Knot Invariants from Four-Dimensional Gauge Theory

Understand

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.