Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks

Abstract

We show how to map Grothendieck's dessins d'enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d $\mathcal{N}=2$ supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.

arXiv comment55 pages; dedicated to the memory of Prof. Omar Foda; v3: minor corrections

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• Reflexions on Mahler: Dessins, Modularity and Gauge Theories