Geometric and Representation-Theoretic Aspects of Quantum Integrability: August 29-October 21, 2022

Participant List

Organized by:

Peter Koroteev (University of California, Berkeley),

Elli Pomoni (DESY),

Benoit Vicedo (University of York),

Dmytro Volin (NORDITA),

Anton Zeitlin (Louisiana State University)

Talk Schedule:

 

 

Time Title Speaker Location
Thursday September 8
2:00pm Opers — what they are and what they are good for?

Abstract

Peter Koroteev SCGP 313
Friday September 9
2:00pm Conserved charges in the quantum simulation of integrable spin chains

Abstract

Ryo Suzuki SCGP 313
Monday September 12
1:30pm Solutions of the Bethe Ansatz Equations as Spectral Determinants

Abstract

Davide Masoero SCGP 313
Wednesday September 14
10:30am A geometric realization of the center of the small quantum group

Abstract

Eric Vasserot SCGP 313
Friday September 16
10:30am Gaudin models associated with Lie superalgebras

Abstract

Evgeny Mukhin SCGP 313
Monday September 19
1:30pm On a deformation of quantum groups and their tensor categories

Abstract

Azat Gainutdinov SCGP 313
Wednesday September 21
10:30am Macdonald Polynomials with elliptic coefficients and analog of the Cherednik’s operators for them

Abstract

Andrey Grekov
SCGP 313
Friday September 23
10:30am Bootstrapping Quantum Spectral Curves

Abstract

Simon Ekhammar SCGP 313
Monday October 3
1:30pm Reduction of cluster structure

Abstract

Alexander Shapiro SCGP 313
Wednesday October 5
10:30am Line operators and Cherkis Bows

Abstract

Nafiz Ishtiaque SCGP 313
Friday October 7
9:30am The Quantum Inverse Scattering Method from 4d Chern-Simons Theory

Abstract

Meer Ashwinkumar SCGP 313
Monday October 10
1:30pm 6D SCFTs, Long Quivers, and (Super-)Spin Chains

Abstract

Florent Baume SCGP 313
Wednesday October 12
10:30am Graded Characters, Quantum Q-systems, and spherical DAHA

Abstract

Rinat Kedem SCGP 313
Friday October 14
10:30am
From Koornwinder operators to cluster algebra: Proof of the Macdonald-Q-system conjecture
Philippe Di Francesco SCGP 313
Monday October 17
1:30pm LOG-CANONICAL COORDINATES ON POISSON-LIE GROUPS

Abstract

Michael Gekhtman SCGP 313
Wednesday October 19
10:30am SCGP 313
Friday October 21
10:30am Andrii Liashyk SCGP 313

Quantum integrable systems (QIS) keep reemerging in various areas of modern mathematics and theoretical physics. They were an inspiration for the creation of the theory of quantum groups and play an important role in the understanding of enumerative geometry, representation theory, and quantum field theory/string theory. The goal of this program is to enhance communication between different communities by bringing together experts in the relevant domains of geometric representation theory, algebraic geometry, and mathematical physics to discuss current developments in the various aspects of QIS. Major directions include:1) The integrability approach to the AdS/CFT correspondence offers a non-perturbative description of the spectrum of planar N = 4 SYM and cousin models in terms of the Quantum Spectral Curve (QSC). A yet unresolved puzzle is that AdS/CFT integrability does not fit into the standard rational-trigonometric-elliptic families of quantum algebras (except at the leading perturbative order where it is Yangian), however the QSC differs from its quantum algebra counterparts only by analytic properties. A bona fide description of AdS/CFT integrability within the general mathematical framework of QIS is the target of recent developments.2) The Bethe/gauge correspondence cemented connections between enumerative geometry, geometric representation theory, and QIS. The Bethe ansatz equations, which determine the spectrum of QIS of spin chain type, emerge as twisted chiral rings describing the vacua of supersymmetric gauge theories. A mathematical interpretation of this phenomenon lies within the framework of enumerative and quantum geometry of symplectic resolutions, in particular, of Nakajima quiver varieties where the Bethe equations play the role of relations in quantum cohomology/K-theory rings. Moreover, it turns out that the machinery of differential/difference equations inherited from the QIS theory, e.g. quantum Knizhnik-Zamolodchikov equations, provides a powerful toolkit to study the enumerative geometry of symplectic resolutions and related questions within the realm of geometric representation theory. This includes recent progress in the understanding of symplectic duality and 3D mirror symmetry.3) The geometric Langlands correspondence became a central topic in mathematics since the early 90s. The first nontrivial example involves the correspondence between oper connections and integrable Gaudin models, and recent developments study various deformations. One such deformation follows the direction of the quantum q-Langlands correspondence, which is tightly tangled with the enumerative geometry of symplectic resolutions. Another follows the q-deformation of the oper/Gaudin duality, leading to the notion of a q-oper. The relation between q-opers and QIS of the spin chain type is achieved via QQ-systems which are also studied within the context of the representation theory of quantum affine algebras and Yangians and through the ODE/IM correspondence which relates QQ-systems with the spectral determinants of differential operators realizing affine opers. These studies explicitly demonstrate that the Q-functions of QQ-systems are charged with respect to the Langlands dual of the corresponding affine Lie algebra.4) Recent discoveries in 4d Chern-Simons (CS) theory have opened up an exciting new direction in classical and quantum integrability by offering a unified framework for constructing, studying, and classifying integrable models. This theory is also connected with the ODE/IM correspondence through its intimate relationship with affine Gaudin models. Most recently the Q-functions of QQ-systems were interpreted as t’Hooft operators of 4d CS theory. Given that electric-magnetic duality can be viewed through the lens of the geometric Langlands correspondence, this statement ties perfectly with the above topics.This program will also be hosting a workshop: Geometric Representation Theory, Integrability, and Supersymmetric Gauge Theories: September 26 – September 30, 2022