The elliptic Mathai–Quillen form and an analytic construction of the complexified string orientation
We construct a cocycle representative of the elliptic Thom class using analytic methods inspired by a 2-dimensional free fermion field theory. This produces the complexified string orientation in elliptic cohomology, and hence determines a pushfoward for families of rational string manifolds. We construct a second pushforward motivated by the supersymmetric nonlinear sigma model studied by Witten in relation to the Dirac operator on loop space. We show that these two pushforwards agree. Analogous constructions in 1-dimensional field theories produce the Mathai–Quillen Thom form in complexified K-theory and the -class for a family of oriented manifolds.
The last 35 years have seen rich cross-fertilization between topology, geometry, analysis and quantum field theory. A key aspect is rooted in the connection between supersymmetric mechanics and the Atiyah–Singer index theorem [Wit82, AG83, Get88]. Witten’s work from the late 80s [Wit87, Wit88] points towards a generalization of the index theorem associated with the analysis of 2-dimensional quantum field theory on the one hand and the algebraic topology of elliptic cohomology on the other—specifically, the string orientation of topological modular forms (TMF), c.f., [AHS01, AHR10]. Such an index theorem, if it existed, would probe some very subtle analytical and differential geometric aspects of these field theories, mimicking the known intricacies in TMF [Hop02]. In so doing, it would also provide differential-geometric tools for studying powerful homotopy invariants.
This paper offers a glimpse at this picture by putting the geometry of 2-dimensional field theories in direct contact with the string orientation of and proving a baby case of this hoped-for index theorem. We work with a version of the differential cocycle model for developed in [BE13], wherein functions on a certain super double loop space furnish cocycles. Analyzing two types of classical field theories and their 1-loop quantum partition functions yields three main results: (1) an analytic construction of elliptic Mathai–Quillen Thom forms for , (2) an analytic construction of the Witten class of a family of string manifolds, and (3) an index theorem equating the pushforwards associated to these two constructions.
This settles a simplified version of the program initiated by Segal [Seg88, Seg04] and Stolz–Teichner [ST04, ST11] which we paraphrase as follows. Techniques in 2-dimensional field theories construct the string orientation of , and there is an index theorem equating analytic and topological pushforwards. The topological pushforward comes from the partition function of a modified free fermion field theory; this constructs a cocycle representative of the elliptic Thom class. The analytic pushforward comes from the 1-loop partition function of a families-version of the supersymmetric nonlinear supersymmetric sigma model related to Witten’s Dirac operator on loop space [Wit88, Wit99]. These beginnings for an index theory associated with 2-dimensional quantum field theory have obvious enhancements from physics: there is much more structure to a quantum field theory than its partition function. The hope is that these enhancements will continue to encode interesting topology. One possible path is the Stolz–Teichner program; we explain in §1.8 how to view their conjectures as categorifications of the results of this paper.
To the best of our knowledge, there is no clear physical argument for why the analytic and topological pushforwards constructed below agree. Of course, one might have anticipated this in analogy to the situation in K-theory, with the topological pushforward being constructed from Thom classes and analytic pushforward from the index of the Dirac operator. To make this analogy as explicit as possible, we construct these pushforwards in complexified K-theory from a 1-dimensional version of the constructions for . This gives a new construction of the Mathai–Quillen form in using techniques from path integrals and 1-loop partition functions which might be of independent interest. We emphasize that the resulting equality of pushforwards in is distinct from the standard physical proof of the local index theorem [AG83]: the usual argument identifies two calculations for the partition function of supersymmetric quantum mechanics, one in the Hamiltonian and the other in the Lagrangian formulation of the theory. Below we identify the partition functions for two different field theories, both computed in the Lagrangian formulation.
In the next subsection we introduce the minimal ingredients necessary to state our main results, and in the remainder of the section we some provide background and outline the key arguments.
1.1. Statement of results
The differential cocycle model for starts with the super double loop stack of , denoted . It has objects for a 2-dimensional lattice defining a super torus and a smooth map. There is a substack consisting of those maps invariant under the precomposition action of (ordinary) translations . This is the substack of constant super tori in . A line bundle on comes from a square root of the Hodge bundle on the moduli stack of elliptic curves. It has a type of complex structure on sections; denote holomorphic sections of the tensor power by . To identify sections with cocycles, note that is equivalent to ordinary cohomology with values in the graded ring of weak modular forms.
The assignment defines a sheaf of graded algebras on the site of smooth manifolds, and there is an isomorphism of sheaves
with closed differential forms valued in weak modular forms. Hence, is a differential cocycle model for in the sense of Hopkins–Singer [HS05].
The above is essentially Theorem 1.1 in [BE13], with a mild repackaging intended to clarify the connection with super double loop spaces.
A geometric family of oriented manifolds determines a vector bundle whose fiber at is the orthogonal complement of the constant sections in for the tangent bundle of the fibers. The classical action for the nonlinear supersymmetric sigma model is a function on whose Hessian on determines a quadratic function on sections
where and are complex vector fields on . We’ll find that is determined by a deformation of a Laplacian on tori by the curvature of ; to emphasize dependence on we write . The formalism of 1-loop quantization points towards evaluation of the functional integral on the left hand side
which we make rigorous via the -super determinant on the right hand side. This determinant defines a line bundle with section over .
represents the twisted Witten class of the family as a differential cocycle in , for a normalization that is essentially the Dedekind -function.
The line bundle is concordant to the trivial line bundle if and only if the family has a rational string structure, and a choice of rational string structure with specifies a concordance between and a function representing the modular Witten class in .
Next we describe the construction of the Thom cocycle. For an oriented real vector bundle , the pullback over itself has a canonical section . Define a vector bundle whose fiber at is where is the parity reversal of . If we equip with a metric and compatible connection, we obtain the function on sections
where and are the pullback of the metric and connection of along , and is the volume of the torus . This is a 2-dimensional generalization of the classical action studied by Mathai and Quillen [MQ86]. Similar -determinant techniques allow one to rigorously define the functional integral involving ; see (22). The relevant family of operators is denoted , and is a deformation of a family of Dirac operators on tori by the curvature of . The associated determinant line bundle is denoted .
represents the twisted Thom class of in associated with the complexified string orientation of , where is as in Theorem 1.3.
The line bundle is concordant to the trivial line bundle if and only if has a rational string structure. A choice of rational string structure with picks out a concordance between and a section that represents the (untwisted) Thom class in .
Call the differential cocycle the elliptic Mathai–Quillen form. It determines a differential refinement of the string orientation of . For a Riemannian embedding , we get an embedding with normal bundle . The Thom isomorphism for together with the inverse to the suspension isomorphism defines the topological pushforward, denoted . The analytic pushforward, denoted , uses the Witten class for the family to modify a canonical volume form on the fibers coming from the integration of differential forms.
Let be a geometric family of oriented manifolds with fiber dimension . There is a canonical isomorphism of super determinant line bundles for the families of operators and over compatible with their respective super determinant sections. This implies that the analytic and topological pushforwards on differential cocycles agree for geometric families of rational string manifolds,
As we’ll explain shortly, the above theorem is more of a geometric rephrasing of the relevant Riemann–Roch factors in the index theorem over rather than a new proof. However, this rephrasing makes direct contact with the geometry of field theories, pointing towards generalizations in extended (functorial) field theories as we shall explain in §1.8.
1.2. Mathai–Quillen forms
The vector bundle can be pulled back over itself,
and the pullback has a tautological section . If we equip with a metric and compatible connection with curvature , the Mathai–Quillen Thom form in ordinary cohomology is the Berezinian integral (c.f. [BGV92] §1.6)
where here and throughout the subscript denotes compact vertical support (or rapidly decreasing forms) in the fiber directions of , and denotes the sheaf of closed differential forms. By the Riemann–Roch theorem, the complexification of the K-theoretic Thom class fits into the commuting diagram
where is the K-theoretic Thom class of associated to the spin orientation, is 2-periodic cohomology over , and is the inverse of the -class of , which is a characteristic class associated with the power series
Mathai and Quillen constructed a differential form representative of by a careful study of trace maps for the Clifford modules that can be used to construct the class in K-theory. Below we offer a different approach, constructing this form from a -regularized super determinant of a family of deformed Dirac operators on . This perspective has a natural generalization to deformed Dirac operators on . We get a differential form refinement of the vertical right arrow in the commutative diagram
for vector bundles with string structure In the above, is cohomology with coefficients in weak modular forms, is the TMF Thom class of associated to the string orientation [AHS01, AHR10], is the Thom class in ordinary cohomology, and is the Witten class class of associated with the power series
where is the Eisenstein series (see §A.4). We will also have use for the class associated to the power series,
which we shall apply to vector bundles equipped with a 3-form satisfying , i.e., a (geometric) rational string structure. The existence of the 3-form means that and are cohomologous; the point is that (6) clearly gives a class with values in modular forms, as the Eisenstein series are modular forms for .
1.3. Analytic and topological pushforwards over
Let be family of spin manifolds with fiber dimension . Choose an embedding , which determines an embedding . Let be the normal bundle of for of dimension . Then the topological pushforward in K-theory sits in the diagram on the left
where is the inverse to the suspension isomorphism. The diagram on the right is the complexification of the one on the left, defining the topological pushforward in complexified K-theory. In this case, the inverse to the suspension isomorphism is simply the integration over the fibers of .
The string orientation of TMF gives a completely analogous story for a family of string manifolds. We get
and this defines the topological pushforward and its complexification.
The analytic pushforward in K-theory (i.e., index of the Dirac operator) gives a wrong way map , and the Riemann–Roch theorem states that the diagram commutes:
where is the -form of the vertical tangent bundle of . The families index theorem states the the topological and analytic pushforwards agree, .
Witten’s physical reasoning [Wit99] leads one to hope for a TMF-generalization of the Dirac operator and analytic pushforward. Although it remains a difficult and open problem to construct such an operator (often referred to as the Dirac operator on loop space), for a family with fiberwise string structures, any candidate analytic pushforward should sit in the diagram
Below we use analytic techniques to construct a differential cocycle refinement for the vertical arrow on the right via a 1-loop quantization procedure for the supersymmetric nonlinear sigma model with target . This is the same physical theory studied by Witten in his construction of the Witten genus.
In spite of the absence of an analytic pushforward in , there is an index theorem of sorts for the pushforwards in , which amounts to commutativity of the diagram
Furthermore, given a differential cocycle model it makes sense to ask for a differential refinement of this diagram. This isn’t terribly deep: it is equivalent to refining
to the level of differential cocycles. Any differential cocycle model is isomorphic to the differential form model for , so (9) can always be rephrased as an equality of differential forms. Such an equality is not hard to cook up; in our case all one needs is for the embedding to be compatible with the Riemannian structure on that defines the Pontryagin forms. The more interesting part of the story (in our view) is in the analytic construction of the differential forms themselves. This comes from analysis of families of operators over certain super stacks we review presently.
Hereafter we will use the notation , , , , , and to denote differential refinements of the classes , , , , , and . Such refinements typically depend on choices of metric and connection that define the Pontryagin forms representing the Pontryagin classes.
1.4. A brief description of the differential cocycle models
To emphasize certain structural aspects of the story, we will prove analogs of Theorems 1.1-1.5 for de Rham cohomology and K-theory with complex coefficients. In this subsection we quickly review the relevant cocycle models with details in §2.1, §3.1 and §4.1.
For and a smooth manifold, define a stack whose objects111For simplicity, we omit family parameters (i.e., -points) throughout the introduction. are super tori with a map to , and whose morphisms are isometries compatible with the maps to . The stack of constant -dimensional super tori is the full substack , for which is invariant under the pre-composition action of by translation. For each there are line bundles over , and the assignment defines a sheaf on the site of smooth manifolds. There are isomorphisms of sheaves
The computations leading to the above isomorphisms make use of preferred atlases for the stacks given by
where is the space of lattices in (see §A.4). Indeed, such covers come from identifying a super circle or super torus with a quotient for or , and a map invariant under the -action with a map , using that . The isomorphism shows that functions on the stacks , and are differential forms on with values in , and (for ) that are invariant under isomorphisms in the relevant stack.
1.5. Mathai–Quillen Thom forms from families of operators over
For a real vector bundle and , define a vector bundle whose fiber at a map is , where denotes the parity reversal functor. This bundle is of infinite rank for . Define a functional on sections
for is the volume of ,
is an odd vector field on , and we use the Berezinian measure on determined by a volume form on . This action is essentially a super-space (or worldsheet) version of the one studied by Mathai and Quillen; see also [Wu05] for this worldsheet point of view. A consequence of Mathai and Quillen’s work is the following.
The (finite-dimensional) Berezin integral
constructs the Mathai–Quillen Thom form in de Rham cohomology.
Although the calculation above is well-trodden terrain, we re-prove the result in §2 to set the stage for the generalization when . The generalization uses -regularization techniques inspired by physics to define the infinite-dimensional analog of the integral in Proposition 1.7; we explain this physical motivation in §1.7.