Quantum Riemann Surfaces in Chern-Simons Theory

TL;DR

This paper develops a method to construct quantum operators that annihilate Chern-Simons theory wavefunctions for knot complements, providing a new perspective on quantization and a finite-dimensional model for holomorphic blocks.

Contribution

It introduces a first-principles construction of the quantum A-polynomial operator and a novel state integral model for Chern-Simons partition functions.

Findings

Constructed the operator 'A-hat' from first principles.

Linked gluing in TQFT to symplectic reduction.

Provided a finite-dimensional state integral model.

Abstract

We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m). The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in TQFT 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.