The quantum group quantization of the moduli space of flat SU(2)-connections on a surface determines the Reshetikhin-Turaev representation of the mapping class group
by
Razvan Gelca
Texas Tech University

The colored Jones polynomials of a knot are usually approached through a technique that originates in statistical mechanics and is based on interpreting the knot diagram as the evolution of a quantum mechanical system. This is the framework of the Reshetikhin-Turaev theory and is done either with quantum groups or with skein modules. The colored Jones polynomials can also be interpreted as the coordinates of the vector associated to the knot complement by the Reshetikhin-Turaev topological quantum filed theory. This theory has as an essential part a projective linear representation, for each surface, of the mapping class group.

On the other hand, the theory of the Jones polynomial, as outlined by Witten, is related to a quantization of the moduli space of flat SU(2)-connections on a surface. The Hilbert space of the quantization has been extensively studied in relation to the Verlinde formula. The operators on the Hilbert space are under current investigation by the speaker, with the case of the torus being now fully understood.

The work presented in this talk shows that the quantum group quantization of the moduli space of flat SU(2)-connections on a surface determines the Reshetikhin-Turaev representation of the mapping class group.

Date received: May 2, 2007