Abstract
In this paper we will study perturbative quantum gravity on supermanifolds with both noncommutative and non-anticommutative coordinates. We shall first analyses the BRST and the anti-BRST symmetries of this theory. Then we will also analyze the effect of shifting all the fields of this theory in background field method. We will construct a Lagrangian density which apart from being invariant under the extended BRST transformations is also invariant under on-shell extended anti-BRST transformations. This will be done by using the Batalin-Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost term for this theory can be elegantly written down in superspace with two Grassmann parameters.
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