In field theory, covariant derivative is something like $$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$ while in differential geometry, covariant derivative is something like $$D_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma^{\nu}_{\mu\lambda}V^{\lambda}$$ the gauge field and Levi-Civita connection appear in the same place. So is there some connection between them?
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Both are connections on a principal fibre bundle, but on different principal fibre bundles.
For Yang--Mills theory with structure group $G$, there is a principal $G$-bundle $\pi:P\rightarrow M$ over the spacetime manifold $M$, with its bundle $\mathrm{Con}(\pi)\rightarrow M$ of principal connections, which can be interpreted as an affine bundle over $M$. The Yang--Mills connection $D^{\mathrm{YM}}$ is a smooth section of $\mathrm{Con}(\pi)\rightarrow M$, which is dynamically generated as a solution of the Yang--Mills equation.
For general relativity, the principal bundle is $L(M)\rightarrow M$, the bundle of linear frames over $M$, which is unlike the principal bundle used in Yang--Mills theory is a so-called natural fibre bundle, it is canonically (i.e. functorially) associated to each manifold $M$. The bundle $\mathrm{Con}(M)\rightarrow M$ of principal connections on $L(M)\rightarrow M$ is also a natural bundle. The LC connection in particular is then canonically associated to each metric. The underlying mechanism is that $\mathrm{Met}(-)$ is a natural bundle functor that associates to each manifold its bundle of smooth metrics, and we can compose it with the jet functors to get also a natural bundle $J^1\mathrm{Met}(-)$. Then the LC connection is a natural transformation $$ D^{\mathrm{LC}}: J^1\mathrm{Met}(-)\rightsquigarrow \mathrm{Con}(-). $$
It is the metric then which is dynamically generated from the Einstein field equations over $M$, and the LC connection is then associated to the metric via this natural transformation of functors.
