Communications in Number Theory and Physics

Volume 1 (2007)

Number 2

Children’s drawings from Seiberg–Witten curves

Pages: 237 – 305



Sujay K. Ashok (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)

Freddy Cachazo (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)

Eleonora dell’Aquila (NHETC, Department of Physics, Rutgers University, New Jersey)


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.

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