${\cal N}=1$ supersymmetry in $4D$ gravity with torsion and non-metricity

TL;DR

This paper explores how ${ m extbf{N}}=1$ supersymmetry can be incorporated into 4D gravity theories with torsion and non-metricity, analyzing the closure of symmetry transformations and constraints on the connection and Lagrangian.

Contribution

It demonstrates that supersymmetry constrains the form of the spin connection and the bosonic Lagrangian in 4D gravity with torsion and non-metricity, regardless of the deformations.

Findings

Supersymmetry restricts the spin connection to depend only on the vielbein.

The form of the bosonic Lagrangian is constrained by supersymmetry.

Special cases with vanishing curvature, non-metricity, or torsion are analyzed.

Abstract

We study the inclusion of fermionic degrees of freedom and ${\cal N}=1$ supersymmetry in 4-dimensional manifolds with arbitrary torsion and non-metricity tensors deforming the connection. We inspect the closure of local diffeomorphism, Lorentz and supersymmetric transformations on the vielbein and gravitino at leading order in fermions. The closure of a pair of supersymmetric transformations acting on the vielbein restricts the form of the spin connection. The dependency of the latter on the fundamental fields is exclusively given by the vielbein, up to Lorentz and/or diffeomorphisms brackets, independently of the torsion or non-metricity deformations of the affine connection, which is not restricted by supersymmetry at this order. We have then studied the couple of the Rarita-Schwinger theory to a generic gravitational Lagrangian; the supersymmetric variation of the former restricts the form of the bosonic Lagrangian in order to accomplish invariance. Special cases where $R_{\mu \nu}{}^{ab}=0$, $Q_{\mu \nu \rho}=0$ or $T_{\mu \nu}{}^{\rho}=0$ are provided.