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Comments on the links between \(su(3)\) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards. (English) Zbl 0895.14009

This work is devoted to explain the relationship between invariants appearing in different subjects in mathematics. The first field are the modular invariants of the partition function of a rational conformal field theory (RCFT) on a torus, specially in the case of \(su(3)\) models.
In the second section, the authors discuss modular invariance for \(su(3)\) theories. The partition function of an RCFT of height \(n\) on a torus \({\mathbb{C}}/({\mathbb{Z}}+\tau{\mathbb{Z}})\), \({\mathfrak I}\tau>0\), is determined by a matrix of non-negative integers \(N:=\{N_{p,p'}\}_{p,p'\in B_n}\), where \[ B_n:=\{p:=(r,s,t)\in{\mathbb{Z}}^3\mid r,s,t\geq 1, r+s+t=n\}. \] The modular action of \(PSL_2({\mathbb{Z}})\) on \(\{{\mathfrak I}\tau>0\}\) induces two matrices \(S\), \(T\) which reflect the isomorphism of the corresponding elliptic curves. A function determined by a matrix \(N\) is a modular invariant if and only if \(N\) commutes with \(S\) and \(T\). The matrices \(S\) and \(T\) are rational combinations of \(3n\)-roots of unity and Galois theory is involved [see A. Coste and T. Gannon, Phys. Lett. B 323, 316-321 (1994)]. This theory produces a parity selection rule which makes the computation of modular invariants easier.
In the third section, the authors consider the Jacobian of a Fermat curve \(x^n+y^n=z^n\). The first coincidence with previous results is that a basis of holomorphic differentials is parametrized in a natural way by \(B_n\); in fact the Jacobian is isogenous to a product of Abelian varieties also indexed by \(B_n\). These factors possess complex multiplication (CM) and the general theory for simpleness of abelian varieties with CM may be applied, see the Shimura-Taniyama theorem [G. Shimura and Y. Taniyama, “Complex multiplication of abelian varieties and its application to number theory”, Publ. Math. Soc. Jap. 6 (1961; Zbl 0112.03502)].
In the fourth section, combinatorial groups for triangulated surfaces are studied, following Grothendieck dessins d’enfants. The starting point is the standard triangulation of the Riemann sphere with vertices \(0,1,\infty\); if \(h\) is a meromorphic map from \(\Sigma\) ramified at \(0,1,\infty\), the standard triangulation induces a special one of \(\Sigma\) which may be encoded by the cartographic group. The universal cartographic group is the modular group and any subgroup of finite index of the uniformizing group \(\Gamma_2\) determines up to isomorphism a pair \((\Sigma,h)\). The Kummer projection defines a cartographic group for Fermat curves which is also related with \(B_n\). A rational triangular billiard associated to a triangle of angles \(\pi/r\), \(\pi/s\), \(\pi/t\) with \(r+s+t=n\) is the classical phase space of a particle moving in the corresponding orbifold. The trajectories determine a closed Riemann surface \(C_{r,s,t}\) (called triangular curve) which projects onto the triangle and produce a cartographic group. It may happen that a Fermat curve projects onto a triangular curve.
In the fifth section, the authors study the Riemann surface of a RFCT on a torus. The goal is to show that a compact Riemann surface can be associated with any RCFT. The authors show for example that the curve associated with \(su(3)_1\) is a triangular curve admitting as covering the Fermat curve of degree \(12\).
The paper finishes with some conclusions and questions about the relationships stated in the paper and two appendices.

MSC:

14H40 Jacobians, Prym varieties
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14K22 Complex multiplication and abelian varieties

Citations:

Zbl 0112.03502
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