General covariance from the viewpoint of stacks

Abstract

General covariance is a crucial notion in the study of field theories on curved spacetimes. A field theory on a manifold X 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.

Appendix

1.1 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 $\textbf{R}^{n}$ and consider an action $P: G \rightarrow \text {Diff}(\textbf{R}^{n})$ for a finite-dimensional Lie group G. The total derivative of this map is $\rho : \mathfrak {g} \rightarrow \text {Vect}(\textbf{R}^{n}) \cong C^{\infty }(\textbf{R}^{n}) \otimes \textbf{R}^{n}$, which for $\alpha \in \mathfrak {g}$ has some coordinate expression:

$$\begin{aligned} \alpha \mapsto \sum _{i=1}^{n} f(x_{i}, \alpha )\partial _{i}, \end{aligned}$$

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

$$\begin{aligned} \alpha \mapsto \sum _{i=1}^{n} \hat{f}_{0}(x_{i}, \alpha )\partial _{i}, \end{aligned}$$

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

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 coordinates, we can write $C^{\bullet }(\mathfrak {g}, \textbf{R} \llbracket \textbf{x} \rrbracket )$ as $\textbf{R} \llbracket \alpha ^{1}, \ldots , \alpha ^{m}, x_{1}, \ldots , x_{n} \rrbracket$, where the $\alpha ^{k}$ 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 $\alpha ^{k} \otimes x_{l}$, for $\alpha ^{k} \in \mathfrak {g}^{\vee }[-1]$ and $x_{l} \in \textbf{R} \llbracket \textbf{x} \rrbracket$, to classify its behavior. Momentarily viewing $\alpha ^{k}$ as a degree 1 element of just $C^{\bullet }(\mathfrak {g}) = \text {Sym}(\mathfrak {g}^{\vee }[-1])$, and noting that $d_{CE}: \mathfrak {g}^{\vee }[-1] \rightarrow \text {Sym}^{2}(\mathfrak {g}^{\vee }[-1])$ is dual to the bracket $[-,-]: \text {Sym}^{2}(\mathfrak {g}^{\vee }[-1]) \rightarrow \mathfrak {g}^{\vee }[-1]$, we have:

$$\begin{aligned} d_{CE}\alpha ^{k} = \frac{-1}{2} \sum _{i,j = 1}^{m} c^{k}_{ij} \alpha ^{i} \wedge \alpha ^{j}, \end{aligned}$$

where $c^{k}_{ij}$ are the structure constants for $\mathfrak {g}$. Concurrently, for $x_{l} \in \textbf{R} \llbracket \textbf{x} \rrbracket$,

$$\begin{aligned} d_{CE}x_{l} = \sum _{i=1}^{m} \alpha ^{i} \otimes \alpha _{i} \cdot x_{l}. \end{aligned}$$

Therefore, by requiring the usual derivation rules, we get

$$\begin{aligned} d_{CE}(\alpha ^{k} \otimes x_{l}) = \frac{-1}{2} \sum _{i,j = 1}^{m} c^{k}_{ij} \alpha ^{i} \wedge \alpha ^{j} \otimes x_{l} + \sum _{i = 1}^{m} \alpha ^{k} \wedge \alpha ^{i} \otimes \alpha _{i} \cdot x_{l}, \end{aligned}$$

which we extend to the rest of $C^{\bullet }(\mathfrak {g}, \textbf{R} \llbracket \textbf{x} \rrbracket )$ with the Leibniz rule. A coordinateless way of writing this is $d_{CE} = [-,-]_{\mathfrak {g}}^{\vee } + \rho ^{\vee }_{0}$, where $\rho ^{\vee }_{0}$ encodes a dual to the action map $\rho _{0}: \text {Vect}(\widehat{\textbf{R}}^{n}_{0}) \rightarrow \mathfrak {g}^{\vee } \otimes \text {Vect}(\widehat{\textbf{R}}^{n}_{0})$ as described implicitly above. In this example, d is in fact a vector field, specifically

$$\begin{aligned} d_{CE} = \frac{-1}{2}c^{k}_{ij}\alpha ^{i} \wedge \alpha ^{j} \frac{\partial }{\partial \alpha ^{k}} + \alpha ^{i} \otimes (\alpha _{i} \cdot x_{l}) \frac{\partial }{\partial x_{l}}, \end{aligned}$$

(50)

on the formal neighborhood of 0 in the stack $[\textbf{R}^{n}/G]$, where we have used the Einstein summation convention over repeated indices in the last step.

To make things easier to grasp, let us consider the case of SO(2) acting on $\textbf{R}^{2}$ via rotations. Then $\mathfrak {g} = \mathfrak {so}(2)$ and the Taylor series ring about the origin is $\textbf{R} \llbracket x,y \rrbracket$. The representation map $\rho _{0}: \mathfrak {so}(2) \rightarrow \text {Der}(\textbf{R} \llbracket x,y \rrbracket ) \cong \textbf{R} \llbracket x,y \rrbracket \otimes \textbf{R}^{2}$ is

$$\begin{aligned} \begin{pmatrix} 0 &{}\quad -1\\ 1 &{}\quad 0 \end{pmatrix} \mapsto y\partial _{x} - x\partial _{y}, \end{aligned}$$

from which we can define $C^{\bullet }(\mathfrak {so}(2), \textbf{R} \llbracket x,y \rrbracket )$. We will leave it as an exercise to the reader to show that $H^{0}(\mathfrak {so}(2), \textbf{R} \llbracket x,y \rrbracket )$ is the set of rotation-invariant Taylor series around 0. This is not surprising: more generally, in the case of SO(n) acting on $\textbf{R}^{n}$, $H^{0}(\mathfrak {so}(n), \textbf{R} \llbracket x_{1}, \ldots , x_{n} \rrbracket )$ is the set of SO(n)-invariant Taylor series around the origin in $\textbf{R}^{n}$.

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 $\textbf{R} \llbracket x-x_{0} \rrbracket$. Notice there that the vector fields coming from the action at these non-fixed points have constant coefficient terms.

1.2 A remark on higher orders

We would like to make sense of Theorem 4.24 in the case that we do not cut off the orders of $\varepsilon$ 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 $\varepsilon$ 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 $\{S_{g+ \varepsilon h},-\}$ in more powers of $\varepsilon h$.

Construction 5.1

A concrete jumping-off point here would be to consider that for a generally covariant theory, we have

$$\begin{aligned} \frac{\text {d}^{k}}{\text {d}t^{k}}\int _{X} (f_{t}^{*}\varphi )\Delta _{ f_{t}^{*}g} (f_{t}^{*}\varphi )\text {vol}_{f_{t}^{*}g} \Big |_{t=0} = 0 \end{aligned}$$

(51)

for any $k > 0$. This is the general form of Eq. (47). For now, let us stick with $k=2$. The above should have an analogous unpacking to the one following Equation (47); however, we then need to make sense of $\frac{\textrm{d}^{2}}{\textrm{d}t^{2}}(f_{t}^{*}\varphi ) \big |_{t=0}.$ A short exercise in differential geometry gives us that

$$\begin{aligned} \frac{\textrm{d}^{2}}{\textrm{d}t^{2}}(f_{t}^{*}\varphi ) \big |_{t=0} = \frac{1}{2} L_{V} (L_{V}\varphi ), \end{aligned}$$

(52)

as one might expect; the same equation holds for any $k > 0$, and the right side is in fact equal to $\frac{1}{k!}L_{V}^{k}\varphi$, where $L_{V}^{k}$ 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 4.19, replacing $\mathbb {D}_{2}$ with $\mathbb {D}_{3}:= \textbf{R}[\varepsilon ]/(\varepsilon ^{3})$ 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 $\varepsilon$ it is associated with. The result is the following:

Lemma 5.2

The above square commutes if we choose

$$\begin{aligned} D_{2} = \frac{1}{2}\big [L_{V}^{2}, Q_{g}\big ] - \big [L_{V}, Q_{g}\big ]L_{V}. \end{aligned}$$

Moreover, $\widetilde{\mathscr {F}}_{g}:= (\mathscr {F}_{g} \otimes \mathbb {D}_{3}, Q_{g})$ and $\widetilde{\mathscr {F}}_{g + \varepsilon L_{V}g}:= (\mathscr {F}_{g} \otimes \mathbb {D}_{3}, Q_{g} + \varepsilon [L_{V}, Q_{g}] + \varepsilon ^{2}(\frac{1}{2}[L_{V}^{2}, Q_{g}] - [L_{V}, Q_{g}]L_{V}) )$, are cochain isomorphic via the map $\textrm{Id} + \varepsilon L_{V} + \frac{\varepsilon ^{2}}{2}L_{V}^{2}$.

Remark 5.3

The operator $D_{2} = \frac{1}{2}[L_{V}^{2}, Q_{g}] - [L_{V}, Q_{g}]L_{V}$ 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 $\nabla ^{\mu }T_{\mu \nu } = 0$): otherwise, we would not have this cochain isomorphism of field theories. However, it would be harder to pin down a physical interpretation of the associated conservation law.

Remark 5.4

Additionally, we could now update Theorem 4.24 so that it holds up to second order in $\varepsilon$: the isomorphism of observables is induced from the isomorphism $\textrm{Id} + \varepsilon L_{V} + \frac{\varepsilon ^{2}}{2}L_{V}^{2}$ of the field theories. The proof is otherwise the same. It may be clear to the reader by now that these results can be generalized to arbitrarily high orders of $\varepsilon$. In that case, we can expand the differential as

$$\begin{aligned} Q_{g + \varepsilon L_{V}g} = Q_{g} + \varepsilon \big [L_{V}, Q_{g}\big ] + \varepsilon ^{2} \left( \frac{1}{2}\big [L_{V}^{2}, Q_{g}\big ] - \big [L_{V}, Q_{g}\big ]L_{V}\right) + \varepsilon ^{3}D_{3} + \cdots . \end{aligned}$$

on the fields—as long as the metric perturbation is induced by a vector field—and pick out $D_{k}$ for all $k \in \textbf{N}$ so that we get an analogous commutative square, with the isomorphism

$$\begin{aligned} \textrm{Id} + \sum _{k=1}^{\infty } \frac{\varepsilon ^{k}}{k!}L_{V}^{k}. \end{aligned}$$

Thus, the isomorphism of observables $\textrm{Obs}^{\textrm{cl}}(X,\widetilde{\mathscr {F}}_{g}) \cong \textrm{Obs}^{\textrm{cl}}(X,\widetilde{\mathscr {F}}_{g + \varepsilon L_{V}g})$ remains true to all orders of $\varepsilon$, since we see from the above that all of the appropriate $D_{k}$ must exist, regardless of how difficult they are to compute or interpret physically. To see more explicitly, one would need to consider expansions of $\{S_{g + \varepsilon h}, - \}$ to all orders of $\varepsilon h$: a first step in this case would be to generalize Lemma 4.11 to higher derivatives with respect to t.

Remark 5.5

In addition to 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. Ricci flow $\partial _{t}g = -\textrm{Ric}(g)$ is itself a “generally covariant flow” in the sense that the equation is diffeomorphism equivariant²³: I would be interested to see how viewing it as a flow on the moduli stack of metrics modulo diffeomorphism would provide some advantages.