From gauges to gaugeoids: connections and transformations

29.05.2020, 11:00
Arbeitsgruppenseminar Analysis

Sylvie Paycha (Potsdam University)

Two essential ingredients in classical gauge field theory are  gauge fields given by  connections on a principal bundle  and gauge transformations given by automorphisms of the bundle.

Principal bundles are in one to one correspondence with gauge groupoids and connections on principal bundles amount  to infinitesimal connections on their corresponding gauge groupoid. We study gauge groupoids equipped with direct connections, which one can roughly view as enhanced infinitesimal connections in so far as they can  be built from a torsion free connection on the underlying manifold combined with an infinitesimal connection.
Direct connections occur in  various disguises in the mathematical literature, and specifically in Hairer's regularity structures where they are called re-expansion maps.

Gauge groupoids behave functorially under jet  prolongation, a useful fact to have in mind in the context of regularity structures. We shall argue that this functoriality carries out to flat direct connections when  the underlying manifold is affine.

Gauge transformations of a principal bundle induce transformations of the corresponding gauge groupoid, which  for short, we call gaugeoid transformations. Not all gaugeoid transformations come from a gauge transformation, which gives rise to interesting cohomological obstructions and led us  to investigating higher analogues of gauge groupoids and their transformations.

This is based on ongoing joint work with S. Azzali and A. Frabetti  together with Y. Boutaib and A. Garmendia.

Meeting ID: 498 775 1829
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