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# Communications in Number Theory and Physics

## Volume 1 (2007)

### Number 2

### Children’s drawings from Seiberg–Witten curves

Pages: 237 – 305

DOI: http://dx.doi.org/10.4310/CNTP.2007.v1.n2.a1

#### Authors

#### Abstract

We consider $\CN=2$ supersymmetric gauge theories perturbed bytree level superpotential terms near isolated singular points inthe Coulomb moduli space. We identify the Seiberg-Witten curveat these points with polynomial equations used to construct whatGrothendieck called “dessins d’enfants” or “children’sdrawings” on the Riemann sphere. From a mathematical point ofview, the dessins are important because the absolute Galoisgroup ${\rm Gal}(\bar \Bbb{Q}/\Bbb{Q})$ acts faithfully on them.We argue that the relation between the dessins andSeiberg–Witten theory is useful because gauge theory criteriaused to distinguish branches of $\CN=1$ vacua can lead tomathematical invariants that help to distinguish dessinsbelonging to different Galois orbits. For instance, we show thatthe confinement index defined in hep-th/0301006 is a Galoisinvariant. We further make some conjectures on the relationbetween Grothendieck’s program of classifying dessins intoGalois orbits and the physics problem of classifying phases of$\CN=1$ gauge theories.