Holomorphic field realization of SH$^c$ and quantum geometry of quiver gauge theories

TL;DR

This paper reformulates the SH$^c$ algebra using holomorphic fields to simplify the analysis of instanton partition functions in 4D $ ext{N}=2$ supersymmetric gauge theories, revealing connections to quantum geometry and integrable systems.

Contribution

It introduces a holomorphic field realization of SH$^c$ algebra and expresses instanton partition functions in terms of fundamental building blocks, advancing the understanding of quantum geometry in quiver gauge theories.

Findings

Holomorphic field realization simplifies SH$^c$ algebra and its representations.

Partition functions expressed as products of fundamental blocks.

Polynomiality of qq-characters and relation to Seiberg-Witten geometry.

Abstract

In the context of 4D/2D dualities, SH$^c$ algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of $\mathcal{N}=2$ supersymmetric gauge theories. In this paper, we rewrite the SH$^c$ algebra in terms of three holomorphic fields $D_0(z)$, $D_{\pm1}(z)$ with which the algebra and its epresentations are simplified. The instanton partition functions for arbitrary $\mathcal{N}=2$ super Yang-Mills theories with $A_n$ and $A^{(1)}_n$ type quiver diagrams are compactly expressed as a product of four building blocks: Gaiotto state, dilatation, flavor vertex operator and intertwiner which are written in terms of SH$^c$ and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the Carlsson-Okounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions defined by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qq-characters which have been introduced as a deformation of the Yangian characters. These relations define a second quantization of the Seiberg-Witten geometry, and, accordingly, reduce to a Baxter TQ-equation in the Nekrasov-Shatashvili limit of the Omega-background.