Describing metric-affine theories anew: alternative frameworks, examples and solutions

In this work we describe metric-affine theories anew by making a change of field variables. A series of equivalent frameworks is presented and identifications are worked out in detail. The advantage of applying the new frameworks is that any MAG theory can be handled as a Riemannian theory with additional fields. We study the Hilbert-Palatini action using the new field variables and disclose interesting symmetries under $SO$ transformations in field space. Then, we use solvable and suitable Riemannian theories as seed models for solvable MAG theories, restricting ourselves to three examples. We present a black hole solution with torsion and non-metricity which under a certain tuning acquires a regular core. A de Sitter universe with the expansion powered by 3-form torsion, is also reported.