ^{d-2}(M,{\mathrm {ad}}^*P)\) and define the BF action by

$$\begin{aligned} S(A,B):=\int _{\Sigma _d} \langle B,F_A\rangle , \end{aligned}$$
(175)

where \(\langle ,\rangle \) denotes the extension of the adjoint and coadjoint type for the canonical pairing between \({\mathfrak {g}}\) and \({\mathfrak {g}}^*\) to differential forms.

Remark 9.1

(Abelian BF theory). The abelian BF action, i.e., the action for the case where \({\mathfrak {g}}={\mathbb {R}}\), in fact arises as the unperturbed part of many different AKSZ theories such as the Poisson sigma model or Chern–Simons theory. In fact, for the abelian case we have \((A,B)\in \Omega ^\bullet (\Sigma _d)[1]\oplus \Omega ^\bullet (\Sigma _d)[d-1]\) such that \(F_A={\mathrm {d}}A\) and thus we get an action \(S=\int _{\Sigma _d}B\wedge {\mathrm {d}}A\).

The solutions to the Euler–Lagrange equations \(\delta S=0\) for S defined as in (226) are given by

$$\begin{aligned} {\mathrm {M}}_{{\mathrm {cl}}}=\left\{ (A,B)\in {\mathscr {A}}\times \Omega ^{d-2}(\Sigma _d,{\mathrm {ad}}^*P)\,\big |\, F_A=0,{\mathrm {d}}_AB=0\right\} \end{aligned}$$
(176)

Remark 9.2

One can check that the BF action is invariant under the action of

$$\begin{aligned} {{\widetilde{\mathscr {G}}}}:={\mathscr {G}}\rtimes \Omega ^{d-3}(\Sigma _d,{{\,\mathrm{ad}\,}}^* P), \end{aligned}$$
(177)

where \({\mathscr {G}}\) acts on \(\Omega ^{d-3}(\Sigma _d,{{\,\mathrm{ad}\,}}^* P)\) by the coadjoint action. For \((g,\sigma )\in {\widetilde{\mathscr {G}}}\) and \((A,B)\in {\mathscr {A}}\times \Omega ^{d-2}(\Sigma _d,{{\,\mathrm{ad}\,}}^*P)\) we have an action

$$\begin{aligned} A\mapsto A^g,\qquad B\mapsto B^{(g,\sigma )}={{\,\mathrm{Ad}\,}}^*_{g^{-1}}B+{\mathrm {d}}_{A^g}\sigma . \end{aligned}$$
(178)

It is then easy to check that \(S(A^g,B^{(g,\sigma )})=S(A,B)\).

Consider an embedded submanifold \(i:\Sigma _{d-2}\hookrightarrow \Sigma _d\) and consider the pullback bundle of P by i according to the diagram

figure a

We can now formulate an important type of classical action which is important for the study of higher-dimensional knots [27].

Definition 9.3

(Wilson surface action). The Wilson surface action is given by

$$\begin{aligned} W(\alpha ,\beta ,A,B;i):=\int _{\Sigma _{d-2}}\langle \alpha ,{\mathrm {d}}_{i^*A}\beta +i^*B\rangle , \end{aligned}$$
(179)

where \(\alpha \in \Omega ^0(\Sigma _{d-2},{{\,\mathrm{ad}\,}}i^*P)\) and \(\beta \in \Omega ^{d-3}(\Sigma _{d-2},{{\,\mathrm{ad}\,}}^*i^*P)\).

Definition 9.4

(Wilson surface observable). The Wilson surface observable is given by

$$\begin{aligned} {\mathcal {W}}_{\Sigma _{d-2}}(A,B;i):=\int {\mathscr {D}}[\alpha ]{\mathscr {D}}[\beta ]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }W(\alpha ,\beta ,A,B;i)} \end{aligned}$$
(180)

Remark 9.5

The expectation values of Wilson surface observables in fact give certain higher-dimensional knot invariants [26]. These invariants are based on the construction of invariants by Bott [14] giving the generalization to a family of isotopy invariants for long knots \({\mathbb {R}}^{n}\hookrightarrow {\mathbb {R}}^{n+2}\) for odd \(n\ge 3\), which are based on constructions involving combinations of configuration space integrals. In [57] it was proven that these invariants are of finite type for the case of long ribbon knots and that they are related to the Alexander polynomial for these type of knots. Further generalizations based on this construction, in particular for rectifiable knots, have been given in [43, 44].

BV Formulation of BF Theory

We can consider BF theory in terms of its BV extension. The BV space of fields is given by

$$\begin{aligned} {\mathcal {F}}_{\Sigma _d}=\Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\oplus \Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}^*P)[d-2], \end{aligned}$$
(181)

where \({\mathscr {A}}=\Omega ^1(\Sigma _d,{{\,\mathrm{ad}\,}}P)\). We will denote the superfields in \({\mathcal {F}}_{\Sigma _d}\) by \(({\varvec{A}},{\varvec{B}})\). Note that there is an induced Lie bracket \([[,]]\) on \(\Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\) which is induced by the Lie bracket on \({\mathfrak {g}}\).

Remark 9.6

If we consider local coordinates on \({\mathfrak {g}}\) with corresponding basis \((e_i)\), we have

$$\begin{aligned} {[}[a,b]]=(-1)^{{\mathrm {gh}}(a)\deg (b)}a^ib^jf_{ij}^ke_k, \end{aligned}$$
(182)

where \(f_{ij}^k\) denotes the structure constants of \({\mathfrak {g}}\).

Moreover, for \({\varvec{A}}\in \Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\) we get the curvature

$$\begin{aligned} {\varvec{F}}_{{\varvec{A}}}=F_{A_0}+{\mathrm {d}}_{A_0}{\varvec{a}}+\frac{1}{2}[[{\varvec{a}},{\varvec{a}}]], \end{aligned}$$
(183)

where \(A_0\) is any reference connection and \({\varvec{a}}:={\varvec{A}}-A_0\in \Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\).

Definition 9.7

(BV action for BF theory). The BV action for BF theory is defined by

$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}({\varvec{A}},{\varvec{B}})=\int _{\Sigma _d}\langle \langle {\varvec{B}},{\varvec{F}}_{{\varvec{A}}}\rangle \rangle , \end{aligned}$$
(184)

where \(\langle \langle ,\rangle \rangle \) is the extension to forms of the adjoint and coadjoint type of the canonical pairing between \({\mathfrak {g}}\) and \({\mathfrak {g}}^*\). For two forms ab we have

$$\begin{aligned} \langle \langle a,b\rangle \rangle =(-1)^{{\mathrm {gh}}(a)\deg (b)}\langle a,b\rangle , \end{aligned}$$
(185)

We can see that

$$\begin{aligned} {\mathcal {F}}_{\Sigma _d}=T^*[-1]\Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1], \end{aligned}$$
(186)

hence we have a canonical symplectic structure \(\omega _{\Sigma _d}\) on \({\mathcal {F}}_{\Sigma _d}\). Similarly as before, let us denote the odd Poisson bracket induced by \(\omega _{\Sigma _d}\) by \(\{,\}_{\omega _{\Sigma _d}}\) and note that \({\mathcal {S}}_{\Sigma _d}\) satisfies the CME

$$\begin{aligned} \left\{ {\mathcal {S}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d}\right\} _{\omega _{\Sigma _d}}=0. \end{aligned}$$
(187)

The cohomological vector field \(Q_{\Sigma _d}\) is given as the Hamiltonian vector field of \({\mathcal {S}}_{\Sigma _d}\), thus \(Q_{\Sigma _d}=\left\{ {\mathcal {S}}_{\Sigma _d},\right\} _{\omega _{\Sigma _d}}\). Note that

$$\begin{aligned} Q_{\Sigma _d}({\varvec{A}})=(-1)^d{\varvec{F}}_{{\varvec{A}}},\qquad Q_{\Sigma _d}({\varvec{B}})=(-1)^d{\mathrm {d}}_{{\varvec{A}}}{\varvec{B}}. \end{aligned}$$
(188)

If we choose a volume element \(\mu \) which is compatible with \(\omega _{\Sigma _d}\), we can define the BV Laplacian by

$$\begin{aligned} \Delta :f\mapsto \frac{1}{2}{{\,\mathrm{div}\,}}_\mu \{f,\}_{\omega _{\Sigma _d}}. \end{aligned}$$
(189)

Then we can show that the QME holds:

$$\begin{aligned} \delta _{{\mathrm {BV}}}{\mathcal {S}}_{\Sigma _d}=\left\{ {\mathcal {S}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d}\right\} _{\omega _{\Sigma _d}}-2{\mathrm {i}}\hbar \Delta {\mathcal {S}}_{\Sigma _d}=0. \end{aligned}$$
(190)

This is in fact true since \(\Delta {\mathcal {S}}_{\Sigma _d}=0\). Moreover, as expected, we have \(\delta _{{\mathrm {BV}}}^2=0\).

Formal Global BF Theory from the AKSZ Construction

Let us consider the case of abelian BF theory. Note that in this case the Wilson surface action is given by

$$\begin{aligned} W(\alpha ,\beta ,A,B;i):=\int _{\Sigma _{d-2}} \alpha ({\mathrm {d}}\beta +i^*B), \end{aligned}$$
(191)

where \({\mathrm {d}}\) is the de Rham differential on \({\mathbb {R}}\). Solving the Euler–Lagrange equations for \(\delta W=0\), we get that the ciritical points are solutions to

$$\begin{aligned} {\mathrm {d}}\alpha&=0, \end{aligned}$$
(192)
$$\begin{aligned} {\mathrm {d}}\beta +i^*B&=0. \end{aligned}$$
(193)

We want to deal with B perturbatively, that means we can consider solutions to \({\mathrm {d}}\alpha ={\mathrm {d}}\beta =0\) instead and hence we look at solutions of the form \(\alpha =const\) and \(\beta =0\). This means that the constant field \(\alpha \) is going to take the place of the background field. The Wilson surface observable is then given by

$$\begin{aligned} {\mathcal {W}}_{\Sigma _{d-2}}(A,B;i)=\int {\mathscr {D}}[\alpha ]{\mathscr {D}}[\beta ]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }\int _{\Sigma _{d-2}}\alpha {\mathrm {d}}\beta }\int _{x\in {\mathbb {R}}}\mu (x){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }x\int _{\Sigma _{d-2}}i^*B}, \end{aligned}$$
(194)

where \(\mu \) is a volume element on the moduli space of classical solutions for the auxiliary theory which is given by

$$\begin{aligned} {\mathrm {M}}_{{\mathrm {cl}}}=\left\{ (\alpha ,\beta )\in \Omega ^0(\Sigma _{d-2})\oplus \Omega ^{d-3}(\Sigma _{d-2})\,\big |\, \alpha =const,\, \beta =0\right\} \cong {\mathbb {R}}. \end{aligned}$$
(195)

By abbuse of notation we will also denote the perturbation of \(\alpha \) around \(x\in {\mathbb {R}}\) by \(\alpha \). Moreover, if we assume that P is a trivial bundle, not for the abelian case, we get

$$\begin{aligned} {\mathcal {F}}_{\Sigma _d}&\cong \Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}[1]\oplus \Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}^*[d-2] \end{aligned}$$
(196)
$$\begin{aligned}&\cong {{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,{\mathfrak {g}}[1]\oplus {\mathfrak {g}}^*[d-2]). \end{aligned}$$
(197)

Remark 9.8

The assumption that P is trivial is similar to a formal lift, whereas the background field is given by a constant critical point of the form (x, 0) with constant background field \(x:T[1]\Sigma _d\rightarrow {\mathfrak {g}}[1]\oplus {\mathfrak {g}}^*[d-2]\). In fact, it induces a linear split theory as in Definition 6.7.

Remark 9.9

(CE complex and \(L_\infty \)-structure). Let \({\mathfrak {g}}\) be a Lie algebra and consider the differential graded algebra

$$\begin{aligned} {\mathrm {CE}}({\mathfrak {g}}):=\left( \bigwedge ^\bullet {\mathfrak {g}}^*,{\mathrm {d}}_{{\mathrm {CE}}}\right) \cong \left( C^\infty ({\mathfrak {g}}[1]),Q\right) . \end{aligned}$$
(198)

This is called the Chevalley–Eilenberg algebra of \({\mathfrak {g}}\) [28]. The real valued Chevalley–Eilenberg complex is given by

$$\begin{aligned} 0\rightarrow {{\,\mathrm{Hom}\,}}\left( \bigwedge ^0{\mathfrak {g}},{\mathbb {R}}\right) \xrightarrow {{\mathrm {d}}_{{\mathrm {CE}}}}{{\,\mathrm{Hom}\,}}\left( \bigwedge ^1{\mathfrak {g}},{\mathbb {R}}\right) \xrightarrow {{\mathrm {d}}_{{\mathrm {CE}}}}{{\,\mathrm{Hom}\,}}\left( \bigwedge ^2{\mathfrak {g}},{\mathbb {R}}\right) \xrightarrow {{\mathrm {d}}_{{\mathrm {CE}}}}\cdots \end{aligned}$$
(199)

endowed with the Chevalley–Eilenberg differential

$$\begin{aligned} {\mathrm {d}}_{\mathrm {CE}}:{{\,\mathrm{Hom}\,}}\left( \bigwedge ^{n}{\mathfrak {g}},{\mathbb {R}}\right) \rightarrow {{\,\mathrm{Hom}\,}}\left( \bigwedge ^{n+1}{\mathfrak {g}},{\mathbb {R}}\right) \end{aligned}$$

given by

$$\begin{aligned}&({\mathrm {d}}_{\mathrm {CE}}F)(X_1,\ldots ,X_{n+1}) \nonumber \\&\quad :=\sum _{j=1}^{n+1}(-1)^{j+1}X_iF(X_1,\ldots ,{\widehat{X}}_j,\ldots ,X_{n+1}) \nonumber \\&\qquad +\sum _{1\le j<k\le n+1}(-1)^{j+k}F([X_j,X_k],X_1,\ldots ,{\widehat{X}}_j,\ldots ,{\widehat{X}}_k,\ldots ,X_{n+1}), \end{aligned}$$
(200)

where the hat means that these elements are omitted. Denote by \((\xi ^{i})\) the coordinates on \({\mathfrak {g}}[1]\) of degree \(+1\). Then Q has to be of the form

$$\begin{aligned} Q=-\frac{1}{2}f_{ij}^k\xi ^{i}\xi ^{j}\frac{\partial }{\partial \xi ^{k}}, \end{aligned}$$

where \(f_{ij}^k\) are the structure constants of \({\mathfrak {g}}\). Note that a function \(F\in {{\,\mathrm{Hom}\,}}\left( \bigwedge ^n{\mathfrak {g}},{\mathbb {R}}\right) \) corresponds to an element in \(C_n^\infty ({\mathfrak {g}}[1])\) such that the Chevalley–Eilenberg differential is indeed mapped to Q under the isomorphism

$$\begin{aligned} F(X_{j_1}\wedge \ldots \wedge X_{j_n})=:F_{j_1\ldots j_n}\longleftrightarrow \frac{1}{n!}\xi ^{j_1}\cdots \xi ^{j_n}F_{j_1\ldots j_n}. \end{aligned}$$

In fact, for a graded vector space \({\mathfrak {g}}=\bigoplus _{k\in {\mathbb {Z}}}{\mathfrak {g}}_k\), the differential graded algebra \((C^\infty ({\mathfrak {g}}),Q)\) corresponds to an \(L_\infty \)-algebra which is actually given by the Chevalley–Eilenberg algebra \({\mathrm {CE}}({\mathfrak {g}}[-1])\) of the \(L_\infty \)-algebra \({\mathfrak {g}}[-1]\). The dual of the cohomological vector field Q is given by a codifferential D of homogenous degree \(+1\) on \({\widehat{{\text {Sym}}}}({\mathfrak {g}})\cong {\widehat{{\text {Sym}}}}({\mathfrak {g}}[-1])\). The isomorphism is induced by the shift isomorphism \(s:{\mathfrak {g}}\xrightarrow {\sim } {\mathfrak {g}}[1]\). The codifferential D decomposes into a sum \(D=\sum _{j\ge 1}{\bar{D}}_j\) such that the restrictions

$$\begin{aligned} D_j:={\bar{D}}_j\big |_{{\widehat{{\text {Sym}}}}^j({\mathfrak {g}})}:{\widehat{{\text {Sym}}}}^j({\mathfrak {g}})\rightarrow {\mathfrak {g}} \end{aligned}$$

satisfy

$$\begin{aligned} \ell _j=(-1)^{\frac{1}{2}j(j-1)+1}s^{-1}\circ D_j\circ s^{\otimes j},\qquad \forall j\ge 1. \end{aligned}$$

Note that since \(Q^2=0\), we get \(D^2=0\). Such a codifferential induces a classical Grothendieck connection as in Sect. 6.1.

Remark 9.10

(\(L_\infty \)-structure on \(\Omega ^\bullet \)). If \({\mathfrak {g}}\) is endowed with a (curved) \(L_\infty \)-structure, we can view

$$\begin{aligned} \Omega ^\bullet (\Sigma _{d},{\mathfrak {g}})=\bigoplus _{\begin{array}{c} r+j=k \\ 0\le r\le d\\ j\in {\mathbb {Z}} \end{array}}\Omega ^r(\Sigma _d)\otimes {\mathfrak {g}}_{j} \end{aligned}$$

as a (curved) \(L_\infty \)-algebra. The \(L_\infty \)-structure arises as the linear extension of the higher brackets

$$\begin{aligned} {\hat{\ell }}_1(\alpha _1\otimes X_1)&:={\mathrm {d}}_{\Sigma _d}\alpha _1\otimes X_1+(-1)^{\deg (\alpha _1)}\alpha _1\otimes \ell _1(X_1) \end{aligned}$$
(201)
$$\begin{aligned} {\hat{\ell }}_n(\alpha _1\otimes X_1,\ldots ,\alpha _n\otimes X_n)&:=(-1)^{n\sum _{j=1}^{n}\deg (\alpha _j)+\sum _{j=0}^{n-2}\deg (\alpha _{n-j})\sum _{k=1}^{n-j-1}\deg (X_k)} \nonumber \\&\quad \times (\alpha _1\wedge \cdots \wedge \alpha _n)\otimes \ell _n(X_1,\ldots ,X_n) \end{aligned}$$
(202)

for \(n\ge 2\), \(\alpha _1,\ldots ,\alpha _n\in \Omega ^\bullet (\Sigma _d)\) and \(X_1,\ldots ,X_n\in {\mathfrak {g}}\). If \({\mathfrak {g}}\) is cyclic, and \(\Sigma _d\) is compact, oriented without boundary, there is a natural cyclic inner product on \(\Omega ^\bullet (\Sigma _d,{\mathfrak {g}})\) given by

$$\begin{aligned}&\langle \alpha _1\otimes X_1,\alpha _2\otimes X_2\rangle _{\Omega ^\bullet (\Sigma _d,{\mathfrak {g}})} \nonumber \\&\quad =(-1)^{\deg (\alpha _2)\deg (X_1)}\int _{\Sigma _d}\alpha _1\wedge \alpha _2\langle X_1,X_2\rangle _{{\mathfrak {g}}} \end{aligned}$$
(203)

for \(\alpha _1,\alpha _2\in \Omega ^\bullet (\Sigma _d)\) and \(X_1,X_2\in {\mathfrak {g}}\).

BV Extension of Wilson Surfaces

We will now construct the BV extended observable for the auxiliary codimension 2 theory in the case where P is a trivial bundle. Let

$$\begin{aligned} {\mathcal {F}}_{\Sigma _{d-2}}\cong \Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}[1]\oplus \Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}^*[d-2] \end{aligned}$$
(204)

endowed with the symplectic form \(\omega _{\Sigma _{d-2}}\) which induces the corresponding BV bracket \(\{,\}_{\omega _{\Sigma _{d-2}}}\). For auxiliary superfields \(({\varvec{\alpha }},{\varvec{\beta }})\in {\mathcal {F}}_{\Sigma _{d-2}}\) and ambient superfields \(({\varvec{A}},{\varvec{B}})\in {\mathcal {F}}_{\Sigma _d}\) we have the following definition:

Definition 9.11

(BV extended Wilson surface action). The BV extended Wilson surface action is given by

$$\begin{aligned} {\varvec{W}}^0_{\Sigma _{d-2}}({\varvec{\alpha }},{\varvec{\beta }},{\varvec{A}},{\varvec{B}};i)=\int _{\Sigma _{d-2}}\langle \langle {\varvec{\alpha }},{\mathrm {d}}_{i^*{\varvec{A}}}{\varvec{\beta }}+i^*{\varvec{B}}\rangle \rangle . \end{aligned}$$
(205)

Remark 9.12

As it was shown in [27], we can extend \({\varvec{W}}^0_{\Sigma _{d-2}}\), regarded as a function on embeddings \(i:\Sigma _{d-2}\hookrightarrow \Sigma _d\), to a form-valued function \({\varvec{W}}_{\Sigma _{d-2}}\) on these embeddings by setting

$$\begin{aligned} {\varvec{W}}_{\Sigma _{d-2}}({\varvec{\alpha }},{\varvec{\beta }},{\varvec{A}},{\varvec{B}};i):=\pi _*\langle \langle {\varvec{\alpha }},{\mathrm {d}}_{{\mathrm {ev}}^*{\varvec{A}}}{\varvec{\beta }}+{\mathrm {ev}}^*{\varvec{B}}\rangle \rangle , \end{aligned}$$
(206)

where \({\mathrm {ev}}\) denotes the evaluation map of embeddings \(\Sigma _{d-2}\hookrightarrow \Sigma _d\) and \(\pi _*\) denotes the integration along the fiber \(\Sigma _{d-2}\).

Proposition 9.13

([27]). The Wilson surface action satisfies a modified version of the dCME, i.e., we have

$$\begin{aligned} {\mathrm {d}}{\varvec{W}}_{\Sigma _{d-2}}-(-1)^d\left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _d}}-\frac{1}{2}\left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _{d-2}}}=0\nonumber \\ \end{aligned}$$
(207)

Remark 9.14

Proposition 9.13 is a consequense of the fact that

$$ {\mathrm {d}}\int _{\Sigma _{d-2}}=(-1)^d\int _{\Sigma _{d-2}}{\mathrm {d}}$$

and (188).

Denote by \(\Delta _{\Sigma _{d-2}}\) the BV Laplacian for the auxiliary theory. Then we get the following proposition.

Proposition 9.15

([27]). Define the vector field

$$\begin{aligned} {\varvec{Q}}_{\Sigma _{d-2}}=\left\{ {\varvec{W}}_{\Sigma _{d-2}},\right\} _{\omega _{\Sigma _{d-2}}}, \end{aligned}$$
(208)

which acts on generators by

$$\begin{aligned} {\varvec{Q}}_{\Sigma _{d-2}}({\varvec{\alpha }})=(-1)^d{\mathrm {d}}_{{\mathrm {ev}}^*{\varvec{A}}}{\varvec{\alpha }},\qquad {\varvec{Q}}_{\Sigma _{d-2}}({\varvec{\beta }})=(-1)^d({\mathrm {d}}_{{\mathrm {ev}}^*{\varvec{A}}}{\varvec{\beta }}+{\mathrm {ev}}^*{\varvec{B}}).\nonumber \\ \end{aligned}$$
(209)

Assume that the formal measure \({\mathscr {D}}[{\varvec{\alpha }}]{\mathscr {D}}[{\varvec{\beta }}]\) is invariant with respect to the vector fields (209). Then we have

$$\begin{aligned}&{\mathrm {d}}{\varvec{W}}_{\Sigma _{d-2}}-(-1)^d\left( \delta _{{\mathrm {BV}}}{\varvec{W}}_{\Sigma _{d-2}}+\frac{1}{2}\left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _d}}\right) \nonumber \\&\quad +\frac{1}{2}\left( \left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _{d-2}}}-2{\mathrm {i}}\hbar \Delta _{\Sigma _{d-2}}{\varvec{W}}_{\Sigma _{d-2}}\right) =0 \end{aligned}$$
(210)

Remark 9.16

Note that the assumption of invariance of the formal measure implies that \(\Delta _{\Sigma _{d-2}}{\varvec{W}}_{\Sigma _{d-2}}=0\).

Definition 9.17

(BV extended Wilson surface observable). We define the BV extended Wilson surface observable as

$$\begin{aligned} {\varvec{{\mathcal {W}}}}_{\Sigma _{d-2}}({\varvec{A}},{\varvec{B}};i)=\int {\mathscr {D}}[{\varvec{\alpha }}]{\mathscr {D}}[{\varvec{\beta }}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\varvec{W}}_{\Sigma _{d-2}}({\varvec{\alpha }},{\varvec{\beta }},{\varvec{A}},{\varvec{B}};i)} \end{aligned}$$
(211)

Formulation by Hamiltonian Q-Bundles

Let \({\mathcal {M}}={\mathfrak {g}}[1]\oplus {\mathfrak {g}}^*[d-2]\). Denote by \(x:{\mathfrak {g}}[1]\rightarrow {\mathfrak {g}}\) the degree 1 \({\mathfrak {g}}\)-valued coordinate on \({\mathfrak {g}}[1]\) and let \(x^*:{\mathfrak {g}}^*[d-2]\rightarrow {\mathfrak {g}}^*\) be the \({\mathfrak {g}}^*\)-valued coordinate on \({\mathfrak {g}}^*[d-2]\) of degree \(d-2\).

Then we can consider a trivial Hamiltonian Q-bundle over \({\mathcal {M}}\) given by the fiber data

$$\begin{aligned} {\mathcal {N}}&={\mathfrak {g}}\oplus {\mathfrak {g}}^*[d-3], \end{aligned}$$
(212)
$$\begin{aligned} {\mathcal {V}}_{\mathcal {E}}&=\left\langle [x,y],\frac{\partial }{\partial y}\right\rangle +\left\langle {{\,\mathrm{ad}\,}}_x^*y^*,\frac{\partial }{\partial y^*}\right\rangle +(-1)^d\left\langle x^*,\frac{\partial }{\partial y^*}\right\rangle , \end{aligned}$$
(213)
$$\begin{aligned} \omega _{\mathcal {N}}&=\left\langle \delta y^*,\delta y\right\rangle , \end{aligned}$$
(214)
$$\begin{aligned} \alpha _{\mathcal {N}}&=\langle y^*,\delta y\rangle , \end{aligned}$$
(215)
$$\begin{aligned} \Theta _{{\mathcal {E}}}&=\left\langle y^*,[x,y]\right\rangle +\langle x^*,y\rangle , \end{aligned}$$
(216)

where y is the \({\mathfrak {g}}\)-valued coordinate of degree 0 on \({\mathfrak {g}}\) and \(y^*\) is the \({\mathfrak {g}}^*\)-valued coordinate of degree \(d-3\) on \({\mathfrak {g}}^*[d-3]\). For an embedding \(i:\Sigma _{d-2}\hookrightarrow \Sigma _d\) we get the auxiliary theory

$$\begin{aligned} {\mathcal {F}}^{\mathcal {N}}_{\Sigma _{d-2}}&=\Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}\oplus \Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}^*[d-3], \end{aligned}$$
(217)
$$\begin{aligned} \omega ^{\mathcal {N}}_{\Sigma _{d-2}}&=(-1)^d\int _{\Sigma _{d-2}}\langle \delta {\varvec{y}}^*,\delta {\varvec{y}}\rangle , \end{aligned}$$
(218)
$$\begin{aligned} {\mathcal {S}}^{\mathcal {N}}_{\Sigma _{d-2}}&=\int _{\Sigma _{d-2}}\langle {\varvec{y}}^*,{\mathrm {d}}_{\Sigma _{d-2}}{\varvec{y}}\rangle +\int _{\Sigma _{d-2}}\langle {\varvec{y}}^*,[i^*{\varvec{A}},{\varvec{y}}]\rangle +\int _{\Sigma _{d-2}}\langle i^*{\varvec{B}},{\varvec{y}}\rangle . \end{aligned}$$
(219)

Note that \({\mathcal {M}}\) is a differential graded symplectic manifold with the following data:

$$\begin{aligned} Q_{\mathcal {M}}&=\left\langle \frac{1}{2}[x,x],\frac{\partial }{\partial x}\right\rangle +\left\langle {{\,\mathrm{ad}\,}}_x^*x^*,\frac{\partial }{\partial x^*}\right\rangle , \end{aligned}$$
(220)
$$\begin{aligned} \omega _{\mathcal {M}}&=\langle \delta x^*,\delta x\rangle , \end{aligned}$$
(221)
$$\begin{aligned} \alpha _{\mathcal {M}}&=\langle x^*,\delta x\rangle , \end{aligned}$$
(222)
$$\begin{aligned} \Theta _{\mathcal {M}}&=\frac{1}{2}\left\langle x^*,[x,x]\right\rangle . \end{aligned}$$
(223)

Hence the ambient theory is given by

$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}&=\Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}[1]\oplus \Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}^*[d-2]\ni ({\varvec{A}},{\varvec{B}}), \end{aligned}$$
(224)
$$\begin{aligned} \omega _{\Sigma _d}&=\int _{\Sigma _d}\langle \delta {\varvec{B}},\delta {\varvec{A}}\rangle , \end{aligned}$$
(225)
$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}&=\int _{\Sigma _d}\left\langle {\varvec{B}},{\varvec{F}}_{{\varvec{A}}}\right\rangle =\int _{\Sigma _d}\left\langle {\varvec{B}},{\mathrm {d}}_{\Sigma _d}{\varvec{A}}+\frac{1}{2}[{\varvec{A}},{\varvec{A}}]\right\rangle . \end{aligned}$$
(226)

Note that (226) is exactly the BF action as in Definition 9.7. In the case of abelian BF theory, i.e., when \({\mathfrak {g}}={\mathbb {R}}\), we get that \(Q_{\mathcal {M}}=0\), \(\Theta _{\mathcal {M}}=0\) and the ambient theory

$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}&=\Omega ^\bullet (\Sigma _d)[1]\oplus \Omega ^\bullet (\Sigma _d)[d-2], \end{aligned}$$
(227)
$$\begin{aligned} \omega _{\Sigma _d}&=\int _{\Sigma _d}\delta {\varvec{B}}\wedge \delta {\varvec{A}}, \end{aligned}$$
(228)
$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}&=\int _{\Sigma _d}{\varvec{B}}\wedge {\mathrm {d}}_{\Sigma _d}{\varvec{A}}. \end{aligned}$$
(229)

Remark 9.18

The constructions presented in this paper are expected to extend to manifolds with boundary. Using the constructions as in [25] together with the quantum BV-BFV formalism [21, 23], one can show how the formal global observables for split AKSZ sigma models on the boundary induce a more general gauge condition as the dQME which is called modified differential Quantum Master Equation (mdQME). This condition also handles the boundary part which arises as the ordered standard quantization \(\Omega \) of the boundary action \({\mathcal {S}}^\partial \) of dgeree \(+1\), induced by the underlying BFV manifold, plus some higher degree terms. The mdQME is then given as some annihilation condition for the formal global boundary observable \({\mathcal {O}}^\partial \). In fact, it is annihilated by the quantum Grothendieck BFV operator \(\nabla _{\mathsf {G}}:={\mathrm {d}}-{\mathrm {i}}\hbar \Delta +\frac{{\mathrm {i}}}{\hbar }\Omega \) (see [24, 25]), which means that \(\nabla _{\mathsf {G}}{\mathcal {O}}^\partial =0\).

Notes

  1. Typically, this is an infinite-dimensional manifold. However, there are certain cases where this is a finite-dimensional manifold, e.g., if we consider the moduli of flat connections on a compact, oriented 2-manifold with holonomies on the boundary according to Atiyah and Bott [2] which is of importance regarding BF theory.

  2. Usually, the BV space of fields is given by the \((-1)\)-shifted cotangent bundle of the BRST space of fields, i.e., \({\mathcal {F}}_{{\mathrm {BV}}}=T^*[-1]{\mathcal {F}}_{\mathrm {BRST}}\).

  3. We want the space of fields \({\mathcal {F}}\) to be endowed with a natural measure.

  4. A cyclic \(L_\infty \)-algebra [40] is an \(L_\infty \)-algebra \({\mathfrak {g}}\) endowed with a non-degenerate, symmetric, bilinear pairing \(\langle ,\rangle _{\mathfrak {g}}:{\mathfrak {g}}\oplus {\mathfrak {g}}\rightarrow {\mathbb {R}}\) such that

    $$\begin{aligned} \langle X_1,\ell _{n+1}(X_2,\ldots ,X_{n+1})\rangle _{\mathfrak {g}}= & {} (-1)^{n+n(\deg (X_1)+\deg (X_{n+1}))+\deg (X_{n+1})\sum _{j=1}^n\deg (X_j)} \\&\langle X_{n+1},\ell _{n}(X_1,\ldots ,X_n)\rangle _{\mathfrak {g}}, \end{aligned}$$

    for \(X_1,\ldots ,X_{n+1}\in {\mathfrak {g}}\) and where \((\ell _n)\) denote the n-ary brackets on \({\mathfrak {g}}\). In the case of a Q-manifold the cyclic inner product corresponds to a symplectic structure.

  5. This is the theory induced by the action term \({\mathcal {S}}_{\mathrm {MC}}(\Psi )=\sum _{j\ge 1}\frac{1}{(j+1)!}\langle \Psi ,\ell _j(\Psi ,\ldots ,\Psi )\rangle _{\mathfrak {g}}\) for a cyclic \(L_\infty \)-algebra \({\mathfrak {g}}\) endowed with an inner product \(\langle ,\rangle _{\mathfrak {g}}\). Here \((\ell _j)\) denotes the family of j-ary brackets on \({\mathfrak {g}}\). The stationary locus of this action is given by solutions of the homotopy Maurer–Cartan equation \(\sum _{j\ge 1}\frac{1}{j!}\ell _j(\Psi ,\ldots ,\Psi )=0\). In fact, the deformed Lagrangian (still classical)

    $$\begin{aligned} {\mathcal {S}}(\Psi )=\frac{1}{2}\langle \Psi ,Q(\Psi )\rangle _{\mathfrak {g}}+\sum _{j\ge 1}\frac{1}{(j+1)!}\langle \Psi ,\ell _j(\Psi ,\ldots ,\Psi )\rangle _{\mathfrak {g}} \end{aligned}$$

    satisfies the CME.

  6. More precisely, \({{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}\) denotes the right adjoint functor to the Cartesian product in the category of graded manifolds with a fixed factor. On objects XYZ we have \({{\,\mathrm{Hom}\,}}(X,{{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(Y,Z))={{\,\mathrm{Hom}\,}}(X\times Y,Z)\), where \({{\,\mathrm{Hom}\,}}\) denotes the set of graded manifold morphisms.

  7. These are background choices for classical fields that are not fixed by the boundary conditions and the Euler–Lagrange equations.

  8. An \(L_\infty \)-algebra \({\mathfrak {g}}\) is called curved if there exists an operation \(\ell _0:{\mathbb {R}}\rightarrow {\mathfrak {g}}\) of degree 0. In particular, the strong homotopy Jacobi identity implies that \(\ell _1\circ \ell _1=\pm \ell _2(\ell _0,)\), meaning that the unary bracket \(\ell _1\) does not square to zero anymore, as it is the case for usual \(L_\infty \)-algebras. In this case we say that \(\ell _1\) has non-vanishing curvature, thus the name “curved”.

  9. Since \(\Gamma \left( \bigwedge ^\bullet T^*M\otimes {\widehat{{\text {Sym}}}}(T^*M)\right) \) is the algebra of functions on the formal graded manifold \(T[1]M\oplus T[0]M\), the differential D turns this graded manifold into a differential graded manifold. In particular, since D vanishes on the body of the graded manifold, we can linearize at each \(x\in M\) and obtain an \(L_\infty \)-structure on \(T_xM[1]\oplus T_xM\).

  10. Note that extending to the case with auxiliary fields \({\mathcal {F}}\times {\mathcal {F}}^{{\mathrm {aux}}}\), we can extend \(\pi \) to a map \(\pi ^{\mathcal {N}}=\pi \times {\mathrm {id}}_{{\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}}:{\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}\times {\mathcal {F}}^{\mathcal {N}}_{\Sigma _k}\rightarrow {{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _k,{\mathcal {E}})\).

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Acknowledgements

Open access funding provided by University of Zurich I would like to thank Alberto Cattaneo for useful comments and remarks on a first draft of these notes.

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Moshayedi, N. Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces. Ann. Henri Poincaré 21, 2951–2995 (2020). https://doi.org/10.1007/s00023-020-00944-y

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