Supersymmetric Field Theories from Twisted Vector Bundles

We give a description of the delocalized twisted cohomology of an orbifold and the Chern character of a twisted vector bundle in terms of supersymmetric Euclidean field theories. This includes the construction of a twist functor for 1|1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1\vert1}$$\end{document}-dimensional EFTs from the data of a gerbe with connection.


Introduction
In this paper, we explore a twisted version of the Stolz-Teichner program on the use of supersymmetric Euclidean field theories (EFTs) as geometric cocycles for cohomology theories [26]. We focus on twisted 1|1 and 0|1-dimensional EFTs over an orbifold X; the corresponding cohomology theories are (twisted) K -theory and delocalized de Rham cohomology. One of our main goals is to describe the Chern character ch : K α (X) → H ev deloc (X, α) of a twisted vector bundle in terms of dimensional reduction of field theories. (For compact X, this Chern character map provides an isomorphism after complexification; thus, delocalized cohomology, which we recall below, is a stronger invariant than regular de Rham cohomology.) Here, the twist is α ∈ H 3 (X; Z). Thus, on the field theory side, our first task is to construct from α a Euclidean twist functor (or anomaly) cohomology such field theories capture-but the orbifold perspective is important to deal with twists, even if X is equivalent to a manifold. We begin our study with a classification of 0|1-dimensional twists for EFTs over an orbifold (Sect. 2). For a particular T α ∈ 0|1-ETw( X), concordance classes of twisted EFTs over the inertia X (the orbifold of "constant loops") are in natural bijection with the delocalized twisted cohomology H * deloc (X, α). Then, turning to 1|1-dimensional considerations, we construct TX ∈ 1|1-ETw(X) taking as input α ∈ H 3 (X; Z), or, rather, a (C × -)gerbe with connectionX → X representing that class (Sect. 4). This is an extension of the transgression construction for gerbes [8] in the sense that it produces, in particular, a line bundle on the stack K(X) of supercircles over X; that stack is a super analogue of LX/ / Diff + (S 1 ), and the line bundle we obtain is a super analogue of the usual transgression of the gerbe.
It is now reasonable to conjecture that 1|1-EFT TX (X)/concordance ∼ = K α (X), (1.1) but this question is open even in the case where X is a manifold and TX is trivial, so we will not dwell on it here. Instead, we will demonstrate the meaningfulness of our construction by associating a twisted field theory E V to anyX-twisted vector bundle V, and identifying its dimensional reduction. Here again, the partition function of E V is the super counterpart of a classical construction, namely the trace of the holonomy, which in this case is not a function but rather a section of the transgression ofX.

Dimensional reduction and the Chern character.
Dimensional reduction is, intuitively, the assignment of a (d −1)-dimensional theory to a d-dimensional theory induced by the functor of bordism categories S 1 ×-: (d −1)-Bord → d-Bord. For field theories over an orbifold, the action of the circle group T on the inertia X can be used to refine this to (partial) assignments 1|1-ETw(X) → 0|1-ETw( X), 1|1-EFT T (X) → 0|1-EFT T ( X).
Our dimensional reduction procedure was developed in [23] and is recalled in Sect. 5.1. It is given by the pull-push operation along functors between certain variants of the corresponding Euclidean bordism categories 0|1-EBord( X) ← 0|1-EBord T ( X) → 1|1-EBord(X). (1. 2) The lack of direct map from left to right is due to certain subtleties concerning Euclidean supergeometry, as explained in the reference above. The results of this paper can be summarized in the following statement. commutes. Here, α ∈ H 3 (X; Z) is the Dixmier-Douady class ofX and T X is the dimensional reduction of TX.
Since the right vertical map is a bijection on concordance classes (Theorem 2.11), this gives a geometric interpretation of the twisted orbifold Chern character.
This theorem generalizes results of Han [14] and Dumitrescu [10] for the untwisted, non-equivariant case. In a different direction, we point out that Berwick-Evans and Han [6] extended that story to the equivariant case for Lie group actions. We also point out that, in the case of a global quotient orbifold X/ /G and twist coming from a central extension of G, Berwick-Evans [5] obtained a somewhat different field-theoretic interpretation of K α (X/ /G) ⊗ C. In fact, he also found a description of TMF ⊗ C in terms of "simple" 2|1-EFTs, and it would be interesting to investigate if these are obtained, in our language, as the dimensional reduction of full-blown 2|1-EFTs.

Notation and conventions.
We generally follow Deligne and Morgan's [9] treatment of supermanifolds. We also use Dumitrescu's [11] notion of super parallel transport (for connections), with the difference that we always transport along the left-invariant vector field D = ∂ θ − θ∂ t of R 1|1 . For the notion of Euclidean structures (in dimension 1|1 and 0|1), see [23, appendix B].
Most manipulations in this paper happen in the bicategory of stacks (Grothendieck fibrations satisfying descent) over the site of supermanifolds (see e.g. Behrend and Xu [3] for details). Every Lie groupoid presents a stack, and we use, concretely, the stack of torsors as a model. An orbifold is a stack presented by a proper étale Lie groupoid. We fix, once and for all, an étale Lie groupoid presentation for our orbifold X, s, t : X 1 ⇒ X 0 .
This determines presentationŝ of X (the inertia orbifold), T X (the stack of maps R 0|1 → X), and T X respectively. There are obvious maps that we often leave implicit. We fix also a gerbe with connectionX → X and, when needed, anX-twisted vector bundle V; those are assumed to come with presentations as well, with the notation introduced in "Appendix B" section. Note that the chosen presentation X 1 ⇒ X 0 of X must be such thatX admits a presentation as a central extension L → X 1 , which is a nontrivial condition. For instance, when X is just a manifold, we need, in general, to choose as presentation theČech groupoid i, j U i ∩ U j ⇒ i U i of some open cover.

Twisted 0|1-EFTs and de Rham Cohomology
In this section, we extend in two directions the results of Hohnhold et al. [16] on the relation between 0-dimensional supersymmetric field theories over a manifold and de Rham cohomology. First, we replace the target manifold by an orbifold, and, second, we provide a classification of twists. This provides, in particular, a field-theoretic description of the delocalized twisted de Rham cohomology of an orbifold, which is isomorphic, via the Chern character, to complexified twisted K -theory [1,27].
We denote by B(X) the stack of fiberwise connected bordisms in 0|1-EBord(X), which can be described concretely as When X is a manifold, the mapping stack Fun SM (R 0|1 , X) is represented by the parityreversed tangent bundle, so we will in general write T X = Fun SM (R 0|1 , X) for the stack of superpoints. If the stack X admits a Lie groupoid presentation X 1 ⇒ X 0 , then T X can be presented by the Lie groupoid T X 1 ⇒ T X 0 ; in particular, if X is an orbifold, T X is again an orbifold.
We define the groupoid of Euclidean 0|1-twists over X to be and, for each T ∈ 0|1-ETw(X), the corresponding set of T -twisted topological respectively Euclidean field theories over X to be the set of global sections of T : In these definitions, Vect can be the stack of real or complex super vector bundles, but ultimately we are interested in the complex case. We recall the construction, in Hohnhold et al. [16, definition 6.2], of the twist This functor is entirely specified by the requirement that the point pt maps to the odd complex line C, and by a group homomorphism Isom(R 0|1 ) → GL(0|1) ∼ = C × , which we take to be the projection onto Z/2 = {±1}. We set T n = T ⊗n 1 , and use the same notation for the pullback of those line bundles to B(X). [20], we define a superconnection A on a Z/2-graded complex vector bundle V → X to be an odd operator (with respect to the total Z/2-grading) on * (X ; V ) satisfying the Leibniz rule

Superconnections and twists. Following Quillen
Here, ω ∈ * (X ) and f ∈ * (X ; V ). It follows that A is entirely determined by its restriction to 0 (X ; V ); denoting by A i , i ≥ 0, the component 0 (X ; V ) → i (X ; V ), we find that A 1 is an affine (even) connection and all other A i are 0 (X )-linear odd homomorphisms. The even operator A 2 : * (X ; V ) → * (X ; V ) is * (X )-linear, and is called the curvature of A. In particular, a flat superconnection is a differential on * (X ; V ). Now, let V 0 , V 1 → X be complex super vector bundles and A i , i = 0, 1, superconnections. Then there exists a superconnection A on the homomorphism bundle Hom(V 0 , V 1 ) → X , characterized by for any section of * (X ; Hom(V 0 , V 1 )) of parity | | and f ∈ * (X ; V 0 ). We define Vect A to be the prestack on Man whose objects over X are vector bundles with superconnection (V, A), and morphisms (V 0 , A 0 ) → (V 1 , A 1 ) are sections ∈ * (X ; Hom(V 0 , V 1 )) of even total degree satisfying A( ) = 0. This turns out to be a stack.
There is a nice interpretation of superconnections in terms of Euclidean supergeometry. Consider the pullback bundle π * V → T X along π : T X → X . Its sections on an open U ⊂ X are given by , and to say that a given odd, fiberwise linear vector field A on π * V is π -related to the de Rham vector field d on the base is precisely the same as saying that Eq. (2.1) holds. Thus a superconnection on V gives π * V the structure of an Isom(R 1|1 )-equivariant vector bundle over T X, where the action on the base is via the projection Isom(R 1|1 ) → Isom(R 0|1 ) and the identification T X = SM(R 0|1 , X ). The superconnection is flat if and only if this action factors through Isom(R 0|1 ). There is also a converse statement.
As usual, we extend the above definitions by saying that a vector bundle with superconnection on a stack X is a fibered functor V : X → Vect A ; it is flat if it takes values in the substack Vect A of flat superconnections. With this in place, we can return to our discussion of twisted field theories.

Proposition 2.3. For X a differentiable stack, there is a natural equivalence of groupoids
Proof. There exists a bisimplicial manifold { T X j ×Isom(R 0|1 ) ×i } i, j≥0 whose vertical structure maps give nerves of Lie groupoids presenting T X×Isom(R 0|1 ) ×i and whose horizontal structure maps give nerves of presentations of T X j / / Isom(R 0|1 ). Applying Vect, we get a double cosimplicial groupoid . . . . . .
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 proposition 8 of [23], On the other hand the limit of each row is equivalent to Vect( T X i / / Isom(R 0|1 )), and the stack map Vect A → Vect( T -/ / Isom(R 0|1 )) of Theorem 2.2 gives us a levelwise equivalence of simplicial groupoids Taking limits, we get an equivalence Vect A (X) → 0|1-ETw(X). Remark 2.5. We can drop the flatness condition by considering vector bundles on T X/ / Isom(R 1|1 ).

Concordance of flat sections.
The goal of this section is to identify concordance classes of twisted 0|1-EFTs. This is an 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 with differentiable stacks and the trivial flat line bundle with an arbitrary flat superconnection. Fix a differentiable stack X with presentation X 1 ⇒ X 0 and let T ∈ 0|1-ETw(X) be the twist associated to the flat superconnection (V, A) on X. Proof. The vector bundle T : B(X) → Vect determines a sheaf T on B(X), assigning 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 (2.4), representing an object in the limit of that diagram, and a global section is specified by a coherent family of sections.
Similarly, the superconnection A determines a sheaf * A on X whose sections over f : S → X are the super vector space of forms in * (S, f * V ) annihilated by A. Global sections of * A are the super vector space * . Now, the data of (V, A) is determined, by hypothesis, by the same coherent family of objects in (2.4) as T . Suppose we are given a coherent family of sections. Individually, the bottom row of (2.4) specifies an element of * (X 0 , V ) which is invariant under the Isom(R 0|1 ) ∼ = R 0|1 Z/2-action; this means it is even and closed, i.e., a section of 0 A (X 0 ). Similarly, the second row by itself specifies a section of 0 A (X 1 ), and the coherence conditions involving vertical maps say these two things determine a section of 0 A (X). The correspondence between sections of T and 0 A is clearly bijective. Replacing T with T ⊗ T 1 in the above argument amounts to replacing V with its parity reversal V (cf. Hohnhold et al. [16, proposition 6.3]).
Recall that a flat superconnection defines a differential on the space of forms with values in the corresponding vector bundle.

Proposition 2.7. Concordance classes of EFTs are in bijection with cohomology classes:
Proof. By naturality of the correspondences in the previous proposition, it suffices to show that n A (X)/concordance ∼ = Hn( * (X, V ), A). Suppose, first, that the closed forms ω 0 , ω 1 ∈ * (X, V ) are cohomologous, i.e., ω 1 − ω 0 = Aα. Then satisfies i * j ω = ω j , j = 0, 1. (Here, we use the same notation for an object over X and its pullback via pr 1 : X × R → X; as usual, t is the coordinate on R.) Conversely, suppose we are given a closed form ω ∈ * (X × R, V ) with ω j = i * j ω, j = 0, 1. We need to find a form α ∈ * (X, V ) such that ω 1 − ω 0 = Aα. Schematically, it will be α = − X×[0,1]/X ω. More precisely, we need to define α f ∈ * (S, f * V ) for each S-point f : S → X. That will be given by the fiberwise integral 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. Thus Applying the theorem to the trivial twist T , we get the following.

Corollary 2.8. For any differentiable stack X, there is a natural bijection
between T n -twisted 0|1-EFTs over X and closed differential forms of parityn on X. If X is an orbifold, passing to concordance classes gives an isomorphism with Z/2-graded delocalized cohomology 0|1-EFT T n [ X] ∼ = Hn deloc (X).
Remark 2.9. Replacing B(X) with the stack of connected 0|1-dimensional manifolds over the orbifold X, we arrive at the notion of topological twists and (twisted) field theories. The basic twist T 1 lifts in a natural way to B top (X), and in this case T n in fact depends on n, and not just on its parity. In an entirely analogous way to Hohnhold et al. [16], one can show that where the latter identification requires the assumption that X is an orbifold.

Twisted de Rham cohomology for orbifolds.
In this section, we review the construction of delocalized twisted de Rham cohomology for orbifolds due to Tu and Xu [27], and show that it can, in fact, be interpreted as concordance classes of suitably twisted 0|1-EFTs. In view of Proposition 2.7, this is not necessarily surprising; the point here is to give explicit descriptions allowing us to show, in Sect. 5, how the relevant twist arises, in a natural way, from dimensional reduction. Let X be an orbifold andX a gerbe with Dixmier-Douady class α ∈ H 3 (X, Z), both with presentations as in "Appendix B" section. Then the (Z/2-graded) delocalized twisted cohomology groups H * deloc (X, α) are defined to be the cohomology of the complex (2.10) Here, * stands for the Z/2-graded de Rham complex, X is the inertia orbifold, is the 3-curvature ofX pulled back to X, and (L , ∇ ) is a line bundle with flat connection we will describe below. (This differs from the definition of Tu and Xu [27, section 3.3] in that we perform the usual trick to convert between Z/2-graded and 2-periodic Z-graded complexes, and we have chosen a more convenient constant in front of , which produces an isomorphic chain complex. We also remark that changing the gerbe with connective structure representing the class α produces a noncanonically isomorphic complex; a specific isomorphism between the complexes depends on the choice of isomorphism between the gerbes.) The line bundle L on the inertia groupoidX 1 ⇒X 0 is as follows: (1) the underlying line bundle L is the restriction of L toX 0 ⊂ X 1 , with the restricted connection ∇ ; (2) the isomorphisms s * L → t * L overX 1 are described fiberwise by the composition where f ∈ X 1 induces a morphism f : (x, g) → (x , g ) inX 1 ⇒X 0 , and we used the canonical map C → L f ⊗ L f −1 .
It is an exercise to check that ∇ is invariant, and flat (provided our central extension admits a curving, which we always assume). Now, the operator ∇ + ∧ · on * ( X, L ) is a flat superconnection on L ∈ Vect( X) (since ∇ is flat, d = 0 and ∧ = 0), and therefore gives rise to a twist T α : B( X) → Vect. Combining Proposition 2.7 with the main result of Tu and Xu [27], we obtain a field-theoretic interpretation of complexified twisted K -theory. (The compactness assumption can be dropped by using field theories and de Rham cohomology with compact support.) Theorem 2.11. For every compact orbifold X and α ∈ H 3 (X, Z), there are natural bijections Remark 2.12. 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 standard in the literature.
Then L is topologically trivial and the connection ∇ is d + A|X 0 ; flatness is due to the fact that d A|X 0 = (t * B − s * B)|X 0 = 0 since s = t onX 0 . To describe the isomorphism s * L → t * L , we use as input h ∈ C ∞ (X 2 , C × ), and we just need to specify a C ×valued function H onX 1 , and we find thatf v = ( f g, zwh( f, g)) and Thus, using the cocycle condition (4.8) for the triple (g , f, f −1 ), we get (2.13)

Torsors and Bordisms Over an Orbifold
In this section, we provide a manageable model of the bordism category 1|1-EBord(X) as a category internal to symmetric monoidal stacks. This will also fix notation used in the remainder of the paper.

Basic definitions.
We start recalling the construction of Euclidean bordism categories over a manifold X , then note that it immediately generalizes to the case of bordisms over a stack X, and finally recast the result in the language of torsors for a given Lie groupoid presentation of X. Given integers d, δ ≥ 0 (subject to certain conditions) and a manifold X , Stolz and Teichner [26] construct a bordism category d|δ-EBord(X ), which we briefly review here. It is a category internal to the category of symmetric monoidal stacks; that is, it is given by symmetric monoidal stacks d|δ-EBord(X ) i , i = 0, 1, called the stack of objects and the stack of morphisms respectively, together with functors standing for composition, source, target and unit, satisfying the expected conditions up to prescribed natural transformations (associator and left and right unitors, similar to the data of a bicategory). In the stack of objects d|δ-EBord(X ) 0 , an object lying over S is given by the following collection of data: (1) a submersion Y → S with d|δ-dimensional fibers and Euclidean structure (in the sense of [26, section 4.2] or, more succinctly, [23, appendix B]); The S-family Y c is called the core. A morphism in the stack of objects is given by a germ (around the cores) of (G, M)-isometries respecting the maps to X . Thus, Y ± should be thought as germs of collar neighborhoods of the core; they are a technical device needed, among other things, to define the composition functor c. In the stack of morphism d|δ-EBord(X ) 1 , an object lying over S is given by the following collection of data: (1) a submersion → S with d|δ-dimensional fibers and Euclidean structure; (2) a map f : → X ; The maps of item (4) are "parametrizations of the boundary", and are subject to certain conditions formalizing this idea. A morphism in the stack of morphisms is given by (1) isomorphisms in the object stack between the respective incoming and outgoing boundaries and (2) an isometry between the 's (or, more precisely, germs of isometries around their cores, that is, the region between Y c in and Y c out ), respecting the maps to X and the parametrizations of the boundaries. The symmetric monoidal structures in the stacks of objects and morphisms are given by fiberwise disjoint union. Now, turning to aspects more specific to this paper, we note that is easy to extend the above definition of bordism category to the case where X is replaced by a "generalized manifold", or stack X: an object in d|δ-EBord(X) 1 is given by an object in d|δ-EBord(pt) 1 together with an object of X (which, by the Yoneda lemma, corresponds to a map ψ : → X in the realm of generalized manifolds) and the corresponding boundary information, which we will not detail here. A morphism over f : S → S in the stack of bordisms is determined by a fiberwise isometry F : → covering f (and suitably compatible with the boundary information) together with a morphism ξ between objects of X as indicated in the diagram below.
The bordisms-over-stacks point of view is very natural from the perspective of geometric structures. The treatment of rigid geometries in [26], and in particular Euclidean structures, can be interpreted as the definition of a stack of atlases. Then, letting X denote the stack of Euclidean atlases, we recover 1|1-EBord from the plain topological bordism category as 1|1-Bord(X). We will further develop this idea elsewhere.
Finally, we assume X is a differentiable stack with Lie groupoid presentation X = (X 1 ⇒ X 0 ), and recall a convenient way to describe maps → X, namely as Xtorsors. (See Behrend and Xu [3, section 2.4] for a full account of the theory of torsors.) An X -torsor over is given by (1) a submersion π : U → , (2) an anchor map ψ 0 : U → X 0 , and (3) an action map μ : The conditions required of μ make it equivalent to the data of a map ψ 1 such that is an internal functor satisfying the condition that is a diffeomorphism; in that case, μ can be recovered as the inverse to the above followed by projection onto the second factor. We will often denote the torsor simply by ψ, and write ( , ψ) for a bordism in 1|1-EBord(X).
A morphism of torsors is an equivariant map between the corresponding U 's. Thus, given a second bordism ( , ψ ) equipped with an X -torsor, which, more specifically, is given by the data → covering f and compatible with the boundary data, together with an equivariant map λ : U → U covering F and compatible with the anchor maps: ψ 0 •λ = ψ 0 (this, again, is taken up to a suitable germ equivalence relation). When π = π , the datum of λ is equivalent to an internal natural transformation between the internal functors determined by ψ, ψ ; namely we set : U → X 1 to be the composition The stack of objects 1|1-EBord(X) 0 has an analogous description. To fix our notation, which closely follows the previous discussion, an object here is given by (1)

Skeletons.
To get an intuitive understanding of bordism categories over a stack X, avoiding torsors, we can think as follows. First, fix a Lie groupoid X 1 ⇒ X 0 presentation of X. Then some bordisms and isometries between them can be represented by pictures like the following.
In fact, the ψ and ξ above relate to their counterparts in (3.1) by postcomposition with the atlas X 0 → X respectively whiskering with the natural transformation between the two maps X 1 ⇒ X. Not every bordism is of this form, but in a fully extended framework it is intuitively clear that every bordism can be expressed as a composition of those; it is also not hard to conceive relations between those basic building blocks. The notion of skeletons, which we introduce now, is essentially a way of dealing systematically with these generators and relations in our case of interest. From now on, we assume that X is an orbifold groupoid, so that the submersions π underlying all X -torsors are étale. We then define a skeleton of a fiberwise connected S-family (Y, ψ) ∈ 1|1-EBord(X) 0 to be given by a map ι : S × R 0|1 → U such that the composition π • ι gives a Euclidean parametrization of the core Y c . In general, a skeleton is given by a skeleton for each connected component.
A skeleton of a fiberwise connected S-family ( , ψ) ∈ 1|1-EBord(X) 1 is given by skeletons for the incoming and outgoing boundary components, together with the following: (1) A collection I i , 0 ≤ i ≤ n, of S-families of superintervals, as defined in "Appendix A" section. We denote the inclusion of the outgoing and incoming boundary components by It is not required that these intervals have strictly positive length. (2) For each i, an embedding I i → U , by which we mean a Euclidean map from a neighborhood of the "core" Here and in what follows, we write We require the following conditions of the above data: (2) if is a family of supercircles, then we also have π • b n = π • a 0 ; (3) if has boundary, then the maps a 0 and b n , together with the parametrization of the boundary, induce skeletons on each boundary component; we require that those agree with the initially given skeletons.
These conditions mean, intuitively, that the superintervals I i → U π − → prescribe an expression of as a composition of shorter pieces (right elbows and length zero left elbows) in 1|1-EBord, together with expressions of each of these pieces in the form (3.3).
We will use the shorthand notation I = {I i } to refer to the skeleton, and ( , ψ, I ) to refer to a bordism with a choice of skeleton.
Notice that b i−1 and a i are isomorphic in the groupoid of (S × R 0|1 )-points of U × U ⇒ U , since their images in agree; we denote by j i : S × R 0|1 → U × U the unique morphism b i−1 → a i , that is, the unique map such that We denote by 1|1-EBord(X) skel i , i = 0, 1, the variants of 1|1-EBord(X) i where each bordism and boundary component comes with a choice of skeleton; morphisms in these stacks are just morphisms in the old variants, after forgetting the skeleton. There is a canonical choice of skeleton on the composition of bordisms with skeleton. With this observation, we obtain an internal category 1|1-EBord(X) skel .
This is clear, since all spaces of skeletons are contractible. It is also clear that 1|1-EBord(X) does not depend on the choice of a Lie groupoid presentation for X, since it only makes reference to torsors over it. On the other hand, the definition of 1|1-EBord(X) skel does make explicit reference to the groupoid X 1 ⇒ X 0 , so the notation is slightly abusive. This is harmless, as shown by the previous proposition.
Remark 3.6. Evidently, we can form a pullback of ( , ψ, I ) ∈ 1|1-EBord(X) skel 1 via a map f : S → S by simply choosing a cartesian morphism λ : ( , ψ ) → ( , ψ) covering f in 1|1-EBord(X) 1 and any skeleton I for ( , ψ ). However, we note that there is a canonical choice to be made: we ask that I, I have the same indexing set and is cartesian for all i. We denote that skeleton by λ * I , the endpoints a i by λ * a i , etc.
We call the collection of maps I i → U π → the associated triangulation of the skeleton I . Triangulations such that the diagrams Finally, suppose we have ( , ψ) ∈ 1|1-EBord(X) 1 and two skeletons. Then we say that I is refinement of I if there is a surjective map r : I → I, such that, for each i ∈ I, r −1 (i) indexes a collection I i 1 , . . . , I i n ⊂ U where b i k = a i k+1 for each 1 ≤ k < n and a i 1 = a i , b i n = b i ; in words, the I i k , 1 ≤ k ≤ n, are adjacent subintervals whose concatenation is precisely I i . We denote by R I I : (ψ, I ) → (ψ, I ) the morphism having the identity as its underlying torsor map.

The globular subcategory; the superpath stack. Denote by
the sub-prestack with the same objects but containing only those morphisms (F, λ): the sub-prestack containing only those morphisms that map into 1|1 -EBord(X) glob 0 via the source and target functors. These two objects fit together into a "globular" internal category 1|1-EBord(X) glob , which can be thought of as a smooth bicategory.
For each test manifold S, we obtain from each of the above variants a category internal to symmetric monoidal groupoids. Those internal categories are fibrant in the sense of Shulman [22], and they clearly determine the same symmetric monoidal bicategory. Thus, the inclusion 1|1-EBord(X) glob → 1|1-EBord(X) skel ought to be considered as an equivalence of internal categories. Since we do not know of a comprehensive theory of internal categories to quote from, we will leave this as an informal statement. Our construction of twists and twisted field theories below will be based on the globular variant. This provides some slight simplifications and allows us to focus on the more conceptual side of the discussion. More specifically, in order to extend our construction of the twist functor T in Sect. 4 from 1|1-EBord(X) glob to 1|1-EBord(X) skel , it is necessary (and sufficient) to choose a stable trivialization of the gerbeX (passing, if needed, to a finer Lie groupoid presentation of X), as the reader unsatisfied with the argument of the previous paragraph should be able to verify.

Some examples.
We give here an essentially complete description of 1|1-EBord = 1|1-EBord(pt). Examples of objects and morphisms in 1|1-EBord(X) can then be constructed by pulling back to a general base space S and choosing a torsor representing a map to X.
The object stack 1|1-EBord 0 contains an object sp given by the manifold There is a similar (but nonisomorphic) object sp with Y + and Y − interchanged. Both of them have Z/2 as automorphism group, generated by the flip fl : R 1|1 → R 1|1 which acts by −1 on odd functions.
Objects in 1|1-EBord 1 are given by fiberwise disjoint unions of one of four kinds of basic bordisms: superintervals, left and right elbows, and supercircles. The basic building blocks are as follows.
(1) The left elbow of length 0, L 0 ∈ 1|1-EBord(sp sp, ∅), has R 1|1 as underlying manifold. The boundary parametrization L 0 ← sp sp, in terms of the underlying manifolds, is the map id id. (2) The right elbows R r ∈ 1|1-EBord(∅, sp sp) form a R 1|1 >0 -family, parametrizing the "super length" r . The underlying family of Euclidean manifolds is R , and the parametrization of the boundary is where T r : (r, s) → (r, r · s) is the translation on affine Euclidean space R 1|1 by the amount specified by the R 1|1 >0 parameter.
Some isomorphisms in 1|1-EBord 1 are listed below. They restrict to the identity on the boundaries. Isomorphisms (1) and (4)  (1) The last of them is an isomorphisms of (R 1|1 >0 × R 1|1 >0 )-families, and r 1 , r 2 indicate the coordinate function of each factor; more formally, the R's indicate pullbacks of R r along the multiplication respectively projection maps R >0 . Similar bordisms L,R, etc., are obtained by reversing the roles of sp and sp. They satisfy analogous relations to the above, and there are also isomorphisms A family I r ∈ 1|1-EBord(sp, sp) of intervals of length r is obtained by composing L 0 and R r along the common sp boundary: Similarly, a family K r ∈ 1|1-EBord(∅, ∅) of supercircles of length r is obtained from elbows and the braiding isomorphism in the way indicated in Fig. 1. There are stacks P and K of superintervals respectively supercircles. More generally, we write K(X), P(X) → 1|1-EBord(X) glob 1 for the stacks of supercircles respectively superintervals over X.
Remark 3.8. Hohnhold et al. [15, theorem 6.42] provide generators and relations for a variant of 1|1-EBord (in their case, unoriented and satisfying a positivity condition, but, more importantly, not extended up to include isometries of bordisms). Since our goal is to give some examples of field theories, and not a classification, we will be satisfied with a somewhat informal approach to constructing functors between internal categories. In fact, we will explain our constructions in detail on superintervals, and we will be less explicit about extending fibered functors on P(X) to full-blown internal functors. Such details are usually easy to guess, and compatibility with relations (1)-(4) above will be easy to verify.

Twists for 1|1-EFTs from Gerbes with Connection
Let X be an orbifold andX → X be a gerbe with connective structure, presented by a Lie groupoid X = (X 1 ⇒ X 0 ) respectively a central extension having L → X 1 as the underlying line bundle with connection, as in "Appendix B" section. The goal of this section is to associate toX a Euclidean 1|1-twist . This construction is drastically simplified by the fact that it takes values in the subcategory BLine → Alg, where Line denotes the symmetric monoidal stack of complex line bundles and BLine the internal category having trivial stack of objects and Line as stack of morphisms. In other words, the only relevant algebra in this construction is the monoidal unit C ∈ Alg, and the only relevant modules are the invertible ones.

Construction of the twist functor.
At the level of object stacks, there is no work to do. We just need to describe a map 1|1-EBord(X) glob 1 → Line of symmetric monoidal stacks, which by abuse of notation we still call T , together with natural isomorphisms μ and , the compositor and unitor (cf. [26, defintion 2.18]). We start discussing the underlying fibered functor T .
Fix a fiberwise connected S-family ( , ψ, I ) of bordisms with skeleton. Our goal is to describe a line bundle T ( , ψ, I ) over S. Recall (3.4) that the skeleton I determines a collection of maps j i : S × R 0|1 → U × U . We set Here and in what follows, we write, for any map f : As to morphisms, we initially consider two cases.
(1) λ : ( , ψ, I ) → ( , ψ , I ) is a refinement of skeletons, i.e., the underlying torsors are equal, the torsor map λ is the identity, and I is a refinement of I .  (2) is more interesting. We denote by SP i : Lã i → Lb i the super parallel transport of ψ * 1 L along J i , and by the gerbe multiplication map, or any other map obtained by adjunction. Then we finally consider the composition Since we are working with globular bordisms,ã 0 andb n are identities, so Lã 0 , Lb n are trivial and we let be adjoint to the above. Proof. Initially, we will assume we can pick compatible refinements for (families of) triangulations of bordisms whenever needed, and explain at the end of the proof how to deal with the fact that such refinements do not always exist. Fix a morphism λ : ( , ψ, I ) → ( , ψ , I ) in 1|1-EBord(X) glob and compatible refinements for the triangulations of and . This yields refinementsĪ ,Ī of I respectively I . We can then express λ as the composition corresponding to the same torsor map as λ, but relating objects with different skeletons. This fixes T (λ).
Of course, we need to check that taking this as a definition for L(λ) is consistent, that is, independent on the choice ofĪ andĪ . Verifying this in the case when all triangulations involved admit compatible refinements boils down to checking that if the original triangulations of and were already compatible, applying formula (4.2) would give But this is easy to see. When calculating T (λ), each appearance of SP i in (4.1) gets replaced by a composition where SP i j , 0 ≤ j ≤ k, denotes parallel transport along a subdivision J i j ⊂ J i and the h i j are tautological identifications involving L i j = C. Thus our claim follows from compatibility of super parallel transport with gluing of superintervals. Next, we verify that L respects compositions of isometries, at least when compatible refinements can be chosen. So let us fix composable morphisms as in the diagram below.
We can assume the all skeletons are compatible. Using (4.1) and the structure maps of the gerbe, we see that (up to braiding),

It follows that
Next, suppose we have a morphism λ : (ψ, I ) → (ψ , I ) where compatible refinements of the underlying triangulations fail to exist. Since every morphism in 1|1 -EBord(X) skel 1 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. Write c i for the common value of π • b i−1 = π • a i : S × R 0|1 → , and define c i similarly. Then the nonexistence of a common refinement means that there is a pair c i , λ * c i : S × R 0|1 → "crossing over" one another; more precisely, in a Euclidean local chart, neither (c i ) red ≤ (λ * c i ) red nor the opposite holds. It suffices to define T (λ) in a small neighborhood S p of each point p ∈ S where that happens; assuming, for simplicity, that the triangulations {c j }, {λ * c j } are identical except for the problematic index i, it suffices to analyze the situation in a small neighborhood in of the point x = c i ( p) = λ * c i ( p). Then we can choose d 1 : S p × R 0|1 → | S p sufficiently close to c i satisfying either d( p) < c i ( p), λ * c i ( p) or the opposite inequality. Denote by I 1 , I 1 the modifications of I , I obtained by replacing c i , λ * c i with d 1 . Then of course I and I 1 admit a common refinement, and so do I and I 1 ; moreover, I 1 and I 1 are based on the same triangulation of . We have a commutative square ( , ψ, and this stipulates the value of T on λ : ( , ψ, I ) → ( , ψ , I ); 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 d 1 . Suppose we repeat the above procedure using a different choice d 2 : S p × R 0|1 → | S p ; to compare them, we can use a third d 3 : S p × R 0|1 → | S p (restricting, perhaps, to a smaller neighborhood S p ) which stays away from both d 1 and d 2 . Thus, we can assume without loss of generality that d 1 and d 2 stay away from one another. We have a commutative diagram ( , ψ, I 1 ) t t j j j j where the skeletons appearing in each triangle and the in middle square admit common refinements, and it follows that T (λ), as prescribed by (4.4), is independent on the choice of d 1 . Similarly, we can reduce the verification that T respects composition of morphisms to the case where all triangulations involved admit compatible refinements. So far, we have defined T on fiberwise connected families of bordisms. Since the symmetric monoidal structure of 1|1-EBord(X) glob 1 is free, we are done.
The functor T is compatible with composition of bordisms in an obvious way, and we will not spell out the definition of the compositor and unitor promoting T to a functor of internal categories.

On the choice of presentations.
We need to argue that T , as constructed above, depends only on the gerbe with connectionX → X, and not on the chosen Lie groupoid presentations for X andX. To formulate this statement more precisely, we introduce some notation. Recall that Proposition 3.5 justified the lack of reference to X 1 ⇒ X 0 in the notations 1|1-EBord(X) skel and 1|1-EBord(X) glob . In this subsection, we must be explicit about the choices of presentations, so we will write 1|1-EBord(X • ) = 1|1-EBord(X) glob .
Then, what we have achieved in Sect. 4.1 is the construction, from a Lie groupoid X 1 ⇒ X 0 and a central extension L → X 1 , of a functor of internal categories Suppose now that X 1 ⇒ X 0 and the central extension L → X 1 provide a second presentation of X and the gerbeX. By [3, proposition 4.15], there exists a Lie groupoid X 1 ⇒ X 0 and a central extension L → X 1 together with a Morita morphism X • → X • and a Morita morphism of central extensions L → L, as well as similar data for L → X 1 ⇒ X 0 . These Morita morphisms are uniquely determined, up to unique natural isomorphism, by the requirement that they induce the identity ofX → X [3, section 2.6].
We now have a diagram as follows.
T L t t j j j j j j j j j j j Alg The claim that T L determines a twist functor T = TX depending only on the gerbeX over X is formalized by the following statement. Proposition 4.6. In the above situation, there exist canonical 2-morphisms making (4.5) commute.
The proof is just a verification that the construction of T L is natural with respect to internal functors between Lie groupoids (of which Morita morphisms are particular cases), so we will omit further details.
Remark 4.7. Not every gerbeX over X is necessarily presentable as a central extension of a given presentation X 1 ⇒ X 0 ; this condition is equivalent toX admitting a trivialization when restricted to X 0 [3, proposition 4.12]. Thus, givenX, we need to pick a sufficiently fine groupoid presentation of X. In this paper, we never letX vary or make structural statements about the groupoid of twists 1|1-ETw(X), so we are allowed to fix, once and for all, a Lie groupoid X 1 ⇒ X 0 suitable forX.

The restriction to K(X).
We denote by K(X) the substack of closed and connected bordisms in 1|1-EBord(X) glob . The twist functor TX determines, by restriction, a line bundle Q on K(X). Our goal in this section is to give a detailed description of Q, in terms of aČech cocycle for Deligne cohomology representing the gerbeX → X.
Let us start fixing some notation. The orbifold X will be presented, as before, by an étale Lie groupoid X 1 ⇒ X 0 , and the gerbeX → X will be presented by aČech 2-cocycle for groupoid cohomology with coefficients in the Deligne complex C × (3), More explicitly, this cocycle is given by a triple where X n = X 1 × X 0 · · · × X 0 X 1 is the space of sequences of n composable morphisms.
To an object (ψ, I ) ∈ K(X) S as above the fibered functor assigns the trivial line bundle over S; the interesting discussion, of course, concerns morphisms. Fix a second object (ψ , I ) ∈ K(X) S and a morphism λ : (ψ, I ) → (ψ , I ) over f : S → S . To that Q assigns a linear map between the corresponding lines, which we identify with a function Q(λ) : S → C × . We consider the two special cases of Sect. 4.1, using the notation fixed there.
. (4.12) Here, D ∈ C ∞ (T U ) is a choice of Euclidean vector field for the Euclidean structure induced by π : U → K , and vol D the corresponding volume form (cf. "Appendix A" section). Moreover, ψ 2 : U × K U × K U → X 2 denotes the map induced by ψ 1 .
Proof. The first claim is obvious. As to the second claim, each h i in (4.1) contributes one factor in the product, and each SP i contributes one term in the summation. In fact, we see easily from Proposition A.1 that super parallel transport on along a superinterval J with respect to the connection form d − A is given by exp ( J vol D D, A ). All terms of the from h( f,

Dimensional Reduction of Twists
In this section, we show that dimensional reduction of the 1|1-twists from Sect. 4 recovers the 0|1-twists described in Sect. 2.3. So we start with the twist functor TX ∈ 1|1-ETw(X) associated to a gerbe with connectionX, and describe its pullback to 0|1-EBord T ( X) by the functor in (1.2). As we will see below, that corresponds to the data of a line bundle with superconnection on X. We will then find that it is flat, and hence, by Proposition 2.3, descends to a line bundle T X on B( X) = T X/ / Isom(R 0|1 ), or, in other words, a 0|1-dimensional Euclidean twist over X. We call T X the dimensional reduction of TX.
Theorem 5.1. Let X be an orbifold,X a gerbe with connection, and α ∈ H 3 (X; Z) its Dixmier-Douady class. Then the twist T X ∈ 0|1-ETw( X) obtained by dimensional reduction of the twist TX ∈ 1|1-ETw(X) is isomorphic to the twist T α from Theorem 2.11.
This means, in particular, that and suggests that TX is the correct 1|1-twist to represent α-twisted K -theory in the sense of (1.1). The proof of the theorem (and the flatness claim necessary to state it) will occupy the remainder of this section. Before getting started, we record a technical lemma. Lemma 5.2. Let X be an ordinary manifold, ev : T X × R 0|1 → X be evaluation map, and writeω for the function on T X corresponding to the differential form ω ∈ n (X ). Then we have where the sign ± is −1 if n ≡ 2, 3 mod 4 and +1 otherwise.
Proof. It suffices to prove the lemma for ω = f 0 d f 1 . . . d f n , where f i ∈ C ∞ (X ). The action μ : T X × R 0|1 → T X is given by the formula where D denotes the de Rham vector field on T X. Thus Contracting an expression like the one under the summation sign with ∂ θ produces as many new terms as there are F i 's, and, in each of those, one of the F i 's get converted into a D f i . Note also that commuting i ∂ θ with any F i or D f i produces a minus sign. Iterating this process, we find Corollary 5.3. For D the de Rham vector field on T X, μ : T X × R 0|1 → T X the action map, π : T X → X the projection, and the remaining notation as in the lemma, μ * D ∧n , π * ω = ±n!(ω + θ dω).

Review of dimensional reduction.
In this subsection we recall the main points about our dimensional reduction procedure, also fixing the notation. See [23] for the complete story.
It is sufficient to describe the effect of the internal functors (1.2) on the corresponding stacks of closed and connected bordisms, which we denote The left stack was already introduced in Sect. 2, and is given by The middle stack is defined as and the map L is induced by the group homomorphism Next we turn to a brief description of the map R, focusing on our case of interest. We fix an étale Lie groupoid presentation X 1 ⇒ X 0 for X, so that we also get presentations of T X, X and T X respectively. Then we have an atlaš Now, we want to describe the TX 0 -family classified by the map R •x, which we will denote Kx . Chasing through the construction of [23], we find that is given, in the language of torsors, by the following data: (1) the trivial family K = TX 0 × T 1|1 of length 1 supercircles, together with the standard covering U = TX 0 × R 1|1 → K , (2) the map ψ 0 : U → X 0 given by the composition the map ψ 1 : U × K U → X 1 which, over the component of points differing by n units, is the n-fold iterate of wherex is the versal family for X.
In this picture, the map L simply forgets P. The subtle aspect about R is that the map ψ : K → X is not simply the composition but, rather, is given by a descent construction using the canonical automorphism of the inertia X.
Eventually, we will need to understand the action of R 1|1 ⊂ Isom(R 1|1 ) on Kx by rotations. More precisely, the natural isomorphism between the stack maps leads to an isomorphism This will be described later (cf. Fig. 2), but we would like to notice two useful facts now. First, R has image in the substack of length 1 supercircles, so that the R 1|1 -action comes from the action on T 1|1 by rotations. Second, an expression of Kx as a composition of more basic bordisms (left and right elbows and braiding) is obtained as follows. Let

The underlying line bundle.
Our goal for the remainder of this proof is to understand the restriction of the line bundle Q from Sect. 4.3, which we call Thus, Q is identified with line bundle with superconnection on X (cf. Remark 2.5).
Let us not worry about the superconnection for now and simply describe the underlying line bundle. Thus our goal is to describe the line bundle on TX 0 (which we still call Q ) induced by the atlasx and the isomorphism H between its two pullbacks via the source and target maps TX 1 ⇒ TX 0 .
As a warm-up, pick a point (x ∈ X 0 , g : x → x) ∈ TX 0 . It determines a point of K(X) consisting of the length 1 constant superpath x in X with its endpoints glued together via g. Thus Q (x,g) = L g . From this we already conclude that Q → TX 0 is trivial, since we assume the same of L. Now fix a second point (x , g ) ∈ TX 0 and a compatible morphism f : x → x ; meaning that g = f g f −1 . This gives rise to a morphismf in K(X) which, in the same vein as above, we can describe as being the constant f . From (4.12), we conclude that the identification H ( f ) : which agrees with (2.13).
Since we worked above with points, we have only shown that the restriction of Q to X → T X agrees with the L from Sect. 2.3. We want a slightly stronger statement, namely we want to identify Q with the a pullback of L via T X → X.
To do this, we need to fully describe the isomorphism H , which we see as a function H : TX 1 → C × . Consider again the versal familyx : TX 0 → B T (X). Then H is determined by the composition TX 0x 6 6 n n n n From (4.12) and the above description of Kx = R •x, we see that where β is the composition the integral is fiberwise over TX 1 , and f , g are similar to the paragraph above (except now they stand forX 1 -points of X 1 instead of pt-points). From Lemma 5.2, we have so that Seeing exp(−B) : TX 0 → C × as an isomorphism exp(−B) : Q → π * L , (5.6) of trivial line bundles, H gets identified with the defining datum (2.13) of L , as desired.

The superconnection.
A superconnection on the line bundle L : X → Vect consists of a superconnection A on the underlying line bundle L →X 0 whose two pullbacks overX 1 are identified with one another through the isomorphism s * L → t * L . Since we just want to describe the superconnection onX 0 (which we know a priori to be invariant), it suffices to look at the versal family and its image in K(X); nothing here will involveX 1 . Now, the superconnection we are seeking to describe is geometrically encoded by the R 1|1 -action on TX 0 and the line bundle Q over it. More specifically, the operator A is the infinitesimal generator associated to the vector field D = ∂ θ − θ∂ t . Thus, we need to understand the action of R 1|1 by rotations of the TX 0 -family Kx , i.e., the isomorphism and its image under TX, where μ, pr : TX 0 ×R 1|1 → TX 0 are the action respectively projection maps. The isomorphism (5.7) can be expressed as a composition of simpler steps as indicated in Fig. 2. There, the left elbows always represent the bordism Lx , or, more precisely, its pullback via pr : TX 0 ×R 1|1 → TX 0 ; the right elbows [a,b] represent bordisms R [a,b] x with the indicated skeleton [a, b]; straight and crossing lines denote appropriate identities and braidings or, more precisely, their avatars as thin bordisms. Thus, for instance, the second picture represents the composition (leaving implicit, as usual, pullbacks via projection maps). The isometries between successive pictures are the obvious ones. For example, the second isomorphism is the semigroup property of right elbows and the fourth uses the symmetry condition of the braiding. Now, the image of each of the intermediate steps under TX is a canonically trivial line bundle (over TX 0 × R 1|1 ), and, with the exception of the fifth step, the corresponding isomorphism of line bundles is the identity. In fact, under the canonical trivilizations of all line bundles in question, we have an identity The right-hand side, once identified with a C-valued function on TX 0 , can be calculated from (4.12) and is given by   Here, the second equality uses Corollary 5.3 and the fact that d A vanishes, the integral being fibered over TX 0 .
Thus, the R 1|1 -action on Q = TX 0 × C, expressed as an algebra homomorphism and the infinitesimal action sends The superconnection corresponding to this odd, fiberwise linear vector field on the total space of Q is given by the formula d + A and, applying the gauge transformation (5.6), we find that the superconnection on L is given by the formula which agrees with (2.10).

1|1-EFTs and the Chern Character of Twisted Vector Bundles
Fix an orbifold X, a gerbe with connectionX and anX-twisted vector bundle with connection V, with the usual notation of "Appendix B" section. In this section, we associate to V a 1|1-dimensional T = TX-twisted field theory E = E V 1|1-EBord(X) glob 1 % % T 9 9 E Alg and show that its dimensional reduction provides a geometric interpretation of the twisted orbifold Chern character.
IfX is trivial, and thus V is just a usual vector bundle over X, the basic idea behind the construction of the field theory E = E V ∈ 1|1-EFT(X) is the following. To a posivitely oriented superpoint of X, specified by a map to the atlas X → X, we assign the vector space V x(0) . To a superinterval as in (3.3), we assign the super parallel transport along ψ * V . Orientation-reversed manifolds map to the dual vector spaces and maps, and the image of elbows is determined by the duality pairing. We need to check that this is consistent with isometries between bordisms, in particular those of the form (3.3). But, indeed, the data of the vector bundle V on X induces a superconnection-preserving bundle map ψ * V → (ψ • F) * V ; this ensures that E V is an internal functor. Now, if V is twisted by a nontrivial gerbe, then ψ * V and (ψ • F) * V only become isomorphic after tensoring with an appropriate line bundle, namely ξ * L. As we will see, this deviation from functoriality is expressed by the fact that E V is a twisted field theory, and the relevant twist is the TX from Sect. 4.
Remark 6.1. The natural transformation E is not invertible, so we must choose between the lax or oplax variants. We make the choice that better fits with our conventions for twisted vector bundles. Now, restricting to the moduli stack of closed, connected bordisms K(X), the twist T gives us a line bundle, and we made the convention that E maps 1 → T , and not the opposite, so that its partition function (i.e., the restriction E| K(X) ) determines a section of T | K(X) , and not of its dual. More specifically, E assigns to an S-family Y in the domain a left T (Y )-module E(Y ); thus, by construction of T , E(Y ) is nothing but a vector bundle over S. When is a positively oriented superpoint, we set where, for any f : S × R 0|1 → U , we write If Y is negatively oriented, then E(Y ) is the dual of the above. To a morphism λ : Y → Y over f : S → S , we assign the obvious identification which makes sense in view of condition (3.7). All the remaining data is determined, uniquely up to unique isomorphism, by the symmetric monoidal requirement for E.
At the level of morphism stacks, E assigns, to each S-family of bordisms from Y 0 to Y 1 , a map of T (Y 1 )-modules; using the fact that the only algebra in sight is C, we find that E( ) is a linear map of vector bundles over S. Now there are several kinds of bordisms to consider. For the sake of clarity, we focus on the superpath stack P(X) → 1|1-EBord(X) glob 1 . Fix ∈ P(X), and recall our usual notation fixed in Sects. 3.2 and 4.1. Instead of describing E( ), it is more convenient to describe its inverse which we set to be the composition where id k denotes the identity map of k<i≤n L j i , E 0 = SP 0 , and E i is the composition Here, SP i denotes the super parallel transport of ψ * 0 V along I i . Finally, there are certain conditions on the above data that need to be verified. It is clear that Y → E(Y ) is a fibered functor, so it remains to verify that → E( ) determines a natural transformation between appropriate fibered functors, and moreover that this is compatible with compositions of bordisms and identity bordisms, that is, commutativity of diagrams (3.5) and (3.6) in [18].

Proposition 6.3. The assignment → E( ) respects compositions of intervals. That is, given bordisms
in P(X) and writing = 2 Y 1 1 , the diagrams Proof. This is clear from the definitions. Proposition 6.4. The assignment → E( ) is natural in ∈ P(X), that is, for each morphism λ : → lying over f : S → S , the diagram

commutes.
Here, λ i : Y i → Y i denotes the image of λ in 1|1-EBord(X) 0 via the source and target functors for i = 0, 1 respectively.
Proof. If λ is a refinement of skeletons, the claim follows immediately from the gluing property of super parallel transport. Next, let us consider the case where the skeletons of , are compatible and the base map f : S → S is the identity. For simplicity, we may assume that the skeletons comprise three intervals I 0 , I 1 , I 2 , and that the first and last of them have length zero. The general case can be deduced by induction, using also the compatibility with compositions.
In this particular situation, (6.5) corresponds to the outer square in the following diagram.
Here, each morphism is tensored with an appropriate identity, which we leave implicit. We notice that each of the inner diagrams commutes: (a) and (f) due to the compatibility between h and m (cf. (B.1)), (b), (c) and (e) because the maps in question act independently on the various tensor factors, and (d) due to the compatibility between the connections of L and V . Thus the outer square commutes, as claimed.
Finally, we briefly explain how to extend E from P(X) to the whole of 1|1-EBord(X). First, we require the so-called spin-statistics relation, that is, that the flip of 1|1-EBord(X) maps to the grading involution of Vect. Second, recall that in Stolz and Teichner's definition, the bordism category does not admit duals, since a length zero right elbow R 0 is not allowed; this is done because they want to allow field theories with infinite dimensional state spaces. In our example, we could introduce those additional morphisms R 0 , and then E would be uniquely determined by the above prescriptions, the requirement that duals map to duals, and the symmetric monoidal property. Concretely, suppose that is a family of length zero left elbows, that is where Y denotes an S-family of positive superpoints with a choice of skeleton, say ι : S × R 0|1 → U andȲ denotes its orientation reversal, with the same underlying skeleton. Then we define E(Ȳ ) = V ∨ i and to be the evaluation pairing. The image of other kinds of bordisms is determined similarly.
Remark 6.6. A trivialization of L → X 1 allows us to extend, in a fairly obvious way, the construction above to 1|1-EBord(X) skel .

Dimensional reduction and the Chern character.
In this subsection, we finish the proof of Theorem 1.3, by showing that the diagram indeed commutes. So our goal is to study the dimensional reduction of the twisted field theory E associated to theX-twisted vector bundle V. Let us denote it by and recall, from Proposition 2.6 and Theorem 5.1 that E determines and is completely determined by an even, closed form ω ∈ * ( X; L ). The underlying form on the atlaŝ X 0 , which by abuse of notation we still denote ω, corresponds to the section under the isomorphism (5.6) and usual identification C ∞ ( TX 0 ) ∼ = * (X 0 ).
Proof. It suffices to verify that the underlying forms onX 0 agree. Our dimensional reduction procedure dictates that E (L •x) = E(Kx ), where Kx ∈ K(X) is the special TX 0 -family from Sect. 5.1. By (6.2), E(Kx ) is obtained from the linear map by taking supertrace in the End(π * V ) component. (For clarity, we are leaving pullbacks by p and i implicit.) Here, SP denotes the super parallel transport along the superinterval which Dumitrescu [10] identified with exp(−∇ 2 V ). Thus, the above homomorphism of vector bundles on TX 0 is given by Using (5.6), our chosen identification exp(−B) : Q → π * L , we get ω = str(m −1 • exp(∇ 2 V − B)) ∈ * (X 0 ; L ), which agrees with the definition (B.2) of ch(V).

Appendix A. A Primitive Integration Theory on R 1|1
Integration of compactly supported sections of the Berezinian line bundle is relatively simple to define [9]. The notion of domains with boundary requires more care, as shown by the following paradox, known as Rudakov's example: on R 1|2 with coordinates t, θ 1 , θ 2 , we consider a function u with ∂ θ 1 u = ∂ θ 2 u = 0. Then [0,1]×R 0|2 [dtdθ ] u = 0, but performing the change of coordinates t = t + θ 1 θ 2 we get 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 |. With this proviso, an integration theory featuring the expected Stokes formula still exists [4], and we would like to describe it concretely in a very special case.
Given a, b : S → R 1|1 , we define the superinterval [b, a] ⊂ S ×R 1|1 to be the domain with boundary prescribed by the embeddings We think of i a as the incoming and i b as the outgoing boundary components. To be consistent with the usual definition of 1|1-EBord, we will to assume that, modulo nilpotents, a ≥ b (cf. Hohnhold et al. [15, definition 6.41]). The fiberwise Berezin integral of a function u = f + θg ∈ C ∞ (S × R 1|1 ) on [b, a] will be denoted [b,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 ∂ t G = g, then It is also clear that any two primitives differ by a constant. We have a fundamental theorem of calculus.
To clarify the meaning of the right-hand side, when using a, b : S → R 1|1 , etc., as arguments to a function, we implicitly identify them with maps S → S × R 1|1 , to avoid convoluted notation like v(id S , b).
Proof. Using partitions of unity, it suffices to prove the analogous statement for the half-unbounded interval [b, +∞], namely Thus, the domain of integration is picked out by the equation t ≥ b 0 + b 1 θ . Performing the change of coordinates t = t − b 0 − b 1 θ , whose Berezinian is 1, we get t≥b 0 +b 1 θ As we noticed in the proof, translations on R 1|1 preserve the canonical section [dtdθ ] of the Berezinian line; the flips θ → −θ of course do not. Thus, an abstract Euclidean 1|1-manifold X does not come with a canonical section of Ber( 1 X ), but the choice of a Euclidean vector field D fixes a section, which we denote vol D . We can then restate the proposition in a coordinate-free way as follows: for any S-family of superintervals [b, a] with a choice of Euclidean vector field D, [b,a] vol D Du = u(b) − u(a).

Appendix B. Gerbes, Twisted Vector Bundles and Chern Forms
A central extension of the Lie groupoid X 1 ⇒ X 0 is given by (1) a complex line bundle L → X 1 with connection ∇ L , (2) a (connection-preserving) isomorphism h : pr * 2 L ⊗ pr * 1 L → c * L over the space X 2 = X 1 × X 0 X 1 of pairs of composable morphisms, (3) a form B ∈ 2 (X 0 ) (called curving).
In more friendly notation, for composable (S-points) f, g ∈ X 1 , h is an operation h f,g : L f ⊗ L g → L f •g .
The multiplication h must satisfy the natural associativity condition, and the curvature of L the equation ∇ 2 L = t * B −s * B. Note that d B is invariant, and therefore determines a form ∈ 3 (X), called 3-curvature. Also, there are canonical isomorphisms L| X 0 ∼ = C and L f −1 ∼ = L ∨ f . For better legibility, we will typically use L ∨ f instead of L f −1 . There is an appropriate notion of Morita equivalence for central extensions [3, section 4.3]. Then, just like differentiable stacks are Lie groupoids up to Morita equivalence, a gerbe with connectionX over X can be defined as a central extension up to Morita equivalence. Gerbes over an orbifold X are classified by classes in H 3 (X; Z), and [ ] is the image in de Rham cohomology.
If L → X 1 ⇒ X 0 is a presentation of the gerbeX, then anX-twisted vector bundle V is presented by (1) a (complex, super) vector bundle V → X 0 with connection ∇ V and (2) an isomorphism m : L ⊗ s * V → t * V of vector bundles with connection over X 1 (where the domain is endowed with the tensor product connection ∇ L ⊗ 1 + 1 ⊗ ∇ V ) satisfying certain natural conditions, namely the commutativity of the following diagrams, where x, y, z and f : x → y, g : y → z denote generic (S-)points of X 0 respectively X 1 .
The Chern character form of V is calculated from the above presentation as follows: (Recall that we write i :X 0 → X 1 for the inclusion and p :X 0 → X 0 for the map s|X 0 = t|X 0 .) Let us describe the underlying L = L|X 0 -valued differential form onX 0 in more detail. The isomorphism m : L ⊗ s * V → t * V gives us an identification exp(∇ 2 L )s * exp(∇ 2 V ) = t * exp(∇ 2 V ).
Using the fact that ∇ 2 L = t * B − s * B, we get so this defines an End( p * V )-valued form onX 0 . Now, i * m is an isomorphism i * L ⊗ p * V → p * V , and the form ch(V) is obtained by composing the coefficients of exp(∇ 2 Proof. This is easy to check directly. It also follows from Theorem 5.1 and Proposition 6.7, since we know a priori that the form ω in the statement of that proposition is even and closed with respect to the relevant differential. Remark B.4. If X 1 ⇒ X 0 is the groupoid of a finite group action on a manifold and the central extension is trivial, then V is just the data of an equivariant vector bundle. In this case, ch(V) represents the equivariant Chern character of Baum and Connes [2]. If X 1 ⇒ X 0 is Morita equivalent to a manifold, then V is what is traditionally called a twisted vector bundle, and ch(V) agrees with the definition of Bouwknegt et al. [7], Park [19], and others.
Remark B.5. Finite-dimensional twisted vector bundles only exist when the twisting gerbe represents a torsion class. Thus, it would be interesting to allow a more general target category and investigate 1|1-EFTs twisted by non-torsion classes.
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