Quantum Modularity in 3d Quantum Field Theory
I will discuss a set of inter-related phenomena in physics, topology, and number theory. A natural playground in which to explore these connections is the so-called 3d-3d correspondence, which relates Chern-Simons theory on a 3-manifold M_3 to a 3d N=2 theory T[M_3] which only depends on the topology of M_3. Both sides of this correspondence exhibit a fascinating number theoretic phenomenon: quantum modularity. Quantum modular forms, introduced by Zagier, are functions defined only at rational numbers, and in the most general cases are neither analytic nor modular. It is still an open question to develop a general theory which encompasses their behavior. I will overview these relations and discuss recent advances which may shed light on some of these questions, including relations to objects such as: false theta functions, mock theta functions, and 2d logarithmic conformal fields theories.
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