On the 6d origin of discrete additional data of 4d gauge theories

TL;DR

This paper explores how 4d gauge theories' additional data, such as gauge groups and theta angles, are represented in 6d, revealing connections to Z_N symmetry and refined superconformal indices.

Contribution

It demonstrates the 6d origin of discrete data in 4d gauge theories and links Z_N symmetry and superconformal index refinements to 2d q-deformed Yang-Mills.

Findings

Z_N symmetry influences 6d representation of 4d data

Superconformal index can be refined for different gauge groups

Connection established between 6d theories and 2d q-deformed Yang-Mills

Abstract

Starting with a choice of gauge algebras, specification of a 4d gauge theory involves additional data, namely the gauge groups and the discrete theta angles. Equivalently, one needs to specify the set of charges of allowed line operators. In this note, we study how these additional data are represented in 6d, when the 4d theory in question is an N=4 super Yang-Mills theory or an N=2 class S theory. We will see that the Z_N symmetry of the so-called T_N theory plays an important role. As a byproduct, we will find that the superconformal index of class S theories can be refined so that it can give 2d q-deformed Yang-Mills theory with different gauge groups associated to the same gauge algebra.