Elliptic Virasoro Conformal Blocks

TL;DR

This paper explores elliptic Virasoro conformal blocks by connecting six-dimensional brane web theories with elliptic deformations of algebraic structures, providing new computational tools for instanton counting.

Contribution

It introduces elliptic deformations of Ding-Iohara algebra and computes elliptic Dotsenko-Fateev integrals matching six-dimensional instanton partition functions.

Findings

Elliptic deformation of Ding-Iohara algebra derived.

Elliptic Dotsenko-Fateev integrals reproduce 6D instanton counting.

Established duality between brane webs and elliptic algebra structures.

Abstract

We study certain six dimensional theories arising on $(p,q)$ brane webs living on $\mathbb{R}\times S^1$. These brane webs are dual to toric elliptically fibered Calabi-Yau threefolds. The compactification of the space on which the brane web lives leads to a deformation of the partition functions equivalent to the elliptic deformation of the Ding-Iohara algebra. We compute the elliptic version Dotsenko-Fateev integrals and show that they reproduce the instanton counting of the six dimensional theory.