Twists of elliptic cohomology and K-theory

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Many subtle issues about elliptic cohomology concern its twists. For example, the representations of loop groups arise in algebraic topology in twisted equivariant elliptic cohomology (Grojnowski-Ando) and in twisted equivariant K-theory (Freed-Hopkins-Teleman). Ando proposes to construct a comparison map between twisted elliptic cohomology and twisted K-theory which explains why the representations of loop groups appear on both theories. For another example, in recent years there has been spectacular progress in the study of topological field theories, and it is of fundamental importance to understand how these vary in families. Ando will investigate how the most important twists of elliptic cohomology are related to the problem of constructing certain families of two-dimensional topological field theories. It is hoped that this will give insight into how elliptic cohomology itself varies in families. Elliptic cohomology is the locus of a deep and mysterious interaction between topology, number theory, and physics. In his 1998 Gibbs Lecture to the American Mathematical Society, the physicist Edward Witten called this interaction "a little piece of 21st century mathematics which happened to land in the 20th century", and proposed that understanding it would be an interesting and important problem for the 21st century. Recent work has shown that one of the fundamental pieces of structure of elliptic cohomology are its "twists": they crop up again and again each time there is important progress in the subject. Ando's project will focus directly on these twists, exploring their manifestations in algebra, topology, and physics.