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You can take a global symmetry and promote it to a local gauge symmetry by introducing an appropriate gauge field and upgrading the partial derivative to a covariant derivative. The photon field arises from global $U(1)$ symmetry, the gluon field from $SU(3)$ and even gravity shows up this way (though it's more elaborate since the symmetry group of general coordinate transformations is infinite and compact, differently from your usual $SU(N)$).

What gauge field do I get from Lorentz symmetry?

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    $\begingroup$ Since the obvious answer is "A $\mathrm{SO}(1,3)$ gauge field", could you make more precise what you mean with this question? $\endgroup$
    – ACuriousMind ♦
    Commented Apr 11, 2015 at 18:11
  • $\begingroup$ Yes, but what physical, measurable particle corresponds to the $SO(1,3)$ gauge field? We have photon for $U(1)$, gluon for $SU(3)$, etc. $\endgroup$
    – Koaaala
    Commented Apr 11, 2015 at 18:15
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    $\begingroup$ None. The Standard model does not have the Lorentz group as a gauge group. $\endgroup$
    – ACuriousMind ♦
    Commented Apr 11, 2015 at 18:19
  • $\begingroup$ See well-ansered question. $\endgroup$
    – Cosmas Zachos
    Commented Apr 9, 2016 at 21:16

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It is the celebrated spin connection on the tangent space, gauging Lorentz rotations so you can take Lorentz covariant derivatives on spinors---you would not be able to do Supergravity without it.

As you see, however, $\omega_\mu^{ab}$ is a composite gauge field, that is, it is is an elaborate function of Vierbeine (or Vielbeine) and their derivatives, ensuring tangent space Lorentz invariance, and not a fundamental field. No matter, it is necessary, and GR fermions live by it!

You might enjoy this review.

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