From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

TL;DR

This paper connects hermitian matrix model correlators to counting holomorphic maps with three branch points, revealing links to algebraic numbers, Galois groups, and Grothendieck's Dessins d'Enfants, with implications for string theory and algebraic geometry.

Contribution

It establishes a novel correspondence between matrix model correlators, Hurwitz numbers, and algebraic curves over number fields, highlighting the Galois group's action on Feynman diagrams and Dessins d'Enfants.

Findings

Explicit formulae for Hurwitz numbers derived from matrix models.

Identification of the Galois group's action on matrix model diagrams.

Introduction of colored-edge Dessins as invariants related to Galois action.

Abstract

We show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers $\bmQ$. This shows that the string theory dual of the one-matrix model at generic couplings has worldsheets defined over the algebraic numbers and a target space $ \mP^1 (\bmQ)$. The absolute Galois group $ Gal (\bmQ / \mQ) $ acts on the Feynman diagrams of the 1-matrix model, which are related to Grothendieck's Dessins d'Enfants. Correlators of multi-matrix models are mapped to the counting of triples of permutations subject to equivalences defined by subgroups of the permutation groups. This is related to colorings of the edges of the Grothendieck Dessins. The colored-edge Dessins are useful as a tool for describing some known invariants of the $ Gal (\bmQ / \mQ) $ action on Grothendieck Dessins and for defining new invariants.