Vector stability in quadratic metric-affine theories

In this work we study the stability of the four vector irreducible pieces of the torsion and the nonmetricity tensors in the general quadratic metric-affine Lagrangian in 4 dimensions. This highly constrains the theory reducing the parameter space of the quadratic curvature part from 16 to 5 parameters. We also study the sub-case of Weyl-Cartan gravity, proving that the stability of the vector sector completely fixes the dynamics of the full Lagrangian to just an Einstein-Proca theory or pure General Relativity.Comment: 21 pages, no figures, no table