Refined 3D index

TL;DR

This paper introduces a refined 3D index for 3-manifolds, enhancing the original by capturing additional symmetries, providing explicit formulas, and enabling finer distinctions among manifolds and gauge theory phases.

Contribution

It develops a new refined index based on Dehn surgery and triangulations, with explicit formulas, invariance checks, and a computational tool for evaluation.

Findings

Derived an explicit infinite-sum formula for the refined index.

Confirmed invariance under triangulation and presentation changes.

Enabled finer distinctions among 3-manifolds and IR phases.

Abstract

We introduce a refined version of the 3D index for 3-manifolds, building on the construction of the 3D $\mathcal{N}=2$ gauge theory $T[M]$ by Dimofte-Gaiotto-Gukov and Gang-Yonekura. The refined index is a superconformal index of $T[M]$ equipped with additional gradings that capture enhanced flavor symmetries of the effective theory. Our construction is based on a Dehn surgery presentation of $M$ in terms of an ideally triangulated link complement $N$. We derive an explicit infinite-sum formula for the refined index and provide nontrivial checks in representative examples, supporting its invariance under changes of triangulation, Dehn surgery presentation, and other auxiliary data. As a strictly stronger invariant, the refined index enables finer distinctions among 3-manifolds and among distinct IR phases of the associated gauge theories. We also introduce a computational tool, \textsc{Refined Index Calculator}, for its explicit evaluation.