A Mathematical Analysis of the Axial Anomaly

As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We use the Batalin-Vilkovisky (BV) formalism and the methods of equivariant quantization of Costello and Gwilliam to produce a new, mathematical derivation of this result. Using these methods, we formalize two conventional interpretations of the axial anomaly, the first as a violation of current conservation at the quantum level and the second as the obstruction to the existence of a well-defined fermionic partition function. Moreover, in the formalism of Costello and Gwilliam, anomalies are measured by cohomology classes in a certain obstruction-deformation complex. Our main result shows that---in the case of the axial symmetry---the relevant complex is quasi-isomorphic to the complex of de Rham forms of the spacetime manifold and that the anomaly corresponds to a top-degree cohomology class which is trivial if and only if the index of the corresponding Dirac operator is zero.


INTRODUCTION
1.1. Background. The aim of this paper is to explain the relationship between the axial anomaly of the massless free fermion and the index of Dirac operators as arising from the general formalism of Batalin-Vilkovisky (BV) quantization. While this result is well-known to physicists (see, e.g., Chapter 22.2 of [Wei05]), we give a mathematically precise version of the anomaly calculation within the framework for the study of anomalies developed by Costello and Gwilliam in [CG].
Given a generalized Dirac operator D on a Z/2-graded vector bundle V → M over a Riemannian manifold M, the massless free fermion is described by the equation of motion for φ a section of V. Consider the operator Γ which is the identity on even sections of V and minus the identity on odd sections of V. Then, Γ anti-commutes with D, since D is odd for the Z/2-grading. It follows that if Dφ = 0, then In other words, the operator Γ preserves the equations of motion, so is a symmetry of the classical theory. This symmetry is known as the axial symmetry. One can ask whether this symmetry persists after the massless free fermion is quantized. This is in general not the case; the axial anomaly measures the obstruction to the promotion of this classical symmetry to a quantum one. More generally, we say that a classical symmetry is anomalous if it does not persist after quantization. Anomalies in fermionic field theories have been an object of renewed recent interest ( [Wit16] and [Fre14]), in part because of their relevance to topological phases of matter.
The question of what it means mathematically to quantize a field theory is still an open one. However, the formalism of Costello and Gwilliam ([CG17], [CG], [Cos11]) is one approach to the perturbative quantization of field theories which has been able to reproduce many properties of quantum field theory-especially those pertaining to the observables of quantum field theories-that physicists have long studied. This formalism naturally includes a framework for studying symmetries, and it is within this framework that we prove our results.
From another (heuristic) vantage point, a fermionic anomaly is an obstruction to the existence of a well-defined fermionic partition function, in the following sense. To compute the partition function of a quantum field theory involving fermionic fields, one performs a path integral over the space of all quantum fields. To do this, one first fixes a number of pieces of geometric and topological data, such as a manifold M, a metric on M, a spin structure on M, and so forth. Let B denote the "space" of all relevant pieces of geometric and topological data. The scare quotes are there because we might need a more abstract notion of space (such as a stack) to appropriately describe the situation. Given a point x ∈ B , we can form the space of fields, which is a product B x × F x of the fermionic fields F x and the non-fermionic fields B x at x. As x varies, these spaces of fields fit together into something like a product B × F of fibrations over B . In a given fiber B x × F x , one can choose to perform the path integral by first integrating over F x , choosing a fixed b ∈ B x . In this way, the fermionic integration produces something which varies over B, i.e. depends both on the fixed non-fermionic field b ∈ B x and the point x ∈ B . However, the fermionic integration does not canonically produce a number in general; it produces an element of a (super-)line depending on the background data. In other words, there is a line bundle This is called the determinant line bundle, because its fiber over a point y ∈ B is the natural home for the determinant of the Dirac operator corresponding to the massless free fermion theory with background data encoded in y. The fermionic path integral produces a section σ of this bundle, which deserves to be called "the partition function" only when L is endowed with a trivialization, for in that case, one can compare σ to this trivialization to get an honest function.
There is a well-developed mathematical literature addressing many of the relevant issues in the case where one chooses B = B and restricts attention to a subspace of B which is an actual space (see [BF86a], [BF86b], [Fre86]). These ideas make maneuvers of physicists (e.g. zeta-function regularization) precise in the special context of families of massless free fermionic field theories. In the present article, we allow B → B to be non-trivial, fix x ∈ B, and study the dependence of L only over the fiber B x . In other words, we study the pullback We will let B x be a space of perturbative gauge fields. We note that the present work differs from the mathematical treatments in the vein of [Fre86] in two ways. The first is that the "space" B x is best understood as an object of formal derived deformation theory ( [Lura]). In [CG], the authors explain how perturbative field theory can be described with the language of formal derived deformation theory; the essence of this relationship is that, for the purposes of quantum field theory, formal derived deformation theory provides a nice unification of perturbation theory and gauge theory. In the present work, we will mainly use formal derived deformation theory for the geometric intuition it provides. This will allow us to interpret algebraic results as geometric ones. In particular, we will be able to describe what is meant by a line bundle, or more generally a vector bundle, over a formal derived deformation problem.
The second way in which our work differs from existing treatments of anomalies is that we focus on a general approach to field theories and their quantization known as the Batalin-Vilkovisky (BV) formalism. In the work of Costello and Gwilliam, this formalism is used to handle the quantization of the observables of field theories and their symmetries. Our work shows that, simply by turning the crank on the machine of BV quantization, one naturally recovers the axial anomaly. In other words, we study the axial anomaly using a framework that was not hand-made for the case of the massless free fermion.

Presentation of Main Results
. This paper has two main results. First, we construct the bundle L → B x . Second, we compute the axial anomaly of the massless free fermion and show that it is given by the index of the associated Dirac operator. The following Idea gives a slightly more concrete statement of the first result: Idea. Let L be a DGLA acting on the massless free fermion by symmetries, C • (L ) be the corresponding Chevalley-Eilenberg cochain complex, and O(L [1]) the underlying graded vector space of C • (L ). The equivariant quantum observables of the massless free fermion with action of L are quasi-isomorphic to a C • (L )-module P whose underlying graded vector space is O(L [1]) ⊗ . This module is isomorphic to the trivial such module if and only if the anomaly vanishes. Here, is a line canonically associated to the massless free fermion. This Idea will be made precise in Theorem 4.2 and Lemma 4.14. As will be explained in Sections 3 and 4, the cochain complex P should be interpreted as the space of sections of a line bundle L over the formal pointed moduli space encoded in L . The fact that the differential on P differs from the differential on C • (L ) ⊗ describes the possibility that L is non-trivial. Indeed, we will also show that this bundle can be trivialized when the index of the Dirac operator defining the fermionic theory is zero. More precisely, there is an obstruction theory associated to the question of the triviality of L : there is an obstruction class Obstr (see Definition 3.45 and Lemma 3.52) in the obstruction-deformation complex C • red,loc (L ) whose cohomological triviality determines whether or not L is trivializable. This statement is proved in 4.14. Thus, the obstruction class is precisely the anomaly of the corresponding symmetry.
However, the result is of little practical significance if one has limited knowledge of the structure of C • loc,red (L ). For the case of the axial symmetry, which corresponds to the DGLA L = Ω • dR , we show the following two results: Proposition 1.1 (Cf. Proposition 6.1). There is a canonical quasi-isomorphism Here Ω • dR is the abelian elliptic DGLA encoding the axial symmetry of the fermionic theory.
Main Theorem (Cf. Theorem 6.2). The cohomology class of Obstr is equal to the cohomology class of where Φ is the quasi-isomorphism of Proposition 1.1 and dVol g is the Riemannian volume form on (M, g).
We will see in Section 6.3 how this obstruction theory leads to the impossibility of lifting the classical conserved axial vector current to a quantum observable.
In fact, we will have very similar theorems in case a Lie algebra g acts on the fermions in a way that commutes with the corresponding Dirac operator. We prove the corresponding generalizations of the above results in Section 7.
We note that the obstruction-theoretic approach to anomalies is known also to physicists: see, e.g. Chapter 22.6 of [Wei05]. In the present work, we give a mathematically precise version of this approach; in particular, we use the renormalization techniques of [Cos11].
Let us summarize. In this paper, we focus on two perspectives on anomalies in fermionic theories. In the first, anomalies are obstructions to the construction of a well-defined fermionic partition function, which perspective is justified by Lemma 4.14. In the second perspective, anomalies are the obstruction to the persistence of a symmetry after quantization. This perspective is justified by the arguments of Section 6.3, using the obstruction theoretic result of the Main Theorem.
1.3. Future Directions. In this subsection, we outline a few natural extensions of the present work.
One of the novel realizations of Costello and Gwilliam is that the observables of a quantum field theory fit together into a local-to-global, cosheaf-like object known as a factorization algebra. In this paper, we use only the global observables. However, the factorization-algebraic structure of observables often reproduces familiar algebraic gadgets which mix geometry and algebra: locally constant factorization algebras, as shown by Lurie ( [Lurb]), are equivalent to E n -algebras, and holomorphic factorization algebras produce vertex algebras (See chapter 5 of [CG17] for the general construction, and [GGW] for an example of the construction). It would be interesting to understand the analogous sort of structure in the case at hand.
Another interesting local-to-global aspect of field theories in the formalism of Costello and Gwilliam is that given a theory on a manifold M, the formalism naturally produces a sheaf of theories on M. Moreover, the construction of the massless free fermion on M depends only a metric, a Z/2-graded vector bundle on M, and a generalized Dirac operator D. All three objects are local in nature, so that it is to be expected that the massless free fermion can be extended to a sheaf of theories on an appropriate site of smooth manifolds (i.e., manfiolds equipped with a metric, Z/2-graded vector bundle, and Dirac operator). We would like to give sheaf-theoretic extensions of the main theorems of this paper.
In yet another direction, we would like to use the BV formalism to make analogous constructions to those in [Fre86]. Namely, in the case where we have a family of Dirac operators parametrized by some space B, we expect the BV formalism to produce a line bundle over B just as we saw in the Idea of the previous subsection. We hope to give a BV-inspired construction of a metric and compatible connection, which can be used to probe the line bundle.
Finally, we would like to compute the partition function, as introduced in Section 4.
1.4. Plan of the Paper. The plan for the rest of the paper is as follows. In the next section, we provide a lightning review of basic concepts from the theory of generalized Dirac operators sufficient to define the theory of massless free fermions in the BV formalism. In Section 3, we review the techniques of equivariant quantization developed by Costello and Gwilliam. Next, in Section 4, we outline the sense in which the BV quantization procedure produces a line bundle over the formal pointed moduli space corresponding to L . In Section 5, we perform the crucial Feynman diagram computation that we will use in the following sections. This section will provide the first appearance of the index of a Dirac operator in the context of fermionic field theory. We will show that this computation fully characterizes the anomaly in Section 6 by proving more precise versions of Proposition 1.1 and the Main Theorem. Also in Section 6 (Subsection 6.3), we bring our discussion closer to the physics literature by describing the relationship between the axial anomaly and the existence of conserved currents in the massless free fermion quantum field theory. Finally, in Section 7, we prove slight generalizations of the preceding results. 1.6. Notation and Conventions. We assemble here many conventions and notations that we use throughout the remainder of the text. Some may be explained where they first appear, but we believe it to be useful to the reader to collect them in one place.
• The references we cite are rarely the original or definitive treatments of the subject; they are merely those which most directly inspired and informed the work at hand. We have made every attempt to indicate which results are taken or modified from another source. • If a Latin letter in standard formatting is used to denote a vector bundle, e.g. E, then that same letter in script formatting, e.g. E , is used to denote its sheaf of sections. Letters in script formatting, except O, will always denote sheaves of sections of vector bundles.
If M 3 is not explicitly mentioned, it is assumed to be pt. • We will use ⊗ to denote the completed projective tensor product of topological vector spaces as well as the algebraic tensor product of finite-dimensional vector spaces. This product has the nice characterization that if V (M) and W (N) are vector spaces of global sections of vector bundles V → M and W → N, then |v| and π v refer to the Z and Z/2-degrees of v, respectively. We will also call |v| the ghost number of v and π v the statistics of v. This gives a convenient terminology for distinguishing between the various types of grading our objects will have; it is a terminology that conforms loosely to that of physics. We will sometimes also refer to ghost number as cohomological degree.
• If M is compact and Riemannian with metric g and Riemannian volume form dVol g , then for a one-form α, Here, g −1 is the metric on T * M induced from g. There is a similar definition if α is a section of a vector bundle V → M with metric (·, ·).
• When dealing with Z × Z/2-graded objects, we will always say "graded (anti)symmetric" if we wish Koszul signs to be taken into account. If we simply say (anti)-symmetric, we mean that this notion is to be interpreted without taking Koszul signs into account. • If v ∈ Sym(V) for some vector space V, then v (r) denotes the component of v in Sym r (V). • Let V be a (non-graded) vector space and φ : V → V a linear map. We denote by φ 0→1 the cohomological degree 1 operator and similarly for φ 1→0 . We denote by φ 0→0 the cohomological degree 0 operator on V ⊕ V[−1] which acts by φ on V and by 0 on V[−1]; similarly, φ 1→1 is the operator on V ⊕ V[−1] which acts by 0 on V and by φ on V[−1]. • If L is an elliptic differential graded Lie algebra, then we will use d L to denote the differential on the complex of Chevalley-Eilenberg cochains of L .

GENERALIZED LAPLACIANS, HEAT KERNELS, AND DIRAC OPERATORS
We present here a list of definitions and results relevant to our work, taking most definitions and results from [BGV92]. Throughout, (M, g) is a closed Riemannian manifold of dimension n with Riemannian volume form dVol g . We let V → M be a Z/2-graded vector bundle with V + , V − the even and odd bundles, respectively. We let V be the sheaf of smooth sections of V. We always use normal-font letters for vector bundles and script letters for the sheaves of sections of the corresponding vector bundles.
where we are thinking of C ∞ functions as operators given by multiplication by those functions.
Remark 2.3: The above equation is a coordinate-free way of saying that H is a secondorder differential operator on V whose principal symbol is just the metric, i.e., that in local coordinates, H looks like g i j ∂ i ∂ j + lower-order derivatives. ♦ Definition 2.4 (Definition 3.36 of [BGV92]). A Dirac operator on V is a grading-reversing differential operator D : V ± → V ∓ such that D 2 is a generalized Laplacian. If V is a Z/2-graded metric bundle with inner product (·, ·), then we say that D is formally self-adjoint if for all s, r ∈ V , M (s, Dr)dVol g = M (Ds, r)dVol g .
Because we never require any other notion of self-adjointness of a Dirac operator, we will simply use the terminology "self-adjoint" instead.
Example 2.5: Let V = Λ • TM, with the Z/2-grading given by the form degree mod 2. Then the operator d + d * is a generalized Dirac operator since its square (d + d * ) 2 is the Laplace-Beltrami operator. Here, d * is the formal adjoint of d, characterized by It is clear from this description that d + d * is formally self-adjoint, since it is the sum of an operator with its formal adjoint. Via Hodge theory, the kernel of d + d * is canonically identified with the de Rham cohomology H • dR (M). ♦ There are numerous other examples of interest-for example, the traditional Dirac operator on a spin manifold and √ 2 ∂ +∂ * on a Kähler manifold-and our formalism will work for these as well. We refer the reader to Section 3.6 of [BGV92] for the details of the construction of these operators.
Definition-Lemma 2.6. [Proposition 3.38 of [BGV92]] If V is a Z/2 graded bundle over (M, g) a Riemannian manifold and D a Dirac operator on V, then the Clifford action of 1-forms on V c : is defined on exact forms by and extended to all 1-forms by C ∞ -linearity. This action is well-defined.
Theorem 2.7 (The Heat Kernel; Section 8.2.3 of [CG]). Let V be a Z/2-graded metric bundle with metric (·, ·) and Dirac operator D. Write H := D 2 for the generalized Laplacian corresponding to D. Let π 1 , π 2 : M × M × R >0 → M denote the projections onto the first and second factors, respectively. Then there is a unique heat kernel where the limit is uniform over M and is taken with respect to some norm on V. The heat kernel is the integral kernel of the operator e −tH in the sense that y∈M id ⊗ (·, ·)(k t (x, y) ⊗ s(y))dVol g (y) = (e −tH s)(x).
Example 2.9: The index of the Dirac operator from Example 2.5 is where χ(M) is the Euler characteristic of M. ♦ The last definition we need to understand the statement of the McKean-Singer formula is the following, found in the discussion preceding Proposition 1.31 of [BGV92] Definition 2.10. If φ : V → V is a grading-preserving endomorphism of the super-vector space V, then the supertrace of φ Str(φ) is Let Γ : V → V denote the operator which is the identity on V + and minus the identity on V − . Then Str(φ) = Tr(φΓ).
With these definitions in place, we can finally state the following Theorem 2.11 (Theorem 3.50 of [BGV92]). Let V be a Hermitian, Z/2-graded vector bundle on a closed Riemannian manifold M, with dVol g the Riemannian volume form on M. Let D be a self-adjoint Dirac operator on V, with k t the heat kernel of D 2 . Then This is the McKean-Singer Theorem.
We will also need a few technical results that will be useful especially in Section 7. The first is Lemma 2.12. If D is a self-adjoint Dirac operator over a compact manifold, then where the decomposition is orthogonal with respect to the metric on V induced from (·, ·).
Remark 2.13: This is a special case of a general Hodge decomposition that applies for any elliptic complex with a metric on the corresponding bundle of sections. ♦ The second result is Proposition 2.37 of [BGV92], as used in the proof of the McKean-Singer Theorem.
Lemma 2.14. If D is a self-adjoint Dirac operator, and if P is the orthogonal projection onto ker(D), then for t large, where λ 1 is the smallest non-zero eigenvalue of D 2 and C is a universal constant.
Finally, we present useful analytic facts about the spectrum of a Laplacian on a compact manifold.
Lemma 2.15 (Cf. Proposition 2.36 of [BGV92]). If H is a formally self-adjoint generalized Laplacian on a compact manifold, then H has a unique self-adjoint extensionH with domain a subspace of L 2 sections of V;H has discrete spectrum, bounded below, and each eigenspace is finite dimensional, with smooth eigenvectors. Letting φ j be an eigenvector forH with eigenvalue λ j , we have Moreover, e −tH is bounded.

THE EQUIVARIANT QUANTIZATION OF FREE FIELD THEORIES
3.1. Introduction. In this section, we study what is in physics called an anomaly. Roughly speaking, an anomaly is the failure of a symmetry of a classical field theory to persist after quantization. Symmetries are usually encoded in actions of a Lie algebra-or more generally a differential graded Lie algebra (DGLA), or even more generally an L ∞ -algebra-on the space of fields of the theory. We develop this perspective below, following [CG] and [Gwi12], which provide a general framework for treating actions of an L ∞ -algebra on a quantum field theory. We will not need the full generality of that framework below-restricting ourselves to DGLA actions on free field theoriesthough by illuminating the relationship of this special case to index theory, we hope to provide motivation for the study of the more general case.
There are multiple additional, complementary perspectives on symmetries, which we will discuss below. First, we will see that the action of a DGLA g on a classical or quantum field theory can be interpreted as providing a family of classical or quantum theories parametrized by the formal pointed moduli problem Bg corresponding to g. This family is twisted in an interesting way relative to a trivial Bg-family of theories. This twisting is measured by an interaction term I in the action of the theory. I furnishes the last perspective on symmetries: in physics terms, it represents a coupling of the physical fields to background gauge fields. Moreover, I is required to satisfy a Maurer-Cartan equation of its own which contains the content of the fact that g acts on the corresponding theory. This equation is called the classical master equation (see Remark 3.28).
To summarize, there are three interrelated perspectives on the notion of a symmetry of a classical theory. We can view a symmetry as (1) an action of a DGLA g on the space of fields of a field theory, or (2) a family of classical field theories parametrized by Bg, or (3) a g-dependent interaction term in the field theory. The elements of g are considered to be background fields.
In the quantum case, there will be analogues to the latter two. The analogue of the classical master equation will be called the weak quantum master equation, and we will see that an interaction satisfying the weak quantum master equation will give a family of quantum field theories over Bg. However, we are also interested in understanding when this family is actually trivial. The equation determining whether or not this can be done is called the strong quantum master equation. The goal of the rest of the paper is to motivate the strong master equation and explain its relationship to the index of a self-adjoint Dirac operator.
3.2. Free BV Theories. Free BV theories are the starting point for quantization, since in the perturbative formalism we think of interacting theories as perturbations of free ones. We will also in this paper be mostly concerned with a particular family of free theories. However, the formalism for families of free theories requires us to consider interactions, even if the individual theories in the family are free.
The following is Definition 7.0.1 of Chapter 5 of [Cos11].
Definition 3.1. A free classical BV theory on a compact manifold N consists of the following data: (1) A Z × Z/2-graded R−vector bundle F over N. We will call the sheaf of sections F the space of fields of the theory.
(2) A graded-anti-symmetric map of vector bundles ·, · loc : F ⊗ F → Dens N of degree (−1, 0). ·, · loc is non-degenerate on each fiber. ·, · loc induces an integration pairing ·, · on the space of global sections F (N) given by By graded anti-symmetry, we mean that if f 1 , f 2 have Z degrees | f 1 | and | f 2 | and Z/2 degrees π f 1 and π f 2 , then All notions of graded symmetry and anti-symmetry should be analogously understood. We also note that since ·, · is of degree -1, it is only non-zero on pairs f 1 and f 2 with | f 1 | and | f 2 | of opposite parity, so that we can rewrite the above equation as (3) A differential operator Q : F → F of degree (1,0) making F (N) into an elliptic complex. Q must be graded skew self-adjoint for the pairing ·, · , with the same signs as discussed above.
Remark 3.2: The elliptic complex (F , Q) is a derived version of H 0 (F ) that encodes the information of the linear PDE Qφ = 0. The zeroth cohomology of (F , Q) is the space of solutions to this PDE modulo the identification of solutions differing by an exact term; physically, the zeroth cohomology is the space of solutions of the equations of motion modulo the identification of physically indistinguishable (gauge-equivalent) configurations. ♦ In order to be able to quantize, we need a free BV theory to also have a gauge fixing: Definition 3.3 (Definition 7.4.1 of Chapter 5 of [Cos11]). A gauge fixing Q GF for a free BV theory is a degree (−1, 0) differential operator F → F , such that [Q, Q GF ] is a generalized Laplacian for some metric on N and Q GF is graded self-adjoint for the pairing ·, · .
Remark 3.4: In physics, the Z-degree of an element of F is called the "ghost number," while its Z/2-degree is often called its "statistics;" we say that an element of F has bosonic or fermionic statistics according as to whether it has grading 0 or 1, respectively. It should be noted that this Z/2 grading has nothing to do with the Z/2 grading of the bundle V in the previous section. In that section, the Z/2-grading was a convenient bookkeeping device to keep track of a particular decomposition of the bundle V with respect to which most of the constructions of that section were well behaved.
Here, the Z/2-grading is much more profound: it affects whether objects commute or anti-commute. ♦ As a very simple example of a free field theory, we have Example 3.5: Given an elliptic complex (E , P), the cotangent theory T * [−1]E to E has space of fields E ⊕ E ! [−1], with all fields having odd Z/2-grading. We define ·, · to be the natural pairing on E ⊕ E ! [−1], and Q to be P + P ! . Here, E ! is the sheaf of sections of the bundle E ∨ ⊗ Dens. ♦ As another simple example, we have the massless scalar boson theory: ♦ Finally, we present the example with which we will be working throughout the rest of this paper: Example 3.7: Let (M, g) be, as in Section 2, a Riemannian manifold with density dVol g , V a Z/2-graded metric bundle on M with metric (·, ·), and D a self-adjoint Dirac op- , with-just as in the cotangent theory example-all fields having fermionic statistics, and when either | f 1 |= 0 and | f 2 |= 1 or vice versa. ·, · is symmetric as a consequence of the symmetry of (·, ·); this is consistent with Equation 3.1 since all fields have fermionic statistics. Let D 0→1 denote the operator D understood as a cohomological degree +1 operator on S , and similarly for D 0←1 as a cohomological degree -1 operator. Then, for Q we take D 0→1 Γ and for Q GF we take Γ D 0←1 . It can be verified directly that Q and Q GF satisfy the necessary symmetry conditions as a result of the fact that Γ and D are self-adjoint for (·, ·) and anti-commute with each other. Using this anticommutation relation, along with the equation where the subscripts on D 2 are there as reminders of which copy of V in S the corresponding operator acts on. Thus, [Q, Q GF ] is a generalized Laplacian, as desired. This free theory is called the massless free spinor theory associated to D. ♦ Remark 3.8: The metric on V can be used to identify V with V ∨ , and the Riemannian density can be used to trivialize the bundle of densities on M, so that V ! ∼ = V. As a result, it can be shown that the the free spinor theory is isomorphic, under the obvious notion of isomorphism of free theories, to the cotangent theory to the elliptic complex We will use this characterization in the sequel. ♦ Notation 3.9. In the sequel, E will denote the elliptic complex and E the elliptic complex Then the space of fields of the free spinor theory is E ⊕ E and Q is the differential on this sum of complexes. This decomposition will be useful to us because elements of E pair non-trivially only with elements of E under ·, · . In a related vein, E ∼ = E ! [−1], so that under the isomorphism of the free spinor theory with the cotangent theory to E , E corresponds to the fiber direction and E to the base direction.

Actions of an Elliptic Differential Graded Lie Algebra on a Free Theory.
We would like to understand what it means for a differential graded Lie algebra (DGLA) to act on a free theory. To begin with, we should specify the particular class of DGLAs suited to the analytic nature of quantum field theory, and the associated notions of Chevalley-Eilenberg cochains.

Elliptic Differential Graded Lie Algebras and their Chevalley-Eilenberg Cochains.
Definition 3.10 (Definition 7.2.1 of [Gwi12]). An elliptic differential graded Lie algebra L is a sheaf of sections of a Z-graded vector bundle of L → N which is also a sheaf of DGLAs. The bracket and differential must be poly-differential operators and the differential must give L the structure of an elliptic complex.
Example 3.11: For a manifold M and a Lie algebra g, the sheaf Ω • dR ⊗ g is an elliptic DGLA with differential given by the de Rham differential and bracket given by the wedge product on the differential form factor and Lie bracket on the g factor. ♦ An elliptic DGLA has the standard Chevalley-Eilenberg (CE) cochain complex, as well as a cochain complex of functionals that are more nicely behaved and are defined using that L is a sheaf of smooth sections of a vector bundle.
Definition 3.12. Let L be an elliptic DGLA. The Chevalley-Eilenberg cochain complex of L is the commutative differential graded algebra equipped with the Chevalley-Eilenberg differential defined by analogy with the finite-dimensional case. Here, we use ∨ to denote continuous linear dual, and the Sym is taken with respect to the projective tensor product of Nuclear spaces. Namely, where the subscript S n denotes coinvariants of the S n action which permutes the tensor factors. See Appendix 2 in [Cos11] and Appendix B.1 in [CG] for more details. We will often use the notation O(L [1]) to describe the underlying graded vector space.
Remark 3.13: L defines a sheaf of pointed formal moduli problems BL , whose value on an open set and a differential-graded Artinian algebra A has for its k-simplices the set of Maurer-Cartan elements of the DGLA One should think of the algebra C • (L ) as the space of functions on BL (N). ♦ Definition 3.14 (Definition 7.2.7 of [Gwi12]). The cochain complex C • loc (L ) of local Chevalley-Eilenberg cochains of the DGLA L is the subcomplex of C • (L ) given by sums of functionals of the form where each D i is a differential operator from L to C ∞ N and dµ is a smooth density on N. We view F n as a coinvariant of the S n action. We can also study C • loc,red (L ), which is the quotient of C • loc (L ) by the subcomplex R (which lives in Sym-degree 0). We also have C • red (L ), defined analogously.
Remark 3.15: Here, we are using that N is compact to be able to integrate densities. If N is not compact, or if we want to think of C • loc (L ) as a local-to-global object and therefore have local cochains on open subsets of N, we should forget about the integration and think of a local CE cochain as taking in elements of L and giving a density. In this case, we cannot think of local cochains as living inside the space of all CE cochains. ♦ In light of the previous remark, we would like to have an alternative characterization of the local Chevalley-Eilenberg cochain complex that is naturally sheaf-like. To do this, we introduce the language of jets: Definition 3.16 (Cf. Section 5.6.2 of [Cos11]). The bundle of jets J(L) of the elliptic DGLA L is the vector bundle of DGLAs over N whose fiber at a point x ∈ N is the space of formal germs at x of sections of L.
The DGLA structure on the fiber is induced from the same stucture on L and the fact that the DGLA operations of L are polydifferential operators. Moreover, as discussed in Section 6.2 of Chapter 5 of [Cos11], J(L) has a natural (left) D N -module structure. Furthermore, J(L) is an inverse-limit of finite-dimensional C ∞ N -modules; we therefore endow it with the topology of the inverse limit.
N means continuous homomorphisms of C ∞ N modules (in particular, those that respect the inverse-limit topology on J(L)). Let also Note that O(J(L)) has a natural fiberwise differential encoding the Chevalley-Eilenberg differential on C • loc (L ). Moreover, O(J(L)) has the structure of a left D N -module.
Lemma 3.17 (Lemma 6.6.1 of Chapter 5 of [Cos11]). For L an elliptic DGLA, there is a canonical isomorphism of cochain complexes ). Here, Dens N is given the right D N -module structure induced from considering densities as distributions.
Remark 3.18: The object on the right hand side of the isomorphism of Lemma 3.17 is manifestly a sheaf. We therefore will, in the sequel, abuse notation and use C • loc,red (L ) to refer also to the sheaf Dens N ⊗ D N O red (J(L)). ♦ Lemma 3.19 (Lemma 6.6.2 of Chapter 5 of [Cos11]). For L an elliptic DGLA, there is a quasi-isomorphism . Putting this together with the previous lemma, we see that the actual tensor product is a model for the derived tensor product.

3.3.2.
Actions of an Elliptic DGLA on a Free Theory. In this subsubsection, we review the main results concerning the action of an elliptic DGLA on a free BV theory. However, most of what we do works in a far more general context of elliptic L ∞ -algebras acting on a general BV theory. We will point out the few instances in which results are specific to the case we consider.
Definition 3.20 (Definition 11.1.2.1 of [CG]). If L is an elliptic DGLA, and E is an elliptic complex, then an action of L on E is the structure of an elliptic DGLA on E [−1] ⊕ L such that the maps in the linear exact sequence as well as the map L → E [−1] ⊕ L , are openwise DGLA maps (i.e. they commute with brackets). We are thinking of E [−1] as an elliptic abelian DGLA.
In other words, an action of L on E is just a map ρ : L ⊗ E → E satisfying certain coherence relations encoding the fact that the differential on E [−1] ⊕ L is a derivation for ρ and that the bracket on E [−1] ⊕ L satisfies the Jacobi identity. We will also use [·, ·] to denote ρ. , and L E [−1] to denote the resulting DGLA. ♦ Finally, we define what it means for an elliptic DGLA to act on a free BV theory: Definition 3.22 (Definition 11.1.2.1 of [CG]). Let (F , Q, ·, · ) be a free BV theory and L an elliptic DGLA. An action of L on (F , Q, ·, · ) is an action of L on the elliptic complex F such that the map ρ leaves the pairing ·, · invariant in the sense that ρ(α, ·) is a graded derivation with respect to ·, · for all α ∈ L . This means that for all f 1 , f 2 ∈ F . Moreover, we require the action of L on F to be even with respect to the Z/2 grading of F in the sense that Remark 3.23: The requirement that the differential on F [−1] L act as a derivation for . This is the sense in which we should think of a DGLA action as a symmetry; since the cohomology of F [−1] is to be thought of as the moduli space of classical solutions, the fact that H • L acts on H • F [−1] is a precise way of saying that L acts on F [−1] in a way that (cohomologically) preserves the equations of motion. ♦ Example 3.24: Recall the massless free spinor theory of Example 3.7 and Notation 3.9. We let L R = Ω • dR,M , with the de Rham differential and trivial Lie bracket.
Then, the derivation property of a DGLA requires that Assuming that |ϕ|= 0, it follows that We may therefore choose to define for all 1-forms α. Furthermore, since a form in Ω ≥2 would have to act by a cohomological degree ≥ 2 operator, and since F is concentrated in degrees 0 and 1, all such forms must act trivially on F . Equations 3.4 and 3.5 therefore suffice to give a map [·, ·] : Ω • dR,M ⊗ F → F . It can be verified that the Jacobi identity is satisfied and that [·, ·] is a derivation for ·, · , so that L R acts on the free spinor theory. This action is known in the physics literature as the axial symmetry of the free spinor. Notice that the constant functions act by a multiple of Γ on the degree 0 fields in S . This is the sense in which the symmetry is axial: it acts by opposite factors on V + and V − . ♦ Returning now to the general context of an elliptic DGLA acting on a free theory F , we note that F encodes the action functional and the action of L on F provides the following deformation of the action functional: ) has a degree +1 Poisson bracket {·, ·} induced from ·, · , whose precise construction can be found in Section 5.3 of [Cos11]. {·, ·} can be extended to all of Moreover, as discussed in Section 11.1 of [CG], we can equivalently think of an action of L on F as a degree zero element which equation is also known as the classical master equation. In our special case, we require I to have Sym-degree 1 with respect to L and Sym-degree 2 with respect to F . If I has the correct Sym-degree and satisfies the classical master equation, then it determines the action of L on F and vice versa. Furthermore, we consider the sections of L to be background fields because we have not demanded that L be endowed with a pairing as F [−1] is. This will have dramatic consequences for quantization. Just as we encoded the theory described by the action functional S in the elliptic complex (F , Q) together with the pairing ·, · , the theory with action functional S + I is encoded in a family of elliptic complexes parametrized by the Maurer-Cartan can be shown by direct computation (using the properties of a DGLA action) to also be a differential on F ⊗ A ⊗ Ω • dR (∆ k ). Moreover, by the requirement that [X, ·] be invariant for the pairing ·, · , [X, ·] is skew self-adjoint. Thus, , ·, · ) is also a free BV theory.
Notice that the interaction I(X, φ), though of cubic-and-higher order with respect to X and φ together, is only quadratic in φ. This is particular to our case; if we were to use the full language of L ∞ -algebras, we would be able to describe deformations of F into non-free theories, in which case we would allow cubic and higher-order interactions in F fields.
It will be useful to us to think of the interaction I as depicted by a vertex with two straight half-edges and one wavy half-edge. The former type of edge represents F inputs and the latter type represents L inputs. See Figure 1. This convention is consistent with the practice in physics of representing gauge fields by wavy lines; in our context, L represents a background gauge field to which the free theory is coupled. See Chapter 7.1 of [Gwi12] for more on this perspective.
Remark 3.25: If (E , P) is an elliptic complex, and L acts on the cotangent theory to E as an extension of an action of L on E , then the interaction I can be rewritten as where ϕ ∈ E and ψ ∈ E ! [−1] are viewed as the base and fiber components of a general element of T * [−1]E . The factor of 1/2 has disappeared because of the symmetry φ X φ FIGURE 1. A vertex depicting the interaction I.
properties of [X, ·]. In this case, we can add arrows to the vertex to indicate that, with respect to the decomposition of T * [−1]E into base and fiber, I is non-zero only when evaluated on a base field as one input and a fiber field as another. Figure 2 shows this modified graphical interpretation. ♦ ϕ ψ X FIGURE 2. The vertex corresponding to the interaction I for a cotangent theory.
Remark 3.27: Recall that Chevalley-Eilenberg cochains of a DGLA M are, as a vector space, Sym(M ∨ [1]); since we let F be Z × Z/2 graded, we view L (N) ⊕ F [−1](N) as a Z × Z/2 graded object and the symmetrization is with respect to the Koszul sign rules for Z × Z/2 graded objects. ♦ Remark 3.28: It can be shown (see Chapters 5.4 and 6.1 in [CG]) that the differential on this cochain complex can be written as d L + Q + {I, ·}, where I is the interaction in Equation 3.7 and Q is the differential on F extended to be a derivation with respect to F -variables. Moreover, we should think of C • (L ) and C • (F [−1]) as the algebras of functions on the formal pointed moduli spaces BL (N) and The term {I, ·} in the differential on Obs cl therefore represents an interesting way in Alternatively, we can think of Obs cl as the space of sections of a vector bundle over BL . The fiber of this vector bundle is simply O(F ), together with a differential which varies over the base space BL (N). Moreover, the inclusion C • (L ) → Obs cl together with the algebra structure on Obs cl endows Obs cl with the structure of a differentialgraded C • (L )-module, as is to be expected from a vector bundle over BL (N). ♦ 3.4. Equivariant Quantization. In this subsection, we continue the notation of the previous subsection, letting L act on the free classical theory F . Since ·, · gives a sort of L -equivariant symplectic structure on F , we would hope that it would induce a Poisson bracket and Laplacian ∆ on Obs cl , as can be done in the finite-dimensional case. If this were the case, we could define Obs q to be O ] but with the differential from Obs cl deformed by the termh∆. This is what is considered BV quantization in the finite-dimensional case. However, the Poisson bracket is only defined if one of the arguments is local, and the Laplacian is undefined. This is because the naïve way to define these operations involves pairing distributions with each other, an operation which is ill-defined.
The solution to this problem is renormalization, whose essence, as developed in [Cos11] (especially Chapters 2 and 5), is to replace the interaction I with a family I[t], one for each t ∈ R + , and to correspondingly define a Poisson bracket {·, ·} t and BV Laplacian ∆ t for every t. We can then define Obs q as a graded vector space to be Obs cl [[h]], but with putative differential Q + d L + {I[t], ·} t +h∆ t . However, the putative differential may not square to zero. We will first address the procedure of renormalization, then the question of the existence of the desired differential.
3.4.1. Renormalization and the BV Formalism. We assume that F has been equipped with a gauge-fixing Q GF ; then H := [Q, Q GF ] is a generalized Laplacian, and so H has an integral heat kernel k t . We will actually use a heat kernel more suited to the study of BV theories: Definition 3.29 (Section 8.3 of Chapter 5 of [Cos11]). The scale t BV heat kernel K t is the unique degree 1 element of for all f ∈ F . The non-degeneracy property of the pairing ·, · , along with the existence of the heat kernel k t , guarantees that K t exists.
Definition 3.30 (Section 9.1 of Chapter 5 of [Cos11]). The scale t BV Laplacian ∆ t is the Definition 3.31 (Section 9.2 of Chapter 5 of [Cos11]). The scale t Poisson bracket {·, ·} t is the map In other words, the Poisson bracket measures the failure of the BV Laplacian to be a derivation for the commutative algebra O(L [1] ⊕ F ). Figure 3 gives a schematic depiction of the Poisson bracket and BV Laplacian. Notice that, based on the diagrammatic depiction of ∆ t , {J, J } t is 0 if either J or J has a dependence only on L and not on F , since K t has components only in F ⊗ F . The same is true for ∆ t J.  The following proposition will be useful for computational reasons in the sequel. It can be verified by explicit computation.
Lemma 3.32. ∆ t has the following two properties: (1) ∆ t squares to zero.
We have addressed two of the three scale t objects that are needed for our purposes: we just need to explain how I gives rise to the family I[t]. This is by far the most subtle part of renormalization, and we refer the reader to [Cos11] for the full treatment of this subject. We will content ourselves to present the results of the renormalization procedure as applied in the case of L acting on a free theory. To do this, we first need the following few definitions: Definition 3.33 (Section 8.3 of Chapter 5 of [Cos11]). The propagator from scale t to t is denoted P(t, t ) and is the unique element of F ⊗ F satisfying This tells us that, in particular, P(t, t ) is smooth so long as both t and t are positive. In the limit as t → 0, P(t, t ) becomes distributional. We will on occasion need to consider P(0, t). where P is the operator whose kernel is P(t, t ). ∂ P is defined analogously to ∂ K t (see Definition 3.30).
Remark 3.35: The RG flow operator has a natural interpretation in terms of Feynman diagrams, and we refer the reader to [Cos11] Chapter 2.3 for details. We comment only that we will use the word "leg" where [Cos11] uses the word "tail" to describe external edges of Feynman diagrams, i.e. edges that end at a univalent vertex. ♦ As mentioned above, we would like to replace I with a family of interactions {I[t]}, one for each t > 0; however, we will do so in a way that the I[t] are related by the RG flow equation . This is the main purpose of the operator W(P(t, t ), ·), but we will have other uses for it in the sequel, e.g. the following  Notice that since the propagator P(t, t ) connects only F edges, all of the trees contributing to I tr have only two F legs. Thus, ∆ t I tr belongs to Sym(L [1] ∨ ).
If the limit as t → 0 of W(P(t, t ), I) (not just theh 0 part) existed, we wouldn't need renormalization; this is, however, not the case. So, we must take the renormalized limit of the operator W. The precise procedure for doing this in general is spelled out in Chapter 2 of [Cos11], and the result of this procedure in the case at hand is discussed in [Gwi12]. We note only that in the case of an action of L on a free theory, this procedure produces a family I wh [t] of interactions schematized by Figure 5 (the subscript wh stands for "wheel," describing the corresponding Feynman diagrams). We will not need to know much about I wh [t] except that it's an element of C • (L ). With these data in place, we can describe the promised family I[t].
Definition 3.38 (Lemma 7.5.3 of [Gwi12]). The scale t renormalized interaction for L acting on F is denoted I[t] and given by Remark 3.39: The renormalized interaction takes this form only in the context of an elliptic DGLA (or more generally an elliptic L ∞ -algebra) deforming a free theory into a family of free theories. If the original theory is interacting, or if the L ∞ -algebra deforms free theories into interacting ones, the structure of I[t] becomes more complicated, and in particular includes contributions at all powers ofh. . Not depicted is the diagram for the counter-term necessary to make this wheel diagram finite.
The only issue is that this differential may not square to zero. In fact, Lemma 3.40. The square of the putative differential d Proof. Direct computation, using the fact that ∆ t is a derivation for {·, ·} t .
Therefore, d 2 t = 0 if and only if is in the {·, ·} t -center of Obs q [t], or equivalently if O(t) lies in C • red (L ). We will see below that d 2 t will always be 0 for an action of L on a free theory. However, we would like to demand something stronger, which we codify in the second part of the following definition.
Definition 3.41. The weak scale t quantum master equation (wQME) is The strong scale t quantum master equation (sQME) is If the generalized Laplacian [Q, Q GF ] is self-adjoint for some metric on F, then we will also be able to define scale ∞ observables because the operator exp −t[Q, Q GF ] will be bounded and have a t → ∞ limit. ♦ The following lemma tells us that the sQME interacts as we would like with the RG flow operator. It is Lemma 9.2.2 of Chapter 5 of [Cos11].
Lemma 3.44. If I satisfies the scale t sQME, then W(P(t , t), I) satisfies the scale t sQME.
Let us examine the failure of the sQME to be satisfied in a bit more detail. Since I was required to satisfy (d L + Q)I + 1 2 {I, I} = 0, the general machinery of renormalization implies that theh 0 part of the sQME is satisfied (see, e.g. Lemma 9.4.1. of Chapter 5 of [Cos11]). Thus, we make the following definition: Definition-Lemma 3.45 (Cf. Corollary 5.11.1.2 of [Cos11]). The scale t obstruction to the L -equivariant quantization of F is The obstruction is a closed, cohomological degree 1 element of Obs q [t].
Moreover, the sQME measures the failure of this L -equivariant quantization to be trivial over C • (L ), as we make precise in the following sense. First, we make the following Definition 3.46. The scale t fundamentally quantum equivariant observables, denoted Obs together with the same differential as on Obs q [t]. Similarly, let Obs q h −1 ,0 denote the same graded vector space but with differential d L + Q +h∆ t ; this is the analogous construction for the trivial action of L on F .

In other words, Obs
where we have allowed finite powers of h −1 at every Sym-degree; this in particular allows for arbitrarily high negative powers ofh as long as they accompany successively higher Sym-powers. These are "fundamentally quantum" observables because specialization to the classical (h = 0) theory is no longer possible now thath is invertible. We introduce this notation because it allows us to inverth in a way that guarantees (1) Obs  represents the observables of a theory in which the L fields are "decoupled" from the F fields. Mathematically, this is encoded in the fact that the differential on Obs The following is Lemma 7.5.4 of [Gwi12]. It gives us an explicit formula for the scale t obstruction.
Lemma 3.49. For the action of an elliptic DGLA on a free theory, Obstr[t] depends only on L , and hence is also a closed, ghost number 1 element of C • red (L ). Proof. Recall that  We have also the following lemma, which describes the relationship between the obstruction at various length scales. It is a modification of Lemma 11.1.1 from Chapter 5 of [Cos11] to the case where the obstruction lives only at orderh, which we know to be the case by the previous lemma.

Obstr[t]
exists and is a closed, degree 1 element of C • loc,red (L ). ♦ The following consequence of the above lemmas will be crucial in the sequel. Proof. Let's start with the first assertion: where we are using that J doesn't depend on F , so that d F J = ∆ t J = {J, ·} t = 0. Thus, I[t] −hJ solves the sQME. Conversely, the above equation shows that if there exists a J ∈ C • red (L ) such that I[t] −hJ satisfies the scale t sQME, then Obstr[t] is exact.

BV QUANTIZATION, THE DETERMINANT LINE, AND THE PARTITION FUNCTION
In this section, we explain the sense in which the BV quantization of the massless free fermion with an action of an elliptic DGLA L produces a line bundle over BL (N). In the first subsection, we prove the main theorem of the section (Theorem 4.2) and comment on its rough interpretation as the incarnation of the Idea from the Introduction. In the second subsection, we do a bit more interpretive work and show that the scale-infinity obstruction Obstr[∞] is precisely the obstruction to the triviality of this bundle.

The Determinant Line Bundle via the Homological Perturbation lemma.
We adopt the notations of Example 3.7 and Notation 3.9, i.e. the notations relating to the massless free fermion; we will assume that an elliptic DGLA L acts on the free theory S . This could be the the complex L R from Example 3.24, but our results will hold in the more general case. First, let us define the line that appeared in the Idea of the Introduction.
We also need to modify C • (L ) slightly to keep track of the fact that we have madē by analogy with Obs q h −1 . Finally, we note that, because D is self-adjoint, we can use the characterization of the heat kernel from Lemma 2.14. This implies, in particular, that we can define the scale infinity BV heat kernel K ∞ and propagator P(0, ∞), and therefore the scale-infinity equivariant quantum observables Obs q [∞].
Theorem 4.2. If an elliptic DGLA L acts on the massless free fermion theory S , then there is a homotopy equivalence of C • (L )-modules Specifically,η is a degree -1 map,ι andπ are quasi-isomorphisms, and the three maps satisfỹ δ is a perturbing differential on C The term δ in the differential represents a "twisting" of the trivial module structure on C • h −1 (L ) ⊗ (det D + ) ∨ , so represents the possibility that the corresponding line bundle is non-trivial. We are being a bit sloppy about the formal variableh here; see the next subsection for more precise details on this point. ♦ Remark 4.4: The mapπ is to be understood as the expectation value map; it takes an observable to its quantum expectation value. ♦ In the course of the proof of Theorem 4.2, we will see that if L acts on S trivially, then δ = 0, so that the corresponding line bundle is trivial. Letπ 0 denote the map π given by Theorem 4.2 for the trivial L action on S . Then, using Proposition 3.47, satisfies the scale-∞ sQME. Then the next corollary follows from Proposition 3.47.

Corollary. If the scale-∞ interaction I[∞] arising from the action of L on S satisfies the scale-∞ sQME, then Obs q [∞] is quasi-isomorphic to the trivial line bundle C
is closed. This element will be called the partition function.
Remark 4.5: The partition function is a (det D + ) ∨ ((h))-valued function on BL (N). Up to a choice of basis element of (det D + ) ∨ , then, it should be understood as a function on BL (N). ♦ Remark 4.6: The conventional definition of the partition function is as the expectation value of the observable 1 in the theory with background fields turned on. Equivalently, the partition function is the expectation value of exp (I[∞]/h) in the theory without background fields. Butπ 0 is precisely the expectation value map, so our definition of the partition function coincides with the conventional one. ♦ The proof of Theorem 4.2 is a minor modification of arguments in Chapters 2.5-2.6 of [Gwi12], and it uses the Hodge decomposition of Lemma 2.12 and properties of homotopy equivalences, including the homological perturbation lemma. Before we begin the proof, we will spell out some details of the relevant background.
The following is the main theorem discussed in [Cra]; it is a generalization of Theorem 2.5.3 of [Gwi12].
Armed with the preceding results on homotopy equivalences, we can proceed to the proof of Theorem 4.2. The main idea of the proof is repeated application of the homological perturbation lemma. We will break the proof up into small pieces.
Let us first establish some notation that we will use throughout the remainder of this section. Let O denote the underlying graded vector space of Obs q h −1 . Let us also abbreviate Sym((H • S ) ∨ ) to W, letting W ( j) denote the Sym j piece of W. Similarly, let The Hodge decomposition of Lemma 2.12 tells us that the cokernel of D is isomorphic to its kernel. This allows us to decompose S as follows: (4.1) ker D ⊕ Im D ker D ⊕ Im D. Q It follows that H • S = ker D ⊕ (ker D[−1]), and we denote by H • S ⊥ the cochain complex Im D Q −→ Im D; this notation is appropriate because Im D = ker D ⊥ . It follows that W = Sym(ker D ∨ ⊕ ker D ∨ [1]) and similarly for W ⊥ . Recall that the differential on Obs q h −1 can be written as a sum We will treat the termsh∆ ∞ and {I[∞], ·} ∞ as successive deformations of the differential Q + d L on O, starting from a homotopy equivalence of (O, Q + d L ) with a smaller subspace. The cochain complex (O, Q + d L ) describes the classical observables of the massless free fermion with trivial L action.
Proposition 4.10 (Cf. Proposition 2.5.5 and Theorem 2.6.2 in [Gwi12]). There is a homotopy equivalence Here, the notation¯h ⊗ reminds us that in each component Sym i (L [1] ∨ ) ⊗ W ( j) , we are allowed to take rational functions ofh.
Proof. As described in Section 2.6 of [Gwi12], the Hodge decomposition leads to the desired homotopy equivalence, with maps π, ι, and η defined as follows. Using Equation 4.1, Now, π is just projection onto the subspace of O for which k = 0, and ι is the inclusion of this subspace into O. η is a bit more subtle. We let P denote the degree -1 operator on S whose kernel is P(0, ∞) (see Definition 3.33). On V [−1] ⊂ S , P acts by 0 on ker D and by D −1 on Im D. P also induces an operator on S ∨ , and we can also extend it C • (L )-linearly to O as a derivation for the multiplication in W ⊥ ; we will abuse notation and call all three operators P. We define η to be 0 on the k = 0 component of O and P/k on the k = 0 components of O. Note that η acts only on the W ⊥ part of O. The fact that this is a homotopy equivalence is verified in [Gwi12]. Now, we turn on the deformationh∆ ∞ to compute the quantum observables of the theory with trivial L action: Proposition 4.11. There is a homotopy equivalence In other words, the deformationh∆ ∞ of the differential on O induces the same deformation on Proof. We use the homological perturbation lemma, though we need to check that the hypotheses of that lemma apply. First, note the operator Q + d L +h∆ ∞ is a differential, since it is the differential on Obs q h −1 ,0 , i.e. it is the differential on O for the trivial action of L on S . Moreover, (1 −h∆ ∞ η) is invertible, as we now show. ∆ ∞ is contraction with K ∞ ; based on the description of K t in the proof of Proposition 5.1 in the next section, and using the characterization of the heat kernel immediately preceding Proposition 2.37 in [BGV92], it follows that Here, {φ i } is an orthonormal basis for ker D sitting in degree 0 and ξ i is φ i but living in degree 1. Let {φ * i } and {ξ * i } denote the dual bases for ker D ∨ . It follows that So ∆ lowers Sym-degree in W by 2. On the other hand, η doesn't change Sym-degree in W ⊥ . More important, no element of O can have a dependence of degree greater than dim ker D on the φ i , since the φ i are of ghost number 0 and have fermionic statistics and so anti-commute with each other. It follows that (∆ ∞ η) dim ker D+1 = 0 so that and the homological perturbation lemma applies to our situation.
The perturbation δ ∆ ∞ of the differential on C • (L )¯h ⊗ Sym(H • S ) ∨ ) is given by the formula however, since η acts trivially on elements of C • (L )¯h ⊗ W and ∆ ∞ preserves C • (L )¯h ⊗ W ⊂ O, the formula for δ ∆ ∞ simplifies: The proposition follows. Now, we'd like to understand the chain complex (C • (L )¯h ⊗ W, d L +h∆ ∞ ) a bit better. We continue to use the notation ξ * i , φ * i from the proof of the previous proposition.
Proof. ι and π will be, just like ι and π, inclusion and projection operators; we simply need to identify (det D + ) ∨ inside of W. To this end, let {φ * be an orthonormal basis for (ker D + ) ∨ and {φ * i } dim ker D − j=1 be an orthonormal basis for (ker D − ) ∨ . ker D − , by Lemma 2.12, is naturally identified with coker D + , and we can use the metric (·, ·) to identify coker D + with its dual. In short, there is an isomorphism Now, ι can be verified directly to be a cochain map, while to verify that π is a cochain map, we must show that if This follows from the fact that d L +h∆ ∞ preserves the filtration of W by total degree with respect to the φ * i . Moreover, π ι = id, by construction.
It remains to construct η and verify that η is a chain homotopy between ι π and id. We can write , where the φ * i have ghost number 0 and are fermionic, while the ξ * i have ghost number -1 and are bosonic. Thus, the ξ * i commute with each other and with the φ * i , while the φ * i anti-commute among themselves. Now, we let I = (i 1 , · · · , i dim ker D ) be a multiindex, J ⊂ (1, · · · , dim ker D), and α ∈ C • h −1 (L ); we consider J to be a multi-index with all indices zero or 1. Moreover, define σ I,J to be the cardinality of the set {i j | j ∈ J, i j = 0}, and denote by |I| the total degree of I, that is Here, |J c | is the cardinality of the complement of J in {1, · · · , dim ker D}. We let 1/(|J c |+σ I,J ) = 0 when both summands in the denominator are zero. Let us verify, using the notation δ jk for the Kronecker delta: whence the proposition follows.
Finally, to prove Theorem 4.2, we turn on the perturbation {I[∞], ·} ∞ : Proof of Theorem 4.2. The three preceding Propositions, together with Proposition 4.9, can be combined to yield a homotopy equivalence We need only to verify that the perturbation {I[∞], ·} ∞ satisfies the hypotheses of the homological perturbation lemma. We have seen that the operator , ·} ∞ is a differential (satisfies the wQME), so it remains to show that To do this, we must understand η a bit better. In the notations of Propositions 4.11 and 4.12, η = η + ι η π . Let us explicitly compute ι , η , and π . Letting ι, η, and π be as in Proposition 4.10. Then but η acts by 0 on this subspace. To show that π = π, note thath∆ ∞ η commutes with ιπ because both ∆ ∞ and η preserve W. Moreover, ηιπ = 0, whence Recall also that the sum in η is finite, since η and ∆ ∞ commute and ∆ dim ker D + +1 ∞ = 0. We will need to understand the effect that various operators have on the Symgrading andh-powers in O. The results are depicted in Figure 6. The table should be interpreted as follows: a 0 means that the operator preserves the corresponding degree, a > means the operator has only terms which increase the corresponding degree, a ≥ means the operator has terms which increase the corresponding degree but none that decrease it; < and ≤ have analogous meanings. In all cases except the >, the operators only have a finite number of terms increasing or decreasing the corresponding degree.
We would like to define ]. This is well-defined; however, the operation of multiplication by ∑ ∞ i=0h i is not defined on the space of Laurent series, as a consequence of the general fact that Laurent series cannot be multiplied. This is because, if p = ∑ i∈Z p ih i , then the putative value of Since j ranges over Z, the sum in parentheses is infinite, so is ill-defined in general. We need to make sure a similar problem doesn't happen when we attempt to define and let ζ denote the putative value of (1 − {I[∞], ·} ∞ • η ) −1 θ, with ζ (i, j,k,m) defined similarly. The C • (L ) column of Figure 6 tells us that ζ (i, j,k,m) contains only sums of terms like for l + i ≤ i, i.e. only a finite number of i contribute to ζ (i, j,k,m) . Moreover, since in all spaces but the >, the corresponding operator has only a finite number of terms changing degree, at a fixed l and i , there is a maximum amount by which ({I[∞, ·} ∞ η ) l can change j , k , and m . Thus, ζ (i, j,k,m) only has contributions from when l + i ≤ i and | j − j|, |k − k|, and |m − m| are small enough for the given l and i . It follows that (1 − {I[∞], ·} ∞ η ) −1 is well-defined, whence the Theorem.

The Obstruction and the Determinant Line Bundle.
In Section 3 and the previous subsection, we saw some indication that the obstruction Obstr is related to the triviality of the equivariant quantum observables Obs q over BL (N) (see, e.g. 3.47 and the Corollary to Theorem 4.2). In this subsection, we show that Obstr is precisely the measure of the triviality of the corresponding line bundle. We begin with a reformulation of Theorem 4.2: Corollary B. Choose the same hypothesis as in Theorem 4.2, and recall the notation Obs q h −1 ,Obstr from Proposition 3.47. There is a homotopy equivalence Here, Obstr[∞]· denotes multiplication by Obstr[∞] in the underlying symmetric algebra of Obs q [∞].
Proof. From the proof of Theorem 4.2, we have a homotopy equivalence The differential on Obs q h −1 ,Obstr [∞] is a pertrubation of d L + Q +h∆ ∞ by the operator hObstr[∞]·. For the same reasons as in the proof of Theorem 4.2, this perturbation satisfies the hypotheses of the Homological Perturbation Lemma. Therefore, all we need to check is that the corresponding differential on C • (L )¯h−1 ⊗(det D + ) ∨ is as specified. To see this, recall that the perturbation of the differential on C • (L )¯h−1 ⊗(det D + ) ∨ is given by the formula Moreover, almost by construction, η is 0 on this subspace. The Lemma follows.
Definition 4.13. The image of 1 ∈ Obs · is the partition section. This generalizes the definition of the partition function in Corollary A.

Together with the isomorphism Obs
, Corollary B gives us an explicit characterization of the differential on C • h −1 (L )⊗(det D + ) ∨ . Moreover, this differential is directly related to the obstruction. Indeed, let us note that C • h −1 (L ) is precisely the Chevalley-Eilenberg cochain complex of the elliptic DGLA L ⊗ C((h)) over the field C((h)), so that should be interpreted precisely as the sheaf of sections of a line bundle over over the base field C((h)), and C • h −1 as the space of functions on this formal moduli problem over C((h)) . The following Lemma shows that the cohomology class of Obstr[∞] measures precisely whether or not this bundle is trivializable.
Lemma 4.14. The determinant bundle Since none of the differentials use the particular structure of det D + , we can replace it with C and prove the analogous results.
Suppose Obstr[∞] = dα for α ∈ C 0 h −1 (L ). Then multiplication by e¯h α gives a cochain isomorphism On the other hand, suppose given a cochain isomorphism of C • h −1 (L )-modules The statement that this map intertwines differentials implies, in particular, that Since Φ is an isomorphism, Φ(1) must be invertible in the symmetric algebra C • h −1 (L ). It follows thath . Remark 4.15: This is a precise way of saying, in the case we are studying, that the anomaly is the failure to construct a well-defined fermionic partition function. If the obstruction is cohomologically non-trivial, then the procedure of path integration, as incarnated in the BV formalism, produces the line bundle of Corollary B and the partition section of Definition 4.13. However, unless Obstr is trivial, this section cannot be made into a function. ♦

THE MAIN FEYNMAN DIAGRAM COMPUTATION
The plan for the next three sections is as follows. In this section, we perform an essential computation that will show the first incarnation of the relationship between index theory and the obstruction theory described in Section 3. In the next section, we provide a more complete picture of the obstruction by characterizing the local Chevalley-Eilenberg cochain complex of de Rham forms. Finally, in Section 7, we prove generalizations of the results of Section 6 in the case that a Lie algebra g acts on V in a way that commutes with D.
Throughout this section, we use the same notation as in Example 3.7, Notation 3.9, and Example 3.24, namely the notation associated to the massless free fermion theory and its axial symmetry. Our main goal here is to prove the following Proposition 5.1. Let (M, g) be a Riemannian manifold, V → M a Z/2-graded metric vector bundle with metric (·, ·), and D a self-adjoint Dirac operator. Denote by L R the DGLA Ω • dR . (These data define a free classical fermion theory S , together with an action of L R on the theory, which has an obstruction Obstr.) Then, Obstr[t], when evaluated on a constant function λ, satisfies, where k t is the heat kernel for the generalized Laplacian D 2 .
Remark 5.2: Compare this formula to Equations 22.2.10 and 22.2.11 in [Wei05], which provide an unregulated formula for the anomaly, i.e. the author computes the t → 0 limit of the above expression by computing the supertrace of the t → 0 limit of k t . This is ill-defined: since the t → 0 limit of k t is the kernel of the identity operator, this limit is singular on the diagonal in M × M. In [Wei05], the author regulates those expressions by introducing a mass scale and taking that scale to infinity. On the other hand, in the present work, we have used the heat kernel techniques of [Cos11] to make sense of the ill-defined expressions. ♦ The proof of Proposition 5.1 is not tremendously difficult; it simply requires a careful accounting of signs. One major component of this accounting is in determining the precise relationship between the BV heat kernel K t and the heat kernel k t as defined in Theorem 2.7, and the next lemma is devoted to specifying this relationship. Lemma 5.3. Let k t,1 denote the heat kernel for the generalized Laplacian D 2 , viewed as an element of V ⊗ (V [−1]) ⊂ S ⊗ S . Similarly, let k t,2 denote the heat kernel viewed as an element of (V [−1]) ⊗ V . Then, the BV heat kernel satisfies the following equation In particular, K t is anti-symmetric under interchange of its two factors.
Proof. Recall that, by Definition 3.30, K t ∈ S ⊗ S satisfies where f ∈ S . We need to make precise meaning of the expression on the left hand side of Equation 5.1, keeping track of all the signs that arise from the Koszul sign rule. This is a slightly subtle issue, so we will first consider a simple example to illustrate why this subtlety arises.
Let us consider the symmetric monoidal category of Z × Z/2-graded vector spaces where the braiding is given by Given W in this category, we ask what the natural way to pair an element ω 1 ⊗ ω 2 ∈ W ∨ ⊗ W ∨ with an element w 1 ⊗ w 2 ∈ W ⊗ W is, assuming ω 1 , ω 2 , w 1 , and w 2 are homogeneous. To do this, we compute the image of ω 1 ⊗ ω 2 ⊗ w 1 ⊗ w 2 under the following composition of maps: where ev : W ∨ ⊗ W → R is the natural pairing of a vector space with its dual and µ is multiplication of real numbers. In short, by (ω 1 ⊗ ω 2 )(w 1 ⊗ w 2 ), we mean (−1) |w 1 ||ω 2 |+π w 1 π ω 2 ω 1 (w 1 )ω 2 (w 2 ). Analogously, we understand the term on the left hand side of Equation 5.1 as the image of −id ⊗ ·, · ⊗ K t ⊗ s under the following chain of compositions: where the first map uses the Koszul braiding on our symmetric monoidal category to move the first factor of S past Bilin(S ) and the second map is a tensor product of evaluation maps Hom(S , S ) ⊗ S → S and Bilin(S ) ⊗ S ⊗ S → R. It follows that K t = −k t,1 + k t,2 , with the relative minus sign arising because the first factors of k t,1 and k t,2 have ghost numbers of opposite parity.
Proof of Proposition 5.1. Recall (Lemma 3.49) that We note that the term d L R I wh [t](λ) is zero because it evaluates I wh with dλ = 0 in one of the slots, so the only term we have to consider is ∆ t I tr [t](λ).
Let us study the Feynman diagrams appearing in I tr [t] with a λ input into each   The tadpole diagram, evaluated on λ, gives where τ is the Koszul braiding on the symmetric monoidal category of Z × Z/2graded (topological) vector spaces. Relative to the corresponding notion without Koszul signs, τ K t has (1) no sign arising from ghost number considerations, since K t is of degree 1 and therefore belongs to V ⊗ (V [−1]) ⊕ (V [−1]) ⊗ V , and (2) a sign arising from the fact that all elements of S have fermionic statistics.
The last equality in Equation 5.2 then follows because by construction ·, [λ, ·] is antisymmetric in its two arguments.
We need to show that which will complete the proof in light of Equation 5.2. Heuristically, the term should compute the trace of K t , since λ simply acts by multiplication by ±λ and ·, · (K t ) pairs the part of K t in E with the part of K t in E ∼ = E ! [−1], which is precisely what one does when one computes the trace of an endomorphism of a vector space.
To compute − ·, [λ, ·] (K t ), we take K t (x, x) (which is an element of the fiber of V ⊗ V over x), let λ act on the second factor, pair the two factors using (·, ·), and integrate the resulting function over M against the Riemannian density. We also use the description of K t from Lemma 5.3. Taking into account all of these comments, we which is precisely what we claimed; this completes the proof.
Remark 5.4: The proof of the McKean-Singer formula in [BGV92] has two parts: one first shows that the supertrace of the heat kernel is independent of t, then one shows that the t → ∞ limit of the supertrace is ind(D). Our work has given a physical, Feynman-diagrammatic interpretation of the first part of that proof. Namely, Lemmas 3.54 and Proposition 5.1 together imply that is independent of t. One can take the t → ∞ limit, using Lemma 2.14, to get ind(D). ♦

THE OBSTRUCTION AND THE OBSTRUCTION COMPLEX
In this section, we use the explicit computation of the previous section to study and characterize Obstr in its entirety. In Section 6.1, we give an abstract characterization of the cohomology class of Obstr (Theorem 6.2). This is the main theorem of the paper. In Section 6.2, we tie up some loose computational ends by computing the obstruction as evaluated on one-forms. Finally, in Section 6.3, we connect our approach to the more traditional Fujikawa approach, as discussed in [Wei05].
6.1. The Main Results. We continue to use the notation of Example 3.24, Example 3.7, and Notation 3.9; in short, we continue to let S be the space of fields for the free spinor theory, with L R := Ω • dR acting by the axial symmetry. So far, we have only studied the obstruction when evaluated on a constant function λ; we would like to understand Obstr as a cohomology class of C • loc,red (L R ) in more detail. In fact, the following nice characterization of the reduced obstruction complex C • loc,red (L R ) will allow us, in Theorem 6.2, to characterize the obstruction entirely.

Proposition 6.1 (Obstruction Complex). If M is oriented, there is a map of complexes of sheaves
Φ is a quasi-isomorphism.
Having established a nice characterization of the obstruction complex, we can state the Main Theorem of this paper: Theorem 6.2. Let (M, g) be a Riemannian manifold of dimension ns, V → M a Z/2-graded vector bundle with metric (·, ·), and D a self-adjoint Dirac operator. Denote by L R the DGLA Ω • dR . (These are the data of the massless free fermion and its axial symmetry, which has an obstruction Obstr.) Under the quasi-isomorphism Φ of Proposition 6.1, the cohomology class of Obstr corresponds to (−1) n+1 2 Str k t (x, x)dVol g (x). More precisely, in cohomology. This statement is also true if M is not oriented. Remark 6.3: Top-degree classes in de Rham cohomology on an oriented manifold are determined by their integral; the integral of Obstr is Thus, the cohomology class of Obstr coincides with the class of (−1) n+1 2 ind(D) vol(M) dVol g , as stated in the Introduction. ♦ Proof of Proposition 6.1. Our line of attack will be to first prove that the two complexes have the same cohomology, and then to use the well-understood structure of the cohomology of the de Rham complex to show that Φ is a quasi-isomorphism. This first step is a codification of ideas implicit in Chapter 5.6 of [Cos11], and its proof sets to paper ideas that have been informally passed around the factorization algebra community.
We first invoke Lemma 3.17, which tells us that , where the jet bundle J(Ω • dR ) was defined in Definition 3.16 and O red (J(Ω • dR )) was defined in Equation 3.3. We will replace both terms in the derived tensor product above with quasi-isomorphic complexes. Let's start with the map of bundles of DGLAs and of D M -modules ι : C ∞ M → J (Ω • dR ) which in the fiber over x ∈ M takes c ∈ R to the jet at x of the constant zero-form c. The Poincaré lemma can be used to show that this is a quasi-isomorphism of D M modules. Namely, since the differential arises from a fiberwise endomorphism of J, we can in each fiber use the Poincaré lemma to show that the fiber is quasiisomorphic to R in degree 0. In particular, the only jets of a de Rham form at x which are closed but not exact are the jets of constant functions. It follows that ι is a fiberwise quasi-isomorphism, so it is also a quasi-isomorphism on any open R n ⊂ M and therefore a quasi-isomorphism of complexes of sheaves. Moreover, ι induces a quasiisomorphism of D M -modules We consider next the Dens M factor in the local Chevalley-Eilenberg cochains. Since M is Riemannian, Dens M is trivial as a bundle, and its (right) D M -module structure is induced from viewing a density as a distribution. This D M -module structure encodes the fact that total derivatives integrate to zero, and it has the following projective resolution: where the differential δ is given by the formula δ(α ⊗ f X 1 · · · X s ) = d( fα) ⊗ X 1 · · · X s + (−1) |α| fα ∧ ∇(X 1 · · · X s ), where ∇ is the natural flat connection on D M and the map Ω n dR ⊗ D M → Dens M is given by the action of D M on densities, which happen to also be n forms because M is oriented. It is a direct computation to check that on a local coordinate patch U, this is a Koszul resolution of Dens M (U) with respect to the regular sequence {∂ i } n i=1 of D M , and so gives a quasi-isomorphism Ω • dR ⊗ C ∞ M D M → Dens M . As a result, we have the following equivalence . We now need to identify this last complex with Ω • dR . Consider the following map It is immediate that this map is well-defined and intertwines differentials. We claim, moreover, that it is an isomorphism. To see this, note that, in a coordinate neighborhood U of M, Ω • dR ⊗ C ∞ M D M (U) is, ignoring the differential, a free D M (U)-module on the generators 1 ⊗ dx I as I ranges over all multi-indices. Thus, when we tensor with C ∞ M (U) over D M (U), the result is a free C ∞ M (U)-module on the same generators, namely the de Rham forms Ω • dR (U). The above map takes generators to generators, so is an isomorphism. This completes the proof that the two complexes of sheaves of our lemma are quasi-isomorphic.
We need only to show that Φ itself is a quasi-isomorphism, which can be checked on open balls. Suppose that U is homeomorphic to R n ; then by the Poincaré Lemma, the cohomology of Ω • dR (U)[n − 1] is just an R concentrated in degree −n + 1 and spanned by the constant function 1. Thus, it suffices to check that the following ghost number −n + 1 element (which is precisely the image of 1 where on the left hand side we are thinking of ω as a collection of jets of n forms on U and on the right hand side we are thinking of ω as a density on U. Ξ is linear in the jets of its input, and since we're dealing with an abelian DGLA, the differential d L R preserves the Sym grading. Therefore, if Ξ were exact in C • loc,red (L R )(U), it would be d L R J for J some degree −n element of Dens M ⊗ D M Hom C ∞ M (J(Ω • dR )[1], C ∞ M ); but the degree −n elements of this latter space are zero, and Ξ = 0, so Ξ is not exact. Thus, However, if any of these forms has degree higher than 1 in L R , then Obstr will evaluate to zero because Obstr only uses forms through their action on S and these higher forms act trivially on S . Thus, Obstr (r) [t] is non-zero only when evaluated on a collection of one-forms and functions whose total ghost number in L R [1] is -1. This can only be accomplished if all but one of the forms is a one-form.
Thus, to understand Obstr[t] explicitly, we need only to evaluate it on collections of de Rham forms of the sort described in the preceding lemma. In fact, because a symmetric function is determined by its value when all of its inputs are identical, it suffices to consider Obstr[t](λ + α), where α is a 1-form and λ is a function, and we are most interested in the case that α and λ are closed. The following corollary then follows from Theorem 6.2 and suffices to characterize Obstr[t] evaluated on any collection of closed forms.
Corollary. Let α be a closed 1-form and λ a closed 0-form. Then Together with Lemma 6.6, this characterizes the value of Obstr[t] on all closed forms.
Proof. First note that both Obstr[t] and ∆ t I for f ∈ C ∞ (M). This is manifestly local. Furthermore, ∆ t I (1) tr is closed because it is an element of Sym 1 (L ∨ R [−1]) of top ghost number and d L R preserves Sym-degree. Thus, d L R ∆ t I (1) tr is of Sym-degree 1 and of ghost number one greater than ∆ t I (1) tr , hence is 0. The crucial observation is that Obstr[t](λ) = ∆ t I (1) tr (λ), as we saw in the proof of Proposition 5.1. Then, by the proof of Theorem 6.2, it follows that Obstr[t] and ∆ t I (1) tr represent the same cohomology class in H • (C • loc,red (L R )(M)). Since evaluation on cocycles in H • dR is independent of the choice of representative of a class in H • (C • loc,red (L R )(M)), the two cocycles take the same value when evaluated on closed de Rham forms, so that where the last equality follows from the linearity of I (1) tr in its L R arguments. But, by Lemma 6.6, ∆ t I (1) tr (α) = 0, so that as desired.
6.3. The Anomaly as a Violation of Current Conservation. In this subsection, we try to make contact with the traditional (and familiar to physicists) notion that an anomaly is the failure of a current which is conserved classically to be conserved at the quantum level. The discussion will be relatively informal, but we will secretly be using a very simplified version of the Noether formalism of Chapter 12 of [CG].
Before we begin, we should establish some notation. So far, we have worked only with the equivariant observables. In this subsection, we will work also with the nonequivariant observables; that is, we will work with the observables of the theory defined by S . In particular, we will let Obs cl (S ) denote the classical observables of S The following proposition gives an interpretation of the interaction I as giving rise to symmetries of the classical theory: Lemma 6.7. The map Proof. Direct computation, using the properties of an action of a DGLA on a free field theory.
Remark 6.8: Recall that we defined I to be an element of L R [1] ∨ ⊗ Obs cl (S ). We abuse notation and refer to the corresponding map L R [1] → Obs cl (S ) also as I. We will often use the notation I X for the observable I(X). ♦ Remark 6.9: We should think of the above map as defining a conserved vector current because it (in particular) takes in a one-form and gives a degree zero observable. More precisely, let us "define" a vector current for all one-forms α. j is conserved in the following sense. The above Lemma tells us that if α = d f , then is an exact observable. Unpacking the definition of exactness in Obs cl (S ), we discover that I α is zero when evaluated on fields φ, ψ which satisfy the equation of motion Qφ = Qψ = 0. In other words, the following observable M d f ( j)dVol g = − M f div( j)dVol g is zero when evaluated on fields satisfying the equations of motion. This implies that div( j) = 0 when evaluated on the solutions of the equations of motion, which is the usual statement of current conservation. ♦ Now, we would like to see whether we can define an analogous map L R [1] → Obs q [t](S ). We will see that this is obstructed precisely by the supertrace of the heat kernel. In particular, suppose we had a map whoseh 0 component is I. Let us write To say that I is a cochain map, we need to show that Theh 0 part of this equation is satisfied by the previous Lemma. However, theh 1 coefficient is: It follows after a bit of reflection that I (1) satisfies this equation if and only if for some linear map J : Ω • dR [1] → R. But this last equality is equivalent to the cohomological triviality of Obstr, as we have seen. Moreover, we can simply take I ( j) = 0 for j > 1. We have therefore shown the following Proposition: Lemma 6.10. There exists a lift in the diagram Obs cl (S ) modh I I if and only if Obstr is cohomologically trivial in C • red,loc (L R ). Remark 6.11: In the proof of the previous lemma, we found that the lift was obstructed by the one-leg term of Obstr (the one represented by the tadpole diagram of Figure  7). However, Obstr has higher-leg contributions which made no appearance in the proof. Although these higher-leg terms are exact as a corollary of Theorem 6.2, it is still worth wondering why those terms did not appear in the proof of the lemma. The answer lies in the fact that we have restricted ourselves only to consider maps of L R [1] → Obs q . If we wanted to extend this map to a map C • (L R ) → Obs q out of the Chevalley-Eilenberg chains of L R , we would find the higher-leg contributions to the anomaly making an appearance. ♦ Remark 6.12: We have not discussed this here, but the exactness of an observable in Obs q is equivalent to that observable having expectation value zero (see [CG17] for details on this point). Thus, the lift I (when it exists) should be viewed as giving a current whose expectation value is conserved. More precisely, just as we should think of I dA as the divergence of the conserved vector j, I dA is a quantization of the divergence of j. The fact that I dA is exact says that the quantum expectation value of j vanishes. ♦

EQUIVARIANT GENERALIZATIONS
We continue to use the notation of Example 3.7 and Notation 3.9. In other words, we are still studying the massless free fermion theory S ; however, we would like to replace the Lie algebra L R from Example 3.24 with a slightly more general algebra, in the following way. Suppose we have an ordinary Lie algebra g acting on V and that this action commutes both with the C ∞ M action and with D. In other words, we have a map of Z/2-graded Lie algebras ρ : g → Γ(End(V)) such that ρ(γ)(Dφ) = Dρ(γ)(φ) for all φ ∈ V , f ∈ C ∞ (M), and γ ∈ g. What we have done so far is the special case that g = R acting by scalar multiplication, and in fact the general case proceeds very similarly, as we will see.
Remark 7.1: The physical interpretation of this g-action is as symmetries of the massless free fermion: since elements of g commute with D, they infinitesimally preserve the equation of motion Dφ = 0. ♦ We will usually just denote the action of an element γ ∈ g on a section φ ∈ V by γφ. Since γ commutes with D, it preserves ker D + and coker D + , so we are free to make the following Definition 7.2. Let g act on V as in the first paragraph of this section. Then, the equivariant index of D is the following function on g: ind(γ, D) = Tr(γ | ker D + ) − Tr(γ | coker D + ).
With this definition in hand, we can now state the equivariant McKean-Singer theorem: Theorem 7.3. Let (M, g) be a Riemannian manifold, V → M a Z/2-graded metric vector bundle, and D a self-adjoint Dirac operator on V. Suppose given an action ρ of the Lie algebra g on V which commutes with the Dirac operator. Then, Here, dVol g is the Riemannian density.
Note that in the special case where g = R and the action is given by scalar multiplication, this theorem just reproduces the regular McKean-Singer formula.
The relevant elliptic DGLA in this context is L g := g ⊗ Ω • dR . We will see that assuming that g acts as above, L g acts on the free spinor theory. Let us first make the following notational convention. If γ ∈ g, we denote by γ ‡ the operator which acts by γ on E and −γ T on E . In other words, E is the given representation ρ of g and E is the dual representation. Now, we can construct the action of L g on the theory S . Lemma 7.4. Let σ : L g ⊗ S → S be given by . σ is an action of L g on the massless free fermion theory. Here, c(α) denotes the Clifford action of Definition-Lemma 2.6.
Proof. As above, we will often use [·, ·] to denote the bracket on S [−1] ⊕ L g provided by σ. We need to check that [α, ζ]] for all forms α, β and all ζ ∈ S , i.e.
the Jacobi identity is satisfied, and (3) [α, ζ], η + (−1) |α||ζ| ζ, [α, η] = 0 for all forms α, ζ, η ∈ S , i.e. [α, ·] is a derivation for the pairing ·, · . The terms in the first identity are non-zero only if α has degree zero in L g and ζ has ghost number zero in S ; in that case, the identity reads Here, the question mark over the equals sign means that the equality does not necessarily hold, and is rather an equality whose truth we want to check. We note the following identities The first holds because γ and γ T preserve V + and V − ; the second identity holds because by assumption γ commutes with D and so too does γ T by the self-adjointness of D; the third identity holds because c(d f ) is defined in terms of only D and f , both of which commute with γ and γ T by assumption. Using these identities, Equation 7.2 becomes D 0→1 f γ ‡ Γ 0→0 ζ ? = c(d f ) 0→1 γ ‡ Γ 0→0 ζ + f D 0→1 γ ‡ Γ 0→0 ζ, which is true by the definition of c(d f ).
Now we turn to the verification of the Jacobi identity. If α and β are both of nonzero cohomological degree, then both sides of the Jacobi identity are automatically Proof. Just as in the proof of Proposition 5.1, only one diagram contributes to the Feynman diagrammatics, and that diagram gives a contribution − M Tr S (γ ‡ (x)k t (x, x))dVol g (x) = − M Tr V + (ρ(γ)k t (x, x))dVol g (x) + M Tr V − (ρ(γ) T k t (x, x))dVol g (x) − (+ ↔ −) = −2 M Str(ρ(γ)(x)k t (x, x))dVol g (x).
In the last equality, we use the fact that k t = k T t as a result of the self-adjointness of D, so that we have Tr V − (γ T k t (x, x)) = Tr V − ((k t (x, x)γ) T ) = Tr V − (γk t (x, x)). The result follows.
Remark 7.6: Just as in Section 5, the t-independence of the obstruction, together with Proposition 7.5, show the t-invariance of the quantity appearing in that Lemma. Moreover, one can take the t → ∞ limit of this quantity (see [BGV92]). As is to be expected, the result is ind(γ, D), the global quantity appearing in the equivariant McKean-Singer formula. ♦ We also have results analogous to those of Section 6. Proposition 7.7. Let M be a manifold, g a Lie algebra, and L g := g ⊗ Ω • dR . If M is orientable, there is a map Φ : C • red (g) ⊗ Ω • dR [n] → C • loc,red (L g ) given by Φ(ω ⊗ α) (γ 1 ⊗ α 1 , · · · , γ r ⊗ α r ) = (−1) |α| ω(γ 1 , · · · , γ r )α ∧ α 1 ∧ · · · ∧ α r , and Φ is a quasi-isomorphism. Here, C • loc,red (L g ) is the cochain complex of reduced, local Chevalley-Eilenberg cochains. There is an analogous statement if M is not orientable using twisted de Rham forms.
Proof. The proof is similar to that of Proposition 6.1. Letting J g refer to the jet bundle of Ω • dR ⊗ g, we have a quasi-isomorphism of DGLAs C ∞ M ⊗ g → J g , and then C • red (J g ) is quasi-isomorphic to C ∞ M ⊗ C • red (g). Then, just as in Proposition 6.1, we can show that both the source and target of Φ have the same cohomology groups.
It remains to show that the map specified above is a quasi-isomorphism. We show this by a spectral sequence argument. Both source and target of Φ are filtered by Sym degree and Φ preserves the filtrations. Therefore, the cohomology of both can be computed by a spectral sequence and there is a map of spectral sequences between the two. The first page of the spectral sequence just computes the cohomology of both sides with respect to the differential induced by the de Rham differential; on both sides, this is Sym >0 (g[1] ∨ ) ⊗ H • dR . This is obvious on the source side, while on the target side, this is a minor modification of the argument of Proposition 6.1. The induced map on E 1 pages is induced from Φ, where we think of both source and target as having only the truncated, Sym-degree-preserving differential. A small modification of the argument from Proposition 6.1 can be used to show that the map on E 1 pages is indeed an isomorphism. Namely, on a neighborhood U of M homeomorphic to R n , the cohomology of Sym >0 (g[1] ∨ ) ⊗ Ω • dR (U) is Sym >0 (g[1] ∨ ). To prove that the induced map on E 1 pages is an isomorphism, we need to show that the images of elements of Sym ≥0 (g[1] ∨ ) under Φ are not exact in the cochain complex (O loc,red (g ⊗ Ω • dR )(U)[1], d dR ) .
This shows that the map induced on E 1 -pages is locally an isomorphism, so it is an isomorphism of sheaves. It follows that the induced map on E 2 -pages is an isomorphism. But the E 2 page is also the E ∞ page, so we see that Φ indeed induces an isomorphism on cohomology.
We also have an analogue to Theorem 6.2; however, this proof requires a slightly more sophisticated touch, since the appropriate generalization of Diagram 6.1 is not immediately obvious.
Theorem 7.8. Let (M, g) be a Riemannian manifold, V → M a Z/2-graded vector bundle with metric (·, ·), and D a self-adjoint Dirac operator. Let g act on V D-equivariantly and denote by L g the DGLA g ⊗ Ω • dR . (These are the data defining a free classical fermion theory S , together with an action of L g on the theory, which has an obstruction Obstr.) Under the quasi-isomorphism Φ of Proposition 7.7, the cohomology class of Obstr as an element of C • loc,red (g ⊗ Ω • dR (M)) coincides with the cohomology class of T : γ → (−1) n+1 2 Str(ρ(γ)k t )dVol g in C • red (g) ⊗ Ω • dR (M)[n].
Proof. Consider the following diagram: Only the downward pointing arrows have not yet been discussed. PD, whose letters stand for "Poincaré Duality," is a composition given by first taking the projection of Ω • dR onto H • dR given by Hodge theory and then taking the Poincaré duality map. The diagonal arrow is a bit more involved. Let p be an element of Sym k ((g ⊗ Ω • dR (M)) ∨ ). The diagonal map sends is given by res(p)(β, γ 1 , · · · , γ k ) = p(γ 1 ⊗ β, γ 2 ⊗ 1, · · · , γ k ⊗ 1), where on the right hand side we have used the harmonic representative for the cohomology class β to understand β as a de Rham form. We are also thinking of the right hand side of the above expression as an element of the coinvariants with respect to the S k action of permutation of the γ i 's. It is a quick verification that the diagonal map is a chain map and that the diagram commutes. Now, since the two maps out of C • red (g) ⊗ Ω • dR (M) are quasi-isomorphisms, it follows that the composite map η := res • i is a quasi-isomorphism as well. In particular, this tells us that the inclusion of local Chevalley-Eilenberg cochains into the full space of Chevalley-Eilenberg cochains is injective on cohomology, i.e. if two local cochains differ by an exact term in the full space, they also differ by an exact term in the local complex.
We claim that [η(Obstr[t])] = [id ⊗ PD(T)] in cohomology. Here, it is crucial to us that η factors through the full Chevalley-Eilenberg cochain complex, since this allows us to treat the d L g I wh [t] and ∆ t I tr [t] separately. (Recall that we know only that the sum d L g I wh [t] + ∆ t I tr [t] is local.) First of all, this allows us to note that the term d L g I wh [t] in the scale t obstruction is exact in C • red (g ⊗ Ω • dR ), so that it does not contribute to the class [η(Obstr[t])]. Moreover, the one-leg term in ∆ t I tr [t] is Φ(T) by Lemma 6.6 and a slight modification of Proposition 7.5 to evaluate also on non-constant functions γ ⊗ f . Hence, it suffices to show that all higher leg contributions to ∆ t I tr [t] go to something exact under res. To see this, note that-by Lemma 6.6-any term with r legs is non-zero only when it has one zero-form input and r − 1 one-form inputs. This implies, in particular, that η takes any term in ∆ t I tr [t] with more than two legs to zero. This is because such diagrams are zero when evaluated on anything with more than one zero-form input, and η evaluates a Sym-degree r element of C • loc,red on r − 1 zero-forms.
We are left with just the task of showing that the two-leg term in ∆ t I tr [t] goes to something exact under η. We will do this in the t → ∞ limit, after first establishing some notation. Let Z t denote where P t is the operator which appears in Definition 3.33 for the propagator P(0, t). We will let P denote the t → ∞ limit of P t ; P is characterized by the fact that it is 0 on ker D[−1] ⊂ S and D −1 on Im D[−1] ⊂ S . We therefore allow t to take the value ∞ in Z t . Finally, let H := [Q, Q GF ] be the generalized Laplacian of the gauge-fixed massless free fermion.