Abstract: Two-dimensional integrable field theories are characterised by the existence of infinitely many integrals of motion. Recently, two unifying frameworks for describing such theories have emerged, based on four-dimensional Chern-Simons theory in the presence of surface defects and on Gaudin models associated with affine Kac-Moody algebras. I will explain how these formalisms can be used to construct infinite families of … Read More

We present a construction of topological quantum gravity, which connects three previously unrelated areas: (1) Topological quantum field theories of the cohomological type, as developed originally by Witten; (2) the mathematical theory of the Ricci flow on Riemannian manifolds in arbitrary spacetime dimension, developed originally by Hamilton and later by Perelman in his proof of the Poincare conjecture; and (3) … Read More

A Looijenga pair is a pair (X,D) with X a smooth complex projective surface and D a singular anticanonical divisor in X. I will describe a series of correspondences relating five different classes of string-theory motivated invariants specified by the geometry of (X,D):– the log Gromov–Witten theory of (X,D),– the Gromov–Witten theory of X twisted by the sum of the dual line bundles to … Read More

For any complex reductive group G one can associate a geometric object called the Affine Grassmannian. The transversal slices are interesting affine Poisson varieties appearing in this geometry. In this talk I’ll focus on the equivariant quantum cohomology of these spaces. In particular, I’ll show how the quantum differential equation in types ADE can be identified with the trigonometric Knizhnik-Zamolodchikov … Read More

Abstract : I will review Braverman-Finkelberg’s geometric Satake correspondence conjecture for affine Lie algebras via instanton moduli spaces on C^2/(Z/\ell) and their refinement by the use of Coulomb branches of affine quiver gauge theories. I will briefly explain proofs of most of the statements in type A, which I believe work in general. Then I would like to spend most … Read More

Deligne categories are tensor categories, introduced by P. Deligne, which provide a formal way to interpolate representation-theoretic structures attached to classical groups and supergroups (such S_n, GL(n),Sp(2n),O(n),GL(n|m),OSp(n|2m),etc.) to complex values of the integer “rank parameter” n. I will first review them and then explain how to use them to construct and study deformed double current algebras which have recently become … Read More

I will discuss constructions of supersymmetric interfaces in quantum field theories with eight supercharges, both within one theory and between different theories, that provide physical realizations of certain geometric constructions in the context of quiver varieties (such as stable envelopes and actions of Yangians/quantum loop/quantum elliptic algebras). The basic building blocks are supersymmetric Janus interfaces that realize stable envelopes introduced … Read More

I will discuss supergroup gauge theory with emphasis on the following aspects: instanton counting, Seiberg-Witten geometry, brane construction, realization in topological string and intersecting defects, Bethe/Gauge correspondence, and quantum algebraic structure. Based on joint works with H.-Y. Chen, N. Lee, F. Nieri, V. Pestun, and Y. Sugimoto (mostly summarized in https://arxiv.org/abs/2012.11711).

Abstract: Given a 3d N=4 supersymmetric quantum field theory, there is an associated Coulomb branch, which is an important reflection of the A-twist of this theory. In the case of gauge theories, this Coulomb branch has a description due to Braverman-Finkelberg-Nakajima; I’ll discuss how we can generalize this geometric description in order to construct non-commutative resolutions of Coulomb branches (giving … Read More