Lisa Jeffrey (Toronto), February 16, 1:30 pm EST

Real Polarizations and the Verlinde Formula

Abstract:

The Verlinde formula specifies the dimension of the quan- tization of the moduli space of flat G-connections on a closed oriented 2-manifold or an oriented 2-manifold with boundary where the holo- nomy around each boundary component takes a fixed value. It was proved in the late 1980’s using algebraic geometric methods, related to quantization using a Ka ̈hler polarization. When G = SU(2), a different interpretation was given by LJ and J. Weitsman, who constructed a real polarization (a Hamiltonian torus action on an open dense sub- set of the moduli space). The quantization is identified by finding the Bohr-Sommerfeld points (points with integer values of the moment map. The number of such points turned out to to equal the Verlinde dimension formula.

For G = SO(4) it is also possible to construct a Hamiltonian torus action on an open dense set, and the number of Bohr-Sommerfeld points again equals the Verlinde dimension. This is due to LJ and her PhD student Kaidi Ye.