Factorising the 3D Topologically Twisted Index

TL;DR

This paper derives a factorized formula for the 3D topologically twisted index of $ ext{TTCSM}$ on various manifolds, revealing new constraints on auxiliary fields in the process.

Contribution

It provides a novel factorization of the 3D topologically twisted index for $ ext{TTCSM}$ on different geometries, including a new constraint on the auxiliary field in the full product manifold case.

Findings

Derived the index as a convolution of TTCSM on $ ext{S}_2 imes ext{S}_1$ segments.

Expressed the index on $ ext{S}_2 imes ext{S}_1$ with a puncture.

Discovered the auxiliary field $D$ is anti-hermitian in the full product case.

Abstract

We explore the path integration -- upon the contour of hermitian (non-auxliary) field configurations -- of topologically twisted $\mathcal{N}=2$ Chern-Simons-matter theory (TTCSM) on $\mathbb{S}_2$ times a segment. In this way, we obtain the formula for the 3D topologically twisted index, first as a convolution of TTCSM on $\mathbb{S}_2$ times halves of $\mathbb{S}_1$, second as TTCSM on $\mathbb{S}_2$ times $\mathbb{S}_1$ -- with a puncture --, and third as TTCSM on $\mathbb{S}_2 \times \mathbb{S}_1$. In contradistinction to the first two cases, in the third case, the vector multiplet auxiliary field $D$ is constrained to be anti-hermitian.