Modular transformations of homological blocks for Seifert fibered homology $3$-spheres

TL;DR

This paper derives explicit modular transformation formulas for homological blocks of Seifert fibered integral homology 3-spheres and uses them to provide new asymptotic expansions for Witten-Reshetikhin-Turaev invariants, confirming a version of the Witten asymptotic conjecture.

Contribution

It introduces explicit modular transformation formulas for homological blocks of Seifert fibered homology 3-spheres and applies these to derive asymptotic expansions of quantum invariants.

Findings

Explicit modular transformation formulas for homological blocks.

Asymptotic expansion formulas for Witten-Reshetikhin-Turaev invariants.

A new proof of a version of the Witten asymptotic conjecture.

Abstract

In this article, for any Seifert fibered integral homology 3-sphere, we give explicit modular transformation formulas of homological blocks introduced by Gukov-Pei-Putrov-Vafa. Moreover, based on the modular transformation formulas, we have explicit asymptotic expansion formulas for the Witten-Reshetikhin-Turaev invariants which give a new proof of a version by Andersen of the Witten asymptotic conjecture.