Metric-Affine Vector-Tensor Correspondence and Implications in $F(R,T,Q,\mathcal{T},\mathcal{D})$ gravity

TL;DR

This paper develops a quadratic Metric-Affine Gravity theory that is equivalent to a Vector-Tensor theory and explores its implications within a generalized $F(R,T,Q, abla T, abla Q)$ gravity framework.

Contribution

It introduces a new quadratic gravity action involving torsion and non-metricity, establishing its equivalence to a Vector-Tensor theory and analyzing special cases and implications.

Findings

The theory is equivalent to a Vector-Tensor model on-shell.

Special cases with simplified quadratic terms are identified.

Implications for extended $F(R,T,Q, abla T, abla Q)$ gravity are discussed.

Abstract

We extend the results of antecedent literature on quadratic Metric-Affine Gravity by studying a new quadratic gravity action in vacuum which, besides the usual (non-Riemannian) Einstein-Hilbert contribution, involves all the parity even quadratic terms in torsion and non-metricity plus a Lagrangian that is quadratic in the field-strengths of the torsion and non-metricity vector traces. The theory result to be equivalent, on-shell, to a Vector-Tensor theory. We also discuss the sub-cases in which the contribution to the Lagrangian quadratic in the field-strengths of the torsion and non-metricity vectors just exhibits one of the aforementioned quadratic terms. We then report on implications of our findings in the context of $F(R,T,Q,\mathcal{T},\mathcal{D})$ gravity.