What gauge field can be constructed from Lorentz symmetry?

Question

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?

Since the obvious answer is "A $\mathrm{SO}(1,3)$ gauge field", could you make more precise what you mean with this question? — ACuriousMind, Apr 11 '15 at 18:11
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. — Koaaala, Apr 11 '15 at 18:15
None. The Standard model does not have the Lorentz group as a gauge group. — ACuriousMind, Apr 11 '15 at 18:19

score 6 · Answer 1 · answered Apr 09 '16 at 21:13

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!

What gauge field can be constructed from Lorentz symmetry?

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