Quadratic metric-affine gravity: solving for the affine-connection

We consider the most general 11 parameter parity even quadratic Metric-Affine Theory whose action consists of the usual Einstein–Hilbert plus the 11 quadratic terms in torsion, non-metricity as well as their mixing. By following a certain procedure and using a simple trick we are able to find the unique solution of the affine connection in terms of an arbitrary hypermomentum. Given a fairly general non-degeneracy condition our result provides the exact form of the affine connection for all types of matter. Subsequently we compute the forms of torsion and non-metricity in terms of their sources (hypermomentum tensor) and also express the metric field equations in effectively Einstein’s GR with modified source terms that depend on the hypermomentum and its derivatives. We show that in the absence of matter the Theory always reduces to GR. Finally we generalize our result and find the form of the connection for a wider class of quadratic Theories.