Integrability, Enumerative Geometry and Quantization: August 22nd – September 23rd, 2022

Participant List

Gaëtan Borot (Humboldt University of Berlin, Germany),

Alexandr Buryak (Higher School of Economics, Russia), 

Chiu-Chu Melissa Liu (Columbia University),

Nikita Nekrasov (SCGP),

Paul Norbury (University of Melbourne, Australia),

Paolo Rossi (University of Padua, Italy)

Additional Talks: 
Tuesday, September 13⋅1:00 – 2:30pm, SCGP room 313: David Holmes
Title: Variations on double ramification cycles
Abstract

Notes by Patrick Lei: https://www.math.columbia.edu/~plei/docs/ISEG.pdf

The role played by integrable systems in enumerative geometry has been first observed by physicists at the beginning of the nineties: the initial observations by Witten on the relation between the partition function of 2d topological gravity, matrix models and intersection theory of the moduli space of curves and the Korteweg-de Vries equation has developed into a rich theory relating $(1+1)$-dimensional classical integrable systems to cohomological field theories on the moduli space of stable curves. This has, in turn, become a powerful tool for probing the topology of spaces of curves and maps to target varieties.

The last few years have seen dramatic development in our understanding of the tautological ring of the moduli space of curves, its intersection theory and the role played by natural geometric cycles therein. The study of cohomological field theories (systems of cohomology classes compatible with the strata structure of the moduli spaces) and double ramification cycles (loci of curves whose marked points support principal divisors) has provided new results both towards describing (conjecturally all) tautological relations and towards constructing and quantizing integrable field theories.

Chekhov-Eynard-Orantin topological recursion has become a unifying tool embracing intersection theory on the moduli space of curves, B-model quantization on Landau-Ginzburg models, integrable systems, where the role of infinite-dimensional symmetries such as W-algebras has been recently clarified, opening the way to understanding better its connection with 4d supersymmetric gauge theories where many of the aforementioned geometric and algebraic structures come into play.

Correspondingly, in the physics community, there has been a renewal of interest for two-dimensional quantum gravity thanks to the Sachdev-Ye-Kitaev model, its conjectured connections to Jackiw–Teitelboim gravity and the connections of the latter to the geometry of the moduli space of curves via the topological recursion and matrix models. Moreover, moduli spaces of super-Riemann surfaces, moduli spaces of Riemann surfaces with boundary and more generally open Gromov-Witten theory, which is being rigorously constructed, also appear to show beautiful connections with integrability and topological recursion.

Some of these connections may be broader than currently known. On the one hand, the appearance of more general quantum algebras (Hecke algebras, Yangians, quantum toroidal algebras, etc.) in 5d gauge theories, in the quantization of character varieties, and in matrix models is well-documented, and it would be desirable to extend the relation to topological recursion and enumerative geometry in this direction. On the other hand, in the quantization of integrable field theories coming from the double ramification cycles, the analog of Virasoro/W-algebras symmetries and the relation to physical theories remain to be explored.

As it often happens when research fields grow and specialize, the communities behind these developments tend to become distinct, especially from the geometry vs. physics viewpoint. The idea of the workshop is to bring together experts from these different communities, including several leaders of the respective fields, to exchange views and ideas and initiate fruitful collaborations.