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

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.

54 pages, 17 figures

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• Les Houches lectures on matrix models and topological strings
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Cited by in corpus (12)

• On KP-integrable Hurwitz functions
• On the complete perturbative solution of one-matrix models
• Correlators in tensor models from character calculus
• Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models
• What is the Simplest Gauge-String Duality?
• On determinant representation and integrability of Nekrasov functions
• Complete solution to Gaussian tensor model and its integrable properties
• CFT4 as SO(4,2)-invariant TFT2
• Topological String Correlators from Matrix Models
• Gauge Theories, Tessellations & Riemann Surfaces
• Holography for Tensor models
• Holomorphic primary fields in free CFT4 and Calabi-Yau orbifolds