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[M3] 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.

Highlights

  • The main goal of this paper is to describe the structural properties and explicit computations of 3-manifold homological invariant, HN∗,∗(M3) (1.1)whose graded Euler characteristic gives quantum sl(N ) invariant of M3

  • 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 [M3] on a Riemann surface with defects

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Summary

Introduction

The main goal of this paper is to describe the structural properties and explicit computations of 3-manifold homological invariant, HN∗,∗(M3). In order to develop a picture analogous to figure 1 and to tackle 3-manifold homologies (1.1) by a variety of methods that were so successful for knots, we first need to realize the N = 0 theory in the physical setup similar to (1.4): doubly-graded space-time: R × T ∗M3 × T ∗Σ n M5-branes: R × M3 × Σ transition In some ways this setup is simpler than (1.4); e.g. it does not require extra ingredients (branes) associated with knots and links. This vantage point clarifies the connection to Seiberg-Witten invariants and homology groups (1.5) and leads to yet another way of computing them, which we call a “refinement” of the Rozansky-Witten theory.

Categorification of a 2d A-model
General A-model and a refinement
The two bases
Twists on M3
Orders of compactification
Deformations and spectral sequences
Turaev torsion
Invariants for general plumbed 3-manifolds
S2 with defects
A different type of example
Triangulations
Reversing the order of compactification
UV: SW invariants and Floer Homology
A “refinement” of the Rozansky-Witten theory
Twists on M4
VW partition function as a CS wave function
Khovanov homology for 3-manifolds
Mock modularity and homological blocks
Back to Heegaard Floer homology
Full Text
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