Manifestly-invariant action for self-dual supersymmetric Yang-Mills theories coupled to auxiliary supergravity

Abstract

We propose a new action-formulation for the N=(1,1) multiplet (AμI,λI) of a self-dual (SD) supersymmetric Yang-Mills (YM) theory in D=2+2 dimensions using an auxiliary supergravity multiplet. We first consider non-supersymmetric case with constraint lagrangians that restrict the metric gμν to be (ημν)=diag.(+,+,−,−). We add the kinetic-term of a purely-bosonic YM-field, and show that the YM-field in the D=2+2 space-time allows only SD or anti-SD (ASD) configurations. In other words, the condition of vanishing energy-momentum (EM) tensor restricts the YM-field to be either SD or ASD, but not an admixture of them. Finally, we generalize this result to N=(1,1) supersymmetric YM multiplet (AμI,λI;DI) in D=2+2 with the off-shell supergravity multiplet (eμm,ψμ;S,P,am) as an ‘auxiliary’ multiplet without its kinetic terms. Special constraint lagrangians restrict the supergravity multiplet to be auxiliary, i.e., both the Riemann-tensor and the gravitino field strength to vanish. We show that the eμm and ψμ-field equations produce the vanishings of both the EM-tensor and spinor-current, which in turn lead only to either SD or ASD supersymmetric YM multiplet, as the only Lorentz and gauge-covariant supersymmetric solutions. A broad class of similar action-formulations with auxiliary supergravity multiplets yielding SD or ASD systems in D=2+2 or higher-dimensions may be constructed in a similar fashion.