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

TL;DR

This paper connects Grothendieck's dessins d'enfants to algebraic Seiberg-Witten curves, using mirror and AGT maps to derive instanton partition functions and conformal blocks, revealing dualities and model correspondences.

Contribution

It introduces a novel mapping from dessins d'enfants to Seiberg-Witten curves and explores their relation to conformal blocks and gauge theory dualities.

Findings

Explicitly constructed 6 trivalent dessins with 4 punctures

Identified duality relations between parametrizations from dessins

Suggested correspondence between dessins and conformal blocks in minimal models

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.