Mathematical Physics

   

On (An)Abelian Geometry and Deformation Theory with Applications to Quantum Physics

Authors: Lucian M. Ionescu

The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic fundamental group (Anabelian Geometry), in analogy with the classical context of Hurewicz Theorem. To investigate this idea, the article considers an "Abstract Galois Theory", as a separated abstract structure from, yet compatible with, the Theory of Schemes, which has its historical origin in Commutative Algebra and motivation in the early stages of Algebraic Topology. The approach to Motives via Deformation Theory was suggested by Kontsevich as early as 1999, and suggests Formal Manifolds, with local models formal pointed manifolds, as the source of motives, and perhaps a substitute for a "universal Weil cohomology". The proposed research aims to gain additional understanding of periods via a concrete project, the discrete algebraic de Rham cohomology, a follow-up of author's previous work. The connection with Arithmetic Gauge Theory should provide additional intuition, by looking at covering maps as flat connection spaces, and considering branching covers of the Riemann sphere as the more general case. The research on Feynman/Veneziano Amplitudes and Gauss/Jacobi sums, allows to deepen the parallel between the continuum and discrete frameworks: an analog of Virasoro algebra in finite characteristic. A larger project is briefly considered, consisting in deriving Motives from the Theory of Deformations, as suggested by Kontsevich. Following Soibelman and Kontsevich, the idea of defining Formal Manifolds as groupoids of pointed formal manifolds (after Maurer-Cartan ``exponentiation''), with associated torsors as ``gluing data'' (transition functions) is suggested. This framework seems to be compatible with the ideas from Theory of Periods, sheaf theory / etale maps and Grothedieck's development of Galois Theory (Anabelian Geometry). The article is a preliminary evaluation of a research plan of the author. Further concrete problems are included, since they are related to the general ideas mentioned above, and especially relevant to understanding the applications to scattering amplitudes in quantum physics.

Comments: 7 Pages.

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[v1] 2019-12-25 10:12:16

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