Sequencing BPS Spectra

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.

160 pages, 69 figures, 4 tables; Don't be intimidated by the length - each section can be read more or less independently.; v2 and v3, minor corrections, a Mathematica file is attached as an ancillary file

References in corpus (4)

• The Refined Topological Vertex
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
• B-type defects in Landau-Ginzburg models

Cited by in corpus (26)

• Fivebranes and 3-manifold homology
• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• BPS states, knots and quivers
• Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants
• Tabulating knot polynomials for arborescent knots
• A slow review of the AGT correspondence
• Topological strings, strips and quivers
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• BPS counting for knots and combinatorics on words
• Holomorphic field theories and Calabi--Yau algebras
• Twisted characters and holomorphic symmetries
• Rectangular superpolynomials for the figure-eight knot
• Kerov functions revisited
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• Refined large N duality for knots
• DAHA and skein algebra on surface: double-torus knots
• q-series and tails of colored Jones polynomials
• Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
• Learning knot invariants across dimensions
• Musings on SVD and pseudo entanglement entropies
• Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again
• On knots, complements, and 6j-symbols
• Fusion of interfaces in Landau-Ginzburg models: a functorial approach
• The quantum UV-IR map for line defects in $\mathfrak{gl}(3)$-type class $S$ theories