# 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.