New trends in QFT, modularity, resurgence

mercredi 6 nov. 2024, 13:30 → 17:30 Europe/Paris

Maryam Mirzakhani (a.k.a. Salle 201 on 2nd floor) (Institut Henri Poincaré)

The Seed seminar of mathematics and physics is a seminar series that aims to foster interactions between mathematicians and theoretical physicists, especially among young researchers. It is structured into three-month thematic periods, the autumn 2024 one being on New trends in QFT, modularity, resurgence.

We open this thematic trimester with an in-person kick-off event at the Institut Henri Poincaré with contributions from Antoine Tilloy, Campbell Wheeler and Claudia Rella.

Registration for attending the event in person is free but mandatory, see Registration in the indico menu.

If you cannot attend the event in person but are interested in following the talks online, please subscribe here to the Seed seminar mailing list, on which Zoom links will be shared for this event and future ones.

1 A variational approach to QFT in low dimensions?

Massive quantum field theories in 1+1 dimensions are interesting in that they are fairly easy to define rigorously but still very difficult to solve (except at some integrable points). In the past few years, my collaborators and I have introduced and developed a variational method to solve them. The variational ansatz is based on a combination of continuous matrix product states and Bogoliubov transform. It is mathematically interesting in that it is in principle arbitrarily precise, works in the continuum and in the thermodynamic limit directly, and gives rigorous bounds to the energy density of the vacuum. I will motivate this variational class, explain what models we already applied it to, and list the many open problems ahead to make it a fully general method to solve generic QFT in 1+1d.

Orateur: Antoine Tilloy (LPENS & Mines Paris & INRIA)

14:30 Pause café

2 Matrix cocycles and Borel resummation

One of the many miracles of modular forms is that their q-series have convergent asymptotics as q approaches roots of unity. This comes by analysing the action of SL(2,Z) and the knowledge of the behaviour as q tends to zero.
In Ramanujan's last letter to Hardy, he famously introduced the mysterious mock modular forms. These were q-series with asymptotics that look like those of a modular form at roots of unity. However, after removing the leading order, there was a new divergent series that appeared, which obstructed modularity.
Thanks to work of Zwegers, it has long (at least implicitly) been known that this failure of modularity can be packaged into an SL(2,Z) cocycle that gives rise to the Borel resummation of the associated asymptotic series. More recently, Garoufalidis-Zagier studied quantum modular forms where similar statements were then conjectured by Garoufalidis-Gu-Mariño. I will outline a proof that cocycles associated to quantum modular forms are the Borel re-summation of associated asymptotic series in an infinite family of examples. This proves that asymptotic series associated to the 4_1
and 5_2 knots are Borel summable. This is based on joint work with Veronica Fantini.

Orateur: Campbell Wheeler (IHES)

16:00 Pause café

3 The arithmetic of resurgent topological strings

Quantising the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent series in the Planck constant and its inverse. These are captured by the Nekrasov-Shatashvili and standard topological strings via the TS/ST correspondence. In this talk, I will discuss the resurgence of these dual asymptotic series and present an exact solution for the spectral trace of local P^2. A full-fledged strong-weak symmetry exchanges the perturbative/nonperturbative contributions to the holomorphic and anti-holomorphic blocks in the factorisation of the spectral trace, and it builds upon the interplay of the L-functions with coefficients given by the Stokes constants and the q-series acting as their generating functions. Guided by this crucial example, I will propose a new perspective on the resurgence of particular formal power series, which are conjectured to possess specific summability and quantum modularity properties, leading us to introduce the general paradigm of modular resurgence. This talk is based on arXiv:2212.10606, 2404.10695, and 2404.11550.

Orateur: Claudia Rella (IHES)