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.