General Covariance from the Viewpoint of Stacks

Filip Dul111Department of Mathematics, Rutgers – New Brunswick, Piscataway NJ
08854, United States of America.
ORCID: 0000-0001-8623-0293.
Author email address<EMAIL_ADDRESS>

###### Abstract

General covariance is a crucial notion in the study of field theories on
curved spacetimes. A field theory defined with respect to a semi-Riemannian
metric is generally covariant if two metrics on $X$ which are related by a
diffeomorphism produce equivalent physics. From a purely mathematical
perspective, this suggests that we try to understand the quotient stack of
metrics modulo diffeomorphism: we will use the language of groupoids to do
this concretely. Then we will inspect the tangent complex of this stack at a
fixed metric, which when shifted up by one defines a differential graded Lie
algebra. By considering the action of this Lie algebra on the observables for
a Batalin-Vilkovisky scalar field theory, we recover a novel expression of the
stress-energy tensor for that example, while describing how this works for a
general class of theories. We will describe how this construction nicely
encapsulates but also broadens the usual presentation in the physics
literature and discuss applications of the formalism.

Key words: Stacks, Formal Derived Geometry, Curved Spacetimes,
Batalin-Vilkovisky Formalism, Gravitation, Conserved Quantities.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Future Directions
  2. 2 Bundles of Batalin-Vilkovisky field theories
    1. 2.1 Introduction to the BV Formalism
    2. 2.2 Equivariant Vector Bundles
    3. 2.3 General Covariance
      1. 2.3.1 An important remark on functoriality
    4. 2.4 Groupoids and Stacks
  3. 3 Formal Stacks
    1. 3.1 Tangent Complexes
    2. 3.2 Chevalley-Eilenberg Cochains as Rings of Functions
    3. 3.3 Vector Bundles over a Formal Stack
  4. 4 Field Theories as Bundles over Formal Stacks
    1. 4.1 Equivariant Observables
    2. 4.2 The Stress-Energy Tensor
  5. 5 Appendix
    1. 5.1 A detailed example
    2. 5.2 A remark on higher orders
    3. 5.3 Acknowledgements
    4. 5.4 Statements and Declarations

##  1 Introduction

###  1.1 Overview

Over a hundred years ago, when Albert Einstein and a group of others were
laying the foundations of general relativity, general covariance became an
essential ingredient in formulating physics in curved spacetimes. Roughly, a
field theory coupled to a background metric on a spacetime $X$ is said to be
generally covariant if the diffeomorphism group of $X$ is a symmetry of the
theory. Physicists usually interpret diffeomorphisms as coordinate changes, so
they may say that a theory exhibits general covariance if it is coordinate-
invariant: i.e. a theory may superficially change to a distinct one if the
coordinates are changed, but if it is generally covariant, then those “two”
theories are equivalent in a way which we will make rigorous. Although general
covariance can be understood in the context of all field theories, it is often
considered in the context of field theories coupled to semi-Riemannian
metrics: this is the particular case we will focus on in this paper, but we
will moreover argue that this particular case is of central importance.

The primary aim of this paper is to package general covariance in the Batalin-
Vilkovisky formalism for classical field theories, especially as it is
presented in [8]. Equivariant vector bundles are an appropriate geometric tool
for understanding families of field theories parameterized by a space of
Riemannian or Lorentzian metrics. We therefore review the global theory of
such bundles, explain its relevance for general covariance in Section 2.3, and
then cast much of it in the language of stacks in Section 2.4. Stacks provide
a generalized notion of space which allows us to deal with quotient spaces
that may be singular or otherwise forget interesting information about the
original space. We will describe how a bundle of generally covariant classical
Batalin-Vilkovisky (BV) field theories over the space of metrics on $X$,
denoted $\mathscr{M}$, descends to a bundle over the quotient stack
$[\mathscr{M}/\mathscr{D}]$ of the metrics modulo the diffeomorphism group of
$X$, denoted $\mathscr{D}$. Our definition for general covariance is
equivalent to the following:

######  Definition 1.1.

A bundle $\pi:(\mathscr{F},\\{S,-\\})\to\mathscr{M}$ of Batalin-Vilkovisky
field theories on a compact manifold $X$ is generally covariant if it descends
to a bundle of stacks

$\pi:([\mathscr{F/D}],\\{S,-\\})\to[\mathscr{M/D}].$

Before and after introducing this definition, we discuss in detail how both
scalar field theory and Yang-Mills theory are generally covariant in our sense
as BV field theories. Indeed, one of our central results is Theorem 2.70, in
which we show that Yang-Mills theory is generally covariant, because it serves
as a nexus for views on covariance from other sources in the literature, as we
discuss in the surrounding commentary. We make remarks on functorial aspects
of our work (mostly in Subsection 2.3.1) which are important in their own
right and also serve as points of comparison to the prevailing literature–both
to the factorization algebra framework (as seen in [7] and [8]) the author was
trained in, but also to the AQFT framework (as seen in [19]) the author
desires to better understand.

Much of our concrete computations are in the regime of perturbative field
theories, so we consider formal neighborhoods in the quotient stack
$[\mathscr{M}/\mathscr{D}]$, which in turn lead us to understanding the field
theories as examples in derived deformation theory: in brief, we associate to
a generally covariant theory a formal moduli problem, as defined in [15], by
pulling back the above bundle of stacks over the formal neighborhood of a
metric $[g]\in[\mathscr{M/D}]$. We then compute the function ring for this
pullback over a formal stack and show that it gives us a ring of equivariant
classical observables, as defined in [8]: Proposition 4.3 is thus one of the
primary results of this paper, in that it explicitly links the stacky geometry
presented earlier with the usual factorization algebra framework of Noether’s
Theorem presented in Costello and Gwilliam’s books. This perspective is a
beautiful fusion of Emmy Noether’s foundational work in both homological
algebra and symmetries in physics: homological algebra allows us to put
external symmetries (perturbations) and internal symmetries (isometries) on
equal footing, so that we can state a more fully encompassing form of
Noether’s Theorem.

In Remark 4.9 and Section 4.2, we consider the conservation of the stress-
energy tensor $T_{\mu\nu}$, the conserved quantity associated to general
covariance via Noether’s Theorem, in derived deformation theoretic terms:
essentially, $T_{\mu\nu}$ tells us how the aforementioned formal stack acts on
the field theories it is coupled to, in the language of $L_{\infty}$ algebras.
In Theorem 4.24, we expound on the above by computing a perturbative
equivalence of observables when the theory is deformed by a vector field, and
make a few remarks on how this might be relevant at higher orders in
perturbation theory in Appendix 5.2. One of the objectives of Section 4.2 is
to provide a potential pathway for physicists to link their tools with ours.

###  1.2 Future Directions

Much of this paper serves as a set-up for a few distinct projects. My primary
motivation looking forward is the subject of anomalies in perturbative quantum
field theory. Anomalies arise when the quantization of families of field
theories over a parameter space with some classical symmetry does not
necessarily respect that symmetry. In [17], Rabinovich computes the BV
quantization of families of free fermions parameterized by gauge fields and by
this process recovers the axial anomaly. The anomaly is then explicitly
quantified cohomologically by viewing the background gauge fields
perturbatively (much in the way we consider metrics modulo diffeomorphisms
perturbatively in Section 4): computations from BV quantization allow
Rabinovich to equate the anomaly with the index of the original Dirac
operator, as per the Atiyah-Singer Families Index Theorem.

Much of the work in the current paper is motivated by the desire to reproduce
similar computations to those in [17] when replacing connections modulo gauge
by metrics modulo diffeomorphism. Before diving into quantization, we must
first understand both the global and perturbative nature of the stack
$[\mathscr{M/D}]$ of metrics modulo diffeomorphism as it parameterizes
classical theories. In the case of free fermions parameterized by
$[\mathscr{M/D}]$, we hope to reproduce a version of results stated in [18]:
there, Rabinovich connects his work in [17] to defining a determinant line
bundle $\mathrm{Det}(D)$ (à la Quillen) over some parameter space $B$ via BV
quantization. In our case, we would let $B=[\mathscr{M/D}]$ and then the
anomaly would constitute the first Chern class of the determinant line bundle
over this.

Another goal is to understand how Wald’s results on viewing black hole entropy
as Noether charge (as in [24]) might be feasible within the BV framework. The
study and physical interpretation of black hole thermodynamics is as popular
now as it has ever been; however, it remains elusive in many regards. Wald’s
work connecting black hole entropy to Noether’s Theorem may well serve as a
point of connection to our work: in particular, the BV version of Noether’s
Theorem is put into detail and application in the second half of [8] and
explicit computations in the case of metrics modulo diffeomorphism (a central
object in Wald’s paper) are provided in this article. It is also advantageous
that Wald focuses on structural and algebraic aspects, so that porting it all
over into the BV framework might be somewhat natural.

##  2 Bundles of Batalin-Vilkovisky field theories

We will begin by introducing the Batalin-Vilkovisky (BV) formalism: the
purpose of the following narrative is to show how classical field theory is
very naturally expressed in this formalism, especially in the context of
diffeomorphism equivariance. The basic ingredients required from the outset
are a space of fields, which define the kinematics of a physical model, and an
action functional, which fixes the dynamics of that model. The fields on a
space (or spacetime) $X$ are sections of some bundle $F\to X$ (usually a
vector bundle), denoted $\mathscr{F}:=\Gamma(X,F)$. The action functional is a
function $S:\mathscr{F}\to\mathbf{R}$ whose critical locus
$\mathrm{Crit}(S)$222This is computed via variational calculus, and described
for example in Appendix E of [23]. is the set of $\phi\in\mathscr{F}$ that
satisfy the Euler Lagrange equations associated to $S$ via functional
differentiation. To truly begin a discussion of the BV formalism, we must
begin by making precise the notion of a functional.

######  Remark 2.1.

So far, our bundle $F\to X$ is not graded: as we unfold what it means to
define a BV theory, $F$ will be replaced by a differential graded bundle, but
the notation will not change.

###  2.1 Introduction to the BV Formalism

The space of fields is usually infinite dimensional, which means we cannot
take the usual algebraic symmetric powers of $\mathscr{F}$ to define their
space of functions. Thus, we have the following definitions which play an
identical role, but for the infinite dimensional case. Much of what follows is
from Chapter 5, Section 3 of [6] and Chapter 3, Section 5 and Appendix B of
[7].

######  Definition 2.2.

(Defined in Section 3.5.7 of [7]) The algebra of functionals on $\mathscr{F}$
is

$\mathscr{O}(\mathscr{F}):=\prod_{k\geq 0}\mathrm{Hom}(\mathscr{F}^{\otimes
k},\mathbf{R})_{S_{k}}.$

We may sometimes denote this ring as $\mathrm{Sym}(\mathscr{F}^{\vee})$.

######  Remark 2.3.

To be fully precise, if $X$ is compact, $\mathscr{F}=\Gamma(X,F)$ is a nuclear
Fréchet space, where $\otimes$ denotes the completed projective tensor
product, so that

$\mathscr{F}^{\otimes k}:=\Gamma(X\times\cdots\times
X,F\boxtimes\cdots\boxtimes F),$

meaning each $\mathrm{Hom}(\mathscr{F}^{\otimes k},\mathbf{R})_{S_{k}}$ is a
space of continuous multilinear functionals endowed with the strong dual
topology: i.e. a space of distributions. The literature mentioned above
defines all of this for a slightly broader class of spaces than Fréchet
spaces, but that is enough for us. In particular, we have the following fact,
from page 1 of [22]:

######  Example 2.4.

Let $X$ be a smooth, compact, finite dimensional manifold, and let $F\to X$ be
a vector bundle with space of sections $\Gamma(X,F)=:\mathscr{F}$. Choose
Riemannian metrics and connections on $TX$ and $F$, let $\nabla^{i}\phi$
denote the $i^{th}$ covariant derivative of $\phi\in\mathscr{F}$, and set

$||f||_{n}:=\sum_{i=0}^{n}\mathrm{sup}|\nabla^{i}\phi(x)|.$

By means of the topology defined by the sequence of norms $\\{||-||_{n}\\}$,
$\mathscr{F}$ is a Fréchet space. Clearly, we can define differential graded
Fréchet spaces as well, as will soon be relevant.

######  Definition 2.5.

The space of local functionals, denoted
$\mathscr{O}_{\mathrm{loc}}(\mathscr{F})$, is the linear subspace of
$\mathscr{O}(\mathscr{F})$ spanned by elements of the form

$F_{k}(\phi)=\int_{X}(D_{1}\phi)(D_{2}\phi)\ldots(D_{k}\phi)\textrm{vol},$

for fields $\phi\in\mathscr{F}$ and differential operators $D_{i}$ on $X$.

######  Lemma 2.6.

([6], Ch. 5, Lemma 6.6.1) There is an isomorphism of cochain complexes

$\mathscr{O}_{\mathrm{loc}}(\mathscr{F})\cong\mathrm{Dens}_{X}\otimes_{D_{X}}\mathscr{O}_{\mathrm{red}}(\mathscr{J}(F)),$

where $\mathscr{J}(F)$ denotes sections of the $\infty$-jet bundle
$\mathrm{Jet}(F)\to X$, and $\mathscr{O}_{\mathrm{red}}(\mathscr{J}(F))$ is
the quotient of
$\mathscr{O}(\mathscr{J}(F))=\mathrm{Sym}(\mathscr{J}(F)^{\vee})$ by the
constant polynomial functions.

######  Remark 2.7.

Sections of $\mathscr{O}_{\mathrm{loc}}(\mathscr{F})$ are exactly elements of
the preceding form, and integration defines a natural inclusion:

$\iota:\mathscr{O}_{\mathrm{loc}}(\mathscr{F})\to\mathscr{O}_{\mathrm{red}}(\mathscr{F}).$

This lemma shows that $\mathscr{O}_{\mathrm{loc}}(\mathscr{F})$ is the space
of Lagrangian densities modulo total derivatives: this is desirable because
adding a total derivative to a Lagrangian density does not affect the dynamics
described in the equations of motion. Local functionals are also more
manageable in terms of functional analysis; for example, the action functional
$S$ is always an element of $\mathscr{O}_{\mathrm{loc}}(\mathscr{F})$, and
local functionals are key in defining the Poisson bracket, as we will see
below.

######  Definition 2.8.

For $F\to X$ a graded vector bundle, a constant coefficient $k$-shifted
symplectic structure is an isomorphism

$F\cong_{\omega}F^{!}[k]:=(\mathrm{Dens}_{X}\otimes F^{\vee})[k]$

of graded vector spaces that is graded antisymmetric.

######  Remark 2.9.

It stands to reason that a symplectic structure on a space defines a Poisson
bracket on its space of functions: this is indeed the case for
$\mathscr{O}_{\mathrm{loc}}(\mathscr{F})\subset\mathscr{O}(\mathscr{F})$. This
is not the case however for all of $\mathscr{O}(\mathscr{F})$, for functional
analytic reasons which are outside the scope of this paper.333Details about
this can be found in Chapter 4 of [8]. We will denote the Poisson
(anti-)bracket induced by $\omega$ as $\\{-,-\\}$.

######  Definition 2.10.

A Batalin-Vilkovisky classical field theory $(\mathscr{F},\omega,S)$ on a
smooth manifold $X$ is a differential $\mathbf{Z}$-graded vector bundle $F\to
X$ equipped with a $-1$-shifted symplectic structure $\omega$ and an action
functional $S\in\mathscr{O}_{\mathrm{loc}}(\mathscr{F})$ such that:
(1) $S$ satisfies the classical master equation (CME): $\\{S,S\\}=0$.
(2) $S$ is at least quadratic, so that it can be written uniquely as
$S(\varphi)=\omega(\varphi,Q\varphi)+I(\varphi)$, where $Q$ is a linear
differential operator and $I\in\mathscr{O}_{\mathrm{loc}}(\mathscr{F})$ is at
least cubic.
A free theory is one in which $I=0$: i.e. the action functional $S$ is purely
quadratic.

######  Remark 2.11.

Although $\\{-,-\\}$ is not a Poisson bracket on $\mathscr{O}(\mathscr{F})$,
bracketing with a local functional like
$S\in\mathscr{O}_{\mathrm{loc}}(\mathscr{F})$ defines a derivation

$\\{S,-\\}:\mathscr{O}(\mathscr{F})\to\mathscr{O}(\mathscr{F})[1]$

regardless of whether or not the BV theory is free. For a free theory, it can
be shown that $\\{S,-\\}=Q$ on $\mathscr{O}(\mathscr{F})$, where the
differential $Q$ on $\mathscr{F}$ is extended to $\mathscr{O}(\mathscr{F})$ as
a derivation. For an interacting theory, $\\{S,-\\}$ is prescribed by an
$L_{\infty}$ algebra structure on $\mathscr{F}$, which we will describe in
Definition 2.17 and provide examples of thereafter. The ellipticity or
hyperbolicity of $(\mathscr{F},Q)$ is sometimes assumed: this will be
commented on later.

######  Definition 2.12.

Let $F\to X$ be a differential graded vector bundle with differential $Q$ on
its sheaf of sections $\mathscr{F}$. Then the dg commutative ring of (global)
classical observables for the theory defined by $(\mathscr{F},Q)$ is

$\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}):=(\mathscr{O}(\mathscr{F}),\\{S,-\\}).$

######  Remark 2.13.

We will briefly describe how $\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F})$ is to
be understood as the dg ring of functions on the derived critical locus of the
action functional $S:\mathscr{F}^{0}\to\mathbf{R}$, where the degree zero part
of the dg fields is the “naïve” original space of fields, following [8].

The ordinary critical locus $\mathrm{Crit}(S)$ is the intersection of the
graph

$\Gamma(dS)\subset T^{\vee}\mathscr{F}^{0}$

with the zero section $\mathscr{F}^{0}\subset T^{\vee}\mathscr{F}^{0}$. We
thus get its commutative algebra of functions to be

$\mathscr{O}(\mathrm{Crit}(S))=\mathscr{O}(\Gamma(dS))\otimes_{\mathscr{O}(T^{\vee}\mathscr{F}^{0})}\mathscr{O}(\mathscr{F}^{0}).$

However, $\mathrm{Crit}(S)$ can be singular (e.g. it may be a non-transverse
intersection), so we follow the philosophy of derived (algebraic) geometry and
replace the above critical locus with the derived critical locus
$\mathrm{Crit}^{h}(S)$, which has ring of functions

$\mathscr{O}(\mathrm{Crit}^{h}(S))=\mathscr{O}(\Gamma(dS))\otimes_{\mathscr{O}(T^{\vee}\mathscr{F}^{0})}^{\mathbb{L}}\mathscr{O}(\mathscr{F}^{0}).$

This is now a commutative dg algebra instead of an ordinary commutative
algebra, and it can be realized as the complex

$\mathscr{O}(T^{\vee}[-1]\mathscr{F}^{0})=\Gamma(\mathscr{F}^{0},\Lambda^{\bullet}T\mathscr{F}^{0}).$

Now we see how the Batalin-Vilkovisky dg fields $(\mathscr{F},Q)$ arise
naturally from a derived geometric perspective as
$T^{\vee}[-1]\mathscr{F}^{0}$; moreover, the differential on
$\Gamma(\mathscr{F}^{0},\Lambda^{\bullet}T\mathscr{F}^{0})$ is contracting
with the 1-form $dS\in\Omega^{1}(\mathscr{F}^{0})$, and this can be shown to
be equivalent to $\\{S,-\\}$, as we would expect.

######  Example 2.14.

A running example through much of this text will be scalar field theory. We
will consider the free case first. Fix a semi-Riemannian manifold $(X,g)$ and
consider its space of smooth functions
$\Gamma(X,\underline{\mathbf{R}})=C^{\infty}(X)$: these are the a priori
fields. The action functional is

(1)
$S_{g}(\varphi)=\frac{-1}{2}\int_{X}\varphi\Delta_{g}\varphi\mathrm{vol}_{g},$

where $\varphi\in C^{\infty}(X)$, $\mathrm{vol}_{g}$ is the volume form
associated to the metric $g$, written in coordinates as $\sqrt{\det
g}dx_{1}\wedge\ldots\wedge dx_{n}$, and the Laplace-Beltrami operator
$\Delta_{g}$ associated to $g$ should not be mistaken for the BV Laplacian
discussed in related literature. The Euler-Lagrange equation here is Laplace’s
equation, $\Delta_{g}\varphi=0$, so that $\mathrm{Crit}(S)$ is the set of
harmonic functions.

By the above, the derived critical locus is then

(2) $\mathscr{F}_{g}=C^{\infty}(X)\xrightarrow{Q_{g}}\mathrm{Dens}(X)[-1],$

where $\mathrm{Dens}(X)$ is the appropriate dual to $C^{\infty}(X)$ and
$Q_{g}\varphi=\Delta_{g}\varphi\mathrm{vol}_{g}$ is the differential, which
imposes the Euler-Lagrange equations: it is written so as to take values in
$\mathrm{Dens}(X)$ but also to capture all of the dependence on
$g\in\mathrm{Met}(X)$ in the action functional. The symplectic structure
$\omega$ on
$\mathscr{F}_{g}=C^{\infty}(X)\xrightarrow{Q_{g}}\textrm{Dens}(X)[-1]$ is

$\omega(\varphi,\mu)=\int_{X}\varphi\mu,$

for $\varphi$ and $\mu$ in degrees 0 and 1, respectively. We can thus write
$S_{g}(\varphi)$ as $\omega(\varphi,Q_{g}\varphi)$.

For $\mathscr{F}_{g}=C^{\infty}(X)\xrightarrow{Q_{g}}\mathrm{Dens}(X)[-1]$,
the underlying graded ring of $\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g})$
is $\mathscr{O}(\mathscr{F}_{g})$, so that it is concentrated in nonpositive
degrees, as Definition 2.2 implies. The action functional $S_{g}(\varphi)$
defined in Equation (1) is a degree 0 element of
$\mathscr{O}(\mathscr{F}_{g})$, but also defines a degree 1 differential on
$\mathscr{O}(\mathscr{F}_{g})$ as $\\{S_{g},-\\}$: thus, $\\{S_{g},S_{g}\\}$
must be a degree 1 element of $\mathscr{O}(\mathscr{F}_{g})$. Since in this
example $\mathscr{O}(\mathscr{F}_{g})$ is concentrated in nonpositive degrees,
the classical master equation holds vacuously. Thus, the free massless scalar
field with metric background $g$ defines a free BV classical field theory,
since the other requirements are easily satisfied.

######  Remark 2.15.

It is implied here that $g$ is a Riemannian metric, because the associated
partial differential operator is the elliptic Laplace-Beltrami operator. If
$g$ were Lorentzian, then we would instead have the hyperbolic d’Alembertian
$\Box_{g}$. For further details comparing these two regimes for the free
scalar field, one should consult the thorough reference [12].

######  Remark 2.16.

An advantage to shifting from the “ordinary” fields $C^{\infty}(X)$ to the
derived critical locus $\mathscr{F}_{g}$ is that there now is an explicit
dependence in the fields on the metric
$g\in\mathrm{Met}(X)=:\mathscr{M}$.444We will denote $\mathrm{Met}(X)$ as
$\mathscr{M}$ when $X$ is implicit. This will allow us to define a
differential graded vector bundle $\pi:\mathscr{F}\to\mathscr{M}$: the base
space is the space of all (semi-)Riemannian metrics on $X$ and the fibers
$\pi^{-1}(g)=\mathscr{F}_{g}$ are field theories depending on the fixed $g$.
This opens up the possibility of seeing how varying the background metric
effects the field theory. We have such a dg vector bundle only when the theory
is free (i.e. $S$ is quadratic in $\varphi$): for an interacting theory, we
will require the notion of an $L_{\infty}$ algebra and bundles thereof.

######  Definition 2.17.

An $L_{\infty}$ algebra over $R$ is a $\mathbf{Z}$-graded, projective
$R$-module $\mathfrak{g}$ with a sequence of multilinear maps of cohomological
degree $2-n$:

$\ell_{n}:\mathfrak{g}\otimes_{R}\ldots\otimes_{R}\mathfrak{g}\to\mathfrak{g},$

where $n\in\mathbf{N}$, such that all $\ell_{n}$ are (1) graded antisymmetric
and (2) satisfy the $n$-Jacobi rule.555We are sweeping details for this rule
under the rug: Definition A.1.2 in [8] is the whole megillah.

######  Example 2.18.

The most natural example of an $L_{\infty}$ algebra for us comes from encoding
nonlinear partial differential equations: i.e. those associated to an
interacting field theory, with a degree three or higher action functional.

For example, say we want to encode
$\Delta_{g}\varphi+\frac{1}{3!}\varphi^{3}=0$, the Euler Lagrange equation
associated to the action functional

$S_{g}(\varphi)=\frac{-1}{2}\int_{X}\varphi\Delta_{g}\varphi\mathrm{vol}_{g}+\frac{1}{4!}\int_{X}\varphi^{4}\mathrm{vol}_{g}.$

The pertinent $L_{\infty}$ algebra has underlying cochain complex

$\mathscr{L}=C^{\infty}(X)[-1]\to\mathrm{Dens}(X)[-2],$

where the differential is $Q_{g}\varphi=\Delta_{g}\varphi\mathrm{vol}_{g}$ and
the only higher bracket is $\ell_{3}:C^{\infty}(X)^{\otimes
3}\to\mathrm{Dens}(X)$, defined as
$\ell_{3}:\varphi_{1}\otimes\varphi_{2}\otimes\varphi_{3}\mapsto\varphi_{1}\varphi_{2}\varphi_{3}\mathrm{vol}_{g}$.
Letting $(R,\mathfrak{m}_{R})$ be a nilpotent Artinian ring in degree $0$, we
get that $\varphi\in C^{\infty}(X)\otimes\mathfrak{m}_{R}$ satisfies the
Maurer-Cartan equation $\mathscr{L}$ if and only if

$Q_{g}\varphi+\frac{1}{3!}\varphi^{3}\mathrm{vol}_{g}=0,$

which recovers the desired partial differential equation (with values in
densities). Thus we see how an $L_{\infty}$ algebra quantifies how a given
equation fails to be linear (a free theory has only nontrivial $\ell_{1}$, and
so only requires a dg structure to be described). Moreover, $\mathscr{L}$ is
an even more particular object, which we now define.

######  Definition 2.19.

A local $L_{\infty}$ algebra on a manifold $X$ is:
(1) A graded vector bundle $L\to X$, where we denote the sections as
$\mathscr{L}$,
(2) a differential operator $d:\mathscr{L}\to\mathscr{L}$ of cohomological
degree 1 such that $d^{2}=0$, and
(3) a collection of polydifferential operators $\ell_{n}:\mathscr{L}^{\otimes
n}\to\mathscr{L}$ for $n\geq 2$ which are alternating, of cohomological degree
$2-n$, and which make $\mathscr{L}$ an $L_{\infty}$ algebra.
If the local $L_{\infty}$ algebra $(\mathscr{L},d)$ is an elliptic complex, we
call it an elliptic $L_{\infty}$ algebra.

The $L_{\infty}$ algebra $\mathscr{L}$ of our ongoing example is an elliptic
$L_{\infty}$ algebra. One advantage of introducing this notion is the
following definition of observables for perturbative field theories, which we
will employ in Section 4 when discussing formal computations:

######  Definition 2.20 (Definition 5.1.1 in [8]).

The observables with support in the open subset $U$ are the commutative dg
algebra

(3) $\mathrm{Obs}^{\mathrm{cl}}(U):=C^{\bullet}(\mathscr{L}(U)),$

where $C^{\bullet}(\mathscr{L})$ denotes Chevalley-Eilenberg cochains. The
factorization algebra of observables for this classical field theory, denoted
$\mathrm{Obs^{cl}}$, assigns $\mathrm{Obs}^{\mathrm{cl}}(U)$ to each open
$U\subset X$.

######  Remark 2.21.

The computations in Example 2.18 work just fine if we replace the elliptic
operator $\Delta_{g}$ with the hyperbolic wave operator $\Box_{g}$, so it
would be convenient to specify the dynamics of the Lorentzian analogue of
$\varphi^{4}$ theory with an $L_{\infty}$ algebra, too. However, as commentary
in Gwilliam and Rejzner’s paper [12] suggests, comparisons between the
Lorentzian and Riemannian settings get stickier when considering interacting
theories. A precise definition of the correct notion of a hyperbolic
$L_{\infty}$ algebra is presented in the recent paper [2].

######  Remark 2.22.

In Example 2.14 of the free scalar field, both components of the graded space
of fields have an action by the diffeomorphism group of $X$–denoted
$\mathscr{D}$ when unambiguous–via pullback: for $f\in\mathscr{D},\varphi\in
C^{\infty}(X),$ and $\mu\in\textrm{Dens}(X)$,
$f\cdot\varphi=f^{*}\varphi=\varphi\circ f$ and $f\cdot\mu=f^{*}\mu$.
Additionally, $\mathscr{D}$ acts on $\textrm{Met}(X)$ via pullback: $f\cdot
g=f^{*}g$. What is special about this example is that the differential $Q_{g}$
commutes with diffeomorphisms in the following sense:
$f^{*}(Q_{g}\varphi)=f^{*}(\Delta_{g}\varphi\textrm{vol}_{g})=\Delta_{f^{*}g}(f^{*}\varphi)\textrm{vol}_{f^{*}g}=Q_{f^{*}g}(f^{*}\varphi)$.
This result is equivalent to the fact that the Laplacian commutes with
diffeomorphisms777This is computed by expressing
$\Delta_{g}=\mathrm{div}_{g}\mathrm{grad}$, and is done explicitly in notes by
Y. Canzani, available at:
https://www.math.mcgill.ca/toth/spectralgeometry.pdf.. This suggests that if
we parameterize families of free scalar BV theories by $\textrm{Met}(X)$, the
result will be a “$\mathscr{D}$-equivariant bundle”. In fact, we can show how
this can work for interacting theories: but first, we must make precise the
idea of a differential graded equivariant bundle.

###  2.2 Equivariant Vector Bundles

To discuss general covariance, we must first understand what an equivariant
vector bundle is and how to use one to specify a family of field theories
parameterized by semi-Riemannian metrics. Once that is done and we make the
connection with general covariance, we will see how groupoids and stacks
naturally arise in this context and provide additional advantages.

######  Definition 2.23 (Definition 1.5 of [4]).

Let $G$ be a Lie group. A smooth fiber bundle $\pi:E\to M$ is said to be
$G$-equivariant if: (i) both $E$ and $M$ are left $G$-spaces, and (ii)
$\pi:E\to M$ is a $G$-equivariant map. If $E=V$ is a vector bundle, we also
require that for all $g\in G$ and $p\in M$, $g:V_{p}\to V_{g\cdot p}$ is a
linear transformation, where $V_{p}:=\pi^{-1}(p)$.

One must be mindful that within the vector bundle part of this definition is
packaged the information that the fibers $V_{p}$ of $V\to M$ could themselves
be $G$-spaces over fixed point sets; however, if there are no fixed points of
$G$ acting on $M$, we immediately get the following.

######  Theorem 2.24.

For $M$ a smooth $G$-space on which the Lie group $G$ acts freely, there is an
equivalence of categories between vector bundles on $M/G$ and $G$-equivariant
vector bundles:

(4) $\textup{VectBun}(M/G)\xrightarrow{\cong}\textup{VectBun}_{G}(M).$

######  Remark 2.25.

This theorem nicely encapsulates how we might keep track of linear data
parameterized by an underlying space which has some symmetries: we simply
quotient the underlying space out by its symmetries and look at the vector
bundle over that. However, this is where the problem becomes apparent: if
there are any points in $M$ which are stabilized by $G$ or any of its
nontrivial subgroups, then $M/G$ is no longer a smooth manifold at those
points. This makes it more difficult to associate to it any structures which
depended on the differentiability of $M$, like its ring of smooth functions or
sections of certain bundles. Stacks deal with those issues nicely, and provide
an analogous theorem in the case the action $M\operatorname{\
\rotatebox[origin={c}]{90.0}{$\circlearrowleft$}\ }G$ is not free.

There is one more distinction which is significant for this work, which we
will define here: the fibers of the vector bundles we want to consider are
differential graded as well as equivariant.

######  Definition 2.26.

A differential graded vector bundle is a vector bundle $V\to M$ whose fibers
$V_{p}$ are $\mathbf{Z}$-graded vector spaces with a smoothly varying
differential $Q_{p}^{i}:V_{p}^{i}\to V_{p}^{i+1}$.

We will usually abbreviate “differential graded” as “dg”, and such a vector
bundle may sometimes be denoted $V^{\bullet}\to M$ or $(V^{\bullet}\to M,Q)$,
depending on what we would like to emphasize within certain contexts. We have
a similar definition when the fibers are $L_{\infty}$ algebras:

######  Definition 2.27.

A bundle of (elliptic) $L_{\infty}$ algebras is a $\mathbf{Z}$-graded vector
bundle $\pi:(V,\ell)\to M$888We may sometimes omit this notation as a pair if
the $L_{\infty}$ structure is implicit. whose fibers
$(V_{p},\ell^{p}):=\pi^{-1}(p)$ are (elliptic) $L_{\infty}$ algebras, such
that the $L_{\infty}$ structure varies smoothly over $M$.

######  Definition 2.28.

A dg vector bundle $(V^{\bullet}\to M,Q)$ is $G$-equivariant if: (i) each of
the $V^{i}\to M$ is $G$-equivariant in the usual sense and (ii) the action by
$G$ induces a cochain map between fibers, i.e. for $g\in G$ and for
$i\in\mathbf{Z}$, the following square commutes:

${V^{i}_{p}}$${V^{i+1}_{p}}$${V^{i}_{g\cdot p}}$${V^{i+1}_{g\cdot
p}.}$$\scriptstyle{Q_{p}^{i}}$$\scriptstyle{g\cdot}$$\scriptstyle{g\cdot}$$\scriptstyle{Q_{g\cdot
p}^{i}}$

A totally analogous definition holds for $G$-equivariant bundles of
$L_{\infty}$ algebras.

######  Remark 2.29.

It might be the case that the differential (or $L_{\infty}$ structure) does
not depend on $p\in M$ for trivial $V\to M$: in this case, the bundle is still
$G$-equivariant, in a rather trivial way. However, it is easy to find examples
in which there is such a dependence, as this the case for our version of
general covariance.

###  2.3 General Covariance

To state general covariance rigorously in our sense, we must first introduce a
few facts about Fréchet manifolds. The space of metrics on a smooth manifold
and its group of diffeomorphisms are infinite dimensional, so defining vector
bundles or other structures which depend on the differentiability of
$\textrm{Met}(X)$ will require us to consider a special class of manifolds
called Fréchet manifolds, which we now define (much of what we state is
adapted from [22]999Another helpful reference is
https://ncatlab.org/nlab/show/Fr%C3%A9chet+manifold.).

######  Definition 2.30 (Adapted from Definition 1.3 in [22]).

A Fréchet manifold is a Hausdorff topological space with an atlas of
coordinate charts taking values in Fréchet spaces (i.e. complete, Hausdorff,
metrizable, locally convex vector spaces) such that the transition functions
are smooth maps between Fréchet spaces.

######  Example 2.31.

$\Gamma(X,\mathrm{Sym}^{2}(T^{\vee}_{X}))$, of which $\mathrm{Met}(X)$ is an
open submanifold, is a Fréchet manifold if $X$ is compact, and similarly, the
diffeomorphism group $\mathrm{Diff}(X)$ is a Fréchet Lie group as long as $X$
is compact [22]. Thus, we will usually assume that $X$ is compact or even
closed in much of what follows, even though in the Lorentzian case, $X$ is
usually not compact. However, many physically relevant Lorentzian manifolds
are assumed to have the simple form $\Sigma\times\mathbf{R}$, for $\Sigma$ a
spacelike compact submanifold and $\mathbf{R}$ the time direction. This is the
path through which many Riemannian results are translated into the Lorentzian
regime.

######  Definition 2.32 (Adapted from Definition 1.7 of [22]).

The space $\mathrm{Met}(X)=\mathscr{M}$ of all Riemannian metrics on a compact
$X$ is the Fréchet manifold defined as the subspace of
$\Gamma(X,\mathrm{Sym}^{2}(T^{\vee}_{X}))$ consisting of all sections which
are Riemannian metrics on $X$, equipped with the smooth topology of uniform
convergence on compact subsets.

Since $\mathscr{M}$ is a Fréchet manifold and since any space of fields
$\mathscr{F}$ on a compact $X$ for a BV classical field theory is a dg Fréchet
manifold by means of being a dg Fréchet space (by Definition 2.10 and Example
2.4), we can bring to fruition Remark 2.16:

######  Proposition 2.33.

Any BV classical field theory for which the action functional $S$ depends on
the metric $g\in\mathscr{M}$ defines a dg Fréchet vector bundle
$\pi:(\mathscr{F},\\{S,-\\})\to\mathscr{M}$ for a free theory or a dg Fréchet
bundle of $L_{\infty}$ algebras for an interacting theory, with fibers
$\pi^{-1}(g)=(\mathscr{F}_{g},\\{S_{g},-\\})$.

###### Proof.

Because $\mathscr{M}$ is always contractible, the underlying graded vector
bundle is $\mathscr{F}\times\mathscr{M}$, where $\mathscr{F}$ is Fréchet by
Example 2.4. A product of Fréchet manifolds is once again Fréchet, and the
assignment of a dg or $L_{\infty}$ structure is smooth. ∎

The computation in Remark 2.22 along with Proposition 2.33 allow us to state
the following:

######  Lemma 2.34.

For the free scalar BV theory defined in Example 2.14, any diffeomorphism
$f\in\mathscr{D}$ defines a cochain map between fibers of the dg Fréchet
vector bundle $(\mathscr{F},Q)\to\mathscr{M}$:

${\mathscr{F}_{g}=C^{\infty}(X)}$${\mathrm{Dens}(X)[-1]}$${\mathscr{F}_{f^{*}g}=C^{\infty}(X)}$${\mathrm{Dens}(X)[-1],}$$\scriptstyle{Q_{g}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{Q_{f^{*}g}}$

which implies that $(\mathscr{F},Q)\to\mathscr{M}$ is a
$\mathscr{D}$-equivariant differential graded vector bundle.

For the remainder of this article, we will sometimes drop the term “Fréchet”
when it is contextually implied, unless attention is otherwise drawn to it.
This result also implies a significant and useful corollary:

######  Corollary 2.35.

If $g\in\mathscr{M}$ is a fixed point of $f\in\mathscr{D}$ (i.e. if $f$ is an
isometry of $g$) and if $Q_{g}\varphi=0$, then $Q_{g}(f^{*}\varphi)=0$. In
other words, isometries of the metric $g$ act on the space of solutions to
$\Delta_{g}\varphi=0$ (Laplace’s equation).

Of course, this corollary holds for any generally covariant BV field theory:
we bring special attention to it in this case because it is a “gold standard”
result when learning PDE for the first time, and thus serves as a touchstone
for the value of the preceding perspective.

######  Remark 2.36.

The differential $Q_{g}$ of the differential graded scalar fields has a very
clear dependence on the base space $\mathscr{M}$. In fact, as a topological
space, the bundle is trivial, as it is
$(C^{\infty}(X)\oplus\textrm{Dens}(X)[-1])\times\mathscr{M}$: the differential
$Q_{g}$ defines any nontriviality as a differential graded vector bundle.

######  Example 2.37.

Lemma 2.34 holds for a particular case in which the BV theory is both free and
non-perturbative: i.e. the Euler-Lagrange equations are linear in the fields
$\phi\in\mathscr{F}_{g}$ and we are not choosing a fixed solution to perturb
around, so that the observables are polynomial functions of the fields as
opposed to Taylor series.

We will now consider an example of an interacting theory. The bundle
$(\mathscr{L},\\{S,-\\})\to\mathscr{M}$101010Note that the notation has
changed since the perturbative space of fields is
$\mathscr{L}=\mathscr{F}[-1]$. representing the family of theories is no
longer just a dg vector bundle, but a bundle of elliptic $L_{\infty}$ algebras
over $\mathscr{M}$. Heuristically speaking, we will no longer view the family
as a collection of vector spaces varying over $\mathscr{M}$, but rather as a
collection of formal neighborhoods varying over $\mathscr{M}$: although the
underlying graded structure is still a vector bundle, the geometry encoded in
the $L_{\infty}$ structures on distinct fibers implies this shift in
perspective.

Let us return to Example 2.18. Recall that the equation of motion in that
instance is:

$Q_{g}\varphi+\frac{1}{3!}\varphi^{3}\mathrm{vol}_{g}=0.$

If we fix a diffeomorphism $f\in\mathscr{D}$, we see that the Euler-Lagrange
form satisfies:

(5)
$f^{*}(Q_{g}\varphi+\frac{1}{3!}\varphi^{3}\mathrm{vol}_{g})=Q_{f^{*}g}(f^{*}\varphi)+\frac{1}{3!}(f^{*}\varphi)^{3}\mathrm{vol}_{f^{*}g}.$

The equivariance property for the first summand is precisely what is shown in
Lemma 2.34, and the second summand (the interaction term) is equivariant
because polynomial functions of the fields are patently equivariant in this
way. Equation (5) can then be reformulated in terms of the brackets on the
elliptic $L_{\infty}$ algebra of Example 2.18 as:

(6)
$f^{*}\big{(}\ell_{1}^{g}(\varphi)+\frac{1}{3!}\ell_{3}^{g}(\varphi,\varphi,\varphi)\big{)}=\ell_{1}^{f^{*}g}(f^{*}\varphi)+\frac{1}{3!}\ell_{3}^{f^{*}g}(f^{*}\varphi,f^{*}\varphi,f^{*}\varphi),$

where we have included the dependence of the brackets $\ell_{k}$ on the
underlying metric $g\in\mathscr{M}$ as a superscript. The above equation is
the $\mathscr{D}$-equivariance property we desire in the Euler-Lagrange term
which implies that the family of theories defined by $\varphi^{4}$ theory as
in Example 2.18 is generally covariant.

This generalizes naturally to the case in which the interaction term is any
polynomial in $\varphi$ times $\mathrm{vol}_{g}$. In that case,
$\ell_{1}=Q_{g}$ and $\ell_{n}:C^{\infty}(X)[-1]^{\otimes
n}\to\mathrm{Dens}(X)[-2]$ for $n\geq 2$ is:

$\ell_{n}:\varphi_{1}\otimes\ldots\otimes\varphi_{n}\mapsto\lambda_{n}\varphi_{1}\ldots\varphi_{n}\mathrm{vol}_{g},$

where the $\lambda_{n}$ are constants. Similarly to Equation (5), it is quick
to show that:

(7) $f^{*}\big{(}\ell_{1}^{g}(\varphi)+\sum_{n\geq
2}\frac{\lambda_{n}}{n!}\ell_{n}^{g}(\varphi,\ldots,\varphi)\big{)}=\ell_{1}^{f^{*}g}(f^{*}\varphi)+\sum_{n\geq
2}\frac{\lambda_{n}}{n!}\ell_{n}^{f^{*}g}(f^{*}\varphi,\ldots,f^{*}\varphi).$

Thus, any scalar field theory with action functional

$S_{g}(\varphi)=\int_{X}(\frac{-1}{2}\varphi\Delta_{g}\varphi+V(\varphi))\mathrm{vol}_{g},$

where $V(\varphi)$ is a polynomial “potential” in $\varphi$, is generally
covariant.

######  Lemma 2.38.

Let $\pi:(\mathscr{L},\\{S,-\\})\to\mathscr{M}$ be a family of perturbative
Batalin-Vilkovisky classical scalar field theories with polynomial potential.
Any $f\in\mathscr{D}$ defines an $L_{\infty}$ map between fibers of
$\pi:(\mathscr{L},\\{S,-\\})\to\mathscr{M}$:

${\mathscr{L}_{g}=C^{\infty}(X)[-1]}$${\mathrm{Dens}(X)[-2]}$${\mathscr{L}_{f^{*}g}=C^{\infty}(X)[-1]}$${\mathrm{Dens}(X)[-2].}$$\scriptstyle{\ell^{g}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{\ell^{f^{*}g}}$

In other words, $\pi:(\mathscr{L},\\{S,-\\})\to\mathscr{M}$ is a
$\mathscr{D}$-equivariant bundle of $L_{\infty}$ algebras.

An analogous version of Corollary 2.35 holds here, and follows by a nearly
identical computation. We can now state a first definition for general
covariance:

######  Definition 2.39.

Let $\pi:(\mathscr{F},\\{S,-\\})\to\mathscr{M}$ define a family of BV field
theories on $X$ parameterized by the space of metrics on $X$. If it is
$\textrm{Diff}(X)=:\mathscr{D}$-equivariant as a differential graded vector
bundle or as a bundle of $L_{\infty}$ algebras (depending on whether the
theories are free or perturbative/interacting), we call the theory generally
covariant.

######  Remark 2.40.

Field theories which satisfy general covariance are therefore not sensitive to
all of $\mathscr{M}$, but only to the moduli space of metrics modulo
diffeomorphism, $\mathscr{M}/\mathscr{D}$. Although many physically relevant
metrics have many isometries, the coarse quotient $\mathscr{M}/\mathscr{D}$
“forgets them”: the need for a more general concept of a space which remembers
them is desirable, and this is where stacks will become useful.

######  Example 2.41.

A tangible example of $\mathscr{M}/\mathscr{D}$ with such a singular point is
the Riemannian manifold $X=\mathbf{R}^{n}$ along with the flat metric $\eta$.
It is well known that the isometry group of $(\mathbf{R}^{n},\eta)$ is
$O(n)\ltimes\mathbf{R}^{n}$, where the $\mathbf{R}^{n}$ in the semidirect
product is the additive group of translations of $\mathbf{R}^{n}$. In
particular, $O(n)\ltimes\mathbf{R}^{n}$ is a subgroup of
$\mathrm{Diff}(\mathbf{R}^{n})$ which stabilizes
$\eta\in\mathscr{M}(\mathbf{R}^{n})$, meaning that the corresponding point in
the quotient is singular. Moreover, $O(n)\ltimes\mathbf{R}^{n}$ acts on the
space of solutions to any generally covariant theory on
$(\mathbf{R}^{n},\eta)$ . Definition 2.39 therefore “enlarges” our usual idea
of equivalence beyond isometries.

####  2.3.1 An important remark on functoriality

Strictly speaking, the claims of Lemmas 2.34 and 2.38 are true in a broader
sense. Instead of assuming that $f:X\to X$ is a diffeomorphism, we let $f:U\to
X$ be an isometric embedding. In other words, consider the category
$\mathbf{Riem}_{n}$ whose objects are Riemannian $n$-folds and whose morphisms
are isometric embeddings: $f:(U,g^{\prime})\to(X,g)$ so that
$f^{*}g=g^{\prime}$. In the case of the free scalar field in Lemma 2.34, the
commutative square is replaced by

${{C^{\infty}(X)}}$${{\mathrm{Dens}(X)[-1]}}$${{C^{\infty}(U)}}$${{\mathrm{Dens}(U)[-1]},}$$\scriptstyle{f^{*}}$$\scriptstyle{Q_{g^{\prime}}=Q_{f^{*}g}}$$\scriptstyle{Q_{g}}$$\scriptstyle{f^{*}}$

which commutes by the very same computation. This implies that the assignment
of a free BV theory is a contravariant functor
$(\mathscr{F},Q):\mathbf{Riem}_{n}\to\mathbf{dgVect}$ from the site of
Riemannian $n$-folds to the category of cochain complexes. We can call this
more general notion “very general covariance” or keep it simply as “general
covariance”. The computation from Lemma 2.38 implies that the above works out
just as well for interacting theories: in that case, the target must be
$L_{\infty}\mathbf{Alg}$, the category of $L_{\infty}$ algebras.121212We will
stick with the broader category of $L_{\infty}$ algebras for the rest of this
section. This suggests something deeper about the physics: not only are the
computations invariant with respect to coordinates choices, but also “manifold
choices” more broadly.

We can compose the preceding functor with the functor
$L_{\infty}\mathbf{Alg}\to\mathbf{dgAlg}$131313As it stands, the target
category can be $\mathbf{dgCAlg}$ (dg commutative algebras), but we leave it
as is because we may lose commutativity after quantization. which takes an
$L_{\infty}$ algebra $\mathscr{L}$ and outputs its Chevalley-Eilenberg
cochains $C^{\bullet}(\mathscr{L})$. Then the composite functor

(8) $\mathrm{Obs}^{\mathrm{cl}}:\mathbf{Riem}_{n}\to\mathbf{dgAlg}$

is covariant, as indeed it should be if we would like to make a factorization
algebra from it (as is done in [7] and [8]). This is a point of connection
with the definition of covariance presented in [10]. In that work, Fewster
outlines a broad framework to understand the idea of “same physics in all
spacetimes” (SPAS) in which he defines (Definition 3.1) a locally covariant
theory to be a covariant functor
$\mathfrak{A}:\mathbf{BkGrnd}\to\mathbf{Phys}$ from some category of
“background geometries” to an appropriate category of “physical quantities”,
like observables. Our preceding construction clearly falls into this class of
objects.

In the study of Algebraic Quantum Field Theory (AQFT), a common choice for
$\mathfrak{A}$ is

(9) $\mathfrak{A}:\mathbf{Loc}_{n}\to C^{*}\text{-}\mathbf{Alg}.$

$\mathbf{Loc}_{n}$ is the category of oriented, time-oriented, and globally
hyperbolic $n$-dimensional Lorentzian manifolds whose morphisms are isometric
embeddings which respect orientations and time-orientations, and
$C^{*}$-$\mathbf{Alg}$ is the category of $C^{*}$ algebras, to which the
observables of a quantum field theory (usually) belong. The work of this
article pertains to classical observables, and much of the focus is on the
first part of the composite functor $\mathrm{Obs^{cl}}$: but once the full
composition is made, the comparison with AQFT is apparent. Further details
concerning this subject are provided in great detail in [19] (in particular
Section 2.5).

######  Remark 2.42.

To summarize, the contents of this paper are presented for a fixed smooth
manifold $X$, its space of metrics, its diffeomorphism group, and various
fields defined on it because focusing on the “smaller problem” made it easier
to manage the functional analytic constructions presented earlier and invoked
later on.

Because the aforementioned fields and groups are sheaves on $X$, it is already
apparent that all of the work lifts to the level of the slice category
$\mathbf{Riem}_{n}/X$,141414This is a “little site” built from the site
$\mathbf{Riem}_{n}$. Note that once $X$ is fixed, $\mathrm{Diff}(X)$ acts on
this site. whose objects isometrically embed into $X$ and whose morphisms $f$
are specified by commutative triangles

${U}$${V}$${X.}$$\scriptstyle{f}$$\scriptstyle{\iota_{V}}$$\scriptstyle{\iota_{U}}$

From here, it is not a stretch to see that our constructions lift to
$\mathbf{Riem}_{n}$. In particular, this means we have the composition of
functors

(10)
$\mathbf{Riem}_{n}\xrightarrow{\mathscr{F}}L_{\infty}\mathbf{Alg}\xrightarrow{\mathrm{Obs^{cl}}}\mathbf{dgAlg}.$

Even better, since we take for granted Costello and Gwilliam’s result that
$\mathrm{Obs^{cl}}:\mathbf{Disj}_{X}\to\mathbf{dgAlg}$ defines a factorization
algebra for a fixed $X$, the above composition ultimately allows us to state
the following:

######  Proposition 2.43.

Any generally covariant BV field theory $(\mathscr{F},\omega,S)$ defines a
functor

(11) $\mathrm{Obs^{cl}}(-,\mathscr{F}):\mathbf{Riem}_{n}\to\mathbf{dgAlg}$

which constitutes a factorization algebra on the site $\mathbf{Riem}_{n}$.

######  Remark 2.44.

Concretely, once we input some $X\in\mathbf{Riem}_{n}$, the output is a
factorization algebra on $X$. Roughly, a prefactorization algebra
$\mathcal{F}$ on $X$ is a functor which takes disjoint opens $U_{i}$ as
subsets of some larger open $V\subseteq X$ and outputs multiplication maps
$\bigotimes_{i}\mathcal{F}(U_{i})\to\mathcal{F}(V)$. A factorization algebra
is a prefactorization algebra which satisfies a particular (co)descent axiom.
Further details can be found in Sections 3.1 and 6.1 of [7].

For the rest of the paper, we make constructions and compute results within
the framework of stacks and formal stacks: one of the ultimate motivations is
to step back and notice that all of the results hold at the level of
generality specified in this subsection. An eventual goal is to make
connections with other literature on functorial perspectives in field theory.
An example of such literature linking AQFTs and factorization algebras is [3]
(Theorem 4.7 in particular).

###  2.4 Groupoids and Stacks

Stacks provide a wonderful packaging of a quotient space, but before diving
into them, we must quickly review groupoids, which are the cornerstone of the
theory of stacks and allow us to do concrete computations with them. For the
most part, we follow the constructions outlined in [5] and [13].

######  Definition 2.45.

A groupoid $\mathcal{G}$ is a small category in which all arrows are
invertible. Common notation is
$\mathcal{G}=\mathcal{G}_{1}\mathrel{\mathop{\vbox{
\offinterlineskip\halign{\hbox to\dimexpr\@tempdima+1em{#}\cr
0.28127pt{\rightarrowfill\cr\kern 1.50694pt\cr
0.28127pt{\rightarrowfill\cr}}}\limits^{\\!s}_{\\!t}}\mathcal{G}_{0}}}$, where
$\mathcal{G}_{1}$ is the set of arrows and $\mathcal{G}_{0}$ the set of
objects; $s$ sends an arrow to its source object, and $t$ sends it to its
target. Every such $\mathcal{G}$ has an identity map
$e:\mathcal{G}_{0}\to\mathcal{G}_{1}$ sending an object to its identity arrow,
an inverse map $i:\mathcal{G}_{1}\to\mathcal{G}_{1}$ sending an arrow to its
inverse, and a multiplication map
$m:\mathcal{G}_{1}\times_{\mathcal{G}_{0}}\mathcal{G}_{1}\to\mathcal{G}_{1}$
that concatenates arrows. $s,t,e,i,$ and $m$ are called the structure maps of
$\mathcal{G}$.

######  Example 2.46.

A premier example of a groupoid is the action groupoid which can be associated
to any smooth $G$-space $M$. Its set of objects is $\mathcal{G}_{0}=M$ and its
set of arrows is $\mathcal{G}_{1}=M\times G$, so that we can write it as

$M\times G\mathrel{\mathop{\vbox{ \offinterlineskip\halign{\hbox
to\dimexpr\@tempdima+1em{#}\cr 7.00002pt{\rightarrowfill\cr\kern 1.50694pt\cr
7.00002pt{\rightarrowfill\cr}}}\limits^{\\!}_{\\!}}M=:\mathcal{M}_{\mathcal{G}}}}$

In this case, $s(p,g)=p$ and $t(p,g)=g\cdot p$. Common notation for the action
groupoid is $M//G$: the action groupoid is defined as a first step toward
understanding coarse quotients which forget stabilizers or may not even be
smooth, in the sense that the action of $G$ could fix certain points in $M$
and so $M/G$ could be singular.

######  Definition 2.47.

A Lie groupoid $\mathcal{G}=\mathcal{G}_{1}\mathrel{\mathop{\vbox{
\offinterlineskip\halign{\hbox to\dimexpr\@tempdima+1em{#}\cr
0.28127pt{\rightarrowfill\cr\kern 1.50694pt\cr
0.28127pt{\rightarrowfill\cr}}}\limits^{\\!s}_{\\!t}}\mathcal{G}_{0}}}$ is a
groupoid such that both the space of arrows $\mathcal{G}_{1}$ and space of
objects $\mathcal{G}_{0}$ are smooth manifolds, all structure maps are smooth,
and the source and target maps $s,t:\mathcal{G}_{1}\to\mathcal{G}_{0}$ are
surjective submersions. (In other words, a Lie groupoid is a groupoid internal
to the category of smooth manifolds.)

######  Remark 2.48.

Moreover, if $\pi:V\to M$ is a $G$-equivariant vector bundle, then we could
also define its action groupoid $V\times G\mathrel{\mathop{\vbox{
\offinterlineskip\halign{\hbox to\dimexpr\@tempdima+1em{#}\cr
7.00002pt{\rightarrowfill\cr\kern 1.50694pt\cr
7.00002pt{\rightarrowfill\cr}}}\limits^{\\!}_{\\!}}V=:\mathcal{V}_{\mathcal{G}}}}$.
Both $\mathcal{V}_{\mathcal{G}}$ and $\mathcal{M}_{\mathcal{G}}$ are in fact
Lie groupoids, and $\mathcal{V}_{\mathcal{G}}$ is a vector space object over
$\mathcal{M}_{\mathcal{G}}$ in the category of Lie groupoids. Thus, by some
abuse of notation, we can view
$\pi:\mathcal{V}_{\mathcal{G}}\to\mathcal{M}_{\mathcal{G}}$ as a vector
bundle.

To consider the above definitions for infinite dimensional manifolds, we need
to choose the right category: in our case, it is the category of Fréchet
manifolds.

######  Definition 2.49.

A Fréchet Lie groupoid is a groupoid internal to the category of Fréchet
manifolds: in other words, $\mathcal{G}_{1}$ and $\mathcal{G}_{0}$ are Fréchet
manifolds and $s$ and $t$ are smooth maps of Fréchet manifolds. We denote
their associated category as $\mathrm{FrLieGpd}$.

######  Example 2.50.

Combining Proposition 2.33 with the above definition implies that for a
compact smooth manifold $X$,

(12) $\mathrm{Met}(X)\times\mathrm{Diff}(X)\mathrel{\mathop{\vbox{
\offinterlineskip\halign{\hbox to\dimexpr\@tempdima+1em{#}\cr
7.00002pt{\rightarrowfill\cr\kern 1.50694pt\cr
7.00002pt{\rightarrowfill\cr}}}\limits^{\\!}_{\\!}}\mathrm{Met}(X)=:\mathscr{M}//\mathscr{D}}}$

is a Fréchet Lie groupoid. By Definition 2.39 and the preceding remark, any
generally covariant BV field theory constitutes a dg vector bundle or
$L_{\infty}$ bundle of Fréchet Lie groupoids:

(13) $\pi:(\mathscr{F}//\mathscr{D},\\{S,-\\})\to\mathscr{M}//\mathscr{D}.$

######  Remark 2.51.

Groupoids are a cornerstone in the definition of stacks, which are the spaces
which we eventually would like to replace ordinary manifolds with for the
purpose of including quotient data in the space. The soul of the matter lies
within the definition of a prestack, which is motivated by the functor of
points perspective. The difference is that instead of a functor from
$\mathbf{Mfd}^{op}$ (or $\mathbf{CRing}$ for an algebraic geometer) to
$\mathbf{Set}$, the functor lands in $\mathbf{Gpd}$, which contains any
“equivalence data” specific to the model at hand. We will define a stack and
then quickly move onto the key example, to avoid unnecessary generalities.

######  Definition 2.52 (Definition 1.1 in [13]).

A prestack is a 2-functor $\mathfrak{X}:\mathbf{Mfd}^{op}\to\mathbf{Gpd}$.
A prestack $\mathfrak{X}$ over $\mathbf{Mfd}^{op}$ is a stack if for any
$N\in\mathbf{Mfd}^{op}$ and open cover $\\{U_{i}\\}$ of $N$, it satisfies
descent, in other words: (1) Given objects $P_{i}\in\mathfrak{X}(U_{i})$ and
isomorphisms $\varphi_{ij}:P_{i}|_{U_{i}\cap U_{j}}\to P_{j}|_{U_{i}\cap
U_{j}}$ such that $\varphi_{jk}\circ\varphi_{ij}=\varphi_{ik}|_{U_{i}\cap
U_{j}\cap U_{k}}$, there is an object $P\in\mathfrak{X}(N)$ and isomorphisms
$\varphi_{i}:P|_{U_{i}}\to P_{i}$ such that
$\varphi_{ij}=\varphi_{j}\circ\varphi_{i}^{-1}$. This is called effective
descent data.
(2) Given $P,P^{\prime}\in\mathfrak{X}(N)$ and isomorphisms
$\varphi_{i}:P|_{U_{i}}\to P^{\prime}|_{U_{i}}$ such that
$\varphi_{i}|_{U_{i}\cap U_{j}}=\varphi_{j}|_{U_{i}\cap U_{j}}$, there is a
unique map $\varphi:P\to P^{\prime}$ such that $\varphi_{i}=\varphi|_{U_{i}}$.

######  Remark 2.53.

As we have defined it, the above is actually a differentiable stack; however,
because this is the only kind of stack we need, we usually drop the adjective.

######  Example 2.54.

A fundamental example of a stack over $\mathbf{Mfd}^{op}$ is an ordinary
manifold. For such a manifold $M$, we can define the stack $\underline{M}$ as
$\underline{M}(N):=\mathrm{Map}(N,M)=C^{\infty}(N,M)$ for $N\in\mathbf{Mfd}$.
This stack is presented by the “discrete groupoid”
$M\times\\{1\\}\mathrel{\mathop{\vbox{ \offinterlineskip\halign{\hbox
to\dimexpr\@tempdima+1em{#}\cr 7.00002pt{\rightarrowfill\cr\kern 1.50694pt\cr
7.00002pt{\rightarrowfill\cr}}}\limits^{\\!}_{\\!}}M}}$, whose objects are the
points of $M$ and the only morphisms are the identities for those points. This
embeds $\mathbf{Mfd}$ into the category $\mathbf{Stk}$ of differentiable
stacks. We also get the following essential lemma.

######  Theorem 2.55 (Yoneda Lemma for Stacks: Lemma 1.3 in [13]).

Let $\mathfrak{X}$ be a stack and let $M$ be a manifold. We have the following
equivalence of categories:

$\mathfrak{X}(M)\cong\mathrm{Mor}_{\mathbf{Stk}}(\underline{M},\mathfrak{X}).$

Since stacks are designed to generalize the notion of an ordinary manifold,
there should be an analogous notion of an atlas for stacks.161616One could
even define stacks by first constructing atlases, as is done for manifolds. We
may sometimes denote $\underline{M}$ simply as $M$, when it’s implicit in
context.

######  Definition 2.56.

An atlas (or covering) for a stack $\mathfrak{X}$ is a manifold $X$ and map
$p:X\to\mathfrak{X}$ such that (1) for any manifold $Y$ and
$Y\to\mathfrak{X}$, the stack $X\times_{\mathfrak{X}}Y$ is a manifold, and (2)
$p$ is a submersion, i.e. for all $Y\to\mathfrak{X}$, the projection
$Y\times_{\mathfrak{X}}X\to Y$ is a submersion.

We now specify the most important example of a stack for this paper.

######  Definition 2.57.

Given a smooth $G$-manifold $M$, the associated quotient stack is the functor
$[M/G]:\mathbf{Mfd}^{op}\to\mathbf{Gpd}$ such that the objects of $[M/G](N)$
are pairs $(P\xrightarrow{\pi}N,P\xrightarrow{\alpha}M)$, $\pi$ being a
principal $G$-bundle over $N$ and $\alpha$ being a $G$-equivariant map, and
the arrows are isomorphisms of principal $G$-bundles over $N$ commuting with
the $G$-equivariant maps to $M$.

Note that $[M/G]$ evaluated on a point recovers $M//G$, so that $[M/G]$
rightly gives a natural generalization of $M//G$. An atlas for $[M/G]$ is
$M\to[M/G]$. Much like how we use atlases to define principal and vector
bundles over an ordinary manifold, we use atlases to define such bundles over
stacks, as follows.

######  Definition 2.58.

A principal $G$-bundle $\mathscr{P}\to\mathfrak{X}$ is given by a $G$-bundle
$\mathscr{P}_{X}$ over an atlas $X\to\mathscr{P}$ with an isomorphism of the
two pullbacks
$p_{1}^{*}\mathscr{P}_{X}\xrightarrow{\simeq}p_{2}^{*}\mathscr{P}_{X}$ from
$X\times_{\mathfrak{X}}X\to X$ satisfying the cocycle condition on
$X\times_{\mathfrak{X}}X\times_{\mathfrak{X}}X$.

######  Remark 2.59.

The definition of a vector bundle over a stack $\mathfrak{X}$ is completely
analogous to this. Of course, one could instead invoke that a vector bundle
$V\to\mathfrak{X}$ of rank $n$ is equivalent to a principal
$GL(n,\mathbf{R})$-bundle and then use the preceding definition, anyway.

######  Example 2.60.

An essential example derived from the above definitions is that of
$[\mathrm{pt}/G]$, for $\mathrm{pt}$ a point. Applying the definition shows
that $[\mathrm{pt}/G](X)$ is precisely $\textrm{Bun}_{G}(X)$, the category of
principal $G$-bundles over $X$ (any morphism of $G$-bundles over the same base
space is necessarily an isomorphism). Because of this, it is common to
identify $[\mathrm{pt}/G]$ with $BG$, since $[X,BG]$ is equivalent to
$\textrm{Bun}_{G}(X)$ modulo bundle ismomorphisms.

Notice moreover that defining a vector bundle $V\to[\mathrm{pt}/G]$ amounts to
fixing the vector space $V$ (a vector bundle over the point) as well as a
representation $\rho:G\to\mathrm{End}(V)$. In other words, we have an
equivalence of categories:

$\mathrm{VectBun}([\mathrm{pt}/G])\cong\mathrm{Rep}(G).$

######  Remark 2.61.

The preceding example is a simple but beautiful illustration of how specifying
a vector bundle $V$ over a quotient stack $[M/G]$ is equivalent to specifying
a $G$-equivariant vector bundle over a $G$-manifold $M$. We thus get the
following fact.

######  Theorem 2.62 (Adapted from Example 4.5 in [13]).

For $M$ a smooth $G$-space, we have the following equivalence of categories:

$\mathrm{VectBun}([M/G])\cong\mathrm{VectBun}_{G}(M).$

This is stated in [13] for cartesian sheaves on $[M/G]$ and we are
representing a vector bundle by its space of sections to deduce the above, so
the statement in [13] holds for a larger class of objects. Additionally,
although it is outside the scope of this paper, it is worth mentioning that a
quotient stack $[M/G]$ contains “all possible ways” in which it could have
been defined starting with an action groupoid: more than one groupoid could
present a stack (this is described and expounded on in [5]), so it is
reassuring that the stack itself contains this data.

A deeper level of care must be taken for our motivating example
$[\textrm{Met}(X)/\textrm{Diff}(X)]=[\mathscr{M/D}]$, which is presented by
the Fréchet Lie groupoid $\mathscr{M//D}$. Smooth maps from an ordinary finite
dimensional manifold to a Fréchet manifold are well-defined, and so the
associated maps from one to the other when viewed as their respective discrete
groupoids are also well-defined. This allows us to formulate the following
definition:

######  Definition 2.63.

For a compact manifold $X$, let
$[\textrm{Met}(X)/\textrm{Diff}(X)]=[\mathscr{M/D}]:\mathbf{Mfd}^{op}\to\mathbf{Gpd}$
be the functor such that the objects of $[\mathscr{M/D}](N)$ are pairs
$(P\xrightarrow{\pi}N,P\xrightarrow{\alpha}\mathscr{M})$, $\pi$ being a
principal $\mathscr{D}$-bundle over $N$ and $\alpha$ being a
$\mathscr{D}$-equivariant map, and the arrows are isomorphisms of principal
$\mathscr{D}$-bundles over $N$ commuting with the $\mathscr{D}$-equivariant
maps to $\mathscr{M}$: $[\mathscr{M/D}]$ is the moduli stack of metrics modulo
diffeomorphism.

######  Lemma 2.64.

For a compact manifold $X$, the Fréchet Lie groupoid $\mathscr{M//D}$ presents
the Fréchet moduli stack $[\mathscr{M/D}]$.

######  Remark 2.65.

Definition 2.63 is stated in the “ordinary” sense, so that we don’t specify
the Fréchet nature of the manifolds. Then, Lemma 2.64 implies that
$[\mathscr{M/D}]$ is represented via the canonical functor
$\mathbf{FrGpd}\to\mathbf{Stk}$ sending a Fréchet Lie groupoid to its
associated differentiable stack. The very detailed paper [20]–in particular
Sections 2 and 5–provides additional details and examples for these
definitions, and is what we primarily relied on above.

In light of Definition 2.39 and Theorem 2.62 as well as the preceding
definition, we have:

######  Proposition 2.66.

Any generally covariant family $\pi:(\mathscr{F},\\{S,-\\})\to\mathscr{M}$ of
BV field theories descends to a Fréchet dg vector bundle or $L_{\infty}$
bundle of stacks:

(14) $\pi:([\mathscr{F/D}],\\{S,-\\})\to[\mathscr{M/D}].$

######  Remark 2.67.

Conversely, any such bundle defines a generally covariant theory: in this
sense, Proposition 2.66 can be taken as the definition of a generally
covariant theory. Moreover, we mindfully dropped the notation of a fixed
smooth manifold $X$ in the statement of this proposition: in the long run, we
would like to better understand what kind of functor
$([\mathscr{F/D}],\\{S,-\\})\to[\mathscr{M/D}]$ constitutes from
$\mathbf{Riem}_{n}$ to the category of $L_{\infty}$ bundles over stacks. More
will be said on this in the following example.

######  Example 2.68 (Perturbative Yang-Mills Theory).

The advantages of the stacky formulation of general covariance may be more
convincing when considering theories which have more data involved; e.g. those
with local symmetries. As an example, let us consider Yang-Mills theory: to
begin, let $(X,g)$ be an oriented, $n$-dimensional Riemannian manifold, and
let $G$ be a compact Lie group whose Lie algebra $\mathfrak{g}$ has a
nondegenerate invariant pairing, $\langle-,-\rangle_{\mathfrak{g}}$. To
minimize any topological complications, fix a trivial principal $G$-bundle
$P\to X$.

In this instance, the fields for Yang-Mills theory are the connection one-
forms $A\in\Omega^{1}(X,\mathfrak{g})=\Omega^{1}(X)\otimes\mathfrak{g}$
associated to $P$, which constitute an affine Fréchet space. To such a field,
we can associate its curvature form
$F_{A}:=dA+\frac{1}{2}[A,A]\in\Omega^{2}(X,\mathfrak{g})$. Letting
$\langle-,-\rangle$ denote the pairing on $\Omega^{\bullet}(X,\mathfrak{g})$
defined by

(15) $\langle\omega_{1}\otimes E_{1},\omega_{2}\otimes
E_{2}\rangle:=\int_{X}\omega_{1}\wedge\omega_{2}\langle
E_{1},E_{2}\rangle_{\mathfrak{g}},$

the Yang-Mills action functional can be written as

(16) $S_{YM}(A)=\frac{1}{2}\langle F_{A},\star F_{A}\rangle,$

where $\star$ denotes the Hodge star operator. The Euler-Lagrange equations
for this action are

(17) $d_{A}\star F_{A}=0,$

where $d_{A}=d+A$ is the exterior covariant derivative associated to $A$.
Alternatively, this can be written as $(d_{A}\star d_{A})A=0$.

To move toward the derived-geometric set up in the BV formalism, we must also
consider that there is an action of the gauge group $C^{\infty}(X,G)$ on the
fields $\Omega^{1}(X,\mathfrak{g})$ defined such that for $g\in
C^{\infty}(X,G)$, $g\cdot A$ is $A^{g}:=g^{-1}Ag+g^{-1}dg$. $S_{YM}(A)$ is
invariant under this action, and so the Yang-Mills equations are covariant
with respect to it. Moreover, the infinitesimal gauge action is: for
$\alpha\in C^{\infty}(X)\otimes\mathfrak{g}=\Omega^{0}(X,\mathfrak{g})$,
$A\mapsto d_{A}\alpha=d\alpha+[A,\alpha]\in
T_{A}\Omega^{1}(X,\mathfrak{g})\cong\Omega^{1}(X,\mathfrak{g})$, the tangent
space to the space of connection one-forms at $A$. This action suggests that
we consider the tangent complex181818This will be defined precisely later, in
Definition 3.2. to $A$ as a point in the stack of connections modulo gauge:

(18)
$\mathbf{T}_{A}[\Omega^{1}(X,\mathfrak{g})/\Omega^{0}(X,\mathfrak{g})]\cong\Omega^{0}(X,\mathfrak{g})[1]\xrightarrow{d_{A}}\Omega^{1}(X,\mathfrak{g}).$

We can begin to define a BV theory for perturbative Yang-Mills about a fixed
solution $A$ by computing the $-1$-shifted cotangent bundle of the above:

(19)
$\Omega^{0}(X,\mathfrak{g})[1]\xrightarrow{d_{A}}\Omega^{1}(X,\mathfrak{g})\xrightarrow{d_{A}\star_{g}d_{A}}\Omega^{n-1}(X,\mathfrak{g})[-1]\xrightarrow{d_{A}}\Omega^{n}(X,\mathfrak{g})[-2]=:\mathscr{E}_{(g,A)}.$

The shifted symplectic pairing comes from (15) and the differential between
$\Omega^{1}(X,\mathfrak{g})$ and $\Omega^{n-1}(X,\mathfrak{g})$ comes from the
equations of motion (17). Also, we are being pedantic in that we are labeling
the Hodge star with the metric used to define it.

######  Remark 2.69.

There is a dependence on the metric in the middle differential (the Yang-Mills
term), but we could in principle compute whether or not the entire complex is
diffeomorphism equivariant: this amounts to checking whether or not the
infinitesimal gauge invariance–described by the differential $d_{A}$ between
$\Omega^{0}(X,\mathfrak{g})[1]$ and $\Omega^{1}(X,\mathfrak{g})$ and also
between $\Omega^{n-1}(X,\mathfrak{g})[-1]$ and
$\Omega^{n}(X,\mathfrak{g})[-2]$–is also diffeomorphism equivariant. Put
plainly, showing the diffeomorphism equivariance of $\mathscr{E}_{(g,A)}$
proves that perturbative Yang-Mills theory is generally covariant as a theory
which also depends on connections modulo (infinitesimal) gauge.

To show that $\mathscr{E}_{(g,A)}$ is $\mathscr{D}$-equivariant, we must show
that the diagram

${\Omega^{0}(X,\mathfrak{g})[1]}$${\Omega^{1}(X,\mathfrak{g})}$${\Omega^{n-1}(X,\mathfrak{g})[-1]}$${\Omega^{n}(X,\mathfrak{g})[-2]}$${\Omega^{0}(X,\mathfrak{g})[1]}$${\Omega^{1}(X,\mathfrak{g})}$${\Omega^{n-1}(X,\mathfrak{g})[-1]}$${\Omega^{n}(X,\mathfrak{g})[-2]}$$\scriptstyle{d_{A}}$$\scriptstyle{d_{A}\star_{g}d_{A}}$$\scriptstyle{d_{A}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{d_{f^{*}A}}$$\scriptstyle{d_{f^{*}A}\star_{f^{*}g}d_{f^{*}A}}$$\scriptstyle{d_{f^{*}A}}$

commutes, for any diffeomorphism $f\in\mathscr{D}$. Notice that in the lower
complex, the Hodge star is defined by the metric $f^{*}g$ and the fixed
connection form is $f^{*}A$. To begin, let
$\alpha\in\Omega^{0}(X,\mathfrak{g})[1]$. We get

(20)
$f^{*}(d_{A}\alpha)=f^{*}(d\alpha+A\wedge\alpha)=d(f^{*}\alpha)+f^{*}A\wedge
f^{*}\alpha,$

because the exterior derivative $d$ is manifestly covariant and pullbacks
commute with wedge products, even if the forms have $\mathfrak{g}$
coefficients: this equation is then equal to
$(d+f^{*}A)(f^{*}\alpha)=d_{f^{*}A}(f^{*}\alpha)$, which proves that the first
square commutes. Moreover, this same computation shows that the last square
commutes, too.

Next, let $\omega\in\Omega^{1}(X,\mathfrak{g})$. Then:

(21) $\displaystyle(d_{A}\star_{g}d_{A})\omega$
$\displaystyle=(d_{A}\star_{g}(d\omega+A\wedge\omega))$ (22)
$\displaystyle=(d+A)(\star_{g}d\omega+\star_{g}(A\wedge\omega))$ (23)
$\displaystyle=(d\star_{g}d\omega+d\star_{g}(A\wedge\omega)+A\wedge\star_{g}d\omega+A\wedge\star_{g}(A\wedge\omega)).$

Before we consider the diffeomorphism action, notice that the $L_{\infty}$
structure can be read off from the last expression. Pulling the above back
along $f\in\mathscr{D}$ results in

(24)
$f^{*}((d_{A}\star_{g}d_{A})\omega)=f^{*}(d\star_{g}d\omega)+f^{*}(d\star_{g}(A\wedge\omega))+f^{*}(A\wedge\star_{g}d\omega)+f^{*}(A\wedge\star_{g}(A\wedge\omega)),$

which, when considering that the pullback commutes with the wedge product and
the manifest covariance of the Hodge star, is equal to

(25) $d\star_{f^{*}g}d(f^{*}\omega)+d\star_{f^{*}g}(f^{*}A\wedge
f^{*}\omega)+f^{*}A\wedge\star_{f^{*}g}d(f^{*}\omega)+f^{*}A\wedge\star_{f^{*}g}(f^{*}A\wedge
f^{*}\omega).$

From here, we see that this is equal to
$(d+f^{*}A)\star_{f^{*}g}(d+f^{*}A)(f^{*}\omega)$, which is what we wanted. We
have therefore shown the following:

######  Theorem 2.70.

The bundle of $L_{\infty}$ algebras
$\mathscr{E}(X)\to\mathrm{Met}(X)\times\Omega^{1}(X,\mathfrak{g})$ with fibers

$\mathscr{E}_{(g,A)}(X)=\Omega^{0}(X,\mathfrak{g})[1]\xrightarrow{d_{A}}\Omega^{1}(X,\mathfrak{g})\xrightarrow{d_{A}\star_{g}d_{A}}\Omega^{n-1}(X,\mathfrak{g})[-1]\xrightarrow{d_{A}}\Omega^{n}(X,\mathfrak{g})[-2],$

is $\mathrm{Diff}(X)$-equivariant. In other words, perturbative Yang-Mills
theory is generally covariant, and the above bundle descends to a bundle of
stacks:

(26)
$\mathscr{E}(X)\to[(\mathrm{Met}(X)\times\Omega^{1}(X,\mathfrak{g}))/\mathrm{Diff}(X)].$

######  Remark 2.71.

We would like to remind the reader that in the style of Subsection 2.3.1, we
can abandon the specific choice of $X$ in (26) and the result is a functor
from $\mathbf{Riem}_{n}$ to the category of bundles of $L_{\infty}$ algebras
over stacks. In this case, what replaces the post-composed functor
$\mathrm{Obs^{cl}}:\mathbf{Riem}_{n}\to\mathbf{dgAlg}$ of Proposition 2.43? To
be more specific, what happens when we input a bundle of $L_{\infty}$ algebras
over a stack to output something in $\mathbf{dgAlg}$ instead of a lone
$L_{\infty}$ algebra? It is of great interest to elaborate more on these
functors in future work.

######  Remark 2.72.

Each of the fibers $\mathscr{E}_{(g,A)}$ describes the formal neighborhood of
a solution $A$ to the Yang-Mills equations as an element of the stack
$[\Omega^{1}(X,\mathfrak{g})/C^{\infty}(X,G)]$ of connections modulo gauge
transformation, and with background metric $g\in\mathscr{M}$: in this sense,
we can view the preceding bundle as parameterizing formal stacks describing
solutions to Yang-Mills modulo gauge living over the stack of metrics modulo
diffeomorphism. More on formal neighborhoods in
$[\Omega^{1}(X,\mathfrak{g})/C^{\infty}(X,G)]$ can be found in the paper [9]
on spontaneous symmetry breaking.

However, we should note that the preceding equivariance computations work out
perfectly well if we don’t treat them perturbatively: after all, the Yang-
Mills term in (17) is diffeomorphism equivariant with respect to both
connection one-forms and metrics. The caveat is that by using $L_{\infty}$
algebras, we are implictly invoking Theorem 2.0.2 in [15], in which the
correspondence between $L_{\infty}$ algebras and formal moduli spaces is
specified; however, if we formally substitute $\omega=A$ in the above
computations, the equivariance property holds for what is evidently the
nonperturbative case. I am interested to see how this can be remedied further
to have a globalized version of Theorem 2.70.

######  Remark 2.73.

Theorem 2.70 states a version of general covariance in which additional
physical fields are inextricably linked to $\mathrm{Met}(X)$ in the moduli
stack. Indeed, any tensor field (even when taking values in some Lie algebra,
for example) is defined with regard to its behavior under diffeomorphisms. So
then why do we state general covariance in terms of metrics modulo
diffeomorphism? The answer is that in the development of the theory of general
relativity, a key observation was Einstein’s equivalence principle.

In general relativity, metrics represent the dynamical variables of the
gravitational field, but any freely falling observer in a gravitational field
can choose coordinates so that they are in an inertial frame: in other words,
any metric can be altered by some diffeomorphism to be locally Euclidean or
Lorentzian. In this sense, metrics and diffeomorphisms are intimately related
when specifying gravitational dynamics, and so we use the associated stack as
a baseline for quantifying general covariance. Further details are provided in
Section 3 of [16].

##  3 Formal Stacks

In Section 4, we will make a connection to the version of the classical
Noether’s Theorem as described in Chapter 12 of [8]. However, we must first
cross the bridge from the world of global stacks as we defined them in Section
2.4 to the world of formal moduli spaces, which are examples of formal stacks.
A key step is to associate to a (differential graded) equivariant vector
bundle a vector bundle over a formal moduli space:191919Formal moduli spaces
are alternatively named “formal moduli problems”. in our case, this formal
moduli space is a formal neighborhood in a quotient stack. In Section 4, this
formal moduli space will be the formal neighborhood of a fixed metric in the
moduli stack of metrics modulo diffeomorphism.

###  3.1 Tangent Complexes

The goal of the next portion is to understand what object we can associate to
a point in a stack that plays the analogous role of a tangent space to an
ordinary manifold. These “tangent complexes” are necessary to compute function
rings on formal neighborhoods in stacks, making them locally ringed spaces.

######  Construction 3.1.

Let $M$ be a smooth $G$-space and let $\textrm{Stab}(p)\subseteq G$ be the
stabilizer subgroup of $p\in M$. The $G$-orbit of $p$ thus looks like a copy
of $G/\textrm{Stab}(p)$ lying in $M$. If we consider the map $t_{p}:G\to M$
defined as $t_{p}(g)=g\cdot p$,202020$t_{p}$ is in fact the target map for the
Lie groupoid $M\times G\rightrightarrows M$ with $p\in M$ fixed in $M\times
G$. then its differential $dt_{p}$ can be used to define a 2-term cochain
complex of vector spaces:

(27) $0\to\mathfrak{g}[1]\xrightarrow{dt_{p}}T_{p}M\to
0=:\mathbf{T}_{p}[M/G],$

where $\mathfrak{g}$ is in cohomological degree $-1$ and $T_{p}M$ is in degree
$0$. Alternative notation is $\mathbf{T}_{p}[M/G]=(\mathfrak{g}[1]\oplus
T_{p}M,dt_{p})$. Note that $\textrm{Stab}(p)$ could be discrete here, although
that is not seen in $\mathbf{T}_{p}[M/G]$. We can also compute

$\textrm{ker}(dt_{p})=H^{-1}(\mathbf{T}_{p}[M/G])=\textrm{Lie(Stab}(p)).$

Thus, if $H^{-1}(\mathbf{T}_{p}[M/G])=0$, then the coarse quotient $M/G$ is an
ordinary manifold at that point, since the action is free nearby it.
$H^{0}(\mathbf{T}_{p}[M/G])$ is the quotient of $T_{p}M$ by
$\textrm{im}(dt_{p})$: it is the usual tangent space of the coarse quotient at
points $p\in M$ where the action is free. As it turns out, this is exactly the
tangent object we are looking for, as the notation suggests: a precise
statement and further details are wonderfully detailed in [1].

######  Proposition 3.2.

The tangent complex to the quotient stack $[M/G]$ at a point $[p]$ is exactly
$\mathbf{T}_{p}[M/G]$ as defined in equation (27).

This inspired the saying that “smooth stacks are geometric spaces whose
tangent spaces are complexes concentrated in nonpositive cohomological
degree”. In the case of quotient stacks, we’re lucky to have a concrete way of
realizing their associated tangent complexes.

######  Remark 3.3.

If we take the union of all the complexes $\mathbf{T}_{p}[M/G]$ over all $p\in
M$, we get a complex of vector bundles over $M$:

$0\to\underline{\mathfrak{g}}\xrightarrow{dt}TM\to 0,$

where $\underline{\mathfrak{g}}=M\times\mathfrak{g}$, considering that the
base space $M$ is implicit. $\underline{\mathfrak{g}}$ is called the Lie
algebroid associated to the action Lie groupoid $M//G$, and $dt$ is called the
anchor map of the Lie algebroid. This is a primordial example of a Lie
algebroid.

######  Example 3.4.

Consider the natural action of the group of diffeomorphisms $\mathscr{D}$ of a
manifold $X$ on the space of Riemannian metrics $\mathscr{M}$:
$t_{g}(f)=f^{*}g$. According to [14], the Lie algebra of $\mathscr{D}$ at the
identity diffeomorphism is $\mathrm{Vect}(X)$, the set of vector fields: this
will be the degree $-1$ part of our tangent complex.

We know that $T_{g}\mathscr{M}\cong\Gamma(X,\textrm{Sym}^{2}(T_{X}^{\vee}))$,
so that we can compute

(28)
$\mathbf{T}_{g}[\mathscr{M}/\mathscr{D}]=(\Gamma(X,T_{X})[1]\oplus\Gamma(X,\textrm{Sym}^{2}(T_{X}^{\vee})),dt_{g}).$

Then, given $V\in\Gamma(T_{X})$, $dt_{g}(V)=L_{V}g$, where $L_{V}g$ is the Lie
derivative of $g$ along $V$: one can see this by considering the one-parameter
family of diffeomorphisms $f=\textrm{exp}(tV)$–i.e. letting $V$ be the
infinitesimal generator of $f$–and computing the derivative at $t=0$ of the
action of $f$ on $g$. Not all diffeomorphisms can be written this way: after
all, $\mathscr{D}$ is not even a simply connected Lie group. Even worse, there
are diffeomorphisms which are infinitesimally close to the identity
diffeomorphism which cannot be written as $\textrm{exp}(tV)$ for some $V$
[14]; however, we need not worry about this in what is to come, as will be
explained in Section 4.

######  Remark 3.5.

We will now show the relevance of the above for field theories by introducing
notation and a lemma: we will perform the relevant computations for the
example of the $\mathscr{D}$-equivariant differential graded vector bundle
$(\mathscr{F},Q)\to\mathscr{M}$ with fibers
$\mathscr{F}_{g}=C^{\infty}(X)\xrightarrow{Q_{g}}\textrm{Dens}(X)[-1]$ and
differential $Q_{g}\varphi=\Delta_{g}\varphi\mathrm{vol}_{g}$. let us call the
actions described in Lemma 2.34
$\tau_{\mathscr{M}}:\mathscr{D}\to\mathrm{Diff}(\mathscr{M})$ and
$\tau_{\mathscr{F}}:\mathscr{D}\to\mathrm{Diff}(\mathscr{F})$. There is also
an “action” on the differential, sending $Q_{g}$ to $Q_{f^{*}g}$ for
$f\in\mathscr{D}$: it clearly comes from the action of $\mathscr{D}$ on
$\mathscr{M}$, but also nicely intertwines with the input and output of the
differential, as described in general covariance.

To get the infinitesimal version of these actions we use computations similar
to those in Example 3.4, keeping in mind that
$\mathrm{Lie}(\mathscr{D})\cong\mathrm{Vect}(X)$. The map
$\tau_{\mathscr{M}_{g}}:\mathrm{Vect}(X)\to T_{g}\mathscr{M}$ is what we
already considered earlier, namely $V\mapsto L_{V}g$, and the fibers are
similar:

######  Lemma 3.6.

The action of $\mathscr{D}$ on the underlying graded vector space of
$\mathscr{F}_{g}$ defines an action of $\mathrm{Vect}(X)$ on
$\mathscr{F}_{g}$. It has a degree 0 part
$\tau_{\mathscr{F}_{g}}:\mathrm{Vect}(X)\to T_{\varphi}C^{\infty}(X)\cong
C^{\infty}(X)$ and a degree 1 part $\tau_{\mathscr{F}_{g}}:\mathrm{Vect}(X)\to
T_{\mu}\mathrm{Dens}(X)\cong\mathrm{Dens}(X)$; they are, respectively,
$V\mapsto L_{V}\varphi$ and $V\mapsto L_{V}\mu$.

Although we provided this example for clarity, such a lemma holds for any
generally covariant BV field theory as we defined it in Definition 2.39, since
tangent vectors can be defined for a Fréchet manifold by means of it being
locally modeled by Fréchet spaces. The most interesting and physically
relevant detail which must be addressed is what happens to the differential
$\\{S_{g},-\\}$ under this infinitesimal action: this will be the content of
Section 4.

###  3.2 Chevalley-Eilenberg Cochains as Rings of Functions

We start with an action of a finite dimensional Lie group $G$ on a finite
dimensional manifold $M$, and then specialize to the case of
$M=\mathbf{R}^{n}$ to consider some concrete computations. In the example of
diffeomorphisms of a manifold $X$ acting on the space of Riemannian metrics on
$X$, $\textrm{Met}(X)=\mathscr{M}$ is a convex cone in
$\Gamma(X,\textrm{Sym}^{2}(T^{\vee}X))$, so that we will be eventually
specializing these constructions to vector spaces or “nice” subsets thereof
anyway.

######  Construction 3.7.

Let $\widehat{M}_{p}$ denote the formal neighborhood of $p\in M$, defined so
that its ring of functions $\mathscr{O}(\widehat{M}_{p})$ is the jets of
$\mathscr{O}(M):=C^{\infty}(M)$ at $p$, and denote the inclusion map
$\hat{p}:\widehat{M}_{p}\to M$: this is equivalent to the restriction map
$\mathscr{O}(M)\to\mathscr{O}(\widehat{M}_{p})$. It is known that
$\mathscr{O}(\widehat{M}_{p})\cong\widehat{\textrm{Sym}}(T_{p}^{\vee}M)$, the
Taylor series ring around $p\in M$, although this isomorphism is not
canonical. We will use the latter, and call the Taylor series ring
$\widehat{\mathscr{O}}_{p}$ when unambiguous.

The action of $G$ on $M$ is defined by a map $P:G\to\textrm{Diff}(M)$. Taking
its total derivative gives us a map $\rho:\mathfrak{g}\to\textrm{Vect}(M)$ of
Lie algebras, where we choose to view $\textrm{Vect}(M)$ as derivations of
$\mathscr{O}(M)$. We then restrict the action of $\textrm{Vect}(M)$ on
$C^{\infty}(M)$ to get an action of $\textrm{Vect}(\widehat{M}_{p})$ on
$C^{\infty}(\widehat{M}_{p})\cong\widehat{\mathscr{O}}_{p}$. The differential
on $\mathbf{T}_{p}[M/G]$ encodes $\rho:\mathfrak{g}\to\textrm{Vect}(M)$ at the
point $p$ and thus on the formal neighborhood $\widehat{M}_{p}$ of $p$ since
$\rho$ is a map of Lie algebras: this gives us
$\mathfrak{g}\to\textrm{Vect}(\widehat{M}_{p})$. Noting that
$\textrm{Vect}(\widehat{M}_{p})\cong\textrm{Der}(\widehat{\mathscr{O}}_{p})$
recovers the action of $\mathfrak{g}$ on $\widehat{\mathscr{O}}_{p}$ via
derivations, this allows us to define
$C^{\bullet}(\mathfrak{g},\widehat{\mathscr{O}}_{p})\cong
C^{\bullet}(\mathfrak{g})\otimes\widehat{\mathscr{O}}_{p}$ in the traditional
way.

######  Lemma 3.8.

Chevalley-Eilenberg (CE) cochains of the differential graded Lie algebra
defined by shifting $\mathbf{T}_{p}[M/G]$ up one degree, denoted
$C^{\bullet}(\mathfrak{g}\xrightarrow{dt_{p}}T_{p}M[-1])$, and
$C^{\bullet}(\mathfrak{g},\widehat{\mathscr{O}}_{p})$ are isomorphic as
differential graded commutative algebras. Moreover,
$C^{\bullet}(\mathfrak{g},\widehat{\mathscr{O}}_{p})$ is the ring of functions
on the formal neighborhood of $[p]\in[M/G]$.

###### Proof.

It is a quick exercise to show that the underlying graded commutative algebras
of $C^{\bullet}(\mathfrak{g}\xrightarrow{dt_{p}}T_{p}M[-1])$ and
$C^{\bullet}(\mathfrak{g},\widehat{\mathscr{O}}_{p})$ are identical, as long
as one is careful to employ the noncanonical isomorphism
$\mathscr{O}(\widehat{M}_{p})\cong\widehat{\textrm{Sym}}(T_{p}^{\vee}M)$. From
there, it is sufficient to show that the Chevalley-Eilenberg differentials are
equivalent, which is also left as a brief exercise. ∎

######  Remark 3.9.

This lemma implies that the dg Lie algebra

(29) $\mathfrak{g}_{p}:=(\mathfrak{g}\oplus
T_{p}M[-1],dt_{p},[-,-]_{\mathfrak{g}})$

is of utmost importance. To say more about this, we must state a definition:

######  Definition 3.10 (Definition 3.1.2 in [8]).

A formal (pointed) moduli problem over $k$ is a functor of simplicially
enriched categories

$F:\mathbf{dgArt}_{k}\to\mathbf{sSets},$

where $\mathbf{dgArt}_{k}$ is the category of (local) Artinian dg algebras
over $k$ and $\mathbf{sSets}$ the category of simplicial sets, which
satisfies: (1) $F(k)$ is contractible. (2) $F$ takes surjective maps in
$\mathbf{dgArt}_{k}$ to fibrations in $\mathbf{sSets}$. (3) For
$A,B,C\in\mathbf{dgArt}_{k}$ and surjections $B\to A$ and $C\to A$ (meaning we
can define the fiber product $B\times_{A}C$), we require that the following
natural map is a weak equivalence:

$F(B\times_{A}C)\to F(B)\times_{F(A)}F(C).$

Clearly, this can be viewed as a “localization” of the traditional algebro-
geometric definition of a stack as a functor $\mathbf{CRing}\to\mathbf{Gpd}$
satisfying descent. What follows in the rest of this section and in Section 4
depends on the following theorem, which allows us to connect the above objects
to the more concrete dg Lie algebras and $L_{\infty}$ algebras we use for
computations:

######  Theorem 3.11 (Theorem 2.0.2 in [15]).

There is an equivalence of $(\infty,1)$-categories between the category
$\mathbf{Lie}_{k}$ of differential graded Lie algebras over a characteristic
zero field $k$ and the category $\mathbf{Moduli}_{k}$ of formal pointed moduli
problems over $k$.

The homotopy category of $L_{\infty}$ algebras is equivalent to the homotopy
category of dg Lie algebras, so that the above remains true in that case (as
is relevant for us). Theorem 3.11 confirms that the dg Lie algebra
$\mathfrak{g}_{p}$ completely defines the data of the formal neighborhood of
$[p]$ in $[M/G]$, as we suspected from Lemma 3.8.

######  Remark 3.12.

Much like how a quotient stack “builds in” group action data into its
definition, functions on a formal neighborhood $\widehat{[M/G]}_{p}$ in the
stack, namely $C^{\bullet}(\mathfrak{g},\hat{\mathscr{O}}_{p})$, have “built
into” them all of the $\mathfrak{g}$-invariant data. Concretely,
$C^{\bullet}(\mathfrak{g},\hat{\mathscr{O}}_{p})$ has the usual ring of
functions $\hat{\mathscr{O}}_{p}$ as a subset: tensoring with
$\textrm{Sym}(\mathfrak{g}^{\vee}[-1])$ and imposing the Chevalley-Eilenberg
differential remembers the data of $\mathfrak{g}$ acting on $\widehat{M}_{p}$,
and therefore on $\mathscr{O}(\widehat{M}_{p})\cong\hat{\mathscr{O}}_{p}$ as
well. To see how these ideas unfold in action, we refer the reader to Appendix
5.1.

######  Example 3.13.

In light of this lemma and Example 3.4, the (Fréchet) dg Lie algebra we must
consider in the context of general covariance is thus

(30)
$\mathfrak{g}_{g}:=\mathfrak{g}_{g}(X):=\Gamma(X,T_{X})\xrightarrow{L_{\bullet}g}\Gamma(X,\mathrm{Sym}^{2}(T^{\vee}_{X}))[-1].$

Note that if don’t specify evaluation of $\mathfrak{g}_{g}$ on all of $X$, it
becomes a sheaf on the site $\mathbf{Riem}_{n}$ introduced in Subsection
2.3.1. The metric $g$ should also not be specified in that case, but we leave
it in the notation to distinguish the above from a generic dg Lie algebra.

By applying Lemma 3.8, we see that the ring of functions on the formal
neighborhood of $[g]\in[\mathscr{M}/\mathscr{D}]$ is
$C^{\bullet}(\mathrm{Vect}(X),\mathscr{O}(\widehat{\mathscr{M}}_{g}))\cong
C^{\bullet}(\mathrm{Vect}(X))\otimes\widehat{\mathrm{Sym}}(T_{g}^{\vee}\mathscr{M})$,
which we interpret as the derived invariants of
$\mathscr{O}(\widehat{\mathscr{M}}_{g})$ with respect to the
$\mathscr{D}$-action. Our definition of general covariance from earlier when
properly “localized” would imply that the observables of such a field theory
$\mathscr{F}_{g}$ over $g\in\mathscr{M}$ form a module over
$C^{\bullet}(\mathrm{Vect}(X),\mathscr{O}(\widehat{\mathscr{M}}_{g}))=C^{\bullet}(\mathfrak{g}_{g})$:
this is exactly what is shown in Proposition 4.3.

######  Remark 3.14.

It should be noted that because $\mathrm{Vect}(X)$ and
$T_{g}^{\vee}\mathscr{M}$ are infinite dimensional, the definition of
$C^{\bullet}(\mathfrak{g}_{g})\cong
C^{\bullet}(\mathrm{Vect}(X))\otimes\widehat{\mathrm{Sym}}(T_{g}^{\vee}\mathscr{M})$
is not precisely the one from the finite dimensional case: in particular, we
have shown that $\mathrm{Vect}(X)$ is a Fréchet Lie algebra and
$T_{g}^{\vee}\mathscr{M}$ is a Fréchet vector space, and so the tensor product
used for the preceding objects is the completed projective tensor product used
in Definition 2.2. In this way, $C^{\bullet}(\mathfrak{g}_{g})$ represents the
same data as it would if its inputs were finite dimensional, but we are just a
bit more careful with functional analytic issues to ensure that all of the
rings are well defined.

###  3.3 Vector Bundles over a Formal Stack

Now that we have made things concrete with an example, we’d like to understand
vector bundles in this context. We’re primarily concerned with perturbative
computations (those in the style of Construction 3.7); however, we will
present the global picture first, since general covariance is first presented
in such a context.

######  Construction 3.15.

Let $V$ be a $G$-equivariant vector bundle over $M$, for which the action
$\tau_{M}:G\to\mathrm{Diff}(M)$ is not necessarily free. Call the action on
the total space of $V\to M$ $\tau_{V}:G\to\mathrm{Diff}(V)$. Recall from
Example 2.46 that we get the pair of Lie groupoids $\mathcal{V}_{\mathcal{G}}$
and $\mathcal{M}_{\mathcal{G}}$ with a map
$\pi:\mathcal{V}_{\mathcal{G}}\to\mathcal{M}_{\mathcal{G}}$ between them. This
information in turn presents a pair of stacks, and the projection gives us a
map $\pi:[V/G]\to[M/G]$ between those stacks. Here, $[V/G]$ is a vector space
object in the category of stacks over the stack $[M/G]$, much like how
$\mathcal{V}_{\mathcal{G}}$ is a vector space object in the category of Lie
groupoids over the Lie groupoid $\mathcal{M}_{\mathcal{G}}$.

The action of a finite dimensional Lie group $G$ on a finite dimensional $M$
restricts to an action of the formal group
$\widehat{G}\overset{\textrm{exp}}{\cong}\mathfrak{g}$ (defined as the formal
neighborhood of the identity in $G$) on $\widehat{M}_{p}$, the formal
neighborhood of $p\in M$. This defines a formal Lie groupoid which then
presents the stack $[\widehat{M}_{p}/\widehat{G}]\cong\widehat{[M/G]}_{p}$,
whose rings of functions we computed earlier to be
$C^{\bullet}(\mathfrak{g}_{p})$, so that $\mathfrak{g}_{p}$ is the dg Lie
algebra associated to the formal moduli problem $\widehat{[M/G]}_{p}$.

We can pull back the $G$-equivariant vector bundle $V\to M$ along
$\hat{p}:\widehat{M}_{p}\to M$ to get a $\mathfrak{g}$-equivariant vector
bundle $\hat{p}^{*}V\to\widehat{M}_{p}$. Topologically, the total space of
$\hat{p}^{*}V$ is the formal neighborhood of the entire fiber
$\pi^{-1}(p)=V_{p}$, which we can think of heuristically as
$V_{p}\times\widehat{M}_{p}$. Both parts of this product have an action of
$\widehat{G}$, even though one of the directions is a formal space and the
other a vector space which is not necessarily viewed as formal (i.e. its ring
of functions may be polynomials, and not power series). Thus, we can consider
the associated formal Lie groupoid here as well, and it presents the stack
$[(\hat{p}^{*}V)/\widehat{G}]$.

The vector bundle which plays the local role of the global stack
$[V/G]\to[M/G]$ is therefore

$[(\hat{p}^{*}V)/\widehat{G}]\to[\widehat{M}_{p}/\widehat{G}].$

On account of $C^{\bullet}(\mathfrak{g}_{p})$ being the space of functions on
$[\widehat{M}_{p}/\widehat{G}]$, we see that a section
$\sigma:[\widehat{M}_{p}/\widehat{G}]\to[(\hat{p}^{*}V)/\widehat{G}]$ is an
element of $C^{\bullet}(\mathfrak{g}_{p})\otimes V_{p}$. This is the
stackified and deformation-theoretic version of a section of $V\to M$ being an
element of $C^{\infty}(M)\otimes V_{p}$ in local coordinates near $p$.
Moreover, this reasoning results in the following lemma.

######  Lemma 3.16.

The ring of functions on
$[(\hat{p}^{*}V)/\widehat{G}]\cong[(\hat{p}^{*}V)/\mathfrak{g}]$ is
$C^{\bullet}(\mathfrak{g},\widehat{\mathscr{O}}_{p}\otimes\mathrm{Sym}(V_{p}^{\vee}))\cong
C^{\bullet}(\mathfrak{g}_{p},\mathrm{Sym}(V_{p}^{\vee}))$, which is isomorphic
as a graded ring to
$C^{\bullet}(\mathfrak{g}_{p})\otimes\mathrm{Sym}(V_{p}^{\vee})$.

###### Proof.

The definition of $[(\hat{p}^{*}V)/\mathfrak{g}]$ implies that
$\mathscr{O}([(\hat{p}^{*}V)/\mathfrak{g}])$ must be the derived
$\mathfrak{g}$-invariant functions on the space $\widehat{M}_{p}\times V_{p}$.
Given that $\mathscr{O}(\widehat{M}_{p}\times
V_{p})=\widehat{\mathscr{O}}_{p}\otimes\mathrm{Sym}(V_{p}^{\vee})$ and that
both parts of this tensor product are $\mathfrak{g}$-modules, we can define
the CE cochains
$C^{\bullet}(\mathfrak{g},\widehat{\mathscr{O}}_{p}\otimes\mathrm{Sym}(V_{p}^{\vee}))$.
In conjunction with Lemma 3.8, these are the derived $\mathfrak{g}$-invariant
functions we are looking for. To see that the differential graded rings are
isomorphic, we simply note that the CE differential on both is

$d_{CE}=[-,-]^{\vee}_{\mathfrak{g}}+\tau_{M_{p}}^{\vee}+\tau_{V_{p}}^{\vee},$

where $\tau_{M_{p}}^{\vee}$ and $\tau_{V_{p}}^{\vee}$ are the “duals” (as in
Appendix 5.1) to the induced actions $\tau_{M_{p}}$ and $\tau_{V_{p}}$ on
$\widehat{\mathscr{O}}_{p}$ and $\mathrm{Sym}(V_{p}^{\vee})$, respectively. ∎

##  4 Field Theories as Bundles over Formal Stacks

###  4.1 Equivariant Observables

Next we shall consider how $Q_{g}$ and more generally $\\{S_{g},-\\}$ behave
under arbitrary perturbations $g+\varepsilon h$, for $h\in T_{g}\mathscr{M}$,
and then invoke that $h=L_{V}g$ comes from a vector field $V$ to see what the
effect is. But we already have the machinery to do this! The preceding
sentence amounts to pulling back the dg vector or $L_{\infty}$ bundle
$(\mathscr{F},\\{S,-\\})\to\mathscr{M}$ to be over the formal neighborhood of
$g\in\mathscr{M}$, and seeing what the “full differential”
$\\{S_{g+\varepsilon h},-\\}$ looks like over this formal neighborhood.

######  Remark 4.1.

Although in finite dimensions we have
$\widehat{G}\overset{\textrm{exp}}{\cong}\mathfrak{g}$, we mentioned in
Example 3.4 that it was no longer the case that there is a bijection between
the formal neighborhood of the identity diffeomorphism in $\mathrm{Diff}(X)$
and its Lie algebra $\mathrm{Vect}(X)$ of vector fields: this could ostensibly
be cause for alarm. However, by Lurie’s Theorem 3.11 it remains true even in
the infinite dimensional case that the dg Lie algebra $\mathfrak{g}_{g}$
introduced earlier is in correspondence with the formal neighborhood of
$[g]\in[\mathscr{M/D}]$. This will be an assurance in what follows.

######  Construction 4.2.

A family $(\mathscr{F},\\{S,-\\})$ of BV field theories defined as a dg vector
or $L_{\infty}$ bundle over $\mathscr{M}$ pulls back to an appropriate bundle
over the formal neighborhood of $g\in\mathscr{M}$, denoted
$\widehat{\mathscr{M}}_{g}$, where
$\mathscr{O}(\widehat{\mathscr{M}}_{g})=\widehat{\mathscr{O}}_{g}\cong\widehat{\mathrm{Sym}}(T_{g}^{\vee}\mathscr{M})$.
Heuristically, this pullback looks like
$\widehat{\mathscr{M}}_{g}\times\mathscr{F}_{g}$.

We get an analogous pullback of stacks when the theory is generally covariant.
In this case, the $\mathscr{D}$-equivariant bundle
$(\mathscr{F},\\{S,-\\})\to\mathscr{M}$ is equivalent to a bundle of stacks
$([\mathscr{F}/\mathscr{D}],\\{S,-\\})\to[\mathscr{M}/\mathscr{D}]$. If we
consider an equivalence class of metrics $[g]\in[\mathscr{M}/\mathscr{D}]$ and
fix its formal neighborhood, we can pull back
$([\mathscr{F}/\mathscr{D}],\\{S,-\\})$ over this formal neighborhood. We
denote the total space of this pullback as
$\widehat{[\mathscr{F}/\mathscr{D}]}_{g}$. We can then conclude:

######  Proposition 4.3.

For a generally covariant family
$\pi:([\mathscr{F}/\mathscr{D}],\\{S,-\\})\to[\mathscr{M}/\mathscr{D}]$ of BV
classical field theories and for a fixed $[g]\in[\mathscr{M}/\mathscr{D}]$, we
have

(31) $\mathscr{O}(\widehat{[\mathscr{F}/\mathscr{D}]}_{g})\cong
C^{\bullet}(\mathfrak{g}_{g},\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g})).$

###### Proof.

By the equivalence of categories in Theorem 3.11 which we are taking for
granted, the formal moduli space defined by the formal neighborhood of
$[g]\in[\mathscr{M/D}]$ is equivalent to the dg Lie algebra

$\mathfrak{g}_{g}=\Gamma(X,T_{X})\xrightarrow{L_{\bullet}g}\Gamma(X,\mathrm{Sym}^{2}(T^{\vee}_{X}))[-1].$

The dg algebra of functions on this formal neighborhood is thus

$C^{\bullet}(\mathfrak{g}_{g})\cong
C^{\bullet}(\mathrm{Vect}(X),\widehat{\mathrm{Sym}}(T_{g}^{\vee}\mathscr{M})).$

The ring of functions on the fiber part of the pullback is simply
$C^{\bullet}(\mathrm{Vect}(X),\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g}))$,
since it is $\mathscr{O}(\mathscr{F}_{g})$ with the differential
$\\{S_{g},-\\}$ and the implicit action of $\mathrm{Vect}(X)$ on the theory
and thus on its observables. Hence, the underlying dg ring of functions on
$\widehat{[\mathscr{F}/\mathscr{D}]}_{g}$ is the underlying dg ring of
$C^{\bullet}(\mathfrak{g}_{g},\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g}))$.
Both dg rings have Chevalley-Eilenberg differential

(32)
$d_{CE}=[-,-]^{\vee}_{\mathrm{Vect}(X)}+\tau_{\mathscr{M}_{g}}^{\vee}+\tau_{\mathscr{F}_{g}}^{\vee}+\\{S_{g},-\\}.$

Here, the first three terms are the usual Chevalley-Eilenberg differential
concerned with the dual of the bracket on $\mathrm{Vect}(X)$ and the actions
of $\mathrm{Vect}(X)$ on $\widehat{\mathrm{Sym}}(T_{g}^{\vee}\mathscr{M})$ and
$\mathscr{O}(\mathscr{F}_{g})$, and the fourth term is the differential on the
free field theory over $[\widehat{\mathscr{M}}_{g}/\mathrm{Vect}(X)]$. Since
the underlying rings agree and the CE differentials do, too, this gives the
result. ∎

######  Remark 4.4.

As in the case of ordinary manifolds, the ring of functions on the bundle is a
module over the ring of functions on the base space. In fact, the veracity of
the above claim can almost be taken as a definition: in the case where we
treat the BV field theory $\mathscr{F}$ perturbatively, Proposition 4.3 simply
computes the function ring of the formal moduli problem representing
perturbative fields parameterized by a formal neighborhood in
$[\mathscr{M/D}]$. As it stands, the statement also includes polynomial
functions of nonperturbative free fields along the fibers.212121The fibers
thus constitute a “non-formal” part of the total moduli problem.

Proposition 4.3 implicitly supplies a natural action of $\mathfrak{g}_{g}$ on
$\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g})$: this allows us to conclude
that Noether’s Theorem as presented in Theorem 12.4.1 of [8] applies. We have
not given the precise details of a “full” $L_{\infty}$ action of
$\mathfrak{g}_{g}$ on $\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g})$, but its
existence is implicit in the theorem: $\\{S_{g},-\\}$ as part of $d_{CE}$
above contains information about the formal neighborhood of
$[g]\in[\mathscr{M/D}]$. We will provide a thorough description of how this
goes momentarily.

######  Remark 4.5 (Further remarks on functoriality).

Before getting into explicit computations, we would like to mention in the
vein of Subsection 2.3.1 that because $\mathfrak{g}_{g}$ is a sheaf on
$\mathbf{Riem}_{n}$ (its diffeomorphism equivariance can quickly be checked),
the equivariant observables similarly define a factorization algebra, as in
Proposition 2.43:

$C^{\bullet}(\mathfrak{g}_{g}(-),\mathrm{Obs^{cl}}(-,\mathscr{F})):\mathbf{Riem}_{n}\to\mathbf{dgVect}.$

This provides yet another factorization algebra when evaluated on the site of
Riemannian manifolds. Thus, considering the stacky geometry of
$[\mathscr{M/D}]$ for a fixed $X$ and invoking $\mathbf{Riem}_{n}$-naturality
after the fact once again provides an interesting construction (and
generalization) of objects introduced, for example, in [8], while
simultaneously opening avenues of comparison to prevailing literature.

Note: From here on out, we will be treating both free222222Any dg Lie algebra
is an $L_{\infty}$ algebra where the only nontrivial bracket is $\ell_{1}$.
and interacting theories as $L_{\infty}$ algebras, since the reliance on
$L_{\infty}$ structures for defining actions becomes more important. In
practice, this means we will be using $\mathscr{L}$ (alias $\mathscr{F}[-1]$)
to denote the fields.

######  Remark 4.6.

Recall that when $\mathfrak{g}$ is a Lie algebra and $R$ is a
$\mathfrak{g}$-module, $H^{0}(C^{\bullet}(\mathfrak{g},R))=R^{\mathfrak{g}}$,
the $\mathfrak{g}$-invariants of $R$. Analogously, albeit with slightly more
care to compute, we have:

(33)
$H^{0}(C^{\bullet}(\mathfrak{g}_{g},\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{L}_{g})))=\\{F\in\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{L}_{g}):F(g+\varepsilon
L_{V}g)-F(g)=0\\}.$

A prime example of such an $F$ is the action functional $S_{g}$ of any
generally covariant theory. Moreover, if $V$ is a Killing field for $g$, the
equality in the conditions on the right side above holds trivially: this is a
shadow of the fact that a moduli stack “remembers” stabilizers where the
coarse quotient would not.

Although it is meaningful (and a good sanity check) to compute cohomology
groups, we stick to Noether’s philosophy of focusing on the cochain complexes
themselves. In our case, this means understanding what the equivariant
observables are providing. There is a guiding definition which, when unpacked
carefully, tells us the value of what we found above:

######  Definition 4.7 (Definition 12.2.12 in [8]).

For $\mathfrak{g}$ a dg Lie or $L_{\infty}$ algebra and $\mathscr{L}$ an
(elliptic) $L_{\infty}$ algebra encoding a Batalin-Vilkovisky classical field
theory, an action of $\mathfrak{g}$ on $\mathscr{L}$ is any of the following
data: (i) An $L_{\infty}$ structure on $\mathfrak{g}\oplus\mathscr{L}$ such
that the exact sequence

(34) $\mathscr{L}\to\mathfrak{g}\ltimes\mathscr{L}\to\mathfrak{g}$

is a sequence of maps of $L_{\infty}$ algebras, the structure maps
$\mathfrak{g}^{\otimes n}\otimes\mathscr{L}^{\otimes m}\to\mathscr{L}$ are
polydifferential operators on the $\mathscr{L}$-variables, and the action
preserves the pairing $\omega$.
(ii) An $L_{\infty}$ morphism $\mathfrak{g}\to
C^{\bullet}_{\mathrm{loc}}(\mathscr{L})[-1]$.
(iii) A degree 1 element $S^{\mathfrak{g}}$ in the dg Lie algebra

$\mathrm{Act}(\mathfrak{g},\mathscr{L}):=C^{\bullet}_{\mathrm{red}}(\mathfrak{g})\otimes
C^{\bullet}_{\mathrm{loc}}(\mathscr{L})[-1]$

which satisfies the Maurer-Cartan equation

(35)
$d_{\mathfrak{g}}S^{\mathfrak{g}}+d_{\mathscr{L}}S^{\mathfrak{g}}+\frac{1}{2}\\{S^{\mathfrak{g}},S^{\mathfrak{g}}\\}=0:$

this can be viewed as an equivariant classical master equation.

######  Remark 4.8.

By $C^{\bullet}_{\mathrm{loc}}(\mathscr{L})$ above we mean observables for
$\mathscr{L}$ that are local in the sense of Definition 2.5:
$C^{\bullet}_{\mathrm{loc}}(\mathscr{L})[-1]$ is the formal moduli version of
symplectic vector fields, which control symmetries and deformations of a
classical field theory. $C^{\bullet}_{\mathrm{red}}(\mathfrak{g})$ is defined
as the kernel of the augmentation map
$C^{\bullet}(\mathfrak{g})\to\mathbf{R}$.232323Thorough details concerning
these two rings is provided in Chapters 3 and 4 of [8].

Moreover, since $S^{\mathfrak{g}}$ is local in the fields $\mathscr{L}$, it
defines a derivation of $\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{L})$ via
$\\{S^{\mathfrak{g}},-\\}$: this is precisely what is used to define the
action of $\mathfrak{g}_{g}$ on
$\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{L}_{g})$ when computing the equivariant
classical observables in Proposition 4.3.

######  Remark 4.9.

The facet of the preceding definition we will hone in on is the third one:
finding a functional $S^{\mathfrak{g}}$ which satisfies an equivariant
classical master equation provides a concrete computational representation of
the action of $\mathfrak{g}$ on $\mathscr{L}$ and a more complete picture of
the Chevalley-Eilenberg description of how the formal moduli stack acts on the
theory.

We would like to encode both deformations by $h\in
T_{g}\mathscr{M}=\Gamma(X,\mathrm{Sym}^{2}(T_{X}^{\vee}))$ and an action of
vector fields $V\in\mathrm{Vect}(X)$ on the BV field theory, since these are
the degree 1 and 0 parts (respectively) of the dg Lie algebra
$\mathfrak{g}_{g}$ representing the formal neighborhood of $g$ as an element
of the stack $[\mathscr{M/D}]$. Any action functional $S_{g}$ for a BV theory
can be written as

$S_{g}(\phi)=\int_{X}\phi D_{g}(\phi),$

where the differential operator may be a nonlinear function in $\phi$.
Denoting the antifields for the theory as $\psi$, we thus posit the following:

(36) $S^{\mathfrak{g}_{g}}=\int_{X}\phi\big{(}D_{g+\varepsilon
h}-D_{g}\big{)}(\phi)+\int_{X}(L_{V}\phi)\psi.$

On the right side, we interpret $D_{g+\varepsilon h}$ as a formal power series
in the metric perturbation $h$, and by $L_{V}$ we mean the “natural action” of
vector fields on the fields, which are usually tensorial in nature (hence the
notation). In accordance with (ii) in Definition 4.7, the $L_{\infty}$
morphism $\mathfrak{g}\to C^{\bullet}_{\mathrm{loc}}(\mathscr{L})[-1]$ is thus
given by sending $(V,h)\in\mathfrak{g}_{g}$ to:

$\\{S^{\mathfrak{g}_{g}},-\\}\in
C^{\bullet}_{\mathrm{loc}}(\mathscr{L})[-1]\subset\
\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{L}_{g})[-1],$

where the latter is interpreted as symplectic vector fields on $B\mathscr{L}$,
the formal derived critical locus as seen in Remark 2.13. By means of general
covariance, which implies that either the dg or $L_{\infty}$ structure
prescribed by $D_{g}(\phi)$ is diffeomorphism equivariant,
$S^{\mathfrak{g}_{g}}$ satisfies the necessary classical master equation.
Strictly speaking, what we need in the preceding is equivariance with respect
to the action by vector fields: in the case of the free scalar field, this
comes from the fact that the Laplace-Beltrami operator satisfies (modulo
$\varepsilon^{2}$)

$\Delta_{g+\varepsilon L_{V}g}=\Delta_{g}+\varepsilon[L_{V},\Delta_{g}].$

######  Remark 4.10.

What we have presented so far allows us to provide a more precise and
meaningful expression for the Chevalley-Eilenberg differential for
$C^{\bullet}(\mathfrak{g}_{g},\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{L}_{g}))$
in Proposition 4.3:

(37)
$d_{CE}=[-,-]^{\vee}_{\mathrm{Vect}(X)}+\\{S^{\mathfrak{g}_{g}},-\\}+\\{S_{g},-\\}.$

This provides the usual dual to the action of vector fields on themselves;
however, we have here the $L_{\infty}$ action of $\mathfrak{g}_{g}$ on the
observables as well as the usual differential $\\{S_{g},-\\}$ of the
observables on their own. If we consider, for example, the interacting scalar
field with polynomial potential as in Example 2.37, the latter two brackets
above would combine to result in bracketing with:

(38) $S^{\mathrm{tot}}:=S_{g}+S^{\mathfrak{g}_{g}}=\int_{X}\varphi
Q_{g+\varepsilon h}\varphi+\sum_{n\geq
2}\int_{X}\frac{\lambda_{n}}{n!}\varphi^{n}\mathrm{vol}_{g+\varepsilon
h}+\int_{X}(L_{V}\varphi)\psi.$

In Definitions 1 and 2 of [11], Getzler defines covariance by supplying
something like $S^{\mathrm{tot}}$ above and demanding that it satisfy a
Maurer-Cartan equation for a curved Lie (super) algebra: I would be curious to
see how the connection between the two could be made completely precise.

To expound more on all of the above, we must introduce the stress-energy
tensor.

###  4.2 The Stress-Energy Tensor

We’d like to consider the stress-energy tensor for the free scalar BV theory:
its generalization to the polynomial self-interaction in Lemma 2.38 is
computationally simple. This section is intended to see how an example of
Definition 4.7 plays out as well as connect between the above work to how
things are “usually done” in physics.

To begin, let us consider an arbitrary first order deformation of the
Laplacian $\Delta_{g}$ on a Riemannian manifold $(X,g)$: in other words, let
$g_{t}$ be a one-parameter family of metrics such that $g_{0}=g$ and let us
compute

$\frac{d}{dt}\Delta_{g_{t}}\varphi\Big{|}_{t=0}.$

Writing $\Delta_{g_{t}}$ in coordinates and not evaluating at $t=0$ for now,
we have:

(39)
$\frac{d}{dt}\Big{(}\frac{1}{\sqrt{\mathrm{det}g_{t}}}\partial_{\mu}(\sqrt{\mathrm{det}g_{t}}g_{t}^{\mu\nu}\partial_{\nu}\varphi)\Big{)}.$

Recall that for a one-parameter family of invertible matrices $A(t)$, we have

$\frac{d}{dt}\mathrm{det}A(t)=\operatorname{Tr}(A(t)^{-1}A^{\prime}(t))\mathrm{det}A(t).$

Using this and a few other manipulations, expression (39) reduces to

(40)
$\frac{-1}{2}\operatorname{Tr}(g_{t}^{-1}\partial_{t}g_{t})\Delta_{g_{t}}\varphi+\frac{1}{\sqrt{\mathrm{det}g_{t}}}\partial_{\mu}\Big{(}\frac{\sqrt{\mathrm{det}g_{t}}}{2}\operatorname{Tr}(g_{t}^{-1}\partial_{t}g_{t})g_{t}^{\mu\nu}\partial_{\nu}\varphi+\sqrt{\mathrm{det}g_{t}}\partial_{t}g_{t}^{\mu\nu}\partial_{\nu}\varphi\Big{)}.$

Denote the derivative of $g_{t}$ at $g_{0}=g$ as $\delta
g:=\partial_{t}g_{t}|_{t=0}$ (this is the traditional notation in physics,
although we could call it $h$ as a degree 1 element of $\mathfrak{g}$).
Evaluating at $t=0$ gives:

######  Lemma 4.11.

The first order deformation of the Laplacian $\Delta_{g}$ with respect to the
metric $g$ is

(41)
$\frac{d}{dt}\Delta_{g_{t}}\varphi\Big{|}_{t=0}=\frac{-1}{2}\operatorname{Tr}(g^{-1}\delta
g)\Delta_{g}\varphi+\frac{1}{\sqrt{\mathrm{det}g}}\partial_{\mu}\Big{(}\sqrt{\mathrm{det}g}\big{(}\frac{1}{2}\operatorname{Tr}(g^{-1}\delta
g)g^{\mu\nu}\partial_{\nu}\varphi+\delta
g^{\mu\nu}\partial_{\nu}\varphi\big{)}\Big{)}.$

Moreover, if we assume the deformation $\delta g\in T_{g}\mathscr{M}$ is
induced by an isometry of $g$, then the first order deformation of the
Laplacian is identically zero.

######  Remark 4.12.

Often in the physics literature, we write

$\operatorname{Tr}(g^{-1}\delta g)=g^{\mu\nu}\delta g_{\mu\nu},$

and we may occasionally adopt that notation. Moreover, the above computation
is done with the action functional (1) in mind. Thus, any difference with the
stress-energy tensor computations using the functional

(42)
$\int_{X}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi\mathrm{vol}_{g}=\int_{X}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi\sqrt{\mathrm{det}g}d^{n}x,$

which is just as common in the physics literature, differs only by boundary
terms.

First, we shall provide a general definition of the stress-energy tensor for
any theory.

######  Construction 4.13.

Let $S_{g}\in C^{\bullet}(\mathscr{L}_{g})$ be an action functional for the dg
space of BV fields $\mathscr{L}_{g}$ which depends on a fixed background
metric $g\in\mathscr{M}$. It can thus be written as

$S_{g}(\phi)=\int_{X}L_{g}(\phi),$

where $\phi\in\mathscr{L}$ and $L_{g}(\phi)$ is a Lagrangian density. If we
let $g_{t}$ be a one-parameter family of metrics such that $g_{0}=g$, we can
perform computations similar to those in Lemma 4.11 to compute:

(43) $\frac{\delta}{\delta
g}S_{g}(\phi):=\frac{d}{dt}S_{g_{t}}(\phi)\Big{|}_{t=0}=\int_{X}\frac{d}{dt}L_{g_{t}}(\phi)\Big{|}_{t=0}.$

The notation invoked on the left hand side is common in physics literature,
and defined this way in [23]. Up to boundary terms which we can safely ignore,
(43) can be written as

(44) $\int_{X}\delta g^{\mu\nu}T_{\mu\nu}(g,\phi)\mathrm{vol}_{g},$

for some $T_{\mu\nu}(g,\phi)$ (or simply $T_{\mu\nu})$ which depends on the
fields $\phi$ and on the metrics $g$.

######  Definition 4.14.

$T_{\mu\nu}$ is the stress-energy (or energy-momentum) tensor of a field
theory on $X$ with fields $\phi\in\mathscr{F}$ and action functional $S_{g}$
depending on $g\in\mathrm{Met}(X)$.

######  Remark 4.15.

Before moving on, we must take note of an important fact: the stress-energy
tensor coupled to the metric perturbation as in Equation (44) is precisely the
first order in $\varepsilon$ part of the power series in $h$ found in Equation
(36).

######  Example 4.16.

To compute the stress-energy tensor of free massless scalar field, we will
begin by noting that according to the definition, we must compute

$\int_{X}\varphi\frac{d}{dt}Q_{g_{t}}\varphi\Big{|}_{t=0},$

where $Q_{g}\varphi=\Delta_{g}\varphi\mathrm{vol}_{g}$. Lemma 4.11 is useful,
since we have already done the necessary work on the first piece. However,
note that $Q_{g}\varphi$ is written in coordinates as

$\partial_{\mu}(\sqrt{\mathrm{det}g}g^{\mu\nu}\partial_{\nu}\varphi)d^{n}x,$

so that we have in fact stripped away some of the complexity of the
computation by pairing with the Riemannian volume form. Hence, we can use
Lemma 4.11 and toss away the first term to get

(45)
$\int_{X}\varphi\frac{d}{dt}Q_{g_{t}}\varphi\Big{|}_{t=0}=\int_{X}\varphi\partial_{\mu}\Big{(}\sqrt{\mathrm{det}g}\big{(}\frac{1}{2}\operatorname{Tr}(g^{-1}\delta
g)g^{\mu\nu}\partial_{\nu}\varphi+\delta
g^{\mu\nu}\partial_{\nu}\varphi\big{)}\Big{)}d^{n}x.$

This is not yet in the preferred form in (44), but if we integrate by parts
and invoke that $\operatorname{Tr}(g^{-1}\delta g)=g^{\mu\nu}\delta
g_{\mu\nu}=g_{\mu\nu}\delta g^{\mu\nu},$ the above becomes

(46) $\int_{X}\delta
g^{\mu\nu}\big{(}-\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}g_{\mu\nu}(g^{\rho\sigma}\partial_{\rho}\varphi\partial_{\sigma}\varphi)\big{)}\mathrm{vol}_{g},$

where we changed the labelling of indices in the second term to omit
confusion. Thus, the stress-energy tensor for our example is
$T_{\mu\nu}=-\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}g_{\mu\nu}(g^{\rho\sigma}\partial_{\rho}\varphi\partial_{\sigma}\varphi)$.
We would have computed this without any by-parts maneuvers had we started with
the action functional (42) more common in physics literature, but it is a good
exercise to see how these agree.

The above is the traditional trajectory one takes to finding the stress-energy
tensor; however, since our theory is generally covariant and so we can use
facts about equivariant vector bundles to simplify things, let us consider
what that buys us. To begin, let $f_{t}$ be a one-parameter subgroup of
diffeomorphisms. General covariance implies that

(47)
$\frac{d}{dt}\int_{X}(f_{t}^{*}\varphi)\Delta_{f_{t}^{*}g}(f_{t}^{*}\varphi)\mathrm{vol}_{f_{t}^{*}g}\Big{|}_{t=0}=0.$

Unfolding the left hand side, this equation becomes

$\int_{X}(L_{V}\varphi)\Delta_{g}\varphi\mathrm{vol}_{g}+\int_{X}\varphi\Delta_{g}(L_{V}\varphi)\mathrm{vol}_{g}+\int_{X}\varphi(\frac{d}{dt}\Delta_{f^{*}_{t}g}\big{|}_{t=0})\varphi\mathrm{vol}_{g}+\int_{X}\varphi\Delta_{g}\varphi(\frac{d}{dt}\mathrm{vol}_{f^{*}_{t}g}\big{|}_{t=0})=0.$

Here, we assumed that $V$ generates the flow $f_{t}$, and used the fact that
$\frac{d}{dt}(f^{*}_{t}\varphi)|_{t=0}=L_{V}\varphi$. This equation is an
integrated linear approximation to the equivariance property computed in Lemma
2.34: it states concretely that a simultaneous first order perturbation along
the $\mathscr{D}$-orbit in $\mathscr{M}$ and in $\mathscr{L}_{g}$ is trivial.

######  Remark 4.17.

The third and fourth terms on the left hand side are exactly those that
comprise the integral of the stress-energy tensor in the special case that the
derivative is computed in the direction of the $\mathscr{D}$-orbit. This
grants us two key insights:

(1) Computationally, the above amounts to the metric perturbation (an element
of $T_{g}\mathscr{M}$) coming from an infinitesimal diffeomorphism (i.e. a
vector field). But we have seen this before: this is saying that $\delta g\in
T_{g}\mathscr{M}$ is in the image of the differential in the dg Lie algebra
$\mathfrak{g}_{g}$ in Example 3.13. Hence, $\delta g^{\mu\nu}=L_{V}g^{\mu\nu}$
(the computation works fine even though $g^{\mu\nu}$ is technically the
inverse). With this, Equation (46) becomes:

$\int_{X}L_{V}g^{\mu\nu}T_{\mu\nu}\mathrm{vol}_{g}.$

A standard result from Riemannian geometry is that
$L_{V}g^{\mu\nu}=\nabla^{\mu}V^{\nu}+\nabla^{\nu}V^{\mu}$, and since
$T_{\mu\nu}$ is symmetric by definition, the above must be

$\int_{X}(\nabla^{\mu}V^{\nu})T_{\mu\nu}\mathrm{vol}_{g}=-\int_{X}V^{\nu}(\nabla^{\mu}T_{\mu\nu})\mathrm{vol}_{g},$

where we invoked integration by parts and the fact that
$\nabla^{\mu}\mathrm{vol}_{g}=0$ in the equality. Then, standard computations
for generally covariant theories (which can be found in Appendix E of [23])
show that for on-shell fields (here meaning $\varphi$ such that
$\Delta_{g}\varphi=0$), the above integral is identically zero. For this to be
true, it must be the case that

(48) $\nabla^{\mu}T_{\mu\nu}=0.$

In the language of Noether’s Theorem, the stress-energy tensor $T_{\mu\nu}$ is
the conserved current associated to general covariance, a symmetry of a field
theory coupled to a metric.

In our regime, this implies that the conservation law
$\nabla^{\mu}T_{\mu\nu}=0$ is what is ultimately responsible for allowing us
to define an $L_{\infty}$ action of the dg Lie algebra $\mathfrak{g}_{g}$
associated to the formal neighborhood of $[g]\in[\mathscr{M/D}]$ on
observables $\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{L}_{g})$ for our generally
covariant BV field theory in the sense of Definition 4.7. However, what we
have shown above is only up to first order in the metric perturbation! The
differential $\\{S^{\mathfrak{g}},-\\}$ we defined previously in principle
contains “higher conservation laws” associated to higher $L_{\infty}$ brackets
read off from higher order terms in the power series $h\in T_{g}\mathscr{M}$.
The author would be interested in assigning a physical interpretation to this.

(2) Additionally, since the third and fourth terms are (up to a sign) the same
as the first two, this means considering the first two alone should give us
all the relevant data of the stress-energy tensor for a generally covariant
field theory: we could even find a second order vector field equivariance
property analogous to the one stated at the end of Remark 4.9 (we do just that
in Section 5.2 of the Appendix).

######  Construction 4.18.

Let us consider the “infinitesimal general covariance” property more formally.
Insight (1) suggests that the action functionals $S_{g}(\varphi)$ and

$S_{g+\varepsilon
L_{V}g}(\varphi)=\frac{-1}{2}\int_{X}\varphi\Delta_{g}\varphi\mathrm{vol}_{g}-\frac{\varepsilon}{2}\int_{X}L_{V}g^{\mu\nu}T_{\mu\nu}\mathrm{vol}_{g}=:S_{g}(\varphi)+\varepsilon
I_{g}(L_{V}g,\varphi),$

where this equality holds modulo $\varepsilon^{2}$, should produce the same
dynamics: this is true because for on-shell fields, the second term is zero.
In other words, if we were to make sense of the differential $Q_{g+\varepsilon
L_{V}g}$ for the BV space of fields, it should be appropriately equivalent to
$Q_{g}$. Moreover, $Q_{g}$ induces the differential $\\{S_{g},-\\}$ on
$\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g})$, so that we would like
$\\{S_{g+\varepsilon
L_{V}g},-\\}=\\{S_{g},-\\}+\varepsilon\\{I_{g}(L_{V}g),-\\}$, the induced
differential on $\mathrm{Obs}^{\mathrm{cl}}(X,\mathscr{F}_{g+\varepsilon
L_{V}g})$ from $Q_{g+\varepsilon L_{V}g}$, to be similarly equivalent. To give
all of the above hands and legs, we must rigorously define
$\mathscr{F}_{g+\varepsilon L_{V}g}$ and its observables in the first place.

Let $\mathbb{D}_{2}=\mathbf{R}[\varepsilon]/(\varepsilon^{2})$ denote the
(real) dual numbers. We can tensor
$\mathscr{F}_{g}=C^{\infty}(X)\xrightarrow{Q_{g}}\mathrm{Dens}(X)[-1]$ with
$\mathbb{D}_{2}$ to get $\mathscr{F}_{g}\otimes\mathbb{D}_{2}$, whose elements
can be written as $\varphi_{0}+\varepsilon\varphi_{1}$ in degree 0 and
similarly for degree 1. The differential $Q_{g+\varepsilon L_{V}g}$ looks like

$Q_{g}+\varepsilon D.$

It remains only to find $D$, which will depend on $g$ and $V$ and must be so
that

${\mathscr{F}_{g}\otimes\mathbb{D}_{2}=C^{\infty}(X)\otimes\mathbb{D}_{2}}$${\mathrm{Dens}(X)[-1]\otimes\mathbb{D}_{2}}$${\mathscr{F}_{g}\otimes\mathbb{D}_{2}=C^{\infty}(X)\otimes\mathbb{D}_{2}}$${\mathrm{Dens}(X)[-1]\otimes\mathbb{D}_{2}}$$\scriptstyle{Q_{g}+\varepsilon
0}$$\scriptstyle{\mathrm{Id}+\varepsilon
L_{V}}$$\scriptstyle{\mathrm{Id}+\varepsilon
L_{V}}$$\scriptstyle{Q_{g}+\varepsilon D}$

commutes. The downward-pointing arrows are $\mathrm{Id}+\varepsilon L_{V}$
since we are still assuming the diffeomorphism $f$ is generated by the vector
field $V$: concretely, this is the first order approximation to the commuting
square in Lemma 2.34. Thus, we are trying to suss out a neat form of the
first-order perturbation of $Q_{g}$ with respect to the metric when the
perturbation is along a diffeomorphism orbit. Our computations from Equation
(47) suggest that we try $D=[L_{V},Q_{g}]$.

######  Lemma 4.19.

Let
$\widetilde{\mathscr{F}}_{g}:=(\mathscr{F}_{g}\otimes\mathbb{D}_{2},Q_{g})$
and $\widetilde{\mathscr{F}}_{g+\varepsilon
L_{V}g}:=(\mathscr{F}_{g}\otimes\mathbb{D}_{2},Q_{g}+\varepsilon[L_{V},Q_{g}])$.
Then the map $\mathrm{Id}+\varepsilon
L_{V}:\widetilde{\mathscr{F}}_{g}\to\widetilde{\mathscr{F}}_{g+\varepsilon
L_{V}g}$ is a cochain isomorphism (i.e. it is an equivalence of free BV field
theories).

######  Remark 4.20.

We omit the proof: it is straightforward, albeit tedious. The above is the
perturbative realization of general covariance: intuitively, the free BV
scalar field coupled to a metric is equivalent to the free BV scalar coupled
to an infinitesimally close metric in the same diffeomorphism orbit. This
lemma also states that for the free scalar field with differential $Q_{g}$ on
its BV space of fields, the first order deformation of $Q_{g}$ along the
$\mathscr{D}$-orbit starting at $g\in\mathscr{M}$ is exactly

$D=[L_{V},Q_{g}].$

This provides a nice coordinate-free form of the stress-energy tensor.

######  Remark 4.21.

Such a lemma holds for any BV theory which is generally covariant by our
definition: the only caveat is that the bookkeeping required to prove lemmas
like those above may be more painstaking. The issues are that the Lie
derivative $L_{V}$ manifests differently on different choices of fields, so
one must be careful, and the bookkeeping may be more painstaking with higher
$L_{\infty}$ terms. Additionally, certain BV fields have more than two terms
in their cochain complexes: the computations in that case are more cumbersome,
but only in the sense of needing to check multiple squares commute. This
happens for example in Example 2.68.

Our goal was not only to show that these two “infinitesimally close” spaces of
fields were equivalent, but to show that their associated observables were
similarly equivalent. This is what we do next. We need the following lemma:

######  Lemma 4.22.

If $\alpha:(V,d_{V})\to(W,d_{W})$ is an isomorphism of cochain complexes, then
there is an induced isomorphism
$\alpha:(\mathrm{Sym}(V),d_{V})\to(\mathrm{Sym}(W),d_{W})$ of cochain
complexes, where the differentials $d_{V}$ and $d_{W}$ are extended to the
respective symmetric algebras as derivations.

######  Remark 4.23.

It is similarly true that $\mathrm{Sym}(V^{\vee})$ and
$\mathrm{Sym}(W^{\vee})$ are isomorphic cochain complexes: the differentials
on $V^{\vee}$ and $W^{\vee}$ are induced by those on $V$ and $W$, and using
this lemma once more gives
$(\mathrm{Sym}(V^{\vee}),d_{V})\cong(\mathrm{Sym}(W^{\vee}),d_{W})$. (We abuse
notation so that $d_{V}$ and $d_{W}$ are the differentials induced from those
on $V$ and $W$, respectively.)

One might expect that because the naïve algebraic symmetric powers of
$\mathscr{F}_{g}$ are not what we use to define observables, we should be
wary; however, the completed projective tensor product we used to define
functionals is the necessary one in the case of infinite-dimensional vector
spaces for these constructions to carry over. We can now state a key theorem:

######  Theorem 4.24.

We have the following isomorphism of classical observables:

(49)
$\mathrm{Obs}^{\mathrm{cl}}(X,\widetilde{\mathscr{F}}_{g})\cong\mathrm{Obs}^{\mathrm{cl}}(X,\widetilde{\mathscr{F}}_{g+\varepsilon
L_{V}g}),$

where the isomorphism is induced by the isomorphism $\mathrm{Id}+\varepsilon
L_{V}$ from Lemma 4.19.

###### Proof.

Since Lemma 4.22 holds for infinite dimensional cochain complexes with the
definition of $\mathrm{Sym}(\widetilde{\mathscr{F}}_{g})$ as in Definition 2.2
(i.e. with the completed projective tensor product), we indeed have that the
isomorphism $\mathrm{Id}+\varepsilon L_{V}$ from Lemma 4.19 induces an
isomorphism of $(\mathscr{O}(\widetilde{\mathscr{F}}_{g}),\\{S_{g},-\\})$ and
$(\mathscr{O}(\widetilde{\mathscr{F}}_{g+\varepsilon
L_{V}g}),\\{S_{g},-\\}+\varepsilon\\{I_{g}(L_{V}g),-\\})$. This is the result.
∎

Recall that although we have done the precise computations in the case of the
massless free scalar field, the same statement holds in the case of any BV
theory with differential $Q_{g}$.

######  Remark 4.25.

As mentioned earlier, $\mathfrak{g}_{g}$ is a sheaf on $\mathbf{Riem}_{n}$,
and so identical computations as in Theorem 4.24 imply the analogous
equivalence of factorization algebras for the equivariant observables of
Proposition 4.3.

######  Remark 4.26.

This result follows almost directly from a theory exhibiting general
covariance; however, having isomorphisms written down explicitly and
recognizing their naturality when compared to the non-perturbative definition
of general covariance provides a sanity check, not to mention an enhanced
perspective on quantities like the stress-energy tensor.

Checking this theorem over a fixed $g\in\mathscr{M}$ and invoking
$\mathscr{D}$-equivariance implies that $\\{S_{g},-\\}$ is the differential
over the entire diffeomorphism orbit $\mathscr{D}\cdot g\subset\mathscr{M}$.
Similarly, seeing how $\\{S_{g},-\\}$ varies over a formal neighborhood of $g$
(i.e. expanding $\\{S_{g+\varepsilon h},-\\}$ in consecutive orders of
$\varepsilon h$) really grants us a view of the formal neighborhood of all of
$\mathscr{D}\cdot g$: this is precisely equivalent to considering the formal
neighborhood of $g$ as an element of the quotient stack $[\mathscr{M/D}]$.

######  Remark 4.27.

Note that although Theorem 4.24 is computed on a fixed $X$ for simplicity, it
holds at the level of factorization algebras, in the sense that
$\mathrm{Obs}^{\mathrm{cl}}(-,\widetilde{\mathscr{F}}_{g})$ and
$\mathrm{Obs}^{\mathrm{cl}}(-,\widetilde{\mathscr{F}}_{g+\varepsilon L_{V}g})$
define equivalent factorization algebras
$\mathbf{Riem}_{n}\to\mathbf{dgVect}$.

A few remarks on higher order versions of this theorem are made in Appendix
5.2.

##  5 Appendix

###  5.1 A detailed example

The following is a detailed example of how Chevalley-Eilenberg cochains arise
as functions on a formal neighborhood around a point in a stack: it is meant
to supplement what was shown in Lemma 3.8. Fix coordinates
$(x_{1},\ldots,x_{n})$ on $\mathbf{R}^{n}$ and consider an action
$P:G\to\textrm{Diff}(\mathbf{R}^{n})$ for a finite-dimensional Lie group $G$.
The total derivative of this map is
$\rho:\mathfrak{g}\to\textrm{Vect}(\mathbf{R}^{n})\cong
C^{\infty}(\mathbf{R}^{n})\otimes\mathbf{R}^{n}$, which for
$\alpha\in\mathfrak{g}$ has some coordinate expression:

$\alpha\mapsto\sum_{i=1}^{n}f(x_{i},\alpha)\partial_{i},$

where we use the shorthand $\partial/\partial x_{i}=\partial_{i}$. If we
restrict to a formal neighborhood of the origin,
$\widehat{\mathbf{R}}^{n}_{0}$, and compute its space of functions, we get the
usual Taylor series of functions about the origin,
$C^{\infty}(\widehat{\mathbf{R}}^{n}_{0})\cong\widehat{\textrm{Sym}}(T_{0}^{\vee}\mathbf{R}^{n})\cong\mathbf{R}\llbracket
x_{1},\ldots,x_{n}\rrbracket$, which we will denote
$\mathbf{R}\llbracket\mathbf{x}\rrbracket$ when convenient. Thus, restricting
the preceding derivative to the formal neighborhood of $0$ gives us
$\rho_{0}:\mathfrak{g}\to\textrm{Vect}(\widehat{\mathbf{R}}^{n}_{0})\cong\mathbf{R}\llbracket\mathbf{x}\rrbracket\otimes\widehat{\mathbf{R}}^{n}_{0}$,
which looks like:

$\alpha\mapsto\sum_{i=1}^{n}\hat{f}_{0}(x_{i},\alpha)\partial_{i},$

where $\hat{f}_{0}$ denotes the Taylor expansion of $f$ at $0$. This defines
an action of $\mathfrak{g}$ on $\mathbf{R}\llbracket\mathbf{x}\rrbracket$ by
derivations, and so we can thus define
$C^{\bullet}(\mathfrak{g},\mathbf{R}\llbracket\mathbf{x}\rrbracket)$.

Fixing a basis $\\{\alpha_{1},\ldots,\alpha_{m}\\}$ for $\mathfrak{g}$
(assuming finite dimension $m$), denote the dual basis for
$\mathfrak{g}^{\vee}$ as $\\{\alpha^{1},\ldots,\alpha^{m}\\}$. With these