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.
Highlights
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
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
As we explain in great details link homologies are realized as spaces of BPS states in several M-theory configurations, in which the geometry of certain branes is prescribed by a choice of link L ⊂ S3 [1,2,3,4,5]
Summary
The bulk of this paper deals with the structure of link homologies. As we explain in great details link homologies are realized as spaces of BPS states in several M-theory configurations, in which the geometry of certain branes is prescribed by a choice of link L ⊂ S3 [1,2,3,4,5]. One of the main goals of this paper is to obtain a physical understanding of the spectral sequences between various link homologies These spectral sequences imply relations between the BPS spectra of different configurations of branes in M-theory, or more generally between different physical theories. The first sections of this paper, discuss spectral sequences and deformation problems in supersymmetric theories in very general terms, and give simple examples of how well-known results (ranging from twists of 2d N = (2, 2) theories to specializations of 4d indices) fit into our framework. They can be read independently of the rest of the paper. One of these examples (in the context of Landau-Ginzburg models) will be the relevant case in the context of link homology
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