Index for a Model of 3d-3d Correspondence for Plumbed 3-Manifolds

TL;DR

This paper develops an index for 3d-3d correspondence in supersymmetric theories associated with plumbed 3-manifolds, demonstrating invariance under Kirby moves and connecting to homological blocks.

Contribution

It provides an effective description of the 3d theory T[M_3] from homological blocks, linking the supersymmetric index to topological invariants of plumbed 3-manifolds.

Findings

Index invariance under 3d Kirby moves

Connection between homological blocks and D^2 x_q S^1 partition functions

Effective description of T[M_3] from boundary conditions

Abstract

We consider the $S^2 \times_q S^1$ supersymmetric index of a 3d $\mathcal{N}=2$ theory $T[M_3]$ when $M_3$ is a plumbed 3-manifold. We engineer an effective description of $T[M_3]$ from the expression of the homological block for plumbed 3-manifolds as a $D^2 \times_q S^1$ partition function of a 3d $\mathcal{N}=2$ theory $T[M_3]$ with a boundary condition. We check that the supersymmetric index for such a $T[M_3]$ is invariant under the 3d Kirby moves.