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Three-manifold quantum invariants and mock theta functions

2019; Royal Society; Volume: 378; Issue: 2163 Linguagem: Inglês

10.1098/rsta.2018.0439

1471-2962

Miranda C. N. Cheng, Francesca Ferrari, Gabriele Sgroi,

Advanced Combinatorial Mathematics

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 Σ(2, 3, 7). This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'.