Quantum Riemann surfaces in Chern-Simons theory

Abstract

We construct from first principles the operators ÂM that annihilate the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU (2), SL(2, R), or SL(2, C) on knot complements M .The operator ÂM is a quantization of a knot complement's classical A-polynomial A M ( , m).The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in topological quantum field theory to put them back together.We advocate in particular that, properly interpreted, "gluing = symplectic reduction."We also arrive at a new finite-dimensional state integral model for computing the analytically continued "holomorphic blocks" that compose any physical Chern-Simons partition function.

Locations

• Advances in Theoretical and Mathematical Physics - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF