Twisted equivariant elliptic cohomology with complex coefficients from gauged sigma models
We use the geometry of -dimensional gauged sigma models to construct a cocycle model for twisted equivariant elliptic cohomology with complex coefficients of a smooth manifold with an action by a finite group. Methodology from gauge theory constructs induction functors that we compare with those in the higher character theory of Hopkins, Kuhn and Ravenel.
Geometric cocycles for cohomology theories often reveal interesting equivariant refinements. A prime example is the refinement of K-theory by equivariant vector bundles, whose utility is difficult to overstate. For example, these geometric cocycles interface with representation theory and provide tools for analyzing higher homotopical structures like power operations. The universal elliptic cohomology theory of topological modular forms (TMF) is expected to admit a (non-Borel) equivariant refinement of comparable richness [And00, Gro07, Lur09], though a complete picture has yet to emerge.
Below we construct a twisted equivariant refinement of motivated by the geometry of -dimensional supersymmetric sigma models, building on the non-equivariant construction of [BE13]. In spite of being defined over , we show that it exhibits chromatic height 2 phenomena in the sense of Hopkins–Kuhn–Ravenel character theory [HKR00]. For example, we construct an induction map using quantization techniques from gauge theory, and show that the resulting character formula agrees with the height 2 case of Theorem D in [HKR00]. This both gives a concrete non-Borel equivariant refinement of , and also suggests a deeper connection between these sigma models and elliptic cohomology.
To demonstrate how the dimension of the physical theory matches with the chromatic height, we construct a similar twisted equivariant refinement of using the geometry of -dimensional gauged sigma models. In this case, the induction maps gotten by quantization methods recover the Frobenius character formula of an induced representation.
1.1. Statement of results
Let be a smooth manifold with an action by a finite group . Form the quotient stack , and let denote a version of the super double loop stack whose objects111In this introductory discussion we omit a technically important family parameter, denoted below. are for a lattice defining a super torus and a map. Equivalently, is a principal -bundle with a -equivariant map . The central geometric object in this paper is a substack whose maps are invariant under precomposition with the action by (even) translations of tori, and hence are a super version of constant double loops or the double inertia stack. There are line bundles over associated with the power of a square root of the Hodge bundle on the moduli stack of elliptic curves, and line bundles over the stacks constructed from a cocycle representing a class . These pullback to the stack using the naturality of for maps of stacks , and we denote by the tensor product. There is a type of holomorphic structure on : it makes sense to talk about sections that depend holomorphically on the lattice , and we denote these sections by .
Let be a finite group acting on and a normalized cocycle representative of . There are isomorphisms of graded abelian groups
natural in maps of -manifolds. The tensor product of sections is compatible with the twisted product structure, .
Above, is a (non-Borel) equivariant refinement of cohomology with values in the graded ring of weak modular forms, i.e., .222We avoid the notation because we do not know this to be the complexification of a twisted equivariant theory over . When is trivial, this is the complexification of the theory defined by Devoto [Dev96], and when it is the twisted coefficient ring studied by Ganter [Gan09]. We give a self-contained definition of in §3.1.
The geometric description of this cohomology theory allows one to use ideas from physics to endow with additional structures. The application in this paper concerns induction maps by way of Freed–Quinn quantization in gauge theory with a finite gauge group [FQ93]. Mathematically, this is a weighted sum over principal bundles.
For any homomorphism of Lie groupoids induced by a homomorphism and any 3-cocycle , there is a linear map
When and is induced by an inclusion a formula for coincides with the height 2 Hopkins–Kuhn–Ravenel induction formula [HKR00, Theorem D].
Ganter proved a similar result to the above for by a more algebraic approach motivated by Freed–Quinn quantization formulas; see §5.2 of [Gan09]. The computations going into our result are similar—and indeed, we drew inspiration from Ganter’s argument—but the geometry is rather different. We work with the objects from physics directly, and so the putative quantization procedure literally is our induction map.
A 2-group is a symmetric monoidal groupoid whose monoidal structure has (weak) inverses; these are also called categorical groups. Examples are furnished by a group together with a -valued 3-cocycle . The dependence of on both and seems to be a shadow of Lurie’s 2-equivariant refinement of elliptic cohomology [Lur09], wherein equivariant refinements exist for any (compact, Lie) 2-group. Theorem 1.2 can be viewed as providing induction formulas for faithful functors between essentially finite 2-groups.
1.2. Gauged sigma models and equivariant TMF
Let be a compact Lie group and a manifold with -action. The fields for gauged supersymmetric sigma model form a groupoid whose objects are triples where is a principal -bundle over equipped with a connection and a -equivariant map . When , classical action functionals on this space of fields can include topological terms depending on a 3-cocycle ; such classical actions are not functions on fields, but rather sections of a line bundle. For a finite group , the super double loop stack is a subcategory of these fields, and the restriction of the line bundles in which these classical actions take values is . In this way, Theorem 1.1 provides a connection between the gauged supersymmetric sigma model and equivariant elliptic cohomology over for finite .
To enhance this connection, one might try to categorify the space of fields. When , one such candidate is Stolz and Teichner’s -Euclidean bordism category over , denoted , whose objects are disjoint unions of -dimensional super circles and whose morphisms are -dimensional super Euclidean surfaces. Following the usual rubric of categorification, one then replaces functions on fields valued in with functors from to complex vector spaces. More generally, sections of line bundles over fields are promoted to natural transformations of functors,
for a category of (topological) algebras, bimodules and bimodule maps, and is a twist functor that determines a degree. The category of these natural transformations is denoted . For a fully extended version of (the appropriate definition of which is still under investigation) Stolz and Teichner’s conjecture is
This flavor of categorification generalizes when one replaces bordisms over by bordisms over the stack , where maps are principal -bundles with connection over whose total space is equipped with a -equivariant map to ; see [ST11, §1.7] for the general framework of gauged bordism categories. In the case at hand, the line bundles have a well-known categorification by classical Chern–Simons theory for with level as an extended field theory; explicitly, we obtain twist functors as the composition
where the first arrow is induced from , is the forgetful functor from super Euclidean bordisms to spin bordisms (see [ST11]), and last arrow is the restriction of classical Chern–Simons theory for the group and level to manifolds of dimension . The category of natural transformations ,
shares features with equivariant elliptic cohomology, leading us to hope
for a suitable equivariant refinement of TMF. In the case that is finite, a de-categorification of this conjecture is Theorem 1.1: the restriction of the right hand side of (1) to closed -dimensional bordisms gives a restriction map
that evaluates a field theory on constant super tori over as bordisms from the empty set to itself.
1.3. Notation and conventions
The supermanifolds in this paper have complex algebras of functions, and are called -manifolds in the survey [DM99]; apart from this altered terminology, our notation and conventions agree with theirs. In particular, a vector bundle over a supermanifold is a finitely generated projective module over the structure sheaf . We will denote this module by , which we caution is typically different than the space of sections coming from maps of supermanifolds.
We frequently use the isomorphism between the supermanifold of maps from to and the odd tangent bundle of . For an ordinary manifold, functions on are differential forms on . The precompositon action of odd translations on on corresponds to the de Rham operator on forms.
For a presheaf on (super) manifolds, two sections are concordant if there is a section whose restriction to and is and , respectively. We call a concordance between the sections and . If is a sheaf, then concordance defines an equivalence relation on sections.
Many of our constructions take place in the category of (smooth) stacks, denoted . A stack is a presheaf of groupoids on the site of super manifolds satisfying descent. Throughout, will denote a test supermanifold, and so the -points of a stack form a groupoid natural and local in . Any super Lie groupoid determines a smooth stack, and all our stacks will admit such descriptions (so they are geometric stacks). An orbifold is a stack for which there exists a representing Lie groupoid that is proper and etalé [Ler10]. For stacks and , is the mapping stack that assigns to the groupoid .
It is a pleasure to thank Ralph Cohen, Kevin Costello, Chris Douglas, Owen Gwilliam, André Henriques, Dmitri Pavlov, Charles Rezk, Nat Stapleton, Stephan Stolz, Peter Teichner and Arnav Tripathy for helpful conversations during the development of this work.
2. Warm-up: twisted equivariant -theory over
In this section we prove an analogous result to Theorem 1.1 for twisted equivariant -theory with complex coefficients, denoted , for a manifold with -action and a normalized 2-cocycle. We start by defining the super loop stack of an orbifold , consisting of super circles with a map to . This has a substack of constant super loops invariant under loop rotation. We construct line bundles on this stack for each and for each normalized -valued 2-cocycle on . When , we compute concordance classes of sections of these line bundles.
There is a natural isomorphism of graded abelian groups
Furthermore, the tensor product of line bundles is compatible with the product structure on twisted equivariant -theory, .
We start by reviewing the equivariant cohomology theory . In §2.2 and §2.3, we provide all the definitions in Theorem 2.1. The remaining sections are more technical: we provide groupoid presentations for , analyze the line bundles within these presentations, and compute their sections, proving Theorem 2.1.
2.1. The delocalized Chern character in twisted equivariant -theory
For a finite group acting on a compact manifold and a normalized 2-cocycle, let denote the -twisted -equivariant -theory of . The delocalized Chern character is the natural map
We emphasize that this target differs from Borel equivariant cohomology over . For example, when , the target is the ring of characters of -projective -representations; see Example 2.9. A general description of was given by Adem and Ruan (also see [FHT08], Proposition 3.11) generalizing the untwisted calculation by Atiyah and Segal ([AS89], Theorem 2).
Theorem 2.2 ([Ar03] Theorem 7.4).
Let be a normalized 2-cocycle. For a manifold with -action, the -twisted -equivariant -theory of with complex coefficients can be computed as
where the sum ranges over conjugacy classes of , denotes the centralizer of , and denotes the 1-dimensional representation of given by .
2.2. The constant super loops in an orbifold
For , call the quotient a family of super circles. As reviewed in Appendix A, an -family of rigid conformal isometries between families of super circles is an -point of that acts on the fibers of the family by super translation and dilation.
The super loop stack of an orbifold , denoted , is the stack associated to the prestack whose objects over are pairs where determines a family of super circles and is an object in the groupoid of maps . Morphisms over consist of 2-commuting triangles
where the horizontal arrow is an isomorphism of -families of super rigid conformal -manifolds.
Before defining the substack of constant loops, we consider the similar object for the ordinary (non-super) loop space. By viewing a principal bundle as a quotient , we can repackage a map as an ordinary path together with a group element such that . Being invariant under loop rotation then means that is a constant path such that , i.e., a constant path with image in the fixed point set . Equivalently, such constant loops are those that factor through the quotient by the action of translations on ,
where we have used the equivalence of stacks . The stack is called the the inertia stack of , denoted .
In the case of super paths we take a similar route. Let denote the morphism of stacks
The substack of constant super loops in , denoted , is the full substack of whose objects are maps with a factorization for .
We observe that the assignment is natural in the orbifold .
2.3. Line bundles over
By forgetting the data of and taking the map on reduced supermanifolds associated to , we obtain a morphism of stacks
In particular, any line bundle on can be pulled back to a line bundle on . The transgression map
provides a source of such line bundles from 2-cocycles representing classes, .
For a 2-cocycle , let denote the line bundle on gotten by the pullback of the transgressed line bundle on to .
Consider the morphism of stacks
that over each sends all objects to , and to a morphism remembers whether the orientation on is preserved or reversed, i.e., we compose the map (that is part of the data of an isometry) with the sign map, .
The line bundle over is defined by post-composing (3) with and pulling back the canonical odd line bundle on , using that . Denote the tensor power of by .
Pulling back along the morphism of stacks induced by the canonical map , we get line bundles also denoted on . Together with the line , this associates to pairs the graded vector spaces , and by inspection this assignment is natural for maps of orbifolds with .
When , the transgression of to a line bundle over is the prequantum line bundle for 2-dimensional Yang-Mills theory of , meaning its vector space of global sections over is the value of 2-dimensional Yang-Mills theory on the circle. This space of sections is the vector space underlying the twisted group algebra , which is a Frobenius algebra and hence does indeed define a 2-dimensional field theory. In fact, the vector space of sections of over is isomorphic to this twisted group algebra.
2.4. Groupoid presentations
We can identify objects of over with pairs for and an object in the groupoid of maps . The equivalence along with the hom-tensor adjunction yields
and, given a groupoid presentation , is equivalent data to an -point of . This gives an atlas
which in turn determines a groupoid presentation of whose objects are . We spell this out in the main case of interest, where
where acts by on the fixed point sets by left multiplication, , and the coproduct in the second presentation is indexed by conjugacy classes. We will focus on the first presentation, though similar arguments apply to the second as well.
For each , consider the quotient for the -action generated by
Define a map
where the first arrow is projection and the second one is . We get a map
from the Cartesian product of (6) and
where the first arrow is the projection and the second arrow is the left -action on the disjoint union, and the third arrow is the action by on by precomposition with automorphisms of .
Isomorphisms between -points of come from isomorphisms between -bundles over -families of super circles. Viewing a principal -bundle over as a quotient,
such isometries come from the action by through gauge transformations and isometries in the base. But such an isometry acts by the identity if it is in the image of
where the inclusion sends the generator to . This gives the claimed super manifold of morphisms.
It remains to understand how these isometries act on the super manifold of objects. For an isomorphism determined by , equivariance of the map from the principal bundle to implies that the effect on objects is . In terms of the supermanifold of objects, this left action is . In the case of an isometry determined by , the effect on objects is to dilate the circle and change to the map to by precomposition with the isometry, so we get for the unique arrow making the diagram commute
Comparing with the definition of (7), this completes the proof.∎
2.5. Presentations of line bundles
Using Proposition 2.8, the line bundle over is determined by the map
which in turn defines the morphism of Lie groupoids that coincides with the one in Definiton 2.6.
To clarify the notation with respect to objects and morphisms in the groupoid , we view as the holonomy of a -bundle on the circle with a trivialization over a basepoint (i.e., an object), and as change of basepoint (i.e., a morphism ). Concretely, this cocycle defines the trivial line bundle on whose fiber at is , that is equivariant for the action of on . Associativity of this action follows from the cocycle condition for . We may pullback along the projection
again denoting the resulting line by . When —the constant function on —the line bundle is trivial. Cohomologous cocycles, , give rise to isomorphic line bundles, , and such an isomorphism is specified by a with .
An -projective representation is a map with the property
For such representations, one can compute explicitly that
so that sections of are in bijection (as functions on with properties) with the vector space generated by characters of -projective representations of . One can also view these sections as being spanned by characters of representations of the central extension of defined by , where the center acts by the standard representation of .
2.6. The proof of Theorem 2.1
Proof of Theorem 2.1.
We view sections of over as functions on with properties. Being equivariant under the dilation action by requires that sections come from functions in the subalgebra of generated by
where the parity of agrees with the parity of , is the standard coordinate on and . Since the dependence on is completely determined by the degree of , the functions generated by can be uniquely identified with differential forms on . Invariance under requires that the forms be closed, as the -action on is generated by the de Rham operator. Turning attention to the -action, the equivalence of stacks
shows that sections of are determined by their restriction to the source of this equivalence. Hence, we require even or odd functions on that transform under the -action by the character
The result is precisely a de Rham model for the twisted equivariant K-theory with complex coefficients in the description of Theorem 2.2. Concordance classes of closed forms are de Rham cohomology classes, which completes the proof. ∎
3. Twisted equivariant elliptic cohomology over
Below we define twisted equivariant elliptic cohomology, following Devoto [Dev96] and Ganter [Gan09]. Then we introduce the main geometric objects: the constant super double loop stack of an orbifold and the line bundles over it whose sections give cocycles for twisted equivariant elliptic cohomology. The remainder of the section is the technical work: we choose groupoid presentations and compute these sections explicitly, proving Theorem 1.1.
3.1. Twisted equivariant elliptic cohomology with complex coefficients
Let denote the smooth manifold of based, oriented lattices, meaning ordered pairs of points in whose ratio is in the upper half plane, . Below we will view as a real manifold, and so an -point of is determined by a pair . Let denote holomorphic functions on such that for and acts by dilation and rotation on lattices. Then weak modular forms of weight are the elements of that are invariant under the -action on . A convenient description of elliptic cohomology of a manifold with complex coefficients is
where the -action is trivial on the de Rham cohomology groups, and acts on through its change-of-basis action on .
For a finite group, define to be the set of pairs of commuting elements of . There is an action of on by conjugation, . Let denote the stabilizer of under this action, and denote the quotient, writing for the point in the image of . The sets and carry a right action of determined by
Let be a normalized 3-cocycle. Define the abelian group as the set of holomorphic functions on that are invariant under the diagonal -action and equivariant for the -actions:
For a pair of normalized cocycles, multiplication of functions on gives the graded multiplication
This definition is a repackaged form of the ring defined by Ganter in [Gan09]. In the untwisted case () and when the cardinality of is odd, the following definition reduces to Devoto’s equivariant elliptic cohomology, taken with complex coefficients; compare Part 3 of Corollary 2.7 in [Dev98] and Theorem 5.3 of [Dev96].
For a manifold with the action of a finite group , let
where is the character of defined by
3.2. The stack
A based lattice defines a discrete subgroup of via the identifications . For an -point , we define a -family of super tori by the quotient . As reviewed in §A, the rigid conformal isometry group of is . This group determines the isometries between super tori: we can lift a family of isometries to the universal covers of a family of super tori.
The super double loop stack of an orbifold , denoted , has objects over pairs where defines a family of super tori and is an object in the groupoid of maps . Morphisms over consist of 2-commuting triangles,
where the horizontal arrow is an isometry of -families of rigid conformal -manifolds.
We define define the substack of constant super double loops in complete analogy with the 1-dimensional case. Let be the morphism of stacks
where we have used