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Verlinde modules and quantization

Given a compact simple Lie group G and a primitive degree 3 twist h, we define a monoidal category C(G, h) with a May structure. An object in the category C(G, h) is a pair (X, f), where X is a compact G-manifold and f a smooth G-map from X to G with respect to the conjugation action of G on itself. Such an object determines a module, the equivariant twisted K-homology K^G(X, f^*(h)), for the Verlinde algebra, termed a Verlinde module, where the module action is induced by the G-action on X. In order to understand which objects in C(G, h) can be quantized, we define the closely related monoidal category D(G, h) consisting of equivariant twisted geometric K-cycles, which also has a May structure. There is a forgetful functor from D(G, h) to C(G, h), showing that an object in D(G, h) determines a Verlinde module. Every object in the category D(G, h) also has a quantization, valued in the Verlinde algebra. Finally, an equivalence relation ~ is defined on objects in D(G, h) such that the quantization functor determines an algebra isomorphism between the geometric equivariant twisted K-homology groups K^G_{geo}(G, h) = D(G, h))/~, and the Verlinde algebra.


Introduction
This work is partly inspired by a theorem of Freed, Hopkins and Teleman [9,10,11], which identifies the twisted G-equivariant K-homology group K G (G, η) of a compact Lie group G, with the Verlinde algebra R ℓ (G) of G at a level ℓ determined by the twist η. Firstly, they show that K G (G, η) is an algebra, with product induced by the multiplication m : G × G −→ G on the group G, and that their isomorphism "explains" the combinatorially complicated fusion product in the Verlinde algebra. Another consequence of their theorem is that the Verlinde algebra R ℓ (G) has the same functorial properties as K G (G, η). Recall that the importance of the Verlinde algebra is that it encodes the selection rules for the operator product expansion in certain rational conformal field theories such as the WZW-model. That is, they encode the dimensions of spaces of conformal blocks of these rational conformal field theories, i.e. dimensions of certain spaces of generalized theta functions (cf. [22]). These dimensions and their polynomial behaviour are of fundamental importance in conformal field theory. union and fusion product are compatible, and can be viewed as an analogue of an algebra structure for categories. Here the fusion product ⊛ is defined as (X 1 , f 1 ) ⊛ (X 2 , f 2 ) = (X 1 × X 2 , m • (f 1 × f 2 )), for objects (X j , f j ), j = 1, 2 in C(G, η). Denote the category of Verlinde modules by VMod(R ℓ (G)), which is a subcategory of the category of all modules Mod(R ℓ (G)) over the Verlinde algebra. We establish several interesting properties of VMod(R ℓ (G)). More formally, we show that the lax monoidal functor F : C(G, η) −→ Mod(R ℓ (G)), defined by F(X, f ) = K G (X, f * (η)), is compatible with the May structures on both categories. These can be viewed as the central results in the paper.
Next we want to understand which objects in C(G, η) can be quantized. To achieve this goal, we define a closely related category D(G, η), which objects are triples (X, E, f ) where (X, f ) is an object in the category C(G, η) and E is a G-equivariant (complex) vector bundle over X. In addition, we assume that X has an equivariant twisted Spinc structure, that is, there is a given equivariant isomorphism, where K η is the algebra bundle of compact operators on G determined by η, Cliff(T X) denotes the Clifford algebra bundle associated to the tangent bundle of X, and K denotes the algebra of compact operators on a G-Hilbert space. The objects of D(G, η) are an equivariant analogue of twisted geometric K-cycles in [4,23]. The key observation made here is that in this special case, this category has a richer structure than usual, given by the May structure defined below.
(G, 1, id : G → G) is a final object in the category D(G, η), where 1 is the trivial line bundle over G. The morphisms of D(G, η) are explicitly described in the text. In particular, a compact quasi-Hamiltonian G-manifold (M, ω, Φ) determines the object (M, 1, Φ) in D(G, η), by a result in [2]. Clearly D(G, η) is much larger, and it is closed under disjoint union , a dual operation, the fusion product ⊛ and also G-vector bundle modification, all of which will be explained in the text. Here we mention that the fusion product is for objects (X j , E j , f j ), j = 1, 2 in D(G, η). We verify that D(G, η) is a strict monoidal category that has a May structure given by and ⊛. Every object (X, E, f ) in D(G, η) has a fundamental class [X] ∈ K G (X, Cliff(T X)), by a construction of Kasparov [13]. The quantization Q(X, E, f ) ∈ R ℓ (G) is defined as Q(X, E, f ) = f * ([E] ∩ [X]), which in the special case when X itself is an equivariant Spinc manifold (i.e. W G 3 (X) = W 3 (X G ) = 0), reduces to Q(X, where ð X is the equivariant Spinc Dirac operator on X, and ð X ⊗ E denotes the coupled operator. We will show that the quantization then determines a monoidal functor, respecting the May structures on both categories. In particular, an object (X, f ) in C(G, η) has a quantization if X has an equivariant twisted Spinc structure, in which case (X, 1, f ) defines an object in D(G, η).
Consider the equivalence relation ∼ on objects in D(G, η) generated isomorphism and by the following three elementary moves explained in the text, The geometric equivariant twisted K-homology group K G geo (G, η) is defined to be the abelian group which is the quotient D(G, η)/ ∼, with addition induced by disjoint union-direct sum. Our first observation is that K G geo (G, η) has a ring structure induced by the fusion product ⊛ on D(G, η).
The map induced by quantization is an isomorphism of rings, This is a special case of a more general theorem which will be proved elsewhere, which uses a hybrid of techniques in [4,23] and [5].
An impact of this result is that the May structure on the category D(G, η) induces the algebra structure on K G (G, η), which by [9,10,11] is just the Verlinde algebra R ℓ (G).

The category C(G, η)
Let G be a connected, simply connected and simple compact Lie group with the multiplication map m : G ×G → G. Let G act on G by the conjugation. Then it is known that every cohomology class [η] ∈ H 3 G (G, Z) has a primitive representative, i.e., there is η ∈ Ω 3 Z (G) a closed 3-form on G with integral periods, such that m * (η) = p * 1 (η) + p * 2 (η). Let C(G, η) denote the category, whose objects are pairs (X, f ), where X is a compact G-manifold and f : X → G is a G-equivariant map with respect to the conjugation action of G acting on G.
A morphism θ between the objects (X 1 , f 1 ) and (X 2 , f 2 ) in the category C(G, η) is a (pointed) smooth G-map θ : X 1 → X 2 compatible with the structure maps, i.e., the following In particular, the objects (X 1 , f 1 ) and (X 2 , f 2 ) in the category C(G, η) are said to be isomorphic if there is an isomorphism θ : ( The chosen η determines a canonical class in H 3 (X, Z) for any object (X, f ) given by the pullback along the structure map f * (η). We list below several interesting objects of the category C(G, η). More examples will be given in the appendix.

Example 1. (Trivial action)
Let X be a G-space with the trivial G-action, and f : X → G be a continuous map such that the image of f lies in the centre of G. Then f is equivariant with respect to the adjoint action of G on G, so that (X, f ) ∈ C(G, η) for any G-vector bundle E over X.

Example 2. (Free action)
Let X be a free G-space, and f : X/G → G/G be a continuous map to the space of conjugacy classes G/G in G. Then f lifts to an equivariant mapf : X → G with respect to the adjoint action of G on G, so that (X, f ) ∈ C(G, η).

Example 3. (Hamiltonian G-spaces)
Let (X, ω) be a Hamiltonian G-space with moment map µ : X −→ g * . If G has an Ad-invariant metric, then g * ∼ = g, which will be assumed here.
Then (X, f ) ∈ C(G, η), where f = exp(µ), where exp : g −→ G denotes the exponential map of the Lie group G.

Example 7. (Fixed point set)
Suppose that a compact Lie group K acts by automorphisms on (X, f ) ∈ C(G, η). Then the fixed-point set X K and the restriction of f , f K : X K → G K determines an object (X K , f K ) ∈ C(G K , η K ) and is such that the inclusion ι : X K ֒→ X is a morphism.

Example 8. (Fixed point set -special case)
As a special case of the example above, suppose that (X, f ) is an object in C(G, η) and g ∈ G. Then the components of the fixed point set of g, (X g , f g ) ∈ C(G g , η g ) is such that the inclusion ι : X g ֒→ X is a morphism.

Example 9. (Finite covers)
Suppose that (X, f ) is an object in C(G, η) and p : G → G a finite cover of G. Consider the fibre product X = {(x,g) ∈ X × G f (x) = p(g)} and f = p * 1 f , where p 1 : X → X is the projection onto the first factor. Then ( X, f ) is an object in C(G, η) such that the quotient map X → X is a morphism in C(G, η).

Example 10. (Quasi-Hamiltonian G-spaces and Hamiltonian LG-spaces)
We recall the definition from [1] of the definition of a group-valued moment map for a quasi-Hamiltonian G-space. A quasi-Hamiltonian G-manifold is a G-manifold M with an invariant 2-form ω ∈ Ω 2 (M) G and an equivariant map Φ ∈ C ∞ (M, G) G such that where v ξ is the vector field on M generated by ξ ∈ g, θ is the left invariant Cartan-Maurer form,θ is the right invariant Cartan-Maurer form; (3) At each x ∈ M, the kernel of ω x is given by

The map Φ is called the group valued moment map of the quasi-Hamiltonian
Basic examples of quasi-Hamiltonian G-spaces are provided by products of conjugacy classes C ⊂ G as in [1]. More generally, a bijective correspondence between Hamiltonian loop group manifolds with proper moment map and quasi-Hamiltonian G-manifold is established in that paper. They constitute a large number of objects in C(G, η).

Example 11. (Disjoint union)
The disjoint union of a pair of objects (X i , f i ), i = 1, 2 in C(G, η) is defined as the disjoint union, For example, let W be an equivariant vector bundle over X. Then ( X, f • π) is again an object in C(G, η). Here X denotes the unit sphere bundle S((X × R) ⊕ W ), π : X −→ X is the projection.
1.2. The fusion monoidal structure on C(G, η). Recall that a monoidal category is a category in which associated to each pair of objects A and B there exists a product object A ⊗ B, and there is an identity object 1, such that 1 ⊗ A ∼ = A ∼ = A ⊗ 1, together with associator isomorphisms for any three objects A, B and C, satisfying the pentagonal identity: with each arrow the appropriate map Φ, and the triangle relation: MacLane's coherence theorem ensures that these conditions are sufficient to guarantee consistency of all other rebracketings.
Recall the fusion product in C(G, η) from Example 5. Given (X 1 , f 1 ) and (X 2 , f 2 ) the following diagram describes the fusion product is a strict monoidal category with product given by the fusion product ⊛ (as in Example 5) and identity element (e, I), where e ∈ G is the identity element and I : e ֒→ G is the inclusion map, which is equivariant under the adjoint action of G on itself and the trivial action of G on e.
Proof. We need to verify that C(G, η) satisfies the axioms of a monoidal category.
Let (X, f ) ∈ C(G, η) and consider the fusion product which is clearly isomorphic to (X, f ). Similarly, (e, I) ⊛ (X, f ) is also isomorphic to (X, f ). Therefore (e, I) ∈ C(G, η) serves as an identity in C(G, η). Next, let (X i , f i ) ∈ C(G, η), i = 1, 2, 3, and The LHS is canonically isomorphic to whereas the RHS is canonically isomorphic to follows from the associativity of the multiplication m on G. In particular, Φ 1,2,3 = id and C(G, η) is a strict category. In particular, the pentagonal identity is automatically satisfied, and by MacLane's coherence theorem, C(G, η) is a strict monoidal category.
1.3. May structure. Motivated by the study of infinite loop spaces and algebraic K-theory J. P. May introduced the machinery of (bi)permutative categories [15,16]. These are, roughly, categories equipped with two monoidal structures satisfying certain compatibility conditions. We have seen before the fusion monoidal structure on C(G, η).

Definition 1.2. A May structure on a category C is a pair of monoidal structures
We call a category equipped with a May structure as a May category. A May functor F : C −→ D between May categories is one which is lax monoidal with respect to both ⊕ and ⊗, such that, in addition, the following diagram commutes: Here the vertical maps are the distributivity isomorphisms and the horizontal maps are furnished by the lax monoidal structures.
Example 13. The category C(G, η) endowed with ⊕ = and ⊗ = ⊛ is a May category. We have already seen that disjoint union and fusion product are monoidal structures on C(G, η). The distributivity condition is readily verified.

Continuous trace C * -algebras on G and G-coalgebra structure
Let G be a compact Lie group and let X be a compact G-space. Let K denote the C * -algebra of compact operators on a separable G-Hilbert space. A G-equivariant Dixmier-Douady (DD) bundle on X is a G-equivariant algebra bundle on X, whose fibres are K with the projective unitary group PU as the structure group. Given any G-equivariant principal PU bundle P over X, one can construct an a DD-bundle over X as an associated bundle K P = P × PU K. The equivariant Dixmier-Douady class of a G-equivariant Dixmier-Douady (DD) bundle on X is a cohomology class in H 3 G (X, Z). A recent theorem of Atiyah-Segal [3] says that every equivariant cohomology class in H 3 G (X, Z) determines a G-equivariant PUbundle P over X up to G-equivariant isomorphism, which determines the G-equivariant DDbundle K P , thus establishing the converse. In fact, the G-equivariant isomorphism classes of DD-bundles K P over X form an abelian group under direct sum, called the equivariant Brauer group, Br G (X). Since the equivariant Dixmer-Douady class of a direct sum of Gequivariant DD-bundles is equal to the sum of the equivariant Dixmer-Douady classes of the G-equivariant DD-bundles, there is a natural isomorphism of groups, Br G (X) ∼ = H 3 G (X, Z). Consider a category whose objects are pairs (X, P ), where X is a compact G-space and P is a fixed choice of a G-equivariant principal PU-bundle on X. A morphism (X, P ) → (X ′ , P ′ ) in this category is a continuous G-map f : X → X ′ , such that f * (P ′ ) = P . Let G, X be as above and let P be a G-equivariant principal PU-bundle on X. Let CT(X, P ) denote the stable continuous trace C * -algebra of all continuous sections of the associated equivariant DD bundle K P over X, that vanish at infinity. The induced G-action on CT(X, P ) makes it into a G-C * -algebra, which enables us to construct the crossed product C * -algebras CT(X, P ) ⋊ G. The association (X, P ) → CT(X, P ) ⋊ G is functorial with respect to the above-mentioned morphisms of pairs. Furthermore, if (X, P ) and (X, P ′ ) are two pairs, such that the Dixmier-Douady invariants of P and P ′ determine the same class in H 3 G (X, Z), then the equivariant Dixmier-Douady Theorem says that the C * -algebras CT(X, P ) ⋊ G and CT(X, P ′ ) ⋊ G are stably isomorphic [3].
Given a compact simple Lie group G with multiplication map m : The equivariant Dixmier-Douady class η of a primitive Gequivariant DD-bundle K P satisfies the corresponding primitivity property, m * (η) = p * 1 (η) + p * 2 (η). There are many natural ways to construct primitive G-equivariant DD-bundle K P on G. One of these uses positive energy representations [3,9,10,11] and another more constructive method is in [18]. In this case, we denote the equivariant continuous trace algebra by CT(G, η). If (X, f ) is an object in the category C(G, η), then we denote the corresponding equivariant continuous trace algebra by CT(X, f * (η)). We next prove some fundamental facts about the equivariant continuous trace algebras CT(G, η) and CT(X, f * (η)).
Lemma 2.1. Let G be a compact group, and η be a primitive DD-bundle on G. Then the C * -algebras CT(G, η) and CT(G, η) ⋊ G both carry a natural coalgebra structure induced by the multiplication on G.
Proof. The multiplication map m : G × G → G induces a * -homomorphism Since the multiplication map m and the C * -tensor product ⊗ are associative, it follows that the induced * -homomorphism m * is coassociative, that is, the diagram below commutes, Moreover, the multiplication map m : G × G → G is G-equivariant under the adjoint action of G. Therefore, it induces a * -homomorphism of G-C * -algebras and in turn that of the crossed product algebras, Coassociativity of m * follows by similar reasoning as above.
Using analogous arguments one easily proves the following result: Let (X, f ) be an object in the category C(G, η). Then CT(X, f * (η)) (resp. CT(X, f * (η)) ⋊ G) is a comodule over the coalgebra CT(G, η) (resp. CT(G, η) ⋊ G), which is induced by the group action map.
Proof. The group action map a : G × X → X is a G-map under the adjoint action of G, therefore it induces a * -homomorphism of G-C * -algebras, since a * (f * η) = p * 1 (η) + p * 2 f * (η) by the primitivity assumption on η. The defining property of the action map a shows that a * is a comodule map, that is, the diagram below commutes, CT(X, f * (η)) Moreover, the group action map a : G × X → X is G-equivariant under the adjoint action of G. Therefore, it induces a * -homomorphism of G-C * -algebras and in turn that of the crossed product algebras, Coassociativity of a * follows by similar reasoning as above.
3. The ring structure on K G (G, η) and Verlinde modules We will derive the ring structure on K G (G, η) as a special instance of a more general result. We will also show that for (X, f ) ∈ C(G, η), K G (X, f * (η)) is a module for the ring K G (G, η) as a special case of a more general result. The functor F : C(G, η) −→ Mod(R ℓ (G)) defined by F(X, f ) = K G (X, f * (η)) is shown to be a lax monoidal which respects the May structures on both categories.
The very general result that we will prove in this section this described as follows. Recall that if A is a separable G-C * -algebra, then the equivariant K-homology K G (A) is in general only an abelian group. However, if A is also a G-coalgebra, then we will show that K G (A) is a ring, with product induced by the comultiplication on A. In particular, we deduce that the K-homology of a C * -quantum group is an ring, which appears to be new. We also prove in an analogous way that the equivariant K-theory K G (A) is a coring, with coproduct induced by the comultiplication on A. In particular, we deduce that the K-theory of a C * -quantum group is a coring, which also appears to be new. Finally, if A is also K-oriented in equivariant K-theory, then both K G (A) and K G (A) are bialgebras in a natural way.
A key step in defining the product in K G (A) uses Kasparov's cup-cap product. More precisely, let A 1 , A 2 , D, B 1 , B 2 be separable G-C * -algebras, where G is a locally compact group. Then the cup-cap product, (Definition 2.12 of [13]) Proposition 3.1. Let A be a separable G-C * -algebra which is also a G-coalgebra. Then the abelian group KK G (A, C) admits a ring structure induced by the comultiplication on A.
Proof. When B 1 = C = B 2 = D and A 1 = A 2 = A, the cup-cap product (4) reduces to The comultiplication ∆ on A is a G- * -homomorphism Since equivariant-K-homology is contravariant with respect to G- * -homomorphisms, we get The composition of the morphisms in equations (5) and (7) gives which is a product • on equivariant K-homology KK G (A, C) induced by the comultiplication. The associativity of the product • follows from the associativity of the cup-cap product (Theorem 2.14 of [13]), the coassociativity of the comultiplication ∆ on A and the naturality of the cup-cap product under G-C * -homomorphisms.
More precisely, the associativity of Kasparov's cup-cap product says that the following diagram commutes, ? ?
On the other hand, the coassociativity of the comultiplication ∆ says that the following diagram commutes, ? ?
Therefore one has the commutative diagram, The top left hand square commutes since the cup-cap product is associative, the bottom right hand square commutes since the comultiplication is coassociative, while the remaining squares commute because the cup-cap product is functorial under G-C * -homomorphisms. Therefore one has the commutative diagram, • which precisely says that product • is associative.
The following corollary was was first established in [9,10,11]. See also [21]. Proof. Setting A = CT(G, η) in Proposition 3.1, we need to show that A is a G-coalgebra.
The multiplication map m : G × G → G is equivariant under the adjoint action of G, therefore it induces a * -homomorphism of G-C * -algebras which is a comultiplication, m * : CT(G, η) → CT(G, η) ⊗ CT(G, η).
Since the multiplication map m and the C * -tensor product ⊗ are associative, it follows that the induced * -homomorphism m * is coassociative.
The following is an immediate corollary of Proposition 3.1.

Corollary 3.3. Let A be a compact C * -quantum group. Then the abelian group KK(A, C) admits a ring structure induced by the comultiplication on A.
We next prove another general result.
Proposition 3.4. Let A 1 be a separable G-C * -algebra which is also a G-coalgebra. Let A 2 be another separable G-C * -algebra which is also a G-comodule for A 1 . Then the abelian group K G (A 2 ) = KK G (A 2 , C) is a module for the algebra K G (A 1 ) = KK G (A 1 , C).

Proof. A special case of the cup-cap product in equation (4), reduces when B
The G-comodule action map induces a * -homomorphism of G-C * -algebras. Using the fact that K-homology is contravariant with respect to * -homomorphisms, we get a canonical abelian group homomorphism Composing ⊗ C with ∆ * , we obtain The fact that • is an action follows from the associativity of the cup-cap product (Theorem 2.14 of [13]) and the defining property of the comodule map ∆.
More precisely, the associativity of Kasparov's cup-cap product says that the following diagram commutes, where K G (A) denotes KK G (A, C), A 1 = CT(G, η) and A 2 = CT(X, f * (η)).
On the other hand, the defining property of the comodule map ∆ says that the following diagram commutes, Therefore the induced diagram in equivariant K-homology commutes Therefore one has the commutative diagram, The top left hand square commutes since the cup-cap product is associative, the bottom right hand square commutes since the coaction is coassociative, while the remaining squares commute because the cup-cap product is functorial under G-C * -homomorphisms.
Therefore the product • satisfies the commutative diagram • which precisely says that • is an action.
Proof. Let (X, f ) ∈ C(G, η). Setting A 1 = CT(G, η) and A 2 = CT(X, f * (η)), we see that The group action map a : G × X → X is a G-map under the adjoint action of G, therefore it induces a * -homomorphism of G-C * -algebras, by the primitivity assumption on η. The defining property of the action map a shows that a * is a comodule map. The corollary is proved by applying Proposition 3.4.
Combining the above with the result of Freed-Hopkins-Teleman [9,10,11], we have, Corollary 3.6. Let (X, f ) ∈ C (G, η), where G is a compact simple Lie group. Then K G (X, f * (η)) is a module over the Verlinde algebra R ℓ (G), where ℓ is the level determined by twist η.
We call such a module over R ℓ (G), a Verlinde module. The following is an immediate corollary of Proposition 3.4.
Corollary 3.7. Let A 1 be a compact C * -quantum group and A 2 be a separable C * -algebra which is also a comodule for A 1 . Then the abelian group KK(A 2 , C) is a module for the algebra KK (A 1 , C).

Morphisms of Verlinde modules.
Here we study morphisms in the category C(G, η), and our main result here is that any morphism in the category C(G, η), determines a morphism of Verlinde modules over the Verlinde algebra. a morphism in C(G, η), where G is a compact simple Lie group. Then φ * : ) is a morphism of Verlinde modules.
Proof. Let φ : (X 1 , f 1 ) −→ (X 2 , f 2 ) be a morphism in C(G, η). That is, φ : X 1 → X 2 is an equivariant map such that the following diagram commutes, Then it induces a morphism of groups We will show that φ * is actually a morphism of Verlinde modules over the Verlinde algebra. That is, the following diagram commutes, That is, for ξ ∈ K G (G, η) and x 1 ∈ K G (X 1 , f * 1 (η)), the commutativity of the diagram above says that where we define • j below. Now we have the commutative diagram, Therefore φ • a 1 = a 2 • φ. So the induced maps on equivariant K-homology satisfy φ * • a 1 * = a 2 * • φ * . As observed earlier, since η is primitive, a * j (f * j (η)) = p * 1 (η) + p * 2 (f * (η)), j = 1, 2. Consider the diagram, The commutativity of the right square follows from the commutativity of the diagram in equation (11). The commutativity of the left square follows by naturality of the cup-cap product of Kasparov [13,6]. Therefore we have justified the commutativity of the diagram in equation (10), proving the proposition.
3.2. The May structure on C(G, η) and a lax May functor. Let Mod(R ℓ (G)) denote the abelian category of all modules over the Verlinde ring R ℓ (G). We endow Mod(R ℓ (G)) with the symmetric monoidal structure given by the tensor product ⊗ = ⊗ Z of abelian groups, as well as the symmetric monoidal structure given by the direct sum ⊕, giving it a May structure.
Recall that a functor F : (C, ⊗ C , I C ) −→ (D, ⊗ D , I D ) between strict monoidal categories is called lax monoidal if there are natural maps F (C) ⊗ D F (D) → F (C ⊗ C D) and I D → F (I C ) for every pair of objects C, D ∈ C satisfying certain predictable compatibility conditions. Proof. Let (X 1 , f 1 ), (X 2 , f 2 ) ∈ C(G, η) and recall that F(X i , f i ) = K G (CT(X i , f * i (η))) for i = 1, 2. Recall that K G (X, f * (η)) = K G (CT(X, f * (η))). Then The cup-cap product in equation (8) with A 1 = CT(X 1 , f * 1 (η)) and A 2 = CT(X 2 , f * 2 (η)) gives where we have used the primitivity of η for the last equality. Therefore we get a natural map There is also a canonical map relating the unit objects as follows: Z→K(e, I) = K G (e), which is the unique unital ring homomorphism. Note that Z is the unit object in the monoidal category Mod(R ℓ (G)) with respect to ⊗ Z . Also The distributive property is clear, completing the proof of the theorem.

The category D(G, η) and quantization
In this section, we explore when objects in the category C(G, η) can be quantized. To achieve this goal, we define a closely related category D(G, η), which objects are triples (X, E, f ) where (X, f ) is an object in the category C(G, η) and E is a G-equivariant (complex) vector bundle over X. In addition, we assume that X has an equivariant twisted Spinc structure, that is, the following diagram commutes, Here K(Z, 3) is the 3rd Eilenberg-Maclane space, ν is a continuous map classifying the stable normal bundle of the Borel construction X G = EG × G X or equivalently, classifying the equivariant stable normal bundle of X. Similarly G G = EG × G G where G acts on itself by conjugation and f G is the map induced by f . Moreover π is a continuous map determined by the Stieffel-Whitney class, up to homotopy. Such a choice is fixed. This implies that where η G is the induced twisting on G G . This is the analogue of the Freed-Witten anomaly cancellation condition for D-branes in type II superstring theory, [12]. Geometrically, equation (13) means that there is an equivariant isomorphism, where K η is the algebra bundle of compact operators on G determined by η, Cliff(T X) denotes the Clifford algebra bundle associated to the tangent bundle of X, and K denotes the algebra of compact operators on a G-Hilbert space. The objects of D(G, η) are an equivariant analogue of twisted geometric K-cycles in [4,23]. The key observation made here is that in this special case, this category has a richer structure than usual, given by the May structure.
(G, 1, id : G → G) is a final object in the category D(G, η), where 1 is the trivial line bundle over G. The morphisms of D(G, η) are explicitly described in the text. In particular, a compact quasi-Hamiltonian G-manifold (M, ω, Φ) determines the object (M, 1, Φ) in D(G, η), by a result in [2]. Clearly D(G, η) is much larger, and it is closed under disjoint union , a dual operation, the fusion product ⊛ and also G-vector bundle modification, all of which will be explained in the text. Here we mention that the fusion product is ( We verify that D(G, η) is a strict monoidal category that has a May structure given by and ⊛.
Let (X 1 , E 1 , f 1 ) and (X 2 , E 2 , f 2 ) denote equivariant twisted geometric K-cycles in D(G, η). They are said to be isomorphic if there is an equivariant diffeomeophism φ : X 1 → X 2 such that the following diagram commutes, We now impose an equivalence relation ∼ on D(G, η), generated by isomorphism and the following three elementary relations: (1) Direct sum -disjoint union. Let (X, E 1 , f ) and (X, E 2 , f ) denote equivariant twisted geometric K-cycles in D(G, η) with the same equivariant twisted Spinc structure, then their disjoint union is the equivariant twisted geometric K-cycle given by the direct sum, (2) Equivariant bordism. Given two equivariant twisted geometric K-cycles (X 1 , E 1 , f 1 ) and (X 2 , E 2 , f 2 ) such that there exists an equivariant twisted Spinc manifold with boundary W , an equivariant vector bundle E over W and a G-map f : W → G such that Here −X denotes the G-manifold X with the opposite equivariant twisted Spinc structure. Then (W, E, f ) is said to be an equivariant bordism between the equivariant twisted geometric K-cycles (X 1 , E 1 , f 1 ) and (X 2 , E 2 , f 2 ).
(3) Equivariant Spinc vector bundle modification. Let (X, E, f ) be an equivariant twisted geometric K-cycle and V an equivariant a Spinc vector bundle over X with even dimensional fibers. Denote by R the trivial rank one real vector bundle. Choose an invariant Riemannian metric on V ⊕ R, let be the total space of the sphere bundle of V ⊕ R, which is a G-manifold. Then the vertical tangent bundle T vert ( X) of X admits a natural equivariant Spinc structure with an associated Z 2 -graded equivariant spinor bundle S + V ⊕ S − V . Denote by π : X → X the projection, which is equivariantly K-oriented. Then the equivariant Spinc vector bundle modification of (X, E, f ) along the equivariant Spinc vector bundle V , is the equivariant twisted geometric K-cycle ( X, π * E ⊗ S + V , f • π).
Definition 4.1. Denote by K G geo,• (G, η) = D(G, η)/ ∼ the geometric equivariant twisted Khomology. Addition in K G geo,• (G, η) is given by disjoint union -direct sum relation. Note that the equivalence relation ∼ preserves the parity of the dimension of the underlying equivariant twisted Spinc manifold. Let K G geo (G, η) denote the subgroup of K G geo,• (G, η) determined by all geometric cycles with even dimensional equivariant twisted Spinc manifolds.
Define the fusion product ⊛ of equivariant twisted geometric K-cycles (X 1 , E 1 , f 1 ) and (X 2 , E 2 , f 2 ) as Here m : G × G → G denotes the multiplication on the group, which is an equivariant map with respect to the conjugation action of G on itself. Then (D(G, η), , ⊛) is a May category and we have, There is a quantization functor Q : D(G, η) → K G (G, η) which we recall here. Recall that every equivariant twisted geometric K-cycle (X, E, f ) has a fundamental class [X] ∈ K G (X, Cliff(T X)) as defined in [13], which is a Dirac type operator. Then Q(X, E, f ) = f * ([E] ∩ [X]) ∈ K G (G, η). Then we have is monoidal, respecting the May structures on both categories.
There is also a canonical map relating the unit objects as follows: Q(e, 1, I) = I * (1 ∩ [e]) = 1 which is the unique unital ring homomorphism. Also by the disjoint union-direct sum property, one has Q((X 1 , E 1 , f 1 ) (X 2 , E 2 , f 2 )) = Q((X 1 Finally, the following is a special case of a more general result, This is a special case of a more general theorem which will be proved elsewhere, which uses a hybrid of techniques in [4,23] and [5]. An impact of this result is that the May structure on the category D(G, η) induces the algebra structure on K G (G, η), which by [9,10,11] is just the Verlinde algebra R ℓ (G).

Outlook and questions
For G as in the paper, recall that the Verlinde algebra R ℓ (G) consists of equivalence classes of positive energy representations (at a fixed level ℓ) of the free loop group LG, [19]. The theorem of Freed-Hopkins-Teleman [9,10,11] establishes an explicit isomorphism between K G (G, η) and R ℓ (G), where η is a degree 3-twist on G that determines the level ℓ. This isomorphism is given by a projective family of second quantized supersymmetric Dirac operators, coupled to a positive energy representation.
Given (X, f ) an object in C(G, η), is it possible to construct a representation theoretic group R ℓ (X, f ), which is a module over R ℓ (G), together with an explicit isomorphism between K G (X, f * (η)) and R ℓ (X, f ) that generalizes the Freed-Hopkins-Teleman isomorphism?
The algebraic properties of the Verlinde algebra R ℓ (G) are well understood by now. Given (X, f ) an object in C(G, η), it would be interesting to understand the algebraic properties of the Verlinde module K G (X, f * (η)).
Composing with the map τ above, we see that and for the adjoint action of G on G. Therefore (G/T × T, p) ∈ C(G, η).
The principal torus bundle T → G → G/T has first Chern class a ∈ H 2 (G/T ; T ), where T denotes the Pontryagin dual of T . Also, the universal cover of T determines a class b ∈ H 1 (T ; T ). Then a calculation shows that p * (η) = p * 1 (a) ∪ p * 2 (b) ∈ H 3 (G/T × T ; Z) where we have used the pairing T × T → Z given by the dot product, (n, m) → n.m.