Contemporary Mathematics Volume 193, 1996 HIGHER GENUS MOONSHINE P. BANTAY 1. Introduction Since their discovery, string theory [1] and the closely related Conformal Field Theory [2] have shed light· on interesting new connections beween mathematics and physics. The vertex operator realization of affine Kac-Moody algebras [3] and the quantum field theoretic approach to knot and link invariants [4] are famous examples. A most interesting example is the relation between Moonshine and the so-called orbifold CFT. It was Dixon, Ginsparg and Harvey [5] who first realized, that Nor- ton's 'generalized Moonshine' postulates [6] - apart from the famous 'genus zero' postulate - are essentially equivalent t the existence of a suitable orbifold model, whose twist group is the Fischer-Griess Monster. That such a model does indeed exist follows from the work of Frenkel, Lepowsky and Meurmann [7] who con- structed its untwisted Hilbert-space, the celebrated Moonshine module. The above connection does not only allow for a reinterpretation of Moonshine in physics terms, but it also opens the way for interesting generalizations and ex- tensions. An obvious one is wether Moonshine is a specific feature of the Monster, or there exists some analogous phenomena for other finite groups. This question has been studied for some time [8], and was answered in the affirmative by the con- struction by Dong and Mason of a 'moonshine module' for the Mathieu group M24 [9]. Physical intuition suggests that there should exist moonshine-like phenomena for a large number of finite groups- perhaps for all of them? Another possible extension follows from the observation, that the connection discovered by Dixon, Ginsparg and Harvey relates the Thompson-McKay series entering the formulation of the Moonshine conjectures with physical quantities characterizing the relevant orbifold model defined over a complex torus, i.e. a Riemann surface of genus one. But a CFT may be defined over Riemann surfaces of arbitrary topology, and this allows one to define some kind of higher genus analogues of the ordinary Thompson-McKay series. The title 'higher genus Moonshine' refers to the study of these quantities, and my goal is to sketch the relevant ideas leading to their definition. For a start, let me recall you the 'generalized' Moonshine postulates of Norton [6]: If G is the Fischer-Griess Monster, then for each pair x, y E G of commuting elements, there exists a. Thompson-McKay series Z ( x, y I r) meromorphic on 1i = {r I Im(r) 0}, such that {1) Z(x,yl ~$~) = Z(xayb,xcydlr) for(~~) E SL{2,Z). 1991 Mathematics Subject Classification. ,Primary 81R50, 16W30. Partially supported by OTKA grant No. F-4010. © 1996 American Mathematical Society
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