On the 6d origin of discrete additional data of 4d gauge theories
Yuji Tachikawa
TL;DR
The paper explains how the discrete data needed to specify 4d gauge theories, namely the global gauge group structure and discrete theta angles, are encoded in the 6d N=(2,0) theory via a partition vector and maximal isotropic line-operator charge lattices. It shows that class S theories inherit a universal center symmetry and that a maximal isotropic sublattice L in H^1(C, C) fully captures the discrete data, with the 4d partition function and Hilbert space constructed from a 6d partition vector. For N=4 SYM and class S theories, the framework reproduces known SU(N) vs SU(N)/Z_N structures, S-duality relations, and yields a refined connection between the 4d superconformal index and 2d q-deformed Yang-Mills with gauge groups determined by the discrete data. The results provide a coherent 6d perspective on line-operator charges, global structures, and discrete torsion, with implications for dualities and low-dimensional correspondences. Further work is proposed to extend the formalism to punctured surfaces, torsion settings, and deeper line-operator analyses.
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.