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 Poincaré 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.

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

“…(Other B-twisted Landau-Ginzburg models have also been proposed to describe the same system, e.g. [96][97][98][99]; their relation with T 2d is still unclear. )…”

“…In the presence of nonzero t R , it is natural to deform the boundary conditions by an infinite gradient flow with respect to h t , as first discussed in section 2.5.6 (and in parallel to the abelian Higgs-branch discussion of section 6.2.3). For right boundary conditions, the deformation is achieved by rescaling 97) and sending λ → ∞. (For left b.c., one should send λ → 0 instead.)…”

“…[21,96,97,134,135], cf. the basic idea in [136], all ultimately tracing back to the physics of M2-M5 and related M5-M5 brane systems from [94,95].)…”

BPS states, knots, and quivers

“…can be very effectively computed for various families of knots, using the formalism of differentials [24,25,27,28]. Note that generalizations of the LMOV conjecture to the case of superpolynomials have been considered in [11,29].…”

Checks of integrality properties in topological strings

“…This formalism is successfully developed in [31] and [33] and has already allowed us to find the inclusive Racah matrices for R = [2,2] and even R = [3,1]. In combination with the differential expansion method [142][143][144][145][146][147][148][149][150], this provides extensions to other rectangular representations. Further progress (for other nonrectangular representations) is expected after developing the ∆-technique briefly outlined in [33].…”