Khovanov Homology And Gauge Theory

TL;DR

This paper proposes a novel gauge theory-based approach to Khovanov homology, using solutions of elliptic PDEs on manifolds with boundary, connecting knot invariants with gauge theory and quantum field theory.

Contribution

Introduces a new gauge-theoretic framework for Khovanov homology using elliptic PDEs on manifolds with boundary, inspired by quantum field theory.

Findings

Formulates elliptic PDEs on manifolds with boundary to define knot invariants.

Establishes a formal analogy with Floer and Donaldson theories.

Provides a classical gauge theory perspective on Khovanov homology.

Abstract

In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and five-dimensional manifolds with boundary, with a rather subtle boundary condition that encodes the knots and links. The construction is formally analogous to Floer and Donaldson theory in three and four dimensions. It was discovered using quantum field theory arguments but can be described and understood purely in terms of classical gauge theory. (Based on a lecture at the conference Low-Dimensional Manifolds and High-Dimensional Categories, University of California at Berkeley, June 2011).