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In the case of charge a global $U(1)$ symmetry leads to the conservation of charge, however upgrading the global symmetry to a local symmetry leads to the electromagnetic potential field $A^\mu$ such that: $$D_\mu =\partial_\mu - iqA_\mu.$$ Since a global space time symmetry leads to the consevation of the energy-momentum tensor $T_{\mu\nu}$ does a local space time symmetry lead to the gravitational field? Where upgrading $\phi(x^\mu+\xi^\mu)$ to $\phi(x^\mu+\xi^\mu(x^\nu))$ $$D_\mu=\partial_\mu+\Gamma^\alpha_{\mu\nu}.$$ (Obviously the indicies don't match here but would the result be in a similar form.)

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Short answer: yes, one can view General Relativity (GR) as a gauge theory, see this paper as a reference. It's not straightforward: the fields of GR are tensors, have many more indices, and require more constraints and identities to create a consistent theory of gravity.

Nevertheless, pay attention to the fact that a local symmetry doesn't "lead" to a physical field: not for gravity, nor for electromagnetism (not a priori, at least).
The second chapter of the linked paper provides a detailed explanation of the gauging procedure, but the main idea is: I have some fields ($\phi$), they have a global symmetry, meaning my Lagrangian is invariant under a global transformation, that's cool. I'd like to make my transformation local, my fields seem fine, but my Lagrangian isn't invariant anymore, that's not cool, I have extra terms. If I added another field ($A_\mu$), which transforms in a particular way, I could reabsorb the extra terms, and the said field can be added without bothering what's already in the Lagrangian, everything cool again. Moreover, I can add a kinetic term ($F_{\mu\nu}$) for this field, meaning this field can propagate on its own! Super cool. Still, this doesn't mean that the new field is physical, maybe the theory described by the Lagrangian just isn't a gauge theory and I'm doing everything wrong.
Then someone builds a LHC and finds a particle that could be described by the said field. NOW we can say that this field is physical.

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