Smooth Loops and Fiber Bundles: Theory of Principal Q-bundles

During the last few decades, nonassociative structures have been employed in various fields of modern physics. Among others, one may mention the rise of nonassociative objects such as 3-cocycles, which are linked with violations of the Jacobi identity in anomalous quantum field theory, and quantum mechanics with the Dirac monopole, the appearance of Lie groupoids and algebroids in the context of YangMills theories, and the application of nonassociative algebras to gauge theories on commutative but nonassociative fuzzy spaces [1,2,3,4,5,6,7,8,9,10,11,12,13]. Nonassociative algebraic structures such as quasigroups and loops have considerable potential interests for mathematical physics, especially in view of the appearance of nonassociative algebras, such as the Mal’cev algebra [14], related to the problem of the chiral gauge anomalies, the emergence of a nonassociative electric field algebra in a two-dimensional gauge theory, and so on [15,16,17,18,1]. Quasigroups and loops have recently been employed in general relativity and for the description of the Thomas precession, coherent states, geometric phases, and nonassociative gauge theories [10,19,20,21,22,23,24,25,26,27,28]. In this paper, we give a detailed account of nonassociative fiber bundles leading to “nonassociative” gauge theories [9,10,19]. In general terms, a consequence of nonassociativity is that the structure constants of the gauge algebra have to be changed by structure functions.