Nonassociative geometry in representation categories of quasi-Hopf algebras
Barnes, Gwendolyn Elizabeth
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It has been understood that quantum spacetime may be non-geometric in the sense that its phase space algebra is noncommutative and nonassociative. It has therefore been of interest to develop a formalism to describe differential geometry on non-geometric spaces. Many of these spaces would fi t naturally as commutative algebra objects in representation categories of triangular quasi-Hopf algebras because they arise as cochain twist deformations of classical manifolds. In this thesis we develop in a systematic fashion a description of differential geometry on commutative algebra objects internal to the representation category of an arbitrary triangular quasi-Hopf algebra. We show how to express well known geometrical concepts such as tensor fields, differential calculi, connections and curvatures in terms of internal homomorphisms using universal categorical constructions in a closed braided monoidal category to capture algebraic properties such as Leibniz rules. This internal description is an invaluable perspective for physics enabling one to construct geometrical quantities with dynamical content and con guration spaces as large as those in the corresponding classical theories. We also provide morphisms which lift connections to tensor products and tensor elds. Working in the simplest setting of trivial vector bundles we obtain explicit expressions for connections and their curvatures on noncommutative and nonassociative vector bundles. We demonstrate how to apply our formalism to the construction of a physically viable action functional for Einstein-Cartan gravity on noncommutative and nonassociative spaces as a step towards understanding the effect of nonassociative deformations of spacetime geometry on models of quantum gravity.