A Note on Knot Invariants and q-Deformed 2d Yang-Mills

TL;DR

This paper demonstrates that Wilson loop expectation values in q-deformed 2d Yang-Mills theory on Riemann surfaces serve as knot invariants in 3-manifolds, with loop areas encoding topological data of the extra dimension.

Contribution

It introduces a method to compute knot invariants via q-deformed 2d Yang-Mills and highlights the significance of quantized loop areas in topological encoding.

Findings

Wilson loop expectation values are knot invariants in circle bundle 3-manifolds.

Loop areas are quantized and encode topological information.

The approach connects 2d gauge theory with 3d knot invariants.

Abstract

We compute expectation values of Wilson loops in q-deformed 2d Yang-Mills on a Riemann surface and show that they give invariants of knots in 3-manifolds which are circle bundles over the Riemann surface. The areas of the loops play an essential role in encoding topological information about the extra dimension, and they are quantized to integer or half integer values.