Delocalized Chern character for stringy orbifold K-theory

Abstract

In this paper, we define a stringy product on K * orb (X) ⊗ C, the orbifold K-theory of any almost complex presentable orbifold X. We establish that under this stringy product, the delocalized Chern character, after a canonical modification, is a ring isomorphism. Here H * CR (X) is the Chen-Ruan cohomology of X. The proof relies on an intrinsic description of the obstruction bundles in the construction of the Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K-theory K * G (G) of a finite group G with the conjugation action. It turns out that the stringy product is different from the Pontryagin product (the latter is also called the fusion product in string theory).2010 Mathematics Subject Classification. Primary: 57R19, 19L10, 22A22. Secondary: 55N15,53D45.

A plethora of inertial products

“…A recent paper [HW13] repeats the description of the orbifold product of [JKK07,ARZ08], the formula of [JKK07,EJK10] for the obstruction class, and the Chern character ring homomorphism of [JKK07,EJK10] in the almost-complex setting, as originally described in Section 10 of [JKK07].…”

Equivariant commutative stringy cohomology rings on almost complex manifolds

“…The main ingredient in Chen-Ruan's ring structure is the obstruction bundle whose fiber is interpreted as the cokernel of certain elliptic operator. The computation of the obstruction bundle was later discovered by S. Hu and the first author [10] for abelian case, and by Hu-Wang for general cases in [23]. Their computations are crucial for our construction in this paper.…”

“…Then we get a space We next give another description of O m , which is similar to the K-theory description of the Chen-Ruan obstruction bundle obtained by Chen-Hu [10] and Hu-Wang [23], see also [18,22,25,6,16]. Consider the element (3.1).…”

“…The third equality follows from the fact that all maximal torus are conjugate and a maximal torus intersects with a conjugate class finitely. Similarly, the RHS is 23] with h 2,3 = (h 2 , h 3 ) and h 1,23 = (h 1 , h 2 h 3 ). 23] .…”

The periodic cyclic homology of crossed products of finite type algebras

“…The case when Γ is trivial is due to Feigin and Tsygan [33] (see also Emmanouil's papers [30,31] and the references therein) for the case when V is smooth, see also the paper by Dolgushev and Etinghof [28]. See also [39].…”