Super-${\tau}_{3}$QED and the dimensional reduction of $N$$=$$1$ super-{QED}$_{2+2}$

TL;DR

This paper formulates a gauge-invariant massive N=1 super-QED in Atiyah-Ward space-time, explores gauge invariance via complex superfields, and performs dimensional reduction to connect with super-QED in 1+2 dimensions, revealing a supersymmetric version of ${ au}_{3}$QED.

Contribution

It introduces a novel formulation of massive N=1 super-QED in Atiyah-Ward space-time and demonstrates its reduction to a supersymmetric ${ au}_{3}$QED in 1+2 dimensions.

Findings

Gauge invariance achieved with complex vector superfield.

Dimensional reduction yields N=1 super-QED in 1+2 dimensions.

Identifies the supersymmetric ${ au}_{3}$QED as the reduced theory.

Abstract

In this work the supersymmetric gauge invariant action for the massive Abelian $N$$=$$1$ super-{\QED} in the Atiyah-Ward space-time ({\DDdd}) is formulated. The questions concerning the scheme of the gauge invariance in {\DDdd} by means of gauging the massive $N$$=$$1$ super-{\QED} are investigated. We study how to ensure the gauge invariance at the expenses of the introduction of a complex vector superfield. We discuss the Wess-Zumino gauge and thereupon we conclude that in this gauge, only the imaginary part of the complex vector field, $B_{\m}$, gauges a $U(1)$-symmetry, whereas its real part gauges a Weyl symmetry. We build up the gauge invariant massive term by introducing a pair of chiral and anti-chiral superfields with opposite $U(1)$-charges. We carry out a dimensional reduction {\it{\`a la}} Scherk of the massive $N$$=$$1$ super-{\QED} action from {\DDdd} to {\Ddd}. Truncations are needed in order to suppress non-physical modes and one ends up with a parity-preserving $N$$=$$1$ super-QED$_{1+2}$ (rather than $N$$=$$2$) in $D$$=$$1$$+$$2$. Finally, we show that the $N$$=$$1$ super-QED$_{1+2}$ we have got is the supersymmetric version ofthe ${\tau}_{3}$QED .