Summary
This paper studies the self-dual Einstein-Dirac theory. A generalization is obtained of the Jacobson-Smolin proof of the equivalence between the self-dual and Palatini purely gravitatinal actions. Hence one proves equivalence of self-dual Einstein-Dirac theory to the Einstein-Cartan-Sciam-Kibble-Dirac theory. The Bianchi symmetry of the curvature, core of the proof, now contains a non-vanishing torsion. Thus, in the self-dual framework, the «extra» terms entering the equations of motion with respect to the standard Einstein-Dirac field equations, are neatly associated with torsion.
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Morales-Técotl, H.A., Esposito, G. Self-dual action for fermionic fields and gravitation. Nuov Cim B 109, 973–982 (1994). https://doi.org/10.1007/BF02726144
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DOI: https://doi.org/10.1007/BF02726144