Fivebranes and 3-manifold homology

TL;DR

This paper explores how fivebrane compactifications unify various homological invariants of 3-manifolds, linking physical theories with topological invariants and providing explicit calculations for monopole, Floer, and Khovanov-Rozansky homologies.

Contribution

It introduces a universal physical framework for 3-manifold homologies via fivebrane compactifications and explicitly connects categorification with Eichler integrals in Chern-Simons theory.

Findings

Explicit form of the S-transform derived

Connection established between categorification and Eichler integrals

Concrete calculations for monopole, Floer, and Khovanov-Rozansky homologies

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 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.