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In this work the supersymmetric gauge invariant action for the massiveAbelian $N$$=$$1$ super-{\QED} in the Atiyah-Ward space-time ({\DDdd}) isformulated. The questions concerning the scheme of the gauge invariance in{\DDdd} by means of gauging the massive $N$$=$$1$ super-{\QED} areinvestigated. We study how to ensure the gauge invariance at the expenses ofthe introduction of a complex vector superfield. We discuss the Wess-Zuminogauge and thereupon we conclude that in this gauge, only the imaginary part ofthe complex vector field, $B_{\m}$, gauges a $U(1)$-symmetry, whereas its realpart gauges a Weyl symmetry. We build up the gauge invariant massive term byintroducing a pair of chiral and anti-chiral superfields with opposite$U(1)$-charges. We carry out a dimensional reduction {\it{\`a la}} Scherk ofthe massive $N$$=$$1$ super-{\QED} action from {\DDdd} to {\Ddd}. Truncationsare needed in order to suppress non-physical modes and one ends up with aparity-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 havegot is the supersymmetric version ofthe ${\tau}_{3}$QED .Comment: 23 pages, latex, no figure

SUPER-τ3QED AND THE DIMENSIONAL REDUCTION OF N=1SUPER-QED2+2

SUPER-τ₃QED AND THE DIMENSIONAL REDUCTION OF N=1SUPER-QED₂₊₂