Gravitation can be interpreted as a gauge theory with a spin 2 graviton field. This graviton field in general relativity is also interpreter as a Riemannian metric. Do other gauge theories also have interpretations as geometries? If so, what kinds of geometries?
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``Geometry'' is a very vague term. The usual gauge theories (Yang-Mills theory) is about connections on vector bundles, see https://en.wikipedia.org/wiki/Gauge_theory_(mathematics). If you include spin $3/2$, for which it is also necessary to have spin-$2$ plus possibly other guys, you get supergravities and the associated buzz word is supergeometry. For $s>2$ you cannot have just one of them, all spins are necessary and there are few strange theories of this type called higher-spin gravities, where the geometry is close to a noncommutative one
John
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