Children's Drawings From Seiberg-Witten Curves

TL;DR

This paper explores the connection between Seiberg-Witten curves in supersymmetric gauge theories and Grothendieck's dessins d'enfants, proposing that physical invariants can classify mathematical Galois orbits.

Contribution

It establishes a novel link between gauge theory criteria and mathematical invariants of dessins, aiding in the classification of Galois orbits and phases of gauge theories.

Findings

Confinement index is a Galois invariant.

Dessins correspond to specific gauge theory phases.

Proposes conjectures relating Galois classification to physical phases.

Abstract

We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called "dessins d'enfants" or "children's drawings" on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of N=1 vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck's programme of classifying dessins into Galois orbits and the physics problem of classifying phases of N=1 gauge theories.