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Mock modular forms as Z-invariants

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BLHW01 - Number theory, machine learning and quantum black holes

A new family of topological 3-manifold invariants has been proposed recently, with the property that they are q-series with integrality properties that allow categorification. They have a mathematical definition based on the data which specifies the associated 3-manifold, though this is of limited applicability and restricted to cases which satisfy a certain negativity condition. Aside from their relevance in topology, these invariants have proven to be of broad interest through a web of relations. Physically, they capture the partition functions of certain 3-dimensional SQF Ts, while from a number theory perspective they provide examples of holomorphic quantum modular forms. Here I will discuss an underlying hidden symmetry of these invariants and how considerations of modularity can be leveraged to predict what these should be for manifolds not covered by their original definition, as well as the wider implications of these results. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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