Twisted K-theory of Differentiable Stacks
نویسنده
چکیده
In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure K α ⊗K j β → K i+j α+β are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant Ktheory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twistedK-groups can be expressed by so-called “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C∗-algebras.
منابع مشابه
String topology for stacks
We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. In particular, we give a good notion of a free loop stack, and of a mapping stack Map(Y,X), where Y is a compact space and X a topological stack, which is functorial both in X and Y and behaves well enough with respect to p...
متن کاملThom Isomorphism and Push-forward Map in Twisted K-theory
We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map f : X → Y (not necessarily K-oriented). The push-forward map generalizes the push-forward map in ordinary K-theory for any K-oriented map and the Atiyah-Singer index theorem of Dirac operators on Clifford modules. For D-branes satisfyin...
متن کاملDifferentiable Stacks and Gerbes
We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S-bundles and S-gerbes over differentiable stacks. In particular, we establish the relationship between S-gerbes and groupoid S-central extensions. We define connections and curvings for groupoid S-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S-gerbes ...
متن کاملString Orbifolds and Quotient Stacks
In this note we observe that, contrary to the usual lore, string orbifolds do not describe strings on quotient spaces, but rather seem to describe strings on objects called quotient stacks, a result that follows from simply unraveling definitions, and is further justified by a number of results. Quotient stacks are very closely related to quotient spaces; for example, when the orbifold group ac...
متن کاملOn the de Rham Cohomology of Differential and Algebraic Stacks
We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural wa...
متن کامل