Abstract
This talk discusses the relative merits of using connections in Lie groupoids instead of connections in principal bundles for the description of gauge theories. The basic definitions of groupoids as well as Lie groupoids and Lie algebroids are recalled. After recalling the Atiyah definition of a connection in terms of a splitting of an exact sequence of vector bundles, I review the definitions of connections and curvatures for Lie groupoids. I briefly mention the differences between the group of automorphisms of a Lie groupoid associated to a principal bundle, from the group of gauge transformations of the latter. I discuss the subtle differences arising from this when applied to gauge theories. The final section contains some conjectures and speculations relative to possible use of groupoids in quantizing gauge theories as well as formulating quantum theories on (pseudo-)Riemannian manifolds.
The author would like to apologize to all those who have advocated the use of groupoids in gauge theories, and whom he does not mention explicitly in the references. Time and space limitations have made it impossible to carry out a thorough literature search; this will be corrected in an extended version, to be published elsewhere.
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References
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Mayer, M.E. (1990). Principal Bundles versus Lie Groupoids in Gauge Theory. In: Chau, LL., Nahm, W. (eds) Differential Geometric Methods in Theoretical Physics. NATO ASI Series, vol 245. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-9148-7_78
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