A Non-Abelian Self-Dual Gauge Theory in 5+1 Dimensions

TL;DR

The paper tackles the problem of formulating a non-Abelian gauge theory for a self-dual 2-form in 6D, a key ingredient for describing multiple M5-branes, by introducing nonlocality along a compact dimension and splitting fields into zero modes and Kaluza–Klein (KK) modes. It presents a consistent gauge-algebra with a nonlocal direction, constructs an action S=S^{(0)}+S^{(KK)} that reproduces 5D Yang–Mills in the small-$R$ limit (with $A_i=2\pi R B_{i5}^{(0)}$ and $g=2\pi R$) and reduces to the Abelian 6D self-dual theory in the Abelian limit, while keeping KK modes explicit as carriers of momentum along the compact circle. The framework yields a dual description of 6D physics via a radius-parametrized coupling and suggests that in the large-$R$ decompactification limit the full 6D Lorentz symmetry could emerge, with interactions mediated by a 1-form gauge potential, and it points toward possible supersymmetric extensions and higher-form generalizations. The work provides a concrete, testable bridge between 5D Yang–Mills and 6D chiral gauge theories in the context of M5-branes, highlighting the role of KK modes and nonlocality in circumventing no-go theorems for non-Abelian 2-forms.

Abstract

We construct a non-Abelian gauge theory of chiral 2-forms (self-dual gauge fields) in 6 dimensions with a spatial direction compactified on a circle of radius R. It has the following two properties. (1) It reduces to the Yang-Mills theory in 5 dimensions for small R. (2) It is equivalent to the Lorentz-invariant theory of Abelian chiral 2-forms when the gauge group is Abelian. Previous no-go theorems prohibiting non-Abelian deformations of the chiral 2-form gauge theory are circumvented by introducing nonlocality along the compactified dimension.