# Towards equivariant Yang-Mills theory

F. Bonechi  INFN Sezione di Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino,
Firenze, Italy<EMAIL_ADDRESS>,  A. S. Cattaneo  Institut für
Mathematik, Universität Zürich
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland<EMAIL_ADDRESS>and
M. Zabzine  Department of Physics and Astronomy, Uppsala University, Box 516,
75120 Uppsala, Sweden<EMAIL_ADDRESS>
###### Abstract.

We study four dimensional gauge theories in the context of an equivariant
extension of the Batalin-Vilkovisky (BV) formalism. We discuss the embedding
of BV Yang-Mills (YM) theory into a larger BV theory and their relation.
Partial integration in the equivariant BV setting (BV push-forward map) is
performed explicitly for the abelian case. As result, we obtain a non-local
homological generalization of the Cartan calculus and a non-local extension of
the abelian YM BV action which satisfies the equivariant master equation.

###### Contents

  1. 1 Introduction
  2. 2 Overview of BV formalisms
    1. 2.1 BV
    2. 2.2 AKSZ
    3. 2.3 BV push-forward
  3. 3 Overview of equivariant BV formalisms
    1. 3.1 Equivariant BV
    2. 3.2 Equivariant BV push-forward
  4. 4 BV formalism for DW
  5. 5 BV formalism for YM
    1. 5.1 Costello’s formulation of Yang-Mills
    2. 5.2 BV-pushforward
  6. 6 equivariant BV pushforward map
    1. 6.1 Integration of UV sector
  7. 7 Summary
  8. A Hodge decomposition and relevant operators

##  1\. Introduction

During the last thirty years equivariant localization played an instrumental
role in the derivation of many exact results in quantum field theory (QFT), in
particular in the context of supersymmetric gauge theories. In QFT equivariant
localization is based on a clever combination of the supersymmetry
transformations with BRST symmetry in such fashion that the resulting
transformations are interpreted as the equivariant differential on the space
of fields. In our previous work [6] we proposed a framework for the treatment
of such theories within the equivariant extension of the Batalin-Vilkovisky
(BV) formalism. In the present paper we continue to study this extension and
discuss the equivariant partial BV integration (equivariant BV push-forward)
map. Our main examples will be four-dimensional gauge theories.

We observe that the BV description of Yang-Mills (YM) theory given by Costello
in [12] can be obtained restricting the BV action of Donaldson–Witten (DW)
theory ([6], [8], [9], [13]) to a BV submanifold. Both theories can be recast
in the AKSZ formulation, provided that, in the case of YM, one allows in the
source a differential graded algebra (dga) $\mathcal{A}$ that is not the
external algebra of differential forms. The BV push-forward map from DW theory
to YM theory corrects the Costello’s BV YM action by terms that in the abelian
case involve the zero modes of the ultraviolet (UV) fields but in the non
abelian case are packed in a full perturbative expansion in the YM fields. In
the equivariant case, we perform the explicit BV push-forward map in the case
of abelian theory and obtain an effective action that is non-local and
satisfies the equivariant master equation. From a geometrical point of view we
can see this result as a non-local homological generalization of the Cartan
calculus.

The paper is organized as follows: in Section 2 we recall basic notions within
the BV formalism and the AKSZ construction of the solution of the classical
master equation. We give the formal definition of the BV push-forward map and
list its formal properties. In Section 3 we summarize the equivariant
extension of the BV formalism and the corresponding equivariant version of the
AKSZ construction. We define formally the equivariant analog of the BV push-
forward map and discuss its properties. In Section 4 we define the 4d AKSZ
theory to which we refer as the Donaldson–Witten (DW) theory. We discuss
different formulations of this theory. In Section 5 we discuss the BV
formulation of Yang–Mills (YM) theory and its relation to the DW theory. We
briefly discuss the application of the BV push-forward map to the abelian DW
theory. In Section 6 we introduce the equivariant extension of DW AKSZ theory.
For the case of abelian DW theory we describe explicitly the equivariant BV
push-forward map. The final result is a non-local deformation of abelian YM
theory which satisfies the equivariant master equation. This provides a non-
local Cartan calculus on the dg algebra $\mathcal{A}$. We summarize the paper
and discuss the open questions in 7. Technical details regarding the Hodge
decomposition and the relevant properties of the operators are collected in
appendix A

#### Aknowledgements

We thank Pavel Mnëv for discussions. A.S.C. acknowledges partial support of
SNF Grant No. 200020_192080 and of the Simons Collaboration on Global
Categorical Symmetries. This research was (partly) supported by the NCCR
SwissMAP, funded by the Swiss National Science Foundation. A.S.C. and M.Z.
thank INFN Sezione di Firenze where part of this work was carried out. F.B.
and M.Z. want to thank ESI, Vienna, where part of this work was carried out.

##  2\. Overview of BV formalisms

###  2.1. BV

The BV formalism was introduced in [3, 4, 5] to study the (in)dependence of
the partition function (or, more generally, expectation values of observables)
from the gauge fixing. In this setting, the space of fields is extended to an
odd symplectic manifold (for bookkeeping, it is convenient to have an
additional ${\mathbb{Z}}$-grading with the symplectic form having degree $-1$;
in this paper, we assume that the parity is the modulo two reduction of the
integer degree). The action is extended to a BV action $S$ (of degree $0$)
satisfying the quantum master equation

(1) $\frac{1}{2}\\{S,S\\}-{\mathrm{i}}\hbar\Delta S=0.$

Here $\\{\ ,\ \\}$ denotes the BV bracket, i.e., the odd Poisson bracket
defined by the odd symplectic structure, and $\Delta$ is the BV Laplacian. The
latter is a second-order differential operator which in Darboux coordinates
$p_{1},\dots,p_{n},q^{1},\dots,q^{n}$ (the parity of each $p$ being opposite
to that of a the corresponding $q$) with respect to the standard Berezinian
density $\mu_{\text{stand}}={\mathrm{d}}^{n}q\,{\mathrm{d}}^{n}p$ takes the
form $\Delta=\frac{\partial}{\partial p_{i}}\frac{\partial}{\partial q^{i}}$.

The first theorem of the BV formalism states that, if $f=\Delta g$ and
${\mathcal{L}}$ is a Lagrangian submanifold on which the integral of $g$
converges, then

$\int\limits_{\mathcal{L}}f\mu_{\text{stand}}^{\frac{1}{2}}=0.$

Here $\mu_{\text{stand}}^{\frac{1}{2}}$ denotes the half-density associated to
$\mu_{\text{stand}}$. Its restriction to a Lagrangian submanifold is
canonically a density.

This fact implies the central theorem of the BV formalism which states that,
for a family ${\mathcal{L}}_{t}$ of Lagrangian submanifolds smoothly depending
on $t$,

$I_{t}:=\int\limits_{{\mathcal{L}}_{t}}f\mu_{\text{stand}}^{\frac{1}{2}}$

is constant, under the assumptions that $\Delta f=0$ and that for each $t$ the
integral converges. In fact, one can show that

$\frac{{\mathrm{d}}}{{\mathrm{d}}t}I_{t}=\int\limits_{{\mathcal{L}}_{t}}(\Delta(\dot{\psi}f)+\dot{\psi}\Delta
f)\mu_{\text{stand}}^{\frac{1}{2}}=\int\limits_{{\mathcal{L}}_{t}}\dot{\psi}\Delta
f\mu_{\text{stand}}^{\frac{1}{2}},$

where we have used the previous theorem. Here $\dot{\psi}$ describes the
variation of the gauge fixing. If $\Delta f=0$, we get the result. For
applications in physics, one considers
$f={\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}S}$, and $S$ is called the BV
action.

The application in field theory then consists in replacing the ill-defined
integral on the reference Lagrangian ${\mathcal{L}}_{0}$, which naively
describes the original divergent integral times a vanishing integral over the
ghosts, with a well-defined integral over some nearby ${\mathcal{L}}_{t}$
(gauge fixing).

The formalism can actually be extended globally to an odd symplectic manifold
$\mathcal{M}$. In this case, the definition of the BV Laplacian requires the
choice of a Berezinian density $\mu$ such that the resulting differential
operator squares to zero. Alternatively, one can observe [14, 15] that there
is a canonical Laplacian $\Delta$ acting on half-densities. For every half-
density $\sigma$ of the form $\Delta\tau$ and every Lagrangian submanifold
${\mathcal{L}}$ on which $\tau$ is integrable, one has

(2) $\int\limits_{\mathcal{L}}\sigma=0.$

Moreover, for every half-density $\sigma$ satisfying $\Delta\sigma=0$, one has
that

$I_{t}:=\int\limits_{{\mathcal{L}}_{t}}\sigma$

is constant, under the assumption that for each $t$ the integral converges.
Typically, one chooses $\sigma$ of the form
${\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}S}\sigma_{0}$ with $\sigma_{0}$ a
reference half-density satisfying $\Delta\sigma_{0}=0$. More generally, we
have

(3)
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}I_{t}=\int\limits_{{\mathcal{L}}_{t}}\dot{\psi}\,\Delta\sigma,$

where $\dot{\psi}$ describes the variation of the gauge fixing.

In field theory, all this has to be regularized, as we are in an infinite-
dimensional context, so neither the integral against
$\mu_{\text{stand}}^{\frac{1}{2}}$ nor the BV Laplacian are defined. For the
considerations in the present paper, it is enough to ignore the BV Laplacian,
i.e., to assume that $S$ satisfies the classical master equation

(4) $\\{S,S\\}=0.$

The integral is on the other hand understood in perturbation theory around a
nondegenerate Gaussian (the choice of gauge fixing achieves this goal).

###  2.2. AKSZ

The AKSZ formalism [2] is a general procedure to construct solutions of the
classical master equation (4) (one checks a posteriori, by restriction to the
variables of degree $0$ what physical theory it describes).

The construction, in $d$ dimensions, requires an odd symplectic manifold $Y$,
with exact symplectic form $\omega_{Y}={\mathrm{d}}\theta_{Y}$ of degree
$d-1$, endowed with a solution $S_{Y}$ (of degree $d$) of the associated
classical master equation. Given a closed $d$-manifold $\Sigma_{d}$ (our
space–time), to these data one associates the odd symplectic manifold
$\mathcal{M}={\mathrm{Map}}(T[1]\Sigma_{d},Y)$ and a BV action $S$ satisfying
the classical master equation.

We present the construction explictly for $Y$ a finite-dimensional vector
space and $\theta_{Y}=p_{i}\,{\mathrm{d}}q^{i}$. In this case, $\mathcal{M}$
has Darboux coordinates determined by the superfields ${\mathbf{P}}_{i}$s and
${\mathbf{Q}}^{i}$s, each of which is a sum of differential forms on
$\Sigma_{d}$ of all degrees: if a superfield corresponds to a target variable
of degree $n$, then its $k$-form component is assigned degree $n-k$. The odd
symplectic form on $\mathcal{M}$ is then defined by

(5)
$\omega=\int\limits_{\Sigma_{d}}\delta{\mathbf{P}}_{i}\wedge\delta{\mathbf{Q}}^{i}~{},$

where in each summand the integral selects the $d$-form. The BV action is
defined as

(6)
$S=\int\limits_{\Sigma_{d}}({\mathbf{P}}_{i}{\mathrm{d}}{\mathbf{Q}}^{i}+S_{Y}({\mathbf{P}},{\mathbf{Q}})).$

######  Remark 2.1.

A typical gauge fixing in an AKSZ theory is obtained as follows. One picks a
Riemannian metric $g$ and defines the codifferential ${\mathrm{d}}^{\dagger}$
(see Appendix A). Then one defines ${\mathcal{L}}_{g}$ as the Lagrangian
submanifold obtained by requiring each differential form in the superfields to
be in the image of ${\mathrm{d}}^{\dagger}$.

###  2.3. BV push-forward

The central theorem of the BV formalism can be extended to partial integration
(BV pushforward).111We follow the terminology of [10] and refer to its Section
2.2 for more details. To the best of our knowledge, the BV pushforward was
first used in [11]. It was then realized by A. Losev that it can be used to
define Wilsonian renormalization in a BV compatible way; this was first put to
use, in the case of $BF$ theories, in P. Mnëv’s thesis [17]. It was then
extensively used by K. Costello [12] to study the renormalization of gauge
theories.

Namely, given a product $\mathcal{M}=\mathcal{M}_{IR}\times\mathcal{M}_{UV}$
of odd symplectic manifolds to which we refer as UV and IR respectively and
with total BV Laplacian $\Delta=\Delta_{IR}+\Delta_{UV}$, one has, for every
half-density $\sigma$ on $\mathcal{M}$ satisfying $\Delta\sigma=0$ and for a
family ${\mathcal{L}}_{UV,t}$ of Lagrangian submanifolds of
$\mathcal{M}_{UV}$, that

$\sigma_{IR}(t):=\int\limits_{{\mathcal{L}}_{UV,t}}\sigma$

defines a family of half-densities $\sigma_{IR}(t)$ on $\mathcal{M}_{IR}$
satisfying $\Delta_{IR}\sigma_{IR}=0$ for every $t$. Moreover,
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\sigma_{IR}$ is $\Delta_{IR}$-exact; in
view of (2), this change of $\sigma_{IR}$ is irrelevant in the BV formalism.

These results are obtained from the BV theorems we have listed above. In fact,

$\Delta_{IR}\sigma_{IR}(t)=\int\limits_{{\mathcal{L}}_{UV,t}}\Delta_{IR}\sigma=\int\limits_{{\mathcal{L}}_{UV,t}}\Delta\sigma-\int\limits_{{\mathcal{L}}_{UV,t}}\Delta_{UV}\sigma=-\int\limits_{{\mathcal{L}}_{UV,t}}\Delta_{UV}\sigma=0,$

where we have used (2) on $\mathcal{M}_{UV}$. Similarly, by (3) on
$\mathcal{M}_{UV}$,

$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\sigma_{IR}(t):=\int\limits_{{\mathcal{L}}_{UV,t}}\dot{\psi}\,\Delta_{UV}\sigma.$

Therefore,

(7)
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\sigma_{IR}(t)=\int\limits_{{\mathcal{L}}_{UV,t}}\dot{\psi}\,\Delta_{UV}\sigma=\int\limits_{{\mathcal{L}}_{UV,t}}\dot{\psi}\,\Delta\sigma-\Delta_{IR}\int\limits_{{\mathcal{L}}_{UV,t}}\dot{\psi}\,\sigma,$

which yields the result if $\Delta\sigma=0$.

Typically, we have reference half-densities $\sigma_{0}$ on $\mathcal{M}$ and
$\sigma_{0,IR}$ on $\mathcal{M}_{IR}$, and
$\sigma={\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}S}\sigma_{0}$. In this case,
we write
$\sigma_{IR}={\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}S_{\text{eff}}}\sigma_{0,IR}$
and call $S_{\text{eff}}$ the effective action.

For applications in field theory (in particular, in this paper), the odd
symplectic manifolds are affine spaces (with a given choice of global Darboux
coordinates), and the reference half-densities are the standard ones. The
effective action is computed in perturbation theory around nondegenerate
Gaussians.

##  3\. Overview of equivariant BV formalisms

###  3.1. Equivariant BV

In [6] an equivariant version of the BV formalism was introduced. The first
observation was that, if the quantum master equation is violated,

(8) $\frac{1}{2}\\{S,S\\}-{\mathrm{i}}\hbar\Delta S=:T\not=0,$

then one still has the central theorem for BV integrals and BV pushforwards if
one restricts oneself to integrating on Lagrangian submanifolds on which $T$
vanishes (we call them $T$-Lagrangian submanifolds); see Section 3.2.

In particular, the case of the extension of an AKSZ theory to the Cartan model
for equivariant cohomology was considered. Namely, one starts with an
infinitesimal action of a Lie algebra on the space–time $\Sigma_{d}$. We
denote by $\\{v_{a}\\}$ the vector fields corresponding to some basis
$(e_{a})$ and introduce degree-two variables $u^{a}$. One can then define the
modified action

(9) $S_{eq}=S+u^{a}S_{\widehat{\iota}_{v_{a}}},$

where $S$ is as in (6), and

(10)
$S_{\widehat{\iota}_{v_{a}}}=\int\limits_{\Sigma_{d}}{\mathbf{P}}_{i}\,\iota_{v_{a}}{\mathbf{Q}}^{i}$

with $\iota_{v_{a}}$ denoting the contraction operator and
$\widehat{\iota}_{v_{a}}$ its extension to the space of fields. It then
follows that

(11) $\frac{1}{2}\\{S_{eq},S_{eq}\\}+u^{a}S_{\widehat{L}_{v_{a}}}=0,$

where

(12)
$S_{\widehat{L}_{v_{a}}}=\int\limits_{\Sigma_{d}}{\mathbf{P}}_{i}\,L_{v_{a}}{\mathbf{Q}}^{i}$

with $L_{v_{a}}$ denoting the Lie derivative with respect to $v_{a}$ and
$\widehat{L}_{v_{a}}$ its extension to the space of fields. Assuming $\Delta
S=0$ and $\Delta S_{\widehat{\iota}_{v_{a}}}=0$,222Both conditions are
reasonable, as, formally, $\Delta$ on these functionals corresponds to taking
the trace on $\Omega^{\bullet}(M)$ of operators of degree different from zero.
we get

(13) $T=-u^{a}S_{\widehat{L}_{v_{a}}}.$

It follows that ${\mathcal{L}}_{g}$ as in Remark 2.1 is a $T$-Lagrangian
submanifold if the metric $g$ is invariant.

In this paper, we consider a variation on this theme. Namely, we consider a BV
theory (specifically, abelian Yang–Mills in Costello’s BV formulation, see
Chapter 6 in [12]) which is based on a generalized AKSZ construction. Since
the source algebra admits Lie derivatives $L_{v_{a}}$ and so has lifts
$\widehat{L}_{v_{a}}$ to the space of fields, we look for an action which
satisfies (11) with $S_{\widehat{L}_{v_{a}}}$ as in (12). However since there
is not a standard Cartan calculus in the source and in particular no
contraction vector field $\iota_{v_{a}}$, we cannot use the solution as in
(9). We will rather obtain a solution to (11) that has essentially the same
properties as the one above; in particular, ${\mathcal{L}}_{g}$ is a
$T$-Lagrangian submanifold for every invariant metric $g$. On the other hand,
we will obtain a generalized Cartan calculus that includes a new definition
$\\{S_{\widehat{\iota}_{v_{a}}},\ \\}$ of the contraction operator together
with higher operations, in a way that is compatible with the theory at hand.

###  3.2. Equivariant BV push-forward

Suppose $S$ satisfies (8), i.e.,

$\Delta f=\left(\frac{{\mathrm{i}}}{\hbar}\right)^{2}T\,f$

with $f={\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}S}$. Consider the half-density
$\sigma=f\sigma_{0}$, with $\sigma_{0}$ a reference $\Delta$-closed half-
density. We then have

$\Delta\sigma=\left(\frac{{\mathrm{i}}}{\hbar}\right)^{2}T\,\sigma.$

We now perform the BV pushforward with the extra assumption that
$T=T_{IR}+T_{UV}$ with $T_{IR}$ and $T_{UV}$ functions on $\mathcal{M}_{IR}$
and ${\mathcal{M}}_{UV}$ respectively. The half-density $\sigma_{IR}(t)$ on
$\mathcal{M}_{IR}$ defined by the pushforward along a $T_{UV}$-Lagrangian
${\mathcal{L}}_{UV,t}$ on $\mathcal{M}_{UV}$ then satisfies

$\Delta_{IR}\sigma_{IR}(t)=\int\limits_{{\mathcal{L}}_{UV,t}}\Delta\sigma=\left(\frac{{\mathrm{i}}}{\hbar}\right)^{2}\int\limits_{{\mathcal{L}}_{UV,t}}(T_{IR}+T_{UV})\,\sigma=\left(\frac{{\mathrm{i}}}{\hbar}\right)^{2}T_{IR}\int\limits_{{\mathcal{L}}_{UV,t}}\sigma,$

i.e.,

$\Delta_{IR}\sigma_{IR}(t)=\left(\frac{{\mathrm{i}}}{\hbar}\right)^{2}T_{IR}\,\sigma_{IR}(t),$

which shows that the relation is well-behaved under BV-pushforward (with the
above assumptions).

As for deformations of the $T_{UV}$-Lagrangian submanifold, a direct
computation using (7) simply yields

$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\sigma_{IR}(t)=-\left(\frac{{\mathrm{i}}}{\hbar}\right)^{2}T_{IR}\int\limits_{{\mathcal{L}}_{UV,t}}\dot{\psi}\,\sigma-\Delta_{IR}\int\limits_{{\mathcal{L}}_{UV,t}}\dot{\psi}\,\sigma.$

In view of (2), the $\Delta_{IR}$-exact term is irrelevant in the BV
formalism. As the other term is proportional to $T_{IR}$, it is also
irrelevant, since we are going to consider only $T_{IR}$-Lagrangian
submanifolds for the further gauge fixing.

If we now apply this to the equivariant BV case where $T$ is defined by (13)
and $\mathcal{M}_{IR}$ and $\mathcal{M}_{UV}$ are based on AKSZ superfields,
we see that the assumptions are satisfied and that the effective theory is
precisely as described at the end of Section 3.1. Namely, for the effective
theory we still have

$S^{\text{eff}}_{\widehat{L}_{v_{a}}}=\int\limits_{\Sigma_{d}}{\mathbf{P}}_{i}\,L_{v_{a}}{\mathbf{Q}}^{i}$

where now the ${\mathbf{P}}$’s and ${\mathbf{Q}}$’s are the superfields on
$\mathcal{M}_{IR}$. On the other hand,
$S^{\text{eff}}_{\widehat{\iota}_{v_{a}}}$ will in general have a very
complicated, nonlocal expression.

##  4\. BV formalism for DW

In this section we describe the AKSZ construction [2] for a four-dimensional
field theory to which we refer as the Donaldson-Witten (DW) theory since upon
the appropriate gauge fixing it can be reduced to the standard cohomological
field theory described by Witten in [18]. This DW BV theory has been discussed
previously in [6, 8, 9, 13]

Let $\mathfrak{g}$ be a finite dimensional quadratic Lie algebra with the Lie
bracket $[~{},~{}]$ and with the invariant metric $\langle~{},~{}\rangle$.
Consider the graded vector space

(14) $\displaystyle\mathfrak{g}[1]\oplus\mathfrak{g}[2]~{},$

which is equipped with the symplectic structure of degree $3$

(15) $\displaystyle\omega=\langle\delta c,\delta\phi\rangle~{},$

where $c$ is the coordinate of degree $1$ and $\phi$ is the coordinate of
degree 2. We can introduce the Hamiltonian function of degree $4$

(16)
$\displaystyle\Theta=\frac{1}{2}\langle\phi,\phi\rangle+\frac{1}{2}\langle\phi,[c,c]\rangle~{},$

which generates the following odd vector field

(17) $\displaystyle{\boldsymbol{q}}c=\phi+\frac{1}{2}[c,c]~{},$ (18)
$\displaystyle{\boldsymbol{q}}\phi=[c,\phi]~{},$

such that ${\boldsymbol{q}}^{2}=0$. Following the standard AKSZ construction
[2] we can define the space of maps

(19)
${{\mathcal{M}}_{DW}}=\\{~{}T[1]\Sigma_{4}~{}\longrightarrow~{}\mathfrak{g}[1]\oplus\mathfrak{g}[2]~{}\\}~{},$

where $\Sigma_{4}$ is a $4$–manifold. Denoting by $\mathbf{C}$ and
$\mathbf{\Phi}$ the superfields of degree $1$ and $2$, respectively, the BV
symplectic form is defined as

(20)
$\displaystyle\omega_{DW}=\int\limits_{T[1]\Sigma_{4}}d^{4}xd^{4}\theta~{}\delta\mathbf{C}^{a}\wedge\delta\mathbf{\Phi}^{b}\eta_{ab}$

and the AKSZ action as

(21)
${S_{DW}}=\int\limits_{T[1]\Sigma_{4}}d^{4}xd^{4}\theta~{}\left(\langle\mathbf{\Phi},d\mathbf{C}\rangle+\frac{1}{2}\langle\mathbf{\Phi},\mathbf{\Phi}\rangle+\frac{1}{2}\langle\mathbf{\Phi},[\mathbf{C},\mathbf{C}]\rangle\right)~{}.$

The BV transformations are

(22)
$\displaystyle{\boldsymbol{\delta}}\mathbf{C}=d\mathbf{C}+\mathbf{\Phi}+\frac{1}{2}[\mathbf{C},\mathbf{C}]~{},$
(23)
$\displaystyle{\boldsymbol{\delta}}\mathbf{\Phi}=d\mathbf{\Phi}+[\mathbf{C},\mathbf{\Phi}]~{}.$

We adopt the following convention for the expansion of superfields in terms of
differntial forms

(24) $\displaystyle\mathbf{C}=c_{0}+c_{1}+c_{2}+c_{3}+c_{4}~{},$ (25)
$\displaystyle\mathbf{\Phi}=\phi_{0}+\phi_{1}+\phi_{2}+\phi_{3}+\phi_{4}~{}.$

The BV symplectic structure can then be written as

(26) $\omega_{DW}=\int\limits_{\Sigma_{4}}\left(\delta
c_{0}\wedge\delta\phi_{4}+\delta c_{1}\wedge\delta\phi_{3}+\delta
c_{2}\wedge\delta\phi_{2}+\delta c_{3}\wedge\delta\phi_{1}+\delta
c_{4}\wedge\delta\phi_{0}\right)~{},$

and the action (21) reads

(29) $\displaystyle{S_{DW}}$ $\displaystyle=$
$\displaystyle\int\langle\phi_{0},d_{c_{1}}c_{3}\rangle+\langle\phi_{1},d_{c_{1}}c_{2}\rangle+\langle\phi_{2},F(c_{1})\rangle+\langle\phi_{3},d_{c_{1}}c_{0}\rangle$
$\displaystyle+\langle\phi_{0},\phi_{4}\rangle+\langle\phi_{1},\phi_{3}\rangle+\frac{1}{2}\langle\phi_{2},\phi_{2}\rangle+\langle\phi_{0},[c_{0},c_{4}]\rangle$
$\displaystyle+\frac{1}{2}\langle\phi_{0},[c_{2},c_{2}]\rangle+\langle\phi_{1},[c_{0},c_{3}]\rangle+\langle\phi_{2},[c_{0},c_{2}]\rangle+\frac{1}{2}\langle\phi_{4},[c_{0},c_{0}]\rangle$

where $d_{c_{1}}=d+[c_{1},~{}]$ and
$F(c_{1})=dc_{1}+\frac{1}{2}[c_{1},c_{1}]$. Restricting to the physical (i.e.
degree $0$) fields

(30)
${S_{DW}}|_{cl}=\int\langle\phi_{2},F(c_{1})\rangle+\frac{1}{2}\langle\phi_{2},\phi_{2}\rangle~{},$

where upon the integration of $\phi_{2}$ we obtain the standard topological
term $-\frac{1}{2}\int F(c_{1})^{2}$ on which the original treatment of the DW
theory is based.

The 4D AKSZ action (21) can also be encoded in the following space

(31) $\displaystyle
T[1]\Sigma_{4}\oplus\mathbb{R}[-1]~{}\longrightarrow~{}\mathfrak{g}[1]~{},$

where we denote $\theta$’s as odd coordinates on $T[1]\Sigma_{4}$ and $\xi$ as
an odd coordinate on $\mathbb{R}[-1]$. The new superfield is written in terms
of old as follows

(32) $\displaystyle\mathbf{A}=\xi\mathbf{\Phi}+\mathbf{C}$

The symplectic structure is

(33)
$\displaystyle\omega_{DW}=\int\limits_{T[1]\Sigma_{4}\oplus\mathbb{R}[-1]}d^{4}xd^{4}\theta
d\xi~{}\delta\mathbf{A}^{a}\wedge\delta\mathbf{A}^{b}\eta_{ab}$

and the AKSZ action becomes

(34)
${S_{DW}}=\int\limits_{T[1]\Sigma_{4}\oplus\mathbb{R}[-1]}d^{4}xd^{4}\theta
d\xi~{}\left(\langle\mathbf{A},(d+\frac{\partial}{\partial\xi})\mathbf{A}\rangle+\mathbf{A}\mathbf{A}\mathbf{A}\right)$

and it reduces to (21). It is a Chern-Simons action with source the dg-
manifold $(T[1]\Sigma_{4}\times{\mathbb{R}}[-1],D+\partial/\partial\xi)$ and
degree $-3$ integral $\int=\int d^{4}xd^{4}\theta d\xi$; the dga of global
functions is then $(\Omega(\Sigma_{4})[\xi],D+\frac{\partial}{\partial\xi})$.
The underlying complex can be described as

(35)
${\Omega^{0}(\Sigma_{4})}$${\Omega^{1}(\Sigma_{4})}$${\Omega^{2}(\Sigma_{4})}$${\Omega^{3}(\Sigma_{4})}$${\Omega^{4}(\Sigma_{4})}$${\Omega^{0}(\Sigma_{4})}$${\Omega^{1}(\Sigma_{4})}$${\Omega^{2}(\Sigma_{4})}$${\Omega^{3}(\Sigma_{4})}$${\Omega^{4}(\Sigma_{4})}$

where the horizontal arrows denote de Rham differential $d$ and the diagonal
arrows denote $\partial/\partial\xi$. The field components are defined by
expanding the superfield $\mathbf{A}$ along the following basis of local
coordinates (as a $\Omega^{0}(\Sigma_{4})$-module)

$\displaystyle\xi~{},~{}~{}~{}{\rm degree~{}-1}$ $\displaystyle
1~{},~{}~{}~{}\theta^{i}\xi~{},~{}~{}~{}~{}{\rm degree~{}0}$
$\displaystyle\theta^{i}~{},~{}~{}~{}\theta^{i}\theta^{j}\xi~{},~{}~{}~{}~{}{\rm
degree~{}1}$
$\displaystyle\theta^{i}\theta^{j}~{},~{}~{}~{}\theta^{i}\theta^{j}\theta^{k}\xi~{},~{}~{}~{}~{}{\rm
degree~{}2}$
$\displaystyle\theta^{i}\theta^{j}\theta^{k}~{},~{}~{}~{}\theta^{1}\theta^{2}\theta^{3}\theta^{4}\xi~{},~{}~{}~{}~{}{\rm
degree~{}3}$
$\displaystyle\theta^{1}\theta^{2}\theta^{3}\theta^{4}~{},~{}~{}~{}~{}{\rm
degree~{}4}~{}~{}~{}~{}~{}~{}.$

##  5\. BV formalism for YM

Let us introduce a metric on $\Sigma_{4}$ and let us decompose the two form
components in self-dual and anti-self-dual forms as

(36)
$\phi_{2}=\phi_{2}^{+}+\phi_{2}^{-}~{},~{}~{}~{}~{}~{}c_{2}=c_{2}^{+}+c_{2}^{-}~{}.$

If we denote with

(37)
${{\mathcal{M}}_{IR}}=\\{c_{2}^{-}=c_{3}=c_{4}=\phi_{0}=\phi_{1}=\phi_{2}^{-}=0\\}$

and

(38)
${{\mathcal{M}}_{UV}}=\\{c_{0}=c_{1}=c_{2}^{+}=\phi_{2}^{+}=\phi_{3}=\phi_{4}=0\\}$

we see that ${{\mathcal{M}}_{DW}}$ is decomposed in the direct product of
symplectic manifolds

(39) ${{\mathcal{M}}_{DW}}={{\mathcal{M}}_{IR}}\times{{\mathcal{M}}_{UV}}\;,$

where the symplectic forms read as

(40) $\omega_{IR}=\int\limits_{\Sigma_{4}}\left(\delta
c_{0}\wedge\delta\phi_{4}+\delta c_{1}\wedge\delta\phi_{3}+\delta
c_{2}^{+}\wedge\delta\phi_{2}^{+}\right)~{},$

and

(41) $\omega_{UV}=\int\limits_{\Sigma_{4}}\left(\delta
c_{2}^{-}\wedge\delta\phi_{2}^{-}+\delta c_{3}\wedge\delta\phi_{1}+\delta
c_{4}\wedge\delta\phi_{0}\right)~{}.$

As in Sections 2 and 3 we are using the subscripts $IR$ and $UV$ that stand
for infrared and ultraviolet respectively.

Restricting ${S_{DW}}$ to ${{\mathcal{M}}_{IR}}$ (i.e., setting to zero
$\phi_{2}^{-}$, $c_{2}^{-}$, $\phi_{1}$, $c_{3}$, $\phi_{0}$ and $c_{4}$ in
(29)) we get

(43) $\displaystyle{S_{IR}}$ $\displaystyle\equiv$
$\displaystyle{S_{DW}}|_{{\mathcal{M}}_{IR}}=\int\langle\phi_{2}^{+},F(c_{1})\rangle+\langle\phi_{3},d_{c_{1}}c_{0}\rangle+\frac{1}{2}\langle\phi_{2}^{+},\phi_{2}^{+}\rangle$
$\displaystyle\phantom{S_{AKSZ}|_{{\mathcal{M}}_{YM}}=\int}+\langle\phi_{2}^{+},[c_{0},c_{2}^{+}]\rangle+\frac{1}{2}\langle\phi_{4},[c_{0},c_{0}]\rangle\;.$

It is crucial to observe that we obtain the same formula if we restrict
${S_{DW}}$ to ${{\mathcal{M}}_{IR}}\times{\mathcal{L}}_{UV}$, where
${\mathcal{L}}_{UV}$ is the lagrangian submanifold of ${{\mathcal{M}}_{UV}}$
defined by $\phi_{0}=\phi_{1}=\phi_{2}^{-}=0$ (i.e. keeping $c_{2}^{-},c_{3}$
and $c_{4}$ different from zero). It is easy to see that this implies that

(44) $\\{{S_{IR}},{S_{IR}}\\}=0\;.$

By restricting (43) further to the physical (i.e. degree $0$) fields we get

${S_{IR}}|_{cl}=\int\limits_{\Sigma_{4}}\langle\phi_{2}^{+},F(c_{1})\rangle+\frac{1}{2}\langle\phi_{2}^{+},\phi_{2}^{+}\rangle\equiv
S_{YM}^{fo}$

that is the first order formulation of the Yang-Mills action. Upon the
integration of $\phi_{2}^{+}$ we obtain the standard Yang-Mills action (up to
topological term). This proves that ${S_{IR}}$ is a BV extension of the Yang-
Mills action; we are going to prove that it coincides with Costello’s solution
in Chapter 6.2.1 of [12]. A similar result can be found also in [9].

###  5.1. Costello’s formulation of Yang-Mills

Let us consider now the Chern-Simons formulation of ${S_{DW}}$ written in
(34). The restriction to ${{\mathcal{M}}_{IR}}$ imposes constraints to the
superfield (32) so that $S_{IR}$ is not anymore in the AKSZ form; let us see
how it can still be put in this form.

Let us consider
${\mathcal{A}}_{0}=\\{\omega+(\nu_{2}^{+}+\nu_{3}+\nu_{4})\xi\\}$ that is a
dg-subalgebra of $(\Omega(\Sigma_{4})[\xi],d+\frac{\partial}{\partial\xi})$
together with its dg-ideal
${\mathcal{I}}_{0}=\\{\omega_{2}^{-}+\omega_{3}+\omega_{4}\\}$ (recall that
$\nu_{2}^{+}\wedge\omega_{2}^{-}=0)$. The quotient dga

(45)
$({\mathcal{A}}={\mathcal{A}}_{0}/{\mathcal{I}}_{0},d+\frac{\partial}{\partial\xi})$

encodes the following complex

(46)
${\Omega^{0}(\Sigma_{4})}$${\Omega^{1}(\Sigma_{4})}$${\Omega^{2+}(\Sigma_{4})}$${\Omega^{2+}(\Sigma_{4})}$${\Omega^{3}(\Sigma_{4})}$${\Omega^{4}(\Sigma_{4})}$

where the horizontal map is the de Rham differential (composed with the
projection to self dual forms in degree $1$ and to $0$ in degree $2$ of the
upper line) and the diagonal one is the identity. Analogously, the algebra
structure is the wedge product composed with the same projections when it
involves forms of the upper line. By inspection we check that in (43) there is
no difference if we assume that forms are multiplied according to this exotic
ring structure instead of just with the wedge product. We then showed that
also (43) is an AKSZ solution of the master equation (4) with target
$\mathfrak{g}[1]$ and with source the (spec of the) dg-algebra
$({\mathcal{A}},d+\frac{\partial}{\partial\xi})$ endowed with the degree $-3$
integral $\int:{\mathcal{A}}\rightarrow{\mathbb{R}}$ defined as

$\int(\omega+\nu\xi)=\int\limits_{\Sigma_{4}}\nu\;.$

A local basis of $\mathcal{A}$ (as a $\Omega^{0}(\Sigma_{4})$-module) is

(47) $\displaystyle 1~{},~{}~{}~{}~{}{\rm degree~{}0}$
$\displaystyle\theta^{i}~{},~{}~{}~{}\pi^{+}(\theta^{i}\theta^{j})\xi~{},~{}~{}~{}~{}{\rm
degree~{}1}$
$\displaystyle\pi^{+}(\theta^{i}\theta^{j})~{},~{}~{}~{}\theta^{i}\theta^{j}\theta^{k}\xi~{},~{}~{}~{}~{}{\rm
degree~{}2}$
$\displaystyle\theta^{1}\theta^{2}\theta^{3}\theta^{4}\xi~{},~{}~{}~{}~{}{\rm
degree~{}3}$

where $\pi^{+}$ is the projection on self-dual two forms. We can summarize by
saying that

(48)
${S_{IR}}=\int~{}\langle\mathbf{A},(d+\frac{\partial}{\partial\xi})\mathbf{A}\rangle+\mathbf{A}\mathbf{A}\mathbf{A}~{},$

where now $\mathbf{A}$ is expanded along the local basis (47) and the product
is taken in $\mathcal{A}$. This CS solution of the master equation for Yang-
Mills was first introduced in Lemma 2.1.1 of [12]. In the following sections
we will refer to $({\mathcal{A}},d+\frac{\partial}{\partial\xi})$ as the
Costello dg-algebra.

The extension of the AKSZ construction to the case when the source dg algebra
is not locally freely generated (and so is not the algebra of functions on a
dg manifold) has been discussed in [7].

###  5.2. BV-pushforward

The decomposition (39) defines also the (trivial) fibration
${\mathcal{M}}_{DW}\rightarrow{\mathcal{M}}_{IR}$ where we interpret

(49)
$IR\equiv\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{0}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}\\}\in{\mathcal{M}}_{IR}$

as infrared fields and

(50)
$UV\equiv\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{4}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}}\\}\in{\mathcal{M}}_{UV}$

as ultraviolet fields. We are then in a position to define the BV pushforward
of the topological DW theory to the sector of the physical YM fields.

Let us decompose the full BV action (29) as

${S_{DW}}={S_{IR}}+{S_{UV}}+S_{mixed}$

where

(51)
${S_{UV}}=\int\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}}\rangle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}},d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}}\rangle+\frac{1}{2}\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}}\rangle+\frac{1}{2}\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},[{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}}]\rangle$

and

(54) $\displaystyle S_{mixed}$ $\displaystyle=$
$\displaystyle\int\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},[{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}}]\rangle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}},d_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}\rangle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}},[{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}}]\rangle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}},F({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}})\rangle$
$\displaystyle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}\rangle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\rangle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},[{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{0}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{4}}]\rangle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}},[{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{0}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}}]\rangle$
$\displaystyle+\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}},[{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{0}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}}]\rangle+\frac{1}{2}\langle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},[{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}]\rangle\rangle\;.$

In order to perform the vertical integration it is necessary to separate the
UV fields in cohomology and fluctuations, by using the Hodge decomposition of
forms. Let us define

(55)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h}\,\;\;\;{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma}$

where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h}\equiv\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{2}^{-}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{3}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}}\\}=P({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c}),{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma}\equiv\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{0}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{1}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}\\}=P({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi})$
are harmonic forms ($P$ denotes the projection to harmonic forms) and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}}$
denote the fluctuations. We call
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h}$
the zero modes; they are colored in red because, like the IR fields, they are
not integrated out and must be treated as external fields. The zero modes
symplectic form reads

$\int(\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{0}}\wedge\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}}+\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{1}}\wedge\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{3}}+\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}\wedge\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{2}^{-}})\;.$

The gauge fixing is now imposed on the UV-fluctuations by choosing a
Lagrangian submanifold ${\mathcal{L}}_{UV}$ defined by

(56)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{4}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{2}^{-}}=0\;,\;\;\;{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}\in{\rm
Im}d^{\dagger}\;,$

where $d^{\dagger}$ denotes the codifferential. The effective action is now
defined by

$e^{\frac{i}{\hbar}S_{\text{eff}}}=\int\limits_{{\mathcal{L}}_{UV}}e^{\frac{i}{\hbar}S_{DW}}\;.$

Let us analyze the abelian case first. The explicit computation is postponed
to the next section where the more general equivariant case will be analyzed.
We anticipate here that in the non equivariant abelian case the only
corrections to (43) come from the UV zero modes contributions, i.e.

(57)
$S_{\text{eff}}=S_{IR}+\int\langle\sigma_{0},\phi_{4}\rangle+\langle\sigma_{1},\phi_{3}\rangle+\frac{1}{2}\langle\sigma_{2}^{-},\sigma_{2}^{-}\rangle$

where the last terms come from the evaluation of (51) and (54) on zero modes.
Since all remaning fields are red, we suppressed the coloring. The abelian YM-
BV differential is corrected to

(58)
${\boldsymbol{\delta}}c_{0}=\sigma_{0},\;\;\;{\boldsymbol{\delta}}c_{1}=dc_{0}+\sigma_{1},\;\;\;{\boldsymbol{\delta}}c_{2}^{+}=\pi^{+}dc_{1}+\phi_{2}^{+}$
${\boldsymbol{\delta}}\phi_{2}^{+}=0,\;\;\;{\boldsymbol{\delta}}\phi_{3}=d\phi_{2}^{+},\;\;\;{\boldsymbol{\delta}}\phi_{4}=d\phi_{3}$

The UV zero modes are not integrated out so far and so they survive as
dynamical variables. In particular we have

${\boldsymbol{\delta}}\sigma_{0}={\boldsymbol{\delta}}\sigma_{1}={\boldsymbol{\delta}}\sigma_{2}^{-}=0$
(59)
${\boldsymbol{\delta}}h_{2}^{-}=\sigma_{2}^{-}\,,\;\;\;\;{\boldsymbol{\delta}}h_{3}=P\phi_{3}\,,\;\;\;\;{\boldsymbol{\delta}}h_{4}=P\phi_{4}\;.$

By construction ${\boldsymbol{\delta}}$ satisfies

${\boldsymbol{\delta}}^{2}=0\;.$

######  Remark 5.1.

It is easy to check that the effective action (57) is the CS action (48) with
the replacement ${\mathbf{A}}\rightarrow{\mathbf{A}}+h+\xi\sigma$, where
$h=h_{2}^{-}+h_{3}+h_{4}$ and
${\mathbf{\sigma}}=\sigma_{0}+\sigma_{1}+\sigma_{2}^{-}$ are the superfields
encoding the UV zero modes. The homological vector field
$d+\frac{\partial}{\partial\xi}$ is encoded in the following complex that is
an enlarged version of (46)

(60) ${\Omega^{0}}$${\Omega^{1}}$${\Omega^{2+}\oplus
H^{2-}}$${H^{3}}$${H^{4}}$${H^{0}}$${H^{1}}$${\Omega^{2+}\oplus
H^{2-}}$${\Omega^{3}}$${\Omega^{4}}$$\scriptstyle{d}$$\scriptstyle{\pi^{+}d\oplus
0}$$\scriptstyle{i}$$\scriptstyle{i}$$\scriptstyle{d\oplus
0}$$\scriptstyle{id}$$\scriptstyle{d}$$\scriptstyle{P}$$\scriptstyle{P}$

where $i$ denotes the embedding of de Rham cohomology into differential forms
as harmonic forms.

######  Remark 5.2.

The calculation of $S_{\text{eff}}$ for non-abelian theory involves all orders
in the perturbation theory and will be discussed in a future paper.

##  6\. equivariant BV pushforward map

In the previous section we have constructed 4d DW theory as the standard AKSZ
theory with the source being $T[1]\Sigma_{4}$ and the target
$\mathfrak{g}[1]\oplus\mathfrak{g}[2]$. Let us consider now a vector field
$v\in\Gamma(T\Sigma_{4})$ (possibly a complete one whose flux defines a group
action) and let us consider the equivariant extension of the ASKZ action as
describd in Section 3.

Borrowing the notation from the previous section, the equivariant AKSZ action
is defined on the same space of fields ${{\mathcal{M}}_{DW}}$ as

(61) $\displaystyle S_{eq-DW}$ $\displaystyle=$
$\displaystyle{S_{DW}}+S_{\widehat{\iota}_{v}}$ (62) $\displaystyle=$
$\displaystyle\int\limits_{T[1]\Sigma_{4}}d^{4}xd^{4}\theta~{}\left(\langle\mathbf{\Phi},d_{v}\mathbf{C}\rangle+\frac{1}{2}\langle\mathbf{\Phi},\mathbf{\Phi}\rangle+\frac{1}{2}\langle\mathbf{\Phi},[\mathbf{C},\mathbf{C}]\rangle\right)~{},$

where $d_{v}=d+\iota_{v}$ is the (Cartan) equivariant differential and

(63)
$S_{\widehat{\iota}_{v}}=\int\limits_{T[1]\Sigma_{4}}d^{4}xd^{4}\theta~{}\langle\mathbf{\Phi},\iota_{v}\mathbf{C}\rangle$

is the hamiltonian defining the lift $\widehat{\iota}_{v}$ of the contraction
vector field $\iota_{v}$ to ${{\mathcal{M}}_{DW}}$. For the clarity of
forthcoming formulas here we suppress the formal variable $u$ and assume that
we have just one $v$, in the final result is easy to restore $u^{a}$ and
$v_{a}$. This action satisfies the equivariant master equation

(64) $\frac{1}{2}\\{S_{eq-DW},S_{eq-DW}\\}=-S_{\widehat{L}_{v}}~{},$

where

(65)
$S_{\widehat{L}_{v}}=\int\limits_{T[1]\Sigma_{4}}d^{4}xd^{4}\theta~{}\langle\mathbf{\Phi},L_{v}\mathbf{C}\rangle$

is the hamiltonian defining the lift $\widehat{L}_{v}$ of ${L}_{v}$ to
${{\mathcal{M}}_{DW}}$. In components $S_{eq-DW}$ is given by

(66) $S_{eq-
DW}={S_{DW}}+\int\langle\phi_{1},\iota_{v}c_{4}\rangle+\langle\phi_{2},\iota_{v}c_{3}\rangle+\langle\phi_{3},\iota_{v}c_{2}\rangle+\langle\phi_{4},\iota_{v}c_{1}\rangle$

where the component expression of $S_{DW}$ is given in (29).

Let us choose now a metric on $\Sigma_{4}$ invariant under $L_{v}$ and
consider the splitting
${{\mathcal{M}}_{DW}}={\mathcal{M}}_{IR}\times{\mathcal{M}}_{UV}$ defined in
(39). The restriction $S_{eq-DW}$ to ${{\mathcal{M}}_{IR}}$ does not satisfy
the equivariant master equation (64) (since the lift $\widehat{\iota}_{v}$ to
${{\mathcal{M}}_{DW}}$ of the contraction operator $\iota_{v}\in{\rm
Der}(\Omega(\Sigma_{4})[\xi])$ does not restrict to ${{\mathcal{M}}_{IR}}$).
In other terms, the invariance of the metric guarantees that
${{\mathcal{M}}_{IR}}$ inherits from ${{\mathcal{M}}_{DW}}$ the
$\widehat{L}_{v}$ action but not the Cartan complex for equivariant
cohomology.

######  Remark 6.1.

We saw in Section 5.1 that the infrared action $S_{IR}$, which is a BV
extension of Yang-Mills, is an AKSZ action by itself with source the dg
algebra $\mathcal{A}$ defined in (45). It is natural then to look for the
definition of the Cartan complex for equivariant cohomology on $\mathcal{A}$
and then lifting it to ${{\mathcal{M}}_{IR}}$. It is easy to check that while
the Lie derivative ${L}_{v}\in{\rm Der}(\Omega(\Sigma)[\xi])$ descends to
${\rm Der}({\mathcal{A}})$, the same is not true for $\iota_{v}\in{\rm
Der}(\Omega(\Sigma)[\xi])$. So there is not a natural notion for the
contraction operator.

We are going instead to compute the BV push-forward map for $S_{eq-DW}$ to the
space ${{\mathcal{M}}_{IR}}$; by construction the effective action will
satisfy the equivariant master equation.

The general non-abelian theory would require the full perturbation theory. At
the same time in the abelian case the calculation can be carried out exactly
still being non-trivial. Thus from now on we switch to the abelian equivariant
DW theory.

Borrowing the notations from the previous section we can split the abelian
$S_{eq-DW}$ as follows

(67) $S_{eq-DW}=S_{IR}+S_{UV}+S_{mixed}$

with
$\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{0}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}\\}\in{\mathcal{M}}_{IR}$

(68)
$S_{IR}=\int\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{0}}+\frac{1}{2}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}})^{2}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}}\right)$

and
$\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{4}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}}\\}\in{\mathcal{M}}_{UV}$

(69)
$S_{UV}=\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}}+\frac{1}{2}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}})^{2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}}\iota_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{4}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}}\iota_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}}\right)$

and

(70)
$S_{mixed}=\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}}d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{2}^{-}}d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{0}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{1}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\iota_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{-}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}\iota_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}}\right)\\\
$

We stress that we use the ${L}_{v}$-invariant metric to define all fields.

We decompose the UV fields as in (55) by using the Hodge decomposition with
respect to the invariant metric. We then define the lagrangian submanifold
${\mathcal{L}}_{UV}$ as follows

(71)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{4}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{2}^{-}}=0\;,\;\;\;{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}\in{\rm
Im}\,d^{\dagger}\;,$

and by construction ${\mathcal{L}}_{UV}$ is invariant under the action of
${L}_{v}$. Before imposing the conditions
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}\in{\rm
Im}\,d^{\dagger}$ explicitly let us rewrite a bit $S_{IR}$, $S_{UV}$ and
$S_{mixed}$ assuming just that
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{4}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{2}^{-}}=0$.
Assuming the decomposition (55) $S_{UV}$ is written as

(72)
$S_{UV}=\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}+\frac{1}{2}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}})^{2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}\iota_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{1}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{3}}\right)$

and $S_{mixed}$ as follows

(73) $\displaystyle S_{mixed}=$
$\displaystyle\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{0}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{1}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\iota_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}\right.$
$\displaystyle\left.+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}\iota_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{3}}\right)~{}.$

We are going to integrate out only the blue fields
$\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}\\}$
and the red fields (IR fields and UV zero modes) are treated as external.
Therefore it is useful to identify $S_{0}$ as containing only quadratic terms
in the blue fields

(74)
$S_{0}=\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}\right)=\int\left(\frac{1}{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}+\frac{1}{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}+\frac{1}{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}+\frac{1}{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}(\pi^{-}d){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}\right)~{},$

where $\pi^{\pm}$ are the projectors on $\Omega^{2\pm}$, $S_{1}$ as containing
the linear terms in the blue fields

(75)
$S_{1}=\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}(\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}+\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}+\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\right)$

and new $\tilde{S}_{IR}$ containing only red fields (including UV zero modes)

(76) $\displaystyle\tilde{S}_{IR}$
$\displaystyle=\int\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{0}}+\frac{1}{2}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}})^{2}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}}+\frac{1}{2}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}})^{2}\right.$
$\displaystyle\left.+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{1}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{3}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{0}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{1}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{3}}\right)\;,$

so that $S_{eq-DW}=S_{0}+S_{1}+\tilde{S}_{IR}$. It is convenient to write the
quadratic form appearing in $S_{0}$ as

(86)
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccc}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}\end{array}\right)\left(\begin{array}[]{cccc}0&d&0&0\\\
d&0&0&0\\\ 0&0&0&d\\\
0&0&(\pi^{-}d)&0\end{array}\right)\left(\begin{array}[]{c}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}\end{array}\right)$

where the wedge product is assumed. Analogously the integrand in $S_{1}$ is
written as

(92)
$\displaystyle\left(\begin{array}[]{cccc}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}\end{array}\right)(1-P)\left(\begin{array}[]{c}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}\\\
-\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}-\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}+\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}}\\\
\pi^{-}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\end{array}\right)$

where the wedge product is assumed and $(1-P)$ is the project outside of
harmonic forms (cohomology). The claim is that the quadratic form in (86) is
non degenerate once the gauge fixing conditions (71) are taken into account so
that the UV integration is well defined. This will be discussed in the next
subsection.

###  6.1. Integration of UV sector

The canonical way to restrict fields to ${\mathcal{L}}_{UV}$ in (71) is by
adding a trivial sector of auxiliary fields. Indeed, let us add
$({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi},\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi^{\vee},\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta,\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta^{\vee})$
with
$\deg{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi}=-2$,
$\deg{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi^{\vee}}=1$,
$\deg{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta}=-1$
and
$\deg{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta^{\vee}}=0$,
with
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi}$
being (even) zero form
(${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi^{\vee}}$
is top form) and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta}$
being (odd) zero form
(${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta^{\vee}}$
top form). The BV symplectic form in UV sector is now

(93)
$\tilde{\omega}_{UV}=\int\left(\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{2}^{-}}\wedge\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}+\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}\wedge\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}+\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}\wedge\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{4}}+\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi}\wedge\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi^{\vee}}+\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta}\wedge\delta{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta^{\vee}}\right)$

and the BV action is extended to

(94) $S_{eq-
DW}+\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi^{\vee}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta^{\vee}}{L}_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi}\right)~{}.$

The new fields contain zero modes that must be decoupled from fluctuations, as
we did for the UV fields. It is easy to see that zero modes decouple and so
they can be ignored in the following. The extra term in the action just
modifies the quadratic term (74) in UV fields

(95)
$S_{0}^{\prime}=S_{0}+\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi^{\vee}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta^{\vee}}{L}_{v}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi}\right)$

The following gauge fixing fermion (with ${L}_{v}$ invariant metric)

(96)
$\Psi=\int{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\varphi}}(d\star{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}})$

defines the following ${\mathcal{L}}_{UV}$

(97)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{4}}=0~{},~{}~{}~{}~{}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{2}^{-}}=0~{},~{}~{}~{}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\eta}^{\vee}}=0~{},~{}~{}~{}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{3}}=\star
d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\varphi}}~{},~{}~{}~{}~{}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\varphi}^{\vee}}=d\star{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}~{}.$

Evaluating the quadratic part (95) of the action we obtain

(98)
$S_{0}^{\prime}|_{{\mathcal{L}}_{UV}}=\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}d\star
d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\varphi}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}+d\star{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\eta}}\right)\;.$

Introducing the inner product $(~{},~{})$ as in (111) we can rewrite it in the
following matrix form

(99) $S_{0}^{\prime}|_{{\mathcal{L}}_{UV}}=\frac{1}{2}(x,Dx)$

where
$x=({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\varphi}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\eta}})\in
V\equiv(1-P)\left(\Omega^{0}\oplus\Omega^{0}\oplus\Omega^{1}\oplus\Omega^{2-}\oplus\Omega^{0}\right)$
and $D:V\rightarrow V$ is defined as

(100) $D=\left(\begin{array}[]{ccccc}0&-\Delta&0&0&0\\\ -\Delta&0&0&0&0\\\
0&0&0&-d^{\dagger}&d\\\ 0&0&-\pi^{-}d&0&0\\\
0&0&d^{\dagger}&0&0\end{array}\right)$

The differential operator $D$ is by construction self-adjoint and invertible,
its inverse being as follows

(101) $D^{-1}=\left(\begin{array}[]{ccccc}0&-\Delta^{-1}&0&0&0\\\
-\Delta^{-1}&0&0&0&0\\\ 0&0&0&-2\Delta^{-1}d^{\dagger}&\Delta^{-1}d\\\
0&0&-2\pi^{-}\Delta^{-1}d&0&0\\\
0&0&\Delta^{-1}d^{\dagger}&0&0\end{array}\right)\;.$

In order to check it, one has to use the properties (122) and the explicit
definition of $\Delta$ on zero forms.

Next the restriction of the linear term $S_{1}$ in (75) to
${\mathcal{L}}_{UV}$ gives

(102)
$S_{1}|_{{\mathcal{L}}_{UV}}=\int\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{0}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}}-\star
d{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\varphi}}(\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}+\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\phi}_{1}}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+d{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}+\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{c}_{2}^{-}}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}\right)~{},$

that can be written as as

$S_{1}|_{{\mathcal{L}}_{UV}}=(x,a)$

with $a\in V$ being

$a=(1-P)\left(\star{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{4}},-d^{\dagger}\iota_{v}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{2}^{+}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma_{2}^{-}}),-\star{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}}+d^{\dagger}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{2}^{+}}-\star\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h_{4}},-\pi^{-}\iota_{v}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{3}},0\right)\;.$

Combining these terms together we have the following structure

$\displaystyle(S_{0}^{\prime}+S_{1})_{{\mathcal{L}}_{UV}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(x,Dx)+(x,a)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(x+D^{-1}a,D(x+D^{-1}a))-\frac{1}{2}(a,D^{-1}a)$

so that, after integrating out the fields in UV sector, we obtain

$S_{\text{eff}}=\tilde{S}_{IR}-\frac{1}{2}(a,D^{-1}a)~{}.$

with $\tilde{S}_{IR}$ defined in (76). Upon integration we also obtain the
field independent determinants which however depend on the metric. In our
consideration we ignore those determinants, but for physical discussions
(e.g., like metric dependence etc) they do play an important role. Since we
are left with red fields, IR fields and UV zero modes, from now on we stop the
colour marking of the fields. After some computations one gets the following
effective BV action for the IR fields

$\displaystyle S_{\text{eq-eff}}$
$\displaystyle=\int\left(\phi_{2}^{+}dc_{1}+\phi_{3}dc_{0}+\frac{1}{2}(\phi_{2}^{+})^{2}+\sigma_{0}\phi_{4}+\sigma_{1}\phi_{3}++\frac{1}{2}(\sigma_{2}^{-})^{2}\right.$
$\displaystyle\phi_{3}\iota_{v}c_{2}^{+}+\phi_{4}\iota_{v}c_{1}+\sigma_{1}\iota_{v}h_{4}+\sigma_{2}^{-}\iota_{v}h_{3}+\phi_{3}\iota_{v}h_{2}^{-}+\phi_{2}^{+}\iota_{v}h_{3}$
$\displaystyle\left.-\phi_{4}K\iota_{v}(\phi_{2}^{+}+\sigma_{2}^{-})-2(\phi_{3}+\iota_{v}h_{4})K\pi^{-}\iota_{v}\phi_{3}-2c_{2}^{+}dK\pi^{-}\iota_{v}\phi_{3}\right)\;.$

By construction it satisfies the equivariant master equation (64) with

$\displaystyle
S^{\text{eff}}_{\widehat{L}_{v_{a}}}=\int\left(\phi_{4}L_{v}c_{0}+\phi_{3}L_{v}c_{1}+\phi_{2}^{+}L_{v}c_{2}^{+}+\sigma_{0}L_{v}h_{4}+\sigma_{1}L_{v}h_{3}+\sigma_{2}^{-}L_{v}h_{2}^{-}\right)$

By putting $v=0$ one recognizes the non equivariant effective action in (57)
which coincides, up to zero modes, with the BV extension of abelian Yang-Mills
theory. We can say then that (6.1) is an equivariant extension of the abelian
Yang-Mills BV theory.

We can read off the equivariant BV transformations for IR fields

(104)
$\displaystyle{\boldsymbol{\delta}_{eq}}c_{0}=\iota_{v}c_{1}+\sigma_{0}-K\iota_{v}\phi_{2}^{+}-K\iota_{v}\sigma_{2}^{-}$
$\displaystyle{\boldsymbol{\delta}_{eq}}c_{1}=dc_{0}+\sigma_{1}+\iota_{v}h_{2}^{-}+\iota_{v}c_{2}^{+}-2K(\pi^{-}\iota_{v}\phi_{3})$
$\displaystyle\phantom{{\boldsymbol{\delta}_{eq}}c_{1}=}-2\iota_{v}(\pi^{-}K\phi_{3})+2\iota_{v}(\pi^{-}dKc_{2}^{+})-2\iota_{v}\pi^{-}K\iota_{v}h_{4}$
$\displaystyle{\boldsymbol{\delta}_{eq}}c_{2}^{+}=\pi^{+}\Big{(}dc_{1}+\phi_{2}^{+}+\iota_{v}h_{3}-\iota_{v}(K\phi_{4})\Big{)}$
$\displaystyle{\boldsymbol{\delta}_{eq}}\phi_{2}^{+}=\pi^{+}\Big{(}\iota_{v}\phi_{3}-2dK(\pi^{-}\iota_{v}\phi_{3})\Big{)}$
$\displaystyle{\boldsymbol{\delta}_{eq}}\phi_{3}=d\phi_{2}^{+}+\iota_{v}\phi_{4}$
$\displaystyle{\boldsymbol{\delta}_{eq}}\phi_{4}=d\phi_{3}$

and for UV zero modes

(105)
$\displaystyle{\boldsymbol{\delta}_{eq}}\sigma_{0}=-2P\iota_{v}K\pi^{-}\iota_{v}\phi_{3}+P\iota_{v}\sigma_{1}$
$\displaystyle{\boldsymbol{\delta}_{eq}}\sigma_{1}=P\iota_{v}\phi_{2}^{+}+P\iota_{v}\sigma_{2}^{-}$
$\displaystyle{\boldsymbol{\delta}_{eq}}\sigma_{2}^{-}=P\pi^{-}\iota_{v}\phi_{3}$
$\displaystyle{\boldsymbol{\delta}_{eq}}h_{2}^{-}=\pi^{-}P\iota_{v}(-K\phi_{4}+h_{3})+\sigma_{2}^{-}$
$\displaystyle{\boldsymbol{\delta}_{eq}}h_{3}=P(\phi_{3}+\iota_{v}h_{4})$
$\displaystyle{\boldsymbol{\delta}_{eq}}h_{4}=P\phi_{4}\ $

The reader, with a straightforward but tedious calculation, can check that
${\boldsymbol{\delta}_{eq}}$ satisfies

(106) ${\boldsymbol{\delta}_{eq}}^{2}=-\hat{L}_{v}~{}.$

######  Remark 6.2.

The invariant metric defining the UV gauge fixing is arbitrary. In the
computation of (104) we implicitly assumed that it is in the same conformal
class of the metric defining the IR space (or equivalently the YM theory).
There is then a family of equivariant extensions parametrized by invariant
metrics that are conformally equivalent to the metric used for the definition
of the YM action. From the general discussion of subsection 3.2 it follows
that they give equivalent extensions of abelian YM.

There are terms in ${\boldsymbol{\delta}_{eq}}$ that are quadratic in the
vector field $v$. Restoring the formal parameter $u$ of degree $2$, one can
expand ${\boldsymbol{\delta}_{eq}}$ in powers of $u$ and get

(107)
${\boldsymbol{\delta}_{eq}}={\boldsymbol{\delta}}+u{\boldsymbol{\delta}}_{-1}+u^{2}{\boldsymbol{\delta}}_{-3}~{}~{}~{}~{}.$

From (106) they satisfy the following relations

$\displaystyle{\boldsymbol{\delta}}^{2}=0~{},$
$\displaystyle{\boldsymbol{\delta}}{\boldsymbol{\delta}}_{-1}+{\boldsymbol{\delta}}_{-1}{\boldsymbol{\delta}}=L_{v}~{},$
(108)
$\displaystyle{\boldsymbol{\delta}}{\boldsymbol{\delta}}_{-3}+{\boldsymbol{\delta}}_{-3}{\boldsymbol{\delta}}+{\boldsymbol{\delta}}_{-1}^{2}=0~{},$
$\displaystyle{\boldsymbol{\delta}}_{-1}{\boldsymbol{\delta}}_{-3}+{\boldsymbol{\delta}}_{-3}{\boldsymbol{\delta}}_{-1}=0~{},$
$\displaystyle{\boldsymbol{\delta}}_{-3}^{2}=0$

The first two lines are the standard Cartan relations that make us to intepret
${\boldsymbol{\delta}}_{-1}$ as the contraction operator; the third lines
states that ${\boldsymbol{\delta}}_{-1}$ squares to zero only up to
$\operatorname{ad}({\boldsymbol{\delta}})$. This is an instance of the dgla
introduced in [1] to build differential models of equivariant cohomology and
was used in the BV context in [16].

######  Remark 6.3.

Analogously to Remark 5.1 we can read from (104) and (105) the equivariant
differential on the enlarged Costello complex (60). We can organize it as

(109) $d^{\rm{eff}}_{u}=d_{+1}+ud_{-1}+u^{2}d_{-3}~{},$

where $d_{+1}$ is the degree $1$-differential described in (60). The operation
$d_{-1}$ lowers the degree by $1$ and can be summarized by the following
diagram

${\Omega^{0}}$${\Omega^{1}}$${\Omega^{2+}\oplus
H^{2-}}$${H^{3}}$${H^{4}}$${H^{0}}$${H^{1}}$${\Omega^{2+}\oplus
H^{2-}}$${\Omega^{3}}$${\Omega^{4}}$$\scriptstyle{\iota_{v}}$$\scriptstyle{\iota_{v}(1+2\pi^{-}dK)\oplus\iota_{v}}$$\scriptstyle{\pi^{+}\iota_{v}\oplus\pi^{-}P\iota_{v}}$$\scriptstyle{P\iota_{v}}$$\scriptstyle{P\iota_{v}}$$\scriptstyle{P\iota_{v}\oplus
P\iota_{v}}$$\scriptstyle{-(K\iota_{v}\oplus
K\iota_{v})}$$\scriptstyle{-2(K\pi^{-}\iota_{v}+\iota_{v}\pi^{-}K)}$$\scriptstyle{\pi^{+}(\iota_{v}-2dK\pi^{-}\iota_{v})\oplus\pi^{-}P\iota_{v}}$$\scriptstyle{\iota_{v}}$$\scriptstyle{-(\pi^{+}\iota_{v}K\oplus
P\pi^{-}\iota_{v}K)}$

The last operation $d_{-3}$ (the terms quadratic in $v$) lowers the degree by
$3$ and the only non-trivial operations are given by the following diagram

${\Omega^{0}}$${\Omega^{1}}$${\Omega^{2+}\oplus
H^{2-}}$${H^{3}}$${H^{4}}$${H^{0}}$${H^{1}}$${\Omega^{2+}\oplus
H^{2-}}$${\Omega^{3}}$${\Omega^{4}}$$\scriptstyle{-2\iota_{v}\pi^{-}K\iota_{v}}$$\scriptstyle{-2P\iota_{v}K\pi^{-}\iota_{v}}$

By construction we know that

(110) $(d^{\rm{eff}}_{u})^{2}=(d_{+1}+ud_{-1}+u^{2}d_{-3})^{2}=uL_{v}~{},$

so that we have the same generalized Cartan calculus as in (6.1).

######  Remark 6.4.

Due to the dependence on the homotopy operator $K$, the equivariant operator
$d^{\rm{eff}}_{u}$ is not a derivation. This can be checked on the Costello
complex ignoring the contribution of zero modes. As a consequence, the
effective action (6.1) has non zero bracket with the cubic term
$\int{\mathbf{A}}^{3}$ so that the equivariant extension of the non abelian
case cannot be obtained by the sum of (6.1) and the cubic term. The only
possible approach is the computation of the BV-pushforward, which will involve
all orders in perturbation theory. Due to degree considerations the general
structure of ${\boldsymbol{\delta}_{eq}}$ is the same as in (107), with
${\boldsymbol{\delta}}$ being the effective BV differential defined by the
(non equivariant) BV pushforward.

##  7\. Summary

In this paper we considered two four dimensional BV theories: DW theory
associated to BV manifold ${\mathcal{M}}_{DW}$ and YM theory based on BV
manifold ${\mathcal{M}}_{IR}$ (see the text for the definitions). There exists
a formal push-forward map from ${\mathcal{M}}_{DW}$ to ${\mathcal{M}}_{IR}$
and thus on general ground we expect that the two theories are quasi-
isomorphic at the level of observables. For example, a local simple observable
in one theory can be mapped into something complicated and non-local in
another theory. We do not know how much this formal statement is useful for
actual calculations in one of the theories. In this paper we have performed
the BV push-forward map for the case of abelian theories, and in the case of
equivariant theory we obtain non-local corrections to the BV YM action. The
equivariant extension of the BV formalism was originally introduced in [6] and
in this setting upon an appropriate gauge fixing the residual BRST symmetries
should realize the equivariant differential on the space of fields (or some
version of equivariant differential). Such residual BRST symmetries may allow
to perform the exact calculations for some theories and for some specific
observables. Thus we hope that these new structures presented here can be
useful for the calculations in the same fashion as the equivariant
considerations help calculate the ordinary integrals of the closed
differential forms. But this requires further study.

Another interesting result of this work is the homological generalization of
Cartan calculus with the explicit constructions in terms of non-local
operators. This structure requires further study and better understanding both
at the level of the formal properties and at the level of concrete examples.
It would be interesting to see if any of the standard considerations for the
equivariant cohomology and equivariant localization can be generalized to this
new setting.

##  Appendix A Hodge decomposition and relevant operators

In this appendix we collect the relevant properties of the Hodge decomposition
and the related differential operators. All formulas are written for a 4d
closed manifold $\Sigma_{4}$. Let $d$ be the de Rham operator. We pick up the
metric and introduce the inner product on differential forms

(111) $(\alpha,\beta)=\int\alpha\wedge\star\beta~{},$

where $\star$ is the Hodge star operator with the property

(112) $\star^{2}\omega_{r}=(-1)^{r}\omega_{r}~{},$

where $\omega_{r}\in\Omega^{r}(\Sigma_{4})$. We define the adjoint operator
$d^{\dagger}$ for the de Rham differential

(113) $(\alpha,d\beta)=(d^{\dagger}\alpha,\beta)$

with the explicit definition given by

(114) $d^{\dagger}=-\star d\star~{}.$

Next we define the Laplace operator

(115) $\Delta=dd^{\dagger}+d^{\dagger}d~{},$

which is a self-adjoint operator $\Delta^{\dagger}=\Delta$. The Hodge
decomposition for $\Omega^{r}$ corresponds to the decomposition into three
orthogonal spaces

(116) $\Omega^{r}=({\rm Im}~{}d)\oplus({\rm Im}~{}d^{\dagger})\oplus({\rm
Harm}^{r})~{},$

where $({\rm Harm}^{r})=({\rm ker}~{}\Delta)$ stands for harmonic forms which
can be identified with the de Rham cohomology. We denote by $P$ the projector
on $({\rm Harm}^{r})$ and by $(1-P)$ the projector to $({\rm
Im}~{}d)\oplus({\rm Im}~{}d^{\dagger})$. Away from $({\rm Harm}^{r})$ the
Laplace operator is invertible, thus whenever we write $\Delta^{-1}$ we assume
that it is defined only on the $(1-P)$ space. The operator $\Delta^{-1}$ is
self-adjoint ($\Delta^{-1\dagger}=\Delta^{-1}$) and we have the following
obvious properties

$[d,\Delta]=0~{},~{}~{}[d^{\dagger},\Delta]=0~{},~{}~{}[d,\Delta^{-1}]=0~{},~{}~{}[d^{\dagger},\Delta^{-1}]=0~{},$

where the relations with $\Delta^{-1}$ understood only for the $(1-P)$ space.
It is convenient to define the homotopy operator

(117) $K=\int\limits_{0}^{\infty}dt~{}e^{-t\Delta}d^{\dagger}~{},$

which is defined everywhere and satisfies the property

(118) $Kd+dK=1-P~{}.$

If understood correctly $K$ can be expressed as follows

(119) $K=\Delta^{-1}d^{\dagger}~{}.$

On two forms we have the following projectors

(120) $\pi^{\pm}=\frac{1}{2}(1\pm\star)~{},$

which splits $\Omega^{2}=\Omega^{2+}\oplus\Omega^{2-}$ into self-dual and
anti-self-dual two forms correspondingly. We have the following useful
properties

(121)
$\pi^{\pm}\Delta=\Delta\pi^{\pm}~{},~{}~{}~{}~{}~{}~{}\pi^{\pm}\Delta^{-1}=\Delta^{-1}\pi^{\pm}$

and

(122) $\Delta\pi^{\pm}=2\pi^{\pm}dd^{\dagger}\pi^{\pm}~{}.$

Assume that $\Sigma_{4}$ admits the action of a vector field $v$ (typically
coming from some group action). On the differential forms $v$ acts via the Lie
derivative

(123) ${L}_{v}=d\iota_{v}+\iota_{v}d~{},$

where $\iota_{v}$ is contraction of $v$ with a differential form. We pick an
invariant metric $g$ such that ${\mathcal{L}}_{v}g=0$. In this case we have
the following properties

(124)
$[{L}_{v},d]=0~{},~{}~{}~{}~{}[{L}_{v},d^{\dagger}]=0~{},~{}~{}~{}~{}[{L}_{v},\Delta]=0~{}.$

and thus ${L}_{v}$ preserves the subspaces in the corresponding Hodge
decomposition. Moreover we have the following property

(125) $\displaystyle{L}_{v}^{\dagger}=-{L}_{v}~{}.$

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