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, $V^\natural$, is historically the first example of such a construction being a $Z_2$ orbifolding of the Leech lattice MCFT, $V^\Lambda$. 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 $Z_n$ orbifoldings of $V^\natural$ are $V^\Lambda$ and $V^\natural$ itself (assuming that $V^\natural$ is uniquely determined by its characteristic function $J(\tau)$. We show that these constraints on the possible $Z_n$ orbifoldings of $V^\natural$ are also sufficient to demonstrate the genus zero property for Generalised Moonshine functions in the simplest non-trivial prime cases by considering $Z_p\times Z_p$ orbifoldings of $V^\natural$. 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 $\beta$-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

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

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

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I

We define the partition and $n$-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.Comment: 57 Pages, 5 figures. This is an extended version of roughly one half of arXiv:0712.062

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II

We continue our program to define and study $n$-point correlation functions for a vertex operator algebra $V$ on a higher genus compact Riemann surface obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas for the genus two partition function for free bosonic theories and lattice vertex operator algebras $V_L$. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. We also compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity. We compare our results with those obtained in the companion paper, when a pair of tori are sewn together, and show that the partition functions are not compatible in the neighborhood of a two-tori degeneration point. The \emph{normalized} partition functions of a lattice theory $V_L$ \emph{are} compatible, each being identified with the genus two theta function of $L$.Comment: 51 pages, 3 figure