14,405 research outputs found
Monstrous Moonshine and the uniqueness of the Moonshine module
In this talk we consider the relationship between the conjectured uniqueness
of the Moonshine module of Frenkel, Lepowsky and Meurman and Monstrous
Moonshine, the genus zero property for Thompson series discovered by Conway and
Norton. We discuss some evidence to support the uniqueness of the Moonshine
module by considering possible alternative orbifold constructions from a Leech
lattice compactified string. Within these constructions we find a new
relationship between the centralisers of the Monster group and the Conway group
generalising an observation made by Conway and Norton. We also relate the
uniqueness of the Moonshine module to Monstrous Moonshine and argue that given
this uniqueness, then the genus zero properties hold if and only if orbifolding
the Moonshine module with respect to a Monster element reproduces the Moonshine
module or the Leech theory. (Talk presented at the Nato Advanced Research
Workshop on `Low dimensional topology and quantum field theory`, Cambridge,
6-13 Sept 1992)Comment: 12 pages, DIAS-STP-92-2
Generalised Umbral Moonshine
Umbral moonshine describes an unexpected relation between 23 finite groups
arising from lattice symmetries and special mock modular forms. It includes the
Mathieu moonshine as a special case and can itself be viewed as an example of
the more general moonshine phenomenon which connects finite groups and
distinguished modular objects. In this paper we introduce the notion of
generalised umbral moonshine, which includes the generalised Mathieu moonshine
[Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number
Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data
for it. A central role is played by the deformed Drinfel'd (or quantum) double
of each umbral finite group , specified by a cohomology class in
. We conjecture that in each of the 23 cases there exists a rule
to assign an infinite-dimensional module for the deformed Drinfel'd double of
the umbral finite group underlying the mock modular forms of umbral moonshine
and generalised umbral moonshine. We also discuss the possible origin of the
generalised umbral moonshine
Generalised Moonshine and Abelian Orbifold Constructions
We consider the application of Abelian orbifold constructions in Meromorphic
Conformal Field Theory (MCFT) towards an understanding of various aspects of
Monstrous Moonshine and Generalised Moonshine. We review some of the basic
concepts in MCFT and Abelian orbifold constructions of MCFTs and summarise some
of the relevant physics lore surrounding such constructions including aspects
of the modular group, the fusion algebra and the notion of a self-dual MCFT.
The FLM Moonshine Module, , is historically the first example of
such a construction being a orbifolding of the Leech lattice MCFT,
. We review the usefulness of these ideas in understanding Monstrous
Moonshine, the genus zero property for Thompson series which we have shown is
equivalent to the property that the only meromorphic orbifoldings of
are and itself (assuming that
is uniquely determined by its characteristic function .
We show that these constraints on the possible orbifoldings of
are also sufficient to demonstrate the genus zero property for
Generalised Moonshine functions in the simplest non-trivial prime cases by
considering orbifoldings of . Thus Monstrous
Moonshine implies Generalised Moonshine in these cases.Comment: Talk presented at the AMS meeting on Moonshine, the Monster and
related topics, Mt. Holyoke, June 1994, 16 pp, Plain TeX with AMS Font
Much ado about Mathieu
Eguchi, Ooguri and Tachikawa have observed that the elliptic genus of type II
string theory on K3 surfaces appears to possess a Moonshine for the largest
Mathieu group. Subsequent work by several people established a candidate for
the elliptic genus twisted by each element of M24. In this paper we prove that
the resulting sequence of class functions are true characters of M24, proving
the Eguchi-Ooguri-Tachikawa conjecture. We prove the evenness property of the
multiplicities, as conjectured by several authors. We also identify the role
group cohomology plays in both K3-Mathieu Moonshine and Monstrous Moonshine; in
particular this gives a cohomological interpretation for the non-Fricke
elements in Norton's Generalised Monstrous Moonshine conjecture. We investigate
the intriguing proposal of Gaberdiel-Hohenegger-Volpato that K3-Mathieu
Moonshine lifts to the Conway group Co1.Comment: 38 pages; references added; minor corrections and additions,
including more speculation
Monstrous and Generalized Moonshine and Permutation Orbifolds
We consider the application of permutation orbifold constructions towards a
new possible understanding of the genus zero property in Monstrous and
Generalized Moonshine. We describe a theory of twisted Hecke operators in this
setting and conjecture on the form of Generalized Moonshine replication
formulas.Comment: 14 pages, to appear in Proceedings of the Conference "Moonshine - The
First Quarter Century and Beyond
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