Andrei Negut, MIT
Exts and AGT with matter
The AGT relations for N = 2 super-symmetric U(r) gauge theory with adjoint matter predict a relation between a certain integral over the moduli space of instantons (a geometric object) and the trace of an intertwiner for the W_r algebra (a representation theoretic object). The main purpose of this mini-course is to explain one proof of this statement in the r = 2 case, by showing how to compute the commutation relations between the Carlsson-Okounkov Ext operator and the Heisenberg-Virasoro algebra. The main technical tool is the shuffle algebra incarnation of the Schiffmann-Vasserot Yangian, and in fact, the proof of the relations holds in arbitrary rank. I also wish to ask the audience for help in proving the general r case, which essentially boils down to connecting the W_r algebra with the shuffle algebra. The breakdown of the lectures is the following:
1) An overview of the problem, including certain analogues such as the case of finite-dimensional Lie algebras.
2) The Schiffmann-Vasserot Yangian via shuffle algebras. The Heisenberg and Virasoro subalgebras.
3) The moduli space M of rank r sheaves on the plane and the Ext operator.
4) The shuffle algebra acts on the cohomology ring of M. The Ext operator is an intertwiner.
5) From intertwiners to conformal blocks. Ideas for generalizing to arbitrary W_r.