Three-Manifold Quantum Invariants and Mock Theta Functions

TL;DR

This paper explores the connection between mock modular forms and quantum invariants of three-manifolds, proposing a conjecture on their mock modular properties and providing concrete computations for the Brieskorn sphere.

Contribution

It introduces a new conjecture linking mock modular forms to three-manifold invariants and demonstrates this with explicit calculations for a specific Seifert manifold.

Findings

Conjecture on mock modular properties of quantum invariants for certain three-manifolds

Concrete computations for the Brieskorn sphere $\\Sigma(2,3,7)$

Highlighting the role of mock modular forms in three-manifold topology

Abstract

Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere $\Sigma(2,3,7)$. This note is partially based on the talk by the first author in the conference "Srinivasa Ramanujan: in celebration of the centenary of his election as FRS" held at the Royal Society in 2018.