Two Lectures on Gauge Theory and Khovanov Homology

TL;DR

This paper explores the connection between gauge theory, elliptic PDEs, and knot invariants like the Jones polynomial and Khovanov homology, proposing a geometric and physical perspective on their definitions.

Contribution

It introduces a novel approach to defining knot invariants via counting solutions to elliptic PDEs with specific boundary conditions, linking gauge theory and knot homology.

Findings

Motivates the PDE approach to Khovanov homology

Describes boundary conditions for elliptic PDEs in gauge theory

Connects physical theories with knot invariants

Abstract

In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the solutions of certain elliptic partial differential equations in 4 or 5 dimensions. The second lecture is devoted to a description of the rather unusual boundary conditions by which these equations should be supplemented. An appendix describes some physical background. (Versions of these lectures have been presented at various institutions including the Simons Center at Stonybrook, the TSIMF conference center in Sanya, and also Columbia University and the University of Pennsylvania.)