A Moonshine Dialogue in Mathematical Physics

Abstract

Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive at dessins d'enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell's theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincaré would have been happy to see it.

Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

“…The dessin of Figure 2i has of course trivalent black points; it corresponds to a subgroup of the modular group Γ that is recognized to be the congruence subgroup Γ 0 (6) of Γ depicted in Figure 2j. The normalizer of Γ 0 (6) in Γ is the moonshine group Γ + 0 (6) [13]. The configuration G = [12 6 , 24 3 ] (4) is of rank four, comprises 12 points and 24 lines/triangles with six lines through each point.…”

“…Let us remind how to pass from the topological structure of a modular dessin D to that of a hyperbolic polygon P ( [13], Section 3). There are ν 2 elliptic points of order two (resp.…”

“…Following the impetus given by Grothendieck [16], it is now known that there are various ways to dress a group P generated by two permutations: (i) as a connected graph drawn on a compact oriented two-dimensional surface, a bicolored map (or hypermap) with n edges, B black points, W white points, F faces, genus g and Euler characteristic 2 − 2g = B + W + F − n [17]; (ii) as a Riemann surface X of the same genus equipped with a meromorphic function f from X to the Riemann sphereC unramified outside the critical set {0, 1, ∞}, the pair (X, f ) called a Belyi pair, and in this context, hypermaps are called dessins d'enfants [14,16]; (iii) as a subgroup H of the free group G = a, b (or of a two-generator group G = a, b|rels ) where P encodes the action of (right) cosets of H on the two generators a and b; the Coxeter-Todd algorithm does the job [11]; and finally (iv), when P is of rank at least three, that is of a point stabilizer with at least three orbits, as a non-trivial finite geometry [10][11][12][13]. Finite simple groups are generated by two of their elements [18], so that it is useful to characterize them as members of the categories just described.…”

“…As a bonus, the two-generator permutation groups and their underlying geometries may luckily be considered as dessins d'enfants [14], although this topological and algebraic aspect of the finite simple (or not simple) groups is barely mentioned in the literature. Ultimately, it may be that the Monster group and its structure fits our quantum world, as in Dyson's words [13]. More cautiously, in Section 2 of the present paper, we briefly account for the group concepts involved in our approach by defining a tripod P − D − G. One leg P is a desired two-generator permutation representation of a finite group P [9].…”

“…Only recently, the first author observed that much of the information needed is encapsulated in permutation representations, of rank larger than two, available in the Atlas of finite group representations [9]. The coset enumeration methodology of the Atlas was used by us for deriving many finite geometries underlying quantum commutation and the related contextuality [10][11][12][13]. As a bonus, the two-generator permutation groups and their underlying geometries may luckily be considered as dessins d'enfants [14], although this topological and algebraic aspect of the finite simple (or not simple) groups is barely mentioned in the literature.…”