# BPS Invariants for 3-Manifolds at Rational Level $K$

## TL;DR

This paper explores the structure and asymptotic behavior of Witten-Reshetikhin-Turaev invariants for 3-manifolds at rational levels, revealing new insights into their form and limits at roots of unity.

## Contribution

It provides an expected structural form of invariants at rational levels and discusses their asymptotic expansion around roots of unity, extending previous results.

## Key findings

- Expected form of WRT invariants at rational levels
- Analysis of asymptotic expansion around roots of unity
- Connection to homological blocks at roots of unity

## Abstract

We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition function at or around roots of unity $q=e^{2\pi i \frac{1}{K}}$ with rational level $K=\frac{r}{s}$ where $r$ and $s$ are coprime integers. From the exact expression for the $G=SU(2)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds at other roots of unity obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at other roots of unity. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the standard limit in the volume conjecture.