View source for Moonshine conjectures

In 1978, J. McKay observed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m1302201.png" />. The number on the left is the first non-trivial coefficient of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m1302204.png" />-function, and the numbers on the right are the dimensions of the smallest irreducible representations of the Fischer–Griess Monster <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m1302205.png" /> [[#References|[a10]]] (cf. also [[Sporadic simple group|Sporadic simple group]]). On the one side stands a [[Modular function|modular function]]; on the other, a finite [[Sporadic simple group|sporadic simple group]]. Moonshine is the explanation and generalization of this unlikely connection.

Monstrous moonshine [[#References|[a6]]] conjectured that there is an infinite-dimensional graded [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m1302206.png" />, with the following properties. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m1302207.png" /> carries a finite-dimensional representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m1302208.png" />; write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m1302209.png" /> for its character. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022010.png" />, define the [[Thompson–McKay series|Thompson–McKay series]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022012.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022013.png" /> is a generator ( "Hauptmodul" ) of the field of modular functions for some genus-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022014.png" /> [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022015.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022016.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022017.png" /> as a [[Normal subgroup|normal subgroup]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022018.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022020.png" /> is the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022021.png" />).

There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022022.png" /> distinct Thompson series (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022023.png" /> has only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022024.png" /> conjugacy classes). For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022027.png" /> is the identity. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022028.png" /> has two order-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022029.png" /> conjugacy classes, corresponding to the modular groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022030.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022031.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022032.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022033.png" />-dimensional irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022034.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022037.png" />.

Central to these conjectures is the moonshine module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022038.png" />, constructed in [[#References|[a9]]]. It is an important example of a [[Vertex operator algebra|vertex operator algebra]] (VOA) [[#References|[a2]]], [[#References|[a9]]], and as such possesses infinitely many heavily constrained bilinear products. One of these products makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022039.png" /> into the Griess algebra [[#References|[a10]]] extended by an identity element. The automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022040.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022041.png" />.

The monstrous moonshine conjectures, and, in particular, the identification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022042.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022043.png" />, were proved in [[#References|[a4]]], which also defined vertex operator algebras and a generalization of Kac–Moody algebras called Borcherds or generalized Kac–Moody algebras [[#References|[a3]]] (cf. also [[Kac–Moody algebra|Kac–Moody algebra]]; [[Borcherds Lie algebra|Borcherds Lie algebra]]).

The series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022044.png" /> were generalized by S.P. Norton to commuting pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022045.png" /> [[#References|[a12]]]. In particular, to each such pair there is a [[Modular function|modular function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022046.png" />, invariant under a genus-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022047.png" /> group, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022048.png" /></td> </tr></table>

for some root of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022049.png" />, for any

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022050.png" /></td> </tr></table>

This action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022051.png" /> is related to its natural action on the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022052.png" /> of the torus. The coefficients of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022053.png" />-expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022054.png" /> are characters of the centralizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022055.png" /> evaluated at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022056.png" />. Simultaneous conjugation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022058.png" /> leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022059.png" /> unchanged: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022060.png" />. The [[Thompson–McKay series|Thompson–McKay series]] are recovered by the specialization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022061.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022062.png" />. Only special cases of these generalized moonshine conjectures have been proven.

There are several other conjectures. For example, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022063.png" /> were conjecturally related by the replication formulas [[#References|[a6]]], [[#References|[a1]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022064.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022065.png" /> is the unique polynomial for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022066.png" /> is a power series with only strictly positive powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022067.png" />. The resemblance of (a1) with Hecke operators is not accidental. These formulas are related to identities such as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022069.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022070.png" />, as well as to the existence of several modular equations obeyed by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022071.png" />. These identities played a large role in the proof of moonshine. R. Borcherds proved that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022072.png" /> obey (a1) by using the denominator identities of Borcherds algebras; the resulting modular equations can be used to prove the Hauptmodul property [[#References|[a7]]]. It would be very nice to see other direct connections between the group operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022073.png" /> and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022074.png" />: for instance, are there other relations among the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022075.png" /> which correspond to more general products of conjugacy classes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022076.png" />?

The Hirzebruch prize question [[#References|[a11]]] asks for a compact, differentiable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022077.png" />-dimensional manifold on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022078.png" /> acts by diffeomorphisms and whose twisted Witten genus is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022079.png" />. It is still open (as of 2000).

Vertex operator algebras are a key component in moonshine. There will be some form of moonshine for finite subgroups of the automorphism group of any rational vertex operator algebra obeying a technical (and probably redundant)  "C2 condition"  (see e.g. [[#References|[a18]]]). The resulting functions, however, will not be Hauptmoduls in general. There is no complete understanding yet (2000) of why Hauptmoduls are associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022080.png" />, although it has been conjectured to do with the  "6-transposition property"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022082.png" /> [[#References|[a12]]], or the uniqueness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022083.png" /> [[#References|[a17]]].

Moonshine is related to conformal field theory (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022084.png" /> is the chiral algebra of a holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022085.png" /> theory), in which context many of moonshine's special features appear natural [[#References|[a8]]], [[#References|[a17]]], and this has motivated some of the developments (most significantly the definition of vertex operator algebras).

It still (as of 2000) is unclear why certain Hauptmoduls appear in moonshine and others do not. What form does moonshine (e.g. the replication formulas) take in genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022086.png" /> (see e.g. [[#References|[a16]]])? How about moonshine for other groups (see e.g. [[#References|[a13]]])? In [[#References|[a14]]], a variant called modular moonshine was established, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022087.png" /> is replaced with vertex algebras over characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022088.png" />, and where the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022089.png" /> is taken by centralizers in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022090.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022091.png" /> elements.

For his work in monstrous moonshine and related topics, Borcherds was awarded a Fields Medal in 1998 [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Alexander,   C. Cummins,   J. McKay,   C. Simons,   "Completely replicable functions" , ''Groups, Combinatorics and Geometry'' , Cambridge Univ. Press  (1992)  pp. 87–98</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.E. Borcherds,   "Vertex algebras, Kac–Moody algebras, and the Monster"  ''Proc. Nat. Acad. Sci. USA'' , '''83'''  (1986)  pp. 3068–3071</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.E. Borcherds,   "Generalized Kac–Moody algebras"  ''J. Algebra'' , '''115'''  (1988)  pp. 501–512</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Borcherds,   "Monstrous moonshine and monstrous Lie superalgebras"  ''Invent. Math.'' , '''109'''  (1992)  pp. 405–444</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.E. Borcherds,   "What is Moonshine?" , ''Proc. ICM, Berlin'' , DMV  (1998)  pp. 607–615</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.H. Conway,   S.P. Norton,   "Monstrous moonshine"  ''Bull. London Math. Soc.'' , '''11'''  (1979)  pp. 308–339</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C. Cummins,   T. Gannon,   "Modular equations and the genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130220/m13022092.png" /> property of Moonshine functions"  ''Invent. Math.'' , '''129'''  (1997)  pp. 413–443</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Dixon,   P. Ginsparg,   J. Harvey,   "Beauty and the Beast: superconformal symmetry in a Monster module"  ''Commun. Math. Phys.'' , '''119'''  (1988)  pp. 221–241</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I.B. Frenkel,   J. Lepowsky,   A. Meurman,   "Vertex operators and the monster" , Acad. Press  (1988)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  R.L. Griess,   "The friendly giant"  ''Invent. Math.'' , '''69'''  (1982)  pp. 1–102</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  F. Hirzebruch,   T. Berger,   R. Jung,   "Manifolds and modular forms" , ''Aspects of Math.'' , Vieweg  (1994)  (Edition: Second)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S.P. Norton,   "Generalized moonshine"  ''Proc. Symp. Pure Math.'' , '''47'''  (1987)  pp. 208–209</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  L. Queen,   "Modular functions arising from some finite groups"  ''Math. Comput.'' , '''37'''  (1981)  pp. 547–580</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  A.J. E. Ryba,   "Modular moonshine?" , ''Moonshine, the Monster, and Related Topics'' , ''Contemp. Math.'' , '''193''' , Amer. Math. Soc.  (1996)  pp. 307–336</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  R.E. Borcherds,   "Modular moonshine III"  ''Duke Math. J.'' , '''93'''  (1998)  pp. 129–154</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  G.W. Smith,   "Replicant powers for higher genera" , ''Moonshine, the Monster, and Related Topics'' , ''Contemp. Math.'' , '''193''' , Amer. Math. Soc.  (1996)  pp. 337–352</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  M.P. Tuite,   "On the relationship between Monstrous moonshine and the uniqueness of the Moonshine module"  ''Commun. Math. Phys.'' , '''166'''  (1995)  pp. 495–532</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  Y. Zhu,   "Modular invariance of characters of vertex operator algebras"  ''J. Amer. Math. Soc.'' , '''9'''  (1996)  pp. 237–302</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  C. Dong,   H. Li,   G. Mason,   "Modular invariance of trace functions in orbifold theory"  ''preprint q-alg/9703016''  (1997)</TD></TR></table>