SUPERSYMMETRIC FIELD THEORIES AND ORBIFOLD COHOMOLOGY
A Dissertation
Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by Augusto Stoffel
Stephan Stolz, Director
Graduate Program in Mathematics Notre Dame, Indiana April 2016 SUPERSYMMETRIC FIELD THEORIES AND ORBIFOLD COHOMOLOGY
Abstract by Augusto Stoffel
Using the Stolz–Teichner framework of supersymmetric Euclidean field theories (EFTs), we provide geometric interpretations of some aspects of the algebraic topology of orbifolds. We begin with a classification of 0|1-dimensional twists for EFTs over an orbifold X, and show that the collection of concordance classes of twisted EFTs over the inertia ΛX is in natural bijection with the delocalized twisted cohomology of X (which is isomorphic to its complexified K-theory). Then, turning to 1|1-dimensional considerations, we construct a (partial) twist functor over X taking as input a class in
H3(X; Z). Next, we define a dimensional reduction procedure relating the 0|1-dimensional Euclidean bordism category over ΛX and its 1|1-dimensional counterpart over X, and explore some applications. As a basic example, we show that dimensional reduction of untwisted EFTs over a global quotient orbifold X//G recovers the equivariant Chern character. Finally, we describe the dimensional reduction of the 1|1-twist built earlier, showing that it has the expected relation to twisted K-theory. CONTENTS
ACKNOWLEDGMENTS ...... iv
CHAPTER 1: INTRODUCTION ...... 1 1.1 Supersymmetric field theories and cohomology theories ...... 1 1.2 Field theories over orbifolds ...... 4 1.3 Outline of the dissertation ...... 7
CHAPTER 2: STACKS IN DIFFERENTIAL GEOMETRY ...... 9 2.1 Sheaves and stacks on a site ...... 9 2.1.1 Fibered categories and presheaves ...... 10 2.1.2 Grothendieck topologies and descent ...... 12 2.1.3 The site of a fibration; stacks over stacks ...... 15 2.2 Differentiable stacks ...... 16 2.2.1 Lie groupoids and torsors ...... 17 2.2.2 Bibundles and Morita equivalences ...... 22 2.2.3 Differential geometry of stacks; gerbes ...... 24 2.2.4 Orbifolds; inertia ...... 27 2.3 Sheaf cohomology ...... 29 2.3.1 Calculating cohomology ...... 29 2.3.2 Deligne cohomology ...... 31 2.4 Group actions on stacks ...... 32 2.4.1 Basic definitions ...... 32 2.4.2 Quotient stacks of G-stacks ...... 36
CHAPTER 3: SUPERGEOMETRY ...... 42 3.1 Superalgebra ...... 42 3.2 Supermanifolds ...... 44 3.2.1 Basic definitions ...... 44 3.2.2 The functor of points formalism ...... 47 3.2.3 Calculus on supermanifolds ...... 48 3.2.4 Super Lie groups ...... 49 3.2.5 Superpoints, differential forms, and superconnections . . . . . 51 3.3 Euclidean structures ...... 53 3.3.1 Euclidean structures in dimension 1|1 ...... 54 3.3.2 Euclidean supercircles ...... 59
ii 3.4 Integration on supermanifolds ...... 65 3.4.1 The Berezin integral ...... 65 3.4.2 Domains with boundary ...... 67 3.4.3 A primitive integration theory on R1|1 ...... 68
CHAPTER 4: ZERO-DIMENSIONAL FIELD THEORIES AND TWISTED DE RHAM COHOMOLOGY ...... 71 4.1 Superpoints and differential forms ...... 71 4.2 Superconnections and twists ...... 75 4.3 Concordance of flat sections ...... 76 4.4 Twisted de Rham cohomology for orbifolds ...... 79
CHAPTER 5: TWISTS FOR ONE-DIMENSIONAL FIELD THEORIES . . . 82 5.1 Euclidean supercircles over an orbifold ...... 83 5.2 Gerbes and partial twists ...... 86 5.2.1 A super version of transgression ...... 87 5.2.2 The restriction to K1(X) ...... 93
CHAPTER 6: DIMENSIONAL REDUCTION AND THE CHERN CHARAC- TER FOR ORBIFOLDS ...... 100 6.1 Dimensional reduction ...... 100 6.1.1 R//Z-equivariant bordisms ...... 101 6.1.2 T-equivariant bordisms ...... 103 6.1.3 The map BT(ΛX) → BR//Z(ΛX) ...... 105 6.1.4 Global quotients ...... 110 6.1.5 More general kinds of equivariant bordisms ...... 113 6.2 The Chern character for global quotients ...... 114 6.2.1 The Baum–Connes Chern character ...... 114 6.2.2 Parallel transport and field theories ...... 115 6.2.3 Proof of theorem 6.4 ...... 117 6.3 Dimensional reduction of twists ...... 121 6.3.1 The underlying line bundle ...... 122 6.3.2 The superconnection ...... 123
BIBLIOGRAPHY ...... 128
iii ACKNOWLEDGMENTS
First, I would like express my deep gratitude to my advisor Stephan Stolz for his guidance and patience, and for introducing me to so many exciting new ideas. Many thanks to the mathematics department at the University of Notre Dame, especially the topology and geometry groups, for providing a vibrant research envi- ronment. I profited much from mathematical discussions with my fellow graduate students, among whom I would like to single out Renato Bettiol, Ryan Grady, Santosh Kandel, Leandro Lichtenfelz, and Peter Ulrickson; I would also like to thank them for their friendship. I acknowledge my thesis committee, Mark Behrens, Liviu Nicolaescu, and Larry Taylor for their interest in my work, and I also thank Karsten Grove for the financial support during my sixth year. Finally, I would like to thank my wife and best friend Alana for her love, encour- agement, and support, as well as my family, Jacinta, Ireno, Amanda, and Iria for their constant concern and care, in spite of the distance.
iv CHAPTER 1
INTRODUCTION
The goal of this dissertation is to explore the relation between the algebraic topology of orbifolds and supersymmetric quantum field theories. Intuitively, orbifolds can be thought of as manifolds whose points are allowed to have a finite automorphism group. An example to keep in mind is the global quotient orbifold X//G arising from the action of a finite group G on a compact manifold X; morphisms between such orbifolds are induced not only by equivariant smooth maps, but also by group homomorphisms. Thus, cohomology theories for orbifolds are related to global equivariant homotopy theory, in the sense that we are dealing with invariants which depend functorially on X and G. We will focus on twisted K-theory and delocalized de Rham cohomology, as well as the Chern character relating them. On the field theory side, we employ the framework of supersymmetric Euclidean field theories due to Stolz and Teichner [38]. In our context, these field theories can be thought of as a way to encapsulate and generalize Chern–Weil theory. Before giving a more detailed overview of our result in section 1.2, we outline the main ideas in the Stolz–Teichner program.
1.1 Supersymmetric field theories and cohomology theories
Let d-Bord be the category where objects are closed (d − 1)-manifolds and a morphism between two objects Yin,Yout is a d-manifold Σ together with an identification of its boundary with the disjoint union Yin q Yout, taken up to diffeomorphism relative to the boundary. The operation of disjoint union makes it into a symmetric monoidal
1 category. A topological quantum field theory, as defined by Atiyah [1], Segal [35], and others, is a symmetric monoidal functor Z : d-Bord → C, where the target C is often taken to be the category vector spaces or some other “algebraic” category. This is a very flexible definitions that admits generalizations in many directions. Here, we interested in field theories of the supersymmetric Euclidean flavor. This means, firstly, that all bordisms are supermanifolds—a kind of mildly noncommutative manifold whose algebra of functions is Z/2-graded and commutative in the graded sense; and, secondly, all bordisms are equipped with a Euclidean structure (this induces, in particular, a flat Riemannian metric on the underlying manifold). Supersymmetric Euclidean field theories were first considered by Stolz and Teichner [39], who proposed their use as cocycles for TMF, the universal elliptic cohomology theory of topological modular forms. More specifically, they consider Euclidean field theories (EFTs) of dimension 2|1 (bordisms have local charts with 2 even and 1 odd coordinates) over a fixed manifold X. This means that besides supersymmetry and Euclidean structures, all bordisms are endowed with a smooth map to X. The collection of such objects (in fact, a groupoid) is denoted 2|1-EFT(X). Its formation is natural in X, and it is interesting to consider an equivalence relation weaker than isomorphism: two field theories are called concordant if there is a 2|1-EFT over X × R which, when restricted to X ×{0} respectively X ×{1}, recovers the two EFTs initially given.
Conjecture 1.1 (Stolz and Teichner [39]). There are natural isomorphisms
2|1-EFT∗(X)/concordance ∼= TMF∗(X).
A geometric construction of TMF (which, so far, is only understood in purely homotopy-theoretic terms) would be interesting for several reasons; for instance, it could help making precise heuristics from physics indicating that TMF is the correct
2 home for the T-equivariant index of families of Dirac operators on a loop space. The conjecture above is still wide open. To mention one fundamental difficulty, the excision/Mayer-Vietoris property of a cohomology theory can only possibly be reflected on the field theory side if we deal with extended (or local) field theories. However, the higher-categorical framework in which extended bordism categories are usually discussed is not amenable to supersymmetry, since the latter forces us not to consider individual bordisms, but rather always work with families. It is also interesting to consider supersymmetric EFTs of lower dimension and their relation with more classical cohomology theories. A conjecture analogous to the above states that for 1|1-EFTs the relevant cohomology theory is K-theory. In this case, a direct comparison between the field theory side and the homotopy theory side is possible (a vector bundle with connection gives rise to a supersymmetric field theory over X using the notion of super parallel transport developed by Dumitrescu [16]), but a classification of 1|1-EFTs over a manifold is a surprisingly elusive problem. Still, those ideas can be used to give nice geometric interpretations of classical constructions in algebraic topology, such as the Chern character [20, 15]. Going in a slightly different direction, Hohnhold, Stolz, and Teichner [23] show that it is possible to assemble the K-theory spectrum by gluing together spaces of families of EFTs over a point. Finally, let us discuss the 0|1-dimensional case. Since all 0|1-bordisms are closed, a topological field theory over a manifold X turns out to be just a function on the mapping space Map(R0|1,X) which is invariant under the action of the diffeomorphism group of R0|1 by precompositions. It is a well-known fact that Map(R0|1,X) is a (finite-dimensional) supermanifold, denoted ΠTX, and the algebra of smooth functions on ΠTX can be identified with the algebra of differential forms Ω∗(X). Not as well known, perhaps, is the fact that the action of Diff(R0|1) neatly encodes the de Rham differential and the grading operator on Ω∗(X). From that fact, one can deduce that 0|1-TFTs over X, up to concordance, are exactly the same as de Rham cohomology
3 classes for X (see Hohnhold et al. [22]). The above considerations can be extended to the equivariant case [37]. If X is a manifold acted upon by a Lie group G, then one can consider gauged 0|1-dimensional field theories over X. Here, instead of a map to X, each bordism is endowed with a principal G-bundle with connection together with a G-equivariant map from its total space to X. It turns out field theories of this kind provide a geometric interpretation of the Weil model for equivariant cohomology.
1.2 Field theories over orbifolds
Once the proper setup of EFTs over a manifold is developed, it is immediately clear what an EFT over a generalized manifold, or a stack, is. The gauged EFTs from the previous section, for example, are essentially EFTs over a stack of G-connections. This observation prompted the goal of this thesis: to study EFTs over an orbifold, and their relation to twisted K-theory and delocalized cohomology. Twisted K-theory of orbifolds has been a subject of great interest recently, both on the mathematics and on the physics fronts. Just like usual K-theory is related to ordinary cohomology via the Chern character, twisted and equivariant K-theory is related to a certain kind of de Rham–like cohomology, which we call delocalized cohomology. To give an idea of what those things are, let us consider the case where there is no twist and the orbifold in question is a global quotient X//G. Then K∗(X//G)
∗ is just the equivariant K-theory KG(X), the Grothendieck group of equivariant vector
∗ bundles over X, and delocalized cohomology Hdeloc(X//G) is a variant of the de Rham cohomology designed so that there exists a natural homomorphism
∗ ∗ chG : KG(X) → Hdeloc(X//G), the equivariant Chern character, inducing an isomorphism after tensoring with C. (We
4 could define a naive equivariant Chern character taking values in ordinary equivariant
∗ cohomology HG(X; Q), but, in light of the Atiyah–Segal completion theorem, it would not be a rational isomorphism. Hence, delocalized cohomology is a finer invariant.) There are various construction of delocalized cohomology in the literature, account- ing for successively more general classes of twists and target stack. We will mostly follow the work of Tu and Xu [40]. They consider twists for K-theory geometrically presented in the form a T-gerbe α with connective structure over the orbifold X, and
∗ give a suitable construction of twisted delocalized cohomology groups Hdeloc(X, α) and a Chern character map
∗ ∗ ch: Kα(X) → Hdeloc(X, α) yielding an isomorphism upon complexification. Our first main result (chapter 4) is a field-theoretic interpretation of the twisted de Rham cohomology of Tu and Xu. Fix an orbifold X and denote by ΛX its inertia stack. Then a T-gerbe α with connective structure gives rise to a twist functor (or anomaly) Tα for 0|1-EFTs over ΛX, and enables us to show the following.
Theorem 1.2. There is a natural bijection between concordance classes of Tα-twisted (or anomalous) 0|1-dimensional Euclidean field theories over the inertia of X and classes in the corresponding twisted de Rham cohomology groups,
n,Tα ∼ n 0|1-EFT (ΛX)/concordance = Hdeloc(X, α).
At this point, a natural question is how to describe the Chern character in field-theoretic terms. A well-established idea is to consider a dimensional reduction procedure. Roughly speaking, we can produce from any field theory a new field theory of dimension one less by evaluating the old one on bordism of the form — × S1.A subtle point here is that if Σ is an Euclidean supermanifold, Σ × S1 does not come
5 with a canonical Euclidean structure. Still, a dimensional reduction procedure can be carried out using the notion of T-equivariant bordisms. Those are the middleman in the span 0|1-EBord(ΛX) ← 0|1-EBordT(ΛX) → 1|1-EBord(X), where the map on the left induces an isomorphism on the set of field theories based on each bordism category and the map on the right is based on a descent construction involving the natural action of the 2-group pt//Z on ΛX. One can then consider various constructions on the 1|1-dimensional side and study their image on the 0|1-dimensional side. The simplest questions we can ask concern the untwisted case over a global quotient. Let V → X be G-equivariant vector bundle equipped with an invariant superconnection A (in the sense of Quillen [33]). Then one can construct, using super parallel transport, a 1|1-dimensional EFT Z(V ) over X//G. To a Euclidean supercircle K equipped with a map γ : K → X//G, it assigns the supertrace of the holonomy of the pulled back connection on γ∗V → K.
Theorem 1.3. For X = X//G a global quotient orbifold, dimension reduction of Euclidean field theories captures the equivariant Chern character. More specifically, the diagram
Z 1 1|1-EFT(X) red / 0|1-EFTT(ΛX) A ∼ Vect (X) =conc ch 0 G / ev / KG(X) Hdeloc(X) commutes.
We continue our exploration with a study of twists. Here, we again let X be a general orbifold. The most geometrically meaningful twists for K-theory arise from degree 3 integral cohomology classes; we should then expect the same data to yield a twist for 1|1-EFTs over X. From a gerbe with a given Dixmier–Douady class, we construct, in chapter 5, a line bundle on the stack of closed and connected
6 1|1-dimensional bordisms over X; we refer to that as a partial twist, since it is the kind of data one gets from a full 1|1-twist by forgetting all non-closed bordisms. The construction is a super version of the usual transgression of gerbes to line bundles on the loop space. We then show that the dimensional reduction of this partial twist, which is a 0|1-twist over ΛX, matches the twist Tα mentioned above in connection with twisted delocalized de Rham cohomology. During the time this dissertation was written, two preprints by Daniel Berwick- Evans considering similar questions appeared [9, 8]. Among other (not strictly related) results, they give field-theoretic interpretations to delocalized cohomology of a global quotient X//G with twist determined by a central T-extension of G, and the corresponding Chern character. His approach is based on perturbative sigma models, which, as far as we understand at this point, serve as a substitute for our descent-based dimensional reduction.
1.3 Outline of the dissertation
Chapter 2 is an overview of the theory of stacks and its applications in differential geometry. We introduce differentiable stacks, sheaf cohomology and sketch the basic ideas about group actions on them. Chapter 3 provides the necessary background on supergeometry; we spend some time on the notion of Euclidean structures. These two chapters are almost entirely expository. In chapter 4, we study 0|1-dimensional EFTs over an orbifold. We provide a classification of twists and show the relation between concordance classes of 0|1-EFTs and twisted delocalized cohomology. Chapter 5 concerns the construction of (partial) twists in dimension 1|1. We define a stack K(X) of Euclidean supercircles over an orbifold X, and provide a modification of transgression to the realm of supermanifolds, so that a class in H3(X; Z) determines a line bundle on K(X).
7 Finally, in chapter 6 we develop a notion of dimensional reduction for EFTs (in dimensions 0|1 and 1|1) over a differentiable stack. As an application, we give our geometric interpretation of the equivariant Chern character for global quotients. We then study the relation between the twists constructed in the two previous chapters.
8 CHAPTER 2
STACKS IN DIFFERENTIAL GEOMETRY
In this thesis, we will work extensively with stacks on the site of supermanifolds. They will play two roles for us. First, they organize geometric information (such as vector bundles or families of bordisms over varying base spaces) and keep track of the data and conditions needed to perform gluing constructions; also, they provide the right framework to express the idea that a given functorial construction is smooth. The second role, which will occupy most of this short introduction, is as generalized manifolds. We single out a class “finite dimensional” stacks, called geometric or differentiable, and show how we can do differential geometry with them. In particular, we review the theory of sheaf cohomology and group actions on stacks. As a corollary of the above, much of what we do is situated in a 2-categorical context. See Lack [26] for an overview of the theory and further pointers to the literature. Contrarily to that reference, however, we will follow the “modern” convention of using unqualified terms to refer to weak constructions, adding the adjective strict if relevant; thus, 2-category means weak 2-category or bicategory, 2-functor means pseudofunctor, 2-limit means homotopy limit, or more specifically bilimit, and so forth. We also often omit the prefix 2-.
2.1 Sheaves and stacks on a site
In this section, we sketch the rudiments of the theory of sheaves and stacks on a site. Basic references for this material are Vistoli [42] and the appendix in Hohnhold et al. [22].
9 2.1.1 Fibered categories and presheaves
Let C, S be categories and fix a functor p: C → S; we will usually omit explicit references to p and call C a category over S. A morphism φ: y → x in C mapping to f : T → S under p will be depicted by the diagram
φ y / x _ _
f T / S.
In this situation, we often say that x lies over S, and that the morphism φ lies over, or covers, f. The morphism φ is called cartesian if, given any map g : U → T in S, and any object z of C lying over U together with a morphism ψ : z → x covering f ◦ g, there is a unique ξ : z → y making the diagram
ψ z _ ξ ( y #/ x _ φ _ U # g ' T / S f commute. A category C over S is called a fibered category over S, or Grothendieck fibration, or simply fibration, if given an object x 7→ S in C and a morphism f : T → S in S, there exists a cartesian morphism φ: y → x covering f; y is called a pullback of x along f. A morphism, or fibered functor, between fibered categories p: C → S and q : D → S is a base-preserving functor F : C → D sending cartesian morphisms to cartesian morphisms (by base-preserving we mean that we have a strict equality of functors q ◦ F = p). A 2-morphism, or fibered natural transformation, between F,G: C → D, is a natural transformation α: F → G such that for any x 7→ S in C, the morphism αx : F (x) → G(x) covers the identity map of S. Fibered categories over
10 a fixed S thus form a strict 2-category; the category of morphisms between C and D is denoted FunS(C, D). We call C a category fibered in groupoids over S, or simply a groupoid fibration, if every morphism in C is cartesian. To justify the terminology, consider, for each
S ∈ S, the subcategory CS ⊂ C comprising all objects over S and all morphisms covering the identity of S. It is then a simple exercise to see that CS is a groupoid. A notion closely related to fibered categories is that of a (weak) presheaf of categories, i.e., a (weak 2-)functor F : Sop → Cat taking values in the category of small categories. The adjective weak means that we are thinking of Sop as a weak 2-category (with only identity 2-morphisms). Thus, we do not have a strict equality between F (g ◦ f) and F (g) ◦ F (f), but rather a prescribed 2-isomorphism between them (the compositor), satisfying appropriate coherence conditions. From such an F , we can build a fibered category F → S, where objects are pairs (S, x) with x ∈ F (S), and a morphism (T, y) → (S, x) is given by a morphism f : T → S together with a morphism y → F (f)(x) in F (T ). This is often called Grothendieck construction. Note that the fibration F comes with a choice of pullback for each x and f as above (namely, F (f)x). This extra information is called a cleavage. The space of all cleavages is contractible, and this allows an inverse construction, associating a presheaf Sop → Cat to a fibered category C, to be made uniquely up to 2-isomorphism.
A simple example of presheaf of (discrete) categories is the functor FX represented by an object X ∈ S, i.e., the assignment S 7→ S(S,X). The corresponding fibration is the category S/X of objects over X (often called slice category or overcategory of X) together with the forgetful functor S/X → S.A 2-categorical version Yoneda lemma asserts that there is a natural equivalence of categories
∼ FunS(S/X, C) = CX .
11 Note that the construction of a functor from the right to the left-hand side requires a choice of cleavage, so it is only well defined up to a unique natural isomorphism. Nonetheless, we will systematically blur the distinction between the two sides of this equivalence; we will also blur the distinction between S ∈ S and the fibration S/S, and call CS the groupoid of S-points of C.
Finally, we note that given fibrations C, D, we can form a new fibration FunS(C, D), the mapping fibration or internal hom, by requiring that
FunS(C, D)S = FunS(C × S, D).
For all S ∈ S. The exponential law holds:
∼ FunS(B × C, D) = FunS(B, FunS(C, D)).
(We also remark here that if C and D are stacks, so is the internal hom.)
2.1.2 Grothendieck topologies and descent
A Grothendieck topology on a category S consists of a collection of sets of morphisms {Ui → S}i∈I in S, called coverings of the object S, satisfying the following conditions:
1. if T → S is an isomorphism, then {T → S} is a covering,
2. if {Ui → S}i∈I is a covering and T → S is any morphism, then the fibered
products T ×S Ui exist and the set {T ×S Ui → T }i∈I is a covering, and
3. if {Ui → S}i∈I is a covering and for each Ui we have a covering {Uij → Ui}j∈Ii ,
then the collection of all compositions {U → S} ` is again a covering. ij (i,j)∈ i{i}×Ii
A category endowed with a Grothendieck topology is called a site.
12 If X is a manifold or a topological space, then the collection of all open subsets of X and inclusions between them forms a category X. By assigning to each open U ⊂ X the collection of all open coverings of U, we obtain a Grothendieck topology on X; the corresponding site is called the small site of X. To see the relation between this and the usual notion of a topology, note that if Ui is an open cover of U, then
0 0 0 condition (2) above means that for any U ⊂ U, the collection of all U ×U Ui = U ∩Ui is an open cover of U 0. More interestingly, we can define a topology on the category Man of smooth manifolds by declaring a covering of a manifold X to be a collection {Ui → X} of jointly surjective open embeddings. As a slight variant, we could declare a covering to be a collection of jointly surjective local diffeomorphisms (also called étale maps). These two topologies are equivalent in the sense that they give rise to the same descent theory (we will refrain from discussing that aspect in detail, but notice that a covering in the second sense admits a refinement in the first sense, and vice versa; it is also worth mentioning that what we call a topology here was termed a pretopology by Grothendieck, and two equivalent pretopologies give rise to the same topology in Grothendieck’s sense). Similar definitions can be made for the categories SM of supermanifolds and Top of topological spaces. Here, we notice that although Man and SM do not have arbitrary fibered products, checking the axioms for a topology only require that we take fibered products along submersions, which do exist.
Given a fibered category C → S and a covering {Ui → S}i∈I , we define the descent
category C{Ui→S} of the covering as follows. Objects are collections consisting of an object xi 7→ Ui for each i ∈ I, and a “gluing isomorphism” φij : xi|Uij → xj|Uij for each pair i, j ∈ I. These data are subject to the condition that upon restricting to Uijk
we have an equality φik = φjk ◦ φij. Here, Ui1···in = Ui1 ×S · · · ×S Uin is an “n-fold
0 0 intersection”. A morphism (xi, φij) → (xi, φij) is given by a collection of morphisms
13 0 fi : xi → xi compatible with the gluing data: over Uij, the diagram
f i / 0 xi xi
0 φij φij fj / 0 xj xj
commutes. Restriction from S to each Ui determines a functor CS → C{Ui→S}. We call C a stack if for any covering this functor is an equivalence of categories. A stack whose fibers are discrete categories is called a sheaf. The inclusion of stacks into all fibered categories over S has a left adjoint, called stackification. A typical application of this fact is the following. Suppose we have a stack C and a class of objects U = {U} in S with the property that any covering admits a refinement involving only that kind of objects (e.g., U ⊂ Man the subcategory of contractible manifolds), and suppose we have a fibered functor C0 → C inducing an ∼ equivalence (C0)U = CU for all U ∈ U (think of C0 as a category of “trivial” families).
Then C is the stackification of C0, and in order to define a map from C into another stack, it suffices to describe its behavior on C0. It is also useful to understand the stack condition from the perspective of presheaves of categories. Fix a presheaf F : Sop → Cat, an object S ∈ S, and a covering U → S. Then we can form the simplicial object
← ← ← U ⇔ U ×S U ← U ×S U ×S U ← ···
It admits a canonical map into the constant simplicial object S. Hitting that diagram with F and taking the 2-limit, we get a functor
→ → → F (S) → lim F (U) ⇒ F (U ×S U) → F (U ×S U ×S U) → ··· .
14 It turns out (see Hollander [25]) that the fibered category corresponding to F is a stack if and only if the above is an equivalence of categories. Finally, we quote a handy technical result [22, proposition 7.13].
Proposition 2.1. Let F : Sop → Cat be the presheaf associated to a fibration F, and
op F• : S → sSet the presheaf of simplicial sets obtained by taking nerves. Denote by
Fn, n ≥ 0, the associated (discrete) fibrations. Then for any fibered category C there is an equivalence ∼ FunS(F, C) = lim (FunS(F•, C)) .
2.1.3 The site of a fibration; stacks over stacks
On the category underlying a fibration C → S there is a natural topology: a family of morphisms {xi → x} in C is a covering family if and only if their images in
S form a covering family {Ui → U}. We can therefore talk about sheaves and stacks on C. The set of global sections of a sheaf F : C → Set is the set of sheaf homomorphisms from the trivial sheaf C, whose sections over any object are the one-point set, to F . This defines a functor Γ(C, —) from sheaves over C to Set. For example, the site associated to the slice category S/X is called the big site of X. The big and small sites of a space (or manifold) are equivalent to one another, but occasionally it is important to keep in mind the distinction between them. Finally, we quote a result from Metzler [31, section 4.2].
Theorem 2.2. Fix a site S and a strictly commutative diagram of categories
f C / D
p q Ó S.
If f and q are stacks (the former with respect to the topology on D induced by q), then
15 p is also a stack. If p is a stack and q a prestack, then there exists a stack C0 → D equivalent to C as categories lying over D strictly; if q is a sheaf, we can take C0 = C.
From the above we see that the notion of a stack C over a stack D can be alternatively defined to be simply a map of stacks C → D over the base site S.
2.2 Differentiable stacks
In this thesis, we are mostly interested in stacks on the site of supermanifolds, and in particular in those satisfying a certain finite-dimensionality condition, called geometric or differentiable stacks. Our basic references for differentiable stacks are Behrend and Xu [5] and Metzler [31]. In this section, we collect the most important facts and definitions about differentiable stacks in ordinary or supermanifolds. Throughout this section, S denotes either the site of ordinary smooth manifolds or supermanifolds, and all stacks will be over S. Objects of S will be refered to as “manifolds” even in the super case. Moreover, from now on all stacks will by default be fibered in groupoids. If we want to allow allow general category fibrations, we will mention that explicitly. A morphism of stacks X → Y is called a representable submersion if for any manifold S and map S → Y, the fiber product X×Y S is representable and the natural map from that fiber product into S is a submersion. (Note that a submersion between representable stacks is a representable submersion, by the fact that submersions are preserved by base changes.) A morphism X → Y is an epimorphism if for any manifold S and S → Y, there is a cover T → S and a commutative square (up to a natural transformation)
T / S
zÒ X / Y.
An atlas or presentation of a stack X is given by a manifold X together with a
16 surjective (i.e., epimorphic) representable submersion x: X → X. A stack that admits a presentation is called a differentiable stack, and the corresponding object x ∈ XS is called a versal family.
2.2.1 Lie groupoids and torsors
A (super) Lie groupoid X = (X1 ⇒ X0) is a groupoid internal to the category of
(super)manifolds: it is given by manifolds X0,X1, called space of objects and space of morphisms respectively, together with
1. submersions s, t: X1 → X0 called source and target maps,
2. a unit map u: X0 → X1
3. a composition map c: X = X ×s,t X → X . 2 1 X0 1 1
These maps are required to satisfy the usual properties of a category, encoded by the commutativity of a number of diagrams, as well as the condition that (c, pr2): X2 → X ×s,s X is a diffeomorphism, expressing invertibility of all arrows. (Observe that, 1 X0 1 as indicated by our definition of X2, we use the traditional notation gf, as opposed to the diagrammatic notation fg, for the composition of f : x → y and g : y → z. Contrarily perhaps to most category theorists, we prefer to blame the inadequacies of the notation on our habit of writing rightward-pointing arrows.) A fundamental example of Lie groupoid is the following. If X is a manifold acted upon by a Lie group (or more generally a monoid object) G on the right, then we define the transport groupoid X × G ⇒ X as follows:
s:(x, g) 7→ x · g, t:(x, g) 7→ x, u: x 7→ (x, e),
c:(x, g) × (x · g, h) 7→ (x, gh).
17 Our conventions for s and t are chosen so that the traditional notation for composition of functions matches with multiplication in G:
gh g g◦h x · gh −→ x = x · gh →h x · g → x = x · gh −−→ x .
For left actions, we use the opposite conventions for s an t, so we would write
hg g h◦g x −→ hg · x = x → g · x →h hg · x = x −−→ hg · x .
Another recurring example arises from a countable covering {Ui → X}i∈I of a manifold. In that case, the groupoid has as space of objects qi∈I Ui and as space of morphisms the disjoint union of all double intersections Uij = Ui ×X Uj. Source and target maps are determined by the inclusions s: Uij → Ui, t: Uij → Uj, the unit map is determined by the identity maps Ui → Uii, and composition is determined by the
inclusions of triple intersection into double intersections, Ujk ×Uj Uij → Uik.
A third important example concerns differentiable stacks. If X0 → X is an atlas, we can form the pullback square
s X1 = X0 ×X X0 / X0
t (2.1) X0 / X, and define the two remaining structure maps using the cartesian property. The Lie groupoid X1 ⇒ X0 is called a presentation of X. There is a converse to the above construction, producing a differentiable stack from a Lie groupoid—the stack of torsors. We start by defining a (right) action of the
Lie groupoid X = (X1 ⇒ X0) on a manifold P . This is given by a map a: P → X0,
18 called anchor or moment map, together with an action map
µ: P ×a,t X → P, (p, h) 7→ p · h, X0 1 which is required to satisfy the following conditions:
a(p · h) = s(h), (p · h) · h0 = p · (h ◦ h0), p · u(a(p)) = p,
0 where p ∈ PS, h, h ∈ (X1)S are S-points for which the formulas make sense. To define left actions, we swap the roles of s and t. A smooth map f : P → P 0 between two manifolds with an X-action is equivariant if the corresponding anchor maps a, a0 satisfy a0 ◦ f = a, and f(p) · h = f(p · h) for all composable p ∈ PS, h ∈ (X1)S. An X-torsor over a manifold S is given by a manifold P endowed with an X- action and an X-invariant surjective submersion π : P → S (the structure map), such that the action is free and transitive on the fibers. Here, X-invariance means that
π(p · h) = π(p) for every composable p ∈ PS and h ∈ (X1)S, and the condition of being free and transitive on the fibers can be expressed by saying that the map
P ×X0 X1 → P ×S P, (p, h) 7→ (p, p · h) is a diffeomorphism. A map between X-torsors π : P → S and π0 : P 0 → S0 over f : S → S0 is simply an equivariant map P 0 → P covering f. To understand this definition, it is useful to think of a point p ∈ P as an arrow from a(p) ∈ X0 to π(p) ∈ S; acting on it with an arrow h: x → a(p) in X1 produces
19 an arrow p · h with the same target and new source x.
p h y v π(p) a(p) = t(h) s(h) = a(p · h) i
p·h
We can now state the relationship between differentiable stacks and Lie groupoids. For a proof, we refer the reader to Behrend and Xu [5, section 2.4].
Theorem 2.3. For every Lie groupoid X, the category BX of X-torsors is a dif- ferentiable stack. If X is a differentiable stack, X → X an atlas, and X1 ⇒ X0 the associated Lie groupoid (2.1), we have an equivalence of stacks
∼ X = B(X1 ⇒ X0).
Let us consider two examples of torsors. First, if X = (Y × G ⇒ Y ) is the transport groupoid of a group action, then an X-torsor over S is a principal G-bundle P → S together with a G-equivariant map P → Y . A map between such torsors is a bundle map intertwining the given maps into Y .
For the second example, let X = (X1 ⇒ X0) be arbitrary. Then given any map f : S → X , we can form the trivial torsor π : P → S with P = S ×f,s X , anchor 0 X0 1 0 0 given by the natural map P → X0, and action given by (s, h) · h = (s, h ◦ h ).A morphism of trivial torsors f → f 0 (covering the identity of S) is specified by a
0 map g : S → X1 with s(g) = f, t(g) = f . Thus the category of trivial S-torsors is isomorphic to the set-theoretic groupoid X1(S) ⇒ X0(S). Notice also that for an arbitrary torsor π : P → S, a section of π determines a trivialization. Since π admits local sections, every torsor is locally trivial. In particular, for contractible S,
B(X1 ⇒ X0)S is equivalent to X1(S) ⇒ X0(S).
20 To finish our discussion on presentations of differentiable stacks, we would like to give a slightly different perspective on torsors. Suppose we have an X-torsor
π : U → S with anchor map ψ0 : U → X0 and action map µ: U ×X0 X1 → U. We
first note that we have a Lie groupoid U ×S U ⇒ U (associated to the nerve of the covering U), which should be thought as a “refinement” of S, or rather the trivial
Lie groupoid S ⇒ S (this will be made precise in the next subsection). Second, the ∼ diffeomorphism U ×S U = U ×X0 X1, expressing the free and transitive property of the action, postcomposed with the projection onto X1 yields a map ψ1 which makes the diagram
ψ1 U ×S U / X1
pr2 pr1 s t
ψ0 U / X0 into an internal functor of Lie groupoids. The internal functors obtained in this fashion are characterized by the fact that if certain point of X0 is in its image, then the whole equivalence class of that point is in the image. More concretely, the condition is that
(pr , ψ ): U × U → U ×ψ0,t X . 1 1 S X0 1 is a diffeomorphism; the action map for the torsor is then the inverse to the above followed by projection onto the second factor.
0 0 Next, assume we have a torsor map λ relating torsors (π, ψ0, µ), (π, ψ0, µ ) based on the same structure map π : U → S. Then we can build an internal natural trans- formation Λ: U → X1 relating the corresponding internal functors as the composition
(id,λ) ∼ 0 pr = ψ0,t 2 U −−−→ U ×S U −−−−−→ U ×X X1 −−→ X1. µ0-action 0
21 Conversely, given Λ, we can recover the corresponding λ as
∼ (id,Λ) ψ0,t = pr2 U −−−→ U ×X X1 −−−−→ U ×S U −−→ U. 0 µ-action
2.2.2 Bibundles and Morita equivalences
Now that we are able to manipulate one fixed differentiable stack in terms of a presenting Lie groupoid, we turn to the corresponding description of 1- and 2- morphisms between stacks. Observe that we can speak of internal functors (also called strict morphisms) between Lie groupoids and internal natural transformations between them, and thus obtain a strict 2-category, but this is not the most appropriate thing to look at. In fact, an internal functor f : X → Y which is fully faithful and essentially surjective (meaning, respectively, that the diagram
f1 X1 / Y1
(s,t) (s,t)
f0×f0 X0 × X0 / Y0 × Y0 is cartesian and t ◦ pr : X ×f0,s Y → Y is a surjective submersion) may not have an 2 0 Y0 1 0 inverse up to natural transformation. So we would like to localize the 2-category of Lie groupoids and internal functors at that class of morphisms. The formalism we are about to introduce implements this idea.
Given Lie groupoids X = (X1 ⇒ X0) and Y = (Y1 ⇒ Y0), a bibundle from X to
Y is an Y -torsor π : P → X0 endowed with a commuting action of X having π as anchor map (for this condition to make sense, the anchor map for the Y -action must be X-invariant). We adopt the convention that X acts on the left and Y on the right. We denote this structure by P : X → Y . An isomorphism of bibundles P,Q: X → Y is a morphism of Y -torsors P → Q which is also X-equivariant. For instance, to an internal functor f : X → Y we can associate the bibundle
22 P = X ×f0,t Y (here, f : X → Y is the object component of f). The structure f 0 Y0 1 0 0 0 maps are pr1 : Pf → X0 and s ◦ pr2 : Pf → Y0, and the multiplications maps defined in the natural way. An internal natural transformation gives rise to an isomorphism of bibundles. For a general bibundle P , a section of π : P → X0 determines an isomorphism with a bibundle of the form above, which should be thought of as a “trivial” bibundle. Given bibundles P : X → Y , Q: Y → Z, their composition is the bibundle
Q ◦ P : X → Z whose underlying manifold is (Q ×Y0 P )/Y , endowed with the obvious actions by X and Z. Composition of bibundles is associative up to a canonical isomorphisms, with the bibundle corresponding to the identity functor serving as an identity. A bibundle is (weakly) invertible if and only if it is also an X-torsor over Y0; it is then called a bitorsor or a Morita equivalence; Lie groupoids related by a bitorsor are said to be Morita equivalent. Lie groupoids, bibundles and isomorphisms of bibundles form a (weak) 2-category. With this setup, the problem pointed out earlier, that essentially surjective and fully faithful internal functors need not have a (weak) inverse goes away: they give rise to bitorsors. In fact, two Lie groupoids are Morita equivalent if and only if they are related by a zigzag of essentially surjective, fully faithful functors [5, theorem 2.26], and the 2-category of Lie groupoids and bitorsors can be thought of as a localization of the “naive” 2-category of Lie groupoids and internal functors. Note that the category of X-torsors over S is isomorphic to the category of bibundles (S ⇒ S) → (X1 → X0), functorially (up to 2-isomorphism) in S. Thus, composition of bibundles also allows us to compare the categories of torsors for distinct Lie groupoids. This in fact yields an equivalence of 2-categories
{Lie groupoids and bitorsors} → {differentiable stacks}.
23 By theorem 2.3, this functor is essentially surjective on objects. Moreover, notice that given a map of differentiable stacks f : X → Y and atlases X0 → X, Y0 → Y, a bibundle between the corresponding Lie groupoids can be built from the following 2-pullback
P = X0 ×Y Y0 / Y0
f X0 / X / Y.
The dictionary lemmas in Behrend and Xu [5, section 2.6] express in detail the fact that this gives rise to an equivalence on hom-categories. This is also explained in Blohmann [10, theorem 2.18]
2.2.3 Differential geometry of stacks; gerbes
If, as we suggested, a stack is to be thought of as a generalized manifold, then we should be able to make sense of the usual objects from differential geometry in
∗ this new context. We define the de Rham complex ΩX of a stack X to be the sheaf
∗ ∗ ∗ 0 ∗ assigning Ω (S) to any object x ∈ XS, and the homomorphism f : Ω (S ) → Ω (S) to a morphism x → y in X covering f : S → S0. Its global sections are
∗ ∗ Ω (X) = FunS(X, Ω ), where, on the right-hand side, Ω∗ : Sop → Set denotes the sheaf of differential forms. The same move can be done for any sheaf on S, and is motivated by the 2-Yoneda lemma, which states that, for representable X, this definition is consistent with the original one. Similarly, the groupoid of vector bundles over X is
Vect(X) = FunS(X, Vect).
To add connections, we replace Vect with Vect∇, and so forth.
24 If X is presented by a Lie groupoid X1 ⇒ X0, then proposition 2.1 gives a bijection
∗ ∼ ∗ X1 ∗ ∗ ∗ Ω (X) = Ω (X0) = {ω ∈ Ω (X0) | s ω = t ω}, as well as an equivalence between Vect(X) and the groupoid whose objects are pairs
∗ ∗ (V, φ) with V ∈ Vect(X0) and φ: s V → t V an isomorphism satisfying the condition
expressed, in terms of (S-)points, by φidx = idVx and
φf φg φg◦f f g Vx −→ Vy −→ Vz = Vx −−→ Vz for all x → y → z ∈ X2, and whose morphisms (V, φ) → (V 0, φ0) are bundle maps η : V → V 0 satisfying the natural compatibility condition with φ, φ0. A connection on (V, φ) is a connection on V such that s∗∇ = t∗∇. To illustrate this, if X = X//G is a quotient stack, then
∗ ∼ ∗ G ∼ ∇ ∼ ∇ Ω (X//G) = Ω (X) , Vect(X//G) = VectG(X), Vect (X//G) = VectG(X), the second (respectively, third) item in the list being the groupoid of G-equivariant vector bundles (with invariant connection). Also, if X is a manifold and we calculate
Vect(X) in terms of the groupoid coming from a nice open cover {Ui}, then the resulting groupoid is equivalent to the subgroupoid where all vector bundles are trivial, and this is precisely the category of gluing cocycles. Now, we want to discuss a genuinely higher categorical kind geometric structure: gerbes. To motive this, recall that complex line bundles are classified by their first Chern class, furnishing a geometric interpretation for degree 2 integral cohomology classes. We might then ask what is a geometric interpretation for cohomology classes of higher degree, say degree 3. A stacky answer to this question is to consider bundles whose fibers are the groupoid of complex lines. This can be made sense of in a fashion intrinsic to stacks (see for instance Behrend and Xu [5, section 4]), but we will be
25 contented with a definition in terms of presentations.
× A central C -extension of a Lie groupoid X = (X1 ⇒ X0) is a Lie groupoid ˜ ˜ X = (X1 ⇒ X0) together with an internal functor
˜ π X1 / X1
X0 X0
× ˜ ˜ and a left C -action on X1 making X1 → X1 a principal bundle; we require that × ˜ (wf) ◦ (zg) = wz(f ◦ g) for any z, w ∈ C and composable f, g ∈ X1. It is not hard to define the notion of C×-equivariant bitorsor X˜ → X˜ 0 between central extensions of X, and a C×-gerbe over the differentiable stack X presented by X is defined to be a central C×-extension, up to Morita equivalence. A connection on the central C×-extension X˜ → X is a connection 1-form θ ∈ 1 ˜ × Ω (X1) on the principal C -bundle satisfying
∗ ∗ ∗ 1 ˜ pr1 θ + pr2 θ − c θ = 0 in Ω (X2).
2 ∗ ∗ A curving for the connection θ is a form B ∈ Ω (X0) such that the identity t B−s B = 2 ˜ dθ holds in Ω (X1). Finally, the 3-curvature of a connection with curving is the form
3 X1 3 Ω = dB ∈ Ω (X0) = Ω (X). We will introduce sheaf cohomology for stacks in section 2.3 below. We quote here some classical results on the classification of gerbes [5].
Theorem 2.4 (Giraud). Isomorphism classes of C×-gerbes over X are in one-to-one correspondence with H2(X; C×).
The exponential sequence of sheaves 0 → Z → C → C× → 0 induces a ho- momorphism H2(X; C×) → H3(X; Z). (Here, C and C× denote sheaves of smooth functions, not the locally constant ones.) Given a gerbe on X, the image in H3(X; Z)
26 of the corresponding cohomology class is called the Dixmier–Douady class. If X is an orbifold, we have H≥1(X; C) = 0, so gerbes are classified by H3(X; Z). Moreover, given a connection with curving and 3-curvature Ω, the image of the Diximier–Douady
3 class in de Rham cohomology is [Ω] ∈ HdR(X).
2.2.4 Orbifolds; inertia
Intuitively, an orbifold is a space locally modeled on the quotient of a manifold by a finite group. The original definition was in terms of local charts consisting of a neighborhood in Euclidean space together with a linear action by a finite group. From the classical perspective, it was not so clear what a map of orbifolds should be, and none of the competing definitions was entirely satisfactory. Of course, this makes talking about further geometric structures on orbifolds a pretty hard task. One of the main issues was that, whatever reasonable definition of morphism one picks, gluing maps can only be done properly if orbifolds are seen as objects in a 2-category. A good reference for the modern perspective on orbifolds, which we will summarize below, is Henriques and Metzler [21]. Other useful references include Moerdijk [32] and Lerman [29]. An orbifold X is a differentiable stack equivalent to (the stack presented by) a proper étale Lie groupoid X1 ⇒ X0. Here, étale means that the source and target maps
X1 → X0 are local diffeomorphisms, and proper means that (s, t): X1 → X0 × X0 is proper. Maps of orbifolds are simply maps of stacks. The simplest examples of orbifold are global quotients, i.e., stacks X//G arising from the action of a finite group. The following proposition gives more examples.
Proposition 2.5. Let G be a Lie group acting properly and locally freely on a manifold X. Then X//G is an orbifold.
Orbifolds of the type X//G with G a Lie group are called quotient orbifolds. It
27 is a folk theorem that every effective orbifold is of this type, being presented as the quotient of its frame bundle by the orthogonal group. Let us sketch the proof of the proposition. For each point x ∈ X, the slice theorem gives us a G-invariant neighborhood of the form Ux = (G × Σx)/Gx, where the slice
Σx ⊂ X is a little disk transversal to the orbit x · G on which the stabilizer group Gx
(finite by assumption) acts effectively by diffeomorphisms. Then Ux//G is equivalent to the orbifold Σx//Gx, and X//G can be recovered by gluing together those global quotients. More specifically, we note that Y0 = qx∈X0 Σx → X//G is an atlas (provided we take the disjoint union over a large enough countable collection X0 ⊂ X), and we get a proper étale Lie groupoid presentation Y1 ⇒ Y0 by considering, as in (2.1)
s Y1 = Y0 ×X//G Y0 / Y0
t Y0 / X//G.
Now, let C be a fibered category over a site S. We define the inertia ΛC of C to be the fibration whose S-points given by pairs (x, α) with x ∈ CS and α an automorphism of x. A morphism (x, α) → (x0, α0) is given by a morphism ψ : x → x0 in C such that
0 α ◦ ψ = ψ ◦ α. The inertia can also be defined as the homotopy equalizer of idC and idC, or as the mapping fibration FunS(pt//Z, C) (see Bunke, Schick, and Spitzweck [13] for a detailed discussion). The inertia of a stack is a stack. The the groupoid pt//Z can be though of as an “infinitesimal circle”, and ΛC as an “infinitesimal loop space” of C. Note that there is a canonical automorphism of idΛC, namely the natural transformation assigning to (x, α) the automorphism α. As we will later see, this can also be seen as a strict pt//Z-action on ΛC. ∼ We are particularly interested in the inertia of an orbifold X = (X1 ⇒ X0), an this turns out to be again an orbifold. In fact, ΛX is presented by the inertia groupoid Xˆ Xˆ , where Xˆ ⊂ X is the equalizer of s and t, Xˆ = Xˆ ×s,t X , 1 ⇒ 0 0 1 1 0 X0 1
28 ˆ ˆ ˆ and the structure maps s,ˆ tˆ: X1 → X0, etc., are given by the action of X1 on X0 by conjugation:
tˆ(x, α, ψ) = (x, α), sˆ(x, α, ψ) = (s(ψ), ψ−1 ◦ α ◦ ψ),
ˆ and so forth. Here, we write (x, α) ∈ X0 for the point corresponding to the automor- phism α: x → x.
2.3 Sheaf cohomology
For X a differentiable stack, the global sections functor Γ(X, —) defined on the category Ab(X) of sheaves of abelian groups is left exact and has enough injectives, so there are right derived functors Hi(X, —): Ab(X) → Ab. More generally, there is a total derived functor RΓ(X, —): D+(X) → D+(Ab) defined on the corresponding derived categories of complexes bounded below. If C∗ ∈ D+(X) is a complex of sheaves, the homology of the complex RΓ(X,C∗) is called the hypercohomology of X with values in C∗ and denoted Hi(X,C∗).
2.3.1 Calculating cohomology
From a differentiable stack and an atlas X → X, we can form a Lie groupoid and therefore a simplicial manifold
← ← ← X = X0 ⇔ X1 ← X2 ← ···
All compositions Xp → X are canonically isomorphic; we fix a choice. This defines, for any sheaf of abelian groups F on X and each p, a small sheaf Fp on Xp such that
Fp(Xp) = F (Xp → X). Applying F to the above diagram gives us a cosimplicial
29 abelian group → → → F0(X0) ⇒ F1(X1) → F2(X2) → ···
The cohomology of the associated cochain complex is called the Čech cohomology of
ˇ ∗ F associated to the covering X → X, and denoted H (X•; F ).
q p+q Proposition 2.6. There is an E1 spectral sequence H (Xp; Fp) ⇒ H (X; F ).
i Corollary 2.7. If for every p the sheaf Fp is acyclic, i.e., satisfies H (Xp; Fp) = 0 for i > 0, then the above spectral sequence collapses and we have
∗ ˇ i H (X; F ) = H (X•; F ).
Now, let (C∗, d) be a complex of sheaves on X bounded from below. Then we get a double complex
. . . .O .O .O
∂ ∂ ∂
0 d 1 d 2 d Γ(X2,C ) / Γ(X2,C ) / Γ(X2,C ) / ··· O O O ∂ ∂ ∂ (2.2)
0 d 1 d 2 d Γ(X1,C ) / Γ(X1,C ) / Γ(X1,C ) / ··· O O O ∂ ∂ ∂
0 d 1 d 2 d Γ(X0,C ) / Γ(X0,C ) / Γ(X0,C ) / ···
The cohomology groups of the total complex of this double complex are called Čech
∗ ˇ ∗ • hypercohomology groups of C and denoted H (X•; C ).
i Proposition 2.8. If for every i and p the sheaf Cp on Xp is acyclic, then
∗ • ˇ ∗ • H (X; C ) = H (X•; C ).
∗ The de Rham cohomology HdR(X) of a stack X is defined to be the hypercohomology
30 ∗ ∗ of its de Rham complex ΩX. On the one hand, the Poincaré lemma shows that ΩX is ∗ ∼ ∗ a resolution of the constant sheaf RX, so HdR(X) = H (X, RX). On the other hand, ∗ ∼ ˇ ∗ • the above proposition implies that HdR(X) = H (X•;Ω ).
Corollary 2.9 ([40, corollary 3.2]). Let X1 ⇒ X0 be a proper étale Lie groupoid presenting the orbifold X. Then de Rham cohomology can be calculated by using global sections of Ω∗:
∗ ∗ ∗ ∗ ∗ X1 HdR(X) = H (Ω (X), d) = H (Ω (X0) , d).
2.3.2 Deligne cohomology
Given a stack X and n ≥ 1, we define the Deligne complex C×(n) to be the following complex of sheaves on X:
× d log 1 n−1 C −−→ Ω → · · · → Ω .
× Here, C , the sheaf of smooth C×-valued functions, is placed in degree 0, and differential forms are complex valued. The hypercohomology H∗(X, C×(n)) is called the nth Deligne cohomology of X. What we called a gerbe here is also known, more specifically, as a 1-gerbe (with band C×). Following this nomenclature, a 0-gerbe (with connective structure) is a principal C×-bundle (with connection); one possible definition of higher gerbes with connective structure, in terms of a “gluing cocyle”, is indicated by the theorem below, but we will not dwell on that topic.
Theorem 2.10. Isomorphisms classes of n-gerbes with connective structure on X are in bijection with Hn+1(X, C×(n + 2)).
In degrees other than n + 1, Deligne cohomology does not convey any new informa-
i × ∼ i × i × ∼ tion: we have H (X, C (n + 2)) = H (X, C ) for i > n + 1, and H (X, C (n + 2)) = Hi(X, C×) for i < n + 1.
31 To give rough idea of the proof, note that when X is an orbifold, we can always choose a Lie groupoid presentation X1 ⇒ X0 such that each Xn is a disjoint union of contractible manifolds. In that case, it is easy to see that a central extension with connective structure indeed defines a 2-cocycle in the Čech complex (2.2), and isomorphisms of such objects are in bijection with trivializations of the difference cocycles. Finally, to fix our notation, we notice that a 2-cocyle in the Čech complex (2.2) with coefficients in C×(3) is given by a triple
∞ × 1 2 (h, A, B) ∈ C (X2, C ) × Ω (X1) × Ω (X0) satisfying the cocycle conditions
−1 −1 ∞ × h(a, b)h(a, bc) h(ab, c)h(b, c) = 1 in C (X3, C ),
∗ ∗ ∗ 1 pr2 A + pr1 A − c A = d log h in Ω (X2), (2.3)
∗ ∗ 2 t B − s B = dA in Ω (X1),
2.4 Group actions on stacks
We mostly follow Ginot and Noohi [19]. The original reference for the basic definitions is Romagny [34].
2.4.1 Basic definitions
Let X be a groupoid fibration over a site S and G strict monoid object in the 2-category of fibrations over S. We denote by m: G × G → G and 1: pt → G the multiplication law and unit map of G. A (left) action of G on X is a map of groupoid fibrations µ: G×X → X together with (necessarily invertible) natural transformations
32 α, a as in the diagram below.
µ G × G × X m×id / G × X G × X / X 3; O ; α a id×µ µ 1×id ' id µ G × X / X X
In formulas, given an object x ∈ XS and g, h ∈ GS, and using a dot to denote the group action, we are given natural isomorphisms
x x αg,h : g · (h · x) → (gh) · x, a : 1 · x → x.
This data is required to satisfy compatibility conditions that bear some resemblance to the axioms of a monoidal category. Firstly, a kind of pentagon identity relating the different ways in which the action of three group elements g, h, k ∈ GS can be associated:
x x x g·x αg,hk ◦ g · αh,k = αgh,k ◦ αg,h.
Second, a condition on the two ways of associating the action of the unit and another group element:
x x g·x x g · a = αg,1 and a = α1,g.
It seems appropriate to call α and a the associator and unitor for the action, in analogy to the terminology used in the theory of monoidal categories. We say the action is strict if α, a are both the identity. Now, suppose we are given fibrations with G-action (X, µ, α, a) and (Y, ν, β, b). A G-equivariant map between them is a morphism of fibrations f : X → Y together
33 with a natural transformation
µ G × X / X 7? σ id×f f G × Y ν / Y
satisfying the following compatibility condition: for each x ∈ XS and g, h ∈ GS, we have
x h·x x x f(x) x x f(x) f(αg,h) ◦ σg ◦ g · σh = σgh ◦ βg,h and f(a ) ◦ σ1 = b .
We will avoid the temptation of calling σ the “equivariator”, a term which does not seem to appear in the literature, and instead refer to it as equivariance datum. Finally, a G-equivariant 2-morphism between morphisms (f, σ), (f 0, σ0) as above is given by a 2-morphism τ : f → f 0 between the underlying fibered functors which is compatible with σ, σ0 in the sense that
0x x g·x x σg ◦ g · τ = τ ◦ σg
for any x ∈ XS, g ∈ GS. In terms of pasting diagrams, the conditions on σ are expressed by the commuta- tivity of the cube whose two halves are depicted below,
id ×µ / G × G × X G × G × X ?G G × X m×id α µ m×id % µ % w µ G × X / X G × X / X :B id ×σ σ id ×ν / id ×ν / G × G × Y G × Y @H G × G × Y G × Y m×id β σ ν ν % w G × Y ν / Y Y
34 and commutativity of the prism
G × X 1×id 7 µ a X yÑ /' X id 3; σ G × Y 1×id 8 ν b ' Y t| / Y. id
Here, all vertical maps are products of f and the identity of G. The condition on τ is the commutativity of the following diagram.
µ G × X 4< / X σ0
id ×τ 0 τ 0 id×f +3 id×f f AI +3 f Ó σ Ó / G × Y ν Y
We are mostly interested in the case where G is a (representable) sheaf of groups, but we will also consider the group stack pt//Z, assigning to any S ∈ S the groupoid
Z ⇒ pt. Note that a strict action of pt//Z on a stack X is precisely the data of an automorphism of idX, i.e., a natural choice of automorphism for each object of X. For instance, the inertia stack ΛX comes with a canonical pt//Z-action. We will also make use of a 2-categorical model for the circle group to be denoted R//Z. It is presented by the Lie 2-group Z × R ⇒ R (the transport groupoid of the Z-action on R) endowed with the multiplication map determined by the group structures on the spaces of objects and morphisms, and unit 0 ∈ R. At the Lie 2-group level, there are evident strict homomorphisms
T ← R//Z → pt//Z.
The left map gives us an equivalence of group stacks, but in concrete situations it may be more convenient to consider one model or the other.
35 2.4.2 Quotient stacks of G-stacks
Let X be a stack endowed with a left action of a sheaf of groups G. Then we define a new stack G\\X whose S-points are given by a left G-torsor P → S together with a G-equivariant map ψ : P → X; a morphism (P 0, ψ0) → (P, ψ) covering f : S0 → S is given by a diagram ψ0 P 0 IQ Φ ξ ψ ' P / X 0 S f ' S where Φ is a map of G-torsors and ξ an equivariant 2-morphism. There is a faithful functor i: X → G\\X sending x: S → X to the S-point of G\\X consisting of the trivial G-torsor G × S → S together with the G-equivariant map
µ ψ : G × S −−→id×x G × X −→ X.
This makes the diagram below 2-cartesian.
µ G × X / X
pr2 i X i / G\\X
Now, we can attempt to perform the construction of a transport groupoid G×X ⇒ X internally in the 2-category of stacks. For this to work, we need to define internal categories with the appropriate degree of weakness (e.g., if the action is not strictly unital, the same must be allowed of our internal categories). In any case, it is clear that we get a “nerve”, that is, an augmented (weak) simplicial object
i ← ← ← G\\X ← X ⇔ G × X ← G × G × X ← ··· (2.4)
36 Since the various compositions Gn × X → G\\X are not equal, just isomorphic (with a specified isomorphism), the augmentation depends, strictly speaking, on a choice. For definiteness, we take that to be the composition of i with the projection
n prn+1 : G × X → X.
Proposition 2.11. The above induces an equivalence of stacks
j ← ← ← G\\X ←− colim X ⇔ G × X ← G × G × X ← ··· .
The reader well versed on colimits of categories may be able to interpret the discussion in sections 3.2 and 4.2 of Ginot and Noohi’s paper [19] as a proof, even though it does not use the language of colimits. In any case, we will provide our own argument. Before getting there, we give some background on (homotopy) colimits in Cat. Given a diagram of small categories F : D → Cat indexed by a small 1-category
(with no strictness requirements on F ), we denote by D n F the Grothendieck construction (cf. section 2.1.1). It is the oplax colimit of F , meaning that for each
C ∈ Cat, there is an equivalence between the category of functors D n F → C and the category of lax natural transformations F → constC and modifications between them. The colimit of F is obtained by localizing D n F at the class of opcartesian morphisms. Spelling out the above, the colimit can be described in terms of generators and relations as follows. We write i, j, etc., for objects of D and Ai, Aj for their images via F ; also, we use the same notation both for a morphisms f : i → j in D and its image f : Ai → Aj. To build A = colimD Ai, we start with the disjoint union qi∈DAi and then freely adjoin inverse morphisms
−1 fx : x → f(x), fx : f(x) → x
for each f : i → j in D and x ∈ Ai; finally, we impose a number of natural relations,
37 most notably φ f f f(φ) x −→ y −→y f(y) = x −→x f(x) −−→ f(y) ,
−1 −1 where φ is a morphism in Ai, as well as its counterpart involving fx , fy . This process can be made precise using the free category generated by a directed graph and congruences. For more details, including the proof that this has the desired universal property, see Fiore [17, chapter 4].
Proof of proposition 2.11. Colimits of stacks are obtained by taking colimits objectwise in S and then stackifying. Thus, it suffices to show that, for each S ∈ S,
j ← S ← ← (G\\X)S ←− colim XS ⇔ (G × X)S ← (G × G × X)S ← ···
triv gives an equivalence of the right-hand side with the full subgroupoid (G\\X)S of the left-hand side involving only trivial G-torsors. To simplify the argument, we assume, without loss of generality, that the GS-action on XS is strict [34, proposition 1.5].
triv n Consider the functor l : (G\\X)S → colimn(G × X)S prescribed by the following conditions. First, on XS, seen as a subgroupoid of both the domain (via i: XS ,→
triv (G\\X)S ) and codomain, l is just the identity. Second, to the morphism x → g · x in
triv (G\\X)S determined by g ∈ GS, l associates the morphism
−1 x pr2 µ µg : x −−→ (g, x) −→ g · x in the colimit groupoid. To see that this is well defined and respects compositions, it suffices to check that the outer square of the following diagram in the colimit groupoid
38 commutes, for any g, h ∈ GS and ξ : g · x → y in XS.
ξ g · x / y e ; pr2 pr2
id ×ξ g·x / y µh (h, g · x) (h, y) µh
µ µ y h·ξ # hg · x / h · y
This follows from the fact that each circuit traveling inside the square commutes.
Now, the composition jS ◦ l is equal to the identity, and we claim that the reverse composition is isomorphic to the identity. In fact, l ◦ jS(g1, . . . , gn, x) = x, and we define a natural transformation u: id → l ◦ jS by
u(g1,...,gn,x) = prn+1 :(g1, . . . , gn, x) → x.
Naturality with respect to those morphisms in the colimit groupoid which arise from
n morphisms in (G × X)S is obvious. A general morphism arising from the indexing category ∆op is as in the left vertical arrow of the diagram below,
prn+1 (g1, . . . , gn, x) :/ x pr2 ( µx (gJ , x) gJ
µ prk+1 $ / (gI1 , . . . , gIk , gJ · x) gJ · x
where, I1,...,Ik,J ⊂ [n] are (possibly empty) disjoint and adjacent subsets whose
union contains n, and g{i1,...,ij } = gi1 . . . gij . Its image through l◦jS is the right vertical arrow, and naturality of u, that is, the claim that the outer square commutes, follows from commutativity of the circuits involving (gJ , x). This finishes the proof that jS is
triv an equivalence onto (G\\X)S .
39 Now, given a stack C, applying FunS(—, C) to diagram (2.4) produces a (weak) cosimplicial groupoid. The following descent calculation for G-stacks is then a corollary of proposition 2.11.
Proposition 2.12. For any stack C and G-stack X, diagram (2.4) induces an equiv- alence of groupoids
∼ → FunS(G\\X, C) = lim (FunS(X, C) ⇒ FunS(G × X, C) → ··· ) .
Again, a concrete description of 2-limits in the 2-category of small categories can be found in Fiore [17, chapter 5]. For the convenience of the reader, we give a quick summary here. We fix the same notations as in the discussion of colimits above; in particular, we have a diagram F : D → Cat. Then (a model for) the limit of F is the category whose objects are (pseudo) natural transformations ∆pt → F with domain the constant functor with value the discrete category with one object, and whose morphisms are modifications between them. In concrete terms, an object consists of a collection of objects ai ∈ Ai for each i ∈ D together isomorphisms τf : f(ai) → aj for each morphism f : i → j in D; these data are required to satisfy certain coherence
0 0 conditions. A morphism (ai, τf ) → (ai, τf ) consists of a collection of morphisms
0 ai → ai in Ai for each i ∈ D, subject to appropriate conditions. Finally, we quote a reassuring fact about quotients of differentiable stacks.
Proposition 2.13. If X is a differentiable stack endowed with an action of the Lie group G, then G\\X is again a differentiable stack.
Fix a presentation X1 ⇒ X0 for X and consider the bibundle
E µ1 µ2 { G × X0 X0
40 corresponding to the action map µ: G × X → X. Then a groupoid presentation for
G\\X can be given by a groupoid E ⇒ X0 with structure maps s = µ2 and t = pr2 ◦µ1. The composition law can be deduced by expressing the group action axioms in terms of bibundles.
41 CHAPTER 3
SUPERGEOMETRY
The theory of supermanifolds is a generalization of usual differential geometry where algebras are replaced by super (i.e., Z/2-graded) algebras. In this chapter, we review the basic facts and definitions of supergeometry, and then spend some time on the less well-known concept of Euclidean structures on supermanifolds. The literature is replete with crash courses and incomplete surveys on supermanifolds (see e.g. [14, 24, 28]). Among more comprehensive references, we point the reader to Dimitry Leites’s seminar notes [27].
3.1 Superalgebra
Since super and Z-graded algebra is a familiar topic in algebraic topology, we will only briefly sketch some definitions and highlight our conventions. What follows makes sense in the category of Z/2-graded modules over a ring, but, for definiteness, let us focus on the category SVect of Z/2-graded real vector spaces. We pick, among the two possible choices, the symmetric monoidal structure with braiding isomorphisms V ⊗ W → W ⊗ V given by v ⊗ w 7→ (−1)p(v)p(w)w ⊗ v, where v, w are homogeneous elements of parity p(v), p(w). We attach the adjective “super” to indicate or emphasize that a given construct is internal to SVect, but often omit it. With respect to this symmetric monoidal structure, saying that a (super) algebra A is commutative means, in formulas, that vw = (−1)p(v)p(w)wv for homogeneous v, w ∈ A. Similarly, a (super) Lie algebra is a Z/2-graded vector space g endowed with a bracket [, ]: g ⊗ g → g (of even parity) which is skew-symmetric, i.e., [v, w] =
42 −(−1)p(v)p(w)[w, v], and satisfies the Jacobi identity: for each v ∈ g the operator [v, —] is a derivation of parity p(v) with respect to [, ]. There is a functor Π: SVect → SVect, called parity reversal, that reverses the grading of all objects. The nth symmetric power Symn(V ) of a super vector space
⊗n V is V /Σn, the space of coinvariants with respect to the permutation action. If V is purely even, then this is just the usual symmetric power, and Sym(ΠV ) is the nth exterior power of V , considered as a super vector space of parity n. The direct sum of all symmetric powers forms a commutative algebra Sym(V ). Note that the exterior algebra of a finite dimensional super vector space V (defined as Sym(ΠV )) needs not to be finite dimensional.
Sometimes we need to consider Z-graded objects in the category of super vector spaces. Our sign convention for the tensor product reads
v ⊗ w = (−1)p(v)p(w)+|v||w|w ⊗ v for homogeneous element of parity p(v), p(w) and cohomological degree |v|, |w|.
Finally, we define super versions of trace and determinant. Let V = V0 ⊕ ΠV1 be a finite dimensional super vector space, with Vi purely even. An endomorphism
AB T = CD
of V can be written as a matrix where the entry ij is a linear map Vj → Vi. We define the supertrace of T as str T = tr A − tr D = tr T ◦ , where is the grading automorphism. The fundamental property of str is that it vanishes on (super)commutators. If T is an automorphism, we define its Berezinian
43 to be Ber T = det(A − BD−1C) det(D)−1.
It is analogous to the determinant in that
Ber exp(T ) = exp str(T ).
See Le˘ıtes[28] for more details.
3.2 Supermanifolds
3.2.1 Basic definitions
∞ A supermanifold (X,CX ) of dimension p|q, where p an q are nonnegative integers, is a Hausdorff, paracompact topological space X endowed with a sheaf of Z/2-graded
∞ R-algebras CX , called the structure sheaf or sheaf of smooth functions, which is locally isomorphic, as a ringed space, to the model space Rp|q defined by
p|q p ∞ R = (R ,CRp ⊗ R[θ1, . . . , θq]), θi odd,
∞ in a way that preserves the super algebra structure. We will usually denote (X,CX ) simply by X.
∞ ∞ A smooth map φ: (X,CX ) → (Y,CY ) is simply a morphism in the appropriate category of ringed spaces. For concreteness, let us spell out what this (as well as the “locally isomorphic” condition from the previous paragraph) means: φ is given by a continuous map X → Y , which by abuse of notation we still call φ, together with a
∗ −1 ∞ ∞ homomorphism of sheaves of Z/2-graded R-algebras φ : φ CY → CX (or, in terms
−1 ∞ ∞ of the right adjoint to φ , a homomorphism CY → φ∗CX ). Supermanifolds and smooth maps thus form a category we denote SM. We will generally import terminology from usual differential geometry into this new context.
44 For instance, a diffeomorphism will refer to an isomorphism is SM, and a submersion will refer to a map that admits local sections (the implicit function theorem carries over to supermanifolds, and characterizes this situation in terms of the derivative). We will also often call a supermanifold simply a manifold, especially if that is clear from the context. When topological adjectives, such as “connected” or “contractible” are attached to a supermanifold, they are understood to refer to its underlying topological
∞ space. An open submanifold U ⊂ X of a supermanifold (X,CX ) is given by an open
∞ subset U ⊂ X together with the restriction of the structure CX sheaf to U (talking about submanifold with boundary, on the other hand, is a more subtle issue, as we will see in section 3.4.2).
∞ Note that each stalk (C p|q )x is a local ring, with maximal ideal m generated by the R p ∞ germs of smooth functions on that vanish at x. The image of a germ f ∈ (C p|q )x R R ∞ ∼ in the residue field (C p|q )x/m = can be characterized as the unique λ such that R R ∞ f − λ is not invertible in (C p|q )x. In particular, m can be characterized as the set R of all f such that f − λ is a unit for all λ 6= 0. This property is preserved by any
R-algebra homomorphism, so the usual requirement in the definition of varieties and schemes that all sheaves and homomorphisms are of local rings holds automatically in our context. Notice, however, that contrarily to the case of ordinary manifolds, a function f on a supermanifold is not determined by its values on points (i.e., its image in the residue field of each point): any purely odd function has value zero on points. This is the basic reason why it is necessary to think about supermanifolds with an algebraic geometer’s mindset. The category of smooth manifolds sits inside SM as a full subcategory—the
∞ supermanifolds of odd dimension 0. Given a supermanifold X, denote by Nil ,→ CX the ideal of nilpotent functions; it is generated by the purely odd functions. Then
∞ ∞ ∞ C /Nil is a sheaf of -algebras locally isomorphic to C p , so (X,C /Nil) is an X R R X ordinary smooth manifold, called the reduced manifold of X, and denoted |X|.
45 There is a natural smooth map |X| → X induced by the sheaf homomorphism
∞ ∞ CX → CX /Nil. Further examples of supermanifolds are given by the parity reversal construction. Given a real, even vector bundle V → X over an ordinary manifold, set ΠV = (X, Sym(ΠV)), where V is the sheaf of sections of V and ΠV its parity reversal; vector bundle homomorphisms induce smooth maps in the obvious way, so Π is a functor. (If
V is the trivial rank q bundle over Rp, this construction just recovers the model space Rp|q.) It turns out that every supermanifold is diffeomorphic to one of the above form, but not canonically so [3]. Choosing such a diffeomorphism is equivalent to choosing a splitting of the short exact sequence
∞ ∞ Nil → CX → CX /Nil, or, equivalently, a map X → |X| for which the canonical |X| → X is a section. Note also that not every smooth map, and not even every diffeomorphism, is induced by a map of vector bundles. For instance, the diffeomorphism of R1|2 determined on functions by t 7→ t + θ1θ2, θi 7→ θi does not come from a map of vector bundles (here, t is the standard coordinate on R). A vector bundle V of rank r|s on a supermanifold X is defined to be a sheaf of
∞ CX -modules that is locally free of rank r|s. Given a map f : Y → X, the inverse
∗ ∞ −1 image f V = C ⊗ −1 ∞ f V is a vector bundle on Y , and given a second vector Y f CX bundle W on Y , a map of vector bundles W → V covering f is defined to be a sheaf homomorphism W → f ∗V. We fix a topology on SM by declaring coverings to be local diffeomorphisms (also called étale maps). We are then able, as usual, to express natural constructions in terms of sheaves or stacks; for instance, the above paragraph defines the stack Vect of vector bundles and linear maps on the site of supermanifolds.
46 3.2.2 The functor of points formalism
We already remarked that a function on a supermanifold is not determined by its values on points, and of course a smooth map X → Y is also not. However, by the Yoneda lemma, giving such map is equivalent to giving a natural assignment
{maps S → X} → {maps S → Y } for each supermanifold S. Thus, we call a map S → X an S-point of X, writing
XS for the collection of all S-points, and a map X → Y is now completely specified by its values on S-points. In fancier words, we are thinking about supermanifolds in terms of the presheaf they represent. Describing a map via functors of points is more geometric, and often more convenient, than giving the corresponding sheaf homomorphism. One reason why this is so is the following fact.
Proposition 3.1. The presheaves on SM given by X 7→ C∞(X)ev and X 7→
C∞(X)odd are represented by R respectively R0|1. More generally, an open domain U ⊂ Rp|q represents the presheaf that assigns, to each X, a collection of p even and q odd functions on X taking values in U (which is to say, the restriction of the p even functions to |X| determines a map into |U|).
For a proof, see [28, section 2.1.7]. It is also useful to be able to describe the pullback of functions via the map F : X → U corresponding, via the above proposition, to the even respectively odd functions fi, φj on X, 1 ≤ i ≤ p, 1 ≤ j ≤ q. We write
0 P I fi = fi + ri, where ri is nilpotent, and fix a function u = uI (t1, . . . , tp)θ on U.
Proposition 3.2. In the above situation, we have
X rJ X rJ F ∗u = ∂J u(f 0, . . . f 0, φ , . . . , φ ) = ∂J u (f 0, . . . , f 0)φI . J! 1 p 1 q J! I 1 p J I,J
47 p Here, the summations are over multi-indices J = (j1, . . . , jp) ∈ Z≥0 and I =
q J j1 jp J j1 jp (i1, . . . , iq) ∈ {0, 1} , and we write r = r1 ··· rp , ∂ = ∂t1 ··· ∂tp , J! = j1! ··· jp!.
In the light of proposition 3.1, we write, for V a Z/2-graded vector space, C∞(X; V ) = (C∞(X) ⊗ V )ev. This agrees with the set of smooth maps X → V , where V is seen as a supermanifold with reduced manifold V ev and structure sheaf
∞ odd CV ev ⊗Sym(V ). However, this must no be confused with the space of global sections of the trivial vector bundle with fiber V , which is C∞(X) ⊗ V .
3.2.3 Calculus on supermanifolds
∞ To any supermanifold X is associated the tangent sheaf TX of derivations of CX ; its sections are called vector fields. On Rp|q with coordinates t1, . . . tp, θ1, . . . , θq, there are vector fields ∂/∂ti, 1 ≤ i ≤ p and ∂/∂θj, 1 ≤ j ≤ q, and one can show they form a basis [28, p. 31]. So TX is a vector bundle of rank equal to the dimension of X.
Commutators of vector fields make TX into a sheaf of Lie algebras. Note the sign rule: [v, w] = vw − (−1)p(v)p(w)wv for homogeneous vector fields v, w.
1 The dual vector bundle to TX is denoted ΩX . The pairing between them will be
1 ∞ ∞ 1 denoted h , i: TX ⊗Ω → CX . There is map d: CX → ΩX determined by hv, dfi = v(f) (this explains the choice of notation for the pairing: we want to avoid transposing symbols, which would force us to add signs). It is the universal even derivation
∞ 1 with values in a CX -module. We give ΩX cohomological degree 1 and consider the
∗ ∗ 1 symmetric powers Ω = Sym ∞ (Ω ). Then d extends uniquely to a square zero X CX X ∗ ∗ n degree 1 derivation d: ΩX → ΩX . Sections of ΩX are called differential n-forms. Note
∗ that in positive fermionic dimensions, ΩX is never bounded above. The pairing of
∗ vector fields and 1-forms also extends to a derivation, of degree −1, of ΩX . The operator associated to a vector field v is denoted iv and called contraction with v. One can also define, as in ordinary differential geometry, Lie derivatives of dif- ferential forms (and, more generally, any type of tensor field). Cartan calculus still
48 works, provided the sign rules are taken into account suitably:
[d, iv] = Lv, [Lv, iw] = i[v,w], [Lv,Lw] = L[v,w], [iv, iw] = 0.
3.2.4 Super Lie groups
A super Lie group G is a group object in the category of supermanifolds. As in the classical situation, we are interested in the (super) Lie algebra g = Lie(G) of left-invariant vector fields, i.e., those v ∈ Der(C∞(G)) satisfying
∗ C∞(G) m / C∞(G) ⊗ C∞(G)
v 1⊗v
∗ C∞(G) m / C∞(G) ⊗ C∞(G), where m∗ is the homomorphism associated to the multiplication map G × G → G. To relate this with the usual, pointwise definition for ordinary Lie groups, evaluate each of the left tensor factors at a point g ∈ G; then commutativity of the diagram means that v is Lg-related to itself. A (right) action of a super Lie group G on a supermanifold X is a smooth map µ: X × G → X satisfying the usual conditions. The composition
µ∗ 1⊗(ev ◦v) C∞(X) −→ C∞(X) ⊗ C∞(G) −−−−−→e C∞(X),
∞ where v ∈ g and eve : C (G) → R is the evaluation at the identity, gives a map
∞ ∞ g → Der(C (M)) = C (X,TX ); it is a homomorphism of super Lie algebras. In particular, applying this for X = G with the right action by translation gives an
∞ inverse for the restriction g → TeG = Dereve (C (G), R), showing that g, as a super vector space, has the same dimension as G. Passing from ordinary to super Lie algebras, the simplest new example is the
49 free Lie algebra on one odd generator Q, that is, the vector space spanned by one odd element Q and one even element N with Lie bracket determined by the relation Q2 = 1/2[Q, Q] = N. We should then ask what is the corresponding simply-connected
Lie group. It turns out to be R1|1 endowed with the multiplication determined, on the standard basis t, θ ∈ C∞(R1|1), by the formulas
t 7→ t ⊗ 1 + 1 ⊗ t + θ ⊗ θ, θ 7→ θ ⊗ 1 + 1 ⊗ θ.
1|1 In the S-point formalism, identifying RS with pairs consisting of an even and an odd function on S (proposition 3.1), the group law is described by
(t, θ), (t0, θ0) 7→ (t + t0 + θθ0, θ + θ0).
Since this group law is not abelian, we will use multiplicative notation, in spite of
1|1 this leading us to denote, rather unsatisfactorily, the unit by 1 = (0, 0) ∈ RS and the
1|1 −1 1|1 inverse of u = (t, θ) ∈ RS by u = (−t, −θ). (Notice that R has a commutative ring structure, unrelated to the above, coming from the fact that it represents the functor C∞. Confusion between these two structures is unlikely, but we will insistently use · to indicate the above nonabelian group law.) We pick the following standard generators of Lie(R1|1):
1 D = ∂ − θ∂ ,N = [D,D] = −∂ . θ t 2 t
The Lie algebra of right-invariant vector fields on R1|1 is generated by
1 Q = ∂ + θ∂ , [Q, Q] = ∂ . θ t 2 t
Another basic example is the Lie algebra on one odd generator squaring to zero
50 and the corresponding (abelian) Lie group R0|1. Here, the group law is simply addition of odd functions. We then notice that there is a short exact sequence of super Lie groups
1|1 0|1 1 → R → R → R → 1 and it is not split. Also, the one-to-one correspondence between vector fields on an ordinary manifold and local actions of R extends to the super case if we restrict to even vector fields. Similarly, odd vector fields precisely correspond to local R1|1-actions; restriction to R corresponds to squaring the generating vector field, and such an action factors through R0|1 if and only if the corresponding vector field has square 0 [16, 27].
3.2.5 Superpoints, differential forms, and superconnections
In this section, we recall some important facts about the geometry of the generalized manifold SM(R0|1,X). For a detailed discussion of the first part, see Hohnhold et al. [22, section 3]. The geometric interpretation of superconnections is unpublished work of Schommer-Pries, Stolz, and Teichner.
Proposition 3.3. The presheaf SM(R0|1,X), S 7→ SM(S × R0|1,X) is representable by ΠTX.
The idea of the proof is as follows. Write a given algebra homomorphism
∗ ∞ ∞ ∞ 0|1 ∞ ∗ ∗ φ : C (X) → C (S) ⊗ C (R ) = C (S)[θ] as f 7→ φ0(f) + θφ1(f), where
∗ ∞ i+p(f) ∗ φi (f) ∈ C (S) . Then we can check that φ0 is an algebra homomorphism,
∗ ∞ ∞ and thus defines a map φ0 : S → X, and φ1 : C (X) → C (S) is an odd derivation over φ0, which can be identified, geometrically, with an odd vector field along φ0. An S-point of ΠTX admits exactly the same interpretation, naturally in S. Next, we notice that for an ordinary manifold X, C∞(ΠTX) is naturally isomorphic to the Z/2-graded algebra Ω∗(X) of differential forms on X. (In the supermanifold case, functions on ΠTX are called pseudodifferential forms, and there is an inclusion of
51 ∞ ∗ ∞ 0|1 ∼ 0|1 × sheaves of CX -algebras ΩX → CΠTX .) This induces an action of Diff(R ) = R oR on Ω∗(X) encoding the differential Z-graded algebra structure: a dilation λ ∈ R× sends a differential form ω to λnω if and only if ω is homogeneous of degree n, and the odd vector field induced by the R0|1-action on ΠTX identifies with the de Rham differential. At the Lie group level, the action of R0|1 is given by the formula
∗ ∞ 0|1 ∗ ω ∈ Ω (X) 7→ ω + θdω ∈ C (R ) ⊗ Ω (X). (3.1)
0|1 ∼ Proposition 3.4. For X an ordinary manifold, the identification SM(R ,X) = ΠTX is Diff(R0|1)-equivariant.
From now on, we will always use the more concise notation ΠTX, even when its identification with a mapping space is conceptually the best way to think about this object.
Now, let V → X be a Z/2-graded complex vector bundle. Denote by Ω∗ the sheaf of complex-valued differential forms and by Ω∗(X; V ) the Ω∗(X)-module of differential forms with values in V . Following Quillen [33], we define a superconnection A on V to be an odd operator (with respect to the total Z/2-grading) on Ω∗(X; V ) satisfying the Leibniz rule
|ω| A(ωf) = (dω)f + (−1) ωAf. (3.2)
Here, ω ∈ Ω∗(X) and f ∈ Ω∗(X; V ). From the above, it follows that A is entirely
0 determined by its restriction to Ω (X; V ); denoting by Ai, i ≥ 0, the component
0 i Ω (X; V ) → Ω (X; V ), we find that A1 is an affine (even) connection and all other Ai are Ω0(X)-linear odd homomorphisms. Conversely, the data of an affine connection
0 and an arbitrary collection odd Ω -linear homomorphisms Ai, 1 6= i ≥ 0, determines a superconnection. The even operator A2 : Ω∗(X; V ) → Ω∗(X; V ), which can be checked to be Ω0(X)-linear, is called the curvature of A.
Now, let V0,V1 → X be complex super vector bundles and Ai, i = 0, 1, super-
52 connections. Then there exists a superconnection A on the homomorphism bundle
Hom(V0,V1) → X, characterized by
|Φ| (AΦ)f = A1(Φf) − (−1) Φ(A0f)
∗ ∗ for any section Φ of Ω (X; Hom(V0,V1)) of parity |Φ| and f ∈ Ω (X; V0). We define VectA to be the stack on Man whose objects over X are vector bundles with superconnection (V, A), and morphisms (V0, A0) → (V1, A1) are sections Φ ∈
∗ Ω (X; Hom(V0,V1)) of even total degree satisfying A(Φ) = 0. With this definition, ordinary connections Vect∇ ,→ VectA do not form a full substack, but the inclusion is still injective on isomorphism classes. There is a nice interpretation of superconnections in terms of Euclidean super- geometry (see section 3.3 below for the definitions). Consider the pullback bundle p∗V → ΠTX along p: ΠTX → X. Its sections on an open U 0 = p−1(U), where U ⊂ X
∗ ∞ is open, are given by Ω (U) ⊗C∞(U) C (U; V ) = Ω(U; V ), and to say that a given odd,
fiberwise linear vector field A on p∗V is p-related to the de Rham vector field d on the base is precisely the same as saying that equation (3.2) holds. Thus a superconnection on V gives p∗V the structure of an Isom(R1|1)-equivariant vector bundle over ΠTX, where the action on the base is via the projection Isom(R1|1) → Isom(R0|1) and the identification ΠTX = SM(R0|1,X). The superconnection is flat if and only if this action factors through Isom(R0|1).
Theorem 3.5. The stack map VectA → Vect(ΠT —// Isom(R1|1)) defined above is an [ equivalence. The same is true for the map VectA → Vect(ΠT —// Isom(R0|1)).
3.3 Euclidean structures
The definition of Euclidean structures on supermanifolds follows the philosophy of Felix Klein’s Erlangen program. One starts by fixing a model space and a subgroup of
53 diffeomorphisms, called the isometry group; a Euclidean structure is then a maximal atlas taking values in the model space whose transition maps are isometries. This idea is explained in detail in Stolz and Teichner [38, sections 2.5 and 4.2]. In (real) dimensions 0|1 and 1|1, the model spaces are R0|1 respectively R1|1 with isometry groups
0|1 0|1 1|1 1|1 Isom(R ) = R o Z/2, Isom(R ) = R o Z/2.
In both cases, Z/2 acts by negating the odd coordinate and Rd|1 acts by left multipli- cation. In low dimensions, it is also possible to give ad hoc definitions in terms of sections of certain sheaves. In the remainder of this section, we discuss some of those alternative definitions, and study the stack of 1|1-dimensional closed connected Euclidean supermanifolds, which we will also call Euclidean supercircles. Nothing here is new, but we will provide some proofs we could not find in the literature. Besides the references cited above, parts of our discussion below are inspired by Berwick-Evans [7].
3.3.1 Euclidean structures in dimension 1|1
In Dumitrescu [16], a conformal structure on a 1|1-manifold X is defined to be a distribution D (i.e., a subsheaf of the tangent sheaf TX ) of rank 0|1 fitting in a short exact sequence
⊗2 0 → D → TX → D → 0. (3.3)
A Euclidean structure is then defined to be a choice, up to sign, of an odd vector field D
1|1 generating D. The fundamental example is the vector field D = ∂θ −θ∂t on R . Note
2 that it squares to −∂t, so in fact D, D generate TR1|1 . More generally, conformal and Euclidean structures on a family X → S of 1|1-manifolds are appropriate splittings or sections of the vertical tangent bundle TX/S.
54 We want to show that this is equivalent to the original definition. Denote by E and E0 the stacks of families of 1|1-dimensional Euclidean manifolds according to the chart definition respectively the vector field definition. It is clear that we have a map E → E0, since the transition maps of an Euclidean chart preserve the canonical vector
field D on R1|1 up to sign. Now, given an object in E0, the atlas from proposition 3.8 below is indeed Euclidean, by propositions 3.6 and 3.7. This given an inverse map E → E0.
Proposition 3.6. The subgroup of diffeomorphisms of R1|1 preserving the form ω = dt − θdθ is precisely Isom(R1|1) = R1|1 o Z/2, acting in the standard way on the left.
A correct reading of this assertion requires that we think in families; thus, the claim is that the subsheaf of Diff(R1|1) ⊂ SM(R1|1, R1|1) preserving ω is representable by the Lie group R1|1 o Z/2. Moreover, it will be clear from the proof that the proposition is true locally in R1|1, that is, if U ⊂ R1|1 is a connected domain, then the sheaf of embeddings U → R1|1 preserving ω is R1|1 o Z/2.
Proof. An S-family of diffeomorphisms of R1|1 is given by a diffeomorphism
1|1 1|1 Φ: S × R → S × R commuting with the projections onto S. We can express this diffeomorphism in terms of a map φ: S × R1|1 → R1|1 by the formula
(s, x) 7→ (s, φ(s, x) · x), where s, x should be interpreted as T -points of S and R1|1 for a generic supermanifold T ,
1|1 0|1 and · indicates the usual group operation on R . Writing φ = (r, η) ∈ (R×R )S×R1|1
55 0|1 and x = (t, θ) ∈ (R × R )T in terms of their components, the above formula becomes
(s, t, θ) 7→ (s, t + r(s, t, θ) + η(s, t, θ)θ, η(s, t, θ) + θ).
Hence the equation ω = Φ∗ω reads
dt − θdθ = dt + dr − θdθ − (2θ + η)dη.
To analyze the restrictions imposed by this equation, let us write
∞ i r = r0 + r1θ, η = η1 + η0θ, where ri, ηi ∈ C (S × R) .
Then dr − (2θ + η)dη = 0 gives us
0 = dr0 − η1dη1 (3.4)
+ (dr1 + (2 + η0)dη1 − η1dη0)θ (3.5)
+ (r1 − η1η0)dθ (3.6)
− (2 + η0)η0θdθ. (3.7)
Each individual line above vanishes. From (3.7), we get that η0 = 0 or −2, since either η0 or (2 + η0) has nonzero reduced part and hence is invertible, and (3.6) tells us that r1 = η0η1. Plugging that into (3.5), we get (2 + 2η0)dη1 = 0, so dη1 = 0 since the factor in front of it is a nonzero constant. Finally, (3.4) implies that dr0 = 0. Now, recall that those formulas should be interpreted as equalities of S-families of differential forms on R1|1, i.e., sections of Ω∗(S × R1|1) modulo Ω≥1(S). So in
∞ fact we have r0, η1 ∈ C (S), and there is a locally constant function a = 1 + η0 ∈
(Z/2)S = {±1}S. Therefore the diffeomorphism Φ determines and is determined by
56 1|1 (r0, η1, a) ∈ (R o Z/2)S via the correspondence
1|1 (r0, η1, a) 7→ φr ,η ,a = (r0 + (a − 1)η1θ, η1 + (a − 1)θ) ∈ . 0 1 RS×R1|1
It is simple to check that any choice of (r0, η1, a) as above determines a diffeo- morphism preserving ω, and that the choices (r0, η1, 1) respectively (0, 0, −1) act as translation by (r0, η1) respectively negation of the odd variable. Therefore, to finish the proof, we just need to verify that given a second diffeomorphism Φ0 prescribed, in
0 0 0 a similar way, by (r0, η1, a ), the composition
0 Φ Φ 0 (s, t, θ) 7→ φ (s, θ) · (s, t, θ) 7→ φ 0 0 0 (s, θ ) · φ (s, θ) · (s, t, θ), r0,η1,a r0,η1,a r0,η1,a where θ0 = η + (a − 1)θ is the θ-component of the middle term, agrees with the action
0 0 0 of the product (r0, η1, a ) · (r0, η1, a); more explicitly,
φ 0 0 0 0 0 0 (s, θ) = φ 0 0 (s, η + (a − 1)θ) · φ (s, θ). (r0+r0+a η1η1,η1+a η1,a a) r0,η1,a r0,η1,a
This is a tedious but straightforward calculation.
Proposition 3.7. A diffeomorphism of R1|1 preserves ω = dt − θdθ if and only if it preserves D = ∂θ − θ∂t up to sign.
Proof. If an S-family of diffeomorphisms Φ: S × R1|1 → S × R1|1 preserves ω, then
1|1 it is determined by φ: (R o Z/2)S and it is easy to check that it sends D to either
2 2 D or −D. Conversely, if Φ∗D = ±D, then Φ∗D = (±D) , so that
∗ 2 ∗ 2 hD, Φ ωi = hΦ∗D, ωi = 0, hD , Φ ωi = hD , ωi.
2 ∞ ∗ Since D,D generate T 1|1 as a C -module, if follows that Φ ω = ω. R R1|1
57 Proposition 3.8. Let X → S be an S-family of 1|1-manifolds and D a vertical vector field generating a distribution as in (3.3). Then X admits an atlas such that D can be written locally as ∂θ − θ∂t.
Proof. We apply the Frobenius theorem (Deligne and Morgan [14, lemma 3.5.2]) to the vector field D2. This gives us local charts (t, θ): U ⊂ X → S × R1|1 where D2 gets identified with −∂t. With respect to one of those charts, we can write
D = f∂θ + g∂t, f = f0 + f1θ, g = g1 + g0θ,
∞ i where fi, gi ∈ C (S × R) , so that
2 D = f(∂θf)∂θ + f(∂θg)∂t + g(∂tf)∂θ + g(∂tg)∂t
2 2 (the remaining terms one could expect in this expansion involve ∂θ , g , or [∂θ, ∂t], so they vanish). Inspecting the coefficients of ∂t, θ∂t, ∂θ, and θ∂θ respectively, we get
0 0 0 f0g0 + g1g1 = −1, f1g0 + g1g0 − g0g1 = 0,
0 0 0 −f0f1 + g1f0 = 0, g1f1 + g0f0 = 0.
The first equation implies that f0, g0 are invertible, and the fourth equation implies
0 that g1g0f0 = 0. Multiplying the third equation by g0 gives us g0f0f1 = 0, so f1 = 0.
0 Using again the fourth equation, we conclude that f0 = 0. Therefore (first equation),
0 0 0 g0 is a multiple of g1 and the second equation reduces to g0g1 = 0 = g1. Finally, we learn from the first equation that f0 and −g0 are inverses. To summarize, we have
−1 ∞ even D = f0∂θ − f0 θ∂t, where f0 ∈ C (S) .
−1 Performing the change of coordinates t 7→ t, θ 7→ f0 θ, we can assume f0 = 1, which
58 finishes the proof.
3.3.2 Euclidean supercircles
We are interested in the stack K of closed connected 1|1-dimensional Euclidean
1|1 manifolds. Given a parameter supermanifold S and a map l : S → R>0, we can form
1|1 the S-family of supercircles of length l, Kl = (S × R )/Z, where the generator of the Z-action is described, in terms of T -points of S×R1|1, by (s, u) 7→ (s, l(s)·u). Moreover, given any map r : S → R1|1, the diffeomorphism of S × R1|1, (s, u) 7→ (s, r(s) · u)
1|1 1|1 descends to an isometry Kr−1lr → Kl, and the flip fl: R → R (the diffeomorphism negating the odd coordinate) descends to an isometry Kfl(l) → Kl, since fl is a group automorphism of R1|1. We can assemble this collection of examples into a Lie groupoid as follows. Note that the right R1|1-action on itself by conjugation extends to an action of the semidirect product Z/2 n R1|1 where Z/2 acts via fl. It is then clear that we have a map of
1|1 1|1 stacks R>0//(Z/2 n R ) → K. To the S-point of the domain corresponding to a map 1|1 l : S → R>0, it assigns Kl, and to the morphism corresponding to the S-point (a, r) of
1|1 Z/2 n R , it assigns the isometry Kl·(a,r) → Kl. Notice, however, that this does not give us an isomorphism of stacks, since the S-family of morphisms (0, l): l → l in the domain stack maps to the identity map of Kl. To fix that problem, consider the Lie groupoid X1 ⇒ X0 obtained from the above transport groupoid by modding out the morphisms mentioned above. To be completely explicit, it is the Lie groupoid with
1|1 1. space of objects X0 = R>0,
1|1 2. space of morphisms X1 = (X0 × Z/2 n R )/Z, where the Z-action is generated
by (l, a, r) 7→ (l, a, l · r) (this being, of course, a formula of S-points of X0 ×
Z/2 × R1|1),
3. target map t: X1 → X0 sending the equivalence class of (l, a, r) in (X1)S to
59 l ∈ (X0)S and source map s: X1 → X0 sending the equivalence class of (l, a, r)
−1 a to r · fl (l) · r ∈ (X0)S,
4. composition of morphisms given by (l, a, r)◦(r−1fla(l)r, a0, r0) = (l, (r, a)·(r0, a0)), and identity map l 7→ (l, 0, 0).
Theorem 3.9. The map B(X1 ⇒ X0) → K is an equivalence of stacks.
Proof. The fibered functor in question is faithful by construction, and full by proposition 3.6. So it remains to show it is essentially surjective. Pick any K ∈ KS, and denote by DK a vector field giving its Euclidean structure. Below we will make a series of statements that are only literally true locally in S; each time, we implicitly restrict to a neighborhood in S, which of course is enough for the proof.
First, we note that the action µ: R1|1 × K → K is locally free. In fact, the generators D,D2 of the Lie algebra of R1|1 are µ-related to the linearly independent
2 vector fields DK ,DK , and local freeness follows from the implicit function theorem. 1|1 Now, let us consider the action RT × KT → KT on T -points. We claim that the isotropy subgroup Hx of any x ∈ KT is cyclic. In fact, there is a preorder relation on
Hx induced by the order on R. We pick a minimal element > 0, and claim it generates
Hx. This is because the preorder on Hx is compatible with the group operation, and the same argument that a discrete subgroup of R is cyclic applies. 1|1 Now, choose a basepoint x of K, i.e., a section x: S → K, and let l : S → R>0 be a generator of Hx. Then the map
1|1 id×s 1|1 µ R × S −−→ R × K → K
∼ 1|1 factors through a diffeomorphism Kl = (R × S)/Zl → K.
From the above, we see that an S-family in K is determined, up to isomorphism, by
1|1 1|1 a conjugacy class in RS ; an actual “length” function l : S → R is extra information,
60 determined for instance by a basepoint (i.e., a section of the submersion K → S). In particular, the coarse moduli space of Euclidean supercircles is not a representable supermanifold. We also note that there are full substacks of K associated to each
1|1 conjugation-invariant (generalized) submanifold of R>0. The two most straightforward ∼ 1|1 examples are K1 = pt//(Z/2 n T ), the stack of Euclidean supercircles of length 1, ∼ and Kev = R>0 ×K K, the stack of Euclidean supercircles of purely even length. The latter stack is hardly more general than the former; in fact, since the conjugation
1|1 action of R is trivial on R>0, we have an isomorphism of Lie groupoids
1|1 1|1 (R>0 × Z/2 n R )/Z / R>0 × Z/2 n T
pr1 pr1 id R>0 / R>0
where the domain presents Kev, the codomain is the transport groupoid for the
1|1 trivial action of Z/2 n T on R>0, and the internal functor at the morphism level is determined by the diffeomorphism
−1 −1/2 (l, a, (r0, r1)) 7→ (l, a, (l r0, l r1))
1|1 of R>0×Z/2×R . The fact that this gives a functor (compatibility with compositions)
−1 −1/2 follows from the fact that (r0, r1) 7→ (l r0, l r1) defines a group automorphism of
R1|1 commuting with fl. Thus, we conclude the following:
∼ Theorem 3.10. There is an equivalence Kev = R>0 × K1.
2 A Euclidean vector field DK (respectively, its square DK ) of a supercircle K ∈ KS determines an R1|1-action (respectively, an R-action) on K. Notice that the R-action does not in general descend to a T-action. The above Lie groupoid description of K 1|1 exhibits a second, distinct action on K. Given T -points x = (x0, x1) ∈ RT , t ∈ RT ,
61 1|1 1|1 let us write tx = (tx0, tx1) ∈ RT . Then, given l : S → R>0, the map
1|1 1|1 R × S × R → S × R , (t, s, x) 7→ (s, tl(s) · x)
defines a map T × Kl → Kl (here, tl(s) denotes multiplication of functions, not composition in R1|1). This is easily seen to be a group action, and it is natural in S, so it defines a BT-action on K. The action on each Kl is free: writing the equation kl · x = tl · x, where k ∈ Z, in terms of components, we get
(kl0 + x0 + kl1x1, kl1 + x1) = (tl0 + x0 + tl1x1, tl1 + x1),
thus kl1 = tl1; comparing the first components we get kl0 = tl0 and, since l0 is invertible, t = k represents 0 ∈ TT . Thus, it makes sense to attempt to describe
0|1 the associated orbit space. Let Kl → Σ = S × R be the map induced by the assignment (s, x0, x1) 7→ (s, x1 − x0l1/l0). It is easy to check that it is T-invariant.
0 0 Now, let (s, x0, x1), (s, x0, x1) be two T -points over the same fiber (in Σ). This means
0 0 that x1 − x0l1/l0 = x1 − x0l1/l0. Choosing
x0 − x − x0 x t = 0 0 1 1 , l0 we get tl · x = x0, so the action is fiberwise (in Σ) transitive (to check the claimed
0 identity, note that x1x1 is a multiple of l1.). In short, the above defines a principal
T-bundle structure on K → Σ. Next we notice that the base Σ admits a canonical Euclidean structure provided K is has purely even length. In fact, let r : S → R1|1, so that left multiplication by r induces an isometry
r· Kr−1·l·r / Kl
S × R0|1 / S × R0|1.
62 We can check that the unique dashed arrow fitting the diagram is
(s, x1) 7→ (s, (1 + r1l1/l0)x1 + (r1 − r0l1/l0)),
which is an isometry if the factor in front of x1 equals 1 (for general enough S and arbitrary r, this is only the case if l1 = 0). Since everything we did is natural in S, we can formulate the above constructions as stack maps
K → Btop = {families of connected 0|1-manifolds},
Kev → B = BEucl = {families of connected Euclidean 0|1-manifolds}.
Finally, we turn to the problem of constructing families of Euclidean supercircles.
We start noticing that R0|1 not being a direct factor of R1|1 (as super Lie groups) implies that there is no canonical Euclidean structure on the cartesian product of a Euclidean 1-manifold with a Euclidean 0|1-manifold. Thus we would like to understand exactly what additional data is needed to specify a Euclidean structure on such product. Slightly more generally, we will analyze the case of a principal T-bundle over a Euclidean 0|1-manifold. Let Σ → S be an S-family of Euclidean 0|1-manifolds. The differential form
−θdθ on R0|1 is invariant under Isom(R0|1), and hence defines a canonical fiberwise
0|1 1-form ζ on Σ. Similarly, the vector field ∂θ on R is preserved up to sign, and induces a vertical vector field D0 on Σ, well defined up to sign. Given a principal
T-bundle P → Σ with fiberwise connection ω, it makes sense to require that its curvature coincides with dζ ∈ Ω2(Σ)/Ω≥1(S). The groupoid with such objects and connection-preserving gauge transformations as morphisms will be denoted by PΣ. As Σ varies, they fit into a stack P on the site of B.
Proposition 3.11. For each Σ ∈ B, the fiber of K1 → B at Σ is naturally equivalent
63 to PΣ.
Proof. Given an object of PΣ (a principal T-bundle π : P → Σ with connection form ω), let D be the horizontal lift of D0. We claim D determines a Euclidean structure on the total space P . In fact, we have
hD2, ωi = hD ∧ D, dωi = π∗hD0 ∧ D0, dζi = −1
(the first identity follows from Cartan calculus, and the second is due to the fact that
0 2 D and D are π-related). Thus, D and D are both nowhere vanishing, and span TP/S by dimensional reasons, so D determines a Euclidean structure on P → S. Moreover, connection-preserving gauge transformations of course respect the Euclidean structure.
To see why the functor PΣ → KS thus defined actually lands in K1, suppose that
Σ = S × R0|1 → S and P = S × R0|1 × T → Σ are trivial families. The required curvature condition implies that the connection form ω on P can be written as
ω = dt − θdθ + hdθ where t, θ are the standard coordinates on T respectively R0|1, and h ∈ C∞(S) has odd parity. There is a bundle automorphism of P , described on T -points by (s, θ, t) 7→ (s, θ, t − h(s)θ), under which ω pulls back to the standard dt − θdθ, so every object of PΣ can be represented using the standard connection. To that object, we associate the Euclidean supercircle of length 1. All isometries of a length 1 supercircle
K → S commute with the T-action, and therefore come from connection-preserving gauge transformations (covering some isometry of Σ). Thus the functor PΣ → K1|Σ is full and faithful.
Now, the composition P → B → SM is a stack over SM, and, via the map
K1 → B, the stack K1 → SM is (equivalent to) a stack over B (see theorem 2.2).
64 ∼ The above proposition shows that P = K1 as stacks over B. The next result follows immediately.
Theorem 3.12. The map P → K1 defined above is an equivalence of stacks over SM.
3.4 Integration on supermanifolds
Here, we give a brief and incomplete introduction to the theory of integration on supermanifolds, which we will need in chapter 5. We follow Deligne and Morgan [14, p. 80 ff.] and chapter 7 of the English version of Leites et al. [27].
3.4.1 The Berezin integral
p|q On an open domain U ⊂ R with coordinates (t, θ) = (t1, . . . , tp, θ1, . . . , θq), we define the integral of a compactly supported function u ∈ C∞(U) to be
Z Z Z q(q−1)/2 [dtdθ] u = ∂θ1 ··· ∂θq u(t, θ) dt = (−1) f1,...,1(t) dt. U |U| |U|
P I Here, f1,...,1 is the highest degree coefficient the expression u(t, θ) = θ fI (t), where
q I i1 iq the summation is over all multi-indices I = (i1, . . . , iq) ∈ {0, 1} and θ = θ1 ··· θq .
To make the transition from integration on Rp|q to integration on general superman- ifolds, we need a change of variables formula. It turns out that if φ: U → V is a diffeomorphism between open domains of Rp|q whose reduced part |U| → |V | preserves orientation, then Z Z [dtdθ] u = [dtdθ] Ber(dφ) u ◦ φ. V U
∗ Here, Ber(dφ) is the Berezinian of dφ: TU → φ TV with respect to the canonical bases
∞ ∗ ∂t1 , . . . , ∂θq of the rank p|q free CU -modules TU , φ TV . This means we should interpret
1 ∞ the symbol [dtdθ] as a section of Ber(ΩU ), which we see as a right CU -module, and the change of variables formula tells us that the integral of such sections is invariant
65 under coordinate changes. On a supermanifold X with an orientation of the reduced
1 manifold |X|, we define the integral of compactly supported sections ω of Ber(ΩX ) by requiring that, for any orientation-preserving local chart φ: U ⊂ Rp|q → X with φ∗ω R R ∗ compactly supported, X ω = U φ ω. Using partitions of unity (and linearity), one R can show that the operation ω 7→ X ω is uniquely and consistently defined. We can 1 forgo the choice of an orientation on |X| by twisting Ber(ΩX ) with the orientation bundle or(X). Berezin integration can also be performed along the fibers of a family X → S. In
1 that case, the integrand is a section ω of the relative Berezinian line Ber(ΩX/S). This object is constructed as follows. Locally in X, the submersion X → S looks like a projection onto a direct factor, and we have a splitting
0 → TX/S → TX → TS → 0 (3.8)
of the tangent bundle TX . Dualizing and taking Berezinians (which, analogously to the determinant, turns direct sums into tensor products), we get an isomorphism
∼ −1 Ber(ΩX/S) = Ber(ΩX ) ⊗ Ber(ΩS) . Now, the key observation is that, although the splitting (3.8) is not canonical, the line Ber(ΩX/S) is well defined via the above formula. Writing (locally in X) X = Y × S, the integral of a section
1 ∼ 1 ∞ ω = η ⊗ u ∈ Ber(ΩX/S) = Ber(ΩY ) ⊗ CS , is given by Z Z ω = η u. X/S Y
66 If X is obtained by a base change
φ X / X0
π π0 f S / S0
1 ∞ then for any ω ∈ Ber(ΩX0/S0 ) and u ∈ C (S) we have
Z Z (φ∗ω)(π∗u) = f ∗ ω u. X/S X0/S0
3.4.2 Domains with boundary
1|2 The following is known as Rudakov’s example: on R with coordinates t, θ1, θ2, R we consider a function u with ∂θ1 u = ∂θ2 u = 0. Then [0,1][dtdθ] u = 0, but performing 0 the change of coordinates t = t + θ1θ2 we get
Z Z 0 0 0 [dtdθ] u(t, θ) = [dt dθ] u(t ) + θ1θ2∂tu(t ) = u(1) − u(0). [0,1]×R0|2 [0,1]×R0|2
What this naive calculation shows is that in order to extend Berezin integration to supermanifolds with boundary, additional data is required near the boundary. It turns out that the correct notion of boundary of a domain U in a supermanifold X is a codimension 1|0 submanifold K,→ X whose reduced manifold is the boundary of
|U|. (In the incorrect calculation above, we manipulated the boundary of [0, 1] ⊂ R1|2 as a fermionic codimension 2 submanifold; in fact, our change of coordinates does not restrict to a diffeomorphism of the codimension 1|0 submanifold {0, 1} × R0|2.) Then, one can show that on a neighborhood of a boundary point, there always exists a coordinate system t1, . . . , θq such that the boundary K is singled out by the equation t1 = 0, and t1 < 0 on the interior or the domain. Integration of sections of the Berezinian line with respect to coordinate systems satisfying these conditions is well defined. Using, again, partitions of unity, one extends this definition of integral from
67 sections supported in a chart to sections with compact support in X. The considerations of this subsection are the starting point for a theory of inte- gration on chains, leading to a supermanifold version of the Stokes formula [6], [27, section 7.3 of the English version]. Since we are only interested in a very simple situation—the fundamental theorem of calculus on 1|1-dimensional Euclidean space— we will not discuss the general theory. In any case, it is worthwhile mentioning that differential forms of degree k can be integrated on submanifolds of dimension k|0, but they are not the correct objects to integrate on submanifolds of fermionic codimension 0. In that case, one has to use so-called integral forms.
3.4.3 A primitive integration theory on R1|1
1|1 b 1|1 Given a, b: S → R , we define the superinterval Ia ⊂ S × R to be the domain with boundary prescribed by the embeddings
0|1 1|1 a· 1|1 ia : S × R ,→ S × R −→ S × R ,
0|1 1|1 b· 1|1 ib : S × R ,→ S × R −→ S × R .
We think of ia as the incoming and ib as the outgoing boundary components. To be consistent with the usual definition of 1|1-EBord, we need to assume that, modulo nilpotents, a ≥ b (cf. Hohnhold, Stolz, and Teichner [23, definition 6.41]).
∞ 1|1 b The fiberwise Berezin integral of a function u = f + θg ∈ C (S × R ) on Ia will R b be denoted a [dtdθ] u. Now, notice that we can always find primitives with respect to ∞ the Euclidean vector field D = ∂θ − θ∂t. In fact, if G ∈ C (S × R) satisfies ∂tG = g, then u = D(θf − G).
It is also clear that any two primitives differ by a constant. We have a fundamental theorem of calculus.
68 ∞ 1|1 1|1 Proposition 3.13. Given u, v ∈ C (S×R ) with u = (∂θ−θ∂t)v and a, b: S → R , with a ≥ b modulo nilpotents, we have
Z b [dtdθ] u = v(b) − v(a). a
To clarify the meaning of the right-hand side, when using a, b: S → R1|1, etc., as arguments to a function, we implicitly identify them with S-families of point-points
1|1 S → S × R , to avoid convoluted notation like v(idS, b).
Proof. Using partitions of unity, it suffices to prove the analogous statement for the
b half-unbounded interval I+∞, namely
Z b [dtdθ] u = v(b), +∞ assuming u and v are compactly supported. Writing u = f + θg, we have v = θf − G with G the compactly supported primitive of g. Thus,
v(b) = b1f(b0) − G(b0),
0|1 where b0, b1 are the components of b. On the other hand the embedding ib : S ×R →
1|1 b S × R corresponding to the outgoing boundary of I+∞ is expressed, on T -points, as
(s, θ) 7→ (s, b0 + b1θ, θ + b1).
Thus, the domain of integration is picked out by the equation t ≥ b0 + b1θ. Performing
69 0 the change of coordinates t = t − b0 − b1θ, whose Berezinian is 1, we get
Z Z 0 0 0 [dtdθ] u = [dt dθ] f(t + b0 + b1θ) + θg(t + b0 + b1θ) 0 t≥b0+b1θ t ≥0 Z 0 0 0 = [dt dθ] b1θ∂tf(t + b0) + θg(t + b0) t0≥0
= b1f(b0) − G(b0).
As we noticed in the proof, translations on R1|1 preserve the canonical section [dtdθ] of the Berezinian line; the flips θ 7→ −θ of course do not. Thus, an abstract
1 Euclidean 1|1-manifold X does not come with a canonical section of Ber(ΩX ), but the choice of an Euclidean vector field D fixes a section, which we denote volD. We can then restate the proposition in a coordinate-free way as follows: for any S-family
b of superintervals Ia with a choice of Euclidean vector field D,
Z b volD Du = u(b) − u(a). a
70 CHAPTER 4
ZERO-DIMENSIONAL FIELD THEORIES AND TWISTED DE RHAM COHOMOLOGY
In this chapter, we extend to the case of orbifolds the results of Hohnhold et al. [22] on the relation between 0-dimensional supersymmetric field theories over a manifold and differential forms and de Rham cohomology. This provides, in particular, a field-theoretic description of the twisted delocalized de Rham cohomology of an orbifold, which is isomorphic, via the Chern character, to complexified twisted K- theory. A field-theoretic interpretation of the Chern character itself will be the subject of chapter 6.
4.1 Superpoints and differential forms
0|1 Given any stack X, we can consider the mapping stack ΠT X = FunSM(R , X), which we call the stack of superpoints of X. Our notation stems from the fact that, when X is a manifold, SM(R0|1, X) is represented by the parity-reversed tangent bundle. We will not try to give meaning to the notions of tangent bundle and parity reversal in this more general context, but rather take the above as a definition. We start by noticing that the classes of differentiable stacks and orbifolds are preserved by the functor ΠT .
Proposition 4.1. If the stack X admits the Lie groupoid presentation X1 ⇒ X0, then ΠT X can be presented by the Lie groupoid ΠTX1 ⇒ ΠTX0; in particular, if X is an orbifold, ΠT X is again an orbifold.
71 0|1 0|1 Proof. The groupoid of S-points of FunSM(R , X) is simply FunSM(S × R , X). If S is a contractible supermanifold, then this groupoid is equivalent to
0|1 0|1 SM(S × R ,X1) ⇒ SM(S × R ,X0), which in turn is equivalent to the groupoid of S-points of the stack presented by
ΠTX1 ⇒ ΠTX0. Since all equivalences are natural in S, the result follows.
Our goal now is to extend the discussion of section 3.2.5 to the case of differentiable stacks, casting it in the language of field theories.
Let us denote by Btop(X) the stack of connected 0|1-dimensional manifolds over the orbifold X. A conceptual definition in terms of a certain comma stack can be given, but we can equivalently, and more simply, set the above to mean
0|1 0|1 Btop(X) = FunSM(R , X)//Diff(R ).
Here, we are considering the quotient stack associated to a group action on a stack; see section 2.4 for more details. Similarly, we denote by BEucl(X), or just B(X), the stack of connected Euclidean 0|1-manifolds over the orbifold X, which, for concreteness, we take to mean
0|1 0|1 B(X) = FunSM(R , X)//Isom(R ).
We then define groupoids of topological respectively Euclidean 0|1-twists over X to be
0|1-Tw(X) = FunSM(Btop(X), Vect), 0|1-ETw(X) = FunSM(B(X), Vect), and, for each T ∈ 0|1-Tw(X) or 0|1-ETw(X), the corresponding set of T -twisted topological respectively Euclidean field theories over X to be the space of global
72 sections of T :
T ∞ T ∞ 0|1-TFT (X) = C (Btop(X),T ), 0|1-EFT (X) = C (B(X),T ).
In these definitions, Vect can be the stack of real or complex vector bundles, but ultimately we are interested in the complex case. We recall the construction, in Hohnhold et al. [22, definition 6.2], of the twist
0|1 T1 : Btop(pt) = pt//Diff(R ) → Vect.
This functor is entirely specified by the requirement that the point pt maps to the odd
0|1 ∼ × complex line ΠC, and by a group homomorphism Diff(R ) → GL(0|1) = C ; under 0|1 ∼ 0|1 × the usual identification Diff(R ) = R o R , that is taken to be the projection
× ⊗n onto R . We set Tn = T1 , and use the same notation for the pullback of those line bundles to Btop(X) and B(X) (over the latter stack, Tn only depends on the parity of n).
Theorem 4.2. For any differentiable stack X, there is a natural bijection
Tn ∼ n 0|1-TFT (X) = Ωcl(X)
between Tn-twisted 0|1-TFTs over X and closed differential forms of degree n on X. If X is an orbifold, passing to concordance classes gives an isomorphism with de Rham cohomology
Tn ∼ n 0|1-TFT [X] = HdR(X).
The corresponding statement for Euclidean field theories is the following.
73 Theorem 4.3. For any differentiable stack X, there is a natural bijection
Tn ∼ n¯ 0|1-EFT (X) = Ωcl(X)
between Tn-twisted 0|1-EFTs over X and closed differential forms of parity n¯ on X. If
X is an orbifold, passing to concordance classes gives an isomorphism with Z/2-graded de Rham cohomology
Tn ∼ n¯ 0|1-EFT [X] = HdR(X).
The starting point for the proof is the following observation. If X is a differentiable stack with presentation X1 ⇒ X0, and F a sheaf on X, then there is an equalizer diagram
Γ(X,F ) → F (X0) ⇒ F (X1).
Using the natural isomorphism of sheaves Ω∗(—) and C∞(ΠT —) for manifolds, we get an isomorphism
∗ ∗ ∗ Ω (X) = lim(Ω (X0) ⇒ Ω (X1)) ∼ ∞ ∞ = lim(C (ΠTX0) ⇒ C (ΠTX1)) = C∞(ΠT X).
0|1 ∗ ∼ ∞ R Similarly, one can show that Ωcl(X) = C (ΠT X) , and then argue that closed differential forms on X are concordant if and only if they are cohomologous. Since we are also interested in more general twists, we will give further details in that level of generality.
74 4.2 Superconnections and twists
As usual, we define a vector bundle with superconnection on a stack X to be a fibered functor V : X → VectA; it is flat if it takes values in the substack VectA[ of flat superconnections.
Proposition 4.4. For X a differentiable stack, there is a natural equivalence of groupoids VectA[(X) → 0|1-ETw(X).
0|1 ×i Proof. There exists a bisimplicial manifold {ΠTXj × Isom(R ) }i,j≥0 whose ver- tical structure maps give nerves of Lie groupoids presenting ΠT X × Isom(R0|1)×i and
0|1 whose horizontal structure maps give nerves of presentations of ΠTXj// Isom(R ). Applying Vect, we get a double cosimplicial groupoid
. . O .O O O .O O / / 0|1 (4.1) Vect(ΠTX1) / Vect(ΠTX1 × Isom( )) / ··· O O O O R / / / 0|1 / Vect(ΠTX0) / Vect(ΠTX0 × Isom(R )) / ···
Now we calculate the (homotopy) limit of this diagram in two different ways. Taking the limit of the columns and then the limit of the resulting cosimplicial groupoid, we get, by propositions 2.1 and 2.12,
0|1 → holim FunSM(ΠT X, Vect) ⇒ FunSM(ΠT X × Isom(R ), Vect) → ··· ∼ 0|1 = FunSM(ΠT X// Isom(R ), Vect) = 0|1-ETw(X).
On the other hand, by proposition 2.1, the limit of each row is equivalent to
0|1 A[ 0|1 Vect(ΠTXi// Isom(R )), and the stack map Vect → Vect(ΠT —// Isom(R )) from
75 section 3.2.5 gives us a levelwise equivalence of simplicial groupoids
A[ 0|1 Vect (X•) → Vect(ΠTX•// Isom(R )).
Taking limits, we get an equivalence VectA[(X) → 0|1-ETw(X).
4.3 Concordance of flat sections
The goal of this section is to identify concordance classes of twisted 0|1-EFTs. This is a simple extension of the well-known fact that closed differential forms are concordant through closed forms if and only if they are cohomologous; the extension takes place in two orthogonal directions: we replace manifolds by differentiable stacks and the trivial flat line bundle with an arbitrary flat superconnection. Fix a differentiable stack X with presentation X1 ⇒ X0 and let T ∈ 0|1-ETw(X) be the twist associated to the flat superconnection (V, A) on X.
Proposition 4.5. There are natural bijections
0|1-EFTT (X) ∼= {closed even forms with values in V },
0|1-EFTT ⊗T1 (X) ∼= {closed odd forms with values in V }.
Proof. The vector bundle T : B(X) → Vect determines a sheaf ΓT on B(X), assign- ing to an object f : S → B(X) the complex vector space of sections of f ∗T . The bundle T is specified by a coherent family of objects in the double cosimplicial groupoid (4.1), representing an object in the limit of that diagram, and a global section is specified by a coherent family of sections (see discussion after proposition 2.12 for more details). Similarly, the superconnection determines a sheaf Γ∗ on X whose sections over A A f : S → X are the super vector space of forms in Ω∗(S, f ∗V ) annihilated by A. Global
76 sections of Γ∗ are the super vector space A
∗ X ∗ ∗ ΓA( ) = lim (ΓA(X0) ⇒ ΓA(X1)) .
Now, the data of (V, A) is determined, by hypothesis, by the same coherent family of objects in (4.1) as T . Suppose we are given a coherent family of sections. Individually,
∗ the bottom row of (4.1) specifies an element of Ω (X0,V ) which is invariant under
0|1 ∼ 0|1 the Isom(R ) = R o Z/2-action; this means it is even and closed, i.e., a section of Γ0 (X ). Similarly, the second row by itself specifies a section of Γ0 (X ), and the A 0 A 1 coherence conditions involving vertical maps say these two things determine a section of Γ0 (X). The correspondence between sections of T and Γ0 is clearly bijective. A A
Replacing T with T ⊗ T1 in the above argument amounts to replacing V with its parity reversal ΠV (cf. Hohnhold et al. [22, proposition 6.3]).
Recall that a flat superconnection defines a differential on the space of forms with values in the corresponding vector bundle.
Proposition 4.6. Concordance classes of EFTs are in bijection with cohomology classes:
T ⊗Tn ∼ n¯ ∗ 0|1-EFT [X] = H (Ω (X,V ), A).
Proof. By naturality of the correspondences in the previous proposition, it suffices to show that n¯ X ∼ n¯ ∗ X ΓA( )/concordance = H (Ω ( ,V ), A).
∗ Suppose, first, that the closed forms ω0, ω1 ∈ Ω (X,V ) are cohomologous, i.e., ω1−ω0 =
Aα. Then ∗ ω = ω0 + A(tα) ∈ Ω (X × R,V )
∗ satisfies ij ω = ωj, j = 0, 1. (Here, we use the same notation for an object over X and its pullback via pr1 : X × R → X; as usual, t is the coordinate on R.)
77 ∗ ∗ Conversely, suppose we are given a closed form ω ∈ Ω (X × R,V ) with ωj = ij ω,
∗ j = 0, 1. We need to find a form α ∈ Ω (X,V ) such that ω1 − ω0 = Aα. Schematically, R ∗ ∗ it will be α = − X×[0,1]/X ω. More precisely, we need to define αf ∈ Ω (S, f V ) for each S-point f : S → X. That will be given by the fiberwise integral
Z ∗ αf = − (f × id) ω, S×[0,1]/S which is clearly natural in S. Notice that the vector bundle in which ω takes values comes with a canonical trivialization along the R-direction, so the integral makes sense.
∗ ∗ ∼ ∗ Now, define operators Af = f A ⊗ 1 and dR = 1 ⊗ d on Ω (S × R,V ) = Ω (S,V ) ⊗
∗ ∗ Ω (R). From the derivation property of A, it follows that (f × id) A = Af + dR. Then,
∗ writing ωf = (f × id) ω, we have
Z Z Z ∗ ∗ f A(αf ) = −f A ωf = − Af ωf = dRωf S×[0,1]/S S×[0,1]/S S×[0,1]/S
∗ ∗ = f ω1 − f ω0.
Thus ω1 − ω0 = Aα.
Proof of theorem 4.3. The first claim follows immediately from the above propo- sitions, taking (V, A) to be the trivial complex line bundle with the de Rham differential. The second claim follows from corollary 2.9.
Theorem 4.2 can be proved by adapting the above propositions to the topological context.
78 4.4 Twisted de Rham cohomology for orbifolds
In this section, we review the construction of twisted (delocalized) de Rham cohomology for orbifolds due to Tu and Xu [40], and show that it can, in fact, be interpreted as concordance classes of suitably twisted 0|1-EFTs. In view of the previous section, this is not necessarily surprising; the point here is to give explicit descriptions allowing us to show, in chapter 6, how the relevant twist arises, in a very natural way, from dimensional reduction.
Given an orbifold X and a C×-gerbe X˜ with Dixmier–Douady class α ∈ H3(X, Z), ˜ we fix a presentation X1 ⇒ X0 for X such that X can be represented as a central
˜ 3 X1 T-extension X1 ⇒ X0 with connection θ, curving B and 3-curvature Ω ∈ Ω (X0) (see section 2.2.3 for more details). Then the (Z/2-graded) twisted cohomology groups
∗ with compact support Hc (X, α) are defined to be the cohomology of the complex
∗ 0 0 (Ωc (ΛX,L ), ∇ + 2Ω ∧ ·).
∗ Here, Ωc stands for the Z/2-graded version of the compactly supported de Rham complex, ΛX is the inertia orbifold, and (L0, ∇0) is a line bundle with flat connection we will describe below. (This differs from the definition of Tu and Xu [40, section
3.3] in that we perform the usual trick to convert between Z/2-graded and 2-periodic Z-graded complexes. We have also chosen a more convenient constant in front of Ω, which produces an isomorphic chain complex.) We remark that changing the gerbe with connective structure representing the class α produces a different, but (noncanonically) isomorphic complex; a specific isomorphism between the complexes depends on the choice of isomorphism between the gerbes. While we are at it, it might be enlightening to give a summary of the results in Tu
∞ and Xu’s paper. On the space Cc (X1,L) of compactly supported smooth sections of ˜ the line bundle L associated to X1 → X1, one can define a convolution product and
79 a ∗ operation, and the twisted K-groups K∗+α(X) are defined to be the C∗-algebra
∞ K-theory of a certain completion of Cc (X1,L) [41]. There is a Chern character for K-theory of C∗-algebras (due to Connes and Karoubi) taking values in the periodic
∞ cyclic homology HP ∗(Cc (X1,L)) which yields an isomorphism after tensoring with C. What Tu and Xu in fact do is to show that the twisted de Rham cohomology groups
∞ ∗+α defined above agree with HP ∗(Cc (X1,L)). Thus, K (X) ⊗ C can be described using just differential forms.
0 ˆ ˆ The line bundle L corresponds to the line bundle on the inertia groupoid X1 ⇒ X0 (cf. section 2.2.4) given by
ˆ 0 1. the line bundle on X0 ⊂ X1 (which we abusively denote L ) associated to the ˜ restriction of the principal T-bundle X1 → X1, and
∗ 0 ∗ 0 ˆ 2. isomorphisms s L → t L over X1 given as follows: given an element f ∈ X1 0 0 ˆ ˆ ˜ ˜ inducing a morphism f : (x, g) → (x , g ) in X1 ⇒ X0, pick a lift f ∈ X1 and let f act via the rule 0 ˜ ˜−1 0 v ∈ L(x,g) 7→ fvf ∈ L(x0,g0).
This is clearly independent on the choice of lift f˜.
0 0 ˆ The connection ∇ on L → X0 is simply gotten by restricting the connection on ˜ X1 → X1 (and translating from principal to affine connections). That this defines a connection on the inertia groupoid, i.e., s∗∇0 = t∗∇0, is proven in [40, proposition 3.9]. We note two other important facts, proven in the same place: if the connection on X˜ admits a curving, then ∇0 is flat; moreover, the construction above works for any differentiable stack, but if X is an orbifold then ∇0 is independent on the choice of connection on X˜. Now, the operator ∇0 + 2Ω ∧ · on Ω∗(X,L0) is a flat superconnection on L0 ∈
Vect(ΛX), and therefore gives rise to a twist Tα : B(X) → Vect. The following result is a corollary of proposition 4.6.
80 Theorem 4.7. For every compact orbifold X and α ∈ H3(X, Z), there are natural bijections
0|1-EFTTα [ΛX] ∼= Heven(X, α), 0|1-EFTTα⊗T1 [ΛX] ∼= Hodd(X, α).
To finish this section, let us rephrase the description of (L0, ∇0) in terms of a ˜ Deligne 2-cocycle (h, A, B) on X1 ⇒ X0 representing X (see section 2.3.2 for the definition and notation). Then L0 is topologically trivial and the connection ∇0 is d + A| ; flatness is due to the fact that dA| = (t∗B − s∗B)| = 0 since s = t Xˆ0 Xˆ0 Xˆ0 ˆ ∗ 0 ∗ 0 ∞ × on X0. To describe the isomorphism s L → t L , we use as input h ∈ C (X2, C ), × ˆ and we just need to specify a C -valued function H on X1. Let v = (g, z) ∈ X1 × C ˜ ˜−1 −1 −1 −1 −1 and f = (f, w) ∈ X1 × C. Then f = (f , w h (f, f )), and we find that fv˜ = (fg, zwh(f, g)) and
fv˜ f˜−1 = (fgf −1, zwh(f, g)w−1h(f, f −1)h(fg, f −1))
= (g0, zh(f, g)h−1(f, f −1)h(fg, f −1)).
Thus, using the cocycle condition (2.3) for the triple (g0, f, f −1), we get
f H g → g0 = h(f, g)h−1(f, f −1)h(fg, f −1) (4.2) h(f, g) = . h(fgf −1, f)
81 CHAPTER 5
TWISTS FOR ONE-DIMENSIONAL FIELD THEORIES
Motivated by the conjectural relation between concordance classes of 1|1-EFTs over a (generalized) manifold and K-theory, we should be able to associate, to any K- theory twist, a corresponding twist for 1|1-EFTs. The most geometrically meaningful K-theory twists correspond to degree 3 integral cohomology classes. Such a class can be represented by a PU(H)-bundle (since PU(H) is a K(Z, 2)), and the twisted K-groups are obtained by taking concordance classes of sections of the associated bundle with fibers the space of Fredholm operators on the Hilbert space H, on which PU(H) acts by conjugation [2]. Other models for twisted K-theory take as input a
(C×-banded) gerbe with specified Dixmier–Douady class. When that class is torsion, the twisted K-groups can be described as the Grothendieck group of twisted vector bundles; for the general case, one can use the bundle gerbe to produce a Fredholm bundle and, again, consider concordance classes of sections [11, 18]. There is also an approach based on C∗-algebras [41]. In this chapter, we give a first step towards constructing a twist for 1|1-EFTs over an orbifold associated to a degree 3 integral cohomology class. As input data, we take a gerbe with connective structure over an orbifold X having the specified Dixmier–Douady class, and what we in fact do is to construct a line bundle on the stack of closed, connected 1|1-dimensional bordisms K(X). This is the kind of data one gets by restricting a full twist to the stack of closed connected bordisms, so we will call it a partial twist. The construction is, essentially, a super version of transgression of gerbes on X to line bundles on the free loop stack LX.
82 Our claim that the 1|1-twists from this chapter are the “correct” ones, i.e., are re- lated to the K-theory twists associated to the same degree 3 class, will be substantiated by the dimensional reduction calculations in the next chapter.
5.1 Euclidean supercircles over an orbifold
Let X be a differentiable stack. We denote by K(X) the stack of closed, connected Euclidean 1|1-manifolds over X. An object (K, ψ) of K(X) over S is given by an S- family K of Euclidean supercircles together with a map ψ : K → X, and a morphism (K0, ψ0) → (K, ψ) covering S0 → S is given by a fiberwise isometry F : K0 → K covering S0 → S together with a natural transformation ψ0 → ψ ◦ F ; compositions are performed in the obvious way. In other words, K(X) is the comma stack built out of the stacks K, X, and the “underlying supermanifold” functor K → SM. Our main interest in K(X) is due to the fact that a 1|1-EFT over X determines a function on K(X), the partition function of the theory. This is an immediate consequence of the fact that the empty manifold, being the monoidal unit in the bordism category, is required to map to the vector space C. Similarly, a twist functor determines a vector bundle on K(X) (a line bundle if we restrict to invertible twists, as often done) and a twisted field theory determines a section. To be more concrete, we might want to pick a Lie groupoid presentation X =
(X1 ⇒ X0) for X and represent the map ψ as an X-torsor. Let us spell this out in order to fix the notation. An object of K(X)S is specified by a family of Euclidean supercircles K → S as above together with
1. a submersion π : U → K,
2. an anchor map ψ0 : U → X0,
83 3. an action map µ: U ×X0 X1 → U or, equivalently, a map ψ1 making
ψ1 U ×K U / X1
ψ0 U / X0
an internal functor satisfying the conditions mentioned at the end of section 2.2.1.
We will often denote this object simply ψ. Given a second object
0 0 0 0 0 0 0 0 ψ = (K → S , π : U → K , ψ0, µ ), a morphism λ: ψ → ψ0 in K(X) covering f : S → S0 is given by a fiberwise isometry
0 0 F : K → K over f together with an equivariant map λ0 : U → U covering F and
0 0 compatible with the anchor maps: ψ0 ◦ λ0 = ψ0 (when π = π, this can be interpreted as a natural transformation between the corresponding internal functors; this is also explained at the end of section 2.2.1). Now, assume X is an orbifold, so that π : U → K is étale. To perform the constructions of the next section, certain choices of endpoints in U will be needed; this leads to a “fatter” prestack modeling K(X). In this section we will denote the new model K0(X), but later on we will stop distinguishing notationally between the two models, and mostly use the new one. An S-point of K0(X) is given by the same data
ψ = (K, π, ψ0, µ) ∈ K(X)S as above, together with the following additional data:
1. A collection Ii, 1 ≤ i ≤ n, of S-families of superintervals, as defined in sec- tion 3.4.3. We denote the inclusion of the outgoing and incoming boundary components by ιout ιin 0|1 i 1|1 i 0|1 S × R ,→ S × R ←-S × R .
It is not required that these intervals have strictly positive length.
84 2. For each i, an embedding Ii ,→ U, by which we mean a Euclidean map from a
out in 1|1 neighborhood of the “core” [ιi , ιi ] ⊂ R of Ii into U.
Writing
out 0|1 ιi 1|1 bi : S,→ S × R −−→ S × R ,→ U, in 0|1 ιi 1|1 ai : S,→ S × R −→ S × R ,→ U,
we require that the maps π ◦ bi and π ◦ ai+1 : S → K coincide; we call the common value ci (here and in what follows, subindices are read cyclically, so the above also includes the requirement that π ◦ bn = π ◦ a0). This means, intuitively, that the π superintervals Ii ,→ U −→ K define a triangulation of K. (A triangulation of K can be defined, in precise terms, to be an appropriate expression of K as a composition of right elbows and thin left elbows in the bordism category 1|1-EBord.)
The data above will be called a skeleton for the groupoid U ×K U ⇒ U. We 0 will use the shorthand notation I = {Ii} to refer to it, and (ψ, I) ∈ K (X) to refer
0 to a typical object in K (X). Notice that the S-points bi−1 and ai are isomorphic in the groupoid (U ×K U)S ⇒ US, since their images in KS agree; we denote by ji : S → U ×K U the unique morphism bi−1 → ai. A morphism in K0(X) is just the data of a morphism in K(X), after forgetting the skeleton. Thus, it is clear that the forgetful map K0(X) → K(X) gives an equivalence of stacks. Evidently, we can form a pullback of (ψ, I) ∈ K0(X) via a map f : S0 → S by simply choosing a cartesian morphism λ: ψ0 → ψ covering f in K(X) and any skeleton for ψ0. However, we note that there is a canonical choice to be made. Namely, pick
85 0 0 0|1 0 ci : S × R → K the unique map such that
0 F / KO KO 0 ci ci f S / S
0 0 commutes; similarly, there are canonical choices for ai and bi. We denote the corre-
0 ∗ 0 ∗ sponding skeleton I by λ I, ci by λ ci, etc. Triangulations satisfying the condition expressed by the diagram above will be called compatible. Finally, let us explain what we mean by a refinement of a skeleton. Suppose we have ψ ∈ K(X) and
0 0 I = {Ii}i∈I ,I = {Ii}i∈I0 two skeletons. Then we say that I0 is refinement of I if there is a surjective map
0 −1 0 0 r : I → I, such that, for each i ∈ I, r (i) indexes a collection Ii1 ,...,Iin ⊂ U where b0 = a0 for each 1 ≤ k < n and a = a0 , b = b0 ; in words, the I0 , ik ik+1 i1 i in i ik
1 ≤ k ≤ n, are adjacent subintervals whose concatenation is precisely Ii. We denote
I 0 by RI0 : (ψ, I ) → (ψ, I) the morphism having the identity as its underlying torsor map.
5.2 Gerbes and partial twists
Let X be an orbifold and X˜ → X be a gerbe with connective structure. For us, such object will always be presented by a Čech cocycle in Deligne cohomology with respect to a groupoid presentation of X, as in section 2.3.2. From that data, Lupercio and Uribe [30] construct a line bundle with connection on the loop orbifold LX. In this section, we describe a super analogue of this transgression procedure; namely, we produce, from the same input data, a complex line bundle L: K(X) → Vect.
86 5.2.1 A super version of transgression
Let us start fixing some notation. The orbifold X will be presented by an étale ˜ Lie groupoid X1 ⇒ X0, and the gerbe X → X will be presented by a Čech 2-cocycle for groupoid cohomology with coefficients in the Deligne complex C×(3),
× d log 1 2 C −−→ Ω → Ω .
More explicitly, this cocycle is given by a triple
∞ × 1 2 (h, A, B) ∈ C (X2, C ) × ΩC(X1) × ΩC(X0) satisfying the cocycle conditions
−1 −1 ∞ × h(a, b)h(a, bc) h(ab, c)h(b, c) = 1 in C (X3, C ), (5.1)
∗ ∗ ∗ 1 pr2 A + pr1 A − c A = d log h in Ω (X2), (5.2)
∗ ∗ 2 t B − s B = dA in Ω (X1), (5.3)
where Xn = X1 ×X0 · · · ×X0 X1 is the space of sequences of n composable morphisms. It suffices to discuss the effect of the fibered functor L: K(X) → Vect on S-points for contractible S. To an object (ψ, I) ∈ K(X)S as above, it assigns the trivial line bundle over S; the interesting discussion, of course, concerns morphisms. Fix a second
0 0 0 0 0 object (ψ ,I ) ∈ K(X)S0 and a morphism λ: (ψ, I) → (ψ ,I ) over f : S → S . To that, we need to assign a linear map between the corresponding lines, which is just a map
L(λ): S → C×. We start with two special cases. First, if λ is a refinement of skeletons, we let L(λ) = 1 . Second, suppose the triangulations of K, K0 are compatible, that
∗ 0 is, they have the same cardinality and ci = λ ci for all i. This means that the
∗ 0 endpoints of the superintervals Ii, λ Ii ⊂ U are (uniquely) isomorphic in the groupoid
87 ˜ U ×K U ⇒ U, and we denote by a˜i, bi ∈ (U ×K U)S the corresponding morphisms ∗ 0 ∗ 0 ˜ ai → λ ai, bi → λ bi. Note that a˜i, bi are the endpoints of the superintervals
∗ 0 Ji = Ii ×K λ Ii ⊂ U ×K U.
With this notation in hand, we define L(λ) ∈ C∞(S, C×) to be
Z ! ∗ X ∗ Y ψ2h(˜ai, ji) L(λ) = exp volD hD, ψ Ai . (5.4) 1 ∗ ∗ 0 ˜ 1≤i≤n Ji 1≤i≤n ψ2h(λ ji, bi−1)
Here, D ∈ C∞(TU) a choice of Euclidean vector field for the Euclidean structure induced by π : U → K, and volD the corresponding volume form (cf. section 3.4.3).
Moreover, ψ2 : U ×K U ×K U → X2 denotes the map induced by ψ1.
Proposition 5.1. The prescriptions above uniquely determine a fibered functor L: K(X) → Vect.
Proof. Initially, we will assume we can pick compatible refinements for (families of) triangulations of supercircles whenever needed, and explain at the end of the proof how to deal with the fact that such refinements do not always exist.
0 0 Fix a morphism λ: (ψ, I) → (ψ ,I ) in K(X)S and assume we can choose compatible refinements for the triangulations of K and K0. This yields refinements I¯, I¯0 of I respectively I0. We can then express λ as the composition
¯ ¯0 RI ¯ RI (ψ, I) o I (ψ, I¯) λ / (ψ0, I¯0) I0 / (ψ0,I0) _ _ _ _
f S S / S0 S0 where λ¯ is the morphism in K(X) corresponding to the same torsor map as λ, but relating objects with different skeletons. That forces us to have L(λ) = L(λ¯). Of course, we need to check that taking this as a definition for L(λ) is consistent,
88 that is, independent on the choice of I¯ and I¯0. Verifying this in the case when all triangulations involved admit compatible refinements boils down to checking that if the original triangulations of K and K0 were already compatible, applying formula (5.4) would give L(λ) = L(λ¯). But this is easy to see. When calculating L(λ¯), all additional terms under the product sign involve h(1,... ) or h(..., 1) in both the numerator and denominator and, since we pick h to be a normalized cocycle, this does not change the product. Moreover, for each i ∈ I, we have, by additivity of the Berezin integral,
Z Z ∗ X ∗ volD hD, ψ1Ai = volD hD, ψ1Ai, ¯ Ji j∈r−1(i) Jj so the summation inside the exponential also remains unchanged. Next, we verify that L respects compositions, at least when compatible refinements can be chosen. So let us fix composable morphisms as in the diagram below; we need to show that L(λ00) = L(λ)f ∗L(λ0).
λ00 0 % (ψ, I) λ / (ψ0,I0) λ / (ψ00,I00) _ _ _
f f 0 / 0 / 00 S S 8 S f 00
All the notation fixed in the paragraph preceding (5.4) applies to λ, and we add one or two primes when referring to the corresponding constructs for λ0 respectively λ00. We may assume without loss of generality that the triangulations of K,K0,K00 are all compatible. Thus, the J’s are morphisms
∗ 0 0 0 0 ∗ 00 0 0 0 Ji : Ii → λ Ii in U ×K U ⇒ U, Ji : Ii → (λ ) Ii in U ×K0 U ⇒ U ,
00 00 ∗ 00 Ji : Ii → (λ ) Ii in U ×K U ⇒ U,
89 and we have the identity
00 ∗ 0 Ji = λ Ji ◦ Ji in U ×K U ⇒ U.
∗ Let us denote Hi ⊂ U ×K U ×K U the unique superinterval satisfying pr23 Ji = Hi,
∗ 00 ∗ ∗ 0 ∗ 0 pr13 Ji = Hi, pr12(λ Ji) = Hi. When calculating f L(λ ) using (5.4), the integrals under summation are
Z ! Z Z ∗ 0 ∗ ∗ ∗ ∗ f volD hD, (ψ1) Ai = volD hD, ψ1Ai = volD hD, ψ2 pr1 Ai. 0 ∗ 0 Ji λ Ji Hi
Similarly, the ith integral in the calculation of L(λ) respectively L(λ00) can be written as Z Z ∗ ∗ ∗ ∗ volD hD, ψ2 pr2 Ai, volD hD, ψ2c Ai. Hi Hi Therefore, when calculating L(λ)f ∗(L(λ0))L(λ00)−1, the exponentials in (5.4) contribute
Z ! X ∗ ∗ ∗ ∗ exp volD hD, ψ2(pr1 A + pr2 A − c A)i 1≤i≤n Hi Z ! X ∗ = exp volD hD, ψ2d log hi H 1≤i≤n i (5.5) Y h(ψ2(outgoing boundary of Hi)) = h(ψ (incoming boundary of H )) 1≤i≤n 2 i ∗ ∗˜0 ˜ Y ψ h(λ b , bi) = 2 i . ψ∗h(λ∗a˜0 , a˜ ) 1≤i≤n 2 i i
(The first identity uses (5.2), and the second uses proposition 3.13) The contribution of the terms under the product sign for L(λ)f ∗(L(λ0))L(λ00)−1 is
−1 ∗ ∗ ∗ 0 ∗ 0 ∗ 00 ! Y ψ2h(˜ai, ji) ψ2h(λ a˜i, λ ji) ψ2h(˜ai , ji) ∗ ∗ 0 ˜ ∗ 00 ∗ 00 ∗˜0 ∗ 00 ∗ 00 ˜00 1≤i≤n ψ2h(λ ji, bi−1) ψ2h((λ ) ji , λ bi−1) ψ2h((λ ) ji , bi−1)
90 ∗ 0 and using property (5.1) on the two terms involving λ ji, we can rewrite this as
−1 ∗ ∗ ∗ 0 ∗ 00 ! Y ψ h(˜ai, ji) ψ h(λ a˜ , a˜i ◦ ji) ψ h(˜a , ji) 2 2 i 2 i . ∗ 00 ∗ 00 ∗˜0 ˜ ∗ 00 ∗ 00 ∗˜0 ∗ 00 ∗ 00 ˜00 1≤i≤n ψ2h((λ ) ji ◦ λ bi−1, bi−1) ψ2h((λ ) ji , λ bi−1) ψ2h((λ ) ji , bi−1)
Multiplying with (5.5), and again using property (5.1), this time for the triples
∗ 0 00 ∗ 00 ∗˜0 ˜ (λ a˜i, a˜i, ji) and ((λ ) ji , λ bi−1, bi−1), we get
L(λ)f ∗(L(λ0))L(λ00)−1 = 1 as required. Now, suppose we have a morphism λ: (ψ, I) → (ψ0,I0) where compatible refine- ments of the underlying triangulations fail to exist. Since every morphism in K(X) can be expressed as the composition of a morphism covering the identity and a morphism involving pullback skeletons, it suffices to consider the case when λ covers id: S → S.
∗ 0 This means that there is a pair ci, λ ci : S → K “crossing over” one another; more
∗ 0 precisely, in an Euclidean local chart, neither (ci)red ≤ (λ ci)red nor the opposite holds. It suffices to define L(λ) in a small neighborhood Sp of each point p ∈ S
∗ 0 where that happens; assuming, for simplicity, that the triangulations {cj}, {λ cj} are identical except for the problematic index i, it suffices to analyze the situation
∗ 0 in a small neighborhood in K of the point x = ci(p) = λ ci(p). Then we can choose
1 ∗ 0 d : Sp → K|Sp sufficiently close to ci satisfying either d(p) < ci(p), λ ci(p) or the opposite inequality. Denote by I1, I10 the modifications of I, I0 obtained by replacing
∗ 0 1 1 0 ci, λ ci with d . Then of course I and I admit a common refinement, and so do I and I10; moreover, I1 and I10 are based on the same triangulation of K. We have a commutative square 1 (ψ, I1) λ / (ψ, I10) (5.6) (ψ, I) λ / (ψ0,I0),
91 and this stipulates the value of L on λ: (ψ, I) → (ψ0,I0); here, the unlabeled arrows refer to morphisms whose underlying torsor maps are the identity. We need to see why this is independent on the choice of d1. Suppose we repeat the above procedure using
2 3 a different choice d : Sp → K|Sp ; to compare them, we can use a third d : Sp → K|Sp
1 (restricting, perhaps, to a smaller neighborhood Sp) which stays away from both d and d2. Thus, we can assume without loss of generality that d1 and d2 stay away from one another. We have a commutative diagram
1 (ψ, I1) λ / (ψ0,I10) u ) (ψ, I) (ψ0,I0) i 5 2 (ψ, I2) λ / (ψ0,I20) where the skeletons appearing in each triangle and the in middle square admit common refinements, and it follows that L(λ), as prescribed by (5.6), is independent on the choice of d1. Similarly, we can reduce the verification that L respects composition of morphisms to the case where all triangulations involved admit compatible refinements.
Proposition 5.2. The dependence of L on the Deligne cocycle (h, A, B) is natural, and therefore its isomorphism class only depends on the isomorphism class of the gerbe with connective structure X˜.
Proof. Since the assignment (h, A, B) 7→ L is clearly monoidal, it suffices to assume we have a trivialization
(h, A, B) = (d + δ)(g, F ) = (δg, d log g + δF, dF ), where (g, F ) ∈ C∞(X , ×) × Ω1 (X ), and produce from that a trivialization of L. 1 C C 0
92 Let us calculate L(λ). Each integral in (5.4) contributes
∗ R ∗ Z ˜ exp ∗ 0 volD hD, ψ0F i ∗ ψ1g(bi) λ Ii exp volD hD, ψ1(d log g + δF )i = ∗ R ∗ , ψ g(˜ai) exp volD hD, ψ F i Ji 1 Ii 0 while the terms under the product sign contribute
∗ ∗ −1 ∗ Y ψ g(˜ai)ψ g(˜ai ◦ ji) ψ g(ji) 1 1 1 . ∗ ∗ 0 ∗ ∗ 0 ˜ −1 ∗ ˜ 1≤i≤n ψ1g(λ ji)ψ1g(λ ji ◦ bi−1) ψ1g(bi−1)
The middle terms in the numerator and denominator cancel out, and putting both things together, we get
ψ∗g(λ∗j0)−1 exp R vol hD, ψ∗F i Y 1 i λ∗I0 D 0 L(λ) = i = f ∗s(ψ0,I0)s(ψ, I)−1. ∗ −1 R ∗ ψ g(ji) exp volD hD, ψ F i 1≤i≤n 1 Ii 0
Here, Z ! X ∗ Y ∗ −1 s(ψ, I) = exp volD hD, ψ0F i ψ1g(ji) 1≤i≤n Ii 1≤i≤n defines a nowhere vanishing section of L, thus a trivialization. This is clearly natural with respect to composition of trivializations.
5.2.2 The restriction to K1(X)
The usual transgression of gerbes with connective structure produces, in fact, a Diff+(S1)-equivariant line bundle on the loop space with an invariant connection [12, proposition 6.2.3]. Our construction above gives a line bundle on the moduli stack of supercircles over X, and not just on a “super loop space”, so the super analogue of Diff+(S1)-equivariance is automatically built into our discussion. (Incidentally, we would like to remark that, because Lupercio and Uribe work with one triangulation at a time instead of assembling all triangulations into a stack as we did here, the Diff+(S1)-equivariance of their construction is not manifest, at least for us.) In this
93 subsection, we study in more detail this aspect over the substack
∼ 1|1 1|1 K1(X) = FunSM(T , X)// Isom(T ).
1|1 We write SLX = FunSM(T , X) for the (free) super loop space. Our first goal is to construct a connection on the restriction L: SLX → Vect. As a prestack model for the super loop space, we use an obvious modification of the
0 prestack K (X) of section 5.1: an object in SL(X)S is exactly like above, with trivial
K = S × T1|1; morphisms are like above, with the requirement that F : K → K0 is the identity map. We can in fact pass to a slightly simpler prestack model for SLX where we require that endpoints of the triangulation are induced by constant maps
S → T1|1.
Suppose we have an object (ψ, I) ∈ SL(X)S. We want to describe a connection on the trivial line bundle over S, so it suffices to specify the Christoffel symbol ∆ = ∆ ∈ Ω1 (S). Given a vector field v ∈ C∞(TS), we set (ψ,I) C
Z X ∗ ∗ ∗ hv, ∆i = ji hv, ψ1Ai − volD hD ∧ v, ψ0Bi. (5.7) 1≤i≤n Ii
On the right-hand side, v denotes the vector fields on U respectively U ×K U related to the vector field v ⊗ 1 on S × T1|1.
Proposition 5.3. The above prescription determines a connection on L: SLX → Vect.
Proof. Suppose we have a morphism λ: (ψ, I) → (ψ0,I0) covering f : S → S0, and denote by ∆, ∆0 the corresponding Christoffel symbols. We need to check that
− d log L(λ) = f ∗∆0 − ∆. (5.8)
If λ: ψ → ψ0 has the identity as its underlying torsor map and I a refinement of
94 I0, then it follows from additivity of the integrals and the fact that A is normalized that ∆0 = ∆. Next, we consider the case where the triangulations of K and K0 are compatible, fixing the same notations as in the previous subsection. The contribution of the terms under the product sign in (5.4) to −d log L(λ) is
X ∗ ∗ ∗ 0 ˜ − ψ2d log h(˜ai, ji) − ψ2d log h(λ ji, bi−1) 1≤i≤n
Using (5.2) and deducting the contribution of the terms involving A in (5.7) to f ∗∆0 − ∆, we get
X ∗ ∗ ∗ ∗ ∗ 0 ˜ ∗ ∗ ˜∗ ∗ − a˜i ψ1A − (˜ai ◦ ji) ψ1A + (λ ji ◦ bi−1) ψ1A − bi−1ψ1A 1≤i≤n ˜ bi X ∗ ∗ ˜∗ ∗ X ∗ = − a˜i ψ1A − bi ψ1A = ψ1A a˜ 1≤i≤n 1≤i≤n i
The contribution of the integrals in (5.7) to f ∗∆0 − ∆ is
Z X ∗ ∗ ∗ ∗ − volD hD ∧ v, t ψ0B − s ψ0Bi 1≤i≤n Ji Z Z X ∗ ∗ ∗ X ∗ = − volD hD ∧ v, ψ1(t B − s B)i = − volD hD ∧ v, ψ1dAi 1≤i≤n Ji 1≤i≤n Ji
Therefore, to establish (5.8), we need to show that, for each 1 ≤ i ≤ n,
Z ˜b Z ∗ ∗ i ∗ −Lv volD hD, ψ1Ai + hv, ψ1Ai = − volD hD ∧ v, ψ1dAi. a˜ Ji i Ji
To simplify the notation, we check this identity in a separate lemma.
b Lemma 5.4. Let Ia be an S-family of superintervals with Euclidean vector field D, and
b v a vector field commuting with D. Assume also that Ia pulls back from a pt-family
95 [a, b] ⊂ R1|1. Then for any A ∈ Ω1(U),
Z b b Z b
Lv volD iDA = ivA + volD iDivdA. a a a
Proof. We can move Lv inside the integral without generating boundary terms; this adds a sign (−1)p(v). On the other hand,
p(v) (−1) LviDA = iDLvA = iDivdA + iDdivA = iDivdA + LDivA.
Integrating and using proposition 3.13 gives the result.
We omit the calculations showing that the assignment (h, A, B) 7→ ∆ is natural, i.e., only depends, up to a specified isomorphism, on the cohomology class of (h, A, B).
Proposition 5.5. The two R1|1-actions on the line bundle L: SLX → Vect related to the R1|1-action by rotations on the base, namely
1. the one arising from parallel transport along the orbits of the R1|1-action on SLX and
2. the one arising from L being the restriction of a line bundle on K1(X), in fact coincide. In particular, the connection has trivial holonomy around the orbits.
Proof. The exact meaning of the proposition is that the corresponding statements are true for one S at a time. The orbit of the S-point (K, ψ) of SLX is the R1|1 × S- point ψ˜ consisting of the data summarized in the diagram below, where µ denotes the
96 action map R1|1 × K = S × R1|1 × T1|1 → K = S × T1|1.
1|1 ψ1 (R × K) ×K (U ×K U) / U ×K U / X1
1|1 ψ0 (R × K) ×K U / U / X0
π µ R1|1 × K / K
pr2 R1|1 × S / S
Rewriting the fiber products more simply, this picture becomes diffeomorphic to
1|1 ψ1◦pr2 R × U ×K U / X1
1|1 ψ0◦pr2 R × U / X0
µ−1◦π R1|1 × K
R1|1 × S.
Moreover, the skeleton induced on R1|1 × U is just the pullback of the original one
1|1 via pr2 : R × S → S.
1|1 To the above family, is assigned the trivial line bundle Lψ˜ over S × R . Now, the R1|1-equivariance information coming from the fact that L is a pullback via
SLX → K1(X) gives us a second trivialization of Lψ˜; we claim these two trivializations
1|1 agree. To see this, pick a map ir = (r, id): S → R × S. Then the original ψ and its ∗ ˜ ∗ ˜ rotation by r units can be identified with i0ψ and irψ, and these two S-families are
97 related by a morphism λ whose underlying torsor map is
U id / U
r−1◦π π K r / K.
Thus, L(λ) = 1.
On the other hand, the connection forms on the trivial line bundles Lψ˜ and Lψ
∗ satisfy a simple relation: ∆ψ˜ = pr2 ∆ψ. In particular, parallel sections of Lψ˜ along the R1|1-direction are precisely the fiberwise constant sections.
It is not clear to us whether or not the connection on K1(X) extends to a connection on K(X). We notice that the vector field v ∧ D on S × R1|1 does not determine a
1|1 vector field on Kl = (S × R )/Zl unless v(l) = 0.
1|1 Proposition 5.6. Let v be a vector field on S and l : S → R>0. Then the pushforward of v, thought of as a vector field on S × R1|1, via the translation by l units in S × R1|1 is
2 v + v(l0)D + v(l1)D.
Proof. Along the composition
1|1 (l,id) 1|1 1|1 µ 1|1 φ: S × R −−→ R × R → R , the standard coordinate functions pull back as follows:
t 7→ t1 + t2 + θ1θ2 7→ l0 + t + l1θ,
θ 7→ θ1 + θ2 7→ l1 + θ.
Here, t1 = t ⊗ 1, t2 = 1 ⊗ t, and similarly for θ, and each component li of l is identified
98 with li ⊗ 1. Thus,
∗ ∗ v(φ t) = v(l0) + v(l1)θ2, v(φ θ) = v(l1), which agrees with the action of the above vector field on t and θ.
99 CHAPTER 6
DIMENSIONAL REDUCTION AND THE CHERN CHARACTER FOR ORBIFOLDS
In this chapter, we describe a series of functors between bordism categories related to a stack X
0|1-EBord(ΛX) ← 0|1-EBordT(ΛX) → 0|1-EBordR//Z(ΛX) → 1|1-EBord(X) which implements our notion of dimensional reduction. We then specialize to the case where X is an orbifold and give two applications. First, we give a field-theoretic interpretation for the simplest instance of orbifold Chern character, namely the one concerning untwisted cohomology of global quotients. Second, we show that the partial twists we constructed in chapter 5 indeed have the expected dimensional reduction.
6.1 Dimensional reduction
Since 0|1-dimensional bordism categories have {∅} as their stack of objects, it suffices to discuss the above functors of symmetric monoidal internal categories in terms of the corresponding substacks of (fiberwise) closed and connected families in each stack of morphisms; slightly abusively, we still call those “bordism stacks”. To 0|1-EBord(ΛX) corresponds the stack B(ΛX) from chapter 4, and to 1|1-EBord(X) corresponds the stack K(X) from chapter 5. The two middle bordism categories, or
100 rather the corresponding stacks of closed connected bordisms, as well as the maps
B(ΛX) ← BT(ΛX) → BR//Z(ΛX) → K(X). relating them, will be defined in the ensuing subsections. The first map from left to right is full and essentially surjective, and thus induces bijections on sections of sheaves (and exists for an arbitrary stack Y in place of ΛX). The corresponding statement for field theories is that we have a bijection between the sets of (twisted) field theories with related twists. The second map is an equivalence, but has an obscure inverse. The third map admits a simple and natural geometric description. We will refer to the two middle stacks as the stacks of T-equivariant and R//Z-equivariant bordisms over ΛX. At the end of this section, we specialize these constructions to the case where X = X//G is a global quotient by a finite group, which will hopefully illustrate the ideas. Following the physics jargon, restriction of 1|1-EFTs (or just functions on K(X)) to 0|1-EFTs via the above maps of bordism stacks will be referred to as dimensional reduction. Our motivation for doing this is that the stack K(X) of Euclidean super- circles over X is not differentiable (unless X = pt), since it is, so to speak, infinite dimensional, and therefore unwieldy to analysis; dimensional reduction allows us to probe its geometry by means of 0|1-dimensional gadgets over X.
6.1.1 R//Z-equivariant bordisms
Let ΛX be the inertia of a stack in SM. It is naturally endowed with a pt//Z-action, and hence an induced R//Z-action. We then define a stack BR//Z(ΛX) where an object over S is given by the following data:
1. a family Σ → S of connected Euclidean 0|1-manifolds,
2. a principal T-bundle P → Σ with a fiberwise connection ω whose curvature
101 agrees with the tautological (fiberwise) 2-form on Σ, and
3. an R//Z-equivariant map ψ : P → ΛX, with equivariance datum ρ, where R//Z acts on P via the usual homomorphism R//Z → T.
(Recall that R//Z-equivariance is not just a condition on ψ, but rather extra data encoded by the natural transformation ρ, see section 2.4). We will usually denote this object (Σ, P, ψ, ρ) or, diagrammatically,
ψ P / ΛX R//Z
Σ.
A morphism (Σ0,P 0, ψ0, ρ0) → (Σ, P, ψ, ρ) covering a map of supermanifolds S0 → S is given by
1. a fiberwise isometry F :Σ0 → Σ covering S0 → S,
2. a connection-preserving bundle map Φ: P 0 → P covering F , and
3. a natural transformation ξ : ψ0 ⇒ ψ ◦ Φ compatible with the equivariance data ρ0, ρ.
Compositions are performed as suggested by the geometry.
Now we discuss the map ι: BR//Z(ΛX) → K(X). An object (Σ, P, ψ, ρ) over S is mapped to the supercircle over X consisting of
1. the family of 1|1-dimensional manifolds P endowed with the fiberwise Euclidean structure determined by ω (see proposition 3.11), and
2. the map P → X obtained by composing ψ with the forgetful map ΛX → X.
Notice that this construction forgets the T-action on P as well as the equivariance datum ρ. To define ι at the level of morphisms, recall from section 3.3.1 that a
102 connection-preserving bundle map P 0 → P covering a fiberwise isometry Σ0 → Σ is a fiberwise (over S) isometry with respect to the Euclidean structures on P 0, P .
6.1.2 T-equivariant bordisms
For any stack X, we define BT(X) to be the stack whose S-points are given by an S-family of connected Euclidean 0|1-manifolds Σ → S together with two pieces of data:
1. a principal T-bundle P → Σ with a fiberwise connection ω whose curvature agrees with the tautological 2-form on Σ,
2. a map ψ :Σ → X.
Morphisms between two objects (Σ0,P 0, ψ0) and (Σ, P, ψ) over f : S0 → S consist of a fiberwise isometry F : Σ0 → Σ covering f, a connection-preserving bundle map Φ: P 0 → P covering F , and a natural transformation ξ : ψ0 ⇒ ψ ◦ F . Compositions are performed as suggested by the geometry. The data (1) and (2) are completely unrelated in the sense that
T ∼ T B (X) = B ×B B(X), and in particular the map BT(X) → B(X) we are interested in is simply the projection onto the second component. Our interest in BT(X) is due to the fact that it admits a straightforward quotient stack presentation. Recall that T1|1 = R1|1/Z is the (length 1) super circle group.
Proposition 6.1. There is an equivalence of stacks
1|1 ΠT X// Isom(T ) → BT(X),
103 where the action of Isom(T1|1) on ΠT X is through the quotient
1|1 1|1 0|1 0|1 π : Isom(T ) = T o Z/2 → R o Z/2 = Isom(R )
Proof. Given f : S → ΠT X, we construct an S-point of BT(X) by letting Σ and P be trivial, the connection form on P be the standard one, namely ω = dt − θdθ,
0|1 0 and ψ : Σ = S × R → X be the adjoint to f. A morphism ξ : f → f in (ΠT X)S prescribes a morphism in BT(X) in the obvious way, and this determines a map
F :ΠT X → BT(X). Next, we build a 2-morphism between the two compositions
1|1 ΠT X × Isom(T ) ⇒ ΠT X → BT(X).
1|1 Given f ∈ ΠT XS as above and Φ ∈ Isom(T )S, we get a diagram
Φ S × T1|1 / S × T1|1
π(Φ) ψ S × R0|1 / S × R0|1 / X
S S.
Ignoring the left column we get the data of F (f), and ignoring the right column we get the data of F (f · Φ); the diagram gives us a morphism between them. Naturality in f and S is clear, and so are the additional compatibility conditions that we need in order to specify an object in the 2-limit of
T 1|1 T → FunSM ΠT X, B (X) ⇒ FunSM ΠT X × Isom(T ), B (X) → ···
Using proposition 2.12, this specifies the map ΠT X// Isom(T1|1) → BT(X).
104 T For contractible S, it is clear that any object in B (X)S is equivalent to the one associated to some f as above, so our map of stacks is essentially surjective. Moreover, any morphism in BT(X) between the images of f, f 0 is uniquely prescribed by some ξ : f → f 0 and Φ as above, so the map is full and faithful.
Under the identification of the proposition, the map BT(X) → B(X) becomes the natural map
1|1 0|1 ΠT X//(T o Z/2) → ΠT X//(R o Z/2) induced by the quotient π : T1|1 o Z/2 → R0|1 o Z/2. It follows immediately that this map induces an isomorphism on functions, and more generally on global sections of any sheaf over SM.
6.1.3 The map BT(ΛX) → BR//Z(ΛX)
Let us denote by α the canonical automorphism of the identity of ΛX. It suffices to describe the restriction of the desired map BT(ΛX) → BR//Z(ΛX) to the full prestack
T of trivial objects. To an object (Σ, P, ψ) ∈ B (ΛX)S with
0|1 1|1 Σ = S × R ,P = S × T , ψ :Σ → ΛX,
and the standard Euclidean structures, we want to assign an object (Σ, P, ψ! : P →
R//Z 1|1 ΛX, ρ) ∈ B (X)S. Consider the covering U = S × R → P . Our goal is to descend ˜ ψ : U → ΛX, the pullback of ψ via U → Σ, to a map ψ! : P → ΛX.
ψ˜ ψ & / / / U P Σ Λ: X
ψ!
In order to do that, we need to provide certain isomorphisms over double overlaps and then check a coherence condition on triple overlaps. Denote by pr1, pr2 : U ×P U ⇒ U
105 ˜ ˜ the projections. Then we are looking for a 2-morphism α˜ : ψ ◦ pr1 ⇒ ψ ◦ pr2 (whose domain and codomain happen to be the same map, henceforth denoted ψ ◦ pr). Note that U ×P U breaks up as a disjoint union indexed by Z, where the nth component comprises pairs of the form (x, n·x). On that component, we set α˜ to be the horizontal composition (whiskering) α˜ = αn ◦ (ψ ◦ pr).
Regarding the coherence condition, we need to check that
∗ ∗ ∗ pr13 α˜ = pr23 α˜ ◦ pr12 α,˜ (6.1)
where prij denotes the projection U ×P U ×P U → U ×p U forgetting the third index.
The threefold fiber product breaks up as a disjoint union indexed by Z × Z, where the component (n, m) and its image through the prij are as follows.
(x, n · x, (n + m) · x) , _ pr12 pr23 pr13 u * (x, n · x)(x, (n + m) · x)(n · x, (n + m) · x)
Therefore, on that component,
∗ m ∗ n pr23 α˜ = α ◦ (ψ ◦ pr), pr12 α˜ = α ◦ (ψ ◦ pr),
∗ n+m pr13 α˜ = α ◦ (ψ ◦ pr), and their vertical compositions are as required by (6.1). We thus obtain the desired
ψ! : P → ΛX. Next, we need to provide the R//Z-equivariance datum ρ for ψ!. To
106 analyze the putative square
µ / P × R//Z 5= P
ψ!×id ρ ψ! (6.2) ΛX × R//Z / ΛX we notice that, after a suitable base change, any S-point of P × R//Z can be pulled back from the (P × R)-points in : P × R → P × R, (p, t) 7→ (p, t + n), where n ∈ Z, and any morphism of S-points can be pulled back from m: in → in+m. Thus, we can extract all information encoded by ρ by evaluating the above diagram on in and m. The top-right composition factors through P × T, so every in maps to the same
∗ µ ψ! ∈ ΛXP ×T, and m maps to the identity. The left-bottom composition factors
∗ through ΛX × pt//Z, so, for any n, in maps to pr1 ψ! ∈ ΛXP ×T, and m: in → in+m
∗ m ∗ ∗ maps to pr1 α : pr1 ψ! → pr1 ψ!. For each in, the natural transformation ρ should
∗ ∗ give a morphism ρ(in): pr1 ψ! → µ ψ! fitting in the diagram below.
ρ(i ) ∗ n / ∗ pr1 ψ! µ ψ!
∗ m pr1 α ρ(i ) ∗ n+m / ∗ pr1 ψ! µ ψ!
This means ρ is completely specified by ρ(i0). To provide ρ(i0), it suffices to give a ∗ ˜ ∗ ˜ morphism pr1 ψ → µ ψ, where the latter is the composition
µ ψ U × R → P × R −→ P → Σ −→ ΛX,
∗ ˜ ˜ satisfying appropriate coherence conditions on U ×P U ×R. Since µ ψ = pr1 ψ, we can take that to be the identity. One can check that ρ satisfies the coherence conditions required of the equivariance datum.
The effect of BT(ΛX) → BR//Z(ΛX) on morphisms is also given by descent. Given
107 a morphism in BT(ΛX)
Φ 0 P / P ψ * ΛX ξ < F ÐØ Σ / Σ0 ψ0 where Σ0 = S0 × R0|1, P 0 = S0 × T1|1 are also trivial families, consider the fiberwise universal cover U 0 = S × R1|1 → P 0 and choose a lift Φ:˜ U → U 0. We can then lift ψ, ψ0 and ξ by composing respectively whiskering with U → Σ or U 0 → Σ0
˜ U Φ / U 0 ξ˜ ˜ +3 ˜0 ψ Ò ψ ΛX
˜ ∗ 0 and descend ξ to a morphism ξ! : ψ! → Φ ψ!. To justify that, we need to show that on nth component of U ×P U the diagram
pr∗ ξ˜ ∗ ˜ 1 / ∗ ˜0 ˜ pr1 ψ pr1(ψ ◦ Φ)
αn αn pr∗ ξ˜ ∗ ˜ 2 / ∗ ˜0 ˜ pr2 ψ pr2(ψ ◦ Φ) commutes. (To be precise, αn above stands, respectively, for αn ◦ (ψ ◦ pr), the gluing
∗ 0 isomorphism used to build ψ!, and its counterpart for Φ ψ!.) This follows immediately from the compatibility condition between ξ and α, namely ξ ◦ αψ = αΦ∗ψ0 ◦ ξ. To see ˜ why the ξ! thus obtained does not depend on the choice of lift Φ, note that any two such choices differ by the action of an integer m on U 0, and ξ˜ only depends on ψ˜0 ◦ Φ˜; replacing Φ˜ with m ◦ Φ˜ does not change that composite. We omit the verification that ξ! is compatible with the equivariance data.
Finally, we assign to the morphism in BT(ΛX) prescribed by the data (F, Φ, ξ)
R//Z the morphism in B (ΛX) prescribed by the data (F, Φ, ξ!). That this assignment respects compositions follows from uniqueness for descent of morphisms.
108 Theorem 6.2. The fibered functor BT(ΛX) → BR//Z(ΛX) is an equivalence.
Proof. At the morphism level, the effect of the functor in question was described ˜ in two steps: ξ 7→ ξ 7→ ξ!. This is a one-to-one procedure because the first step is injective (since U → Σ has local sections) and the second step (descent) is in fact bijective. Thus, it remains to show that the fibered functor BT(ΛX) → BR//Z(ΛX) is full and essentially surjective. In order to do that, we will build a prestack Btriv and a factorization Btriv v u (6.3) ~ " BT(ΛX) / BR//Z(ΛX). where u is full and essentially surjective on the groupoid of S-point for any contractible S. The prestack Btriv is defined as follows:
1. an object consists of an object (Σ, P, ψ, ρ) ∈ BR//Z(ΛX) together with a section s:Σ → P , and
2. a morphism (Σ0,P 0, ψ0, ρ0, s0) → (Σ, P, ψ, ρ, s) is a pair consisting of a morphism
(F, Φ, ξ) of the underlying objects in BR//Z(X) together with a map r : Σ0 → R relating s and s0 in the sense that Φ ◦ s0 = (s ◦ F )e2πir.
With a little poetic license, a morphism can be depicted as follows (the square containing r would literally make sense, as a 2-commutative diagram, if we replaced
P with P//R). ψ0 P 0 O Φ ξ ' Ø ψ s0 / X r 3; P O Λ (6.4) 0 s Σ F ' Σ
We define u: Btriv → BR//Z(ΛX) to be the forgetful functor, which simply discards s and r, so it is clearly full and essentially surjective over contractible S as claimed.
109 Next, we construct v : Btriv → BT(ΛX). To an object (Σ, P, ψ, ρ, s) ∈ Btriv, we assign the object (Σ, P, s∗ψ) in BT(ΛX). Now fix a morphism as in (6.4). To define its image in BT(ΛX), the only new data we need to provide is a morphism (s0)∗ψ0 → (s ◦ F )∗ψ, which we take to be the following composition:
−1 (s0)∗ξ ρ (s0)∗ψ0 −−−→ (s0)∗Φ∗ψ ∼= (Φ ◦ s0)∗ψ = ((s ◦ F )e2πir)∗ψ −−−→s◦F,r (s ◦ F )∗ψ.
We omit the verification of functoriality. To finish the proof, we just need to show that (6.3) commutes (up to 2-isomorphism). It suffices to look at (Σ, P, ψ, ρ, s) ∈ Btriv where P and Σ are trivial families, and
∗ pick s to be the unit section; our goal is to produce an isomorphism between (s ψ)! and ψ, natural in the input data (Σ, P, ψ, ρ, s) and compatible with the respective equivariance data. From the discussion leading to the construction of the ρ in (6.2), we see that the data of the present (arbitrarily given) ρ is essentially an isomorphism
∗ ∗ ∗ ρ0 : pr1 ψ → µ ψ in ΛXP ×R. Now, let π ψ be the pullback through π : U → P and
∗ ∗ recall that sgψ is the U-point of ΛX used to put together (s ψ)!. Note that each half of the diagram
sg∗ψ pr1 0|1 s×id / ψ & U = S × × / P × / P / ΛX R R R µ 9
π∗ψ
∗ ∗ ∗ commutes, so ρ0 gives a morphism sgψ → π ψ and, by descent, a morphism (s ψ)! → ψ. We omit the naturality and compatibility checks.
6.1.4 Global quotients
Let us illustrate the above constructions when X = X//G is the quotient orbifold associated to the action of a finite group G on a compact manifold X.
110 We start noticing that a quotient stack presentation for Λ(X//G) can be given as follows. Consider the product X × G with diagonal G-action, where G acts on itself by conjugation. There is an invariant submanifold
Xˆ = {(x, g) ∈ X × G | x ∈ Xg}, and an object over S in the quotient stack X//Gˆ consists of a pair (Q, (f, A)), where Q → S is a principal G-bundle and (f, A): Q → Xˆ ⊂ X × G is a G-equivariant smooth map. Denote by α: Q → Q the bundle automorphism determined by A; on T -points, it is given by
α(q) = qA(q), q ∈ QT .
Notice that this automorphism preserves f, and therefore (Q, f, α) determines an S-point of Λ(X//G). Conversely, given an S-point (Q, f, α) of Λ(X//G), we can specify a G-equivariant map A: Q → G by requiring that the above equation holds, and compatibility between f and α implies that the resulting map (f, A): Q → X × G factors through Xˆ, thus determining an object of X//Gˆ over S.
The translation back and forth between A and α provides a pt//Z-equivariant equivalence between Λ(X//G) and X//Gˆ , compatible with the maps Λ(X//G) → X//G forgetting the prescribed automorphism and X//Gˆ → X//G induced the projection pr1 : X × G → X. We will shift freely between these two formulations. The geometric content of an S-family in BR//Z(X//Gˆ ) is the following:
1. a family Σ → S of connected Euclidean 0|1-manifolds,
2. a principal T-bundle P → Σ with connection ω satisfying the usual curvature condition,
3. a principal G-bundle Q → P ,
111 4. a G-equivariant map (f, A): Q → Xˆ ⊂ X × G; or, equivalently, a bundle automorphism α: Q → Q and a G-equivariant map f : Q → X such that f ◦ α = f, and, finally,
5. a collection of natural isomorphisms of G-torsors
ρp,t : Qp → Qpe2πit
for each pair of T -points q : T → P , t: T → R, intertwining the maps
fp : Qp → X, fpe2πit : Qpe2πit → X
and subject to the condition that for any n: T → Z the diagram
ρp,t Qp / Qpe2πit
n αp (6.5)
ρp,t+n Qp / Qpe2πi(t+n)
commutes.
The last condition means that α agrees with the holonomy of Q around the fibers of P . A morphism in BR//Z(X//Gˆ ) is given by a fiberwise isometry F : Σ0 → Σ, a connection-preserving bundle map Φ: P 0 → P covering F , and a bundle map Q0 → Q covering Φ which is required to be compatible in the obvious way with the data in (4) and (5) above.
The geometric content of an S-family in BT(X//Gˆ ) is the following:
1. a family Σ → S of connected Euclidean 0|1-manifolds;
2. a principal T-bundle P → Σ with a connection ω whose curvature agrees with the tautological 2-form on Σ; and
112 3. a principal G-bundle Q → Σ, and
4. a G-equivariant map f : Q → Xˆ.
A morphism (Σ0,P 0,Q0, f 0) → (Σ, P, Q, f) consists of a fiberwise isometry F : Σ0 → Σ, a connection-preserving bundle map Φ: P 0 → P covering F , and a bundle map Q0 → Q covering F and intertwining the maps f : Q → Xˆ and f 0 : Q0 → Xˆ. From proposition 6.1, it follows that BT(X//Gˆ ) admits the presentation
1|1 ∼ 1|1 (ΠT (X//Gˆ ))// Isom(T ) = ΠT X//ˆ (Isom(T ) × G).
Finally, let us describe the map relating T-equivariant and R//Z-equivariant bor-
T ˆ disms in this special situation. Fix (Σ, P, Q, f) ∈ B (X//G)S and let (Σ,P,Q!, f!, ρ) ∈
R//Z ˆ B (X//G)S be its image. Locally in S, f determines a conjugacy class of G and
Q! → P is the G-bundle with that holonomy around the fibers of P → Σ. More specifically, let us assume P and Q are trivial; if S is connected, then f determines an element g ∈ G, namely the one corresponding to the connected component of
ˆ g X = qg∈GX in which f|Σ×{e} takes values. Then Q! → P is the G-bundle built as a quotient
Q! = (Σ × R × G)/Z → P = Σ × T, where the Z-action is generated by the diffeomorphism prescribed, on T -points, by ˆ (s, t, h) 7→ (s, t + 1, gh). The map f! : Q! → X is induced by the Z-invariant map
(s, t, h) 7→ f(s, e) · h. The automorphism of Q! determined by the G-component of f! can be expressed as (s, t, h) 7→ (s, t, gh).
6.1.5 More general kinds of equivariant bordisms
Using the BT-action on K (i.e., the natural T-action attached to every family of Euclidean supercircles), we could also associate, to any stack Y with a chosen
113 automorphism, bordism stacks KT(Y) and KR//Z(Y) by imposing, on each map K 3 K → Y, the equivariance condition indicated by the superscript. There are also variants for Kev, etc., and in each case we would still have comparison maps as above. Since our goal here is to probe K(X) with the simplest possible objects, we only care about
T ∼ T R//Z ∼ R//Z K1 (ΛX) = B (ΛX), K1 (ΛX) = B (ΛX).
The T-equivariant variations seem to be related to the stacks of classical vacua considered by Berwick-Evans [9].
6.2 The Chern character for global quotients
In this section, we show how to recover, in terms of dimensional reduction of field theories, the (untwisted) delocalized Chern character constructed by Słomińska [36] and Baum and Connes [4] in the case of a finite group G acting on a compact manifold X. We start by briefly recalling their construction.
6.2.1 The Baum–Connes Chern character
ˆ ` g Let X = {(x, g) ∈ X × G | xg = x} = g∈G X . The equivariant Chern character is a ring homomorphism
h iG i i ˆ ∼ M i g chG : KG(X) → HG(X; ) = H (X ; ) . (6.6) C g∈G C
Here, i ∈ Z/2 and ordinary cohomology is Z/2-graded. We recall that the equivariant ordinary cohomology of Xˆ with coefficients in C can be identified with the invariants in its nonequivariant cohomology; this can be deduced from the Serre spectral sequence for the fibration EG ×G X → BG using the fact that the integral reduced cohomology of a finite group is torsion.
i i g Now, let us define, for each g ∈ G, a homomorphism chg : KG(X) → H (X ; C) as
114 the composition
i i g ∼ i g ch ⊗ trg i g KG(X) → Khgi(X ) = K (X ) ⊗ R(hgi) −−−−→ H (X ; Q) ⊗Q C.
(The middle isomorphism is due to the fact that the action of the cyclic group hgi generated by g on Xg is trivial; ch denotes the usual, nonequivariant Chern character, and trg assigns to any representation of hgi the trace of the operator g.) Finally, we
i i let chG : KG(X) → HG(X) be the direct sum of all chg via the identification (6.6).
Concretely, the effect of chg on the K-theory class represented by a G-equivariant
g vector bundle V → X is the following. For each x ∈ X , the fiber Vx is a representation
1 r of the cyclic group generated by g. Let λ1, . . . λr be distinct eigenvalues, and Vx ,...,Vx the corresponding eigenspaces. Each λi is a |g|-root of unity, so V |Xg can be written as direct sum
1 r V |Xg = V ⊕ · · · ⊕ V .
Then
X i ev g chg(V ) = λi ch(V ) ∈ H (X ; C) and G hM ev g i chG(V ) = ⊕g∈G chg(V ) ∈ H (X ; ) . g∈G C
6.2.2 Parallel transport and field theories
Let X be a differentiable stack and V : X → VectA a vector bundle with Quillen superconnection. Then we can construct a 1|1-dimensional EFT over X using parallel transport along superpaths in X. Roughly speaking, this EFT assigns to a superpoint x: spt → X the fiber Vx, and to a superinterval c: It,θ → X the parallel transport map
SP(c): Vc(0,0) → Vc(t,θ) constructed by Dumitrescu [16]. A choice of Hermitian metric is also needed to specify the effect of the right elbow of length 0, R0 : spt q spt → ∅. It is
115 then part of the conjecture of Stolz and Teichner on the relation between 1|1-EFTs and K-theory that, for reasonable X, the field theory above corresponds to the K-theory class represented by V . We note, however, that there are some subtleties that need to be addressed in the above construction. For instance, objects in 1|1-EBord(X) are not merely (S-families of) superpoints, but rather germs of collars S × spt ⊂ S × R1|1; in order to make the sheaf of vector spaces on S associated to such germ independent of the chosen representative, we need to consider flat sections on a neighborhood of the origin in R1|1, and then prove that all linear maps arising from bordisms in fact act on the space of germs of flat sections. In any case, we are presently only interested in the partition function of the field theory above, which admits a straightforward description independently of the details of the construction of the full EFT. To an S-point (Kl, ψ) of K(X), we associate the
∗ supertrace of the holonomy along Kl, i.e., the parallel transport of ψ V → Kl along
1|1 1|1 1|1 the superpath S × R → Kl = (S × R )/Zl with endpoints i0, il : S → S × R .
Proposition 6.3. The above quantity is preserved by isometries of supercircles, and
∞ therefore defines a function ZV ∈ C (K(X)).
Proof. Super parallel transport is invariant under fiberwise isometries of R1|1 and compatible with gluing of superintervals [16, theorem 4.3]. Thus, given an isometry
F : Kl0 → Kl, which for simplicity we assume to cover the identity S → S, and
0 denoting by a: S → Kl, a : S → Kl0 the basepoints of Kl, Kl0 , we have
0 0 ZV (Kl0 , f ◦ F ) = str(SP(a, F (a )) ◦ SP(F (a ), a))
= str(SP(F (a0), a) ◦ SP(a, F (a0)))
= ZV (Kl, f).
∗ Here, SP(a0, a1) refers to parallel transport in ψ V from point a0 to a1 : S → Kl along the positively oriented direction.
116 Theorem 6.4. Let X = X//G be the quotient stack arising from the action of a finite group on a compact manifold. Then the diagram
Z 0 C∞(K(X)) red / 0|1-EFT(ΛX) VectA(X) ch 0 G / ev ˆ / KG(X) HG (X; C) commutes.
Note that the vertical map induces an isomorphism after passing to concordance classes, so the above provides a field-theoretic interpretation of the equivariant Chern character.
6.2.3 Proof of theorem 6.4
As before, let us fix V : X//G → VectA. This map classifies a G-equivariant vector bundle over X, which we still call V , with a G-invariant superconnection A. To get started, we need to describe the pullback of V to a supercircle over X//G.
Proposition 6.5. Fix a supercircle ψ : K → X//G and denote by π : Q → K and f : Q → X the principal G-bundle and G-equivariant map classified by ψ. Then there is a natural superconnection-preserving isomorphism of vector bundles
(f ∗V )/G / ψ∗V
Q/G K.
Proof. Consider the diagram
/ f Q ×K Q / Q / X
π x ψ K / X//G V / VectA.
117 Here, x: X → X//G is the standard atlas and hence V ◦ x classifies the vector bundle with superconnection V → X. Notice that the square 2-commutes. In fact, the top-right composition Q → X//G classifies the trivial G-bundle Q × G → Q, while the left-bottom composition classifies the G-bundle π∗Q → Q (together with the corresponding equivariant maps into X induced by f), and these two Q-points of X//G are isomorphic. Now, the composition V ◦ x ◦ f classifies the vector bundle f ∗V → Q, and the ∼ G-equivariance information provides descent data for the covering Q×K Q = Q×G ⇒ Q → K. The descended vector bundle with superconnection can be described explicit as (f ∗V )/G → K. Thus, 2-commutativity of the square above and the uniqueness property of descent provide a canonical isomorphism ψ∗V ∼= (f ∗V )/G.
∞ ∞ T ˆ To identify the image of ZV ∈ C (K(X)) in C (B (X//G)) with a differential form on Xˆ, following the identifications from section 6.1.2 and chapter 4, we need to consider the versal ΠT Xˆ-family in B(Xˆ)
ev ΠT Xˆ × R0|1 / Xˆ (6.7) ΠT X.ˆ and calculate the corresponding smooth function on the parameter manifold ΠT Xˆ.
The counterpart of the above family in BT(X//Gˆ ) is obtained by adding the trivial principal T-bundle with standard connection over ΠT Xˆ × R0|1, and replacing the map into Xˆ with its postcomposition with the atlas xˆ: Xˆ → X//Gˆ . In turn, the image of that gadget in BR//Z(X//Gˆ ), once restricted to ΠTXg ⊂ ΠT Xˆ, comprises the following data:
1. the family of Euclidean 0|1-manifolds Σ = ΠTXg × R0|1 → ΠTXg,
2. the trivial T-bundle P = ΠTXg × R0|1 × T → Σ, with the standard connection
118 form ω = dt − θdθ,
3. the principal G-bundle Q = (ΠTXg × R1|1 × G)/Z → P , where the Z-action is generated by the map described on S-points by
g 1|1 (x, t, h) ∈ (ΠTX × R × G)S 7→ (x, 1 · t, gh),
ˆ −1 4. the map f : Q → X ⊂ X × G given by (x, t, h) 7→ (ev(t1, x) · h, h gh), which
g is well defined since ev(t1, x) lies in X .
Finally, by proposition 6.5, the image of the above object in K(X//G) can be identified with the ΠTXg-family of length 1 supercicles K = ΠTXg × T1|1 together with the vector bundle with superconnection W = (f ∗V )/G → K. Our task now is to compute the supertrace of the holonomy around K. More pre- cisely, consider the standard superpath c: ΠTXg ×R1|1 → ΠTXg ×T1|1 with endpoints
g g 1|1 ∗ ∗ it : ΠTX → ΠTX × R , x 7→ (x, t), for t = 0, 1, and denote by SP: c0W → c1W the parallel transport operator along that superinterval. There is a slight subtlety to
∗ ∗ notice here. Since the maps c0 = c ◦ i0 and c1 = c ◦ i1 are equal, c0W and c1W are the same vector bundle, but the correct way to identify them is via the action of g. Indeed, let us form the pullback of principal bundles
f˜ & Q˜ / Q / X f
c ΠTXg × R1|1 / K and identify Q˜ = ΠTXg × R1|1 × G. Then the pullback c∗W can be identified with the restriction of the pullback of V to the identity section of Q˜,
∗ ∼ ˜∗ ∼ ˜∗ c W = (f V )/G = (f V )|ΠTXg×R1|1×{e},
119 so we identify
−1 ∗ ˜∗ ˜∗ g ˜∗ ∗ c0Wx = f V(x,0,e) = f V(x,1,g) −−→ f V(x,1,e) = c1Wx.
˜ g Moreover, since f|ΠTXg×R1|1×{e} takes values in X , the group element g acts on the fibers of W˜ , and we can split W˜ = W˜ 1 ⊕ · · · ⊕ W˜ r as a sum of eigenspaces for distinct eigenvalues λ1, . . . , λr (in a Z/2-graded manner). Since the original superconnection on V is G-invariant, it also follows that its pullback to c∗W can be decomposed as a
∗ g 1|1 ˜ i direct sum of operators Ai on each Ω (ΠTX × R , W ). We are finally ready to invoke the calculations of Dumitrescu [15] recovering the usual (nonequivariant) Chen character in terms of parallel transport. Denoting
˜ i ˜ i ˜ i SPi : W |ΠTXg×0 → W |ΠTXg×1 the parallel transport for one unit of time on each W ,
2 the calculations in the above reference show that SPi = exp(−Ai ), so that
2 ch(Ai) = str(exp(−Ai )) = str(SPi).
∗ ∗ The original parallel transport SP: c0W → c1W can be expressed as the composi- tion
∗ ˜ ⊕i SPi ˜ g ˜ ∗ c0W = W(x,0,e) −−−→ W(x,1,e) −→ W(x,1,g) = c0W and we conclude that
X X X str(SP) = str(g SPi) = λi str(SPi) = λi ch(Ai) 1≤i≤r 1≤i≤r 1≤i≤r
is a differential form representative of chg(V ). Next, we notice that the differen- tial forms thus obtained on each ΠTXg determine a G-invariant form on ΠT Xˆ =
g ˆ qg∈GΠTX . This form is precisely the function on ΠT X associated to (6.7) via red(ZV ), and it represents chG(V ) in de Rham cohomology. This finishes the proof of
120 theorem 6.4.
6.3 Dimensional reduction of twists
In this section, we show that dimensional reduction of the (partial) 1|1-twists from chapter 5 recovers the 0|1-twists related to complexified twisted K-theory that we encountered in chapter 4. So we start with the line bundle L = Lα : K(X) → Vect produced from a gerbe with Dixmier-Douady class α ∈ H3(X; Z) and connective struc- ture (or, rather, from the data of a Deligne 2-cocycle (h, A, B) on a nice presentation
X1 ⇒ X0 of the orbifold X) and aim to describe its pullback via the composition
1|1 // ΠT ΛX//Isom(R ) → BT(ΛX) → BR Z(ΛX) → K(X).
Reinterpreting that pullback L0 as a line bundle with superconnection on ΛX, we will
0 ∼ 0|1 find it is flat, and hence descends to a line bundle Tα on B(ΛX) = ΠT ΛX// Isom(R ), or, in other words, a 0|1-dimensional Euclidean twist over ΛX.
3 0 Theorem 6.6. Let X be an orbifold and α ∈ H (X; Z). Then the twist Tα ∈
0|1-ETw(ΛX) obtained by dimensional reduction of the partial 1|1-twist Lα corre- sponding to a gerbe with Dixmier-Douady class α is isomorphic to the twist Tα from theorem 4.7.
This means, in particular, that
Tα⊗Ti ∼ i+α 0|1-EFT [ΛX] = K (X) ⊗ C,
and suggests that Lα is the correct 1|1-twist over X to represent α-twisted K-theory. We also remark that all objects indexed by α actually depend, up to noncanonical isomorphism, on the choice of a gerbe representative and its connective structure; this abuse of language is completely standard in the literature.
121 The proof of the theorem (and the flatness claim necessary to state it) will occupy the remainder of this section.
6.3.1 The underlying line bundle
Let us not worry about the connection for now and simply describe the line bundle on ΛX underlying L0 : ΠT ΛX// Isom(R1|1) → Vect; in other words, we want to further pull back L0 via the map ΛX → ΠT ΛX// Isom(R1|1). Thus our goal is to describe a ˆ line bundle on X0 and an isomorphism between its two pullbacks via the structure ˆ ˆ ˆ maps X1 ⇒ X0. Since X0 is contractible, the line bundle itself is trivializable and all ˆ interesting data are conveyed by the isomorphism over X1. ˆ Fix an S-point (x, g : x → x) of X0. Its image in K(X) consists, informally, of the length 1 constant superpath x in X with its endpoints glued together via g. More formally, it is given by
1. the trivial family of length 1 supercircles K = S × T1|1 → S,
2. the standard covering U = S × R1|1 → K, together with the fiberwise constant x map ψ0 : U → S → X0, and
3. the map ψ1 : U ×K U → X1 which, restricted to the subspace of points differing by n units, is fiberwise constant and equal to gn.
Moreover, we can choose the skeleton to be [0, 1] ⊂ U. Fix a second S-point (x0, g0)
0 ¯ and a compatible morphism f : x → x in (X1)S. This gives rise to a morphism f in K(X)S which, in the same vein as above, we can describe as being fiberwise the constant f. Applying (5.4), we conclude that the identifications between the fibers
0 0 L(x,g) and L(x0,g0) is via
Z 0 ¯ ∗ h(f, g) L (f) = L(f) = exp volD hD, ψ1Ai 0 . [0,1] h(g , f)
122 ∗ Since ψ1A can be pulled back from S, the integrand vanishes and
h(f, g) L0(f) = , h(fgf −1, f) which agrees with (4.2).
6.3.2 The superconnection
A superconnection on the line bundle L0 : ΛX → Vect consists of a superconnection
0 ˆ ˆ A on the underlying line bundle L → X0 whose two pullbacks over X1 are identified with one another through the isomorphism s∗L0 → t∗L0. Since we just want to ˆ ˆ describe the superconnection on X0, it suffices to look at the versal family xˇ: ΠT X0 → 1|1 1|1 ˆ ΠT ΛX//Isom(R ) and the Isom(R )-action on it; nothing here will involve X1. The composition
xˇ 1|1 T ΠTX0 −→ ΠT ΛX// Isom(R ) → B (ΛX) classifies the object given by
ˆ 0|1 ˆ 1. the trivial family of Euclidean 0|1-manifolds Σ = ΠT X0 × R → ΠT X0,
ˆ 1|1 2. the trivial T-bundle P = ΠT X0 × T → Σ with the standard principal connection, and
ˆ 0|1 xˇ×id 0|1 ev 3. the map ΠT X0 × R −−→ ΠT ΛX × R −→ ΛX, which can be identified with ˆ 0|1 ev ˆ xˆ the composition ΠT X0 × R −→ X0 −→ ΛX; here, xˆ is the versal family for ΛX.
Its image in BR//Z(ΛX) consists of
1. the same Σ and P as above, and
2. the map (xˆ ◦ ev)! : P → ΛX gotten by descent, and the associated equivariance datum (which is not important for us now).
123 ˆ 1|1 A picture of the above object, together with its two pullbacks over ΠT X0 × R encoding the R1|1-action, is as follows.
µ×id / (ˆx◦ev)! ˆ 1|1 1|1 ˆ 1|1 / ΠT X0 × R × T / ΠT X0 × T ΛX pr1 × id ˆ 0|1 ΠT X0 × R
µ ˆ 1|1 / ˆ ΠT X0 × R / ΠT X0 pr1
ˆ Further mapping this ΠT X0-point into K(X), and translating the data of (xˆ ◦ ev)! into the language of torsors, we obtain the family consisting of
ˆ 1. the trivial ΠT X0-family K of length 1 supercircles, together with the standard ˆ 1|1 covering U = ΠT X0 × R → K,
ˆ 1|1 ˆ 0|1 ev ˆ 2. the map ψ0 : U = ΠT X0 × R → ΠT X0 × R −→ X0 X0,
3. the map ψ1 : U ×K U → X1 which, over the component of points differing by n units, is the n-fold iterate of
ˆ 1|1 ˆ 0|1 ev ˆ α:ΠT X0 × R → ΠT X0 × R −→ X0 ,→ X1,
4. a skeleton we may choose to be [0, 1] ⊂ U.
Now, the superconnection we are seeking to describe is geometrically encoded by ˆ rotations of the above family. More precisely, the line bundle on ΠT X0 assigned by ∗ 0 ˆ ˆ the functor L to this family is just π L , where π denotes the projection ΠT X0 → X0.
0 ∗ ∗ 0 ∗ ∗ 0 A trivialization of L fixes trivializations of pr1 π L , µ π L , and this gives rise to an R1|1-action on that trivial line bundle. The infinitesimal generator associated to
D = ∂θ − θ∂t is precisely the operator A.
124 By proposition 5.5, this R1|1-action can be described via parallel transport with 1 ˆ respect to the connection d + ∆, where ∆ ∈ Ω (ΠT X0; C) is calculated, according to (5.7), as Z hv, ∆i = hv, α∗Ai| − vol hD ∧ v, ψ∗Bi. (6.8) ΠT Xˆ0×{0} DE E 0 [0,1]
Here, DE stands for the vector field on U giving its Euclidean structure. More ˆ specifically, we are supposed to parallel transport along the vector field Dd on ΠT X0 corresponding to the de Rham differential. So we want to solve the differential equation
∞ ˆ ∗ 0 ∼ ∗ ˆ Ddω + hDd, ∆iω = 0, ω ∈ C (ΠT X0; π L ) = Ω (X0; C).
We will need a little technical lemma.
Lemma 6.7. Let X be an ordinary manifold, ev: ΠTX × R0|1 → X be evaluation map, and write ω˜ for the function on ΠTX corresponding to the differential form ω ∈ Ωn(X). Then we have
∧n ∗ h(∂θ) , ev ωi = ±n!(˜ω + θdωf), where the sign ± is −1 if n ≡ 2, 3 mod 4 and +1 otherwise.
∞ Proof. It suffices to prove the lemma for ω = f0df1 ··· dfn, where fi ∈ C (X). Denote by D the de Rham vector field on ΠTX. Then, by (3.1),
∗ Y ev ω = (f0 + θDf0) d(fi + θDfi) 1≤i≤n Y = (f0 + θDf0) (dfi + dθDfi + θdDfi). 1≤i≤n
125 Thus, writing Fi = dfi + dθDfi + θdDfi, we have
∗ i−1 X i∂θ ev ω = (−1) (f0 + θDf0)F1 ··· Fi−1DfiFi+1 ··· Fn. 1≤i≤n
Contracting an expression like the one under the summation sign with ∂θ produces as many new terms as there are Fi’s, and, in each of those, one of the Fi’s get converted
into a Dfi. Note also that commuting i∂θ with any Fi or Dfi produces a minus sign. Iterating this process, we find
n ∗ 0+1+···+n−1 (i∂θ ) ev ω = (−1) n!(f0 + θDf0)Df1 ··· Dfn
= ±n!(˜ω + θdωf).
Corollary 6.8. For D the de Rham vector field on ΠTX and the remaining notation as in the lemma,
µ∗hD∧n, π∗ωi = ±n!(˜ω + θdωf).
Towards calculating hDd, ∆i, let us fix the notation
ˆ 1|1 ˆ 0|1 µ ˆ ν :ΠT X0 × R → ΠT X0 × R → ΠT X0
and note that Dd, DE and ∂θ (or, more precisely, the vector fields Dd ⊗ 1, 1 ⊗ D and ˆ 1|1 1 ⊗ ∂θ on ΠT X0 × R ) are all ν-related to the de Rham vector field on the codomain,
0 which for clarity we denote Dd. Now, using corollary 6.8 and the fact that α = π ◦ ν, we get ∗ ∗ 0 ∗ ˜ ˜ ˆ hDd, α Ai = ν hDd, π Ai = A + θdAf = A on ΠT X0 × {0}.
We are left with calculating the integral in (6.8). Since ψ0 = s ◦ π ◦ ν, we have
∗ ∗ 0 0 ∗ hDE ∧ Dd, ψ0Bi = ν hDd ∧ Dd, π Bi.
126 ∗ ˜ Applying corollary 6.8, we get hDE ∧ Dd, ψ0Bi = −2(B + θdBf ) so we can finally calculate
Z ˜ ˜ ˜ ˜ ˜ hDd, ∆i = A + 2 [dtdθ] B + θdBf = A + 2dBf = A + 2Ω. [0,1]
ˆ Reinterpreting an odd, fiberwise linear vector field on ΠT X0 × C as a superconnection on L0, we get
A = d + A + 2Ω.
This matches the superconnection described in section 4.4, and finishes the proof of theorem 6.6.
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