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General Covariance from the Viewpoint of Stacks

General covariance is a crucial notion in the study of field theories in curved spacetime. A field theory defined with respect to a semi-Riemannian metric is generally covariant if two metrics which are related by a diffeomorphism produce equivalent physics. From a purely mathematical perspective, this suggest that we try to understand the quotient stack of metrics modulo diffeomorphism. We'll use the language of groupoids to do this concretely. Then we'll 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 the Batalin-Vilkovisky free scalar field theory, we recover a novel expression of the stress-energy tensor for that example. We'll describe how this construction nicely encapsulates but also broadens the usual presentation in the physics literature and discuss applications of the formalism.

1. Introduction 1.1. Overview. About 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 isomorphic 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.
The aim of this paper is to package general covariance in the Batalin-Vilkovisky formalism for classical field theories. 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 and then cast much of it in the language of stacks in Section 2.3: stacks provide a generalized notion of space which allows us to deal with quotient spaces that may be singular. The central idea is that a bundle of generally covariant classical Batalin-Vilkovisky (BV) field theories over the space of metrics on X, denoted M, descends to a bundle over the quotient stack [M/D] of the metrics modulo the diffeomorphism group of X, denoted D. I first define general covariance for a differential graded space of fields and only later explain how a mindful choice of a differential graded structure defines a BV field theory: this serves as an argument for the effectiveness and naturalness of both the BV formalism and the formulation of general covariance posited here.
Much of our concrete computations are in the regime of perturbative field theories, so we must consider formal neighborhoods in the quotient stack [M/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 perturbative field theory a formal moduli problem, in the style of [Lur11]. Along the way, we compute the Chevalley-Eilenberg cochains which are the derived ring of functions for the generally covariant field theory. In Section 4.3.1, we are able to recover and generalize the conservation of the stress-energy tensor, the conserved quantity associated to general covariance via Noether's Theorem, in derived deformation theoretic terms. Additionally, we compute a "perturbative equivalence" of observables when the theory is deformed by a vector field (which generate certain classes of diffeomorphisms), and describe how this equivalence describes the data of "higher" stress-energy tensors.
1.2. Future Directions. In the long run, much of this paper serves as a set-up for a few distinct projects. Firstly, the subject of anomalies in perturbative quantum field theory is concerned with how quantizing families of field theories over a space with some classical symmetry (here, general covariance) does not necessarily respect that symmetry. This relates to the underlying spacetime's topology via the Atiyah-Singer Families Index Theorem.
Secondly, a longer term goal of mine is to understand black hole entropy (and its Riemannian analogue) in the style of [Wald93] with the goal of enhancing it within the BV formalism. This is a pathway to understanding black hole thermodynamics more deeply.
More broadly, understanding physics over special parameter spaces tells us something about those parameter spaces themselves: my hope is that understanding field theories over the quotient stack of metrics modulo diffeomorphisms will help understand the topology of the associated moduli space, an actively researched topic which is summarized nicely in [TW15].
1.3. Acknowledgements. My gratitude primarily goes to my advisor Owen Gwilliam for suggesting this course of study: it was an ideal project, combining elements of my background in geometric relativity while propelling my knowledge of stacks and the BV formalism. I would also like to thank Eugene Rabinovich for his patience in working out many details in the BV framework with me, Andreas Hayash for being a great interlocutor in the subject of stacks, and Nicholas Teh for actively engaging with me about the content of this paper.

Equivariant Vector Bundles, General Covariance, and Stacks
To state a precise definition of general covariance, we must first understand what an equivariant vector bundle is and how we might use one to specify a family of field theories parameterized by some background data. Afterwards, we'll see how groupoids and stacks naturally arise in this context and provide additional advantages.

(Differential Graded) Equivariant Vector Bundles.
Definition 2.1. Let G be a Lie group. A smooth vector bundle π : V → M is said to be G-equivariant if: (i) both V and M are left G-spaces and π : V → M is a G-equivariant map, and (ii) for all g ∈ G and p ∈ M, g : π −1 (p) =: V p → V g·p is a linear transformation.
One must be mindful that in part (i) of this definition is packaged the information that the fibers V p of V → M could themselves be G-spaces over fixed point sets.
Theorem 2.2. 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: Remark 2.3. 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 nontrviail 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 the action M G is not free.
The above is a standard construction, found for example in [Tu20], but there is one more distinction which is significant for this work, which we'll define here: the fibers of the vector bundles we want to consider are differential graded as well as equivariant.
Definition 2.4. A differential graded vector bundle is a vector bundle V → M whose fibers V p are Z-graded vector spaces with a smoothly varying differential Q i p : Differential graded is often abbreviated "dg", and such a vector bundle may sometimes be denoted , depending on what we would like to emphasize within certain contexts.
is G-equivariant in the usual sense and (ii) the action by G induces a cochain map between fibers, i.e. for g ∈ G and for i ∈ Z, the following square commutes: Remark 2.6. It might be the case that the differential does not depend on p ∈ M for trivial V → M: in this case, the dg vector bundle is still G-equivariant, in a rather trivial way. However, it's easy to find examples in which there is such a dependence, and this is what we need to do to introduce our version of general covariance.
2.2. General Covariance. For our purposes, the space of fields F for a field theory is the sheaf of sections of some vector bundle F → X, or a subsheaf thereof, for X the underlying spacetime/manifold. In many instances, the field theory depends on some non-dynamical data, dubbed "background fields". For example, the action functional for a free massless scalar field theory on X is where the scalar fields are ϕ ∈ C ∞ (X) =: F. There is a clear dependence in S on a Riemannian metric g, which is thus the background field in this example. Most of what follows works just as well for the more physically relevant Lorentzian case, in which the Laplacian ∆ g is replaced by the d'Alembertian g .
Construction 2.7. At this point, there is no dependence in F on Met(X). The key maneuver is to upgrade the "naïve" space of fields to a differential graded space of fields, where the differential for a free theory imposes the equations of motion derived from the action functional cohomologically, where the differential depends on Met(X): this upgrade is a centerpiece of the Batalin-Vilkovisky formalism. For the above example, which is indeed a free theory since S is quadratic in ϕ, the "upgraded" fields are where Q g ϕ = ∆ g ϕvol g . We can thus define the differential graded vector bundle π : (F, Q) → M, where we denote Met(X) as M from now on whenever X is implicit, with π −1 (g) = F g . This bundle represents a family of field theories which vary over M. The preceding construction can be done with many field theories which depend on a background metric, and so we will keep related results as general as possible.
In our ongoing example, both components of the graded space of fields have an action by the diffeomorphism group of X-denoted D when unambiguous-via pullback: for f ∈ D, ϕ ∈ C ∞ (X), and µ ∈ Dens(X), f · ϕ = f * ϕ = ϕ • f and f · µ = f * µ. Additionally, D acts on Met(X) via pullback: f · g = f * g. What's special about this example is that the differential Q g commutes with diffeomorphisms in the following sense: . This result is equivalent to the fact that the Laplacian commutes with diffeomorphisms (which is shown in [Can13]). Thus: Lemma 2.8. Any f ∈ D defines a cochain map between fibers of (F, Q) → M: This result also implies a significant and useful corollary: Corollary 2.9. If g ∈ M is a fixed point of f ∈ D (in other words if f is an isometry of g) and if Q g ϕ = 0, then Q g (f * ϕ) = 0. In other words, isometries of the metric g act on the space of solutions to ∆ g ϕ = 0 (Laplace's equation).
Remark 2.10. The differential Q g of the differential graded scalar fields has a very clear dependence on the base space M. In fact, as a topological space, the bundle is trivial, as it is (C ∞ (X) ⊕ Dens(X)) × Met(X): the differential Q g defines any nontriviality as a differential graded vector bundle. Lemma 2.8 agrees with the definitions laid out in the previous section, although evenmtually the infinite-dimensionality of M, D, and F g demand we find a way to make them Fréchet manifolds, so that the finite dimensional constructions hold true.
The above lemma thus shows us how useful it can be to express geometric and analytic information in this language: field theories which are diffeomorphism-equivariant in the way the scalar field theory is above have a special name. We explicitly define what we mean by field theory in Section 4.1, but for now state the key definition with what we have.
Definition 2.11. Let the differential graded vector bundle (F, Q) → M define a family of free field theories on X. If it is a Diff(X) =: D-equivariant as a differential graded vector bundle, we call the theory generally covariant.
Remark 2.12. Field theories which satisfy general covariance are therefore not sensitive to all of M, but only to the moduli space of metrics modulo diffeomorphism, M/D. Because most physically relevant (and interesting) metrics have many isometries, the naïve quotient M/D is a singular space. The need for a more general idea of a space which treats such "manageable" singularities is desirable: this is where stacks will take the spotlight.
Example 2.13. A tangible example of M/D being singular is the one in which the underlying Riemannian manifold is X = R n along with the flat metric η. It's well known that the isometry group of (R n , η) is O(n) ⋉ R n , where the R n in the semidirect product is the additive group of translations of R n . In particular, O(n) ⋉ R n is a subgroup of Diff(R n ) which stabilizes η ∈ M(R n ), meaning that the corresponding point in the quotient is singular. Moreover, O(n) ⋉ R n acts on the space of solutions to any generally covariant theory on (R n , η) . The preceding definition therefore "enlarges" our usual idea of equivalence beyond isometries.
Although we have shown above some properties of a particular free field theory, notice that we have not yet defined a field theory in full generality: we will cover this in Section 4.1.
2.3. Groupoids and Gauge Fields. 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 [Car11].
Definition 2.14. A groupoid G is a small category in which all arrows are invertible. Common where G 1 is the set of arrows and G 0 the set of objects; s sends an arrow to its source object, and t sends it to its target. Every such G has an identity map e : G 0 → G 1 sending an object to its identity arrow, an inverse map i : G 1 → G 1 sending an arrow to its inverse, and a multiplication map m : G 1 × G 0 G 1 → G 1 that concatenates arrows. s, t, e, i, and m are called the structure maps of G.
Example 2.15. 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 G 0 = M and its set of arrows is G 1 = M × G, so that we can write it as In this case, s(p, g) = p and t(p, g) = g · p. Common notation for the action groupoid is M//G: the action groupoid is defined as a first step toward understanding quotient spaces which are not necessarily smooth, in the sense that the action of G could fix certain points in M and so M/G could be singular.
is a groupoid such that both the space of arrows G 1 and space of objects G 0 are smooth manifolds, all structure maps are smooth, and the source and target maps s, t : G 1 → G 0 are surjective submersions. (In other words, a Lie groupoid is a groupoid internal to the category of smooth manifolds.) Moreover, if π : V → M is a G-equivariant vector bundle, then we could also define its action groupoid V × G − → − → V =: V G . Both V G and M G are in fact Lie groupoids, and V G is a vector space object over M G in the category of Lie groupoids. Thus, by some abuse of notation, we can view π : V G → M G as a vector bundle.
Example 2.17. A groupoid which is well known to physicists is that of gauge fields and the gauge transformations between them. The set of objects is the set A of gauge fields-or connection one-forms to the mathematically inclined-and the set of arrows is A × G, where G is the group of gauge transformations. Gauge-invariant theories are those that produce equivalent physics for two gauge fields A and A ′ as long as A g − → A ′ for some g ∈ G. Therefore, it is essential to understand the quotient A/G. Issues arise when particular A ∈ A are fixed points of a subgroup of G: thus, groupoids and in particular stacks become useful tools here.
In the context of general covariance, the gauge fields get replaced by metrics and perhaps additional "matter fields", and so diffeomorphisms play the role of gauge transformations. Due to this, many analogies are usually drawn between the two examples.
Remark 2.18. As the preceding narrative shows, groupoids are very useful when it comes to handling data pertaining to quotient spaces. They are key players in the definition of stacks, which are the spaces which we eventually would like to replace ordinary manifolds with when it comes to including quotient data in the space.
The following depends on a steep increase in categorical language. To motivate this more abstract technology, we'll just say that 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 Mfd op (or CommRing op if one is an algebraic geometer) to Set, the functor lands in Grpd, 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. (1) Given objects P i ∈ X(U i ) and isomorphisms ϕ ij : i . This is called effective descent data.
(2) Given P, P ′ ∈ X(N) and isomorphisms ϕ i : Example 2.21. An essential example derived from the above definition is that of [pt/G], for pt a point. Applying the definition shows that [pt/G](X) is precisely 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 [pt/G] with BG, since [X, BG] is equivalent to Bun G (X) modulo bundle ismomorphisms.
Notice moreover that defining a vector bundle V → [pt/G] amounts to fixing the vector space V (a vector bundle over the point) as well as a representation ρ : G → End(V ). In other words, we have an equivalence of categories: Remark 2.22. 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 2.23. For M a smooth G-space, we have the following equivalence of categories: 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've been defined starting with an action groupoid. More than one groupoid could present a stack (this is described and expounded on in [Car11]), so it is reassuring that the stack itself contains this data.
A deeper level of care must be taken for our motivating example [Met(X)/Diff(X)], which is presented by the Lie groupoid Met(X)//Diff(X). Usually, the space of metrics on a manifold and its group of diffeomorphisms are infinite dimensional, so defining vector bundles or other structures which depend on the differentiability of Met(X) will require us to consider a special class of manifolds called Fréchet manifolds, which we now define (the definitions we give here are adapted from those in [KM96]).
Definition 2.24. 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. Fréchet Lie groupoids are groupoids internal to the category of Fréchet manifolds.
is an open submanifold, is a Fréchet manifold if X is compact, and similarly, the diffeomorphism group Diff(X) is a Fréchet Lie group as long as X is compact [KM96]. 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, it should be noted that many physically relevant Lorentzian manifolds are assumed to have the topological form Σ × R, for Σ a spacelike compact submanifold and R the "time" dimension. This is the path through which many of the Riemannian results are translated into the Lorentzian regime.

Formal Stacks
Much of our work is concerned with perturbative field theory (classical or quantum, although in this paper we are concerned with the former), so a key step will be to associate to a (differential graded) equivariant vector bundle a vector bundle over a formal moduli problem: in our case, this formal moduli problem is a formal neighborhood in a quotient stack. This way, we'll find the right Chevalley-Eilenberg cochains necessary to perform concrete computations, and for that we need to understand tangent complexes.
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.
Construction 3.1. Let M be a smooth G-space and let Stab(p) ⊆ G be the stabilizer subgroup of p ∈ M. The G-orbit of p thus looks like a copy of G/Stab(p) lying in M. If we consider the map t p : G → M defined as t p (g) = g · p, 1 then its differential dt p can be used to define a 2-term cochain complex of vector spaces: where g is in cohomological degree −1 and T p M is in degree 0. Alternative notation is Note that Stab(p) could be discrete here, although that isn't seen in T p [M/G]. We can also compute it's the usual tangent space of the coarse quotient at points p ∈ M where the action is free. As it turns out, this is exactly the tangent object we are looking for, as the notation suggests: further details are wonderfully detailed in [An21].
This inspired the saying that "smooth stacks are geometric spaces whose tangent spaces are complexes concentrated in nonpositive cohomological degree". The above is concentrated in degrees −1 and 0, but if we were to consider higher stacks, there would be data in lower degrees, representing "higher" infinitesimal symmetries. In the case of quotient stacks, we're lucky to have a concrete way of realizing their associated tangent complexes.
Example 3.3. Consider the natural action of the group of diffeomorphisms D of a manifold X on the space of Riemannian metrics M: t g (f ) = f * g. According to [KM96], the Lie algebra of D at the identity diffeomorphism is Vect c (X), the set of compactly-supported vector fields. However, we will use the all of Vect(X) = Γ(X, T X )-which is a Lie algebra in its own right-in our definition of the tangent complex (for a compact manifold this indeed makes no difference).
We know that T g M ∼ = Γ(X, Sym 2 (T ∨ X )), so that we can compute where g = M × g, considering that the base space M is implicit. 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.

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 = 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, Met(X) = M is a convex cone in Γ(X, Sym 2 (T ∨ X)), so that we will be eventually specializing these constructions to vector spaces or "nice" subsets thereof anyway.
, the Taylor series ring around p ∈ M, although this isomorphism is not canonical. We will use the latter, and call the Taylor series ring O p when unambiguous.
Remark 3.6. In the case that the tangent bundle of M is trivial, i.e. T M = M × T p M for any p ∈ M, the preceding non-canonical isomorphism is in fact canonical. Because the space of metrics M is contractible, its tangent bundle is trivial, so that we can rest easy in what follows.
The action of G on M is defined by a map P : G → Diff(M). Taking its total derivative gives us a map ρ  Proof. It is a quick exercise to show that the underlying graded commutative algebras of are identical, as long as one is careful to employ the non- . From there, it is sufficient to show that the Chevalley-Eilenberg differentials are equivalent, which is also left as a brief exercise.
Remark 3.8. This lemma shows clearly that the dg Lie algebra is of utmost importance. In fact, there is an equivalence of (∞, 1)-categories between pointed formal moduli problems and dg Lie algebras, as described in Theorem 2.0.2 in [Lur11]. We will not define formal moduli problems in this paper, but simply choose the relevant side of this equivalence to perform concrete computations and keep the other side in mind for intuition: the equivalence states that the dg Lie algebra L p completely defines the data of the formal Example 3.9. In light of this lemma and Example 3.3, the dg Lie algebra we must consider in the context of general covariance is thus By applying Lemma 3.7, we see that the ring of functions on the formal neighborhood of [g] ∈ [M/D] is C • (Vect(X), O( M g )), which we interpret as the derived invariants of O( M g ) with respect to the D-action. Our definition of general covariance from earlier when properly "localized" would then imply that the observables of such a field theory F g over g ∈ M form a module over C • (Vect(X), O( M g )) = C • (L g ). Before we dig too deeply in that direction, let's consider a finite dimensional example.
Remark 3.10. It should be noted that because Vect(X) and O( M g ) are infinite dimensional, the definition of C • (L g ) is not precisely the one from the finite dimensional case. In a rigorous way, C • (L g ) represents the same data as it would if its inputs were finite dimensional, but we must be careful with functional analytic issues to ensure that all of the rings are well defined. The precise definition is presented in Definition 4.2.
Example 3.11. Fix coordinates (x 1 , . . . , x n ) on R n and consider an action P : G → Diff(R n ) for a finite-dimensional Lie group G. The total derivative of this map is ρ : g → Vect(R n ) ∼ = C ∞ (R n ) ⊗ R n , which for α ∈ g has some coordinate expression: where we use the shorthand ∂/∂x i = ∂ i . If we restrict to a formal neighborhood of the origin, R n 0 , and compute its space of functions, we get the usual Taylor series of functions about the origin, C ∞ ( R n 0 ) ∼ = Sym(T ∨ 0 R n ) ∼ = R x 1 , . . . , x n , which we'll denote R x when convenient. Thus, restricting the preceding derivative to the formal neighborhood of 0 gives us ρ 0 : g → Vect( R n 0 ) ∼ = R x ⊗ R n 0 , which looks like: wheref 0 denotes the Taylor expansion of f at 0. This defines an action of g on R x by derivations, and so we can thus define C • (g, R x ). Fixing a basis {α 1 , . . . , α m } for g (assuming finite dimension m), denote the dual basis for g ∨ as {α 1 , . . . , α m }. With these coordinates, we can write C • (g, R x ) as R α 1 , . . . , α m , x 1 , . . . , x n , where the α k are in degree 1 and the x k in degree 0. Thus, it is sufficient to see what the differential d CE does on an element of the form α k ⊗x l , for α k ∈ g ∨ [−1] and x l ∈ R x , to classify its behavior. Momentarily viewing α k as a degree 1 element of just C • (g) = Sym(g ∨ [−1]), and noting that d CE : where c k ij are the structure constants for g. Concurrently, for x l ∈ R x , Therefore, by requiring the usual derivation rules, we get, which we extend to the rest of C • (g, R x ) with the Leibniz rule. A coordinateless way of writing this is d CE = [−, −] ∨ g + ρ ∨ 0 , where ρ ∨ 0 encodes a dual to the action map ρ 0 : Vect( R n 0 ) → g ∨ ⊗ Vect( R n 0 ) as described implictly above. In this example, d is in fact a vector field, specificially on the formal neighborhood of 0 in the stack [R n /G], where we have used the Einstein summation convention over repeated indices in the last step.
To make things easier to grasp, let's consider the case of SO(2) acting on R 2 via rotations. Then g = so(2) and the Taylor series ring about the origin is R x, y . The representation map from which we can define C • (so(2), R x, y ). We will leave it as an exercise to the reader to show that H 0 (so(2), R x, y ) is the set of rotation-invariant Taylor series around 0. This isn't surprising: more generally, in the case of SO(n) acting on R n , H 0 (so(n), R x 1 , . . . , x n ) is the set of SO(n)-invariant Taylor series around the origin in R n . Another enlightening exercise is to consider the appropriate Chevalley-Eilenberg cochains coming from formal neighborhoods of points away from the origin; e.g. the zeroth cohomology group of the cochains around (x 0 , 0) is isomorphic to R x − x 0 . Notice there that the vector fields coming from the action at these non-fixed points have constant coefficient terms.
Remark 3.12. Much like how a quotient stack "builds in" group action data into its definition, functions on a formal neighborhood [M/G] p in the stack, namely C • (g,Ô p ), have "built into" them all of the g-invariant data. Concretely, C • (g,Ô p ) has the usual ring of functionsÔ p as a subset: tensoring with Sym(g ∨ [−1]) and imposing the Chevalley-Eilenberg differential remembers the data of g acting on M p , and therefore on O( M p ) ∼ =Ô p as well.
3.3. Vector Bundles over a Formal Stack. Now that we've 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 3.5); however, we'll present the global picture first, since general covariance is first presented in such a context. Lemma 3.14. The ring of functions on and that both parts of this tensor product are g-modules, we can define the CE cochains C • (g, O p ⊗ Sym(V ∨ p )). In conjunction with Lemma 3.7, these are the derived 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 where τ ∨ Mp and τ ∨ Vp are the "duals" (as in Example 3.11) to the induced actions τ Mp and τ Vp on O p and Sym(V ∨ p ), respectively.
Remark 3.15. In finite dimensions, the isomorphism [(p * V )/ G] ∼ = [(p * V )/g] holds true, since G exp ∼ = g. However, we mentioned in Example 3.3 that it was no longer the case that there is a diffeomorphism between the formal neighborhood of the identity diffeomorphism in Diff(X) and its Lie algebra Vect c (X) of vector fields with compact support. Therefore from here onwards, we will stick to [(p * V )/g], as it is well-defined in the infinite-dimensional Fréchet manifold case.

Field Theories as a Vector Bundle over a Formal Stack
To return fully to field theories, we must consider the above in the case that the fibers V p are differential graded. To expedite our arrival at the desired destination of understanding general covariance for perturbative field theories, we will perform the relevant computations for the example of the D-equivariant differential graded vector bundle (F, Q) → M with fibers F g = C ∞ (X) Qg − → Dens(X)[−1] and differential Q g ϕ = ∆ g ϕvol g . Thus, for clarity, let's call the actions described in Lemma 2.8 τ M : D → Diff(M) and τ F : D → Diff(F). Remember that there is also an "action" on the differential, sending Q g to Q f * g for f ∈ D: it clearly comes from the action of D on 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 3.3. τ Mg : Vect(X) → T g M is what we already considered earlier, namely V → L V g, and the fibers are similar: Lemma 4.1. The action of D on the underlying graded vector space of F g defines an action of Vect(X) on F g . It has a degree 0 part τ Fg : Vect(X) → T ϕ C ∞ (X) ∼ = C ∞ (X) and a degree 1 part τ Fg : Vect(X) → T µ Dens(X) ∼ = Dens(X); they are, respectively, V → L V ϕ and V → L V µ.
We remind the reader once more that although Lie(D) ∼ = Vect c (X), we define these actions using all of Vect(X) with no apparent issues, and that for compact X, there is no difference.
The most interesting and physically relevant detail which must be addressed is what happens to the differential Q g under this infinitesimal action. For this, we will give the definition of a Batalin-Vilkovisky classical field theory. To do this, we must consider preliminary definitions.
4.1. The Batalin-Vilkovisky Formalism. The infinite dimensionality of the BV fields F means we cannot take the naïve algebraic symmetric powers to define their space of functions. We therefore have the following definition which plays an identical role, but for the infinite dimensional case. The definitions that follow are from Chapter 5, Section 3 of [Cos11]. Let F → X be a graded vector bundle whose sheaf of sections F is the fields for a field theory.  To be fully precise, F = Γ(X, F ) is a nuclear Fréchet space and ⊗ denotes the completed projective tensor product, so that F ⊗k := Γ(X × · · · × X, F ⊠ · · · ⊠ F ), meaning each Hom(F ⊗k , R) S k is a space of continuous functionals endowed with the strong dual topology: i.e. a space of distributions. When unambiguous, we may sometimes denote this ring as Sym(F ∨ ). We now introduce a key related set of functionals.
for fields φ ∈ F and differential operators D i ∈ D X , where D X denotes the differential operators on X.  This lemma shows that O loc (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 O loc (F), and local functionals are key in defining the Poisson bracket, as we will see below.
Definition 4.7. For F → X a graded vector bundle, a constant coefficient k-shifted symplectic structure is an isomorphism of graded vector spaces that is graded antisymmetric.
Example 4.8. For BV field theories, k = −1. A good example is the one we've had all along: the symplectic structure ω on F g = C ∞ (X) for ϕ and µ in degrees 0 and 1, respectively. We can thus write S g (ϕ) as ω(ϕ, Q g ϕ).
Remark 4.9. 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 O loc (F) ⊂ O(F). This is not the case however for all of O(F), for functional analytic reasons which are outside the scope of this paper [Cos11]. Let's denote the Poisson (anti-)bracket induced by ω as {−, −}.
Definition 4.10. A Batalin-Vilkovisky classical field theory on a smooth manifold X is a differential Z-graded vector bundle F → X equipped with a −1-shifted symplectic structure ω and an action functional S ∈ O loc (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(ϕ) = ω(ϕ, Qϕ) + I(ϕ), where Q is a linear differential operator and I ∈ O loc (F) is at least cubic. *(3) The complex (F, Q) is elliptic.
A free theory is one in which I = 0: i.e. the action functional S is purely quadratic. regardless of whether or not the BV theory is free. But for a free theory, it can be shown that In addition, Condition (3) is singled out because the assumption of the ellipticity of (F, Q)being in correspondence with the assumption of the ellipticity of Q as a differential operator-is temporarily optional. In particular, the classical physics which is the content of this paper works just as well for the hyperbolic operators used in the Lorentzian regime; it is when we consider renormalization for the quantized field theory that we may require ellipticity.
Definition 4.12. Let F → X be a differential graded vector bundle with differential Q on its sheaf of sections F. Then the dg commutative ring of (global) classical observables for the theory defined by (F, Q) is Obs cl (X, Remark 4.13. We will briefly describe how Obs cl (X, F) is to be understood as the dg ring of functions on the derived critical locus of the action functional S : F 0 → R, where the degree zero part of the dg fields is the "naïve" original space of fields, following [CG21]. The ordinary critical locus Crit(S) is the intersection of the graph We thus get its commutative algebra of functions to be However, 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 Crit h (S), which has ring of functions This is now a commutative dg algebra instead of an ordinary commutative algebra, and it can be realized as the complex In the preceding constructions, we have a priori chosen (F, Q) so that it is equal to T ∨ [−1]F 0 , but now we see how the Batalin-Vilkovisky dg fields arise naturally from a derived geometric perspective. Moreover, the differential on Γ(F 0 , Λ • T F 0 ) is contracting with the 1-form dS ∈ Ω 1 (F 0 ), and this can be shown to be equivalent to {S, −}, as we would expect.
, the underlying graded ring of Obs cl (X, F g ) is O(F g ), which is concentrated in nonpositive degrees, as Definition 4.2 implies. The action functional S g (ϕ) defined in Equation (2) is a degree 0 element of O(F g ), but also defines a degree 1 differential on O(F g ) as {S g , −}: thus, {S g , S g } must be a degree 1 element of O(F g ). Since in this example O(F g ) is concentrated in nonpositive degrees, the classical master equation holds vacuously. Thus, our ongoing example of the free massless scalar field with metric background g defines a free BV classical field theory, since the other requirements are easily satisfied. Along with noting that Q g = {S g , −}, all computations relevant in Lemma 2.8 carry over to imply that (F, Q) → M exhibits general covariance as a free BV classical field theory. Now that we have seen how the free massless scalar field defines a BV theory, and computations from earlier imply that it is generally covariant, we must return to understanding how the differential Q g behaves under perturbations of g by infinitesimal diffeomorphisms, to complete the picture we started in Lemma 4.1.

The Main
Result. The way to do this is to consider how Q g behaves under arbitrary perturbations g + εh, for h ∈ T g M and only then invoke that h = L V g comes from a vector field V . But we already have the machinery to do this! The preceding sentence amounts to pulling back the dg vector bundle (F, Q) → M to be over the formal neighborhood of g ∈ M, and seeing what the "full differential" Q g+εh looks like on this formal neighborhood.  Proof. For our purposes, we only consider the part of the formal neighborhood of [g] ∈ [M/D] which has the form [ M g /Vect(X)] (we may be losing a bit of data here, but it is a cost worth paying for simplicity and physical relevance of the infinitesimal diffeomorphisms). By the equivalence of categories from [Lur11] which we are taking for granted, this is equivalent to the dg Lie algebra L g = Γ(X, T X ) so that the dg ring of functions on this formal neighborhood is .
The ring of functions on the fiber part of the pullback is simply C • (Vect(X), Obs cl (X, F g )), since it is O(F g ) with the differential {S g , −} and the implicit action of Vect(X) on the theory and thus on its observables. Hence, the underlying dg ring of functions on Tot(g * [F/D]) is the underlying dg ring of C • (L g , Obs cl (X, F g )). Both dg rings have Chevalley-Eilenberg differential Here, the first three terms are the usual Chevalley-Eilenberg differential concerned with the dual of the bracket on Vect(X) and the actions of Vect(X) on Sym(T ∨ g M) and O(F g ), and the fourth term is the differential on the free field theory over [ M g /Vect(X)]. Since the underlying rings agree and the CE differentials do, too, this gives the result.
Remark 4.17. 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. We have not given the precise details of a "full" L ∞ action of L g on Obs cl (X, F g ), but its existence is implicit in the theorem. In Section 4.3.2, we discuss some basics of what such an action might look like concretely.
Remark 4.18. A central expectation rooted in general covariance is that if we perturb the metric by an infinitesimal diffeomorphism-in other words, if we vary g to g + εL V g for a vector field V -then any induced variation on Q g would be trivial. This is precisely the perturbative manifestation of general covariance, and we would like to package it in the language we have thus far described. Moreover, it would imply that {S g , −} is only sensitive to diffeomorphism classes of g at the perturbative level: this is a necessary feature for it as a term in d CE above.
4.3. The Stress-Energy Tensor. Now that we've outlined the big picture, let's get our hands dirty with some concrete computations in the special case of the free massless scalar field to confirm the assertion at the end of the preceding remark.
To begin, let's consider an arbitrary first order deformation of the Laplacian ∆ 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's compute Writing ∆ gt in coordinates and not evaluating at t = 0 for now, we have: Recall that for a one-parameter family of invertible matrices A(t), we have Using this and a few other manipulations, expression (10) reduces to Denote the derivative of g t at g 0 = g as δg := ∂ t g t | t=0 (this is the traditional notation in physics). Evaluating at t = 0 gives the following lemma.
Lemma 4.19. The first order deformation of the Laplacian ∆ g with respect to the metric g is Moreover, if we assume the deformation δg ∈ T g M is induced by an isometry of g, then the first order deformation of the Laplacian is identically zero.
Proof. The first claim follows immediately from the preceding computations and evaluating expression (11) at t = 0. Next, saying that the deformation δg is induced by a diffeomorphism means that δg = L V g, where V is the vector field which generates that diffeomorphism. If V is a Killing field for g, i.e. it generates an isometry of g, then it satisfies L V g = 0. The first two terms in the sum are thus zero, and the third is zero since L V g µν = 0 when L V g µν = 0, which is L V g = 0 with indices.
Remark 4.20. A few remarks are in order here. First of all, the computations above involve a combination of index-dependent and independent notation. For example, often in the physics literature, we write Tr(g −1 δg) = g µν δg µν .
I avoid doing so because the indices are fully repeated, and so not related to the indices left in the expression (which are necessary to retain). Moreover, the above computation is done with the action functional (2) in mind. Thus, any difference with the stress-energy tensor computations using the functional which is more common in the physics literature, differs only by boundary terms, as we'll see.
Indeed, our next step is to understand the stress-energy (alternatively, energy-momentum) tensor for the free massless scalar; however, we give the definition for any such field theory.
Construction 4.21. Let S g ∈ O loc (F) be an action functional for a space of fields F which depends on a fixed background metric g ∈ M. It can thus be written as where φ ∈ F and L g (φ) 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 4.19 to compute: The notation invoked on the left hand side is common in physics literature, and defined this way in [Wald84]. Up to boundary terms which we can safely ignore, (14) can be written as for some T µν (g, φ) (or simply T µν ) which depends on the fields φ and on the metrics g.
Definition 4.22. T µν is the stress-energy (or energy-momentum) tensor of a field theory on X with fields φ ∈ F and action functional S g depending on g ∈ Met(X).
Example 4.23. To compute the stress-energy tensor of free massless scalar field, we'll begin A standard result from Riemannian geometry is that L V g µν = ∇ µ V ν + ∇ ν V µ , and since T µν is symmetric by definition, the above must be where we invoked integration by parts and the fact that ∇ µ vol g = 0 in the equality. Then, standard computations for generally covariant theories (which can be found in Appendix E of [Wald84]) show that for on-shell fields (here meaning ϕ such that ∆ g ϕ = 0), the above integral is identically zero. For this to be true, it must be the case that In the language of Noether's Theorem, the stress-energy tensor T µν is the conserved current associated to general covariance, a symmetry of a field theory coupled to a metric.
(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 stressenergy tensor for a generally covariant field theory. This is what we explore next: we will start to think more homologically, but remember that it is all rooted in the preceding geometry. 4.3.1. A Stress-Energy Theorem. We will recapitulate a few key ideas formulated thus far from a slightly different angle in anticipation of the central statement.
Construction 4.26. Let's consider the "infinitesimal general covariance" property more formally. Insight (1) suggests that the action functionals S g (ϕ) and where this equality holds modulo ε 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+εL 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 Obs cl (X, F g ), so that we would like , −}, the induced differential on Obs cl (X, F g+εL V g ) from Q g+εL V g , to be similarly equivalent. To give all of the above hands and legs, we must rigorously define F g+εL V g and its observables in the first place.
Let D 2 = R[ε]/(ε 2 ) denote the (real) dual numbers. We can tensor F g = C ∞ (X) Qg − → Dens(X)[−1] with D 2 to get F g ⊗ D 2 , whose elements can be written as ϕ 0 + εϕ 1 in degree 0 and similarly for degree 1. The differential Q g+εL V g looks like It remains only to find D, which will depend on g and V and must be so that commutes. The downward-pointing arrows are Id + εL 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 2.8. 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 (18) suggest that we try D = [L V , Q g ].
Lemma 4.27. Let F g := (F g ⊗ D 2 , Q g ) and F g+εL V g := (F g ⊗ D 2 , Q g + ε[L V , Q g ]). Then the map Id + εL V : F g → F g+εL V g is a cochain isomorphism (i.e. it is an equivalence of free BV field theories).
Proof. Let's begin by checking that Id + εL V is indeed a cochain map: in other words, we must show that the preceding square commutes for D = [L V , Q g ]. Let ϕ 0 + εϕ 1 ∈ F 0 g . Then On the other hand, we have Thus, Id + εL V defines a cochain map from F g to F g+εL V g . Now we must check that Id − εL V is an inverse. Let ϕ 0 + εϕ 1 ∈ F 0 g and let's compute: This computation works for elements in F 1 g and F 1 g+εL V g just as well, and so we see that the inverse is what we may have expected. F g and F g+εL V g are thus isomorphic as cochain complexes.
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 D-orbit starting at g ∈ M is exactly This provides a nice coordinate-free form of the stress-energy tensor.
Remark 4.28. For any generally covariant free BV theory with dg space of fields (F g , Q g ), a similar formula holds. The caveat is that the Lie derivative L V manifests differently on different choices of fields, so one must be careful. 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.
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: Remark 4.30. It is similarly true that Sym(V ∨ ) and Sym(W ∨ ) are isomorphic cochain complexes: the differentials on V ∨ and W ∨ are induced by those on V and W , and using this lemma once more gives (Sym(V ∨ ), d V ) ∼ = (Sym(W ∨ ), 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 F g aren't 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 4.31. We have the following isomorphism of classical observables: where the isomorphism is induced by the isomorphism Id + εL V from Lemma 4.27.
Proof. Since Lemma 4.29 holds for infinite dimensional cochain complexes with the definition of Sym( F g ) as in Definition 4.2 (i.e. with the completed projective tensor product), we indeed have that the isomorphism Id + εL V from Lemma 4.27 induces an isomorphism of (O( F g ), {S g , −}) and (O( F g+εL V g ), {S g , −} + ε{I g (L V g), −}). Here, the two differentials are those induced by Q g and Q g + ε[L V , Q g ], as computed in Equation (18) and what followed. This is the result.
Recall that although we've done the precise computations in the case of the massless free scalar field, the same statement holds in the case of any free BV theory with differential Q g .
Remark 4.32. 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.
The preceding theorem gives a concrete description of C • (L g , Obs cl (X, F g )) with differential as a bona fide Chevalley-Eilenberg cochain complex: in particular, checking this theorem over a fixed g ∈ M and invoking D-equivariance implies that {S g , −} is the differential over the entire diffeomorphism orbit D · g ⊂ M. Similarly, seeing how {S g , −} varies over a formal neighborhood of g (i.e. expanding {S g+εh , −} in consecutive orders of εh) really grants us a view of the formal neighborhood of all of D · g: this is precisely equivalent to considering the formal neighborhood of g as an element of the quotient stack [M/D].

4.3.2.
A note on higher orders. We would like to make sense of the main theorem in the case that we do not "cut off" the orders of ε after only a linear perturbation. The linear perturbation gives the necessary data to understand the stress-energy tensor in the usual way; however retaining higher orders of ε to compute "higher" stress-energy tensors may be relevant, and the BV formalism gives an ideal way of interpreting and packaging that data. To put it plainly, we'd like to expand {S g+εh , −} in more powers of εh.
Construction 4.33. A concrete jumping-off point here would be to consider that for a generally covariant theory, we have = 0 for any k > 0. This is the general form of Equation (18). For now, let's stick with k = 2. The above should have an analogous unpacking to the one following Equation (18); however, we then need to make sense of d 2 dt 2 (f * t ϕ) t=0 . A short exercise in differential geometry gives us that as one might expect; the same equation holds for any k > 0, and the right side is in fact equal to 1 k L k V ϕ, where L k V denotes taking the Lie derivative with respect to the vector field V k times. We can now consider a similar computation to the one in Lemma 4.27, replacing D 2 with D 3 := R[ε]/(ε 3 ) and using the above identity, to figure out what operator D 2 makes the following square commute: where we have renamed D = D 1 from above to emphasize the order of ε it is associated to. Although a proof won't be presented here (it is more arduous, though not more difficult than that of Lemma 4.27), we state the following lemma.
Remark 4.35. The operator D 2 = 1 2 [L 2 V , Q g ] − [L V , Q g ]L V thus represents a sort of "higher" stress-energy tensor for a generally covariant theory, in the same way that D 1 = [L V , Q g ] did so in the first order case. It also satisfies some conservation property (analogous to ∇ µ T µν = 0): otherwise, we wouldn't have this cochain isomorphism of field theories. However, it would be harder to pin down a physical interpretation of the associated conservation law.
Remark 4.36. Additionally, we could now update Theorem 4.31 so that it holds up to second order in ε: the isomorphism of observables is induced from the isomorphism Id + εL V + ε 2 2 L 2 V of the field theories. The proof is otherwise the same. Moreover, it might be clear to the reader by now that these results can be generalized to arbitrarily high orders of ε. In that case, we can expand the differential as on the fields-as long as the metric perturbation is induced by a vector field-and pick out D k for all k > 0 so that we get an analogous commutative square, with the isomorphism Thus, the isomorphism of observables (30) Obs cl (X, F g ) ∼ = Obs cl (X, F g+εL V g ) remains true to all orders of ε, since we see from the above that all of the appropriate D k exist, regardless of how difficult they are to compute or interpret. An interesting course of study would be to consider general expansions of {S g+εh , −} in εh: a first step in this case would be to generalize Lemma 4.19 to higher derivatives with respect to t. The expansion of {S g+εh , −} would concretely define the L ∞ action of L g on Obs cl (X, F g ).
Remark 4.37. As well as checking for agreement with a generally covariant theory when h = L V g as we did above, we may want to consider perturbations in the direction of various geometric flows. For example, it would be fruitful to consider field theories over metrics related by the Ricci flow, and a first step here would be to perturb a fixed metric g in the direction of that flow.