Elliptic Quantum Toroidal Algebra $U_{q,t,p}(gl_{1,tor})$ and Affine Quiver Gauge Theories
Abstract
We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized by using the elliptic Ruijsenaars difference operator. Intertwining operators of $U_{q,t,p}(gl_{1,tor})$modules w.r.t. the Drinfeld comultiplication are also constructed. We 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 KimuraPestun. 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 $\mathcal{N}=2^*$ theories and provide a new AldayGaiottoTachikawa correspondence.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.09885
 arXiv:
 arXiv:2112.09885
 Bibcode:
 2021arXiv211209885K
 Keywords:

 Mathematics  Quantum Algebra;
 High Energy Physics  Theory
 EPrint:
 63 pages, 3 figures, to be published in Letters in Mathematical Physics