# Quantum toroidal integrable systems

#### by Jean-Emile Bourgine

Institution: Korea Institute for Advanced Study
Date: Tue 25th July 2017
Time: 1:00 PM
Location: PH213

Abstract: In this talk, I will describe a class of integrable systems characterized by the presence of a quantum toroidal algebra. I will restrict myself to the simplest case of quantum double affine
$\mathfrak{gl}_1$, also known as the Ding-Iohara-Miki (DIM) algebra. These integrable systems can be seen as a kind of spin chains where spins take values in box configurations of Young diagrams. Representations are infinite dimensionnal, and large enough to accommodate the states of a quantum field theory, leading to a notion of double quantization. I will introduce an integral formula for the universal R-matrix, and study its realization in the two main representations of the DIM algebra, namely the horizontal (Fock) and vertical (highest weight) representations. In a mixed
vertical-horizontal representation, the R-matrix provides a Lax matrix that can also be written in terms of Awata-Feigin-Shiraishi intertwiners. The associated T-operators reconstruct the Nekrasov
instanton partition functions of 5d N=1 Super-Yang-Mills, the AGT dual of q-Virasoro/q-Toda conformal blocks.