Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 038, 52 pages      arXiv:1907.13545      https://doi.org/10.3842/SIGMA.2020.038
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Quantum Statistical Mechanics of the Absolute Galois Group

Yuri I. Manin a and Matilde Marcolli bcd
a) Max Planck Institute for Mathematics, Bonn, Germany
b) California Institute of Technology, Pasadena, USA
c) University of Toronto, Toronto, Canada
d) Perimeter Institute for Theoretical Physics, Waterloo, Canada

Received August 01, 2019, in final form April 15, 2020; Published online May 05, 2020

Abstract
We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's dessins d'enfant, the embedding in the Grothendieck-Teichmüller group, and the Drinfeld-Ihara involution.

Key words: quantum statistical mechanics; dessins d'enfant; absolute Galois group; Drinfeld-Ihara involution; quasi-triangular quasi-Hopf algebras.

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