Non-Abelian Electric-Magnetic Duality with Supersymmetry in 4D and 10D

TL;DR

This paper develops a supersymmetric non-Abelian electric-magnetic duality framework in 4D and 10D, introducing multiplets and tensor hierarchy techniques to ensure consistency and supersymmetry closure.

Contribution

It introduces a novel supersymmetric duality formulation for non-Abelian gauge theories in 4D and 10D, incorporating unphysical tensor multiplets and auxiliary fields.

Findings

Established duality between vector multiplets in 4D

Demonstrated role of unphysical tensor multiplet for consistency

Identified auxiliary field K in 10D for supersymmetry closure

Abstract

We present electric-magnetic (Hodge) duality formulation for non-Abelian gauge groups with N=1 supersymmetry in 3+1 (4D) dimensions. Our system consists of three multiplets: (i) A super-Yang-Mills vector multiplet (YMVM) $(A_\mu{}^I, \lambda^I)$, (ii) a dual vector multiplet (DVM) $(B_\mu{}^I, \chi^I)$, and (iii) an unphysical tensor multiplet (TM) $(C_{\mu\nu}{}^I, \rho^I, \varphi^I)$, with the index I for adjoint representation. The multiplets YMVM and DVM are dual to each other like: $G_{\mu\nu}{}^I = (1/2) \epsilon_{\mu\nu}{}^{\rho\sigma} F_{\rho\sigma}{}^I$. The TM is unphysical, but still plays an important role for establishing the total consistency of the system, based on recently-developed tensor-hierarchy formulation. We also apply this technique to non-Abelian electric-magnetic duality in 9+1 (10D) dimensions. The extra bosonic auxiliary field $K_{\mu_1\cdots\mu_6}$ in 10D is shown to play an important role for the closure of supersymmetry on fields.