Sequencing BPS spectra

  • Sergei Gukov
  • Satoshi Nawata
  • Ingmar Saberi
  • Marko Stošić
  • Piotr Sułkowski
Open Access
Regular Article - Theoretical Physics


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 \( \mathcal{N}=2 \) 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.


Differential and Algebraic Geometry Supersymmetry and Duality Topological Field Theories Topological Strings 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sergei Gukov
    • 1
    • 2
  • Satoshi Nawata
    • 1
    • 3
  • Ingmar Saberi
    • 1
  • Marko Stošić
    • 4
    • 5
  • Piotr Sułkowski
    • 1
    • 6
  1. 1.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Centre for Quantum Geometry of Moduli SpacesUniversity of AarhusAarhusDenmark
  4. 4.CAMGSD, Departamento de Matemática, Instituto Superior TécnicoLisbonPortugal
  5. 5.Mathematical Institute SANUBelgradeSerbia
  6. 6.Faculty of PhysicsUniversity of WarsawWarsawPoland

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