On a geometric derivation of Witten's identity for Chern-Simons theory

TL;DR

This paper introduces a geometric approach to derive Witten's identity in Chern-Simons theory, connecting Wilson loop expectations to the Jones polynomial through loop deformation operators.

Contribution

It provides a formal calculational scheme relating Wilson loop expectations to knot polynomials, emphasizing geometric and algebraic structures in Chern-Simons theory.

Findings

Derived conditions on measures for Jones polynomial emergence

Established a geometric framework for Wilson loop deformations

Linked Chern-Simons expectation values to knot invariants

Abstract

We present a formal but simple calculational scheme to relate the expectation value of Wilson loops in Chern-Simons theory to the Jones polynomial. We consider the exponential of the generator of homotopy transformations which produces the finite loop deformations that define the crossing change formulas of knot polynomials. Applying this operator to the expectation value of Wilson loops for an unspecified measure we find a set of conditions on the measure and the regularization such that the Jones polynomial is obtained.