Quantum Statistical Mechanics of the Absolute Galois Group

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.

Intersection theory, characteristic classes, and algebro-geometric Feynman rules

“…We will not review here the detailed properties of Rota-Baxter algebras and Birkhoff factorization, as we will not need them in the rest of the paper, but we refer the readers to [52], [53], [54] for a detailed discussion of Connes-Kreimer renormalization. For examples of this formalism of algebraic renormalization applied outside of the quantum field theory BPHZ renormalization, with other Rota-Baxter structures, see for instance [55], [56].…”

Quantum Statistical Mechanics and the Boundary of Modular Curves

“…Since the introduction of the Bost-Connes system in the mid '90s, [1], a very rich interplay between number theory and quantum statistical mechanics developed, involving the Galois theory of abelian and non-abelian extensions of number fields ( [6], [7], [11], [13], [21], [27]), Shimura varieties ( [4], [11], [14]), L-series and zeta functions ( [3], [8]), etc. On the other hand, the work of Manin and Marcolli, [19], and the subsequent work [12], [20], [22], developed a theory of limiting modular symbols, and a related noncommutative geometry model of the boundary of modular curves.…”