M-Theory, 3 Manifolds and certain SL(2,Z) representations

The M-theory perspective, more specifically the 3d-3d correspondence, leads to interesting predictions for the structure of the Witten-Reshetikhin-Turaev invariants of 3-manifolds and superconformal indices of the 3d SCFTs arising from compactifying five-branes on 3-manifolds. Especially, one expects SL(2,Z) representations  to play an important role. In this talk I will describe  how a natural type of irreducible SL(2,Z) representations give rise to the WRT invariants and govern the structure of flat connections for a specific family of Seifert 3-manifolds, and how mock theta functions are crucial ingredients in understanding the superconformal indices in these examples. The talk is based on joint work in progress with Chun, Ferrari, Gukov and Harrison.