|Title||The $K$-theory of loop spaces and elliptic cohomology|
|Presenter||Matthew James Spong (The University of Melbourne)|
|Author(s)||Mr Matthew James Spong|
|Session||Category Theory, Algebraic Topology, K-Theory|
Let $T$ be a torus. In 1994 Ian Grojnowski gave a construction of a $T$-equivariant elliptic cohomology theory associated to an elliptic curve over the complex numbers. However, as noted by Grojnowski himself, this construction is somewhat ad hoc and unwieldy to work with. Let $M$ be a $T$-space and $LM$ the space of free loops in $M$, so that there is an action of $LT$ on $LM$, where $LT$ is the group of free loops in the torus. Based on work of Nitya Kitchloo, we construct a version of equivariant $K$-theory for $LT$-spaces, and show that the $LT$-equivariant $K$-theory of $LM$ is isomorphic to Grojnowski's theory on the $T$-space $M$. Since the loop space construction is motivated by the idea of fields on a circle, this suggests a physical interpretation of Grojnowski's theory.