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Fivebranes and 3-manifold homology

A preprint version of the article is available at arXiv.

Abstract

Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d \( \mathcal{N}=2 \) theory T[M 3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categorification of Chern-Simons partition function. Some of the new key elements include the explicit form of the S-transform and a novel connection between categorification and a previously mysterious role of Eichler integrals in Chern-Simons theory.

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Gukov, S., Putrov, P. & Vafa, C. Fivebranes and 3-manifold homology. J. High Energ. Phys. 2017, 71 (2017). https://doi.org/10.1007/JHEP07(2017)071

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Keywords

  • Chern-Simons Theories
  • Topological Field Theories
  • M-Theory
  • Topological Strings