Knot Invariants from Four-Dimensional Gauge Theory

TL;DR

This paper investigates the connection between four-dimensional gauge theory equations and the Jones polynomial of knots, providing a direct analysis without relying on quantum dualities, and linking it to conformal blocks and other topics in mathematical physics.

Contribution

It offers a direct verification of the gauge theory approach to knot invariants, clarifying the emergence of the Jones polynomial through solution counting and its relation to conformal blocks.

Findings

Understanding of how gauge theory equations produce the Jones polynomial.

Connection established between four-dimensional gauge theory and Virasoro conformal blocks.

Insights into the relation between gauge theory, integrable models, and M-theory.

Abstract

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the $M$-theory description of BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks.