Sequencing BPS Spectra

TL;DR

This paper explores the behavior of BPS spectra in physics, proposing a spectral sequence framework to relate unperturbed and perturbed states, and studies structural properties of link homologies with applications to Chern-Simons and 3d theories.

Contribution

It introduces a spectral sequence approach to BPS spectra variations and uncovers new structural properties of colored HOMFLY homology, linking physics and knot theory.

Findings

Spectral sequence describes BPS spectrum variations.

Discovery of a novel 'sliding' property in HOMFLY homology.

Identification of modular transformations in related physical theories.

Abstract

This paper provides both a detailed study of color-dependence of link homologies, as realized in physics as certain spaces of BPS states, and a broad study of the behavior of BPS states in general. We consider how the spectrum of BPS states varies as continuous parameters of a theory are perturbed. This question can be posed in a wide variety of physical contexts, and we answer it by proposing that the relationship between unperturbed and perturbed BPS spectra is described by a spectral sequence. These general considerations unify previous applications of spectral sequence techniques to physics, and explain from a physical standpoint the appearance of many spectral sequences relating various link homology theories to one another. We also study structural properties of colored HOMFLY homology for links and evaluate Poincare polynomials in numerous examples. Among these structural properties is a novel "sliding" property, which can be explained by using (refined) modular S-matrix. This leads to the identification of modular transformations in Chern-Simons theory and 3d N=2 theory via the 3d/3d correspondence. Lastly, we introduce the notion of associated varieties as classical limits of recursion relations of colored superpolynomials of links, and study their properties.