Children’s drawings from Seiberg-Witten curves. (English) Zbl 1230.14050
The interest in \(\mathcal N=2\) supersymmetric theories, as well as their tree-level superpotential deformations to \(\mathcal N=1\) is especially due to the seminal work of Seiberg and Witten. Physical questions such as monopole condensation, non-perturbative information, the full moduli space and determination of the coupling constants are encoded in a Riemann surface known as the Seiberg-Witten curve. The factorisation of this curve, itself an interesting algebraic problem, is the key.
A seemingly unrelated subject, important in algebraic geometry, is Grothendieck’s “dessins d’enfants”, or “children’s drawings”. This is a connected bipartite graph on a Riemann surface \(X\), encoding the zero and branch-cut structures. Grothendieck’s correspondence shows that these drawings are pre-images of so-called Belyĭ maps, which are holomorphic maps from \(X\) to the sphere with critical values at \(0,1,\infty\) (Belyĭ’s Theorem states that \(X\) is defined over \(\overline{\mathbb Q}\) instead of \(\mathbb C\) iff such a map exists). Remarkably, the drawings afford a faithful action by the absolute Galois group \(\mathrm{Gal}\,(\overline{\mathbb Q}/\mathbb Q)\), a central object in modern number theory. In the paper under review, the authors connect the above two subjects, especially relating Grothendieck’s classification of the drawings into different Galois orbits with the classification of phases of \(\mathcal N=1\) gauge theories. They show that any (clean) dessin with \(N_c\) edges and \(N_f+1\) faces can be found at an isolated singularity in the moduli space an \(\mathcal N=2\), \(U(N_c)\) gauge theory with \(N_f\) flavours. Certain factorisations, called rigid, of the Seiberg-Witten curve are shown to be determined by the valency lists of the dessin. A generalisation, called non-rigid factorisations, can then be interpreted as \(\mathcal N=1\) vacua. Some conjectures are made; in the strongest form, it is proposed that every Galois invariant of the dessins is a physical order parameter which distinguishes different phases in \(\mathcal N=1\) gauge theory.
A seemingly unrelated subject, important in algebraic geometry, is Grothendieck’s “dessins d’enfants”, or “children’s drawings”. This is a connected bipartite graph on a Riemann surface \(X\), encoding the zero and branch-cut structures. Grothendieck’s correspondence shows that these drawings are pre-images of so-called Belyĭ maps, which are holomorphic maps from \(X\) to the sphere with critical values at \(0,1,\infty\) (Belyĭ’s Theorem states that \(X\) is defined over \(\overline{\mathbb Q}\) instead of \(\mathbb C\) iff such a map exists). Remarkably, the drawings afford a faithful action by the absolute Galois group \(\mathrm{Gal}\,(\overline{\mathbb Q}/\mathbb Q)\), a central object in modern number theory. In the paper under review, the authors connect the above two subjects, especially relating Grothendieck’s classification of the drawings into different Galois orbits with the classification of phases of \(\mathcal N=1\) gauge theories. They show that any (clean) dessin with \(N_c\) edges and \(N_f+1\) faces can be found at an isolated singularity in the moduli space an \(\mathcal N=2\), \(U(N_c)\) gauge theory with \(N_f\) flavours. Certain factorisations, called rigid, of the Seiberg-Witten curve are shown to be determined by the valency lists of the dessin. A generalisation, called non-rigid factorisations, can then be interpreted as \(\mathcal N=1\) vacua. Some conjectures are made; in the strongest form, it is proposed that every Galois invariant of the dessins is a physical order parameter which distinguishes different phases in \(\mathcal N=1\) gauge theory.
Reviewer: Pietro De Poi (Udine)
MSC:
| 14H81 | Relationships between algebraic curves and physics |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 14H57 | Dessins d’enfants theory |
| 12F10 | Separable extensions, Galois theory |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
