878 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
On the Relationship between the Uniqueness of the Moonshine Module and Monstrous Moonshine
We consider the relationship between the conjectured uniqueness of the
Moonshine Module, , and Monstrous Moonshine, the genus zero
property of the modular invariance group for each Monster group Thompson
series. We first discuss a family of possible meromorphic orbifold
constructions of based on automorphisms of the Leech
lattice compactified bosonic string. We reproduce the Thompson series for all
51 non-Fricke classes of the Monster group together with a new relationship
between the centralisers of these classes and 51 corresponding Conway group
centralisers (generalising a well-known relationship for 5 such classes).
Assuming that is unique, we then consider meromorphic
orbifoldings of and show that Monstrous Moonshine holds if
and only if the only meromorphic orbifoldings of give
itself or the Leech theory. This constraint on the
meromorphic orbifoldings of therefore relates Monstrous
Moonshine to the uniqueness of in a new way.Comment: 53 pages, PlainTex, DIAS-STP-93-0
On Representations of Conformal Field Theories and the Construction of Orbifolds
We consider representations of meromorphic bosonic chiral conformal field
theories, and demonstrate that such a representation is completely specified by
a state within the theory. The necessary and sufficient conditions upon this
state are derived, and, because of their form, we show that we may extend the
representation to a representation of a suitable larger conformal field theory.
In particular, we apply this procedure to the lattice (FKS) conformal field
theories, and deduce that Dong's proof of the uniqueness of the twisted
representation for the reflection-twisted projection of the Leech lattice
conformal field theory generalises to an arbitrary even (self-dual) lattice. As
a consequence, we see that the reflection-twisted lattice theories of Dolan et
al are truly self-dual, extending the analogies with the theories of lattices
and codes which were being pursued. Some comments are also made on the general
concept of the definition of an orbifold of a conformal field theory in
relation to this point of view.Comment: 11 pages, LaTeX. Updated references and added preprint n
On the Relationship between Monstrous Moonshine and the Uniqueness of the Moonshine Module
We consider the relationship between the conjectured uniqueness of the Moonshine Module, Vā®, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possible Z meromorphic orbifold constructions of Vā® based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster group M together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that Vā® is unique, we then consider meromorphic orbifoldings of Vā® and show that Monstrous Moonshine holds if and only if the only meromorphic orbifoldings of Vā® are Vā® itself or the Leech theory. This constraint on the meromorphic orbifoldings of therefore relates Monstrous Moonshine to the uniqueness of Vā® in a new way
Optimality and uniqueness of the Leech lattice among lattices
We prove that the Leech lattice is the unique densest lattice in R^24. The
proof combines human reasoning with computer verification of the properties of
certain explicit polynomials. We furthermore prove that no sphere packing in
R^24 can exceed the Leech lattice's density by a factor of more than
1+1.65*10^(-30), and we give a new proof that E_8 is the unique densest lattice
in R^8.Comment: 39 page
On the Y555 complex reflection group
We give a computer-free proof of a theorem of Basak, describing the group
generated by 16 complex reflections of order 3, satisfying the braid and
commutation relations of the Y555 diagram. The group is the full isometry group
of a certain lattice of signature (13,1) over the Eisenstein integers Z[cube
root of 1]. Along the way we enumerate the cusps of this lattice and classify
the root and Niemeier lattices over this ring.Comment: 16 pages; submitte
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