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

Organized by: Peter Koroteev, Elli Pomoni, Benoit Vicedo, Dmytro Volin, Anton Zeitlin

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.