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Generalised Monstrous Moonshine and Genus Two Conformal Field Theory

Introduction

An important trend in recent years has been the growing overlap between areas of interest to both theoretical physicists and pure mathematicians. One example of this trend is in the use of ideas developed in conformal field theory to understand the surprising connections between the Monster finite simple group and the theory of modular forms in pure mathematics known collectively as Monstrous Moonshine. Click here for most of the relevant references. Here is a brief summary of some of the main ideas in this area.

Monstrous Moonshine

The original Moonshine conjectures of Conway and Norton (Conway, J H and Norton, S P, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 308-339 (1979)) concern the relationship between the Monster finite simple group and certain modular forms. The Monster group is the largest finite sporadic simple group and is of order

2^(46).3^(20).5^9.7^6.11^2.13 ^3.17.19.23.29.31.41.47.59.71

The lowest dimensional irreducible representations are of dimension 1, 196883, 21298676,...

In the classical theory of modular forms, the modular invariant J(q) plays a pivotal role. J(q) is invariant under the action of the modular group SL(2,Z)

t->(a t b)/(c t d)

for ad-bc="1" and a,b,c,d integers and is the unique meromorphic function (for t in H, the upper half complex plane with q="exp(2pi " it)) with a simple pole at q="0" and zero constant term. This latter property implies that the compactification of H/SL(2,Z) is a Riemann surface of genus zero. J(q) is said to be the hauptmodul for the genus zero modular group SL(2,Z).

The Laurant expansion for J(q) is

J(q) = q^(-1) 196884 q 21493760 q^2 .

It was observed by Thompson (Thompson, J G, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. Lond. Math. Soc. 11, 352-353 (1979)) that the coefficients of the J(q) series are small positive sums of the dimensions of the irreducible representations of the Monster group e.g. 196884="1 196883, " 21493760="1 196883 21298676," etc. This was greatly generalised by Conway and Norton by considering the McKay-Thompson series Tg(q) for any conjugacy class representative g in the Monster where the dimension of the irreducible representation is replaced by the character of g for that rep. Thus:

Tg(q)="q^(-1) (1 X(g)) " q .,

where X(g) is the character of the 196883 representation etc. The main conjecture of Conway and Norton is that Tg(q) is the hauptmodul for some genus zero modular group which leaves Tg(q) invariant. This allows one to calculate Tg(q) explicitly and to find generating formulas for the coefficients of Tg(q) and to find relationships between different McKay-Thompson series.

The Moonshine Module and Orbifold Constructions

The reason for the original observations of Thompson concerning the J(q) elliptic function later became clear from the construction of the FLM Moonshine Module (Frenkel, I, Lepowsky, J and Meurman, A, Vertex operator algebras and the Monster, Pure and Applied Mathematics, 134. Academic Press, Inc., Boston, MA, 1988). The Monster group is the automorphism group for this vertex operator algebra and the character is the modular invariant J(q) whose coefficients naturally have the observed properties. The Moonshine conjectures were subsequently proved by Borcherds (Borcherds R E, Monstrous moonshine and monstrous Lie superalgebras , Invent. Math., 109, 405-444 (1992)) using Generalised Kac-Moody algebras and an examination of McKay-Thompson series on a case by case basis.

The FLM construction is equivalent to an orbifold construction used by theoretical physicists in conformal field theory for a bosonic string (compactified on a Leech torus which is orbifolded by the lattice reflection symmetry). Then considering the conformal field theory on a genus one Riemann surface leads to the modular invariant partition function J(q) . A more general approach to understanding of the Moonshine properties using orbifolds was then introduced by the present author (Tuite, Michael P., Monstrous Moonshine from orbifolds, Comm. Math. Phys. 146, 277-309 (1992)). The idea is to consider orbifolding of the Moonshine Module with respect to Monster group elements for which the McKay-Thompson series have a particular interpretation. The genus zero property and other new features were then related to the conjectured uniqueness of the Moonshine Module (Tuite, Michael P, On the relationship between Monstrous Moonshine and the uniqueness of the Moonshine module, Comm. Math. Phys . 166, 495-532 (1995)). This is the first time that the genus zero property has been derived from some unifying principle.

Generalised Moonshine

This is a far vaster set of conjectures due to Norton (G. Mason (with an appendix by S.P. Norton), Finite groups and modular functions, Proc. Symp. Pure Math. 47, 181-210, (1987).) and is concerned with McKay-Thompson series for centralisers (and some special extensions thereof) of Monster group elements. It is conjectured that every such modular function is either constant or is the hauptmodul for some genus zero modular group which leaves the series invariant. The constant series cases are understood and can be predicted but the genus zero cases are not yet so. A current area of research is to relate the genus zero property for generalised McKay-Thompson series to the uniqueness of the Moonshine Module by considering abelian orbifolds. An understanding for the simplest prime ordered elements has been provided by the author using such methods (Tuite, Michael P., Generalised Moonshine and abelian orbifold constructions, Contemp. Math., 193, 353-368 (1996.)). Recent results on the Generalised Moonshine appear in

R.I. Ivanov and M.P. Tuite, Generalised Moonshine from Orbifoldings, to appear in Balkan Physical Letters.

R.I. Ivanov and M.P. Tuite, Rational Generalised Moonshine from Abelian Orbifoldings of the Moonshine Module, Nucl.Phys. B635 435-472 (2002), http://arxiv.org/abs/math.QA/0106027.

R.I. Ivanov and M.P. Tuite, Some Irrational Generalised Moonshine from Orbifolds, Nucl.Phys. B635 473-491 (2002), http://arxiv.org/abs/math.QA/0106027.

Genus Two Conformal Field Theory

A conformal field theory can be defined on a Riemann surface of any genus not just genus one. Very little is explicitly known about partition functions even for genus two Riemann surfaces. However, the author has solved certain aspects of this problem in (Tuite, Michael P., Genus Two Meromorphic Conformal Field Theory, CRM Proceedings and Lecture Notes, 30 (2001) 231-251, http://arXiv.org/abs/math/9910136 ).

See also the first in a series of planned papers on genus two and higher n-point functions:

G. Mason and M.P. Tuite, Torus chiral n-point functions for free boson and lattice vertex operator algebras, Commun.Math.Phys., 235 (2003) 47-68. http://xxx.lanl.gov/abs/math.QA/0204323

More Information

Click for some further information on the general area of Monstrous Moonshine .

Applied Mathematics
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