Elliptic Genus Derivation of 4d Holomorphic Blocks

TL;DR

This paper derives elliptic genus formulas for 4d holomorphic blocks by analyzing elliptic vortices via 2d quiver gauge theories, connecting vortex partition functions, surface factorization, and elliptic algebra representations.

Contribution

It introduces a novel derivation of 4d holomorphic blocks using elliptic genus computations of vortex moduli spaces in specific supersymmetric gauge theories.

Findings

Elliptic genus formulas for vortex moduli spaces are obtained.

Partition functions factorize into holomorphic blocks on complex surfaces.

Connections to elliptic Virasoro algebra representations are established.

Abstract

We study elliptic vortices on $\mathbb{C}\times T^2$ by considering the 2d quiver gauge theory describing their moduli spaces. The elliptic genus of these moduli spaces is the elliptic version of vortex partition function of the 4d theory. We focus on two examples: the first is a $\mathcal{N}=1$, $\mathrm{U}(N)$ gauge theory with fundamental and anti-fundamental matter; the second is a $\mathcal{N}=2$, $\mathrm{U}(N)$ gauge theory with matter in the fundamental representation. The results are instances of 4d "holomorphic blocks" into which partition functions on more complicated surfaces factorize. They can also be interpreted as free-field representations of elliptic Virasoro algebrae.