#  Quantum Chern–Simons theories on cylinders: BV-BFV partition functions

Alberto S. Cattaneo  Institut für Mathematik, Universität Zürich
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland<EMAIL_ADDRESS>,
Pavel Mnev  University of Notre Dame  St. Petersburg Department of V. A.
Steklov Institute of Mathematics of the Russian Academy of Sciences
<EMAIL_ADDRESS>and  Konstantin Wernli  Centre for Quantum Mathematics, IMADA,
University of Southern Denmark<EMAIL_ADDRESS>
###### Abstract.

We compute partition functions of Chern–Simons type theories for cylindrical
spacetimes $I\times\Sigma$, with $I$ an interval and $\dim\Sigma=4l+2$, in the
BV-BFV formalism (a refinement of the Batalin–Vilkovisky formalism adapted to
manifolds with boundary and cutting–gluing). The case $\dim\Sigma=0$ is
considered as a toy example. We show that one can identify—for certain choices
of residual fields—the “physical part” (restriction to degree zero fields) of
the BV-BFV effective action with the Hamilton–Jacobi action computed in the
companion paper [16], without any quantum corrections. This Hamilton–Jacobi
action is the action functional of a conformal field theory on $\Sigma$. For
$\dim\Sigma=2$, this implies a version of the CS-WZW correspondence. For
$\dim\Sigma=6$, using a particular polarization on one end of the cylinder,
the Chern–Simons partition function is related to Kodaira–Spencer gravity
(a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the
semiclassical result by Gerasimov and Shatashvili.

###### Key words and phrases:

(Generalized) Hamilton–Jacobi action, Chern–Simons theory, Batalin–Vilkovisky
formalism, Segal–Bargmann transform, Wess–Zumino–Witten model, nonlinear
(Hitchin) phase space polarization, Kodaira–Spencer (BCOV) action.

###### 2020 Mathematics Subject Classification:

57R56, 81T70, 70H20, 81T13 (Primary); 81T18, 53D50, 53D22, 81T40, 81T30,
32Q25, 32G05 (Secondary).

This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss
National Science Foundation. A.S.C. and K.W. acknowledge partial support of
SNF Grants No. 200020_192080 and 200020_172498/1. K. W. also acknowledges
support from a BMS Dirichlet postdoctoral fellowship and the SNF
Postdoc.Mobility grant P2ZHP2_184083, and would like to thank the Humboldt-
Universität Berlin, in particular the group of Dirk Kreimer, and the
university of Notre Dame for their hospitality.

###### Contents

  1. 1 Introduction
    1. 1.1 Chern–Simons theory in the BV-BFV formalism
    2. 1.2 Main results of the paper
      1. 1.2.1 Three-dimensional abelian Chern–Simons theory
      2. 1.2.2 Three-dimensional nonabelian Chern–Simons theory and CS-WZW correspondence
      3. 1.2.3 Seven-dimensional Chern–Simons theory and the CS-BCOV correspondence
    3. 1.3 Structure of the paper
    4. 1.4 Outlook
    5. 1.5 Notations and Conventions
  2. 2 Constrained systems and generalized Hamilton–Jacobi actions
  3. 3 BV-BFV approach warm-up: 1D abelian Chern–Simons
    1. 3.1 Holomorphic-to-holomorphic boundary conditions
    2. 3.2 Antiholomorphic-to-holomorphic boundary conditions
  4. 4 BV-BFV approach to 3D abelian Chern–Simons on a cylinder
    1. 4.1 Holomorphic-to-holomorphic boundary conditions
      1. 4.1.1 Comparison with Hamilton–Jacobi action
      2. 4.1.2 Quantum master equation
    2. 4.2 Antiholomorphic-to-holomorphic boundary conditions
      1. 4.2.1 Partial integral over residual fields and comparison with Hamilton–Jacobi action
      2. 4.2.2 Full integral over residual fields
      3. 4.2.3 Quantum master equation
  5. 5 Chern–Simons theory in “parallel ghost polarization”
    1. 5.1 One-dimensional Chern–Simons theory with values in a cochain complex
      1. 5.1.1 Setup
      2. 5.1.2 Parallel ghost polarization
      3. 5.1.3 Gauge fixing
      4. 5.1.4 Effective action
      5. 5.1.5 Quantum master equation
    2. 5.2 General polarizations
      1. 5.2.1 Splitting the fields
      2. 5.2.2 Effective action
      3. 5.2.3 Modified quantum master equation
    3. 5.3 3D nonabelian Chern–Simons with parallel ghost polarization and antiholomorphic-to-holomorphic polarization in ghost degree zero
      1. 5.3.1 Abelian case
      2. 5.3.2 Nonabelian case
      3. 5.3.3 Group-valued parametrization of the residual field
      4. 5.3.4 A comment on ghost wheels
      5. 5.3.5 Effective action vs. Hamilton–Jacobi
      6. 5.3.6 Quantum master equation
      7. 5.3.7 “Vertical” Wilson lines
      8. 5.3.8 The CS-WZW correspondence: WZW theory as an effective theory of Chern–Simons
      9. 5.3.9 An application: Holomorphic factorization of the WZW theory
    4. 5.4 3D nonabelian Chern–Simons theory in holomorphic-to-holomorphic polarization
  6. 6 BV-BFV approach to higher-dimensional Chern–Simons theories
    1. 6.1 Partition functions in $4l+3$-dimensional CS theories
    2. 6.2 Parallel ghost polarization
      1. 6.2.1 Pushforward over residual fields
    3. 6.3 7D Chern–Simons and Kodaira–Spencer action functional
      1. 6.3.1 General polarizations in $4l+3$-dimensional Chern–Simons theory
      2. 6.3.2 Hitchin polarization on 6-dimensional manifolds and effective action
      3. 6.3.3 Comparison with Kodaira–Spencer gravity
  7. A Segal–Bargmann transform via BV-BFV
    1. A.1 Aside: contour integration in the complexified space of fields, a lattice toy model
  8. B Kodaira–Spencer theory
    1. B.1 Some operations on complex forms
    2. B.2 The generating function for Hitchin polarization
    3. B.3 Deformations of complex structures
    4. B.4 Kodaira–Spencer action

##  1\. Introduction

This paper is a sequel to the paper “Constrained systems, generalized
Hamilton–Jacobi actions, and quantization” [16] by the same authors (but can
be read independently).

As announced in [16], the main result of this paper is the explicit
computation of the perturbative partition functions of Chern–Simons theories
on cylinders $I\times\Sigma$, with respect to various boundary polarizations.
Their restriction to degree zero fields turns out to be the exponential of the
corresponding Hamilton–Jacobi action, defined in [16] and recalled in Section
2, without any quantum corrections.

Interestingly, the Hamilton–Jacobi actions of the theories we consider can be
related to action functionals of conformal field theories on $\Sigma$. This
means that the partition function of Chern–Simons theories (with certain
boundary conditions) can be identified with the partition function of a
conformal field theory (coupled to sources) — a property that one might call
“holographic duality.” In that terminology, among other results, we show the
following:

  * •

The holographic dual theory of 3D abelian Chern–Simons theory is the 2D free
boson CFT, see Section 1.2.1 (while for a different choice of boundary
polarization, we obtain the beta-gamma system as the dual, see (24)).

  * •

The holographic dual of 3D nonabelian Chern–Simons theory is WZW theory (see
Section 1.2.2). In particular, the bulk-boundary version of the
Batalin–Vilkovisky master equation (referred to below as the modified quantum
master equation) corresponds to the Polyakov–Wiegmann formula for the WZW
action functional.

  * •

The holographic dual of 7D Chern–Simons theory is a free 2-form theory for the
“standard” polarization and the Kodaira–Spencer gravity for a particular
nonlinear polarization (see Section 1.2.3).

A remark on the terminology: The term “holographic duality” is often used for
the case where the bulk theory is a theory of quantum gravity, e.g., a string
theory, such as in the celebrated AdS/CFT correspondence [33], [43] and its
more general variant, the gauge/gravity correspondence (see [18] for a
review). The bulk/boundary correspondences we discuss below were, in a
different context, discovered earlier, see for instance [41], [26]. They can
be interpreted as special cases of holography, thinking of Chern-Simons theory
as a string theory [44].

The first motivating point for this paper and its prequel [16], suggested to
us by S. Shatashvili, concerned precisely the last item in the list above:
namely, the systematical understanding of the relation between 7D abelian
Chern–Simons theory and 6D Kodaira–Spencer [31] gravity (otherwise known as
BCOV theory [9]) from the BV-BFV perspective. At the semiclassical level, the
relation is a result of Gerasimov–Shatashvili [26] (see also our review in
[16, Section 7.6]). In this paper, we explore the perturbative BV-BFV
quantization and show that, for an appropriate choice of gauge fixing, no
quantum corrections are added to the semiclassical result. We thus prove the
conjecture put forward by Gerasimov and Shatashvili in their original paper.

There are two other key points that motivated this paper and its companion
paper [16]. Firstly, we were interested in studying in detail the bulk-
boundary or “holographic” correspondences mentioned above. In this paper we
prove that in these special cases the boundary theory is simply an effective
theory of the bulk theory, in the sense that the bulk fields have been
partially integrated out. Given the results in [16], we expect that the
effective action viewpoint can explain more general bulk-boundary
correspondences. To put it in clear words: holographic duality means that the
boundary theory is the semiclassical limit of a certain effective action of
the bulk theory. In the theories we consider in this paper, this semiclassical
limit is exact,111In our framework, this is due to the fact that the
constraints are affine functions of the variables describing the field
configurations on either boundaries (such theories are called “biaffine” in
[16]). but in general there is of course no reason to expect this.

Secondly, partition functions on cylinders can be interpreted as kernels of
generalized Segal–Bargmann transforms (see Appendix A). They are of interest
because, in a $d$-dimensional theory, they describe how a state on a
$(d-1)$-dimensional manifold $\Sigma$ depends on the choice of a polarization.
One way to interpret our results is that in our examples those generalized
Segal–Bargmann transforms (in general, it is only their semiclassical limit)
can be described by another quantum field theory that lives on $\Sigma$.

Our results show that in both those cases – seemingly unrelated at first
glance – the corresponding boundary theory is given by a (generalized)
Hamilton–Jacobi action. The BV-BFV formalism turns out to be a clear
conceptual framework in which one can state and prove those results from first
principles: the only inputs required are those of a local field theory, namely
a space of fields $F_{M}$ and an action functional $S_{M}\colon
F_{M}\to\mathbb{R}$. From a different perspective, this paper can also be
viewed as an invitation to learn the formalism.

Before passing to a detailed description of our results, as a primer on the
BV-BFV formalism we give a brief recollection of abelian Chern–Simons theory
in the BV-BFV formalism, which can be safely skipped by readers familiar with
the subject.

###  1.1. Chern–Simons theory in the BV-BFV formalism

We consider abelian Chern–Simons theory in dimensions $d=4l+3$ with $l$ a
positive integer. For a $d$-dimensional spacetime manifold $N$ (possibly with
boundary), the space of fields is defined as $F_{N}=\Omega^{2l+1}(N)$ and the
action functional is

$S_{N}[A]=\frac{1}{2}\int_{N}A\wedge dA.$

In dimension $d=3$, we also consider nonabelian Chern–Simons theory. Here
there is a structure Lie algebra $\mathcal{G}$ of coefficients endowed with a
nondegenerate invariant pairing $\langle\cdot,\cdot\rangle$. The space of
fields on $N$ is then the space of $\mathcal{G}$-valued 1-forms
$F_{N}=\Omega^{1}(N,\mathcal{G})$ (thought of as the space of connections on a
trivial principal $G$-bundle $N\times G$ with $G$ the connected and simply
connected Lie group integrating $\mathcal{G}$). The action functional is

$S_{N}[A]=\int_{N}\frac{1}{2}\langle A,dA\rangle+\frac{1}{6}\langle
A,[A,A]\rangle.$

Since these theories are gauge theories, to define the perturbative partition
function we need a gauge fixing formalism. In this paper, we will use the BV-
BFV formalism, the modification of the Batalin–Vilkovisky (BV) formalism for
manifolds with boundary introduced by two of the authors together with N.
Reshetikhin in [12, 15]. Let us briefly explain this formalism by means of our
main example.

The BV-BFV extension of abelian Chern–Simons theory has $\mathbb{Z}$-graded
space of fields $\mathcal{F}_{N}=\Omega^{\bullet}(N)[2l+1]$. This notation is
shorthand for saying that a homogeneous form $\omega$ is assigned ghost number
$\mathrm{gh}(\omega)=2l+1-\deg(\omega)$, so that all forms have _total degree_
$\mathrm{gh}+\deg=2l+1$. In particular $\mathcal{F}^{0}_{N}=F_{N}$. The space
$\mathcal{F}_{N}$ is an odd symplectic vector space with odd symplectic form

$\omega_{N}(\mathcal{A},\mathcal{A}^{\prime})=\int_{N}\mathcal{A}\wedge\mathcal{A}^{\prime},$

where $\mathcal{A},\mathcal{A}^{\prime}$ are nonhomogeneous differential forms
and only the top degree part contributes to the integral.222The symplectic
form is odd because it pairs components of
$\mathcal{A}$,$\mathcal{A}^{\prime}$ of opposite parity, which in turn is due
to the fact that $\dim N$ is odd. The BV extended action functional of abelian
Chern–Simons theory is

$\mathcal{S}_{N}[\mathcal{A}]=\frac{1}{2}\int_{N}\mathcal{A}\wedge
d\mathcal{A}.$

In particular, restricting to forms of ghost number 0, we recover the
classical action $S_{N}[A]$.

If $\partial N=\emptyset$, then $(\mathcal{S}_{N},\mathcal{S}_{N})=0$, where
$(\cdot,\cdot)$ denotes the Poisson bracket induced by $\omega_{N}$. This
equation is called classical master equation in the BV formalism, and it
implies $Q_{N}^{2}=0$, where

$Q_{N}=\int_{N}d\mathcal{A}\wedge\frac{\delta}{\delta\mathcal{A}}$

is the odd hamiltonian vector field of $\mathcal{S}_{N}$.

If $\partial N\neq\emptyset$, then we assign additional BFV333BFV is short for
Batalin–Fradkin–Vilkovisky [7, 20].  data to the boundary. The space of
boundary fields is $\mathcal{F}^{\partial}_{\partial
N}=\Omega^{\bullet}(\partial N)[2l+1]$ with even symplectic form

$\omega^{\partial}_{\partial
N}(\mathcal{A},\mathcal{A}^{\prime})=\int_{\partial
N}\mathcal{A}\wedge\mathcal{A}^{\prime}.$

This symplectic form is the de Rham differential (on
$\mathcal{F}^{\partial}_{\partial N}$) of the 1-form

$\alpha^{\partial}_{\partial N}=\frac{1}{2}\int_{\partial
N}\mathcal{A}\wedge\delta\mathcal{A}.$

Finally, using the surjective submersion
$\pi\colon\mathcal{F}_{N}\to\mathcal{F}^{\partial}_{\partial N}$, given by
pullback of differential forms from $N$ to $\partial N$, we can project the
vector field444The vector field $Q_{N}$ is no longer the hamiltonian vector
field of $\mathcal{S}_{N}$. It is instead defined via the formula above.
$Q_{N}$ to $\mathcal{F}^{\partial}_{\partial N}$. One can check that it is
also hamiltonian. For degree reasons it then automatically has a unique odd
hamiltonian function that we denote by $\mathcal{S}^{\partial}_{\partial N}$.
The important structural relation between the boundary BFV data
$(\mathcal{F}^{\partial}_{\partial N},\alpha^{\partial}_{\partial
N},\mathcal{S}^{\partial}_{\partial N})$ and the bulk “broken” BV data
$(\mathcal{F}_{N},\omega_{N},\mathcal{S}_{N},Q_{N},\pi)$ is

(1)
$\delta\mathcal{S}_{N}=\iota_{Q_{N}}\omega_{N}+\pi^{*}\alpha^{\partial}_{\partial
N}.$

The data, together with the structural relation (1), are the content of the
classical BV-BFV formalism. For more details we refer to [12].

For $f$ a function on $\mathcal{F}^{\partial}_{\partial N}$, there is a
symmetry of the data given by shifting
$\mathcal{S}_{N}\to\mathcal{S}_{N}^{f}=\mathcal{S}_{N}+\pi^{*}f$ and
$\alpha^{\partial}_{\partial N}\to\alpha^{\partial,f}_{\partial
N}=\alpha^{\partial}_{\partial N}+\delta f$. Clearly this is a symmetry of
Equation (1).

######  Remark 1.1.

The BV-BFV formulation of abelian Chern–Simons theory can be extended—as a
$\mathbb{Z}_{2}$-graded theory—to dimension $d=1$, see Section 3. Instead of
$\mathbb{R}$-valued forms, there one has to consider forms with values in an
odd vector space $\Pi\mathfrak{g}$, with $\mathfrak{g}$ an ordinary vector
space equipped with an inner product. This is the abelian version of the model
studied in [2].

Let us explain now how to define the BV-BFV partition function. We will be
very brief here; for a detailed exposition we refer to [15]. We will require
some additional pieces of data. The first one is a polarization $\mathcal{P}$
(involutive lagrangian distribution) on $\mathcal{F}^{\partial}_{\partial N}$.
We say that the boundary 1-form $\alpha^{\partial}$ is compatible with
$\mathcal{P}$ if it vanishes on vectors belonging to $\mathcal{P}$. Typically
this is not the case, but it may be achieved by means of the symmetry
$\alpha^{\partial}\to\alpha^{\partial}+\delta f$ discussed above. Denote by
$\mathcal{B}$ the leaf space of the polarization. In the examples of this
paper we actually have $\mathcal{F}^{\partial}_{\partial N}\cong
T^{*}\mathcal{B}$.

######  Remark 1.2.

In the examples in this paper the graded manifold
$\mathcal{F}^{\partial}_{\partial N}$ is actually a vector space, and the
simplest polarizations are splittings into complex lagrangian subspaces
$\mathcal{F}^{\partial}_{\partial
N}\otimes\mathbb{C}=\mathcal{B}\oplus\mathcal{B}^{\prime}$, we call those
_linear polarizations_. However, it is interesting to consider more general
polarizations. An example is the _Hitchin polarization_ on
$\Omega^{3}(M,\mathbb{C})$ explained in subsection 6.3.2.

Next, we require a splitting
$\mathcal{F}_{N}\cong\mathcal{B}\times\mathcal{Y}$ where $\mathcal{Y}$ is also
an odd symplectic vector space. Finally, we choose the data of a gauge fixing
on $\mathcal{Y}$: another splitting
$\mathcal{Y}\cong\mathcal{V}\times\mathcal{Y}^{\prime}$ into odd symplectic
vector spaces and a lagrangian $\mathcal{L}\subset\mathcal{Y}^{\prime}$ such
that $0$ is an isolated critical point of $\mathcal{S}_{N}$ when restricted to
$\mathcal{B}\times\mathcal{V}\times\mathcal{L}\subset\mathcal{B}\times\mathcal{V}\times\mathcal{Y}^{\prime}\cong\mathcal{F}_{M}$,
fiberwise over $\mathcal{B}\times\mathcal{V}$. The odd symplectic space
$\mathcal{V}$ is called the space of residual fields and $\mathcal{L}$ is
called the gauge-fixing lagrangian.

Given all these data, we can define the perturbative partition function as the
integral of the exponentiated BV action over $\mathcal{L}$:

$Z_{N}(\mathsf{A},\mathsf{a})=\int_{\alpha\in\mathcal{L}\subset\mathcal{Y}^{\prime}}\mathcal{D}\alpha\exp\left(\frac{i}{\hbar}\mathcal{S}^{f}_{N}(\mathsf{A},\mathsf{a},\alpha)\right)=\exp\left(\frac{i}{\hbar}S_{\mathrm{eff}}(\mathsf{A},\mathsf{a})\right).$

The partition function $Z$ and the effective action $S_{\mathrm{eff}}$ are
both functions on $\mathcal{B}\times\mathcal{V}$.

The integral is defined as a sum over Feynman diagrams—i.e., modeled on
finite-dimensional Gaussian integrals. As a consequence of the structural
equation (1), one expects $Z_{N}$ to satisfy the modified quantum master
equation (mQME)

(2) $(\Omega_{\mathcal{B}}-\hbar^{2}\Delta_{\mathcal{V}})Z_{N}=0,$

where $\Delta_{\mathcal{V}}$ is the BV operator acting on functions on the odd
symplectic vector space $\mathcal{V}$ of residual fields, given in Darboux
coordinates $(q^{i},p_{i})$ by $\sum_{i}\pm\frac{\partial}{\partial
q^{i}}\frac{\partial}{\partial p_{i}}$, and $\Omega_{\mathcal{B}}$ is a
quantization of the BFV action $\mathcal{S}^{\partial}_{\partial N}$ acting on
functions on $\mathcal{B}$ as a differential operator. If we write
$\mathcal{F}^{\partial}_{\partial N}=T^{*}\mathcal{B}\ni(b,b^{\prime})$, then
$\Omega_{\mathcal{B}}$ is given by $\mathcal{S}^{\partial}_{\partial
N}(b,-i\hbar\frac{\partial}{\partial b})$, with all derivatives to the right.
At lowest order in $\hbar$, we have $\Omega_{\mathcal{B}}^{2}=0$ as a
consequence of $(S,S)=0$. To ensure this to all orders, one might have to add
higher order corrections (although there is no guarantee in general that the
corrections exist). In all problems considered in this paper,
$\Omega_{\mathcal{B}}$ squares to zero without further corrections (see
Theorem A). Since these operators anticommute with each other and square to
zero, there is a double complex where $Z_{N}$ defines a cohomology class
$[Z_{N}]$. This cohomology class is invariant under deformation of the choices
made in the construction. For more details on the mQME (2), we refer to
[13],[15].

######  Remark 1.3 (Choice of residual fields).

The choice of the space $\mathcal{V}\subset\mathcal{Y}$ is not unique. In
fact, there is a partially ordered set of such choices, with maximal element
$\mathcal{Y}$ and a minimal element $\mathcal{V}_{\mathrm{min}}$, and one can
pass from a bigger to a smaller choice by a BV pushforward. A more detailed
discussion can be found in [15, Appendix F]. In this paper, when we deal with
dimensions $d\neq 1$, we usually first have a “big” (infinite-dimensional)
choice of $\mathcal{V}$. In some cases we are able to compute the BV
pushforward to $\mathcal{V}_{\mathrm{min}}$.

###  1.2. Main results of the paper

We are now ready to describe the main results of this paper. We consider only
spacetime manifolds $N$ that are cylinders: $N=I\times\Sigma$. We think of the
interval as $I=[0,1]$, so that $\partial
N=\\{0\\}\times\Sigma\,\sqcup\,\\{1\\}\times\Sigma$, and we denote by
$\Sigma_{\mathrm{in}},\Sigma_{\mathrm{out}}$ the two components. The BFV space
of boundary fields $\mathcal{F}^{\partial}_{\partial N}$ then splits as
$\mathcal{F}^{\partial}_{\partial
N}=\mathcal{F}^{\partial}_{\mathrm{in}}\times\mathcal{F}^{\partial}_{\mathrm{out}}$.

We will consider polarizations of the space of boundary fields
$\mathcal{F}^{\partial}_{\partial N}$ that are products of two polarizations
on the two factors. We will work mostly with linear polarizations, i.e.,
splittings
$\mathcal{F}^{\partial}_{\Sigma}\otimes\mathbb{C}=\mathcal{B}\oplus\mathcal{B}^{\prime}$
where $\mathcal{B},\mathcal{B}^{\prime}$ are complementary complex lagrangian
subspaces of $\mathcal{F}^{\partial}_{\Sigma}\otimes\mathbb{C}$, so that we
have an injection
$\omega_{\Sigma}^{\sharp}\colon\mathcal{B}^{\prime}\to\mathcal{B}^{*}$. We
will then write (suppressing the complexification)
$\mathcal{F}^{\partial}_{\Sigma}\cong T^{*}\mathcal{B}$ and say that we are
using the $\mathcal{B}$-representation.555 A comment on complex vs. real
spaces: by default, spaces of fields and spaces of boundary fields are real
vector spaces. In this paper, we have to complexify them to impose convenient
boundary conditions/polarizations. The splittings involved in the gauge fixing
(and thus in the corresponding spaces $\mathcal{Y}$, $\mathcal{V}$,
$\mathcal{Y}^{\prime}$, $\mathcal{L}$) are only defined over $\mathbb{C}$. The
equations of motion and the evolution relation are defined over $\mathbb{R}$
but need to be complexified to be described by a generating function
(involving a complex polarization). When writing down path integrals an
implied step is a choice of a real contour in the complexified space of
fields, see Appendix A.1 for an illustration of the principle.  In ghost
number 0 we also allow nonlinear polarizations with smooth leaf space
$\mathcal{B}$ such that $F^{\partial}_{\Sigma}\cong T^{*}B$.666In general, one
might have to restrict to neighborhoods in $\mathcal{F}^{\partial}_{\partial
N}$ and $\mathcal{B}$ to achieve this isomorphism.

Consider now a representation $\mathcal{F}^{\partial}_{\partial N}\cong
T^{*}\mathcal{B}_{\mathrm{in}}\times T^{*}\mathcal{B}_{\mathrm{out}}$. Denote
by $\mathcal{EL}$ the zero locus of $Q$. It consists of (nonhomogeneous)
closed forms in the abelian case and of “flat” nonhomogoneous forms in the
nonabelian one. We call the projection $\mathcal{L}\mathrel{\mathop{\mathchar
58\relax}}=\pi(\mathcal{EL})\subset\mathcal{F}^{\partial}_{\partial N}$ the
_BV evolution relation_. Denoting by $F^{\partial}_{\partial N}$ the ghost
number 0 part, we get a product of two ordinary cotangent bundles
$F^{\partial}_{\partial N}\cong T^{*}B_{\mathrm{in}}\times
T^{*}B_{\mathrm{out}}$. We denote the restriction of the graded evolution
relation by $L\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{L}|_{\mathrm{gh}=0}$ and call it simply the _evolution
relation_. One can show that it is a lagrangian subspace and that it consists
of boundary fields that can be extended to solutions of the Euler–Lagrange
equations. A generalized generating function for $L$ is given by the
Hamilton–Jacobi action $S_{\mathrm{HJ}}[b_{\mathrm{in}},b_{\mathrm{out}},e]\in
C^{\infty}(B_{\mathrm{in}}\times B_{\mathrm{out}}\times V_{\mathrm{aux}})$,
where $V_{\mathrm{aux}}$ is a space of additional parameters. The requirement
on $V_{\mathrm{aux}}$ is that in the fiber of $F_{N}$ over any triple
$(b_{\mathrm{in}},b_{\mathrm{out}},e)\in B_{\mathrm{in}}\times
B_{\mathrm{out}}\times V_{\mathrm{aux}}$ there exists a unique solution to the
equations of motion. There is a poset of choices for this space. This is
discussed in detail in the companion paper [16] and recalled in the Section 2
below. A first set of results can then be summarized as follows.

######  Theorem A.

Consider one of the following BV-BFV theories:

  1. (1)

1D abelian Chern–Simons theory with linear or nonlinear polarizations,

  2. (2)

$d=4l+3$-dimensional Chern–Simons theory with linear or nonlinear
polarizations,

  3. (3)

3-dimensional nonabelian Chern–Simons theory with linear polarizations.

Then there exists a space of residual fields $\mathcal{V}$ and a gauge-fixing
lagrangian $\mathcal{L}$ such that the ghost number 0 component of
$\mathcal{V}$ coincides with $V_{\mathrm{aux}}$ and the ghost number 0
component of $S_{\mathrm{eff}}$ coincides with $S_{\mathrm{HJ}}$.

Here the choice of space of residual fields $\mathcal{V}$ is determined in
ghost number 0 by the requirement that it should be isomorphic to
$V_{\mathrm{aux}}$. In all cases in this paper, the gauge-fixing Lagrangian is
the space of 0-forms along the interval $I$ intersected with
$\mathcal{Y}$.777Since we are inverting only the de Rham differential along
the interval $d_{I}$, this is the unique possible choice of gauge-fixing
lagrangian. Invertibility of $d_{I}$ restricted to $\mathcal{L}$ requires that
$\mathcal{L}$ contains no 1-forms. The lagrangian condition then requires that
all 0-forms along $I$ in $\mathcal{Y}$ belong to $\mathcal{L}$. In particular,
there are no quantum corrections to the ghost number 0 part of the effective
action (notice that the HJ action can be computed, as shown in [16, Section
7], completely at the classical level). The leading term in the effective
action was expected to be the Hamilton-Jacobi action from the finite-
dimensional results in the companion paper [16] (see in partiuclar Theorem
11.4 there). This theorem is nothing but an expression of the fact that to
leading order the quantum theory is determined by the Euler-Lagrange locus.
The fact that there are no quantum corrections is probably not surprising for
the abelian Chern-Simons theory. For nonabelian Chern-Simons theory, the
absence of quantum corrections is due to the fact that, given a complex
structure on $\Sigma$, the interaction term is affine in both the holomorphic
and antiholomorphic components of the connection along $\Sigma$, and the fact
that our polarizataion and gauge fixing are compatible with this complex
structure.

Our second main result concerns the mQME.

######  Theorem B.

In all cases of Theorem A with linear polarizations, the BV-BFV partition
functon $Z$ satisfies the modified quantum master equation

$(\Omega-\hbar^{2}\Delta)Z=0$

with
$\Omega=\Omega_{\mathcal{B}_{\mathrm{in}}}+\Omega_{\mathcal{B}_{\mathrm{out}}}$
given by the standard quantization (i.e. with derivatives to the right of
multiplication operators) of the boundary action at both endpoints. For
nonlinear polarizations $F^{\partial}_{\Sigma}=T^{*}B\ni(b,b^{\prime})$, the
mQME is satisfied whenever the constraint $d_{\Sigma}A=0$ is linear in the
momenta $b^{\prime}$.

Again, in this case there are no quantum corrections to $\Omega$. These
theorems summarize the results obtained in the various sections of this paper,
where we discuss the different examples individually. We will outline the
paper in slightly more detail in Section 1.3 below. Before that, let us
comment on some of the more specific results in more detail.

####  1.2.1. Three-dimensional abelian Chern–Simons theory

In three-dimensional Chern–Simons theory, in ghost number 0 we have the
lagrangian splitting

$F^{\partial}_{\Sigma}\otimes\mathbb{C}=\Omega^{1}(\Sigma,\mathbb{C})=\Omega^{1,0}(\Sigma)\oplus\Omega^{0,1}(\Sigma).$

For instance, one can define

$\mathcal{B}_{\mathrm{out}}=\Omega^{0}(\Sigma,\mathbb{C})\oplus\Omega^{1,0}(\Sigma)\ni(\mathsf{A}^{0}_{\mathrm{out}},\mathsf{A}^{1,0}_{\mathrm{out}})$

and

$\mathcal{B}_{\mathrm{in}}=\Omega^{0,1}(\Sigma)\oplus\Omega^{2}(\Sigma,\mathbb{C})\ni(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{2}_{\mathrm{in}}).$

As discussed in Section 4.2, a possible choice for the space of residual
fields is

$\mathcal{V}_{\mathrm{small}}=\\{dt\cdot(\mathsf{A}^{0}_{I\,\mathrm{res}}+\mathsf{A}^{2}_{I\,\mathrm{res}})+(1-t)\cdot\mathsf{A}^{0}_{\mathrm{res}}+t\cdot\mathsf{A}^{2}_{\mathrm{res}}\\}\subset\Omega^{\bullet}(I\times\Sigma,\mathbb{C}),$

where $\mathsf{A}^{k}_{I\,\mathrm{res}},\mathsf{A}^{k}_{\mathrm{res}}$ are
complex valued $k$-forms on $\Sigma$, $t$ is the coordinate on $I=[0,1]$ and
the ghost numbers are
$\mathrm{gh}(\mathsf{A}^{k}_{I\,\mathrm{res}})=-k,\mathrm{gh}(\mathsf{A}^{k}_{\mathrm{res}})=1-k$.
We will denote by $\sigma\mathrel{\mathop{\mathchar
58\relax}}=A^{0}_{I\,\mathrm{res}}$ the only ghost number 0 field in
$\mathcal{V}_{\mathrm{small}}$. The BV-BFV partition function is then computed
as

$Z_{\mathrm{small}}=\exp\frac{i}{\hbar}\Big{(}\underbrace{\int_{\Sigma}\Big{(}\mathsf{A}^{1,0}_{\mathrm{out}}\mathsf{A}^{0,1}_{\mathrm{in}}+(\partial\mathsf{A}^{0,1}_{\mathrm{in}}+\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}})\;\sigma+\frac{1}{2}\sigma\partial\bar{\partial}\sigma\Big{)}}_{S_{\text{HJ}}}+\\\
+\int_{\Sigma}\Big{(}-\mathsf{A}^{0}_{\mathrm{out}}\mathsf{A}^{2}_{\mathrm{in}}+\mathsf{A}^{0}_{\mathrm{out}}\mathsf{A}^{2}_{\mathrm{res}}-\mathsf{A}^{2}_{\mathrm{in}}\mathsf{A}^{0}_{\mathrm{res}}+\frac{1}{2}\mathsf{A}^{2}_{\mathrm{res}}\mathsf{A}^{0}_{\mathrm{res}}\Big{)}\Big{)}.$

In particular, focusing on the summand of the effective action in the first
line, we recognize the Hamilton–Jacobi action from Example 2.2, as an instance
of Theorem A. It is the action functional of a 2D free boson conformal field
theory, coupled to the boundary fields $\mathsf{A}^{1,0}_{\mathrm{out}}$ and
$\mathsf{A}^{0,1}_{\mathrm{in}}$. We arrive at the same result in ghost number
0 in Section 5.3.1, using
$\mathcal{B}_{\mathrm{in}}=\Omega^{0}(\Sigma,\mathbb{C})\oplus\Omega^{0,1}(\Sigma)$.
One can integrate out the remaining residual fields to obtain then the fact
that the partition function of three-dimensional Chern–Simons theory for the
minimal space of residual fields (cf. Remark 1.3) coincides with the partition
function of the 2D free boson CFT. In particular, one can observe the Weyl
anomaly in the 3D Chern–Simons partition function. See Remark 4.3.

####  1.2.2. Three-dimensional nonabelian Chern–Simons theory and CS-WZW
correspondence

The same lagrangian splitting as above can be used to study the 3D nonabelian
Chern–Simons theory—see Section 5.3. The representation we use in that section
is $\mathcal{F}^{\partial}_{\partial N}=T^{*}\mathcal{B}_{\mathrm{in}}\times
T^{*}\mathcal{B}_{\mathrm{out}}$ with

$\mathcal{B}_{\mathrm{out}}=\Omega^{0}(\Sigma,\mathcal{G}_{\mathbb{C}})\oplus\Omega^{1,0}(\Sigma,\mathcal{G})\ni(\mathsf{A}^{0}_{\mathrm{out}},\mathsf{A}^{1,0}_{\mathrm{out}})$

and

$\mathcal{B}_{\mathrm{in}}=\Omega^{0}(\Sigma,\mathcal{G}_{\mathbb{C}})\oplus\Omega^{0,1}(\Sigma,\mathcal{G}).$

As a space of residual fields one can use

$dt\cdot\Omega^{0}(\Sigma,\mathcal{G}_{\mathbb{C}})\oplus\Omega^{2}(\Sigma,\mathcal{G}_{\mathbb{C}})[-1]\ni(dt\cdot\sigma,\mathsf{A}^{*}_{\mathrm{res}})$

with $\mathrm{gh}(\sigma)=0,\mathrm{gh}(\mathsf{A}^{*}_{\mathrm{res}})=-1$. We
compute the effective action in Lemma 5.12 and see that it has a tree part
$S^{\mathrm{eff}(0)}$ and a 1-loop part $\mathbb{W}$:

$S^{\mathrm{eff}}=S^{\mathrm{eff}(0)}-i\hbar\mathbb{W}=S^{\mathrm{eff}(0)}_{\mathrm{ph}}+S^{\mathrm{eff}(0)}_{\mathrm{gh}}-i\hbar\mathbb{W}$

(the subscript $\mathrm{ph}$ denotes the terms involving only fields of ghost
number 0, the subscript $\mathrm{gh}$ denotes terms involving fields with
nonzero ghost number). At first glance the explicit formula (61) seems
obscure, but we observe a number of interesting phenomena:

  1. i)

One has to restrict the residual field $\sigma$ to a certain “Gribov region”
$B_{0}\subset\mathcal{G}_{\mathbb{C}}$—a region where the exponential map
$\exp\colon\mathcal{G}_{\mathbb{C}}\to G_{\mathbb{C}}$ is injective—to make
sure that certain power series appearing in
$S^{\mathrm{eff}(0)}_{\mathrm{gh}}$ converge (Remark 5.13).

  2. ii)

As shown in Lemma 5.14, when we restrict $\sigma$ to $B_{0}$, we can
reparametrize by $g=e^{-\sigma}\colon\Sigma\to G_{\mathbb{C}}$. In this
reparametrization, we can rewrite $S^{\mathrm{eff}(0)}_{\mathrm{ph}}$ as

$S^{\mathrm{eff}(0)}_{\mathrm{ph}}=\int_{\Sigma}\Big{(}\langle\mathsf{A}^{1,0}_{\mathrm{out}},g\,\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}\rangle-\langle\mathsf{A}^{1,0}_{\mathrm{out}},\bar{\partial}g\cdot
g^{-1}\rangle-\langle\mathsf{A}^{0,1}_{\mathrm{in}},g^{-1}\partial
g\rangle\Big{)}+\mathrm{WZW}(g)$

with the Wess–Zumino–Witten term

$\mathrm{WZW}(g)=-\frac{1}{2}\int_{\Sigma}\langle\partial g\cdot
g^{-1},\bar{\partial}g\cdot g^{-1}\rangle-\frac{1}{12}\int_{\Sigma\times
I}\langle dh\cdot h^{-1},[dh\cdot h^{-1},dh\cdot h^{-1}]\rangle$

and $h=e^{(t-1)\sigma}.$ This coincides with the Hamilton–Jacobi action of
Chern–Simons theory, see Example 2.3.

  3. iii)

The term $-i\hbar\mathbb{W}$ in principle violates Theorem A and is divergent.
However, it has a nice interpretation as a change of path integral measure
from $\mathcal{D}\sigma$ to $\mathcal{D}g$, see Section 5.3.4. In particular,
if one interprets $Z$ as a _half-density_ rather than a function on the space
of residual fields, and thus $S^{\mathrm{eff}}$ as a log-half-density, the
effective action has no quantum corrections in the $(g,g^{*})$ coordinates on
$\mathcal{V}$ (here $g^{*}$ is the Darboux coordinate for $g$). It is in this
sense that Theorem A holds.

  4. iv)

In Section 5.3.6 we show that $Z$ satisfies the modified quantum master
equation in the different interpretations of $Z$ (partition function vs. half-
density). Interestingly, in the $(g,g^{*})$ representation the mQME implies
the well-known Polyakov–Wiegmann identity for the WZW action.

We thus observe a strong version of the CS-WZW correspondence: Namely, the
effective theory of nonabelian Chern–Simons theory on $I\times\Sigma$ is a
“gauged WZW theory,” i.e., a WZW theory on $\Sigma$ coupled to chiral gauge
fields $\mathsf{A}^{1,0}_{\mathrm{in}},\mathsf{A}^{0,1}_{\mathrm{out}}$.

We also compute expectation values of vertical Wilson lines (Section 5.3.7)
and show that they are given by field insertions in this WZW theory. This
extends the CS-WZW correspondence to the level of observables. See the
discussion in Section 5.3.8.

Formally, after integrating over the residual group-valued field $g$, the
Chern–Simons partition function agrees with the partition function of the
gauged WZW theory. One can use this to heuristically show the holomorphic
factorization of the WZW model, as argued in Section 5.3.9.

Different versions of the relation between nonabelian Chern–Simons theory and
the WZW model were studied in the literature before. A connection somewhat
close to the one we are discussing appeared in [11, Section 4]; one important
difference is that we are focusing on the homological (BV-BFV) aspects
obtaining WZW as an effective BV theory. The other point is that the logic of
our computation is different (it is a pure perturbative computation; it does
not rely on quantum gauge invariance but has it as a result), see Remark 5.22.

####  1.2.3. Seven-dimensional Chern–Simons theory and the CS-BCOV
correspondence

Finally, let us consider seven-dimensional Chern–Simons theory on a cylinder
$N=I\times M$ with $M$ a Calabi–Yau manifold. In particular, the complex
structure on $M$ defines a lagrangian splitting of
$F^{\partial}_{M}=\Omega^{3}(M,\mathbb{C})$:

$\Omega^{3}(M,\mathbb{C})=X^{+}\oplus X^{-},\qquad
X^{+}=\Omega^{3,0}(M)\oplus\Omega^{2,1}(M),\qquad
X^{-}=\Omega^{1,2}(M)\oplus\Omega^{0,3}(M).$

This lagrangian splitting determines a polarization of $F^{\partial}_{M}$.

On a Calabi–Yau manifold, however, there is another polarization of
$F^{\partial}_{M}$ due to Hitchin [29]. Namely, a complex three-form $A$ on
$M$ which is not itself decomposable, i.e., a wedge product of three 1-forms
on $M$, has a decomposition $A=A^{+,\mathrm{nl}}+A^{-,\mathrm{nl}}$ where
$A^{\pm,\mathrm{nl}}$ are decomposable three-forms uniquely defined up to
exchange of $+$ and $-$. This polarization is discussed in 6.3.2.

We can compute the partition function $Z$ on the cylinder with

$\mathcal{B}_{\mathrm{in}}=\Omega^{\leq 2}(M,\mathbb{C})\oplus
X^{+}\ni(\mathsf{c}_{\mathrm{in}},\mathsf{A}_{\mathrm{in}}^{+,\mathrm{l}})$

and

$\mathcal{B}_{\mathrm{out}}=\Omega^{\leq 2}(M,\mathbb{C})\times
X^{-,\mathrm{nl}}\ni(\mathsf{c}_{\mathrm{out}},\mathsf{A}_{\mathrm{out}}^{-,\mathrm{nl}}).$

In this case, Theorem A holds—as shown in Section 5.2—and Theorem B holds
because the constraint $d_{M}A=0$ is linear in the momentum
$A^{+,\mathrm{nl}}$. Thus, the physical part of the effective action coincides
with the Hamilton–Jacobi action computed in [16, Section 7.6] and is given by

$S_{\mathrm{ph}}[\mathsf{A}^{+,\mathrm{l}}_{\mathrm{in}},\mathsf{A}^{-,\mathrm{nl}}_{\mathrm{out}},\mathsf{A}^{2,0}_{I\,\mathrm{res}},\mathsf{A}^{1,1}_{I\,\mathrm{res}},\mathsf{A}^{0,2}_{I\,\mathrm{res}}]\\\
=\frac{1}{2}\int_{M}\partial\mathsf{A}^{1,1}_{I\,\mathrm{res}}\bar{\partial}\mathsf{A}^{1,1}_{I\,\mathrm{res}}+\int_{M}\mathsf{A}^{+,\mathrm{l}}_{\mathrm{in}}d\mathsf{A}^{0,2}_{I\,\mathrm{res}}+\mathsf{A}^{2,1}_{\mathrm{in}}\bar{\partial}\mathsf{A}^{1,1}_{I\,\mathrm{res}}-G(\mathsf{A}^{+,\l}_{\mathrm{in}}+d\mathsf{A}^{2,0}_{I\,\mathrm{res}}+\partial\mathsf{A}^{1,1}_{I\,\mathrm{res}};\mathsf{A}^{-,\mathrm{nl}}_{\mathrm{out}})$

with no quantum corrections in our choice of gauge fixing. Here
$\mathsf{A}^{p,q}_{I\,\mathrm{res}}$ denote 2-forms of Hodge type $(p,q)$
which are the residual fields of ghost number 0, and
$G(A^{+,\mathrm{l}},A^{-,\mathrm{nl}})$ is the generating function satisfying
$\delta G=A^{-,\mathrm{l}}\delta A^{+,l}-A^{+,\mathrm{nl}}\delta
A^{-,\mathrm{nl}}$. Since the partition function $Z$ satisfies, by Theorem B,
the modified quantum master equation, when changing the gauge fixing the
partition function changes by an $(\Omega-\hbar^{2}\Delta)$-exact term.

The partition function $Z$ can be interpreted as the integral kernel of a
generalized Segal–Bargmann transform, see Appendix A. We thus show that the
approximation used by Gerasimov and Shatashvili in [26]—where they were only
assuming this representation to be true in the semiclassical limit—is exact.
Following [26], we can then relate the Chern–Simons partition function to the
partition function of Kodaira–Spencer or BCOV theory, defined in [9] and
recalled in Appendix B, as follows. One can consider a certain
($\Omega$-closed) state
$\psi(\mathsf{A}_{\mathrm{out}}^{-,\mathrm{nl}},\mathsf{c}_{\mathrm{out}})$ in
the $A^{-,\mathrm{nl}}$-representation. We then apply the operator $Z$ to
$\psi$ — by multiplying and formally integrating over
$\mathcal{B}_{\mathrm{out}}$— and show that the result $Z\cdot\psi$ is still
$(\Omega-\hbar^{2}\Delta)$-closed. Next we identify a gauge-fixing lagrangian
$\mathcal{L}\subset\mathcal{V}$ and compute
$Z^{\prime\prime}[\mathsf{A}^{3,0}_{\mathrm{in}},\mathsf{A}^{2,1}_{\mathrm{in}},c_{\mathrm{in}}]=\int_{\mathcal{L}}Z\cdot\psi.$
One can then show that in ghost number 0

$Z^{\prime\prime}_{\mathrm{ph}}[\mathsf{A}^{3,0}_{\mathrm{in}}=\omega_{0},\mathsf{A}^{2,1}_{\mathrm{in}}=x]\sim
Z_{KS}[x],$

where $\omega_{0}$ is a normalized generator of $H_{\partial}^{3,0}(M)$, $x$
is a $\partial$-harmonic form, and $Z_{KS}[x]$ is the Kodaira-Spencer
partition function with background $x$. For the precise statement see Section
6.3.3. In particular, we see that this statement holds not only in the
semiclassical approximation to $Z_{CS}$ as in [26], but that it is exact. For
general boundary conditions
$\mathsf{A}^{3,0}_{\mathrm{in}},\mathsf{A}^{2,1}_{\mathrm{in}}$, the
Chern–Simons partition function can be computed from the mQME.

###  1.3. Structure of the paper

We summarize the remaining results by outlining the structure of the paper.
In Section 2, we recall the construction of the Hamilton–Jacobi action from
[16], and the important examples (abelian and nonabelian Chern–Simons theory)
from that paper.

In Section 3, we consider as a warm-up the example of the abelian 1D CS
theory. This is the 1D AKSZ theory with target a vector space $\mathfrak{g}$
that we assume to have an inner product and a compatible complex structure
$J$, so that
$\mathfrak{g}\otimes\mathbb{C}=\mathfrak{g}^{+}\oplus\mathfrak{g}^{-}$ splits
into $\pm i$-eigenspaces of $J$. We then compute the partition function for
both $\mathcal{B}_{\mathrm{in}}=\mathcal{B}_{\mathrm{out}}=\mathfrak{g}^{+}$
in Section 3.1 and
$\mathcal{B}_{\mathrm{in}}=\mathfrak{g}^{-},\mathcal{B}_{\mathrm{out}}=\mathfrak{g}^{+}$
in Section 3.2 and comment briefly on the Theorems A and B in this context
(which are in this case rather trivial).

In Section 4, we consider the 3D abelian Chern–Simons theory on
$I\times\Sigma$ as a 1D theory with values in
$\mathfrak{g}=\Omega^{\bullet}(\Sigma)$. Choosing a complex structure on
$\Sigma$, we split $\mathfrak{g}=\mathfrak{g}^{+}\oplus\mathfrak{g}^{-}$ and
consider
$\mathcal{B}_{\mathrm{in}}=\mathcal{B}_{\mathrm{out}}=\mathfrak{g}^{+}$ in
Section 4.1 and
$\mathcal{B}_{\mathrm{in}}=\mathfrak{g}^{-},\mathcal{B}_{\mathrm{out}}=\mathfrak{g}^{+}$
in Section 4.2. In both cases, we comment on the HJ and mQME properties, and
in the second case also the pushforward to the minimal space of residual
fields and the relation to the 2D free boson CFT is discussed.

In Section 5, we consider the case where $\mathcal{B}_{\mathrm{in}}$ and
$\mathcal{B}_{\mathrm{out}}$ both have components only in nonnegative ghost
number, and agree in positive ghost number. We call these “parallel ghost
polarization”. In Section 5.1, we consider 1D Chern–Simons theory with values
in a complex, with opposite linear polarizations in ghost number 0. In Section
5.2, we consider the same theory with a possibly nonlinear polarization on the
$\mathrm{out}$-boundary. These subsections serve as a toy model for the
higher-dimensional Chern–Simons theories considered later. In Section 5.3, we
return to the three-dimensional Chern–Simons theory, with opposite linear
polarization in degree 0. After briefly studying again the abelian case in
Section 5.3.1, we discuss the nonabelian case in more detail, the results are
summarized already in Section 1.2.2 above. Finally in Section 5.4, we consider
the nonabelian theory with parallel polarizations both in the ghost and
physical sectors.

In Section 6, we turn to Chern–Simons theories of arbitrary dimension. We
consider both linear polarizations that are transversal in the ghost sector at
opposite ends (Section 6.1) and parallel in the ghost sector (Section 6.2).
Finally in Section 6.3 we turn our attention to nonlinear polarizations at one
boundary, in particular the 7D case with Hitchin polarization, that was
summarized in Section 1.2.3 above.

The appendices contain some complementary material. In Appendix A we show how
to recover the usual Segal–Bargmann transform as a BV-BFV partition function
on an interval with a particular choice of boundary polarizations. This is an
illustration of the maxim that topological partition functions on cylinders
yield instances of generalized Segal–Bargmann transforms. We also comment on
the contour integration in the complexified space of fields. In Appendix B, we
recall very briefly the Kodaira–Spencer theory of deformations of complex
structures and the BCOV action functional.

###  1.4. Outlook

Finally, let us point out some interesting directions for further research.

  * •

All our partition functions depend nontrivially on the choice of complex
structure on the boundary.888As a matter of fact, they even depend on the
_Riemanian metric_ inducing the complex structure on the boundary, a
phenomenon known as conformal anomaly. See Remark 4.3.(b). This dependence
should be described by extending the partition function to a (projectively
flat) section of a vector bundle over the moduli space of complex structures
on the boundary, for instance the one constructed in [4].

  * •

Recently [34] it has been suggested that the partition function of a 3D $U(1)$
Chern–Simons theory can be computed by averaging over Narain moduli space of
boundary CFT’s. We believe our methods could be generalized to include
nontrivial flat bundles and we plan to investigate this proposal.

  * •

Our results on the CS-WZW correspondence strongly suggest that the space of
$n$-point conformal blocks can be described as the $\Omega$-cohomology (see
Section 5.3.8; the genus-zero case of this statement was a result of [1]).
This would provide an interesting new description of the space of conformal
blocks. We also hope it would lead to a better understanding of the
relationship between Chern–Simons theory and the KZ(B) connection.

  * •

It would be highly interesting to compare our findings on the CS-BCOV
correspondence to other approaches to the subject such as [17].

  * •

Another proposal to compute holographic duals of action functionals from BV-
BFV formalism on manifolds with boundary was made by the second and third
authors together with M. Schiavina in [36]. The point of view there was more
focused on descent equations and extensions to higher codimension, while the
present paper emphasizes the role of the BV effective action. The relationship
between the two constructions needs to be explored.

  * •

In [27], the authors show that there exists a 1-loop exact quantization of
Chern-Simons theory on $\mathbb{R}^{3}$, which is similar to the result that
we obtain here (in our case, the wheels appear only in the ghost sector of the
theory). The gauge fixing they use is different from ours, and the focus there
is not on partition functions, but rather on the anomaly-freeness of the
theory, a problem which does not appear in our gauge fixing. Nevertheless it
would be interesting to investigate this gauge-fixing from the BV-BFV
viewpoint and compare it with our current results.

###  1.5. Notations and Conventions

> _This is a quantum paper and notations fluctuate. Fixing one makes a
> complementary one explode._

In this paper we study field theories on cylinders $N=I\times\Sigma$ from
different viewpoints, with $I=[0,1]$ the interval with its standard
orientation, and a $\Sigma$ a $(d-1)$-dimensional closed oriented manifold.
Notations are adapted to the individual sections.
We are considering Chern–Simons-type theories, in different dimensions and
with different targets. The Chern–Simons superfield is denoted
$\mathcal{A}\in\Omega^{\bullet}(I\times\Sigma,\Pi\mathfrak{g})$.

When we are considering 1-dimensional theories (with a possibly infinite-
dimensional target) as in Sections 3, 5.1, 5.2, we denote the components of
the superfield $\mathcal{A}=\psi+A$, where $\psi\in\Omega^{0}(I,\mathfrak{g})$
and $A\in\Omega^{1}(I,\mathfrak{g})$. Decoration of $\psi,A$ with superscripts
denotes components with respect to a splitting of $\mathfrak{g}$. Decoration
of $\psi,A$ with subscripts denotes components with respect to a splitting of
$\Omega^{\bullet}(I)$. Typical subscripts are $\mathrm{in}$ and
$\mathrm{out}$, denoting fields supported on the $\mathrm{in}$ or
$\mathrm{out}$ boundary (elements of $\mathcal{B}_{\mathrm{in}}$ or
$\mathcal{B}_{\mathrm{out}}$) respectively, $\mathrm{res}$ for residual fields
(elements of $\mathcal{V}$), and $\mathrm{fl}$ for fluctuations (elements of
$\mathcal{L}$).
When we are thinking about higher-dimensional theories (still on cylinders) as
in Section 4 , 6, we denote the components of
$\mathcal{A}=\mathsf{A}+dt\cdot\mathsf{A}_{I}$, with
$\mathsf{A},\mathsf{A}_{I}\in\Omega^{0,\bullet}(I\times\Sigma)$. Superscripts
now denote components of homogeneous form degree in $\Sigma$.
In Sections 5.3, 5.4, it is convenient to revert to a more “traditional”
notation $\mathcal{A}=c+A+A^{*}+c^{*}$, here the nonhomogeneous differential
form is split according to form degree. There we also denote the (finite-
dimensional) coefficient Lie algebra by $\mathcal{G}$.

######  Acknowledgments.

We thank Samson Shatashvili for suggesting the study of 7D abelian
Chern–Simons theory in the quantum BV-BFV formalism, now in Section 6.3, which
was the original motivation out of which this paper and [16] grew. We also
thank Ivan Contreras, Philippe Mathieu, Nicolai Reshetikhin, Pavel Safronov,
Michele Schiavina, Stephan Stolz, Alan Weinstein, Ping Xu and Donald Youmans
for useful discussions. We also thank an unknown referee for very helpful and
inspiring comments and suggestions.

##  2\. Constrained systems and generalized Hamilton–Jacobi actions

We start with a short review of the results of [16] that are relevant for this
paper. We focus on action functionals of the form999Such an action functional
is the classical part of an AKSZ theory [3] on an interval $I$. See also [6]
for the study of such a theory in the Dirac formalism.

$S[p,q,e]=\int_{I}(pdq-\langle H(p,q),e\rangle),$

where $I$ is the interval $[0,1]$, $(p,q)$ are coordinates on a given
cotangent bundle $T^{*}B$ (and, by abuse of notation, also stand for a map
from $I$ to $T^{*}B$), $e$ is a one-form on $I$ taking value in some vector
space $\mathfrak{h}$, and $H$ is a given map $T^{*}B\to\mathfrak{h}^{*}$. The
pairing between the $p$ and the $q$ coordinates is understood, whereas for the
pairing between $\mathfrak{h}$ and its dual we use the notation $\langle\
,\rangle$. In the applications of this paper the space $\mathfrak{h}$ and the
manifold $B$ are infinite-dimensional (typically, Fréchet spaces).

To be more precise, $T^{*}B$ denotes some given vector bundle over $B$ with a
nondegenerate pairing to $TB$; we denote by $\theta$ the canonical one-form on
it (which we will also call the Noether 1-form in the following) and by
$\omega=d\theta$ the canonical symplectic form; by $\mathfrak{h}^{*}$ we
denote a given subspace of the dual of $\mathfrak{h}$ such that its pairing to
$\mathfrak{h}$ is still nondegenerate. The first term in the action can also
be written in coordinate-free way as $x^{*}\theta$ in terms of a path $x\colon
I\to T^{*}B$. For the second term, we assume a given map $X$ from
$\mathfrak{h}$ to the vector fields on $T^{*}B$ and define, up to carefully
chosen constants, the map $H$ by $\iota_{X}\omega=dH$. (Note that $H$ is a map
from $\mathfrak{h}$ to the functions on $T^{*}B$, and we assume that, dually,
it belongs to the chosen subspace $\mathfrak{h}^{*}$.)

######  Example 2.1 (3D Chern–Simons theory).

Consider 3D Chern–Simons theory for a quadratic Lie algebra $\mathcal{G}$ on
$I\times\Sigma$, where $\Sigma$ is a closed oriented surface with a chosen
complex structure. The complexified phase space is
$T^{*}B=\Omega^{1,0}(\Sigma)\otimes\mathcal{G}\oplus\Omega^{0,1}(\Sigma)\otimes\mathcal{G}$
with $B=\Omega^{0,1}(\Sigma)\otimes\mathcal{G}$. We then have
$\mathfrak{h}=\Omega^{0}(\Sigma)\otimes\mathcal{G}$ and
$\mathfrak{h}^{*}=\Omega^{2}(\Sigma)\otimes\mathcal{G}$. The pairings are
induced by the given pairing on $\mathcal{G}$ and by integration on $\Sigma$.
An element of $T^{*}B$ is a connection one-form, the map $X$ yields the gauge
transformations, and $H$ is the curvature two-form.

We split the fields into two classes: the dynamical field (the map $x$ to
$T^{*}B$) and the Lagrange multiplier (the $\mathfrak{h}$-valued one-form
$e$). We accordingly split the Euler–Lagrange (EL) equations into the
evolution equation, the variations with respect to the dynamical field,

$dx=\langle X,e\rangle,$

and the constraints, the variations with respect to the Lagrange multiplier,

$H=0.$

Note that the constraints must be satisfied at every time.

We define the evolution relation $L$ as the possible boundary values (at $0$
and $1$ in $I$) that a solution to the EL equations can have. Assuming it to
be a (possibly immersed) submanifold, $L$ turns out to be an isotropic
submanifold of $\overline{T^{*}B}\times T^{*}B$ [12], where the bar means that
we use the opposite symplectic form. We assume it to be actually split
lagrangian (i.e., for every point $v$ of $L$, its tangent space $T_{v}L$,
which is isotropic in general, must have an isotropic complement).101010In
particular, this implies that $L$ is lagrangian, i.e., that the symplectic
orthogonal of $T_{v}L$ is $T_{v}L$ itself for every point $v$ of $L$. Thanks
to the Hodge decomposition theorem, this assumption is satisfied in all the
examples of this paper.

We then denote by $C$ the projection of $L$ on either factor $T^{*}B$ and we
assume it to be a submanifold. As observed in [13], if $L$ is lagrangian, then
$C$ is coisotropic. In particular, at every point $c\in C$ and for every
$\xi\in\mathfrak{h}$, the vector $\langle X(c),\xi\rangle$ is tangent to $C$.
Moreover, the span of these vectors at each point defines an involutive
distribution on $C$, called the characteristic distribution (the reduced phase
space of the theory is then defined as the reduction of $C$ with respect to
its characteristic distributon).111111In the case of 3D Chern–Simons theory,
Example 2.1, $C$ is the space of flat connections on the surface $\Sigma$, and
the reduced phase space is the space of flat connections modulo gauge
trnasformations.

In the case at hand, we have that $C$ is the zero locus of $H$. The evolution
equation, for a given $e$, is then the hamiltonian evolution for the (time-
dependent) hamiltonian $\langle H,e\rangle$. Since $C$ is coisotropic, this
evolution does not leave $C$—so it is enough to implement the constraint $H=0$
at the initial, or final, endpoint—and lies along the characteristic
distribution. It follows that the evolution relation $L$ consists of pairs of
points on $C$ lying on the same leaf of the characteristic distribution.

Next we are interested in solutions to the EL equations. For this we have to
fix boundary conditions; namely, we have to choose lagrangian submanifolds
$L_{0}$ and $L_{1}$ of $T^{*}B$ at the endpoints of $I$, and we assume that
the intersection of $L_{0}\times L_{1}$ with the evolution relation $L$ is
discrete.121212More precisely, one looks for solutions of the EL equations
that are critical points for the action. This requires changing the boundary
one-form by an exact term in such a way that it vanishes on $L_{0}$ and
$L_{1}$. In particular, this can only happen if $L_{0}$ and $L_{1}$ are
isotropic. Moreover, we want the intersection of $L_{0}\times L_{1}$ with $L$
to be discrete, so that locally the solution is unique. At each intersection
point, the tangent spaces to $L_{0}\times L_{1}$ and to $L$ are then
complementary, which implies that they are not only isotropic but split
lagrangian. We want this to happen for generic boundary conditions. This is
the reason why $L$ is required to be split lagrangian. For simplicity, we will
work with a unique solution. We are also interested in letting boundary
conditions vary, so we consider families of lagrangian submanifolds
(polarizations). Concretely, at the initial endpoint we take the $L_{0}$s to
be the fibers of $T^{*}B$, which we then parametrize by $B$, whereas at the
final endpoint we realize $T^{*}B$ as $T^{*}B^{\prime}$, with $B^{\prime}$ a
possibly different manifold, and take the $L_{1}$s to be the fibers of
$T^{*}B^{\prime}$, which we then parametrize by $B^{\prime}$.131313In the
examples of this paper, $B$ is a vector space, so $T^{*}B$ is of the form
$B^{*}\oplus B$. We are also interested in complex polarizations. In the case
at hand, this simply means allowing $B$ to be a complex vector space. Then the
complexified phase space is $B^{*}\oplus B$.  We want the variations of the
action with the given boundary conditions not to have boundary terms. This is
automatically satisfied at the initial point, where we take the polarization
$T^{*}B$, but we have to adapt the action to the canonical one-form
$\theta^{\prime}$ of $T^{*}B^{\prime}$ at the final endpoint. For this, we
assume that there is a function $f$ on $B\times B^{\prime}$ such that
$\theta=\theta^{\prime}+df$ and we modify the action to

$S^{f}[p,q,e]\mathrel{\mathop{\mathchar
58\relax}}=S[x,e]-f(q(1),Q(p(1),q(1))),$

where $Q$ is the base coordinate of $T^{*}B^{\prime}$.

We define the Hamilton–Jacobi (HJ) action $S^{f}_{\text{HJ}}$ of the theory
with respect to the given polarizations as the evaluation of $S^{f}$ on a
solution (which we assume to be unique) to the evolution equation for each
choice of $e$. Note that $S^{f}_{\text{HJ}}$ is a function on $B\times
B^{\prime}\times\Omega^{1}(I,\mathfrak{h})\ni(q_{\mathrm{in}},Q_{\mathrm{out}},e)$.
Also note that we do not impose the constraints in the definition of
$S^{f}_{\text{HJ}}$. It was proved in [16] $i)$ that $S^{f}_{\text{HJ}}$ is
invariant under certain equivalence transformations of $e$, and $ii)$ that it
is a generalized generating function for the evolution relation $L$ with
respect to the given polarizations.

Let us elaborate on this. As for $i)$, assume for simplicity that, as in every
example of this paper, $\mathfrak{h}$ is actually a Lie algebra and $H$ is an
equivariant momentum map (for the infinitesimal action $X$ of $\mathfrak{h}$
on $T^{*}B$). Then $e$ may be regarded as a connection one-form on $I$. The
equivalence transformations are in this case gauge transformations that are
trivial at the endpoints. As for $ii)$, the statement means that, upon setting
to zero the variation of $S^{f}_{\text{HJ}}$ with respect to (the equivalence
class of) $e$, we recover the final $P$ variables of a solution as the
variation of $S^{f}_{\text{HJ}}$ with respect to $Q_{\mathrm{out}}$ and the
initial $p$ variables of a solution as minus the variation of
$S^{f}_{\text{HJ}}$ with respect to $q_{\mathrm{in}}$.

Explicit examples, relevant for this paper, are discussed in [16, Section 7].
We recall the results.

######  Example 2.2 (Abelian 3D Chern–Simons theory).

We use the notations of Example 2.1, but now with $\mathfrak{g}=\mathbb{R}$.
We take the initial polarization as $T^{*}B$, with $B=\Omega^{0,1}(\Sigma)$,
and the final polarization as $T^{*}B^{\prime}$, with
$B^{\prime}=\Omega^{1,0}(\Sigma)$.141414 This polarization is known in the
literature on Chern–Simons theory. In a context close to the context of the
present paper, it was discussed in [14, Section 2.4.4] and in [1].  We denote
by $\partial$ and $\bar{\partial}$ the Dolbeault operators. The HJ action then
reads

$S^{f}_{\text{HJ}}=\int_{\Sigma}\left(\mathsf{A}^{1,0}_{\mathrm{out}}\mathsf{A}^{0,1}_{\mathrm{in}}+\sigma(\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}}+\partial\mathsf{A}^{0,1}_{\mathrm{in}})+\frac{1}{2}\sigma\partial\bar{\partial}\sigma\right),$

with $\mathsf{A}^{0,1}_{\mathrm{in}}\in B$,
$\mathsf{A}^{1,0}_{\mathrm{out}}\in B^{\prime}$, and
$\sigma\in\Omega^{0}(\Sigma)$.

######  Example 2.3 (Nonabelian 3D Chern–Simons theory).

Again we use the notations of Example 2.1. The initial and final polarizations
now are $T^{*}B$, with $B=\Omega^{0,1}(\Sigma)\otimes\mathcal{G}$, and
$T^{*}B^{\prime}$, with $B^{\prime}=\Omega^{1,0}(\Sigma)\otimes\mathcal{G}$.
We assume the exponential map from $\mathcal{G}$ to the its simply connected
Lie group $G$ to be surjective. In this case the gauge-invariant parameter
$g\in\operatorname{Map}(\Sigma,G)$ is of the form $g=e^{-\sigma}$ with
$\sigma\in\operatorname{Map}(\Sigma,\mathcal{G})$. The HJ action then reads

$S^{f}_{\text{HJ}}=\int_{\Sigma}\Big{(}\langle\mathsf{A}^{1,0}_{\mathrm{out}},g\,\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}\rangle-\langle\mathsf{A}^{1,0}_{\mathrm{out}},\bar{\partial}g\cdot
g^{-1}\rangle-\langle\mathsf{A}^{0,1}_{\mathrm{in}},g^{-1}\partial
g\rangle\Big{)}+\mathrm{WZW}(g)$

with the Wess–Zumino–Witten term

$\mathrm{WZW}(g)=-\frac{1}{2}\int_{\Sigma}\langle\partial g\cdot
g^{-1},\bar{\partial}g\cdot g^{-1}\rangle-\frac{1}{12}\int_{\Sigma\times
I}\langle dh\cdot h^{-1},[dh\cdot h^{-1},dh\cdot h^{-1}]\rangle,$

where $h=e^{(t-1)\sigma}$.151515Here we are using the conventions of Section
5.3 (Lemma 5.14) which are different from the conventions of [16]. Thus, the
HJ action of Chern–Simons theory can be identified with a “gauged WZW action”
(see for instance [23]). This points at a deep relationship between these two
theories.

##  3\. BV-BFV approach warm-up: 1D abelian Chern–Simons

As a warm-up exercise before the BV-BFV treatment of 3D Chern–Simons, let us
consider one-dimensional abelian Chern–Simons theory161616 This is the abelian
version of the theory considered in [2].  on an interval $I=[0,1]$ — the AKSZ
theory with $\mathbb{Z}_{2}$-graded space of BV fields

$\mathcal{F}=\mathrm{Map}(T[1]I,\Pi\mathfrak{g})=\Omega^{\bullet}(I)\otimes\Pi\mathfrak{g}.$

Here $\mathfrak{g}$ is a vector space of coefficients endowed with a
nondegenerate inner product $(,)$ and $\Pi$ is the parity-reversal symbol. A
vector in $\mathcal{F}$ is the superfield $\psi+A$, with $\psi$ a
$\Pi\mathfrak{g}$-valued $0$-form and $A$ a $\mathfrak{g}$-valued $1$-form,
and the BV action is:

(3) $S(\psi+A)=\int_{I}\frac{1}{2}(\psi,d_{I}\psi)$

with $d_{I}=dt\frac{d}{dt}$ the de Rham differential on the interval
$t\in[0,1]$. The odd symplectic form on $\mathcal{F}$ is given by
$\omega=-\int_{I}(\delta A,\delta\psi)$. The cohomological vector field (BRST
operator) $Q$ on $\mathcal{F}$ is defined by $Q\mathrel{\mathop{\mathchar
58\relax}}\psi\mapsto 0,\;A\mapsto d_{I}\psi$.

The BFV phase space assigned to a point is
$\mathcal{F}^{\partial}_{pt}=\Pi\mathfrak{g}$, equipped with Noether $1$-form
$\alpha_{pt^{\pm}}=\pm\frac{1}{2}(\psi,\delta\psi)$ where $\pm$ corresponds to
the orientation of the point; the BFV action is zero,171717It is nonzero in
nonabelian theory: there one has
$S_{pt^{\pm}}=\pm\frac{1}{6}(\psi,[\psi,\psi])$. $S_{pt}=0$. We are using the
following sign convention for the BV-BFV structure relation:

(4) $\delta S=\iota_{Q}\omega+\pi^{*}\alpha_{\partial}.$

Assume that $\mathfrak{g}$ is equipped with a complex structure
$J\in\mathrm{End}(\mathfrak{g})$, $J^{2}=-\mathrm{Id}$, compatible with the
inner product. We have a splitting of the complexification of $\mathfrak{g}$
into $\pm i$-eigenspaces of $J$:

(5) $\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}^{+}\oplus\mathfrak{g}^{-}$

— the “holomorphic” and “antiholomorphic” subspaces of
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}\otimes\mathbb{C}$, which are
lagrangian due to compatibility between $J$ and $(,)$.

###  3.1. Holomorphic-to-holomorphic boundary conditions

Consider the boundary polarization
$\mathrm{Span}(\frac{\partial}{\partial\psi^{-}})$ imposed at both $t=0$ and
$t=1$ (a.k.a. $\psi^{+}-\psi^{+}$ representation, as the partition function
will depend on the boundary value $\psi^{+}_{\mathrm{in}}$ at $t=0$ and
boundary value $\psi^{+}_{\mathrm{out}}$ at $t=1$). For compatibility with
this polarization, we need to modify the action (3) by boundary terms:

(6) $S\mapsto
S^{f}=S+\frac{1}{2}(\psi^{+},\psi^{-})\big{|}_{t=1}-\frac{1}{2}(\psi^{+},\psi^{-})\big{|}_{t=0}.$

Then the corresponding boundary $1$-form is:

$\alpha_{\partial
I}^{f}=\left.\left(\frac{1}{2}(\psi,\delta\psi)+\delta\frac{1}{2}(\psi^{+},\psi^{-})\right)\right|_{t=1}-\left.\left(\frac{1}{2}(\psi,\delta\psi)+\delta\frac{1}{2}(\psi^{+},\psi^{-})\right)\right|_{t=0}\\\
=(\psi^{-},\delta\psi^{+})\big{|}_{t=1}-(\psi^{-},\delta\psi^{+})\big{|}_{t=0}$

— the canonical $1$-form in the chosen representation, as desired (cf. Section
1.1; see [16, Section 9] and references therein for more details). The space
of fields $\mathcal{F}$ is fibered over the base
$\mathcal{B}=\Pi\mathfrak{g}^{+}\oplus\Pi\mathfrak{g}^{+}=\\{(\psi^{+}_{\mathrm{in}},\psi^{+}_{\mathrm{out}})\\}$
with the fiber

$\mathcal{Y}=\Omega^{\bullet}(I,\partial
I;\Pi\mathfrak{g}^{+})\oplus\Omega^{\bullet}(I;\Pi\mathfrak{g}^{-}).$

Here the first summand on the r.h.s. is $\mathfrak{g}^{+}$-valued forms
vanishing on the boundary and the second summand is $\mathfrak{g}^{-}$-valued
forms with free boundary conditions. The cochain complex $\mathcal{Y}$ admits
the following splitting (a Hodge decomposition):

(7) $\mathcal{Y}=\underbrace{\big{(}dt\cdot\mathfrak{g}^{+}\oplus
1\cdot\Pi\mathfrak{g}^{-}\big{)}}_{\mathcal{V}}\bigoplus\\\
\bigoplus\underbrace{\big{(}\Omega^{0}(I,\partial
I;\Pi\mathfrak{g}^{+})\oplus\Omega^{0}_{\int=0}(I;\Pi\mathfrak{g}^{-})\big{)}}_{\mathcal{Y}^{\prime}_{K-ex}}\bigoplus\underbrace{\big{(}\Omega^{1}_{\int=0}(I;\mathfrak{g}^{+})\oplus\Omega^{1}(I;\mathfrak{g}^{-})\big{)}}_{\mathcal{Y}^{\prime}_{d-ex}}.$

Here the first term (“residual fields”) is a deformation retract of
$\mathcal{Y}$ (in this case, in fact, its cohomology). The subscript $\int=0$
means “forms with vanishing total integral” (against $dt$ in $0$-form case).
The two last terms jointly form an acyclic subcomplex $\mathcal{Y}^{\prime}$
of $\mathcal{Y}$, split into a $d$-exact part and its direct complement — the
$K$-exact part, where
$K\colon\mathcal{Y}^{\bullet}\rightarrow\mathcal{Y}^{\bullet-1}$ is the chain
homotopy between identity and projection onto $\mathcal{V}$. Explicitly, $K$
kills all $0$-forms and acts on $\mathfrak{g}^{+}$\- and
$\mathfrak{g}^{-}$-valued $1$-forms as follows:

(8)
$K\colon\begin{array}[]{lcl}dt\,g^{+}(t)&\mapsto&\int_{0}^{t}dt^{\prime}g^{+}(t^{\prime})-t\int_{0}^{1}dt^{\prime}g^{+}(t^{\prime}),\\\
dt\,g^{-}(t)&\mapsto&-\int_{t}^{1}dt^{\prime}g^{-}(t^{\prime})+\int_{0}^{1}dt^{\prime}\,t^{\prime}\,g^{-}(t^{\prime}).\end{array}$

The integral kernel of $K$ is the propagator:

(9)
$\eta(t,t^{\prime})=\pi^{+}\otimes(\theta(t-t^{\prime})-t)+\pi^{-}\otimes(t^{\prime}-\theta(t^{\prime}-t)),$

where $\pi^{\pm}$ are the projectors from $\mathfrak{g}$ to
$\mathfrak{g}^{\pm}$ and $\theta$ is the step function.

The BV-BFV partition function is given by the following path integral (see
[15] for the general construction):

(10)
$Z(\psi^{+}_{\mathrm{in}},\psi^{+}_{\mathrm{out}};\psi^{-}_{\mathrm{res}},A^{+}_{\mathrm{res}})=\\\
=\int_{\mathcal{Y}^{\prime}_{K-ex}\subset\mathcal{Y}^{\prime}}\mathcal{D}\psi^{+}_{\mathrm{fl}}\mathcal{D}\psi^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}S^{f}\left(\widetilde{\psi^{+}_{\mathrm{in}}}+\widetilde{\psi^{+}_{\mathrm{out}}}+\psi^{+}_{\mathrm{fl}}+\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}}+dt\cdot
A^{+}_{\mathrm{res}}\right)}.$

Here the notations are:

  * •

$\widetilde{\psi^{+}_{\mathrm{in}}}$ is the discontinuous extension181818 See
[15] and [16, Section 9] for the details on discontinuous extension of
boundary fields.  of $\psi^{+}_{\mathrm{in}}$ at $t=0$ by zero at $t>0$;
likewise, $\widetilde{\psi^{+}_{\mathrm{out}}}$ is the discontinuous extension
of $\psi^{+}_{\mathrm{out}}$ at $t=1$ by zero at $t<1$;

  * •

the “fluctuation”
$(\psi^{+}_{\mathrm{fl}},\psi^{-}_{\mathrm{fl}})\in\mathcal{Y}^{\prime}_{K-ex}$
is the field we integrate over (while setting to zero the component in
$\mathcal{Y}^{\prime}_{d-ex}$ is the gauge fixing);

  * •

$(\psi^{-}_{\mathrm{res}},dt\cdot A^{+}_{\mathrm{res}})\in\mathcal{V}$, with
$\psi^{-}_{\mathrm{res}}\in\Pi\mathfrak{g}^{-}$ and
$A^{+}_{\mathrm{res}}\in\mathfrak{g}^{+}$, is the residual field.

Continuing the computation (10), we have the Gaussian integral

(11)
$Z=\int\mathcal{D}\psi^{+}_{\mathrm{fl}}\mathcal{D}\psi^{-}_{\mathrm{fl}}\;\exp\frac{i}{\hbar}\Big{(}\underbrace{\frac{1}{2}\int_{I}\big{(}\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}},d_{I}(\widetilde{\psi^{+}_{\mathrm{out}}}+\widetilde{\psi^{+}_{\mathrm{in}}})\big{)}}_{a}+\\\
+\underbrace{\frac{1}{2}\int_{I}\big{(}\widetilde{\psi^{+}_{\mathrm{out}}}+\widetilde{\psi^{+}_{\mathrm{in}}},d_{I}(\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}})\big{)}}_{b}+\underbrace{\frac{1}{2}\int_{I}\big{(}\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}},d_{I}\psi^{+}_{\mathrm{fl}}\big{)}}_{c}+\underbrace{\frac{1}{2}\int_{I}\big{(}\psi^{+}_{\mathrm{fl}},d_{I}(\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}})\big{)}}_{d}+\\\
+\underbrace{\frac{1}{2}(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}}\big{|}_{t=1})}_{e}\underbrace{-\frac{1}{2}(\psi^{+}_{\mathrm{in}},\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}}\big{|}_{t=0})}_{f}\Big{)}\\\
=\int\mathcal{D}\psi^{+}_{\mathrm{fl}}\mathcal{D}\psi^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}\left(\int_{I}(\psi^{-}_{\mathrm{fl}},d_{I}\psi^{+}_{\mathrm{fl}})+(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}}(1))-(\psi^{+}_{\mathrm{in}},\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}}(0))\right)}\\\
=e^{\frac{i}{\hbar}(\psi^{+}_{\mathrm{out}}-\psi^{+}_{\mathrm{in}},\psi^{-}_{\mathrm{res}})}.$

Here the terms in the first expression above are:

  * •

Term $a$ is a pure boundary term, in fact $a=e+f$, which leads to
$\frac{1}{2}$ factors of the boundary terms $e,f$ being doubled and replaced
by $1$ in the second equality in (11).

  * •

$b=0$.

  * •

$c=d=\frac{1}{2}\int_{I}(\psi^{-}_{\mathrm{fl}},d_{I}\psi^{+}_{\mathrm{fl}})$.

###  3.2. Antiholomorphic-to-holomorphic boundary conditions

Next, consider imposing the polarization $\frac{\partial}{\partial\psi^{+}}$
at $t=0$ and $\frac{\partial}{\partial\psi^{-}}$ at $t=1$ (a.k.a.
$\psi^{-}-\psi^{+}$ representation: we are fixing the boundary value
$\psi^{-}_{\mathrm{in}}$ at $t=0$ and $\psi^{+}_{\mathrm{out}}$ at $t=1$). The
“polarized action” (the counterpart of (6)) in this case is:

(12)
$S^{f}=S+\frac{1}{2}(\psi^{+},\psi^{-})\big{|}_{t=1}+\frac{1}{2}(\psi^{+},\psi^{-})\big{|}_{t=0}$

and the corresponding boundary $1$-form is:

$\alpha^{f}_{\partial
I}=\left.\left(\frac{1}{2}(\psi,\delta\psi)+\delta\frac{1}{2}(\psi^{+},\psi^{-})\right)\right|_{t=1}-\left.\left(\frac{1}{2}(\psi,\delta\psi)-\delta\frac{1}{2}(\psi^{+},\psi^{-})\right)\right|_{t=0}\\\
=(\psi^{-},\delta\psi^{+})\big{|}_{t=1}-(\psi^{+},\delta\psi^{-})\big{|}_{t=0}.$

This $1$-form vanishes along the chosen polarization, as desired.

Next, the fiber of the space of fields $\mathcal{F}$ over the base
$\mathcal{B}=\Pi\mathfrak{g}^{-}\oplus\Pi\mathfrak{g}^{+}=\\{(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{out}})\\}$
is the complex

(13)
$\mathcal{Y}=\Omega^{\bullet}(I,\\{1\\};\Pi\mathfrak{g}^{+})\oplus\Omega^{\bullet}(I,\\{0\\};\Pi\mathfrak{g}^{-}),$

which admits the following decomposition:

(14)
$\mathcal{Y}=\underbrace{\big{(}dt\cdot\mathfrak{g}_{\mathbb{C}}\oplus(1-t)\cdot\Pi\mathfrak{g}^{+}\oplus
t\cdot\Pi\mathfrak{g}^{-}\big{)}}_{\mathcal{V}}\bigoplus\\\
\bigoplus\underbrace{\big{(}\Omega^{0}_{\int=0}(I,\\{1\\};\Pi\mathfrak{g}^{+})\oplus\Omega^{0}_{\int=0}(I,\\{0\\};\Pi\mathfrak{g}^{-})\big{)}}_{\mathcal{Y}^{\prime}_{K-ex}}\bigoplus\\\
\bigoplus\underbrace{\begin{array}[]{l}\Big{(}\Big{\\{}g^{+}(t)dt\in\Omega^{1}(I;\mathfrak{g}^{+})\;\Big{|}\;\int_{I}dt\,g^{+}(t)\cdot
t=0\Big{\\}}\oplus\\\
\oplus\Big{\\{}g^{-}(t)dt\in\Omega^{1}(I;\mathfrak{g}^{-})\;\Big{|}\;\int_{I}dt\,g^{-}(t)\cdot(1-t)=0\Big{\\}}\Big{)}\end{array}}_{\mathcal{Y}^{\prime}_{d-ex}}$

Again, this is a splitting of $\mathcal{Y}$ into a deformation retract191919
Note that here we have chosen $\mathcal{V}$ to be larger than cohomology
(which in fact vanishes in this case).  and an acyclic subcomplex, with the
latter split in turn into the $d$-exact part and a direct complement — the
$K$-exact part, with the chain homotopy $K$ taking the form

$K\mathrel{\mathop{\mathchar
58\relax}}\begin{array}[]{lcl}dt\,g^{+}(t)&\mapsto&-\int_{t}^{1}dt^{\prime}\,g^{+}(t^{\prime})+2(1-t)\int_{0}^{1}dt^{\prime}\,t^{\prime}\,g^{+}(t^{\prime}),\\\
dt\,g^{-}(t)&\mapsto&\int_{0}^{t}dt^{\prime}\,g^{-}(t^{\prime})-2t\int_{0}^{1}dt^{\prime}\,(1-t^{\prime})\,g^{-}(t^{\prime}).\end{array}$

Its integral kernel — the propagator — is

(15)
$\eta(t,t^{\prime})=\pi^{+}\otimes\big{(}-\theta(t^{\prime}-t)+2(1-t)\,t^{\prime}\big{)}+\pi^{-}\otimes\big{(}\theta(t-t^{\prime})-2t\,(1-t^{\prime})\big{)}.$

We write an element of the space of residual fields $\mathcal{V}$ as
$(1-t)\cdot\psi^{+}_{\mathrm{res}}+t\cdot\psi^{-}_{\mathrm{res}}+dt\cdot
A_{\mathrm{res}}$, with
$\psi^{+}_{\mathrm{res}}\in\Pi\mathfrak{g}^{+},\psi^{-}_{\mathrm{res}}\in\Pi\mathfrak{g}^{-},A_{\mathrm{res}}\in\mathfrak{g}_{\mathbb{C}}$.

The BV-BFV partition function is:

(16)
$Z(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{out}};\psi^{+}_{\mathrm{res}},\psi^{-}_{\mathrm{res}},A_{\mathrm{res}})=\\\
=\int_{\mathcal{Y}^{\prime}_{K-ex}\subset\mathcal{Y}^{\prime}}\mathcal{D}\psi^{+}_{\mathrm{fl}}\,\mathcal{D}\psi^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}S^{f}\left(\widetilde{\psi^{-}_{\mathrm{in}}}+\widetilde{\psi^{+}_{\mathrm{out}}}+(1-t)\cdot\psi^{+}_{\mathrm{res}}+t\cdot\psi^{-}_{\mathrm{res}}+\psi^{+}_{\mathrm{fl}}+\psi^{-}_{\mathrm{fl}}+dt\cdot
A_{\mathrm{res}}\right)}\\\
=\int\mathcal{D}\psi^{+}_{\mathrm{fl}}\,\mathcal{D}\psi^{-}_{\mathrm{fl}}\;\exp\frac{i}{\hbar}\Big{(}\int(\psi^{-}_{\mathrm{fl}},d_{I}\psi^{+}_{\mathrm{fl}})+\frac{1}{2}(\psi^{-}_{\mathrm{res}},\psi^{+}_{\mathrm{res}})+(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{res}}+\psi^{-}_{\mathrm{fl}}(1))-(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{res}}+\psi^{+}_{\mathrm{fl}}(0))\Big{)}\\\
=\exp{\frac{i}{\hbar}\left(\frac{1}{2}(\psi^{-}_{\mathrm{res}},\psi^{+}_{\mathrm{res}})+(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{res}})-(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{res}})+(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{out}})\right)}.$

Here the last term comes from the simple Feynman diagram with a single
propagator connecting $\psi^{+}_{\mathrm{out}}$ and $\psi^{-}_{\mathrm{in}}$.

######  Remark 3.1.

One can further integrate out $\psi^{\pm}_{\mathrm{res}}$ in (16) resulting in
the partition function

(17)
$Z(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{out}})=e^{\frac{i}{\hbar}(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{in}})}.$

It corresponds to choosing the space of residual fields in (13) to be zero
(which is possible since the full complex $\mathcal{Y}$ is acyclic). Thus,
(17) is the minimal realization of the partition function of the theory on the
interval with prescribed boundary polarizations, and it is the BV pushforward
of the nonminimal realization (16).

######  Remark 3.2.

The exponent
$S_{\mathrm{HJ}}=(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{in}})$ in (17) is
the Hamilton–Jacobi action for the theory: it is the action (12) evaluated on
the (unique) solution of EL equation $\dot{\psi}=0$ satisfying the boundary
conditions $\psi^{-}|_{t=0}=\psi^{-}_{\mathrm{in}}$,
$\psi^{+}|_{t=1}=\psi^{+}_{\mathrm{out}}$. Also, $S_{\mathrm{HJ}}$ is the
generating function for the evolution relation of the theory:

$L_{S_{\mathrm{HJ}}}=\Big{\\{}\psi|_{t=1}=\psi^{+}_{\mathrm{out}}+\frac{\partial
S_{\mathrm{HJ}}}{\partial\psi^{+}_{\mathrm{out}}}=\psi^{+}_{\mathrm{out}}+\psi^{-}_{\mathrm{in}}\;\;,\;\;\psi|_{t=0}=\psi^{-}_{\mathrm{in}}-\frac{\partial
S_{\mathrm{HJ}}}{\partial\psi^{-}_{\mathrm{in}}}=\psi^{-}_{\mathrm{in}}+\psi^{+}_{\mathrm{out}}\Big{\\}}\\\
=\big{\\{}\psi|_{t=1}=\psi|_{t=0}\big{\\}}\quad\subset\;\;\Pi\mathfrak{g}\times\Pi\mathfrak{g}.$

This provides a simple example of Hamilton–Jacobi formalism, see [16] and
Section 2, with the phase space being the symplectic supermanifold
$\Pi\mathfrak{g}$.

Moreover, the exponent in (16) is a generalized generating function for the
evolution relation, with $\psi^{\pm}_{\mathrm{res}}$ the auxiliary parameters.
It can also be seen as the Hamilton–Jacobi action for the action $S^{f}+\int
dt(\lambda,\psi-\frac{1}{2}\psi_{\mathrm{res}})$ with $S^{f}$ as in (12) and
where $\lambda\in\Pi\mathfrak{g}$ (a constant along $I$) is a Lagrange
multiplier.

Likewise, the exponent
$(\psi^{+}_{\mathrm{out}}-\psi^{+}_{\mathrm{in}},\psi^{-}_{\mathrm{res}})$ in
the right hand side of (11) is the generalized generating function for the
same evolution relation, with respect to
$(\psi^{+}_{\mathrm{out}},\psi^{+}_{\mathrm{in}})$-polarization, with
$\psi^{-}_{\mathrm{res}}$ the auxiliary parameter, cf. [16, Section 6.1].

To summarize, in these three cases Threorem A holds:

  * •

Case of (17): 1D abelian Chern-Simons with
$(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{in}})$-polarization at the
endpoints of the interval, with $\mathcal{V}=V_{\mathrm{aux}}=0$.

  * •

Case of (16): 1D abelian Chern-Simons with
$(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{in}})$-polarization, with
$\mathcal{V}$ parametrized by
$(\psi^{+}_{\mathrm{res}},\psi^{-}_{\mathrm{res}},A_{\mathrm{res}})\in\Pi\mathfrak{g}^{+}\oplus\Pi\mathfrak{g}^{-}\oplus\mathfrak{g}_{\mathbb{C}}$
and with $V_{\mathrm{aux}}=\Pi\mathfrak{g}^{+}\oplus\Pi\mathfrak{g}^{-}$
parametrized by $(\psi^{+}_{\mathrm{res}},\psi^{-}_{\mathrm{res}})$.

  * •

Case of (11): 1D abelian Chern-Simons with
$(\psi^{+}_{\mathrm{out}},\psi^{+}_{\mathrm{in}})$-polarization, with
$\mathcal{V}=\Pi\mathfrak{g}^{-}\oplus\mathfrak{g}^{+}$ parametrized by
$(\psi^{-}_{\mathrm{res}},A^{+}_{\mathrm{res}})$ and
$V_{\mathrm{aux}}=\Pi\mathfrak{g}^{-}$ parametrized by
$\psi^{-}_{\mathrm{res}}$.

Note that in these cases $V_{\mathrm{aux}}$ is a direct summand in
$\mathcal{V}$ but it is not singled out by the condition of vanishing ghost
number (rather, it is the _odd_ part of $\mathcal{V}$): space of fields of 1D
abelian Chern-Simons as considered here does not admit a $\mathbb{Z}$-grading.

We also remark that in all these cases Theorem B works trivially: $\Omega=0$
in 1D abelian Chern-Simons and $\Delta$ contains a derivative in
$A_{\mathrm{res}}$ on which $S_{\mathrm{eff}}$ does not depend.

##  4\. BV-BFV approach to 3D abelian Chern–Simons on a cylinder

Consider the $3$-dimensional abelian Chern–Simons theory on a cylinder
$I\times\Sigma$, with $\Sigma$ a closed oriented surface and $I=[0,1]$ the
interval parametrized by the coordinate $t$. The space of BV fields, as given
by the AKSZ construction, is the $\mathbb{Z}$-graded mapping space

$\mathcal{F}=\mathrm{Map}(T[1](I\times\Sigma),\mathbb{R}[1])=\Omega^{\bullet}(I\times\Sigma)[1].$

Exploiting the fact that the source is a cylinder, we can also write it as a
free (i.e., with a quadratic action) $1$-dimensional AKSZ theory with the
target given by forms on $\Sigma$:

$\mathcal{F}=\mathrm{Map}(T[1]I,\mathrm{Map}(T[1]\Sigma,\mathbb{R}[1]))=\Omega^{\bullet}(I,\Omega^{\bullet}(\Sigma)[1]).$

The BV action is:

(18) $S=\int_{I\times\Sigma}\frac{1}{2}\mathcal{A}\wedge
d\mathcal{A}=\int_{I}\frac{1}{2}(\mathcal{A},d_{I}\mathcal{A})+\frac{1}{2}(\mathcal{A},d_{\Sigma}\mathcal{A}).$

Here $d=d_{I}+d_{\Sigma}$ is the de Rham operator on the cylinder splitting
into the surface part and the interval part; the pairing is integration over
the surface: $(u,v)=\int_{\Sigma}u\wedge v$. The field splits into $0$\- and
$1$-form components along $I$ as

$\mathcal{A}=\mathsf{A}+dt\cdot\mathsf{A}_{I}$

with $\mathsf{A},\mathsf{A}_{I}$ two $t$-dependent nonhomogeneous forms on
$\Sigma$; their homogeneous components are prescribed internal
$\mathbb{Z}$-grading (ghost number) as follows:
$\mathrm{gh}(\mathsf{A}^{(p)})=1-p$, $\mathrm{gh}(\mathsf{A}_{I}^{(p)})=-p$.

Comparing to the discussion of Section 3, this theory can be seen as
$1$-dimensional Chern–Simons on $I$ with coefficients in
$\mathfrak{g}=\Omega^{\bullet}(\Sigma)$. Here the fact that $\mathfrak{g}$ is
itself a cochain complex with differential $d_{\Sigma}$ gives rise to an
additional term in the action. Also, the fact that $\mathfrak{g}$ has a degree
$-2$ (rather than degree $0$) graded-symmetric pairing allows one to prescribe
$\mathbb{Z}$-grading to fields (in such a way that the action has degree $0$
and the odd symplectic form has degree $-1$) rather than just
$\mathbb{Z}_{2}$-grading.

The BFV phase space assigned to a boundary surface ($\\{1\\}\times\Sigma$ or
$\\{0\\}\times\Sigma$) is
$\mathcal{F}^{\partial}_{\Sigma}=\Omega^{\bullet}(\Sigma)[1]$ which is
$0$-symplectic, with the Noether $1$-form
$\pm\int_{\Sigma}\frac{1}{2}\mathsf{A}\wedge\delta\mathsf{A}$ where the sign
is $+$ for the out-boundary and $-$ for the in-boundary. The phase space
carries a degree $-1$ BFV action

(19) $S_{\Sigma}=\pm\int_{\Sigma}\frac{1}{2}\mathsf{A}\wedge
d_{\Sigma}\mathsf{A}.$

Next, assume that $\Sigma$ is endowed with a complex structure, so that
complex-valued $1$-forms split as
$\Omega^{1}_{\mathbb{C}}(\Sigma)=\Omega^{1,0}(\Sigma)\oplus\Omega^{0,1}(\Sigma)$.
Then, mimicking (5), we split the (complexified) space of all forms on
$\Sigma$ as follows:

(20)
$\underbrace{\Omega^{\bullet}_{\mathbb{C}}(\Sigma)}_{\mathfrak{g}_{\mathbb{C}}}=\underbrace{(\Omega^{0}_{\mathbb{C}}(\Sigma)\oplus\Omega^{1,0}(\Sigma))}_{\mathfrak{g}^{+}}\bigoplus\underbrace{(\Omega^{0,1}(\Sigma)\oplus\Omega^{2}_{\mathbb{C}}(\Sigma))}_{\mathfrak{g}^{-}}.$

This is, clearly, a splitting into lagrangian subspaces.

###  4.1. Holomorphic-to-holomorphic boundary conditions

Consider the polarization
$\mathrm{Span}\\{\frac{\delta}{\delta\mathsf{A}^{-}}\\}$ on both boundary
surfaces, at $t=0$ and $t=1$, i.e., the one where we prescribe boundary values
$\mathsf{A}^{+}_{\mathrm{in}}$, $\mathsf{A}^{+}_{\mathrm{out}}$. The
corresponding modification of the action by boundary terms adjusting for the
polarization is:

$S^{f}=S+\frac{1}{2}\int_{\\{1\\}\times\Sigma}\mathsf{A}^{+}\mathsf{A}^{-}-\frac{1}{2}\int_{\\{0\\}\times\Sigma}\mathsf{A}^{+}\mathsf{A}^{-}$

and the corresponding Noether $1$-form is:

$\alpha^{f}_{\Sigma\times\partial
I}=\int_{\\{1\\}\times\Sigma}\mathsf{A}^{-}\delta\mathsf{A}^{+}-\int_{\\{0\\}\times\Sigma}\mathsf{A}^{-}\delta\mathsf{A}^{+}.$

The fiber of the (complexified) space of fields over the space of boundary
conditions
$\mathcal{B}=\mathfrak{g}^{+}[1]\oplus\mathfrak{g}^{+}[1]=\\{(\mathsf{A}^{+}_{\mathrm{in}},\mathsf{A}^{+}_{\mathrm{out}})\\}$
is:

$\mathcal{Y}=\Omega^{\bullet}(I,\partial
I;\mathfrak{g}^{+}[1])\oplus\Omega^{\bullet}(I;\mathfrak{g}^{-}[1]).$

Hodge decomposition (7) holds (where one should replace $\Pi$ with degree
shift $[1]$) and the formula for the chain homotopy (8) and the propagator (9)
also. Writing out the projectors $\pi^{\pm}$ explicitly in our case, we obtain
the following formula for the propagator:

(21) $\eta((z,\bar{z},t)\,;\,(z^{\prime},\bar{z}^{\prime},t^{\prime}))=\\\
=\delta^{(2)}(z-z^{\prime})\;\frac{i}{2}\big{(}-dz\wedge
d\bar{z}^{\prime}+dz^{\prime}\wedge
d\bar{z}^{\prime}\big{)}\;\big{(}\theta(t-t^{\prime})-t\big{)}+\\\
+\delta^{(2)}(z-z^{\prime})\;\frac{i}{2}\big{(}d\bar{z}\wedge
dz^{\prime}+dz\wedge
d\bar{z}\big{)}\;\big{(}t^{\prime}-\theta(t^{\prime}-t)\big{)}.$

— This is a distributional $2$-form on $(I\times\Sigma)\times(I\times\Sigma)$.
Here $z$ is the local complex coordinate on $\Sigma$. Our convention for the
normalization of the delta function is: $\int\frac{i}{2}dz\wedge
d\bar{z}\;\delta^{(2)}(z-z^{\prime})=1$.

Note that the propagator (21) is for the $d_{I}$ term in the action (18) only,
whereas the $d_{\Sigma}$ term is treated as a perturbation.

The space of residual fields is:

$\mathcal{V}=dt\cdot\mathfrak{g}^{+}\oplus
1\cdot\mathfrak{g}^{-}=\\{dt\cdot\mathsf{A}^{0}_{I\,\mathrm{res}}+dt\cdot\mathsf{A}^{1,0}_{I\,\mathrm{res}}+\mathsf{A}^{0,1}_{\mathrm{res}}+\mathsf{A}^{2}_{\mathrm{res}}\\},$

where
$\mathsf{A}^{0}_{I\,\mathrm{res}},\;\mathsf{A}^{1,0}_{I\,\mathrm{res}},\;\mathsf{A}^{0,1}_{\mathrm{res}},\;\mathsf{A}^{2}_{\mathrm{res}}$
are $t$-independent forms on $\Sigma$ of de Rham degree $0,(1,0),(0,1),2$,
respectively, with internal degree $0,-1,0,-1$, respectively.

######  Remark 4.1 (Axial gauge).

We call the gauge fixing introduced here the _axial gauge_ : it sets the
“axial” field fluctuations — those which are 1-forms along $I$ and forms of
any degree along $\Sigma$ — to zero.

On the level of homological algebra, for $M,N$ closed manifolds, one can
construct a chain contraction $K$ from $\Omega^{\bullet}(M\times N)$ to
$H^{\bullet}(M)\otimes\Omega^{\bullet}(N)$ of the form
$K=K_{M}\otimes\mathrm{id}_{N}$ with $K_{M}$ a chain contraction from forms on
$M$ to its cohomology (cohomology can be swapped for any deformation retract
of the de Rham complex in the construction). The integral kernel of $K$ — the
propagator — is a distributional form on $(M\times N)^{\times 2}$ containing
the delta form on $N\times N$. A version of the axial gauge for Chern–Simons
theory was first employed in [21]. In our situation, $N=\Sigma$ and $M=I$ is
not a closed manifold and hence the construction has to be adapted for
boundary conditions — which is exactly what we did above. The chain
contraction, corresponding to (21), has the form $K=K_{I,\partial
I}\otimes\mathrm{id}_{\mathfrak{g}^{+}}+K_{I}\otimes\mathrm{id}_{\mathfrak{g}^{-}}$.
We will encounter versions of this construction for different choices of
boundary conditions further in this paper (e.g., in the case of Section 4.2,
the chain contraction has the form
$K_{I,\\{1\\}}\otimes\mathrm{id}_{\mathfrak{g}^{+}}+K_{I,\\{0\\}}\otimes\mathrm{id}_{\mathfrak{g}^{-}}$).202020
Depending on the choice of boundary conditions (e.g., in the case of Section
4.2), the space of forms subject to boundary conditions $\mathcal{Y}[-1]$ may
fail to be a subcomplex of $\Omega^{\bullet}(I\times\Sigma)$ with respect to
the total de Rham differential $d_{I}+d_{\Sigma}$. However, the operator $K$
we are constructing can be seen as a chain contraction for just the “axial”
differential $d_{I}$.

The BV-BFV partition function is readily calculated:

(22)
$Z(\underbrace{\mathsf{A}^{0}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{in}}}_{\mathsf{A}^{+}_{\mathrm{in}}},\underbrace{\mathsf{A}^{0}_{\mathrm{out}},\mathsf{A}^{1,0}_{\mathrm{out}}}_{\mathsf{A}^{+}_{\mathrm{out}}};\underbrace{\mathsf{A}^{0}_{I\,\mathrm{res}},\mathsf{A}^{1,0}_{I\,\mathrm{res}}}_{\mathsf{A}^{+}_{I\,\mathrm{res}}},\underbrace{\mathsf{A}^{0,1}_{\mathrm{res}},\mathsf{A}^{2}_{\mathrm{res}}}_{\mathsf{A}^{-}_{\mathrm{res}}})=\\\
=\int_{\mathcal{Y}^{\prime}_{K-ex}\subset\mathcal{Y}^{\prime}}\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\;\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}S^{f}\left(\widetilde{\mathsf{A}^{+}_{\mathrm{in}}}+\widetilde{\mathsf{A}^{+}_{\mathrm{out}}}+\mathsf{A}^{+}_{\mathrm{fl}}+\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{fl}}+dt\cdot\mathsf{A}^{+}_{I\,\mathrm{res}}\right)}\\\
=\int\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\;\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}\big{(}\int_{I\times\Sigma}\mathsf{A}^{-}_{\mathrm{fl}}d_{I}\mathsf{A}^{+}_{\mathrm{fl}}+\int_{\\{1\\}\times\Sigma}\mathsf{A}^{+}_{\mathrm{out}}\mathsf{A}^{-}-\int_{\\{0\\}\times\Sigma}\mathsf{A}^{+}_{\mathrm{in}}\mathsf{A}^{-}+\int_{I\times\Sigma}\frac{1}{2}\mathcal{A}\,d_{\Sigma}\mathcal{A}\big{)}}\\\
=\int\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\;\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;\exp\frac{i}{\hbar}\Big{(}\int_{I\times\Sigma}\mathsf{A}^{-}_{\mathrm{fl}}\,d_{I}\mathsf{A}^{+}_{\mathrm{fl}}+\int_{\Sigma}\mathsf{A}^{+}_{\mathrm{out}}\,(\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{fl}}\big{|}_{t=1})-\\\
-\int_{\Sigma}\mathsf{A}^{+}_{\mathrm{in}}\,(\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{fl}}\big{|}_{t=0})+\int_{\Sigma}\mathsf{A}^{0}_{I\,\mathrm{res}}\partial\mathsf{A}^{0,1}_{\mathrm{res}}+\int_{I\times\Sigma}dt\;\mathsf{A}^{+}_{I\,\mathrm{res}}\,\bar{\partial}\mathsf{A}^{+}_{\mathrm{fl}}\Big{)}.$

Here we are using the splitting $d_{\Sigma}=\partial+\bar{\partial}$ of de
Rham operator on $\Sigma$ into the holomorphic and antiholomorphic Dolbeault
operators. Finally, computing this Gaussian integral, we obtain

(23)
$Z=\exp\frac{i}{\hbar}\int_{\Sigma}\Big{(}(\mathsf{A}^{+}_{\mathrm{out}}-\mathsf{A}^{+}_{\mathrm{in}})\;\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{0}_{I\,\mathrm{res}}\,\partial\mathsf{A}^{0,1}_{\mathrm{res}}+\frac{1}{2}(\mathsf{A}^{+}_{\mathrm{out}}+\mathsf{A}^{+}_{\mathrm{in}})\;\bar{\partial}\mathsf{A}^{+}_{I\,\mathrm{res}}\Big{)}.$

Here the last term arises from the Wick contraction

$\wick{\big{\langle}\int_{\Sigma}\mathsf{A}^{+}_{\mathrm{out}}\,\c{1}{\mathsf{A}^{-}_{\mathrm{fl}}}\big{|}_{t=1}\;\;\int_{I\times\Sigma}dt^{\prime}\c{1}{\mathsf{A}^{+}_{\mathrm{fl}}}\bar{\partial}\mathsf{A}^{+}_{I\,\mathrm{res}}\,\big{\rangle}}=\underbrace{\int_{I}dt^{\prime}\,(t^{\prime}-\theta(t^{\prime}-t))\big{|}_{t=1}}_{1/2}\int_{\Sigma}\mathsf{A}^{+}_{\mathrm{out}}\,\bar{\partial}\mathsf{A}^{+}_{I\,\mathrm{res}},$

and a similar one with $\mathsf{A}^{+}_{\mathrm{in}}$ talking to
$\bar{\partial}\mathsf{A}^{+}_{I\,\mathrm{res}}$.

Graphically, the diagrams contributing to (23) are:

Figure 1. Feynman diagrams for the abelian theory on a cylinder in
holomorphic-to-holomorphic polarization.

Here the conventions (Feynman rules) are:

  * •

Black dots are vertices, which can be on in- or out-boundary (then they are
univalent, with single incoming half-edge), or in the bulk (then they are
bivalent – with one incoming and one outgoing, or with two outgoing half-
edges).

  * •

Half-edges can be internal (joined into pairs forming an edge, depicted as a
long edge above) or external – depicted as a short edge ending with a white or
a gray blob, depending on orientation.

  * •

Boundary vertices are decorated by $\mathsf{A}^{+}_{\mathrm{out}}$ on the out-
boundary and by $\mathsf{A}^{+}_{\mathrm{in}}$ on the in-boundary.

  * •

White blobs are decorated by $\mathsf{A}^{-}_{\mathrm{res}}$, gray blobs are
decorated by $\mathsf{A}^{+}_{I\,\mathrm{res}}$.

  * •

(Long) edges are decorated by the propagator $\eta$.

  * •

Bulk vertices with one incoming and one outgoing half-edge carry $\partial$;
bulk vertices with two outgoing half-edges carry $\bar{\partial}$.
(Equivalently, one may say that bulk vertices are decorated by $d_{\Sigma}$
independently of orientation.)

  * •

For each connected graph $\Gamma$ in Figure 1, we take the product of
decorations obtaining a differential form on
$\mathrm{Conf}_{\Gamma}=\Sigma^{\\#\mathrm{in-
vertices}}\times(I\times\Sigma)^{\\#\mathrm{bulk\,vertices}}\times\Sigma^{\\#\mathrm{out-
vertices}}$ depending on the residual fields. Then we take integral over
$\mathrm{Conf}_{\Gamma}$, obtaining the value of the diagram.

We will return to the version of the result (23) in the context of nonabelian
Chern–Simons theory in Section 5.4.

####  4.1.1. Comparison with Hamilton–Jacobi action

We can write the result (23) in the form

(24)
$Z=\exp\frac{i}{\hbar}\Big{(}\underbrace{\int_{\Sigma}\Big{(}(\mathsf{A}^{1,0}_{\mathrm{out}}-\mathsf{A}^{1,0}_{\mathrm{in}})\lambda+\lambda\partial\sigma+\frac{1}{2}(\mathsf{A}^{1,0}_{\mathrm{out}}+\mathsf{A}^{1,0}_{\mathrm{in}})\bar{\partial}\sigma\Big{)}}_{S_{\text{HJ}}}+\\\
+\int_{\Sigma}\Big{(}(\mathsf{A}^{0}_{\mathrm{out}}-\mathsf{A}^{0}_{\mathrm{in}})\mathsf{A}^{2}_{\mathrm{res}}+\frac{1}{2}(\mathsf{A}^{0}_{\mathrm{out}}+\mathsf{A}^{0}_{\mathrm{in}})\bar{\partial}\mathsf{A}^{1,0}_{I\,\mathrm{res}}\Big{)}\Big{)},$

where we introduced the alternative notation for degree zero residual fields

$\lambda\mathrel{\mathop{\mathchar
58\relax}}=\mathsf{A}^{0,1}_{\mathrm{res}},\;\;\sigma\mathrel{\mathop{\mathchar
58\relax}}=\mathsf{A}^{0}_{I\,\mathrm{res}}.$

In the first integral in (24) we recognize the Hamilton–Jacobi action [16, Eq.
(48)], which can be seen as the conformal $\beta\gamma$-system coupled to the
boundary fields, while in the second integral we collected the contribution of
nonzero-degree fields.

####  4.1.2. Quantum master equation

The space of states on a surface with $\mathsf{A}^{+}$-fixed polarization is
the space of functions of $\mathsf{A}^{+}$ of the form

(25) $\Psi(\mathsf{A}^{+})=\sum_{n\geq
0}\int_{\Sigma^{n}}\gamma_{n}\pi_{1}^{*}\mathsf{A}^{+}\cdots\pi_{n}^{*}\mathsf{A}^{+}$

where $\\{\gamma_{n}\\}$ are $\hbar$-dependent distributional forms on
$\Sigma^{n}$ and $\pi_{i}\colon\Sigma^{n}\rightarrow\Sigma$ is the projection
to the $i$-th copy of $\Sigma$ (we refer to [15, Section 3.5.1] for details).
The space of states is equipped with the differential (the quantum BFV
operator)

(26)
$\Omega_{\Sigma}^{+}=\int_{\Sigma}\left(-i\hbar\,\partial\mathsf{A}^{0}\,\frac{\delta}{\delta\mathsf{A}^{1,0}}+\epsilon\mathsf{A}^{1,0}\,\bar{\partial}\mathsf{A}^{0}\right)$

with the sign $\epsilon=+1$ for the out-boundary and $\epsilon=-1$ for the in-
boundary;212121The integral over $\Sigma$ in (26) is understood as being with
respect to the “standard” orientation, which coincides with the induced one
from the cylinder on the out-boundary and is opposite to the induced one on
the in-boundary. the superscript in $\Omega^{+}_{\Sigma}$ is a reminder of the
choice of polarization. This operator is the canonical quantization of the
boundary BFV action (19),

(27)
$S_{\Sigma}=\epsilon\int_{\Sigma}\mathsf{A}^{1,0}\,\bar{\partial}\mathsf{A}^{0}+\mathsf{A}^{0,1}\,\partial\mathsf{A}^{0}.$

In the quantization, $\mathsf{A}^{0},\mathsf{A}^{1,0}$ become multiplication
operators and
$\mathsf{A}^{0,1}\mapsto-\epsilon\,i\hbar\frac{\delta}{\delta\mathsf{A}^{1,0}}$,
$\mathsf{A}^{2}\mapsto-\epsilon\,i\hbar\frac{\delta}{\delta\mathsf{A}^{0}}$
become derivations.

######  Lemma 4.2.

The partition function (23) satisfies the BV quantum master equation modified
by the boundary terms (see [15]):

(28)
$\Big{(}\underbrace{\int_{\Sigma}\big{(}-i\hbar\,\partial\mathsf{A}^{0}_{\mathrm{out}}\,\frac{\delta}{\delta\mathsf{A}^{1,0}_{\mathrm{out}}}+\mathsf{A}^{1,0}_{\mathrm{out}}\,\bar{\partial}\mathsf{A}^{0}_{\mathrm{out}}\big{)}}_{\Omega_{\mathrm{out}}^{+}}+\underbrace{\int_{\Sigma}\big{(}-i\hbar\,\partial\mathsf{A}^{0}_{\mathrm{in}}\,\frac{\delta}{\delta\mathsf{A}^{1,0}_{\mathrm{in}}}-\mathsf{A}^{1,0}_{\mathrm{in}}\,\bar{\partial}\mathsf{A}^{0}_{\mathrm{in}}\big{)}}_{\Omega_{\mathrm{in}}^{+}}-\\\
-\hbar^{2}\underbrace{\int_{\Sigma}\frac{\delta}{\delta\mathsf{A}^{-}_{\mathrm{res}}}\;\frac{\delta}{\delta\mathsf{A}^{+}_{I\,\mathrm{res}}}}_{\Delta_{\mathrm{res}}}\Big{)}\;Z=0.$

###### Proof.

One checks this by a direct computation:

(29) $(\Omega_{\mathrm{out}}^{+}+\Omega_{\mathrm{in}}^{+})Z=\\\
=Z\cdot\int_{\Sigma}\Big{(}(\partial\mathsf{A}^{0}_{\mathrm{out}}-\partial\mathsf{A}^{0}_{\mathrm{in}})\,\mathsf{A}^{0,1}_{\mathrm{res}}+\frac{1}{2}(\partial\mathsf{A}^{0}_{\mathrm{out}}+\partial\mathsf{A}^{0}_{\mathrm{in}})\,\bar{\partial}\mathsf{A}^{0}_{I\,\mathrm{res}}+\mathsf{A}^{1,0}_{\mathrm{out}}\,\bar{\partial}\mathsf{A}^{0}_{\mathrm{out}}-\mathsf{A}^{1,0}_{\mathrm{in}}\,\bar{\partial}\mathsf{A}^{0}_{\mathrm{in}}\Big{)}.$

On the other hand,

(30)
$\hbar^{2}\Delta_{\mathrm{res}}Z=Z\cdot\int_{\Sigma}\big{(}\mathsf{A}^{+}_{\mathrm{out}}-\mathsf{A}^{+}_{\mathrm{in}}-\partial\mathsf{A}^{+}_{I\,\mathrm{res}}\big{)}\,\big{(}\partial\mathsf{A}^{-}_{\mathrm{res}}+\frac{1}{2}\bar{\partial}(\mathsf{A}^{+}_{\mathrm{out}}+\mathsf{A}^{+}_{\mathrm{in}})\big{)}.$

Inspecting this expression, we see that it coincides with (29), which proves
(28). ∎

Following the terminology of [15], we call the equation
$(\Omega_{\partial}-\hbar^{2}\Delta_{\mathrm{res}})Z=0$ the modified (by the
boundary term) quantum master equation (mQME).

###  4.2. Antiholomorphic-to-holomorphic boundary conditions

Consider the polarization
$\mathrm{Span}\\{\frac{\delta}{\delta\mathsf{A}^{+}}\\}$ at $t=0$ and
$\mathrm{Span}\\{\frac{\delta}{\delta\mathsf{A}^{-}}\\}$ at $t=1$. I.e., we
prescribe boundary values
$\mathsf{A}^{-}_{\mathrm{in}},\mathsf{A}^{+}_{\mathrm{out}}$. The
corresponding modification of the action by boundary terms adjusting for the
polarization is:

$S^{f}=S+\frac{1}{2}\int_{\\{1\\}\times\Sigma}\mathsf{A}^{+}\mathsf{A}^{-}+\frac{1}{2}\int_{\\{0\\}\times\Sigma}\mathsf{A}^{+}\mathsf{A}^{-}$

and the modified boundary Noether $1$-form is:

$\alpha^{f}_{\Sigma\times\partial
I}=\int_{\\{1\\}\times\Sigma}\mathsf{A}^{-}\delta\mathsf{A}^{+}-\int_{\\{0\\}\times\Sigma}\mathsf{A}^{+}\delta\mathsf{A}^{-}.$

The fiber of the (complexified) space of fields over the space of boundary
conditions
$\mathcal{B}=\mathfrak{g}^{-}[1]\oplus\mathfrak{g}^{+}[1]=\\{(\mathsf{A}^{-}_{\mathrm{in}},\mathsf{A}^{+}_{\mathrm{out}})\\}$
is the complex

$\mathcal{Y}=\Omega^{\bullet}(I,\\{1\\};\mathfrak{g}^{+}[1])\oplus\Omega^{\bullet}(I,\\{0\\};\mathfrak{g}^{-}[1]).$

Hodge decomposition (14) holds (where one replaces $\Pi\rightarrow[1]$) and
the propagator is given by (15) or, more explicitly,

$\eta((z,\bar{z},t)\,;\,(z^{\prime},\bar{z}^{\prime},t^{\prime}))=\\\
=\delta^{(2)}(z-z^{\prime})\frac{i}{2}(-dz\wedge
d\bar{z}^{\prime}+dz^{\prime}\wedge
d\bar{z}^{\prime})\;\big{(}-\theta(t^{\prime}-t)+2(1-t)\,t^{\prime}\big{)}+\\\
+\delta^{(2)}(z-z^{\prime})\frac{i}{2}(d\bar{z}\wedge dz^{\prime}+dz\wedge
d\bar{z})\;\big{(}\theta(t-t^{\prime})-2t\,(1-t^{\prime})\big{)}.$

The space of residual fields is:

(31)
$\mathcal{V}=dt\cdot\mathfrak{g}_{\mathbb{C}}\oplus(1-t)\cdot\mathfrak{g}^{+}[1]\oplus
t\cdot\mathfrak{g}^{-}[1]=\\{dt\cdot\mathsf{A}_{I\,\mathrm{res}}+(1-t)\cdot\mathsf{A}^{+}_{\mathrm{res}}+t\cdot\mathsf{A}^{-}_{\mathrm{res}}\\},$

where $\mathsf{A}_{I\,\mathrm{res}}$, $\mathsf{A}^{+}_{\mathrm{res}}$,
$\mathsf{A}^{-}_{\mathrm{res}}$ are $t$-independent forms on $\Sigma$. The
homogeneous components of these residual fields and their internal degrees
(ghost numbers) are as follows:
$\mathsf{A}_{{I\,\mathrm{res}}}=$ $\mathsf{A}_{I\,\mathrm{res}}^{0}+$
$\mathsf{A}_{I\,\mathrm{res}}^{1,0}+$ $\mathsf{A}_{I\,\mathrm{res}}^{0,1}+$
$\mathsf{A}_{I\,\mathrm{res}}^{2}$ $\mathsf{A}^{+}_{\mathrm{res}}=$
$\mathsf{A}^{0}_{\mathrm{res}}+$ $\mathsf{A}^{1,0}_{\mathrm{res}}$
$\mathsf{A}^{-}_{\mathrm{res}}=$ $\mathsf{A}^{0,1}_{\mathrm{res}}+$
$\mathsf{A}^{2}_{\mathrm{res}}$ $0$ $-1$ $-1$ $-2$ $1$ $0$ $0$ $-1$

The BV-BFV partition function is:

(32)
$Z(\mathsf{A}^{-}_{\mathrm{in}},\mathsf{A}^{+}_{\mathrm{out}};\mathsf{A}_{I\,\mathrm{res}},\mathsf{A}^{+}_{\mathrm{res}},\mathsf{A}^{-}_{\mathrm{res}})=\\\
=\int_{\mathcal{Y}^{\prime}_{K-ex}\subset\mathcal{Y}^{\prime}}\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\,\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}S^{f}\Big{(}\mathsf{A}^{-}_{\mathrm{in}}+\mathsf{A}^{+}_{\mathrm{out}}+(1-t)\cdot\mathsf{A}^{+}_{\mathrm{res}}+t\cdot\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{+}_{\mathrm{fl}}+\mathsf{A}^{-}_{\mathrm{fl}}+dt\cdot\mathsf{A}_{I\,\mathrm{res}}\Big{)}}\\\
=\int\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\,\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}\Big{(}\int_{I\times\Sigma}\big{(}\mathsf{A}^{-}_{\mathrm{fl}}+t\mathsf{A}^{-}_{\mathrm{res}}\big{)}\,d_{I}\big{(}\mathsf{A}^{+}_{\mathrm{fl}}+(1-t)\mathsf{A}^{+}_{\mathrm{res}}\big{)}+\int_{\\{1\\}\times\Sigma}\mathsf{A}^{+}_{\mathrm{out}}\mathsf{A}^{-}-\int_{\\{0\\}\times\Sigma}\mathsf{A}^{-}_{\mathrm{in}}\mathsf{A}^{+}+\int_{I\times\Sigma}\frac{1}{2}\mathcal{A}\,d_{\Sigma}\mathcal{A}\Big{)}}\\\
=\int\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\,\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;\exp\frac{i}{\hbar}\Big{(}\int_{I\times\Sigma}\mathsf{A}^{-}_{\mathrm{fl}}\,d_{I}\mathsf{A}^{+}_{\mathrm{fl}}+\frac{1}{2}\int_{\Sigma}\mathsf{A}^{-}_{\mathrm{res}}\mathsf{A}^{+}_{\mathrm{res}}+\int_{\Sigma}\mathsf{A}^{+}_{\mathrm{out}}(\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{fl}}\big{|}_{t=1})-\\\
-\int_{\Sigma}\mathsf{A}^{-}_{\mathrm{in}}(\mathsf{A}^{+}_{\mathrm{res}}+\mathsf{A}^{+}_{\mathrm{fl}}\big{|}_{t=0})+\frac{1}{2}\int_{\Sigma}\mathsf{A}_{I\,\mathrm{res}}\;d_{\Sigma}(\mathsf{A}^{+}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{res}})\Big{)}\\\
=\exp\frac{i}{\hbar}\int_{\Sigma}\Big{(}-\mathsf{A}^{+}_{\mathrm{out}}\mathsf{A}^{-}_{\mathrm{in}}+\mathsf{A}^{+}_{\mathrm{out}}\mathsf{A}^{-}_{\mathrm{res}}-\mathsf{A}^{-}_{\mathrm{in}}\mathsf{A}^{+}_{\mathrm{res}}+\frac{1}{2}\mathsf{A}^{-}_{\mathrm{res}}\mathsf{A}^{+}_{\mathrm{res}}+\\\
+\frac{1}{2}\big{(}\mathsf{A}^{0,1}_{I\,\mathrm{res}}\partial\mathsf{A}^{0}_{\mathrm{res}}+\mathsf{A}^{1,0}_{I\,\mathrm{res}}\bar{\partial}\mathsf{A}^{0}_{\mathrm{res}}+\mathsf{A}^{0}_{I\,\mathrm{res}}(\partial\mathsf{A}^{0,1}_{\mathrm{res}}+\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{res}})\big{)}\Big{)}.$

Here the first term in the final result is a contribution of the diagram where
$\mathsf{A}^{+}_{\mathrm{out}}$ is contracted by a propagator with
$\mathsf{A}^{-}_{\mathrm{in}}$.

####  4.2.1. Partial integral over residual fields and comparison with
Hamilton–Jacobi action

Motivated by comparison with the Hamilton–Jacobi formalism, we consider the BV
pushforward of the partition function (32) along the odd symplectic fibration

$p\mathrel{\mathop{\mathchar
58\relax}}\mathcal{V}\rightarrow\mathcal{V}_{\mathrm{small}}=\\{dt\cdot(\mathsf{A}^{0}_{I\,\mathrm{res}}+\mathsf{A}^{2}_{I\,\mathrm{res}})+(1-t)\cdot\mathsf{A}^{0}_{\mathrm{res}}+t\cdot\mathsf{A}^{2}_{\mathrm{res}}\\}.$

In its kernel, we choose the gauge-fixing lagrangian subspace $\mathcal{L}$
cut out by equations
$\mathsf{A}^{1,0}_{I\,\mathrm{res}}=\mathsf{A}^{0,1}_{I\,\mathrm{res}}=0$ and
parametrized by
$\mathsf{A}^{1,0}_{\mathrm{res}},\mathsf{A}^{0,1}_{\mathrm{res}}$. The
corresponding BV pushforward is:

(33)
$Z_{\mathrm{small}}=\int\mathcal{D}\mathsf{A}^{1,0}_{\mathrm{res}}\,\mathcal{D}\mathsf{A}^{0,1}_{\mathrm{res}}\;Z=\\\
=\exp\frac{i}{\hbar}\Big{(}\underbrace{\int_{\Sigma}\Big{(}\mathsf{A}^{1,0}_{\mathrm{out}}\mathsf{A}^{0,1}_{\mathrm{in}}+(\partial\mathsf{A}^{0,1}_{\mathrm{in}}+\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}})\;\sigma+\frac{1}{2}\sigma\partial\bar{\partial}\sigma\Big{)}}_{S_{\text{HJ}}}+\\\
+\int_{\Sigma}\Big{(}-\mathsf{A}^{0}_{\mathrm{out}}\mathsf{A}^{2}_{\mathrm{in}}+\mathsf{A}^{0}_{\mathrm{out}}\mathsf{A}^{2}_{\mathrm{res}}-\mathsf{A}^{2}_{\mathrm{in}}\mathsf{A}^{0}_{\mathrm{res}}+\frac{1}{2}\mathsf{A}^{2}_{\mathrm{res}}\mathsf{A}^{0}_{\mathrm{res}}\Big{)}\Big{)}.$

Here we denoted the degree zero scalar residual field by

$\sigma\mathrel{\mathop{\mathchar
58\relax}}=\mathsf{A}^{0}_{I\,\mathrm{res}}\;\;\in\Omega^{0}_{\mathbb{C}}(\Sigma).$

In the first bracket in (33) we recognize the Hamilton–Jacobi action [16, Eq.
(47)] (see also Example 2.2) — the action of a free (conformal) massless boson
interacting with the boundary fields,222222 One can see $S_{\text{HJ}}$ as the
abelian version of the gauged Wess–Zumino–Witten theory, see e.g (2.7) in
[24].  while in the second bracket we collected the contributions of nonzero-
degree fields.

####  4.2.2. Full integral over residual fields

If we wish to integrate out the remaining residual fields completely, we
construct the gauge-fixing lagrangian
$\mathcal{L}_{\mathrm{small}}\subset\mathcal{V}_{\mathrm{small}}$ as follows.
Choose an area form $\mu$ on $\Sigma$. Consider the splitting of $0$-forms
into constants and forms with vanishing integral against $\mu$:
$\mathsf{A}^{0}=\mathsf{A}^{0}_{\mathrm{c}}+\underline{\mathsf{A}^{0}}$. Also,
consider the splitting of $2$-forms into constant multiples of $\mu$ and forms
of vanishing total integral:
$\mathsf{A}^{2}=\mu\cdot\mathsf{A}^{2}_{\mathrm{c}}+\underline{\mathsf{A}^{2}}$.
Then, we define the lagrangian
$\mathcal{L}_{\mathrm{small}}\subset\mathcal{V}_{\mathrm{small}}$ by equations
$\mathsf{A}^{2}_{I\,\mathrm{res}}=\sigma_{\mathrm{c}}=\underline{\mathsf{A}^{2}_{\mathrm{res}}}=0$.
Thus, the lagrangian is parametrized by $\mathsf{A}^{0}_{\mathrm{res}}$,
$\underline{\sigma}$, $\mathsf{A}^{2}_{\mathrm{res},\mathrm{c}}$.232323We have
to split off the constants from $\sigma$, because they are in the kernel of
the Laplacian $\partial\bar{\partial}$ and thus would obstruct the evaluation
of the integral of (33) over $\sigma$.

The resulting full BV integral is:

$Z_{*}=\int_{\mathcal{L}_{\mathrm{small}}\subset\mathcal{V}_{\mathrm{small}}}\mathcal{D}\mathsf{A}^{0}_{\mathrm{res}}\;\mathcal{D}\underline{\sigma}\;\mathcal{D}\mathsf{A}^{2}_{\mathrm{res},\mathrm{c}}\;\;Z_{\mathrm{small}}=\\\
=\delta(\underline{\mathsf{A}^{2}_{\mathrm{in}}})e^{-\frac{i}{\hbar}\int_{\Sigma}\mu\cdot\mathsf{A}^{2}_{\mathrm{in},\mathrm{c}}\mathsf{A}^{0}_{\mathrm{out},\mathrm{c}}}\int\mathcal{D}\underline{\sigma}\;\exp\frac{i}{\hbar}\int_{\Sigma}\Big{(}\mathsf{A}^{1,0}_{\mathrm{out}}\mathsf{A}^{0,1}_{\mathrm{in}}+(\partial\mathsf{A}^{0,1}_{\mathrm{in}}+\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}})\;\underline{\sigma}+\frac{1}{2}\underline{\sigma}\partial\bar{\partial}\underline{\sigma}\Big{)}.$

Further, assume that the area form $\mu=\sqrt{\det{g}}\;d^{2}x$ is the
Riemannian area form associated to a certain metric $g$ on $\Sigma$ inducing
simultaneously the complex structure we use in our polarization. Then the
integral over $\underline{\sigma}$ evaluates finally to

(34)
$Z_{*}=\delta(\underline{\mathsf{A}^{2}_{\mathrm{in}}})\;e^{-\frac{i}{\hbar}\int_{\Sigma}\mu\cdot\mathsf{A}^{2}_{\mathrm{in},\mathrm{c}}\mathsf{A}^{0}_{\mathrm{out},\mathrm{c}}}\;\cdot\big{(}{\det}^{\prime}_{\Omega^{0}(\Sigma)}\Delta_{g}\big{)}^{-\frac{1}{2}}\cdot
e^{\frac{i}{\hbar}\mathbb{I}(\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}})}$

where

  * •

$\Delta_{g}$ is the metric Laplace operator acting on $0$-forms,
${\det}^{\prime}$ means the zeta-regularized product of nonzero eigenvalues.

  * •

The exponent in (34) is

(35)
$\mathbb{I}=\int_{\Sigma}\mathsf{A}^{1,0}_{\mathrm{out}}P_{\mathrm{harm}}(\mathsf{A}^{0,1}_{\mathrm{in}})-i\int_{\Sigma\times\Sigma\;\ni(z,z^{\prime})}\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}}\big{|}_{z}G(z,z^{\prime})\,\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}}\big{|}_{z^{\prime}}+\partial\mathsf{A}^{0,1}_{\mathrm{in}}\big{|}_{z}G(z,z^{\prime})\,\partial\mathsf{A}^{0,1}_{\mathrm{in}}\big{|}_{z^{\prime}}.$

Here $G$ is the Green’s function for $\Delta_{g}$, viewed as a function on
$\Sigma\times\Sigma$ with a logarithmic singularity at the
diagonal.242424Recall that, in terms of Dolbeault operators and the area form,
the metric Laplace operator is:
$\Delta_{g}=\frac{2i}{\mu}\partial\bar{\partial}$. The operator
$P_{\mathrm{harm}}\mathrel{\mathop{\mathchar
58\relax}}\mathsf{A}^{0,1}\mapsto\mathsf{A}^{0,1}-2i\int_{\Sigma\ni
z^{\prime}}\bar{\partial}G(z,z^{\prime})\partial\mathsf{A}^{0,1}\big{|}_{z^{\prime}}$
is the projector onto harmonic $(0,1)$-forms in the Hodge decomposition.

Written in different notations, the exponent in (34) is:

(36)
$\mathbb{I}=\int_{\Sigma}\mathsf{A}^{1,0}_{\mathrm{out}}\,(1-\bar{\partial}(\partial\bar{\partial})^{-1}\partial)\,\mathsf{A}^{0,1}_{\mathrm{in}}-\frac{1}{2}\mathsf{A}^{1,0}_{\mathrm{out}}\,\bar{\partial}(\partial\bar{\partial})^{-1}\bar{\partial}\,\mathsf{A}^{1,0}_{\mathrm{out}}-\frac{1}{2}\mathsf{A}^{0,1}_{\mathrm{in}}\,\partial(\partial\bar{\partial})^{-1}\partial\,\mathsf{A}^{0,1}_{\mathrm{in}}.$

######  Remark 4.3.

  1. (a)

The exponent $\mathbb{I}$ in (34) depends only on the complex structure on
$\Sigma$, not on the particular compatible metric $g$. In other words, it is
invariant under Weyl transformations of the metric $g\mapsto e^{\phi}\,g$.
Weyl-invariance of $\mathbb{I}$ is manifest in the form (36).

  2. (b)

Unlike $\mathbb{I}$, the full quantum answer (34) is not Weyl-invariant, since
the determinant of the Laplacian is not invariant (a phenomenon known as the
“conformal anomaly” or “trace anomaly” of the free scalar field as a conformal
field theory). In addition to that quantum effect, the dependence of $Z_{*}$
on boundary $\mathrm{gh}\neq 0$ fields
$\mathsf{A}_{\mathrm{in}}^{2},\mathsf{A}_{\mathrm{out}}^{0}$ involves the
metric area form $\mu$.

  3. (c)

The lagrangian generated by $\mathbb{I}$ is

$L_{\mathbb{I}}=\left\\{\begin{array}[]{c}\mathsf{A}_{\mathrm{out}}=\mathsf{A}_{\mathrm{out}}^{1,0}+\frac{\delta\mathbb{I}}{\delta\mathsf{A}^{1,0}_{\mathrm{out}}},\\\
\mathsf{A}_{\mathrm{in}}=\mathsf{A}_{\mathrm{in}}^{0,1}-\frac{\delta\mathbb{I}}{\delta\mathsf{A}^{0,1}_{\mathrm{in}}}\end{array}\right\\}=\left\\{\begin{array}[]{c}\mathsf{A}_{\mathrm{out}}=(1-\bar{\partial}(\partial\bar{\partial})^{-1}\bar{\partial})\mathsf{A}^{1,0}_{\mathrm{out}}+P_{\mathrm{harm}}\mathsf{A}^{0,1}_{\mathrm{in}},\\\
\mathsf{A}_{\mathrm{in}}=(1+\partial(\partial\bar{\partial})^{-1}\partial)\mathsf{A}^{0,1}_{\mathrm{in}}+P_{\mathrm{harm}}\mathsf{A}^{1,0}_{\mathrm{out}}\end{array}\right\\}.$

It is easy to see that this lagrangian coincides with the evolution relation
of abelian Chern–Simons theory on the cylinder $I\times\Sigma$,

$L_{CS}=\big{\\{}(\mathsf{A}_{\mathrm{out}},\mathsf{A}_{\mathrm{in}})\;\in\Omega^{1}(\Sigma)\times\Omega^{1}(\Sigma)\;\;\big{|}\;\;d\mathsf{A}_{\mathrm{out}}=0,\;d\mathsf{A}_{\mathrm{in}}=0,\;\mathsf{A}_{\mathrm{out}}-\mathsf{A}_{\mathrm{in}}=d(\cdots)\big{\\}}.$

Thus, $\mathbb{I}$ is a (nongeneralized252525I.e., with no auxiliary fields.)
Hamilton–Jacobi action for the abelian theory on the cylinder.

  4. (d)

Classically, one can obtain $\mathbb{I}$ from the generalized Hamilton–Jacobi
action (Example 2.2) as the conditional extremum of $S_{\text{HJ}}$ in
$\sigma$, with $\mathsf{A}^{1,0}_{\mathrm{out}}$ and
$\mathsf{A}^{0,1}_{\mathrm{in}}$ fixed.

######  Remark 4.4.

To make (b) of Remark 4.3 above more explicit: if
$g_{\tau}=e^{\phi_{\tau}}g_{0}$ is a $\tau$-dependent family of metrics
compatible with the given complex structure on $\Sigma$, one has

(37)
$\frac{d}{d\tau}Z_{*}^{g_{\tau}}=(\Omega^{+}_{\mathrm{out}}+\Omega^{-}_{\mathrm{in}})(\xi
Z_{*}^{g_{\tau}})+Z_{*}^{g_{\tau}}\cdot\frac{1}{48\pi}\int_{\Sigma}\mu_{g_{\tau}}R_{g_{\tau}}\dot{\phi}_{\tau}$

with $R$ the scalar curvature of the metric, $\mu_{g_{\tau}}$ the Riemannian
area form of $g_{\tau}$, the $\Omega$ operators given by (40), (41) below
and262626This is the quantization of the hamiltonian
$H=\mathsf{A}^{2}_{\mathrm{res},c}\int_{\Sigma}\underline{\sigma}\dot{\mu}$
generating the family of lagrangians given by $\mu_{g^{\tau}}$, see [15,
Section 2].
$\xi=\mathsf{A}_{\mathrm{in},c}\int_{\Sigma\times\Sigma}(\bar{\partial}A^{1,0}_{\mathrm{out}}+\partial
A^{0,1}_{\mathrm{in}})_{z}G(z,z^{\prime})\dot{\mu}_{z^{\prime}}$. The second
term in (37) is the trace anomaly. Furthermore, one can compensate the anomaly
term by including the Liouville action as a counterterm,272727We are referring
to the fact that in a conformal field theory with central charge $c$, the
partition function has the following behavior under Weyl transformations of
metric: $Z_{\mathrm{CFT}}^{e^{\phi}g}=Z_{\mathrm{CFT}}^{g}\cdot
e^{\frac{c}{48\pi}\int_{\Sigma}\frac{1}{2}d\phi\wedge*d\phi+R\phi\mu_{g}}$,
see, e.g., [22].  i.e., by introducing

$\widehat{Z}^{g}=Z_{*}^{g}\cdot
e^{-\frac{1}{48\pi}\int_{\Sigma}\frac{1}{2}d\phi\wedge*d\phi+R_{g}\phi\mu_{g}},$

where $\phi$ is defined by $g=e^{\phi}g_{0}$ with $g_{0}$ some “reference”
metric in the same conformal class (e.g., one can take $g_{0}$ to be the
uniformization metric on $\Sigma$ — spherical, flat or hyperbolic metric for
$\Sigma$ of genus $0$, $1$ or $\geq 2$, respectively). Then, for a conformal
variation of metric we have
$\delta_{\varphi}\widehat{Z}^{e^{\varphi}g}=(\Omega^{+}_{\mathrm{out}}+\Omega^{-}_{\mathrm{in}})(\xi
Z_{*}^{g})$.

As an aside, it is tempting to compare the two phenomena:

  1. (i)

The anomalous metric dependence (under a Weyl transformation
$g_{\Sigma}\rightarrow e^{\phi}g_{\Sigma}$) of the partition function on the
cylinder and the cancellation of that dependence by a Liouville action
counterterm.

  2. (ii)

The anomalous metric dependence (under $g_{M}\rightarrow g_{M}+\delta g_{M}$)
of the perturbative Chern–Simons partition function on a closed 3-manifold $M$
and the cancellation of that dependence by the gravitational Chern–Simons
counterterm introducing the dependence on framing $M$, see [41, 5].

But in fact, these effects seem different. In particular, the dependence on
Weyl transformations in (i) rescales $Z$ by a real factor, whereas the
anomalous metric dependence in (ii) affects only the phase of the partition
function.

######  Remark 4.5.

As implied by (33), one can view the “physical part”
$Z_{*}^{\mathrm{ph}}=e^{\frac{i}{\hbar}\mathbb{I}(\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}})}$
of (34) as a generating function for the correlators of chiral currents
$j=i\partial\phi$, $\bar{j}=i\bar{\partial}\phi$ in massless scalar theory
(viewed as abelian WZW model):

(38) $\langle j(z_{1})\cdots
j(z_{n})\bar{j}(w_{1})\cdots\bar{j}(w_{m})\rangle=\\\
=\hbar^{n+m}\frac{1}{Z^{\mathrm{ph}}_{*}}\left.\frac{\delta}{\delta\mathsf{A}^{0,1}_{\mathrm{in}}(z_{1})}\cdots\frac{\delta}{\delta\mathsf{A}^{0,1}_{\mathrm{in}}(z_{n})}\frac{\delta}{\delta\mathsf{A}^{1,0}_{\mathrm{out}}(w_{1})}\cdots\frac{\delta}{\delta\mathsf{A}^{1,0}_{\mathrm{out}}(w_{m})}\right|_{\mathsf{A}^{0,1}_{\mathrm{in}}=\mathsf{A}^{1,0}_{\mathrm{out}}=0}Z^{\mathrm{ph}}_{*}$

Here we are assuming that all points $z_{1},\ldots,z_{n},w_{1},\ldots,w_{m}$
in $\Sigma$ are pairwise distinct (so that we can ignore the term
$\int_{\Sigma}\mathsf{A}^{1,0}_{\mathrm{out}}\mathsf{A}^{0,1}_{\mathrm{in}}$
in (33)). We note that (36) implies a short-distance behavior of these
correlators consistent with the OPEs (operator product expansions)

(39)
$j(z)j(w)\sim\frac{1}{(z-w)^{2}}+\mathrm{reg},\quad\bar{j}(z)\bar{j}(w)\sim\frac{1}{(\bar{z}-\bar{w})^{2}}+\mathrm{reg},\quad
j(z)\bar{j}(w)\sim\mathrm{reg}$

as $z\rightarrow w$ (“reg” stands for the regular part) – the standard
fundamental OPEs of abelian WZW model.

####  4.2.3. Quantum master equation

The space of states on the out-surface with $\mathsf{A}^{+}$-fixed
polarization was discussed in Section 4.1.2: it is the space of functions of
$\mathsf{A}^{+}_{\mathrm{out}}$ of the form (25) with the BFV operator

(40)
$\Omega_{\mathrm{out}}^{+}=\int_{\Sigma}\left(-i\hbar\;\partial\mathsf{A}^{0}_{\mathrm{out}}\frac{\delta}{\delta\mathsf{A}^{1,0}_{\mathrm{out}}}+\mathsf{A}^{1,0}_{\mathrm{out}}\,\bar{\partial}\mathsf{A}^{0}_{\mathrm{out}}\right).$

The space of states on the in-surface with $\mathsf{A}^{-}$-fixed polarization
is the space of functions of $\mathsf{A}^{-}_{\mathrm{in}}$ (defined similarly
to (25)) with the BFV operator

(41)
$\Omega_{\mathrm{in}}^{-}=\int_{\Sigma}\Big{(}-i\hbar\;\partial\mathsf{A}^{0,1}_{\mathrm{in}}\frac{\delta}{\delta\mathsf{A}^{2}_{\mathrm{in}}}+\hbar^{2}\frac{\delta}{\delta\mathsf{A}^{0,1}_{\mathrm{in}}}\,\bar{\partial}\frac{\delta}{\delta\mathsf{A}^{2}_{\mathrm{in}}}\Big{)}.$

This is the quantization of the BFV action (27) where
$\mathsf{A}^{0,1},\mathsf{A}^{2}$ become multiplication operators and
$\mathsf{A}^{1,0}\mapsto-\epsilon\,i\hbar\frac{\delta}{\delta\mathsf{A}^{0,1}}$,
$\mathsf{A}^{0}\mapsto-\epsilon\,i\hbar\frac{\delta}{\delta\mathsf{A}^{2}}$
become derivations, where $\epsilon=-1$ for the in-boundary, as in Section
4.1.2.

The BV Laplacian on residual fields (31) is:282828Here the factor $2$ comes
from the fact that the odd symplectic form on $\mathcal{V}$, induced from the
standard BV 2-form on the space of fields,
$\omega_{\mathcal{F}}=\int_{I\times\Sigma}\frac{1}{2}\delta\mathcal{A}\,\delta\mathcal{A}$,
is
$\omega_{\mathcal{V}}=\frac{1}{2}\int_{\Sigma}\delta\mathsf{A}^{+}_{\mathrm{res}}\,\delta\mathsf{A}^{-}_{I\,\mathrm{res}}+\delta\mathsf{A}^{-}_{\mathrm{res}}\,\delta\mathsf{A}^{+}_{I\,\mathrm{res}}$.
The factor $\frac{1}{2}$ in the latter expression comes from
$\int_{0}^{1}dt(1-t)=\int_{0}^{1}dt\,t=\frac{1}{2}$.

$\Delta_{\mathrm{res}}=2\int_{\Sigma}\frac{\delta}{\delta\mathsf{A}^{-}_{\mathrm{res}}}\frac{\delta}{\delta\mathsf{A}^{+}_{I\,\mathrm{res}}}+\frac{\delta}{\delta\mathsf{A}^{+}_{\mathrm{res}}}\frac{\delta}{\delta\mathsf{A}^{-}_{I\,\mathrm{res}}}.$

######  Lemma 4.6.

Partition function (32) satisfies the modified quantum master equation

$\left(\Omega^{+}_{\mathrm{out}}+\Omega^{-}_{\mathrm{in}}-\hbar^{2}\Delta_{\mathrm{res}}\right)\;Z=0.$

###### Proof.

Indeed, a straightforward computation gives:

$\displaystyle\Omega^{+}_{\mathrm{out}}Z$
$\displaystyle=Z\cdot\int_{\Sigma}\Big{(}-\partial\mathsf{A}^{0}_{\mathrm{out}}\,(\mathsf{A}^{0,1}_{\mathrm{in}}-\mathsf{A}^{0,1}_{\mathrm{res}})+\mathsf{A}^{1,0}_{\mathrm{out}}\;\bar{\partial}\mathsf{A}^{0}_{\mathrm{out}}\Big{)},$
$\displaystyle\Omega^{-}_{\mathrm{in}}Z$
$\displaystyle=Z\cdot\int_{\Sigma}\Big{(}\partial\mathsf{A}^{0,1}_{\mathrm{in}}\,(\mathsf{A}^{0}_{\mathrm{out}}-\mathsf{A}^{0}_{\mathrm{res}})-(\mathsf{A}^{1,0}_{\mathrm{out}}-\mathsf{A}^{1,0}_{\mathrm{res}})\;\bar{\partial}(\mathsf{A}^{0}_{\mathrm{out}}-\mathsf{A}^{0}_{\mathrm{res}})\Big{)},$
$\displaystyle-\hbar^{2}\Delta_{\mathrm{res}}Z$
$\displaystyle=Z\cdot\int_{\Sigma}\Big{(}(\partial\mathsf{A}^{0,1}_{\mathrm{res}}+\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{res}})\,(-\mathsf{A}^{0}_{\mathrm{out}}+\frac{1}{2}\mathsf{A}^{0}_{\mathrm{res}})+$
$\displaystyle+\bar{\partial}\mathsf{A}^{0}_{\mathrm{res}}\,$
$\displaystyle(-\mathsf{A}^{1,0}_{\mathrm{out}}+\frac{1}{2}\mathsf{A}^{1,0}_{\mathrm{res}}+\frac{1}{2}\partial\mathsf{A}^{0}_{I\,\mathrm{res}})+\partial\mathsf{A}^{0}_{\mathrm{res}}\,(\mathsf{A}^{0,1}_{\mathrm{in}}-\frac{1}{2}\mathsf{A}^{0,1}_{\mathrm{res}}+\frac{1}{2}\bar{\partial}\mathsf{A}^{0}_{I\,\mathrm{res}})\Big{)}.$

The sum of these three expressions is zero. ∎

Similarly, one can check the quantum master equation for the “small”
realization (33):

$\left(\Omega^{+}_{\mathrm{out}}+\Omega^{-}_{\mathrm{in}}-\hbar^{2}\Delta_{\mathrm{small}}\right)\;Z_{\mathrm{small}}=0,$

where

$\Delta_{\mathrm{small}}=2\int_{\Sigma}\frac{\delta}{\delta\mathsf{A}^{2}_{\mathrm{res}}}\frac{\delta}{\delta\sigma}+\frac{\delta}{\delta\mathsf{A}^{0}_{\mathrm{res}}}\frac{\delta}{\delta\mathsf{A}^{2}_{I\,\mathrm{res}}}$

is the BV Laplacian on $\mathcal{V}_{\mathrm{small}}$.

Finally, the result of the full integration over residual fields (34)
satisfies the BFV cocycle (gauge-invariance) condition

$\left(\Omega^{+}_{\mathrm{out}}+\Omega^{-}_{\mathrm{in}}\right)\;Z_{*}=0.$

##  5\. Chern–Simons theory in “parallel ghost polarization”

In three-dimensional Chern–Simons theory there is another way of picking a
pair of polarizations on the opposite sides of a cylinder: we can use the
$(\mathsf{A}^{0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{out}})$
representation on the out-boundary surface and the
$(\mathsf{A}^{0}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{in}})$ representation
on the in-surface. Thus, the corresponding polarizations are transversal in
ghost number 0 and parallel in ghost number $\neq 0$. See also the discussion
of quantization of 1D systems with this class of polarizations in [16, Section
11].

###  5.1. One-dimensional Chern–Simons theory with values in a cochain complex

As a warm-up, we consider again the one-dimensional theory, with a slightly
different setup. Fix an odd integer $k$. Let

$\mathfrak{g}=\bigoplus\mathfrak{g}^{i}$

be a graded vector space with a differential $d_{\mathfrak{g}}$ and a
compatible graded symmetric pairing $(\cdot,\cdot)$ of degree $-2k$.292929The
prime example being $\mathfrak{g}=\Omega^{\bullet}(M)$ with $M$ a
$2k$-manifold, $d_{\mathfrak{g}}$ the de Rham differential and
$(\alpha,\beta)=\int_{M}\alpha\wedge\beta$. Now, we let $X=\mathfrak{g}[k]$ \-
this is a 0-shifted graded symplectic vector space. We call the induced
grading on $C^{\infty}(X)$ the ghost number. It is convenient to express
elements of $C^{\infty}(X)$ in terms of the shifted identity map
$\psi\in\mathrm{Hom}(X,\mathfrak{g})$ which has total degree (ghost number +
degree) $k$.303030 If $\\{t_{a}\\}$ is a basis in $\mathfrak{g}$ and
$\\{\psi^{a}\\}$ is the shifted dual basis in $X^{*}$, then we have
$\psi=\sum_{a}t_{a}\psi^{a}\in\mathrm{Hom}(X,\mathfrak{g})$. If the degree of
$t_{a}$ is $|a|$, then the ghost number of $\psi^{a}$ is $k-|a|$, and thus
$\psi$ is indeed an object of total degree $k$.  We denote the ghost number
$l$ component of a field $\varphi$ by $\varphi^{[l]}$. In particular, the
ghost number $l$ component $\psi^{[l]}$ of $\psi$ has degree $k-l$. For
instance, the function

$\Theta(\psi)=\frac{1}{2}(\psi,d_{\mathfrak{g}}\psi)$

has ghost number $+1$. Its hamiltonian vector field $Q$ has ghost number $+1$
and satisfies $Q^{2}=0$, thus $(X,\omega,Q)$ is a BFV vector space.

We split the complexification of the ghost number $0$ component of $X$ as
$X^{[0]}_{\mathbb{C}}=X^{+}\oplus X^{-}$, with $X^{\pm}$ the degree 0 $\pm
i$-eigenspaces of a complex structure $J$ on $X^{[0]}$ compatible with the
pairing. Thus, $X_{\mathbb{C}}$ admits the total decomposition

(42) $X_{\mathbb{C}}=X^{[>0]}_{\mathbb{C}}\oplus X^{+}\oplus X^{-}\oplus
X^{[<0]}_{\mathbb{C}}$

where $X^{[>0]},X^{[<0]}$ are the components of positive (resp. negative)
ghost number.313131I.e., coordinates, e.g., on $X^{[>0]}$ have positive ghost
number so $X^{[>0]}=\bigoplus_{i<k}\mathfrak{g}^{i}[k]$. We also introduce the
notations $d_{\mathfrak{g}}^{+},d_{\mathfrak{g}}^{-}$ for the composition of
the differential $X^{[1]}\to X^{+}\oplus X^{-}$ with projections and similarly
for the restriction of the differential ${X^{-}\oplus X^{+}\to X^{[-1]}}$ (so
that $d_{\mathfrak{g}}^{-}=d_{\mathfrak{g}}\big{|}_{X^{+}}$,
$d_{\mathfrak{g}}^{+}=d_{\mathfrak{g}}\big{|}_{X^{-}}$). We automatically have
$(d_{\mathfrak{g}}^{+})^{2}=(d_{\mathfrak{g}}^{-})^{2}=0$ and
${d_{\mathfrak{g}}^{+}d_{\mathfrak{g}}^{-}=-d_{\mathfrak{g}}^{-}d_{\mathfrak{g}}^{+}}$.323232A
particularly important example is the case $\mathfrak{g}=\Omega^{\bullet}(M)$
where $M$ is a $2k$-manifold, with $k=2l+1$ odd, endowed with a complex
structure that allows us to decompose
$\Omega^{k}_{\mathbb{C}}(M)=\Omega^{+}(M)\oplus\Omega^{-}(M)$ into lagrangian
subspaces with respect to $(\alpha,\beta)=\int_{M}\alpha\wedge\beta$, with the
splitting given by (106) below. In the case $k=1$, we have
$d^{+}_{\mathfrak{g}}=\partial,d^{-}_{\mathfrak{g}}=\bar{\partial}$.

####  5.1.1. Setup

We now consider the 1-dimensional AKSZ theory with target the symplectic
graded vector space $(X,(\cdot,\cdot))$ and hamiltonian $\Theta(\psi)$. The
space of fields is

$\mathcal{F}=\Omega^{\bullet}(I;X).$

It is parametrized by the superfield $\mathcal{A}$ valued in
$\Omega^{\bullet}(I;\mathfrak{g})$. We denote the 0- and 1-form components of
$\mathcal{A}$ by $\psi$ and $A$, respectively. The total degrees of
$\mathcal{A},\psi,A$ are all odd. The action is

$S[\psi+A]=\frac{1}{2}\int_{I}({\mathcal{A}},d_{I}\mathcal{A})+\frac{1}{2}\int_{I}(\mathcal{A},d_{\mathfrak{g}}\mathcal{A})=\frac{1}{2}\int_{I}(\psi,d_{I}\psi)+\int_{I}(A,d_{\mathfrak{g}}\psi).$

The space of boundary fields is

$\mathcal{F}^{\partial}=X_{\mathrm{in}}\times
X_{\mathrm{out}}\ni(\psi_{\mathrm{in}},\psi_{\mathrm{out}}).$

The boundary 1-form is

$\alpha_{\partial
I}=\alpha^{\partial}_{\mathrm{out}}+\alpha^{\partial}_{\mathrm{in}}=\frac{1}{2}(\psi_{\mathrm{out}},\delta\psi_{\mathrm{out}})-\frac{1}{2}(\psi_{\mathrm{in}},\delta\psi_{\mathrm{in}})=\frac{1}{2}(\psi,\delta\psi)\bigg{|}_{t=0}^{t=1}.$

Splitting elements of $X_{\mathbb{C}}$ according to (42),
$\psi=\psi^{[<0]}+\psi^{+}+\psi^{-}+\psi^{[>0]}$, the boundary 1-form splits
similarly:

$\alpha^{\partial}_{\mathrm{out}}=\frac{1}{2}\Big{[}(\psi_{\mathrm{out}}^{[<0]},\delta\psi_{\mathrm{out}}^{[>0]})+(\psi_{\mathrm{out}}^{+},\delta\psi_{\mathrm{out}}^{-})+(\psi_{\mathrm{out}}^{-},\delta\psi_{\mathrm{out}}^{+})+(\psi_{\mathrm{out}}^{[>0]},\delta\psi_{\mathrm{out}}^{[<0]})\Big{]}$

and similarly for $\alpha^{\partial}_{\mathrm{in}}$.

####  5.1.2. Parallel ghost polarization

Let us now consider the case where the polarizations are parallel in the ghost
sector (of the target) and transversal in the physical sector:

$\displaystyle\mathcal{P}$
$\displaystyle=\mathcal{P}_{\mathrm{in}}\times\mathcal{P}_{\mathrm{out}},$
$\displaystyle\mathcal{P}_{\mathrm{in}}$
$\displaystyle=\left\\{\frac{\delta}{\delta\psi^{[<0]}_{\mathrm{in}}},\frac{\delta}{\delta\psi_{\mathrm{in}}^{+}}\right\\},$
$\displaystyle\mathcal{P}_{\mathrm{out}}$
$\displaystyle=\left\\{\frac{\delta}{\delta\psi^{[<0]}_{\mathrm{out}}},\frac{\delta}{\delta\psi_{\mathrm{out}}^{-}}\right\\},$

so that we are using the $(\psi^{[>0]}_{\mathrm{in}},\psi^{-}_{\mathrm{in}})$
representation at $t=0$ and the
$(\psi^{[>0]}_{\mathrm{out}},\psi^{+}_{\mathrm{out}})$ representation at
$t=1$, i.e.:

$\mathcal{B}=\mathcal{B}_{\mathrm{in}}\times\mathcal{B}_{\mathrm{out}},\qquad\mathcal{B}_{\mathrm{in}}=X^{[>0]}_{\mathbb{C}}\oplus
X^{-},\qquad\mathcal{B}_{\mathrm{out}}=X^{[>0]}_{\mathbb{C}}\oplus X^{+}.$

The polarized 1-form is $\alpha^{f}_{\partial
I}=\alpha^{\partial,\mathcal{P}_{\mathrm{out}}}+\alpha^{\partial,\mathcal{P}_{\mathrm{in}}}$
with

$\displaystyle\alpha^{\partial,\mathcal{P}_{\mathrm{in}}}$
$\displaystyle=-(\psi^{[<0]}_{\mathrm{in}},\delta\psi^{[>0]}_{\mathrm{in}})-(\psi_{\mathrm{in}}^{+},\delta\psi_{\mathrm{in}}^{-})=\alpha^{\partial}_{\mathrm{in}}-\delta
f_{\mathrm{in}},$ $\displaystyle\alpha^{\partial,\mathcal{P}_{\mathrm{out}}}$
$\displaystyle=(\psi^{[<0]}_{\mathrm{out}},\delta\psi^{[>0]}_{\mathrm{out}})+(\psi_{\mathrm{out}}^{-},\delta\psi_{\mathrm{out}}^{+})=\alpha^{\partial}_{\mathrm{out}}+\delta
f_{\mathrm{out}},$

where333333Since $\psi$ is odd, we have, e.g.,
$\delta(\psi^{+},\psi^{-})=(\psi^{-},\delta\psi^{+})-(\psi^{+},\delta\psi^{-})$.
$f_{\mathrm{in}}=\frac{1}{2}(\psi^{[>0]}_{\mathrm{in}},\psi^{[<0]}_{\mathrm{in}})-\frac{1}{2}(\psi^{+}_{\mathrm{in}},\psi^{-}_{\mathrm{in}})$,
$f_{\mathrm{out}}=\frac{1}{2}(\psi^{[>0]}_{\mathrm{out}},\psi^{[<0]}_{\mathrm{out}})+\frac{1}{2}(\psi_{\mathrm{out}}^{+},\psi_{\mathrm{out}}^{-})$,
so that $\alpha^{f}_{\partial I}=\alpha_{\partial I}+\delta f$ with

$f(\psi_{\mathrm{out}},\psi_{\mathrm{in}})=f_{\mathrm{out}}(\psi_{\mathrm{out}})-f_{\mathrm{in}}(\psi_{\mathrm{in}})\\\
=\frac{1}{2}(\psi^{[>0]}_{\mathrm{out}},\psi^{[<0]}_{\mathrm{out}})+\frac{1}{2}(\psi_{\mathrm{out}}^{+},\psi_{\mathrm{out}}^{-})-\frac{1}{2}(\psi^{[>0]}_{\mathrm{in}},\psi^{[<0]}_{\mathrm{in}})+\frac{1}{2}(\psi_{\mathrm{in}}^{+},\psi_{\mathrm{in}}^{-}).$

The polarized action is

$S^{f}[\mathcal{A}]=\frac{1}{2}\int_{I}({\mathcal{A}},d_{I}\mathcal{A})+\frac{1}{2}\int_{I}(\mathcal{A},d_{\mathfrak{g}}\mathcal{A})+f(\mathcal{A}).$

####  5.1.3. Gauge fixing

The kernel $\mathcal{Y}$ of the map $\mathcal{F}\to\mathcal{B}$ is

(43) $\mathcal{Y}=\Omega^{\bullet}(I,\partial
I;X^{[>0]}_{\mathbb{C}})\oplus\Omega^{\bullet}(I,\\{0\\};X^{-})\oplus\Omega^{\bullet}(I,\\{1\\};X^{+})\oplus\Omega^{\bullet}(I;X^{[<0]}_{\mathbb{C}}).$

Choosing the minimal space of residual fields

$\mathcal{V}=\langle dt\rangle\otimes X^{[>0]}_{\mathbb{C}}\oplus\langle
1\rangle\otimes X^{[<0]}_{\mathbb{C}}\ni{\mathcal{A}}_{\mathrm{res}}=dt\cdot
A_{\mathrm{res}}+1\cdot\psi_{\mathrm{res}},$

we obtain

$\mathcal{Y}=\mathcal{V}\times\mathcal{Y}^{\prime}\ni(\mathcal{A}_{\mathrm{res}},\mathcal{A}_{\mathrm{fl}})$

with

$\mathcal{Y}^{\prime}=\Omega^{\bullet}(I,\partial
I;X^{[>0]}_{\mathbb{C}})_{\int=0}\oplus\Omega^{\bullet}(I,\\{0\\};X^{-})\oplus\Omega^{\bullet}(I,\\{1\\};X^{+})\oplus\Omega^{\bullet}(I;X^{[<0]}_{\mathbb{C}})_{\int\cdot\wedge
dt=0}.$

Here the notation $\int\cdot=0$ (resp. $\int\cdot dt=0$) denotes acylic
subcomplexes of forms with vanishing integral (resp. forms whose product with
$dt$ has vanishing integral). Choosing an extension

${\widetilde{\psi}}=\widetilde{\psi^{[>0]}_{\mathrm{in}}}+\widetilde{\psi_{\mathrm{in}}^{-}}+\widetilde{\psi^{[>0]}_{\mathrm{out}}}+\widetilde{\psi_{\mathrm{out}}^{+}}$

of boundary fields into the bulk, we obtain a splitting of
$\mathcal{A}=\psi+A$ into

$\psi+A=\widetilde{\psi}+\psi_{\mathrm{res}}+\psi_{\mathrm{fl}}+dt\cdot