Homological algebra of knots and BPS states

TL;DR

This paper explores the algebraic and physical structures underlying knot homologies, revealing new symmetries and properties through the study of BPS states, wall crossing phenomena, and deformations in Landau-Ginzburg models.

Contribution

It introduces a comprehensive framework connecting knot homologies with BPS state algebras, colored differentials, and mirror symmetry, advancing the understanding of categorified knot invariants.

Findings

Existence of colored differentials relating different knot invariants

Formulation of properties for colored HOMFLY homology

Discovery of mirror symmetry between symmetric and anti-symmetric representations

Abstract

It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models. One important application to knot homologies is the existence of "colored differentials" that relate homological invariants of knots colored by different representations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable "mirror symmetry" between these triply-graded theories.