Determine whether the matter ansatz defines a consistent truncation

Determine whether imposing the matter ansatz [?] [?] on the unrestricted vector-spinor field of the parent superconnection theory defines a consistent truncation, rather than merely a reduced theory on the constrained field space, by establishing whether the discarded gamma-traceless EulerLagrange equations vanish whenever the reduced equations hold.

Background

The paper imposes the matter ansatz [?] [?] [?] [?] to restrict an unrestricted vector-spinor one-form to its gamma-trace, spin-[?] component. Varying the action after this restriction produces only the tangent, or gamma-trace, projection of the parent vector-spinor EulerLagrange equation.

For the reduced theory to be a consistent truncation of the parent theory, the complementary gamma-traceless component must also vanish. The paper explicitly observes that the derivative contribution from the spinorial BF term produces a twistor-type condition on the dynamical spinor that is not implied by the reduced Weyl equations. It therefore concludes that generic solutions of the reduced Dirac-type system need not lift to solutions of the unrestricted theory, leaving the consistency question unresolved.