A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY

ALEJANDRO ADEM^{∗}, YONGBIN RUAN^{∗∗}, AND BIN ZHANG

Abstract. In this paper we define an associative stringy product for the twisted orbifold K–theory of a compact, almost complex orbifold 𝒳. This product is defined on the twisted K–theory ^{τ} K_{orb}(∧𝒳) of the inertia orbifold ∧𝒳, where the twisting gerbe τ is assumed to be in the image of the inverse transgression H^{4}(B𝒳, ℤ) → H^{3}(B ∧ 𝒳, ℤ).

1. Introduction

Over the last twenty years, there has been a general trend towards the infusion of physical ideas into mathematics. One of the successful examples in the last few years is the subject of twisted K-theory. Interest in it originates from two different sources in physics, the consideration of a D-brane charge on a smooth manifold by Witten [28] and the notion of discrete torsion on an orbifold by Vafa [27]. In mathematics, there have been important developments connected to this. On the one hand, it inspired a new subject often referred to as stringy orbifold theory. On the other hand, it revitalized and re-established connections to many classical topics such as equivariant K-theory, groupoids, stacks and gerbes. For smooth manifolds, the mathematical foundation of twisted K-theory has been worked out in [6], [7], [20], [4]. For any cohomology class α ∈ H^{3}(X, ℤ), one can associate a twisted K-theory ^{α}K(X). One interesting phenomenon is the difference between a torsion class and a non-torsion one: for torsion α, we have a natural notion of twisted vector bundle or twisted sheaf; for a non-torsion α, there is no geometric notion of vector bundle and one has to use infinite–dimensional analysis.

The case of an orbifold or even a more general singular space is much more interesting. This naturally relates to equivariant theories if we specialize to the case of 𝒳 = [M/G] where M is a smooth manifold and G is a compact Lie group acting almost freely on M. One can consider a cohomology class α ∈ H^{3}(BG, ℤ) and its corresponding twisted K-theory, where BG is the classical classifying space for G. This was the set-up of [3] for orbifold twisted Ktheory ^{α}K_{orb}(𝒳) using discrete torsion. Twisted K-theory has been generalized to K-theory

Date: April 13, 2007.

∗Partially supported by the NSF and NSERC.

∗∗Partially supported by the NSF.

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 2

twisted by gerbes (see [18]), and also using the framework of groupoids (see [16]). In the general case, one can think that the twisting is a cohomology class α ∈ H^{3}(B𝒳, ℤ) where B𝒳 is now the classifying space of the orbifold 𝒳.^{1}

One advantage of working with orbifolds is the nontrivial cohomological counterpart called Chen-Ruan cohomology of orbifolds, H_{CR}^{∗}(𝒳, ℂ). One can use discrete torsion [25] (more generally torsion gerbes [23]) to obtain a twisted Chen-Ruan cohomology H_{CR}^{∗}(𝒳, ℒ_{α}). Moreover, the twisted Chen-Ruan cohomology has an important internal product making H_{CR}^{∗}(𝒳, ℒ_{α}) a ring. On the other hand, the tensor product produces a map

αK_{orb}(𝒳) ⊗ ^{β}K_{orb}(𝒳) → ^{α+β}K_{orb}(𝒳).

Note that it shifts the twisting to α + β; one natural question is if there is an internal ”stringy” product for ^{α}K_{orb}(𝒳)? Freed, Hopkins and Teleman [13] have proved the beautiful result that the twisted equivariant K-theory ^{α}K_{G}(G) for the adjoint action is isomorphic to the Verlinde algebra of representations of the central extension of the loop algebra ℒG for a semi-simple Lie group G. The Verlinde algebra carries a very important ring structure via the Verlinde product, whose structure constant encodes the information for so–called conformal blocks. Using the group structure of G, one can also construct a ring structure (via the Pontryagin product) for ^{α}K_{G}(G); these rings turn out to be isomorphic.

Due to the importance of the Verlinde product in representation theory, the existence of a stringy product on the twisted K-theory for a general space becomes a important question. This is the problem we will address in this article and its sequel.

Since the Verlinde product is ”quantum” in nature, there is a general interest in the field to construct a quantum product for twisted K-theory. Such a quantum product was first constructed for ordinary K-theory on smooth projective manifolds by Lee [17]. The main tool for the construction is Gromov-Witten theory; the main interest now is to generalize it to a setting with group actions or stacks. It remains a distant goal for Artin stacks, where the isotropy groups are higher dimensional. The main problem is the construction of Gromov-Witten theory on Artin stacks, which encounters many serious difficulties. The situation is a lot more hopeful for orbifolds (Deligne-Mumford stacks) where Gromov-Witten theory has been established by Chen-Ruan [10]. Indeed, our stringy product can be thought of as the first step towards such a quantum product. For example, the obstruction bundle appearing in our formula can be viewed as a quantum contribution.

Our main observation is that there is indeed a stringy product for the twisted K-theory of orbifolds. Moreover, the key information determining such a stringy product does not lie in

1To be totally precise, we will actually be twisting with cocycles.

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 3

H^{3}(B𝒳, ℤ) as one conventionally believes; instead, it lies in H^{4}(B𝒳, ℤ). There are two key ingredients to the construction of our product. The first key idea is as follows: given a class ϕ ∈ H^{4}(B𝒳, ℤ), it induces a class θ(ϕ) ∈ H^{3}(B ∧ 𝒳, ℤ) where ∧𝒳 is the inertia stack of 𝒳. Hence we can define a twisted K-theory ^{θ(ϕ)}K(∧𝒳). The inertia stack ∧𝒳 can be viewed as the moduli space of constant loops on 𝒳. Furthermore, there is a key multiplicative formula for θ(ϕ) characterized by the effect of ϕ on the moduli space ℳ of constant morphisms from a Riemann surface. This map, which can be thought of as the inverse of the classical transgression map, appears in [12] for finite group cohomology. Based on this we derive a simple extension for orbifold groupoids and explicitly prove its multiplicative property. A more geometric version of this appears in [19].

Our second ingredient is more subtle. Our experience from Chen-Ruan cohomology tells us that a naive definition does not give an associative product. The reason lies in the fact that the fixed–point sets X_{g}, X_{h} for g ≠ h in general do not intersect each other transversely. It is known that in Chen-Ruan cohomology theory one can correct the naive definition by introducing a certain obstruction bundle. Combining these two ingredients, we obtain an associative product which can be viewed as a K-theoretic counterpart of the Chen-Ruan product for orbifold cohomology.

Theorem 1.1. Let 𝒳 denote a compact, almost complex orbifold, and let τ be a U(1)–valued 2–cocycle for the inertia orbifold ∧𝒳 which is in the image of the inverse transgression. Then there is an associative product on ^{τ}K_{orb}(∧𝒳) which generalizes both the Pontryagin and the orbifold cohomology product.

There has been many discussions about the so–called Pontryagin product on K_{G}(G). In our language, it amounts to, for 𝒳 = ∧[∗/G] in the untwisted case. As an application, we use our construction to clarify the twisted Pontryagin product for the case of finite group G where the situation is rather subtle; it may not always exist, and when it exists, it may not be unique either. We provide an explicit calculation of the inverse transgression map for the cohomology of finite groups, showing that in fact it can be computed using the natural multiplication map ℤ × Z_{G}(h) → G, where Z_{G}(h) denotes the centralizer of h ∈ G. Using this we exhibit a group, G = (ℤ/2)^{3} and an integral cohomology class ϕ ∈ H^{4}(G, ℤ) such that under the inverse transgression it maps non–trivially for every properly twisted sector, yielding an interesting product structure on ^{θ(ϕ)}K_{G}(G).

One of the original motivations for the introduction of the twisted theory in orbifolds was the hope of describing the cohomology of desingularizations of an orbifold. Joyce constructed five classes of topologically different desingularization of T^{6}/ℤ_{4} [15], arising from a representation ℤ/4 ⊂ SU(3). It is known that Joyce’s desingularizations are not captured

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 4

by discrete torsion. For a while, there was the expectation that they may be captured by 1-gerbes. The computation in [2] shows that the high hopes for 1-gerbes is probably misplaced; however, we notice that H^{4}(B(T^{6}/ℤ_{4}), ℤ) seems to contain precisely the information related to desingularization. We hope to return to this question later.

We would like to make a comment about notation: throughout this paper we will be using the stacky language of orbifold groupoids, hence given an orbifold 𝒳 we will be thinking of it in terms of its associated stack-a Morita equivalence class of orbifold groupoids, represented by 𝒢; in this context ^{τ}K_{orb}(𝒳) is interpreted as ^{τ}K(𝒢), using the notion of twisted K–theory of groupoids, which we will summarize in section 3.

The results in this article were first announced by the second author at the Florida Winter School on Mathematics and Physics in December, 2004. Here, we present our construction for the orbifold case. The construction for general stacks will appear elsewhere. The authors would like to thank MSRI for the hospitality during the preparation of this manuscript, and the third author would like to thank the MPI–Bonn for its generous support.

2. Preliminaries on Orbifolds and Groupoids

In this section, we summarize some basic facts about orbifolds, using the point of view of groupoids. Our main reference is the book [1], but [22] is also a useful introduction. Recall that an orbifold structure can be viewed as an equivalence class of orbifold groupoids; we shall present all of our constructions in this framework.

Suppose that 𝒢 = {s, t : G_{1} → G_{0}} is an orbifold groupoid, namely, a proper, étale Lie groupoid. There are several important constructions which play a fundamental role in stringy orbifold theory. Given r > 0 an integer, we can consider the r–tuples of composable arrows in 𝒢, i.e.

G_{r} = {(g_{1}, …, g_{r}) ∈ G_{1}^{r} | t(g_{i}) = s(g_{i+1}), i = 1, …, r}

As is well known, these fit together to form a simplicial space, the classifying space B𝒢 of the groupoid. In our discussion of homological invariants of groupoids, we will be considering cochains arising from this complex. Recall that the inertia groupoid ∧𝒢 is a groupoid canonically associated with 𝒢 which is defined as follows:

Definition 2.1. For any groupoid 𝒢, we can associate an inertia groupoid ∧𝒢 as

(∧𝒢)_{0} = {g ∈ G_{1} | s(g) = t(g)}, (∧𝒢)_{1} = {(a, v) ∈ G_{2} | a ∈ (∧𝒢)_{0}}

where

s(a, v) = a, t(a, v) = v^{−1}av.

More generally, we can define the groupoid of k-sectors 𝒢^{k} as

(𝒢^{k})_{0} = {(a_{1}, a_{2}, ⋯, a_{k}) ∈ G_{1}^{k} | s(a_{1}) = t(a_{1}) = ⋯ = s(a_{k}) = t(a_{k})}
(𝒢^{k})_{1} = {(a_{1}, a_{2}, ⋯, a_{k}, u) ∈ G_{1}^{k+1} | s(a_{1}) = t(a_{1}) = ⋯ = s(a_{k}) = t(a_{k}) = s(u)}

with s(a_{1}, ⋯, a_{k}, u) = (a_{1}, ⋯, a_{k}), t(a_{1}, ⋯, a_{k}, u) = (u^{−1}a_{1}u, ⋯, u^{−1}a_{k}u).

The construction of the inertia groupoid and 𝒢^{k} in general is completely functorial. Namely, a homomorphism of groupoids induces a homomorphism between k-sectors and an equivalence of orbifold groupoids induces an equivalence betweem them. The inertia groupoid can be identified as the space of constant loops on 𝒢; more generally, 𝒢^{k} can be identified as the space of constant morphisms from an orbifold sphere with k-orbifold points to 𝒢. We will come back to these descriptions later.

Another important notion is that of a suborbifold.

Definition 2.2. A homomorphism of orbifold groupoids ϕ : 𝒢 → ℋ is an embedding if • ϕ : G_{0} → H_{0} is an immersion.

• For any x ∈ im(ϕ) ⊂ H_{0}, ϕ^{−1}(U_{x}) is a translation groupoid (⊔_{h}V_{h})/H_{x}, where the V_{h}’s are open subsets of G_{0} and H_{x} acts transitively on the components.

• |ϕ| : |𝒢| → |ℋ| is proper.

𝒢 together with ϕ is called a suborbifold of ℋ.

Remark. Suppose that ϕ : 𝒢 → ℋ is a suborbifold and let y = ϕ(x) for x ∈ G_{0}. Then U_{y} ∩ im(ϕ) = ∪_{g∈H_{y}}gV_{x}, where V_{x} is a neighborhood of x in G_{0}. Note however that |ϕ| : |G| → |ℋ| may not be one-to-one.

The following are important examples of suborbifolds.

Example 2.3. Suppose that 𝒢 = X ⋊ G is a global quotient groupoid (i.e. a quotient by a finite group). We often use the stacky notation [X/G] to denote the groupoid. An important object is the so called inertia groupoid ∧𝒢 = (⊔_{g}X_{g}) ⋊ G where X_{g} is the fixed point set of g and G acts on ⊔_{g}X_{g} as h : X_{g} → X_{hgh^{−1}} by h(x) = hx. By our definition, ϕ : ∧𝒢 → 𝒢 induced by the inclusion map X_{g} → X is an embedding.

Example 2.4. Let 𝒢 be the global quotient groupoid defined in the previous example. We would like to define an appropriate notion of the diagonal Δ for 𝒢×𝒢. The correct definition should be Δ = (⊔_{g}Δ_{g}) ⋊ G × G where Δ_{g} = {(x, gx), x ∈ X}. Our definition of suborbifold includes this example.

More generally, we define the diagonal Δ as the groupoid fibered product 𝒢 ×_{𝒢} 𝒢. One can check that Δ = 𝒢 ×_{𝒢} 𝒢 is locally of the desired form and hence a suborbifold of 𝒢 × 𝒢.

Example 2.5. For l ≤ k, there are natural evaluation morphisms e_{i_{1},··· ,i_{l}} : 𝒢^{k} → 𝒢^{l} given by

e_{i_{1},··· ,i_{l}}(a_{1}, ⋯, a_{k}) = (a_{i_{1}}, ⋯, a_{i_{l}}).

Furthermore, we have e : 𝒢^{k} → 𝒢 given by

e(a_{1}, ⋯, a_{k}) = s(a_{1}) = t(a_{1}) = ⋯ = s(a_{k}) = t(a_{k}).

The latter one corresponds to taking the image of constant morphism. We leave as an exercise for the reader to check that e and the e_{i_{1},··· ,i_{l}} are embeddings and that 𝒢^{k} is a suborbifold of 𝒢^{l}.

One of the main tools is the notion of a normal bundle. If i : 𝒢 → ℋ is an embedding, i^{∗}T ℋ is a groupoid vector bundle over 𝒢 such that T 𝒢 is a subbundle. Then we can define the normal bundle N_{𝒢|ℋ} = i^{∗}T ℋ/T 𝒢. N_{𝒢|ℋ} behaves as the normal bundle does for smooth manifolds. If N_{𝒢|ℋ} is oriented, we can define a Thom form Θ which is Poincaré dual to 𝒢 in the sense that

∫_{𝒢} α = ∫_{ℋ} α ∪ Θ.

Given the notion of a suborbifold, we can talk about transversality.

Definition 2.6. Suppose that f : 𝒢_{1} → ℋ, g : 𝒢_{2} → ℋ are smooth homomorphisms. We say that f × g is transverse to the diagonal Δ if locally f × g is transverse to every component of the diagonal Δ. We say that f, g are transverse to each other if f × g is transverse to the diagonal Δ.

Example 2.7. Suppose that f : 𝒢_{1} → ℋ, g : 𝒢_{2} → ℋ are smooth and transverse to each other. It is clear from the definition that the groupoid fibered product p_{1} × p_{2} : 𝒢_{1} ×_{ℋ} 𝒢_{2} → 𝒢_{1} × 𝒢_{2} is a suborbifold if the underlying map is topologically closed. On the other hand, there is a finite-to-one map from the orbifold fibered product to the ordinary fibered product. The ordinary fibered product is closed and so is p_{1}×p_{2}. We often simply denote it by 𝒢_{1}∩𝒢_{2}.

Suppose that ϕ : 𝒢 → ℋ is a homomorphism and i : 𝒦 → ℋ is a suborbifold, and furthermore assume that ϕ, i are transverse.

Definition 2.8. We define ϕ^{−1}(𝒦) = 𝒢 ×_{ℋ} 𝒦.

It is not hard to see that transversality often fails for groupoids. One important case is

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 7

Definition 2.9. We say that 𝒢_{1} intersects 𝒢_{2} cleanly if 𝒢_{1} ∩ 𝒢_{2} = 𝒢_{1} ×_{ℋ} 𝒢_{2} is smooth and the natural morphism to ℋ is an embedding. Furthermore, T_{x}(𝒢_{1} ∩ 𝒢_{2}) = T_{x}𝒢_{1} ∩ T_{x}𝒢_{2} for x ∈ (𝒢_{1})_{0} ∩ (𝒢_{2})_{0}.

Example 2.10. As we mentioned before, the evaluation map e : ∧𝒢 → 𝒢 is a suborbifold. Then, e with itself forms a clean intersection. Indeed, the question is local, and locally, it corresponds the intersection of fixed point sets V^{g} ∩ V^{h}. This is clearly a clean intersection. More generally, e_{i_{1},··· ,i_{l}} : 𝒢^{k} → 𝒢^{l} is an embedding. Then, two different embeddings e_{i_{1},··· ,i_{l}}, e_{j_{1},··· ,j_{l}} : 𝒢^{k} → 𝒢^{l} form a clean intersection. We leave it as an exercise for our readers.

3. Gerbes and Twisted K–Theory

We now consider the cohomology and K–theory of orbifold groupoids.

Definition 3.1. Let 𝒢 denote a Lie groupoid, then we define the continuous U(1) valued k–cochains on 𝒢 as

C^{k}(𝒢, U(1)) = {ϕ : G_{k} → U(1) | ϕ is continuous}.

The differential on this abelian group (using additive notation) is defined via

δϕ(g_{1}, ⋯, g_{k+1}) = ϕ(g_{2}, ⋯ g_{k+1}) + ∑_{i=1}^{k}(−1)^{i}ϕ(g_{1}, ⋯, g_{i}g_{i+1}, ⋯, g_{k+1}) + (−1)^{k+1}ϕ(g_{1}, ⋯, g_{k}).

By a result due to Moerdijk [21], if 𝒢 is an étale groupoid then the cohomology of this chain complex is the Čech cohomology of B𝒢 with coefficients in the sheaf C(U(1)) of U(1)-valued continuous functions over the classifying space B𝒢. By the exact sequence

0 → ℤ → C(ℝ) → C(U(1)) → 0,

we obtain a long exact sequence in cohomology,

H^{k}(B𝒢, C(ℝ)) → H^{k}(B𝒢, C(U(1))) → H^{k+1}(B𝒢, ℤ) → H^{k+1}(B𝒢, C(ℝ)).

Since C(ℝ) is a fine sheaf, the connecting homomorphism is an isomorphism, and so for k > 0,

H^{k}(B𝒢, C(U(1))) ≅ H^{k+1}(B𝒢, ℤ).

We recall

Definition 3.2. An n-gerbe on 𝒢 is a pair (ℋ, θ) consisting of • a refinement 𝒢 ←^{ϵ} ℋ (i.e., ϵ is an equivalence)

• an (n + 1)–cocycle θ : H_{n+1} → U(1).

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 8

Next we define equivalence of gerbes.

Definition 3.3. Given two n–gerbes (ℋ, θ) and (ℋ′, θ′) on 𝒢 we have the following:

• (ℋ, θ) is equivalent to (ℋ′, θ′) if there is a common refinement ℋ′ ← ℋ′′ → ℋ such that induced (n + 1)–cocycles on ℋ′′ are the same.

• (ℋ, θ) is isomorphic to (ℋ′, θ′) if there is a common refinement ℋ′′ such that the induced (n + 1)–cocycles on ℋ′′ differ by a coboundary.

From the definition, it is clear that an n-gerbe defines a Čech (n + 1)-cocycle for the sheaf C(U(1)) of continuous U(1)–valued functions on the classifying space B𝒢. Hence it will define a cohomology class in H^{n+1}(B𝒢, C(U(1))) ≅ H^{n+2}(B𝒢, ℤ). The image of θ under the connecting homomorphism in H^{n+2}(B𝒢, ℤ) is called its characteristic class or Dixmier-Douady class.

A 1-gerbe can be used to twist orbifold groupoid K-theory. We follow the treatment of [16] to describe this. Let ℍ be a separable Hilbert space; it is well-known that the characteristic class of a principal P U(ℍ)–bundle over 𝒢 also lies in H^{3}(B𝒢, ℤ). Hence, given a 1-gerbe, we should be able to associate a P U(ℍ) bundle with the same characteristic class; in fact we can associate a canonical principal P U(ℍ)–bundle. We outline its construction, where for simplicity we assume that the 2–cocycle θ is defined on 𝒢.

We define a U(1)-central extension R → G_{1} with R = G_{1} × U(1) and

(g_{1}, r_{1})(g_{2}, r_{2}) = (g_{1}g_{2}, θ(g_{1}, g_{2})r_{1}r_{2}).

Then s, t : R → G_{0} defines a Lie groupoid, where s(g, r) = s(g), t(g, r) = t(g). Now let R^{x} = t^{−1}(x); there is a system of measure (Haar system) λ = (λ^{x})_{x∈G_{0}}, where λ^{x} is a measure with support R^{x} such that for all f ∈ C_{c}(R), x → ∫_{g∈R^{x}} f(g)λ^{x}(dg) is continuous. Furthermore, for any γ ∈ R the map L_{γ} : R^{s(γ)} → R^{t(γ)} defined by κ → γκ preserves the measure. U(1) clearly acts on R. A complex-valued function f ∈ C_{c}(R) is said to be equivariant if f(rγ) = r^{−1}f(γ) for any r ∈ U(1), γ ∈ R. By ℒ_{x}^{2}, we denote the space L^{2}(R^{x})^{U(1)} consisting of U(1)-equivariant functions defined on R^{x} which are L^{2} with respect to the Haar measure. Let E_{x} = ℒ_{x}^{2} ⊗ ℍ, E = ⊔_{x}E_{x}. Then E → G_{0} is a locally trivial Hilbert bundle. We want to lift it to a bundle over the groupoid 𝒢. R acts naturally on E: for any element γ ∈ R with x = s(γ) and y = t(γ), and any equivariant function f ∈ E_{x}, the action is given by f ↦ γf, (γf)(γ′) = f(γ^{−1}γ′). Therefore, E is naturally a Hilbert bundle over s, t : R → G_{0}. Notice that U(1) acts as complex multiplication and hence E is not a Hilbert bundle over 𝒢. However, P (E) is a projective bundle over 𝒢 with precisely the same characteristic class of θ. Let ℬ be the principal bundle of orthonormal frames of E; it is a U(ℍ)-principal bundle. By our previous argument, P ℬ is a principal P U(ℍ)-bundle

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 9

over 𝒢. Let α be an equivalence class of 1-gerbe and P_{α} be the associated P U(ℍ)-bundle constructed above. Let F red^{0}(ℍ) be the space of Fredholm operators endowed with the ∗-strong topology and F red^{1}(ℍ) be the space of self-adjoint elements in F red^{0}(ℍ). Let 𝒦(ℍ) be the space of compact operators endowed with the norm-topology. Now consider the associated bundles

F red_{α}^{i}(ℍ) := P_{α} ×_{P U(ℍ)} F red^{i}(ℍ) → G_{0},
𝒦_{α}(ℍ) := P_{α} ×_{P U(ℍ)} 𝒦(ℍ) → G_{0}.

By F_{α}^{i}, we denote the space of norm-bounded, G_{1}-invariant, continuous sections x → T_{x} of the bundle F red_{α}^{i}(ℍ) → G_{0} such that there exists a norm-bounded, G_{1}-invariant, continuous section x → S_{x} of 𝒦_{α} → G_{0} with the property that 1 − T_{x}S_{x} and 1 − S_{x}T_{x} are continuous sections of 𝒦_{α}(ℍ) vanishing at infinity.

Definition 3.4. For any section T of F_{α}^{i}, we define the support supp(T) as the set of points x ∈ |𝒢| such that T_{x′} is not invertible for any x′ in the preimage of x.

Then we have

Definition 3.5. Let 𝒢 be an orbifold groupoid and α be a 1-gerbe. We define its α–twisted K-theory as

αK^{i}(𝒢) = {[T] | T ∈ F_{α}^{i}},

where [T] denotes the homotopy class of T where T is compactly supported.

Note that since the space of invertible operators is contractible, any T with compact support is homotopic to a section which is the identity outside a compact subset. Suppose that f : 𝒢 → ℋ is an open embedding. Using the property above, we have a natural extension

f_{∗} : ^{f^{∗}α}K^{i}(𝒢) →^{α} K^{i}(ℋ).

Remark. Suppose that we have a cocycle α = β + δρ. Then there is a canonical isomorphism between central extensions of groupoids

ψ_{ρ} : R_{α} → R_{β}

given by by ψ_{ρ}(g, r) = (g, ρ(g)r). Hence it induces an isomorphism

P_{α}(E) → P_{β}(E)

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 10

and also a canonical isomorphism

ψ_{ρ} : ^{α}K^{i}(𝒢) → ^{β}K^{i}(𝒢).

Suppose that in fact ρ is a cocycle, i.e., δρ = 0. Then α = α + δρ and hence we have an automorphism ψ_{ρ} : ^{α}K^{i}(𝒢) →^{α} K^{i}(𝒢). Furthermore, if ρ = δγ is a coboundary, then ψ_{ρ} is the identity. Hence H^{1}(B𝒢, U(1)) acts as automorphisms of twisted K-theory. It is easy to check in many examples that they are nontrivial automorphisms. In the literature, twisted K-theory is often referred to as being twisted by a Čech cohomology class or characteristic class of a 1-gerbe. This is a rather ambiguous statement, as cohomologous 1-gerbes induce isomorphic twisted K-theory, but this is not canonical. This observation is particularly important when we define a product structure on twisted K-theory.^{2}

There is a natural addition operator for ^{α}K^{∗}(𝒢) induced by the Hilbert space addition ℍ ≅ ℍ⊕ℍ. On the other hand, the multiplication operation induced by Hilbert space tensor product ℍ ≅ ℍ ⊗ ℍ shifts the twisting

αK^{∗}(𝒢) ⊗ ^{β}K^{∗}(𝒢) → ^{α+β}K^{∗}(𝒢).

We also want to point out that the product of an element in twisted K-theory with a vector bundle is to be understood in the following sense. Let P be a family of Fredholm operators on ℍ parameterized by a space M and E a complex vector bundle over M of finite rank. Then P · E is a family of Fredholm operators on ℍ ⊗ E parameterized by M, which at every point x ∈ M has P · E(x) = P (x) ⊗ Id_{E}. Hence it is easy to see that if P ∈^{θ} K^{i}(𝒢) and E is a 𝒢-bundle, then P · E ∈ ^{θ}K^{i}(𝒢). So even though E may not be an element of K^{0}(𝒢), the product with E makes sense.

To define our stringy product, we will need a version of push-forward map in the context of twisted K-theory. For smooth manifolds, such a push-forward map has already worked out by [9]. In the orbifold case, extra care is need to deal with the orbifold structure. Let 𝒢, ℋ be almost complex orbifold groupoids, f : 𝒢 → ℋ be a homomorphism which preserves the almost complex structures, and α be a 1-gerbe on ℋ. We will define the push-forward map f_{∗} : ^{f^{∗}α}K^{∗}(𝒢) →^{α} K^{∗}(ℋ).

Let 𝒢 = {s, t : G_{1} → G_{0}} be an orbifold groupoid and E a rank n complex vector bundle over 𝒢, i.e. π : E → G_{0} is a complex vector bundle with compatible 𝒢-action. We first establish the Thom homomorphism Φ : ^{α}K^{∗}(𝒢) → ^{π^{∗}α}K^{∗}(𝒢 ⋊ E), where 𝒢 ⋊ E is the transformation groupoid with object set E and arrow set G_{1} ×_{G_{0}} E.

2The second author would like thank P. Deligne for explaining this subtle point of twisted K-theory.

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 11

First notice that there is an invariant hermitian metric on E. Since 𝒢 is étale, there is a Haar system on 𝒢 [26]. If we take any hermitian metric on E and then use this Haar system to average it, we get an invariant hermitian metric on E.

Fix any invariant hermitian metric on E, then for any g ∈ G_{0}, e ∈ E_{g} = π^{−1}(g), we use e^{∗} to denote the dual of e with respect to the fixed hermitian metric.

The complex 𝒢-bundle π : E → G_{0} defines a complex of 𝒢-bundles over E,

λ_{E} = (Λ^{even}π^{∗}E, Λ^{odd}π^{∗}E, ϕ),

where ϕ_{(g,e)} = e ∧ −e^{∗}⌞. Notice that in ordinary K-theory, this is the Thom element.

For any element x ∈ ^{α}K^{∗}(𝒢), it is represented by a 𝒢-equivariant section x : G_{0} → F red_{α}^{∗}(ℍ) with supp(x) compact.

Proposition 3.6. For any 1-gerbe α, there is a K^{0}(𝒢)-module homomorphism

Φ : ^{α}K^{∗}(𝒢) →^{π^{∗}α} K^{∗}(𝒢 ⋊ E)

which is the standard Thom isomorphism in case of equivariant K-theory.

Proof. For any element x ∈ ^{α}K^{i}(𝒢), it is given by a 𝒢-equivariant section x : G_{0} → F red_{α}^{i}(ℍ). Therefore, we have a section π^{∗}(x) : E → π^{∗}F red_{α}^{i}(ℍ). Now supp(π^{∗}(x)) = π^{−1}(supp(x)).

For any point (g, e) ∈ E, let us consider the operator:

D : ℍ ⊗ Λ^{even}E_{g} ⊕ ℍ ⊗ Λ^{odd}E_{g} → ℍ ⊗ Λ^{even}E_{g} ⊕ ℍ ⊗ Λ^{odd}E_{g}

defined by

D_{(g,e)} = ( \begin{matrix}(π^{∗}x)(g) ⊗ 1 & −1 ⊗ ϕ_{(g,e)}^{∗} \\ 1 ⊗ ϕ_{(g,e)} & (π^{∗}x)(g)^{∗} ⊗ 1 \end{matrix} )

where ∗ means the adjoint operator. This is the so-called “graded tensor product” of Fredholm operators. It is easy to check that D_{(g,e)} is Fredholm and if e ≠ 0, then it is invertible.

Globalizing this construction, we have a family D of Fredholm operators parameterized by E. In fact, it is a fiberwise Fredholm operator on Hilbert bundles Λ^{•}π^{∗}E ⊗ ℍ over E.

Notice that • supp(D) = π^{−1}supp(x) ∩ i(G_{0}), where i : G_{0} → E is the zero section.

• D is a section of π^{∗}F red_{α}^{i}(ℍ) ≅ F red_{π^{∗}α}^{i}(ℍ).

• D is 𝒢 ⋊ E-equivariant. Therefore D represents an element of ^{π^{∗}α}K^{i}(𝒢 ⋊ E).

Now we define

Φ : ^{α}K^{∗}(𝒢) → ^{π^{∗}α}K^{∗}(𝒢 ⋊ E)

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 12

x ↦ D

Up to homotopy, it is easy to check that this definition doesn’t depends on any choice, so it is well-defined. Because it is the graded tensor product with λ_{E}, Φ is a K(𝒢)-module homomorphism. Furthermore, by the definition, we see that it is a generalization of the Thom isomorphism in equivariant K-theory.

We need a slightly more general version of this. Let U be an open neighborhood of the zero section, let 𝒢_{U} be the union of all orbits through U. From the definition, Φ(x) is supported on the zero section, so by restriction, we have the following Thom homomorphism.

Proposition 3.7. For any 1-gerbe α, there is a K(𝒢)-module homomorphism

Φ :^{α} K^{∗}(𝒢) →^{π^{∗}α} K^{∗}(𝒢 ⋊ 𝒢_{U})

Remark The Thom homomorphism can be defined in more general setting, but for the purposes of this paper, we only consider complex vector bundles over orbifold groupoids.

Let us prove a result for homomorphisms between Lie groupoids. By direct check, we can prove the following lemma.

Lemma 3.8. Let p : F → G_{0} be a smooth surjective map, then the groupoid F ×_{p} 𝒢 is Morita equivalent to 𝒢, where (F ×_{p} 𝒢)_{1} = (F × F) _{p×p} ×_{s×t} G_{1} and (F ×_{p} 𝒢)_{0} = F, and the new source map is s : (F × F) _{p×p} ×_{s×t} G_{1} → F, ((x, y), g) ↦ x, the new target map is t : (F × F) _{p×p} ×_{s×t} G_{1} → F, ((x, y), g) ↦ y.

For f : 𝒢 → ℋ, if we apply the lemma above to the space F = G_{0} × H_{0}, we can prove the next lemma, where all the maps are the natural ones.

Lemma 3.9. Let f : 𝒢 → ℋ be a homomorphism of Lie groupoids, then there exists a Lie groupoid 𝒦 and homomorphisms g : 𝒢 → 𝒦, h : 𝒦 → ℋ, such that g_{0} : G_{0} → K_{0} is an embedding and h is an equivalence. In other words, any homomorphism is an embedding up to Morita equivalence.

Remark This lemma still holds if we require that all our Lie groupoids are orbifold groupoids, though dimensions may change.

Now we can prove our main result in this section.

Theorem 3.10. If f : 𝒢 → ℋ is a homomorphism between almost complex orbifolds which preserves the almost complex structures, such that |f| : |G| → |H| is proper, and f_{0}(𝒢.x) = ℋ.f_{0}(x) for any x ∈ G_{0}, then there is a push-forward map

f_{∗} : ^{f^{∗}α}K^{∗}(𝒢) → ^{α}K^{∗}(ℋ)

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 13

Proof. Given our last lemma, we may assume that f_{0} : G_{0} → H_{0} is a proper embedding.

By our assumption, the normal bundle of G_{0} in H_{0} is a complex 𝒢-bundle, and we can identify an open neighborhood U of the zero section in the normal bundle with a neighborhood of f_{0}(G_{0}) in H_{0}, i.e. we have an embedding j : U → H_{0} as an open subset. It defines a homomorphism: j_{∗} : ^{j^{∗}α}K^{∗}(𝒢 ⋊ 𝒢_{U}) → ^{α}K^{∗}(ℋ ⋊ ℋ_{j(U)}), because the action of 𝒢 on the normal bundle is induced from the ℋ action, in this case any 𝒢-equivariant section is ℋ-equivariant.

It is clear that fπ is homotopic to j. Therefore, π^{∗}f^{∗}α = j^{∗}α + δρ for some ρ. The choice of ρ is not unique; for example, we can add a 1-cocycle. This corresponds exactly to the non-canonicity of the dependence of twisted K-theory on the cohomology class of a 1-gerbe. However, fπ = j on the zero section; therefore, we can choose ρ such that ρ = 0 on the zero section. Since U deformation retracts to the zero section, it fixes ρ uniquely.

Now we have homomorphisms:

f^{∗}αK^{∗}(𝒢) ⟶^{Φ} ^{π^{∗}f^{∗}α}K^{∗}(𝒢 ⋊ 𝒢_{U}) ⟶^{p_{ρ}} ^{α}K^{∗}(𝒢 ⋊ 𝒢_{U}) ⟶^{j_{∗}} ^{α}K^{∗}(ℋ ⋊ ℋ_{j(U)}) → ^{α}K^{∗}(ℋ)

where the last homomorphism is extension for an open saturated subgroupoid . The composition is the push-forward map f_{∗}.

Given our explicit definition, it is easy to check the following properties of the push-forward map.

Proposition 3.11. Let f : 𝒢 → ℋ as before, then there exists an element c = c(𝒢, ℋ) such that for any a ∈^{f^{∗}α} K^{∗}(𝒢) and b ∈^{β} K^{∗}(ℋ), we have

f^{∗}f_{∗}(a) = a · c
f_{∗}(a · f^{∗}(b)) = f_{∗}(a) · b

In particular for suborbifolds, we have following result.

Corollary 3.12. If i : 𝒢_{1} → 𝒢 is a suborbifold, then there is a push-forward map

i_{∗} : ^{i^{∗}α}K^{∗}(𝒢_{1}) → ^{α}K^{∗}(𝒢)

satisfying the above properties.

For later purposes we would like to introduce

Definition 3.13. If E → 𝒢 is a complex orbifold bundle, then its K–theoretic Euler class e_{K}(E) is defined as i^{∗}λ_{E}, the complex of 𝒢–vector bundles obtained by pulling back the Thom element λ_{E} using the zero–section i : 𝒢 → E.

Note that we can define the product x · e_{K}(E) as

x · e_{K}(E) = x · (∧^{even} E) + (x · ∧^{odd} E)^{∗}.

4. The Inverse Transgression for Groupoids

In order to define the stringy product in twisted K–theory, we will need a cohomological formula to match up the levels. The basic construction is the inverse transgression, which was defined in [12]. We provide a formulation for groupoids inspired by the case of finite groups. We will also provide some explicit calculations. See [18] for a more geometric view on this.

Recall that

(∧𝒢)_{0} = {a ∈ G_{1} | s(a) = t(a)}, (∧𝒢)_{1} = {(a, u_{1}) ∈ G_{1} × G_{1} | s(a) = t(a) = s(u_{1})}.

It is easy to check that

(∧𝒢)_{k} = {(a, u_{1}, ⋯, u_{k}) ∈ G_{k+1} | s(a) = t(a) = s(u_{1}), t(u_{i}) = s(u_{i+1})}.

Definition 4.1. Define θ : C^{k}(𝒢, U(1)) → C^{k−1}(∧𝒢, U(1)) by

θ(ϕ)(a, u_{1}, ⋯, u_{k}) = (−1)^{k}ϕ(a, u_{1}, ⋯, u_{k}) + ∑_{i=1}^{k}(−1)^{i+k}ϕ(u_{1}, ⋯, u_{i}, a_{i}, u_{i+1}, ⋯, u_{k}),

where a_{i} = (u_{1} ⋯ u_{i})^{−1}au_{1} ⋯ u_{i}.

A routine but slightly tedious computation shows that this is in fact a cochain map, i.e. δθ = θδ.

We should note that θ is a natural map defined for all groupoids. For orbifold groupoids it induces a homomorphism

θ_{∗} : H^{k}(B𝒢, U(1)) → H^{k−1}(B ∧ 𝒢, U(1)),

and hence a homomorphism

θ_{∗} : H^{k+1}(B𝒢, ℤ) → H^{k}(B ∧ 𝒢, ℤ).

The cochain map θ and the induced map in cohomology will be called the inverse transgression.

Recall that the moduli space of constant morphisms \overline{ℳ}_{3}(𝒢) from an orbifold sphere with three orbifold points can be identified with the 2-sector orbifold 𝒢^{2}, where

(𝒢^{2})_{0} = {(a, b) ∈ G_{2} | s(a) = t(a) = s(b) = t(b)},

(𝒢^{2})_{k} = {(a, b, u_{1}, ⋯, u_{k}) ∈ G_{k+2} | s(a) = t(a) = s(b) = t(b) = s(u_{1}), t(u_{i}) = s(u_{i+1})}

with

s(a, b, u_{1}, ⋯, u_{k}) = (a, b), t(a, b, u_{1}, ⋯, u_{k}) = (a_{k}, b_{k})

where

a_{i} = (u_{1} ⋯ u_{i})^{−1}au_{1} ⋯ u_{i}, b_{i} = (u_{1} ⋯ u_{i})^{−1}bu_{1} ⋯ u_{i}.

There are three natural evaluation morphisms

e_{1} : 𝒢^{2} → ∧𝒢 by e_{1}(a, b) = a,
e_{2} : 𝒢^{2} → ∧𝒢 by e_{2}(a, b) = b,
e_{12} : 𝒢^{2} → ∧𝒢 by e_{12}(a, b)) = ab.

Furthermore, e_{1}, e_{2}, e_{12} are all embeddings.

In a manner totally analogous to our definition of θ, we can define μ, which now will involve the groupoid 𝒢^{2}.

Definition 4.2. Define μ : C^{k+2}(𝒢, U(1)) → C^{k}(𝒢^{2}, U(1)) by

μ(ϕ)(a, b, u_{1}, ⋯, u_{k}) =

ϕ(a, b, u_{1}, ⋯, u_{k}) + ∑_{{(i,j) | 0≤i≤j≤k (i,j)≠(0,0)}}(−1)^{i+j}ϕ(u_{1}, ⋯, u_{i}, a_{i}, u_{i+1}, ⋯, u_{j}, b_{j}, u_{j+1}, ⋯, u_{k}).

A second key multiplicative formula is given by the equation

μδ + δμ = e_{1}^{∗}θ + e_{2}^{∗}θ − e_{12}^{∗}θ.

Note that the function μ defines a chain homotopy between e_{1}^{∗}θ + e_{2}^{∗}θ and e_{12}^{∗}θ. If ϕ is a cocycle, then θ(ϕ) is a cocycle and the formula above implies that

e_{1}^{∗}θ(ϕ) + e_{2}^{∗}θ(ϕ) = e_{12}^{∗}θ(ϕ) + δμ(ϕ).

In particular we see that the difference between the cocycles is given by a canonical coboundary, expressed explicitly as a function of ϕ. This will be very important when we make our identifications in twisted K–theory.

We will verify and apply this formula in low degree, which is our main interest here.

Proposition 4.3. Let ϕ be an element in C^{3}(𝒢, U(1)). Then

δμ(ϕ) + μδ(ϕ) = e_{1}^{∗}θ(ϕ) + e_{2}^{∗}θ(ϕ) − e_{12}^{∗}θ(ϕ).

Proof. This can be proved by an explicit calculation.

δμ(ϕ)(a, b, u_{1}, u_{2}) = −ϕ(a_{1}, b_{1}, u_{2}) + ϕ(a, b, u_{1}u_{2}) − ϕ(a, b, u_{1}) + ϕ(a_{1}, u_{2}, b_{2}) − ϕ(a, u_{1}u_{2}, b_{2})

+ϕ(a, u_{1}, b_{1}) − ϕ(u_{2}, a_{2}, b_{2}) + ϕ(u_{1}u_{2}, a_{2}, b_{2}) − ϕ(u_{1}, a_{1}, b_{1})

μδ(ϕ)(a, b, u_{1}, u_{2}) = ϕ(b, u_{1}, u_{2}) − ϕ(u_{1}, b_{1}, u_{2}) + ϕ(u_{1}, u_{2}, b_{2}) + ϕ(a_{1}, b_{1}, u_{2}) − ϕ(a_{1}, u_{2}, b_{2})

+ϕ(u_{2}, a_{2}, b_{2}) − ϕ(ab, u_{1}, u_{2}) + ϕ(au_{1}, b_{1}, u_{2}) − ϕ(au_{1}, u_{2}, b_{2}) − ϕ(u_{1}a_{1}, b_{1}, u_{2})
+ϕ(u_{1}a_{1}, u_{2}, b_{2}) − ϕ(u_{1}u_{2}, a_{2}, b_{2}) + ϕ(a, bu_{1}, u_{2}) − ϕ(a, u_{1}b_{1}, u_{2}) + ϕ(a, u_{1}u_{2}, b_{2})
+ϕ(u_{1}, a_{1}b_{1}, u_{2}) − ϕ(u_{1}, a_{1}u_{2}, b_{2}) + ϕ(u_{1}, u_{2}a_{2}, b_{2}) − ϕ(a, b, u_{1}u_{2}) + ϕ(a, u_{1}, b_{1}u_{2})
−ϕ(a, u_{1}, u_{2}b_{2}) − ϕ(u_{1}, a_{1}, b_{1}u_{2}) + ϕ(u_{1}, a_{1}, u_{2}b_{2}) − ϕ(u_{1}, u_{2}, a_{2}b_{2}) + ϕ(a, b, u_{1})
−ϕ(a, u_{1}, b_{1}) + ϕ(a, u_{1}, u_{2}) + ϕ(u_{1}, a_{1}, b_{1}) − ϕ(u_{1}, a_{1}, u_{2}) + ϕ(u_{1}, u_{2}, a_{2})

We now add these two expressions. Using the identities au_{1} = u_{1}a_{1}, bu_{1} = u_{1}b_{1}, a_{1}u_{2} = u_{2}a_{2}, and b_{1}u_{2} = u_{2}b_{2}, cancelling and collecting terms, yields the expression

[μδ + δμ](ϕ)(a, b, u_{1}, u_{2}) = ϕ(a, u_{1}, u_{2}) − ϕ(u_{1}, a_{1}, u_{2}) + ϕ(u_{1}, u_{2}, a_{2}) + ϕ(b, u_{1}, u_{2}) − ϕ(u_{1}, b_{1}, u_{2})

+ϕ(u_{1}, u_{2}, b_{2}) − ϕ(ab, u_{1}, u_{2}) + ϕ(u_{1}, a_{1}b_{1}, u_{2}) − ϕ(u_{1}, u_{2}, a_{2}b_{2})

This expression is exactly e_{1}^{∗}θ + e_{2}^{∗}θ − e_{12}^{∗}θ applied to ϕ, hence the proof is complete.

The inverse transgression formula implies that a 2-gerbe ϕ on an orbifold groupoid 𝒢 induces a 1-gerbe θ(ϕ) on the associated inertia groupoid ∧𝒢. Furthermore, two equivalent (isomorphic) 2-gerbes induce equivalent (isomorphic) 1-gerbes on the inertia groupoid.

Recall that there is an embedding e : 𝒢 → ∧𝒢 by e(x) = 1_{x} where 1_{x} is identity arrow. The image e(𝒢) is often referred as non-twisted sector and other components of ∧𝒢 are called twisted sectors.

Corollary 4.4. If ϕ is a cocycle, then e^{∗}θ(ϕ) is a coboundary.

Proof. e^{∗}θ(ϕ)(u, v) = θ(1, u, v). Using the embedding λ : 𝒢 → 𝒢^{2} given by x → (i_{x}, i_{x}), we can pull back e_{1}^{∗}θ(ϕ) + e_{2}^{∗}θ(ϕ) − e_{12}^{∗}θ(ϕ) in cohomology. Note that

λ^{∗}e_{1}^{∗}θ(ϕ) = λ^{∗}e_{2}^{∗}θ(ϕ) = λ^{∗}e_{12}^{∗}θ(ϕ) = e^{∗}θ(ϕ).

Therefore, e^{∗}θ(ϕ) = δe^{∗}μ(ϕ) is a coboundary. This implies that restricted to the untwisted sector, our cocycle θ(ϕ) gives rise to a trivial cohomology class.

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 17

5. The Inverse Transgression in the Case of a Finite Group

In the case when the original orbifold is [∗/G] where G is a finite group, the inverse transgression has a classical interpretation in terms of shuffle products. Recall that ∧[∗/G] can be thought of in terms of G with the conjugation action; this breaks up into a disjoint union of orbits of the form G/Z_{G}(g), indexed by conjugacy classes. Each of these is in turn equivalent to [∗/Z_{G}(g)]; so we have a Morita equivalence ∧[∗/G] ≅ ⊔_{(g)}[∗/Z_{G}(g)]. Hence we can restrict our attention to these components; in particular we would like to describe each θ_{g} : C^{k}(G, U(1)) → C^{k−1}(Z_{G}(g), U(1)). Now for a finite group G, the cochain complex C^{∗}(G, U(1)) is in fact equal to Hom_{G}(B_{∗}(G), U(1)), where B_{∗}(G) is the bar resolution for G (see [8], page 19).

There is natural homomorphism ρ_{g} : Z_{G}(g) × ℤ → G given by ρ_{g}(x, t^{i}) = xg^{i}, where t is a generator for ℤ; the fact that Z_{G}(g) centralizes g is crucial here. This homomorphism induces a map in integral homology

H_{∗}(Z_{G}(g), ℤ) ⊗ H_{∗}(ℤ, ℤ) ≅ H_{∗}(Z_{G}(g) × ℤ, ℤ) → H_{∗}(G, ℤ).

Classically it is known that multiplication is induced by the shuffle product on the chain groups (see [8], page 117–118); i.e. there is a chain map B_{∗}(Z_{G}(g)) ⊗ B_{∗}(ℤ) → B_{∗}(G) which will induce ρ_{g∗} in homology. Let t denote a generator of the cyclic group ℤ. The shuffle product we are interested in is B_{k}(Z_{G}(g)) ⊗ B_{1}(ℤ) → B_{k+1}(G) given by

[g_{1}|g_{2}| … |g_{k}] ⋆ [t^{i}] = ∑_{σ} σ[g_{1}|g_{2}| … |g_{k}|g_{k+1}]

where g_{k+1} = g^{i}, σ ranges over all (k, 1)–shuffles and

σ[g_{1}|g_{2}| … |g_{k+1}] = (−1)^{sign(σ)}[g_{σ(1)}|g_{σ(2)}| … |g_{σ(k+1)}].

A (k, 1)–shuffle is an element σ ∈ S_{k+1} such that σ(i) < σ(j) for 1 ≤ i < j ≤ k. These are precisely the cycles:

1, (k k + 1), (k − 1 k k + 1), (k − 2 k − 1 k k + 1), …, (1 2 3 … k k + 1).

Note that there are k + 1 of them. This can be dualized, using U(1) coefficients, but for cohomology purposes it’s easier to use integral coefficients. Given a cocycle ϕ ∈ C^{k+1}(G, ℤ), we see that θ_{g}(ϕ) ∈ C^{k}(Z_{G}(g), ℤ) can be defined as

θ_{g}(ϕ)([g_{1}|g_{2}| … |g_{k}]) = ϕ([g_{1}|g_{2}| … |g_{k}] ⋆ [g])

where g_{1}, g_{2}, …, g_{k} ∈ Z_{G}(g).

As a consequence of this we see that θ_{g}^{∗} : H^{k+1}(G, ℤ) → H^{k}(Z_{G}(g), ℤ) is induced by the multiplication map

ρ_{g}^{∗} : H^{k+1}(G, ℤ) → H^{k}(Z_{G}(g), ℤ) ⊗ H^{1}(ℤ, ℤ).

To be precise, if ν is the natural generator for H^{1}(ℤ, ℤ), then

ρ_{g}^{∗}(u) = res_{Z_{G}(g)}^{G}(u) ⊗ 1 + θ_{g}^{∗}(u) ⊗ ν.

This discussion clarifies the geometric arguments in [12], and will also allow us to do some computations in cohomology.

Example 5.1. Finite group cohomology is difficult to compute, especially over the integers. The simple examples such as cyclic and quaternion groups are not so interesting in this context, as their odd dimensional cohomology (with trivial ℤ coefficients) is zero. The first interesting example is G = (ℤ/2)^{2}. In this case H^{∗}(G, 𝔽_{2}) is a polynomial algebra on two degree one generators x, y. In degree four there is a natural basis given by x^{4}, y^{4}, x^{3}y, x^{2}y^{2}, xy^{3}. For an elementary abelian 2–group, the mod 2 reduction map for k > 0 is a monomorphism H^{k}(G, ℤ) → H^{k}(G, 𝔽_{2}), and so we can understand it as the kernel of the Steenrod operation Sq^{1} : H^{k}(G, 𝔽_{2}) → H^{k+1}(G, 𝔽_{2}). Hence we see that H^{4}(G, ℤ) can be identified with the subspace generated by x^{4}, y^{4} and x^{2}y^{2}. These are all squares, hence when we apply θ_{g}^{∗} : H^{4}(G, ℤ) → H^{3}(G, ℤ) for any g ∈ G, the result will always be zero.

Next we consider G = (ℤ/2)^{3}; in this case H^{∗}(G, 𝔽_{2}) is a polynomial algebra on three degree one generators x, y, z. In this case we have an element α = Sq^{1}(xyz) = x^{2}yz + xy^{2}z + xyz^{2} which represents a non-square element in H^{4}(G, ℤ). By analyzing the multiplication map in cohomology we obtain the following.

Lemma 5.2. Let g = x^{a}y^{b}z^{c} be an element in G = (ℤ/2)^{3}, where we are writing it in terms of the standard basis (identified with its dual by abuse of notation). Then

θ_{g}^{∗}(α) = a(y^{2}z + z^{2}y) + b(x^{2}z + xz^{2}) + c(x^{2}y + xy^{2})

and so it is non–zero on every component except the one corresponding to the trivial element in G.

Now for an abelian group, the multiplicative formula implies that for all g, h ∈ G, θ_{g}^{∗}+θ_{h}^{∗} = θ_{gh}^{∗} in cohomology, or up to coboundaries. In particular this implies that the correspondence g ↦ θ_{g}^{∗}(α) defines a homomorphism G → H^{3}(G, ℤ) of elementary abelian groups, in this case an isomorphism.

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 19

6. The Twisted Pontryagin Product for Finite Groups

Let G denote a finite group, and consider the orbifold defined by its action on a point. Then the inertia groupoid ∧𝒢 can be identified with the groupoid determined by the conjugation action of G on itself. In this case the untwisted orbifold K–theory is simply K_{G}(G), which is additively isomorphic to ⨁_{(g)} R(Z_{G}(g)), where as before Z_{G}(g)) denotes the centralizer of g in G, and the sum is taken over conjugacy classes. This group can be endowed with a certain product, known as the Pontryagin product, defined as follows. An equivariant vector bundle over G (with the conjugation action) can be thought of as a collection of finite dimensional vector spaces V_{g} with a G-module structure on ⊕_{g∈G}V_{g} such that gV_{h} = V_{ghg^{−1}}. The product of two of these bundles is now defined as follows:

(V ⋆ W)_{g} = ⨁_{{g_{1},g_{2}∈G, g_{1}g_{2}=g}} V_{g_{1}} ⊗ W_{g_{2}}

This formula has been referred to as the holomorphic orbifold model in the physics literature [11].

This product admits an alternate description, which will admit a geometric generalization. In this case we can identify 𝒢^{2} with the orbifold defined by considering G × G with the conjugation action on both coordinates. Our maps e_{1}, e_{2} and e_{12} correspond to (g, h) ↦ g, (g, h) ↦ h, (g, h) ↦ gh respectively, which are G–equivariant with respect to the conjugation action. Then, if α, β are elements in K_{G}(G), the Pontryagin product can also be defined as

α ⋆ β = e_{12∗}(e_{1}^{∗}(α) · e_{2}^{∗}(β)).

We propose to extend this definition to twisted K–theory, with certain conditions on the twisting cocycle. Note that given a 2–cocycle τ = θ(ϕ) in the image of the inverse transgression, then by our multiplicative formula we have e_{1}^{∗}τ + e_{2}^{∗}τ = e_{12}^{∗}τ + δμ(ϕ).

Definition 6.1. Let τ be a U(1) valued 2–cocycle for the orbifold defined by the conjugation action of a finite group G on itself which is in the image of the inverse transgression. The Pontryagin product on ^{τ}K_{G}(G) is defined by the following formula: if α, β ∈ ^{τ}K_{G}(G), then

α ⋆ β = e_{12∗}(e_{1}^{∗}α · e_{2}^{∗}β)

Note that if τ = θ(ϕ) then by our multiplicative formula we have

e_{1}^{∗}τ + e_{2}^{∗}τ = e_{12}^{∗}τ + δμ(ϕ)

and so the product e_{1}^{∗}(α) · e_{2}^{∗}(β) lies in

e_{1}^{∗}τ+e_{2}^{∗}τK_{G}(G) = ^{e_{12}^{∗}τ+δμ(ϕ)}K_{G}(G) ≅ ^{e_{12}^{∗}τ}K_{G}(G).

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 20

Now applying e_{12∗}, this is mapped to ^{τ}K_{G}(G); and so we have a product on our twisted K–theory; it is elementary to verify that this defines an associative product.

Our approach will define a twisted Pontryagin product for any cocycle in the image of the inverse transgression. This cocycle could very well be a coboundary; but that does not necessarily imply that the corresponding product on K_{G}(G) is the standard Pontryagin product. It is also clear that we may choose twistings which give rise to a twisted K–theory without any product.^{3}

Example 6.2. If G is an abelian group, then what we are doing is using the identification θ(ϕ)_{g} + θ(ϕ)_{h} = θ(ϕ)_{gh} to define a product on the abelian group

𝒳(G) = ∑_{g∈G} ^{θ(ϕ)_{g}}R(G)

via the pairing

θ(ϕ)_{g}R(G) ⊗ ^{θ(ϕ)_{h}}R(G) → ^{θ(ϕ)_{gh}}R(G).

In the case described in 5.1 for G = (ℤ/2)^{3} and a cocycle ϕ representing the cohomology class xy^{2}z+xyz^{2}+x^{2}yz, θ(α) establishes a group homomorphism G → H^{3}(G, ℤ), with image the subgroup generated by xy^{2} + x^{2}y, xz^{2} + x^{2}z and yz^{2} + y^{2}z. In this case we see that for g ≠ 1, ^{θ(ϕ)_{g}}R(G) has rank equal to two, and so 𝒳(G) is of rank equal to twenty-two, with the twisted Pontryagin product described above. This can be made explicit.

The case of the Pontryagin product should be considered as motivation for the case of orbifold groupoids. As long as we twist with a cocycle in the image of the inverse transgression, the levels will match up as required. Hence the main difficulty is geometric–as we shall see in the next section, there is an obstruction bundle which plays an important role.

7. Twisted K–theory of Orbifolds

During the course of our investigation of possible stringy products on the twisted K-theory of orbifolds, we came to realize that one needs to use the very same information required to construct the Chen-Ruan cohomology of orbifolds ([10]). We first briefly recall the situation for orbifold cohomology, and then proceed to develop the tools necessary to deal with twisted K–theory. For a very interesting but different approach we refer the reader to [14].

First we recall that H_{CR}^{∗}(𝒢, ℂ) is additively the same as H^{∗}(∧𝒢, ℂ); what is interesting is the ring structure. Recall that there are three evaluation maps e_{1}, e_{2}, e_{12} : 𝒢^{2} → ∧𝒢. A naive definition of the stringy product for orbifold cohomology (which would generalize the

3For much more on Pontryagin products please consult [13] and its sequels.

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 21

Pontryagin product) would be α ⋆ β = (e_{12})_{∗}(e_{1}^{∗}α ∪ e_{2}^{∗}β). However, one soon discovers that this product is not associative due to the fact that e_{1}, e_{12} are not transverse in general. In fact, the correction term is precisely the obstruction bundle to the transversality of e_{1} and e_{12}. A natural idea was to modify the definition of the product via

α ⋆ β = (e_{12})_{∗}(e_{1}^{∗}α ∪ e_{2}^{∗}β ∪ e(E_{𝒢^{2}}))

where we need to construct a bundle E_{𝒢^{2}} in a fashion that is consistent with the obstruction to transversality of e_{1}, e_{12}, and e(E_{𝒢^{2}}) denotes its Euler class. A key observation is that the obstruction bundle in the construction of the Chen-Ruan product provides such a choice. We will adapt this same idea to K–theory.

Throughout this section, we assume that 𝒢 is a compact, almost complex orbifold groupoid. Then 𝒢^{k} naturally inherits an almost complex structure such that the evaluation map e_{i_{1},··· ,i_{l}} : 𝒢^{k} → 𝒢^{l} is an almost complex embedding. 𝒢^{2} can be identified with the space of constant morphisms from an orbifold sphere with three marked point to 𝒢; we now make this identification precise. Consider an orbifold Riemann sphere with three orbifold points (𝕊^{2}, (x_{1}, x_{2}, x_{3}), (m_{1}, m_{2}, m_{3})); in this context we simply denote it by 𝕊^{2}. Suppose that f is a constant morphism from 𝕊^{2} to 𝒢. Here, the term constant means that the induced map |f| : 𝕊^{2} → |𝒢| is constant. Let y = im(|f|) and U_{y}/G_{y} be an orbifold chart at y. By the results in [1], f is classified by the conjugacy class of a homomorphism π_{f} : π_{1}^{orb}(𝕊^{2}) → G_{y}. Recall that

π_{1}^{orb}(𝕊^{2}) = {λ_{1}, λ_{2}, λ_{3}; λ_{i}^{k_{i}} = 1, λ_{1}λ_{2}λ_{3} = 1},

where λ_{i} is represented by a loop around the marked point x_{i}. π_{f} is uniquely determined by a pair of elements (g_{1}, g_{2}) with g_{i} ∈ G_{y} where g_{i} = π_{f}(λ_{i}); on the other hand, (g_{1}, g_{2}) ∈ 𝒢_{0}^{2}. It is clear that the same method can be used to identify the moduli space of constant morphisms from an orbifold sphere with k + 1 marked points to 𝒢 with the groupoid of k–multisectors 𝒢^{k}.

For any f ∈ 𝒢^{2} viewed as a constant morphism, we can form an elliptic complex

¯∂_{f} : Ω^{0}(f^{∗}T 𝒢) → Ω^{0,1}(f^{∗}T 𝒢),

where f^{∗}T 𝒢 is a complex vector bundle by our assumption. This defines an orbibundle E with E_{𝒢^{2}}|_{f} = Coker ¯∂_{f}. We now examine E_{𝒢^{2}} in more detail. Let g_{1}, g_{2} ∈ 𝒢_{0}^{2}; by the definition, g_{1}, g_{2} ∈ G_{x} for x = s(g_{i}) = t(g_{i}). Let N be the subgroup of G_{x} generated by g_{1}, g_{2}. By Lemma 4.5 in [1], N depends only on the component of 𝒢^{2}. Let e : 𝒢^{2} → 𝒢 be an evaluation map. Clearly N acts on e^{∗}T 𝒢 while fixing T 𝒢^{2}.

There is an obvious surjective homomorphism π : π_{1}^{orb}(𝕊^{2}) → N and kerπ is therefore a subgroup of finite index. Suppose that Σ̃ is the orbifold universal cover of 𝕊^{2}. By [1] (see Chapter II), Σ̃ is smooth. Let Σ = Σ̃/kerπ; then Σ is compact and we have a quotient map Σ → 𝕊^{2} = Σ/N. Since N contains the relation g_{i}^{m_{i}} = 1, Σ is smooth.

It is clear that f lifts to an ordinary constant map f̃: Σ → U_{y}; hence f̃^{∗}T 𝒢 = T_{y}𝒢 is a trivial bundle over Σ. Then we can lift the elliptic complex to Σ

¯∂_{Σ} : Ω^{0}(f̃^{∗}T 𝒢) → Ω^{0,1}(f̃^{∗}T 𝒢).

The original elliptic complex is just the N-invariant part of the current one. However, Ker(¯∂_{Σ}) = T_{y}𝒢 and Coker(¯∂_{Σ}) = H^{0,1}(Σ) ⊗ T_{y}𝒢. Now we vary y in a component 𝒢_{(γ)}^{2} to obtain e_{(γ)}^{∗}T 𝒢 for the evaluation map e_{(γ)} : 𝒢_{(γ)}^{2} → 𝒢 and H^{0,1}(Σ) ⊗ e_{(γ)}^{∗}T 𝒢. N acts on both. It is clear that (e_{(γ)}^{∗}T 𝒢)^{N} = T 𝒢_{(γ)} as we claim. The obstruction bundle E_{(γ)} we want is the invariant part of H^{0,1}(Σ) ⊗_{ℂ} e_{(γ)}^{∗}T 𝒢, i.e., E_{(γ)} = (H^{0,1}(Σ) ⊗_{ℂ} e_{(γ)}^{∗}T 𝒢)^{N}. We remark that E_{(γ)} can have different dimensions at the different components of 𝒢^{2}. We can obviously use the same method to construct a bundle E_{𝒢^{k}} over 𝒢^{k} whose fiber is the cokernel of ¯∂_{f} for f ∈ 𝒢^{k}.

Consider the evaluation maps e_{1}, e_{2}, e_{12} : 𝒢^{2} → ∧𝒢. Suppose that the local chart of 𝒢 is U/G. Then, the local chart of ∧𝒢 is (⊔_{g∈G}U^{g})/G. The local chart of 𝒢^{2} is (⊔_{g_{1},g_{2}∈G}U^{g_{1}} ∩ U^{g_{2}})/G. It is clear that the e_{1}, e_{2}, e_{12} are embeddings and hence 𝒢^{2} can be thought of as a suborbifold of ∧𝒢 via three different embeddings, denoted by e_{1∗}𝒢^{2}, e_{2∗}𝒢^{2}, e_{12∗}𝒢^{2}. It is clear that 𝒢^{3} = 𝒢^{2} ×_{e_{12},e_{1}} 𝒢^{2}. Namely, 𝒢^{3} should be viewed as the analogue of the intersection of the sub-orbifolds e_{12∗}𝒢^{2} ∩ e_{1∗}𝒢^{2}. The problem is that e_{12∗}𝒢^{2}, e_{1∗}𝒢^{2} are not intersecting transversely in general. Let

ν = (e_{12}π_{1})^{∗}T ∧ 𝒢/π_{1}^{∗}T 𝒢^{2} + π_{2}^{∗}T 𝒢^{2}

where π_{i} : 𝒢^{3} → 𝒢^{2} are projection maps. This is the so-called excess bundle of the intersection. A crucial ingredient in the proof of associativity for the Chen-Ruan product is the bundle formula:

Theorem 7.1.

E_{𝒢^{3}} ≅ π_{1}^{∗}E_{𝒢^{2}} ⊕ π_{2}^{∗}E_{𝒢^{2}} ⊕ ν.

The proof is given in [10] by gluing arguments. We shall call this the obstruction bundle equation.

These bundles will play a key role in our definition of the product. It ties in to the transversality question mentioned before via the following lemma, which is an orbifold analogue of the clean intersection formula first described by Quillen [24]:

Lemma 7.2. Suppose that i_{1} : ℋ_{1} → 𝒢 and i_{2} : ℋ_{2} → 𝒢 are suborbifolds of 𝒢 forming a clean intersection ℋ_{3}. Then, if u ∈ ^{α}K(ℋ_{1}) we have

i_{2}^{∗}i_{1}_{∗}u = π_{2}_{∗}[π_{1}^{∗}u · e_{K}(ν)]

where π_{1} : ℋ_{3} → ℋ_{1} and π_{2} : ℋ_{3} → ℋ_{2} are the natural projections, ν is the excess bundle of the intersection, and e_{K}(ν) ∈ K(ℋ_{3}) denotes its Euler class.

Note that this lemma will allow us to connect facts about the geometry of the suborbifolds 𝒢^{2} and 𝒢^{3} in ∧𝒢 with products, and the obstruction bundle E plays a key role here. In fact the only information we need about E is the obstruction bundle equation mentioned above.

Definition 7.3. Suppose that 𝒢 is an almost complex orbifold. Let ϕ denote a U(1)–valued 3–cocycle for 𝒢, and θ(ϕ) its inverse transgression, i.e. a U(1)–valued 2–cocycle for ∧𝒢. For α, β ∈ ^{θ(ϕ)}K(∧𝒢), we define

α ⋆ β = e_{12∗}(e_{1}^{∗}α · e_{2}^{∗}β · e_{K}(E_{𝒢^{2}})).

Our goal is to show that this defines an associative product. However, we want to do this using very general properties of our construction, so that it will be a natural extension of the Chen–Ruan product. We will abbreviate E_{2} = E_{𝒢^{2}}, E_{3} = E_{𝒢^{2}}.^{4}

As with the twisted Pontryagin product, we must explain in what sense this defines a product. Given that e_{1}, e_{2}, e_{12} : 𝒢^{2} → ∧G, then e_{i}^{∗}(α) ∈ ^{e_{i}∗(θ(ϕ))}K(𝒢^{2}) for i = 1, 2. Hence

e_{1}^{∗}(α) · e_{2}^{∗}(β) · e_{K}(E_{𝒢^{2}}) ∈ ^{e_{1}^{∗}θ(ϕ)+e_{2}^{∗}θ(ϕ)}K(𝒢^{2}).

Now here we must be careful. We use the fact that the twisting cocycle is in fact equal to e_{12}^{∗}θ(ϕ) + δμ(ϕ). We have a canonical isomorphism

e_{12}^{∗}θ(ϕ)+δμ(ϕ)K(𝒢^{2}) ≅ ^{e_{12}^{∗}θ(ϕ)}K(𝒢^{2}).

Next we apply the push forward e_{12∗} to our expression, which now lands in ^{θ(ϕ)}K(∧G). Note that the product is clearly commutative; it is associativity that requires a proof.

Theorem 7.4. This product is associative:

(α ⋆ β) ⋆ γ = α ⋆ (β ⋆ γ).

4We have chosen to work only with even dimensional K–theory, but this can be readily extended to odd dimensions; the signs work out appropriately after a tedious computation.

A STRINGY PRODUCT ON TWISTED ORBIFOLD K-THEORY 24

Proof.

We pull back the embeddings e_{1}, e_{2} and e_{12} to 𝒢^{3} as follows: let ẽ_{1} = e_{1}π_{1}; ẽ_{2} = e_{2}π_{1}; ẽ_{3} = e_{2}π_{2} and ẽ_{123} = e_{12}π_{2}. We will also make use of an operator defined for triples, namely let I_{3} : 𝒢^{3} → 𝒢^{3} be defined by

(g_{1}, g_{2}, g_{3}) ↦ (g_{2}, g_{3}, g_{3}^{−1}g_{2}^{−1}g_{1}g_{2}g_{3}).

Note that ẽ_{i}I_{3} is equal to ẽ_{i+1} up to conjugation (modulo three); hence they induce the same map in K-theory. Note that e_{K}(E_{3}) is invariant under I_{3}^{∗} by construction.

We are now ready to start computing:

(α ⋆ β) ⋆ γ = e_{12∗}(e_{1}^{∗}(α ⋆ β) · e_{2}^{∗}γ · e_{K}(E_{2})) = e_{12∗}(e_{1}^{∗}(e_{12∗}(e_{1}^{∗}α · e_{2}^{∗}β · e_{K}(E_{2}))) · e_{2}^{∗}γ · e_{K}(E_{2}))

We now apply the clean intersection formula to this expression, whence we obtain

(α ⋆ β) ⋆ γ = e_{12∗}(π_{2∗}(π_{1}^{∗}(e_{1}^{∗}α · e_{2}^{∗}β · e_{K}(E_{2})) · e(ν)) · e_{2}^{∗}γ · e_{K}(E_{2}))
= e_{12∗}(π_{2∗}(ẽ_{1}^{∗}α · ẽ_{2}^{∗}β · π_{1}^{∗}e_{K}(E_{2}) · e_{K}(ν)) · e_{2}^{∗}γ · e_{K}(E_{2}))
= e_{12∗}π_{2∗}(ẽ_{1}^{∗}α · ẽ_{2}^{∗}β · π_{1}^{∗}e_{K}(E_{2}) · e_{K}(ν) · π_{2}^{∗}e_{2}^{∗}γ · π_{2}^{∗}e_{K}(E_{2}))

Note here that we are using the following property of the pushforward: π_{2∗}(x·π_{2}^{∗}y) = π_{2∗}(x)·y. Thus we have

(α ⋆ β) ⋆ γ = ẽ_{123∗}(ẽ_{1}^{∗}α · ẽ_{2}^{∗}β · ẽ_{3}^{∗}γ · π_{1}^{∗}e_{K}(E_{2}) · π_{2}^{∗}e(E_{2}) · e_{K}(ν))
= ẽ_{123∗}(ẽ_{1}^{∗}α · ẽ_{2}^{∗}β · ẽ_{3}^{∗}γ · e_{K}(E_{3}))

Now we consider

α ⋆ (β ⋆ γ) = (β ⋆ γ) ⋆ α = ẽ_{123∗}(ẽ_{1}^{∗}β · ẽ_{2}^{∗}γ · ẽ_{3}^{∗}α · e_{K}(E_{3}))
= ẽ_{123∗}I_{3∗}I_{3}^{∗}(ẽ_{1}^{∗}β · ẽ_{2}^{∗}γ · ẽ_{3}^{∗}α · e_{K}(E_{3}))
= ẽ_{123∗}(I_{3}^{∗}ẽ_{1}^{∗}β · I_{3}^{∗}ẽ_{2}^{∗}γ · I_{3}^{∗}ẽ_{3}^{∗}α · I_{3}^{∗}e_{K}(E_{3}))
= ẽ_{123∗}(ẽ_{2}^{∗}β · ẽ_{3}^{∗}γ · ẽ_{1}^{∗}α · e_{K}(E_{3}))
= ẽ_{123∗}(ẽ_{1}^{∗}α · ẽ_{2}^{∗}β · ẽ_{3}^{∗}γ · e_{K}(E_{3}))
= (α ⋆ β) ⋆ γ

Hence our proof of associativity is complete.

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Department of Mathematics University of British Columbia, Vancouver, B. C., Canada

E-mail address: adem@math.ubc.ca

Department of Mathematics, University of Michigan, Ann Arbor, MI 14627

E-mail address: ruan@math.wisc.edu

Sichuan University and Max-Planck-Institut für Mathematik, Bonn

E-mail address: binzhang@mpim-bonn.mpg.de