Perturbative quantum gravity on complex spacetime

In this Letter we will study non-anticommutative perturbative quantum gravity on spacetime with a complex metric. After analysing the BRST symmetry of this non-anticommutative perturbative quantum gravity, we will also analyse the effect of shifting all the fields. We will construct a Lagrangian density which is invariant under the original BRST transformations and the shift transformations in the Batalin-Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost term for this shift symmetry invariant Lagrangian density can be elegantly written down in superspace with a single Grassmann parameter.