Stringy Product on Orbifold K-theory Revisited

Documenta Mathematica

Due to the work of many authors in the last decades, given an algebraic orbifold (smooth proper Deligne-Mumford stack with trivial generic stabilizer), one can construct its orbifold Chow ring and orbifold Grothendieck ring, and relate them by the orbifold Chern character map, generalizing the fundamental work of Chen-Ruan on orbifold cohomology. In this paper, we extend this theory naturally to higher Chow groups and higher algebraic K-theory, mainly following the work of Jarvis-Kaufmann-Kimura and Edidin-Jarvis-Kimura. Contents 1. Introduction 1 2. Preliminaries on K-theory and motivic cohomology 4 3. Equivariant K-theory and motivic cohomology 7 4. Orbifold theories: global quotient by a finite group 10 5. Orbifold theories: general setting 17 6. Application: hyper-Kähler resolution conjectures 24 References 25

Withdrawn by the authors because the results of this paper are subsumed within and improved by the two papers 1. A plethora of inertial products and 2. Chern Classes and Compatible Power Operations in Inertial K-theory

Communications in Mathematical Physics, 2003

It was argued in , that in the presence of a nontrivial Bfield, D-brane charges in type IIB string theories are classified by twisted Ktheory. In , it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary Hilbert bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. This paper studies in detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced in [4], extending the construction to the equivariant and the holomorphic cases. Included is a discussion of interesting examples.

2004

Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C * -algebras. Up to equivalence, the twisting corresponds to an element of H 3 (X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H 3 G (X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.

Journal of Noncommutative Geometry, 2013

We construct differential K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct a push-forward map in differential orbifold K-theory. Finally, we construct a non-degenerate intersection pairing with values in C/Z for the subclass of smooth orbifolds which can be written as global quotients by a finite group action. We construct a real subfunctor of our theory, where the pairing restricts to a non-degenerate R/Z-valued pairing.

Mathematical Research Letters, 2007

In this paper we prove that for an almost complex orbifold, its virtual orbifold cohomology [16] is isomorphic as algebras to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

Journal of Topology, 2007

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation, and relate the result to the Verlinde algebra and to the Kac numerator at q = 1. Verlinde's formula is also discussed in this context.

Letters in Mathematical Physics, 2010

There are two approaches to constructing stringy multiplications for global quotients. The first one is given by first pulling back and then pushing forward. The second one is given by first pushing forward and then pulling back. The first approach has been used to define a global stringy extension of the functors K 0 and K top by Jarvis-Kaufmann-Kimura, A * by Abramovich-Graber-Vistoli, and H * by Chen-Ruan and Fantechi-Göttsche. The second approach has been applied by the author in the case of cyclic twisted sector and in particular for singularities with symmetries and for symmetric products. The second type of construction has also been discussed in the de Rham setting for Abelian quotients by Chen-Hu. We give a rigorous formulation of de Rham theory for any global quotient from both points of view. We also show that the pull-push formalism has a solution by the push-pull equations in the setting case of cyclic twisted sectors. In the general, not necessarily cyclic case, we introduce ring extensions and treat all the stringy extension of the functors mentioned above also from the second point of view. A first extension provides formal sections and a second extension fractional Euler classes. The formal sections allow us to give a pull-push solution while fractional Euler classes give a trivialization of the co-cycles of the pull-push formalism. The main tool is the formula for the obstruction bundle of Jarvis-Kaufmann-Kimura. This trivialization can be interpreted as defining the physics notion of twist fields. We end with an outlook on applications to singularities with symmetries aka. orbifold Landau-Ginzburg models.

Asterisque- Societe Mathematique de France

We construct smooth equivariant K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a smooth extension of a cohomology theory. For proper submersions (with smooth fibres) we construct a push-forward map in smooth equivariant K-theory. Finally, we construct a non-degenerate inter-section pairing for the subclass of smooth orbifolds which can be written as global quotients by a finite group action.

Cohomological Methods in Homotopy Theory, 2001

In an earlier paper [10], we showed that for any discrete group G, equivariant K-theory for finite proper G-CW-complexes can be defined using equivariant vector bundles. This was then used to prove a version of the Atiyah-Segal completion theorem in this situation. In this paper, we continue to restrict attention to actions of discrete groups, and begin by constructing an appropriate classifying space which allows us to define K * G (X) for an arbitrary proper G-complex X. We then construct rational-valued equivariant Chern characters for such spaces, and use them to prove some more general versions of completion theorems. In fact, we construct two different types of equivariant Chern character, both of which involve Bredon cohomology with coefficients in the system G/H → R(H). The first, ch

K-Theory and Noncommutative Geometry, 2008

Twisted K-theory has its origins in the author's PhD thesis [27] and in a paper with P. Donovan [19]. The objective of this paper is to revisit the subject in the light of new developments inspired by Mathematical Physics. See for instance E. Witten [42], J. Rosenberg [37], C. Laurent-Gentoux, J.-L. Tu, P. Xu [41] and M.F. Atiyah, G. Segal [8], among many authors. We also prove some new results in the subject: a Thom isomorphism, explicit computations in the equivariant case and new cohomology operations.

2005

We explore the relations of twisted K-theory to twisted and untwisted classical cohomology. We construct an Atiyah-Hirzebruch spectral sequence, and describe its differentials rationally as Massey products. We define the twisted Chern character. We also discuss power operations in the twisted theory, and the role of the Koschorke classes.

K-Theory, 2000

In this paper we study the "holomorphic K -theory" of a projective variety.

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists D β (k[G]), β ∈ Z 3 (G, k *) as defined by Dijkgraaf-Pasquier-Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.

2008

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a degree three 3 Deligne cocycle. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.