On Quantum Modularity for Geometric 3-Manifolds

TL;DR

This paper formulates and proves a strong version of the quantum modularity conjecture for Witten--Reshetikhin--Turaev invariants of geometric 3-manifolds, involving distinguished flat connections and integrality properties, with verified cases including Brieskorn spheres.

Contribution

It introduces a strengthened conjecture relating quantum invariants and modular transformations for all geometric 3-manifolds, extending previous hyperbolic cases.

Findings

Proves the conjecture for Brieskorn homology spheres.

Establishes a connection between quantum invariants and flat connections across geometries.

Discusses the relation to Chern--Simons path integrals at roots of unity.

Abstract

The quantum modularity conjecture, first introduced by Don Zagier, is a general statement about a relation between $\mathfrak{sl}_2$ quantum invariants of links and 3-manifolds at roots of unity related by a modular transformation. In this note we formulate a strong version of the conjecture for Witten--Reshetikhin--Turaev invariants of closed geometric, not necessarily hyperbolic, 3-manifolds. This version in particular involves a geometrically distinguished $SL(2,\mathbb{C})$ flat connection (a generalization of the standard hyperbolic flat connection to other Thurston geometries) and has a statement about the integrality of coefficients appearing in the modular transformation formula. We prove that the conjecture holds for Brieskorn homology spheres and some other examples. We also comment on how the conjecture relates to a formal realization of the $\mathfrak{sl}_2$ quantum invariant at a general root of unity as a path integral in analytically continued $SU(2)$ Chern--Simons theory with a rational level.