From Periods to Anabelian Geometry and Quantum Amplitudes

Abstract

To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, we consider the case of algebraic Riemann Surfaces representable by Belyi maps. The category of morphisms of Belyi ramified maps and Dessins D Enfant, will be investigated in search of an analog for periods, of the Ramification Theory for decomposition of primes in field extensions, controlled by theirs respective algebraic Galois groups. This suggests a relation between the theory of (cohomological, Betti-de Rham) periods and Grothendieck s Anabelian Geometry (homotopical/ local systems), towards perhaps an algebraic analog of Hurwitz Theorem, relating the the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. There seem to be good prospects of better understanding the role of absolute Galois group in the physics context of scattering amplitudes and Multiple Zeta Values, with their incarnation as Chen integrals on moduli spaces, as studied by Francis Brown, since the latter are a homotopical analog of de Rham Theory. The research will be placed in the larger context of the ADE-correspondence, since, for example, orbifolds of finite groups of rotations have crepant resolutions relevant in String Theory, while via Cartan-Killing Theory and exceptional Lie algebras, they relate to TOEs. Relations with the author s reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology and the Arithmetic Galois Theory will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. It will allow an elementary investigation of the main concepts defining periods and algebraic fundamental group, together with their conceptual relation to algebraic numbers and Galois groups. The Riemann surfaces with Platonic tessellations, especially the Hurwitz surfaces, are related to the finite Hopf sub-bundles with symmetries the ``exceptional Galois groups. The corresponding Platonic Trinity leads to connections with ADE-correspondence, and beyond, e.g. TOEs and ADEX-Theory. Quantizing everything (cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubits/baryons) could perhaps be The Eightfold (Petrie polygon) Way to finally understand what quark flavors and fermion generations really are.