Using Floer theory on Hamiltonian bundles, the paper constructs natural homomorphisms from π_m(BG) to categorified K-theory groups K^Cat_m(R) and gives a geometric proof that K^Cat_2(Z) is infinitely generated.
Gauge theory and mirror symmetry
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abstract
Outlined in this paper is a description of \emph{equivariance} in the world of 2-dimensional extended topological quantum field theories, under a topological action of compactLie groups. In physics language, I am gauging the theories --- coupling them to a principal bundle on the surface world-sheet. I describe the data needed to gauge the theory, as well as the computation of the gauged theory, the result of integrating over all bundles. The relevant theories are A-models, such as arise from the Gromov-Witten theory of a symplectic manifold with Hamiltonian group action, and the mathematical description starts with a group action on the generating category (the Fukaya category, in this example) which is factored through the topology of the group. Their mirror description involves holomorphic symplectic manifolds and Lagrangians related to the Langlands dual group. An application recovers the complex mirrors of flag varieties proposed by Rietsch.
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Hamiltonian elements in algebraic K-theory
Using Floer theory on Hamiltonian bundles, the paper constructs natural homomorphisms from π_m(BG) to categorified K-theory groups K^Cat_m(R) and gives a geometric proof that K^Cat_2(Z) is infinitely generated.