Braverman, Finkelberg, and Nakajima define the K-theoretic Coulomb branch of a 3d N=4 SUSY gauge theory as the affine variety M_{G,N} arising as the equivariant K-theory of certain moduli space R_{G,N}, labelled by the complex reductive group G and its complex representation N. It was conjectured by Gaiotto, that (quantized) K-theoretic Coulomb branches bear the structure of (quantum) cluster varieties. … Read More

The 4D/2D correspondence recently discovered by Beem et al. constructs representation theoretical objects, such as representations of affine Kac-Moody algebras, as invariants of 4 dimensional superconformal field theories with N = 2 supersymmetry. Furthermore, it is expected that there is a remarkable duality between the representation theoretical objects constructed in this way and the geometric invariants called Higgs branch of … Read More

Time: 1:10pm Abstract:We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of d = 4, N = 2 supersymmetric field theories of class S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated … Read More

It is a general expectation that the refined topological string theory of local Calabi-Yau 3-folds produces quasimodular forms which are solutions of some refined holomorphic anomaly equation. I will present a mathematical derivation that it is indeed the case for the refined topological string on local P^2 in the Nekrasov-Shatashvili limit. The rather indirect proof exploits mathematical incarnations of various … Read More

We analyze symmetries corresponding to separated topological sectors of 3d N= 4gauge theories with Higgs vacua, compactified on a circle. The symmetries are encoded in Schwinger-Dyson identities satisfied by correlation functions of a certain gauge-invariant operator, the “vortex character.” Such a character observable is realized as the vortex partition function of the 3d gauge theory, in the presence of a … Read More

We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of d = 4, N = 2 supersymmetric field theories of class S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the … Read More

Abstract: Starting with some context and motivation from a mathematical and physical perspective, I will discuss recent work with Raphael Rouquier on a higher tensor product operation for 2-representations of Khovanov’s categorification of U(gl(1|1)^+), examples of such 2-representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, and a tensor-product-based gluing formula for these 2-representations expanding on … Read More

Abstract: Stacks of M2-branes in Omega background lead to a class of Coulomb branch algebras due to Kodera and Nakajima. On the other hand, stacks of M5-branes in Omega background lead to a class of vertex operator algebras due to a version of the AGT correspondence. I will provide a new interpretation of key objects in the theory of vertex … Read More

Knizhnik-Zamolodchikov (KZ) equation was derived in late 1980s from Ward identities of two dimensional conformal field theory with a current algebra symmetry. It is obeyed by the WZNW conformal blocks but remarkably admits analytic continuation in spins, magnetic quantum numbers and the level. In this talk I will review the work of recent years in which this extension meets the … Read More

I will discuss my work on the \Omega-deformation of brane intersections in M-theory and its applications to vertex algebras.