Gauge theories consist of a spinor bundle describing the matter fields, associated to some principal bundle whose gauge group rules the local symmetries. The dynamics of matter fields is given in terms of a covariant derivative induced by the gauge fields, which are the local expressions of a principal connection. Both can be seen as parallel displacements on the fibres along curves on the base manifold and require a physical information beyond (but compatible with) the gauge group. The need for extra structure appears also during the canonical quantization of the system, performed via deformations induced by the Wightman and the Feynman propagators, with BRST and BV tricks on the gauge fields.
Now, the huge wave front sets of these propagators forces a renormalization step which breaks the classical geometrical setting. In this talk I present a generalization of gauge theories based on direct connections on their gauge groupoid, which govern both the local symmetries above each fibre and the parallel displacement among fibres, providing a classical setup to interpretate quantum propagators and, in a long-term aim, to justify renormalization steps geometrically.
The talk is based on a work in progress with S. Azzali, Y. Boutaib, A. Garmendia and S. Paycha.