Quasi-Hopf twist and elliptic Nekrasov factor

TL;DR

This paper explores the elliptic deformation of the quantum toroidal algebra of _1 via a quasi-Hopf twist, revealing connections to elliptic Nekrasov factors and elliptic quantum KZ equations in gauge theory and integrable systems.

Contribution

It introduces the quasi-Hopf twist of the algebra, computes the twisted R-matrix, and links it to elliptic Nekrasov factors and elliptic quantum KZ equations, providing new insights into elliptic deformations.

Findings

The algebra remains unchanged under the twist, but the coproduct is deformed.

The twisted R-matrix relates to the elliptic Nekrasov factor for instanton counting.

The R-matrix appears in the commutation relations of intertwiners, leading to an elliptic quantum KZ equation.

Abstract

We investigate the quasi-Hopf twist of the quantum toroidal algebra of $\mathfrak{gl}_1$ as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter $p$ is identified as the elliptic modulus. Computing the quasi-Hopf twist of the $R$ matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on $\mathbb{R}^4 \times T^2$. The same $R$ matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.