Quasi-Hopf twist and elliptic Nekrasov factor

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.

45 pages, 3 figures

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