The Full Quadratic Metric-Affine Gravity (Including Parity Odd Terms): Exact solutions for the Affine-Connection

We consider the most general Quadratic Metric-Affine Gravity setup in the presence of generic matter sources with non-vanishing hypermomentum. The gravitational action consists of all $17$ quadratic invariants (both parity even and odd) in torsion and non-metricity as well as their mixings, along with the terms that are linear in the curvature namely the Ricci scalar and the totally antisymmetric Riemann piece. Adding also a matter sector to the latter we first obtain the field equations for the generalized quadratic Theory. Then, using a recent Theorem, we successfully find the exact form of the affine connection under some quite general non-degeneracy conditions. Having obtained the exact and unique solution of the affine connection we subsequently derive the closed forms of spacetime torsion and non-metricity and also recast the metric field equations into a GR form with modified source terms that are quadratic in the hypermomentum and linear in its derivatives. We also study the vacuum quadratic Theory and prove that in this instance, or more generally for vanishing hypermomentum, the connection becomes the Levi-Civita one. Therefore, we also find exactly to what does the quadratic vacuum Theory correspond to. Finally, we generalize our result even further and also discuss the physical consequences and applications of our study.