New "metric-affine-like" generalization of Yang-Mills theory

We suggest a new generalization of the $\mathrm{U}(n)$ Yang-Mills theory obtained by relaxing the condition of covariant constancy of the Hermitian form in the fibers, $\nabla_a g_{\alpha\beta'} \ne 0$. This theory is a simpler analogue of the metric-affine gravity with $\nabla_a g_{bc} \ne 0$. In our case, connection $\nabla_a$ and Hermitian form $g_{\alpha\beta'}$ are two independent variables so total curvature and total potential are no longer anti-Hermitian matrices: thus, along with the standard YM potential $\boldsymbol{A}_a$ and field strength tensor $\boldsymbol{F}_{ab}$, it contains non-trivially interacting fields $\boldsymbol{B}_a$, $\boldsymbol{h}$, and $\boldsymbol{G}_{ab}$, $\boldsymbol{N}_a$, forming a non-Abelian generalization of St\"{u}ckelberg theory. Due to the spontaneous symmetry breaking $\mathrm{GL}(n,\mathbb{C}) \to \mathrm{U}(n)$, these new fields can be made massive, and the limit $M\to\infty$ restores the standard YM theory.