Abstract
We derive the component lagrangian for general matter coupling to 4D supergravity of a real linear multiplet, containing an antisymmetric tensor field, which transforms nontrivially under gauge and Lorentz transformations, as it comes from the Green-Schwarz anomaly cancellation mechanism in D = 10. It is found that local supersymmetry requires a third Chern-Simons form related to the axial gauge field of conformal supergravity.
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