Special Colloquium
Ralph Kaufmann
University of Connecticut
Stringy K-theory
Abstract: We define a stringy K-theory for orbifolds/stacks.
There are two versions of this construction. The first is for global
quotients of a variety X by an action of a finite group G. The result
is a rich algebra
structure called a pre-G-Frobenius algebra. These algebras
are in particular algebra objects in the braided monoidal category of
D(k[G])-modules.
The second construction works more generally for stacks, which are
suitably "nice".
It essentially yields a Frobenius algebra. The two versions of
stringy K-theory are related by taking G invariants. The most
important novel ingredient is the definition
of a multiplication. This is governed by an obstruction bundle which
can either be defined via moduli spaces or entirely in terms of
representation theoretic data. The latter point of view makes the
construction topological and hence applicable in the algebraic and the
topological categories, i.e. to the Chow ring, K-theory, Cohomology
and topological K-theory.
We furthermore define Chern characters in the different settings which
are ring homomorphisms.
The motivation for these constructions comes from physics (string
theory and topological field theory with a finite gauge group), the
crepant resolution conjecture and the classical work of Atiyah-Segal
on equivariant K-theory.
Wednesday December 13, 2006 at 3:00 PM in SEO 636