Seminar on quantum modularity and resurgence

In this seminar we will cover topics related to quantum modularity and resurgence. Over the previous decades broader classes of modular functions — such as Ramanujan's mock modular forms — have lead to the discovery of quantum modular forms. In a different direction, many of the asymptotic series related to these special functions were seen to have resurgent structures. Recent, progress has shown that understanding resummation of these asymptotic series is intimately linked to their quantum modularity. These kinds of objects arise in the study of quantum invariants of three manifolds and their structure reflects the geometry of the underlying three manifolds. We will explore various recent developments in these areas with guests speakers around twice a month.

## Upcoming talks

**May 14th 2024 14:30**: Murad Alim

**Title**: Resurgence, BPS Structures and Topological String S-Duality.

**Abstract**: The partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of a Calabi-Yau threefold. I will discuss how the resurgence analysis of the partition function allows one to extract BPS invariants of the same underlying geometry. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. This S-duality leads to a new modular structure in the topological string coupling. I will also discuss relations to difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.

## Previous talks

**May 2nd 2024 14:30**: Sary Drappeau

**Title**: Quantum Modularity for the q-Pochhammer Symbol.

**Abstract**: The talk will focus on quantum modularity relations satisfied by the q-Pochhammer symbol (q)_N at roots of unity. 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.

**April 17th 2024 11:00**: Matthias Storzer

**Title**: Modularity of Nahm sums.

**Abstract**: Nahm sums (a class of q-hypergeometric series) appear in several contexts of mathematics: As characters of VOAs, as knot invariants, and as generating functions for certain partitions. Their modularity is known to be connected to the vanishing of elements in the Bloch group. I will present some applications of Nahm sums and work in progress concerning this connection.

**April 4th 2024**: Kathrin Bringmann

**Title**: Mock theta functions, false theta functions and beyond.

**Abstract**: In my talk I discuss examples of functions that are not quite modular forms but still exhibit nice symmetries. I am, as application, in particular interested in their asymptotic growth.

**March 7th 2024 2:30.**: Claudia Rella

**Title**: Strong-weak duality and quantum modularity of resurgent topological strings.

**Abstract**: Quantizing the mirror curve of a toric Calabi-Yau threefold gives rise to quantum-mechanical operators. Their fermionic spectral traces produce factorially divergent power series in the Planck constant and its inverse, which are conjecturally captured by the Nekrasov-Shatashvili and standard topological strings via the TS/ST correspondence. In this talk, I will discuss a general conjecture on the resurgence of these dual asymptotic series, and I will present a proven exact solution in the case of the first spectral trace of local P^2. A remarkable number-theoretic structure underpins the resurgent properties of the weak and strong coupling expansions and paves the way for new insights relating them to quantum modular forms. Finally, I will mention how these results fit into a broader paradigm linking resurgence and quantum modularity. This talk is based on arXiv:2212.10606 and further work with V. Fantini (available soon).

**Monday February 26th 2024 2:30pm**: Qianyu Hao

**Title**: Exact WKB, Nonabelianization and Conformal Blocks.

**Abstract**: In this talk, I will review the exact WKB approach used in the QFT/ODE correspondence related to the NS phase of the Ω-background. By AGT correspondence, those QFT’s are related to c = ∞ CFTs. In particular, I will focus on the Stokes graph, also known as the spectral network in physics. The spectral network plays an essential role in the exact WKB and the nonabelianization of SL(N,C) flat connections. We find that the very same structure also exists at the self dual phase of the Ω-background, which is the c = 1 Liouville CFT by AGT correspondence. I will introduce our work on nonabilianization of the Virasoro conformal blocks using Heisenberg conformal blocks with the key ingredient of spectral networks. This is analogous to the nonabelianization of SL(2, C) flat connections by the GL(1, C) connections. This is a joint work in progress with Andrew Neitzke.

**Thursday February 15th 2024 2:30pm**: Introduction to quantum modularity and resurgence (Campbell and Veronica) (Our notes can be found here and here respectfully)

## Some references