2,738 research outputs found
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
Some Generalizations of the MacMahon Master Theorem
We consider a number of generalizations of the -extended MacMahon
Master Theorem for a matrix. The generalizations are based on replacing
permutations on multisets formed from matrix indices by partial permutations or
derangements over matrix or submatrix indices.Comment: 16 pages, 4 figure
The Virasoro Algebra and Some Exceptional Lie and Finite Groups
We describe a number of relationships between properties of the vacuum Verma
module of a Virasoro algebra and the automorphism group of certain vertex
operator algebras. These groups include the Deligne exceptional series of
simple Lie groups and some exceptional finite simple groups including the
Monster and Baby Monster.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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
Exceptional Vertex Operator Algebras and the Virasoro Algebra
We consider exceptional vertex operator algebras for which particular Casimir
vectors constructed from the primary vectors of lowest conformal weight are
Virasoro descendants of the vacuum. We discuss constraints on these theories
that follow from an analysis of appropriate genus zero and genus one two point
correlation functions. We find explicit differential equations for the
partition function in the cases where the lowest weight primary vectors form a
Lie algebra or a Griess algebra. Examples include the Wess-Zumino-Witten model
for Deligne's exceptional Lie algebras and the Moonshine Module. We partially
verify the irreducible decomposition of the tensor product of Deligne's
exceptional Lie algebras and consider the possibility of similar decompositions
for tensor products of the Griess algebra. We briefly discuss some conjectured
extremal vertex operator algebras arising in Witten's recent work on three
dimensional black holes.Comment: 13 pages, Talk presented at "Vertex Operator Algebras and Related
Areas" at Illinois State University, July 200
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
Vertex Algebras According to Isaac Newton
We give an introduction to vertex algebras using elementary forward
difference methods originally due to Isaac Newton.Comment: 20 page
Genus Two Meromorphic Conformal Field Theory
We construct the genus two (or two loop) partition function for meromorphic
bosonic conformal field theories. We use a sewing procedure involving two genus
one tori by exploiting an explicit relationship between the genus two period
matrix and pinching modular parameters. We obtain expressions for the partition
function for the chiral bosonic string, even rank lattice theories and
self-dual meromorphic conformal field theories including the Moonshine Module.
In particular, we find that for self-dual theories with central charge 24, the
genus two partition function multiplied by a universal holomorphic function of
the moduli is given by a meromorphic Siegel modular form of weight 2 where this
universal function includes ghost contributions. We also discuss a novel
expansion for certain Siegel modular forms.Comment: 25 pages, AMS Latex2e, 2 figures, Talk presented at Workshop on
Moonshine, CRM, Montreal, May 29 to June 4, 199
Rational Generalised Moonshine from Abelian Orbifoldings of the Moonshine Module
We consider orbifoldings of the Moonshine Module with respect to the abelian
group generated by a pair of commuting Monster group elements with one of prime
order and the other of order for or prime. We show
that constraints arising from meromorphic orbifold conformal field theory allow
us to demonstrate that each orbifold partition function with rational
coefficients is either constant or is a hauptmodul for an explicitly found
modular fixing group of genus zero. We thus confirm in the cases considered the
Generalised Moonshine conjectures for all rational modular functions for the
Monster centralisers related to the Baby Monster, Fischer, Harada-Norton and
Held sporadic simple groups. We also derive non-trivial constraints on the
possible Monster conjugacy classes to which the elements of the orbifolding
abelian group may belong.Comment: 40 pages, Improved versio
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