Supersymmetry for Gauged Double Field Theory and Generalised Scherk-Schwarz Reductions

Core claim

The paper constructs a half-maximal supersymmetric gauged double field theory whose bosonic sector is the DFT-dilaton and the pair of double vielbeins and whose fermionic sector is a gravitino and a dilatino, with action $L=e^{-2d}[2P^{MP}P^{NQ}S_{MN PQ}+4i(bar{rho}gamma^m D_m rho - 2 bar{psi}^{bar m}D_{bar m}rho - bar{psi}^{bar m}gamma^m D_m psi^{bar m})]$ built from the semi-covariant curvature and the gauged Dirac operators. The authors state that this action is invariant, to leading order in fermions, under the variations $delta d = -tfrac{i}{12}bar{epsilon}rho$, $delta V^{Mm} = -i bar{V}^{Mbar q}bar{epsilon}gamma^m psi^{bar q}$, $delta rho = -gamma^m D_m epsilon$, $delta psi^{bar m} = D^{bar m}epsilon$. The second claim is a comparison: the generalised Scherk-Schwarz reduction of the parent ungauged theory produces a connection that differs from the gauged connection by a term involving the twist matrix, and the corresponding spin connections differ by the projected quantity in their equation (4.10); the authors assert that this difference vanishes when contracted into the semi-covariant derivatives, so the Killing spinor equations of the gauged and the Scherk-Schwarz reduced theories coincide exactly.