Noncommutativity and Non-anticommutativity in Perturbative Quantum Gravity

TL;DR

This paper explores perturbative quantum gravity on supermanifolds with noncommutative and non-anticommutative coordinates, analyzing symmetries and invariances using advanced formalism and superspace techniques.

Contribution

It introduces a novel analysis of BRST and anti-BRST symmetries in this context and constructs an invariant Lagrangian using the BV formalism and superspace.

Findings

BRST and anti-BRST symmetries are preserved in this framework

The Lagrangian density is invariant under extended BRST and on-shell anti-BRST transformations

Gauge-fixing and ghost terms are expressed in superspace with two Grassmann parameters

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.