For the first time, we build a generalization of the $U(n)$ Yang-Mills theory obtained by abandoning the condition of covariant constancy of the Hermitian form in the fibers: $\nabla_a g_{ \alpha\beta'} \ne 0$. So this theory is a simpler analogue of the well-known metric-affine gravity with $\nabla_a g_{ bc} \ne 0$. In our case the connection $\nabla_a$ and the Hermitian form $g_{ \alpha\beta'}$ are two independent variables, and the total curvature and the total potential are no longer anti-Hermitian matrices: in addition to the usual $\boldsymbol{ F}_{ ab}$ and $\boldsymbol{ A}_a$ they also obtain new Hermitian parts $\boldsymbol{ G}_{ ab}$ and $\boldsymbol{ B}_a$. It is shown that the Hermitian form $g_{ \alpha\beta'}$ is a Higgs field breaking the general $GL(n,\mathbb{ C})$ gauge symmetry to $U(n)$, and its perturbations are Goldstone bosons which can be eliminated from the theory by redefining other fields. The result is a theory consisting of two non-trivially interacting gauge fields, one of which can be made massive while the other remains massless. Letting the mass of the second gauge field tend to infinity allows one to restore the usual YM.
Categories: hep-th