# MS Seminar (Mathematics - String Theory)

Speaker: Hitoshi Konno (Tokyo Univ. of Marine Science and Technology) Elliptic Quantum Toroidal Algebra $U_{q,t,p}(\gl_{1,tor})$ and affine quiver gauge theories Thu, Jan 20, 2022, 13:30 - 15:00 Zoom 2758.pdf We introduce the elliptic quantum toroidal algebra $U_{q,t,p}(\gl_{1,tor})$. After giving some representations including the level (0,0) representation realized by using the elliptic Ruijsenaars difference operator, we construct intertwining operators of the $U_{q,t,p}(\gl_{1,tor})$-modules w.r.t. the Drinfeld comultiplication. We then show that $U_{q,t,p}(\gl_{1,tor})$ gives a realization of the affine quiver $W$-algebra $W_{q,t}(\Gamma(\widehat{A}_0))$ proposed by Kimura-Pestun. This realization turns out to be useful to derive the Nekrasov instanton partition functions, i.e. the $\chi_y$- and elliptic genus, of the 5d and 6d lifts of the 4d ${\cal N}=2^*$ theories and provide a new Alday-Gaiotto-Tachikawa correspondence.