Khovanov Homology And Gauge Theory

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).

16 pp

References in corpus (2)

• Knot Invariants from Four-Dimensional Gauge Theory
• Gaudin model and opers

Cited by in corpus (16)

• Legendrian knots and constructible sheaves
• PSL(2;C) connections on 3-manifolds with L2 bounds on curvature
• An odd categorification of quantum sl(2)
• Refined Chern-Simons Theory and Knot Homology
• Checks of integrality properties in topological strings
• Static Charges in the Low-Energy Theory of the S-Duality Twist
• Gaussian distribution of LMOV numbers
• Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology
• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Sequences of Nahm pole solutions to the SU(2) Kapustin-Witten equations
• Superopers on Supercurves
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Self-dual gravity via Hitchin's equations
• String Theory and Math: Why This Marriage May Last
• Odd Dunkl Operators and nilHecke Algebras