Twisted K-theory and moduli spaces

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Moduli spaces of flat bundles over surfaces have been intensively studied in both mathematics and physics. On the physics side they appear in conformal field theory, gauge theory and string theory, while mathematically they arise in a variety of contexts in symplectic geometry, algebraic geometry, knot theory, and representation theory. Developments in quantum field theory in the 1990's led to predictions about the geometry and topology of these moduli spaces, some of which were later confirmed by rigorous mathematical arguments. In 1998, a new construction of the moduli spaces was discovered, involving the so-called q-Hamiltonian spaces. In contrast to earlier approaches, this construction is entirely finite-dimensional, but it involves a new type of `nonlinear moment map'. The quantization of such q-Hamiltonian spaces was developed quite recently. The technique of `twisted K-theory' used in this quantization procedure had gained popularity in theoretical physics, and specifically the `D-branes on group manifolds' considered in string theory are very closely related to q-Hamiltonian spaces. The proposed research will study open questions concerning the quantization procedure for q-Hamiltonian spaces, and explore its applications. We want to work out explicit formulas for the quantization of certain classes of moduli spaces, which had been out of reach with the more traditional techniques. We also plan to develop a general framework for nonlinear moment maps, unifying and generalizing the various existing settings. Apart from its intrinsic mathematical interest, this research will contribute to our understanding of 2-dimensional gauge theory, and of moduli spaces arising in mathematics and physics.