# Pure gauge theory for the gravitational spin connection

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Referee this paper: arXiv:2212.05069 by Stephon Alexander, Tucker Manton

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The gravitational spin connection appears in gravity as a non-Abelian gauge field for the Lorentz group SO(3,1), which is non-compact. The action for General Relativity is linear in the field strength associated to the spin connection, and its equation of motion corresponds to the standard metricity constraint. Consequently, the zero-torsion spin connection is never realized as an independent degree of freedom and is determined by the vierbein field. In this work, we take a different perspective and consider a pure Yang-Mills theory for the spin connection coupled to Dirac fermions, resulting in the former being a dynamical field. After discussing various approaches towards managing the pathologies associated with non-compact gauge theories, we compute the tree-level amplitude for fermion scattering via a spin connection exchange. In contrast to integrating out torsion in the presence of fermions, the model induces a chiral four-Fermi like term that involves a right-right current interaction, which is not present in the Standard Model.

requested Dec 15, 2022
summarized
paper authored Dec 9, 2022 to gr-qc
edited Jan 2

This is one of those "gravity as a gauge theory" papers. It caught my eye because I have an interest in whether quantum gauge theories with noncompact gauge groups can ever be viable (that they aren't, has been the most prominent technical criticism of Eric Weinstein's Geometric Unity), and this paper has a section on the issue in which it unearths two old approaches to the problem, one due to Margolin and Strazhev, the other to Kevin Cahill, as well as a possibly new approach. These three approaches are respectively called "BRST approach", "dynamical approach", and "gauge-fixing approach".

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