Geometry of supergravity and the Batalin--Vilkovisky formulation of the $\mathcal N=1$ theory in ten dimensions

TL;DR

This paper develops a comprehensive Batalin--Vilkovisky formulation of ten-dimensional $ =1$ supergravity using generalized geometry, avoiding traditional Lorentz degrees of freedom and clarifying the geometric structure of the theory.

Contribution

It introduces a novel BV formulation of supergravity that omits Lorentz degrees of freedom, emphasizing the fibred geometric structure of the field space.

Findings

Successfully demonstrates the BV action satisfies the classical master equation.

Provides a geometric understanding of supersymmetry algebra variations.

Eliminates the need for orthonormal frame degrees of freedom in the formulation.

Abstract

We provide full details of a BV formulation of $\mathcal N=1$ supergravity in ten dimensions, to all orders in fermions, built from the generalised geometry description of the theory. In contrast to standard treatments, we introduce neither the degrees of freedom corresponding to orthonormal frames for the metric nor the local Lorentz symmetries that remove them again. Instead, we observe that the field space has a fibred structure, with the fermionic degrees of freedom spanning the fibres. We explain in detail how this geometric picture allows one to understand simultaneous variations of spinorial quantities and the metric with respect to which the spinor bundles are defined. This leads to additional terms in certain commutators on field space which account for the Lorentz transformation terms appearing in the calculation of the supersymmetry algebra. Unencumbered by the Lorentz degrees of freedom, we provide an efficient and full demonstration that our action satisfies the classical master equation.