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

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.Comment: 18 pages; v2: minor correction

On W-algebras and the symmetries of defects of 6d N=(2,0) theory

We consider 6d N=(2,0) theory on N M5-branes, together with a 4d defect labeled by a Young diagram Y specifying its global symmetry G_Y. A recent conjecture states that a compactification of this system leads to a 2d theory with W-algebra symmetry depending on Y. We provide a check of the conjecture by reproducing the level of the current subalgebra \hat{G}_Y of this W-algebra from the property of the 4d defect.Comment: 16 pages; v2: Corrections to Sec. 2, conclusions unchanged. Additional reference

Derivation of the linearity principle of Intriligator-Leigh-Seiberg

Utilizing the techniques recently developed for N=1 super Yang-Mills theories by Dijkgraaf, Vafa and collaborators, we derive the linearity principle of Intriligator, Leigh and Seiberg, for the confinement phase of the theories with semi-simple gauge groups and matters in a non-chiral representation which satisfies a further technical assumption.Comment: 7 pages, no figures; v2: a few clarifications and references adde

Magnetic discrete gauge field in the confining vacua and the supersymmetric index

It has recently been argued that the confining vacua of Yang-Mills theory in the far infrared can have topological degrees of freedom given by magnetic $\mathbb{Z}_q$ gauge field, both in the non-supersymmetric case and in the N=1 supersymmetric case. In this short note we give another piece of evidence by computing and matching the supersymmetric index of the pure super Yang-Mills theory both in the ultraviolet and in the infrared.Comment: 12 page

Moduli spaces of SO(8) instantons on smooth ALE spaces as Higgs branches of 4d N=2 supersymmetric theories

The worldvolume theory of D3-branes probing four D7-branes and an O7-plane on $\mathbb{C}^2/\mathbb{Z}_2$ is given by a supersymmetric USp x USp gauge theory. We demonstrate that, at least for a particular choice of the holonomy at infinity, we can go to a dual description of the gauge theory, where we can add a Fayet-Iliopoulos term describing the blowing-up of the orbifold to the smooth ALE space. This allows us to express the moduli space of SO(8) instantons on the smooth ALE space as a hyperk\"ahler quotient of a flat space times the Higgs branch of a class S theory. We also discuss a generalization to $\mathbb{C}^2/\mathbb{Z}_{2n}$, and speculate how to extend the analysis to bigger SO groups and to ALE spaces of other types.Comment: 18 pages, 10 figures; v2: additional references and minor changes; published versio

Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories

Supersymmetric gauge theories in five dimensions often exhibit less symmetry than the ultraviolet fixed points from which they flow. The fixed points might have larger flavor symmetry or they might even be secretly six-dimensional theories on S^1. Here we provide a simple criterion when such symmetry enhancement in the ultraviolet should occur, by a direct study of the fermionic zero modes around one-instanton operators.Comment: v3: 20 pages; an error in Sec. 4.2 corrected, overall conclusion unchanged. published versio