Einstein-Yang-Mills Connections

Question

I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations.

Assuming spherical symmetry, the $SU(2)$ connection can be written \begin{equation} A = \omega(r)\tau_1 d\theta + \omega(r)\sin\theta \tau_2 d\theta + \cos\theta \tau_3 d\phi,\tag{1} \end{equation} where the $\tau_i$ are the generators of the $\mathfrak{su}(2)$ algebra.

A static spherically symmetric metric has the form \begin{equation} ds^2 = -T^{-2}(r)dt^2 + B^{-1}(r)dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2.\tag{2} \end{equation}

The Yang-Mills equations are \begin{equation} D\star F = 0,\tag{3} \end{equation} along with the Bianchi Identity \begin{equation} DF = 0.\tag{4} \end{equation}

Clearly, $D$ contains the usual gauge covariant exterior derivative \begin{equation} DF = dF + [A\wedge F]\tag{5} \end{equation} with respect to the $SU(2)$ connection.

Here is where my question arises: since the Yang-Mills field lives in curved spacetime, shouldn't the gauge covariant exterior derivative include additional terms which describe the usual covariant derivative of $F$ with respect to the Levi-Civita connection on the spacetime manifold?

Answer 1

OP is considering Yang-Mills theory over a curved base space $(M,g)$. If the base space connection is the Levi-Civita connection $\nabla^{LC}=\partial+\Gamma$, then it doesn't matter whether one uses the gauge-covariant derivative $D=\partial+A$ or the full covariant derivative $\nabla=D+\Gamma$ since the Christoffel symbols $\Gamma$ drops out of the Yang-Mills theory and OP's eqs. (3), (4) and (5). This is mainly due to the torsionfreeness of the Levi-Civita connection $\nabla^{LC}$.

Answer 2

Levi-Civita connection makes its appearance only when some quantity is covariant under local Lorentz transformations. But if everything is invariant under local Lorentz transformations, the Levi-Civita connection is irrelevant. The metric and YM fields (which are of concern in Yang-Mills theory in curved space time) are invariant, while Riemann curvature is covariant (which is of concern in general relativity). That is why Christoffel symbol drops out of the Yang-Mills theory and OP's eqs. (3), (4) and (5). This fact has nothing to do with torsionfreeness condition of the Levi-Civita connection as the other answer is suggesting. It's true even under non-zero torsion condition.

A side note: Bianchi Identity is the same regardless of flat or curved space time, since metric does not show up in Bianchi Identity. On the otherhand, the Hodge dual $\star$ in Yang-Mills equation is metric-dependent.