3d-3d Correspondence Revisited

TL;DR

This paper revisits the 3d-3d correspondence in fivebrane compactifications, emphasizing the role of all flat connections in defining effective 3d N=2 theories and connecting to homological knot invariants.

Contribution

It highlights the importance of including all flat connections for accurate theory construction and links these to categorified knot invariants, extending previous results.

Findings

All flat connections are crucial for proper 3d theory definition.

Homological knot invariants help construct Lagrangians with desired properties.

Higgsing recovers previously known 3d theories.

Abstract

In fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2 theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomials. Higgsing the full 3d theories constructed this way recovers theories found previously by Dimofte-Gaiotto-Gukov. We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role played by all flat connections in this operation.