Quantum Modularity for the q-Pochhammer Symbol

by Sary Drappeau (Université d’Aix-Marseille)

Thursday May 2, 2024, 2:30 PM → 4:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Seminar on Quantum Modularity and Resurgence

The talk will focus on quantum modularity relations satisfied by the $q$-Pochhammer symbol  $(q)_N = (1-q) ... (1-q^N)$ at $q=\exp(2 \pi i x)$. These formulas can be interpreted as finite analogues of the usual modularity for the Dedekind eta-function. We'll discuss certain aspects which come very handy upon summing over $N$. We'll explain how these can be used, in the context of Kashaev's invariant of hyperbolic knots, to prove, in a few cases, Zagier's quantum modularity conjecture by means of what we currently know on the Volume Conjecture. This is based on joint work with Sandro Bettin.

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