Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and qq-character for 5d Super Yang-Mills

TL;DR

This paper explores the representation theory of quantum $ ext{W}_{1+ ext{infinity}}$ algebra to analyze 5d super Yang-Mills instanton partition functions, establishing their qq-characters as solutions to Schwinger-Dyson equations and quantum Seiberg-Witten curves.

Contribution

It demonstrates the regularity of 5d qq-characters using quantum algebra representations, linking them to Schwinger-Dyson equations and quantum Seiberg-Witten curves.

Findings

Proves regularity of 5d qq-characters.

Shows qq-characters solve Schwinger-Dyson equations.

Interprets qq-characters as quantum Seiberg-Witten curves.

Abstract

The instanton partition functions of $\mathcal{N}=1$ 5d super Yang-Mills are built using elements of the representation theory of quantum $\mathcal{W}_{1+\infty}$ algebra: Gaiotto state, intertwiner, vertex operator. This algebra is also known under the names of Ding-Iohara-Miki and quantum toroidal $\widehat{\mathfrak{gl}}(1)$ algebra. Exploiting the explicit action of the algebra on the partition function, we prove the regularity of the 5d qq-characters. These characters provide a solution to the Schwinger-Dyson equations, and they can also be interpreted as a quantum version of the Seiberg-Witten curve.