# Geometry and Physics of Quantum Toroidal Algebra (20w5024)

## Organizers

(Fudan University)

Boris Feigin (Higher School of Economics (Moscow))

Hiraku Nakajima (Kavli IPMU, University of Tokyo)

Peng Shan (Tsinghua University)

## Description

The Institute of Advanced Study in Mathematics will host the "Geometry and Physics of Quantum Toroidal Algebra" workshop in Hangzhou, China from September 6 to September 11, 2020.

The theories of rational/trigonometric/elliptic functions were developed as functions over plane/cylinder/torus, respectively, before the 20th century. Among them, the theory of elliptic functions is most interesting, and it can be understood as the parent theory of the rest. During the 20th century, the same rational/trigonometric/elliptic trichotomy has been also found in integrable systems, algebras and cohomology groups. Like the theory of functions, recent development reveals that the relationship among integrable system, algebra, geometry, and physics becomes most profound at elliptic level. In particular, Lie algebras (rational) are promoted to affine Lie algebras (trigonometric) and toroidal Lie algebras (elliptic). The workshop aims at investigating quantum versions of toroidal Lie algebras, called quantum toroidal algebras.

In fact, plenty of recent discovery in both mathematics and physics shows that quantum toroidal algebras are remarkably rich. They appear to act cohomology groups of instanton moduli spaces of four-manifolds. These algebras and their representation theory also arise from moduli spaces of flat connections, $G$-bundles and Higgs bundles over a torus. In physics, 4d/5d/6d dimensional reductions/oxidations of supersymmetric gauge theories are indeed related to the trichotomy of quantizations of toroidal Lie algebras. Therefore, studying quantum toroidal algebras is indispensable to understand the dynamics of the mysterious 6d (2,0) theory on M5-branes. Moreover, physics potentially provides a powerful framework for organizing and connecting known these mathematical constructions and motivates surprising new ideas. Hence, the goal of the proposed workshop is to uncover the deeper geometric and categorical structures underlying quantum toroidal algebras by exchanging ideas between mathematicians and physicists.

The Institute of Advanced Study (IAS) in Hangzhou, China, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).