Mathematical Physics

In this lecture I'll discuss various aspects of 4d N=2 and 5d N=1 supersymmetric QFT's in the 1/2 Omega-background (and along the way try to emphasize some relations to the 3d N=2 theories discussed in this workshop). Central to this story is the Nekrasov instanton partition function (or topological string partition function) in this background, which we will obtain through abelianization as an integral of a ratio of Wronskians of certain special solutions to the relevant Schrodinger equation. We will argue that a slight generalization of the above partition function solves an associated Riemann-Hilbert problem and defines a section of a distinguished line bundle over the moduli space of flat connections.

3d mirror symmetry predicts an equivalence between A- and B-twists of a pair of dual 3d N=4 theories. Essentially the strongest invariants one can produce of the resulting 3-dimensional topological field theories are their 2-categories of boundary conditions. The B-side 2-category was first described by Kapustin-Rozansky-Saulinas, but the 2-categorical structure on A-side boundary conditions has not previously been understood. For abelian gauge theories with matter, we propose a model for the 2-category of A-type boundary conditions using Kapranov-Schechtman's "perverse schobers," and we prove a 3d mirror equivalence between dual 2-categories. By reducing to lower-dimensions, we can recover both the BFN construction and the BLPW Koszul duality for hypertoric categories O. This is joint work with Justin Hilburn and Aaron Mazel-Gee.

I will continue the discussions on line defects and surface defects in class S theories, making connections to the construction of the quantum trace map, as well as to the exact WKB method for higher order ODEs.

In the first part of my talk I'll briefly review some aspects of the relations between N=4, d=4 SYM and vertex operator algebras (VOAs) discussed in recent work of Gaiotto and collaborators. The resulting picture predicts conjectural generalisations of the geometric Langlands correspondence. We will focus on a class of examples figuring prominently in recent work of Creutzig-Dimofte-Garner-Geer, labelled by parameters n (rank) and k. For the case k=1,n=2 we will point out that the conformal blocks of the relevant VOA, twisted by local systems, represent sections of natural holomorphic line bundles over the moduli spaces of local systems closely related to the isomonodromic tau functions. Observing the crucial role of (quantised) cluster algebras in the definition of the holomorphic line bundles suggests natural generalisations of this story to higher values of the parameters k and n.

I will give an introduction to 4d N=2 class-S theories. I will describe the construction of such theories, the roles played by extended defects such as line defects and surface defects, as well as connections to Hitchin systems.