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

On Exceptional Vertex Operator (Super) Algebras

We consider exceptional vertex operator algebras and vertex operator superalgebras with the property that particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendents of the vacuum. We show that the genus one partition function and characters for simple ordinary modules must satisfy modular linear differential equations. We show the rationality of the central charge and module lowest weights, modularity of solutions, the dimension of each graded space is a rational function of the central charge and that the lowest weight primaries generate the algebra. We also discuss conditions on the reducibility of the lowest weight primary vectors as a module for the automorphism group. Finally we analyse solutions for exceptional vertex operator algebras with primary vectors of lowest weight up to 9 and for vertex operator superalgebras with primary vectors of lowest weight up to 17/2. Most solutions can be identified with simple ordinary modules for known algebras but there are also four conjectured algebras generated by weight two primaries and three conjectured extremal vertex operator algebras generated by primaries of weight 3, 4 and 6 respectively.Comment: 37 page

Three-dimensional AdS gravity and extremal CFTs at c=8m

We note that Witten's proposed duality between extremal c=24k CFTs and three-dimensional anti-de Sitter gravity may possibly be extended to central charges that are multiples of 8, for which extremal self-dual CFTs are known to exist up to c=40. All CFTs of this type with central charge 24 or higher, provided that they exist, have the required mass gap and may serve as candidate duals to three-dimensional gravity at the corresponding values of the cosmological constant. Here, we compute the genus one partition function of these theories up to c=88, we give exact and approximate formulas for the degeneracies of states, and we determine the genus two partition functions of the theories up to c=40.Comment: 17 pages, harvmac; v2: references added, version accepted in JHE

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and $n$-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szeg\"o kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all $n$-point functions in terms of a genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.Comment: A number of typos have been corrected, 39 pages. To appear in Commun. Math. Phy

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, ${\cal V}^\natural$, 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 $Z_n$ meromorphic orbifold constructions of ${\cal V}^\natural$ 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 ${\cal V}^\natural$ is unique, we then consider meromorphic orbifoldings of ${\cal V}^\natural$ and show that Monstrous Moonshine holds if and only if the only meromorphic orbifoldings of ${\cal V}^\natural$ give ${\cal V}^\natural$ itself or the Leech theory. This constraint on the meromorphic orbifoldings of ${\cal V}^\natural$ therefore relates Monstrous Moonshine to the uniqueness of ${\cal V}^\natural$ in a new way.Comment: 53 pages, PlainTex, DIAS-STP-93-0

Torus n-Point Functions for R\mathbb{R}-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay's trisecant identity for elliptic functions.Comment: 50 page

Generalized Moonshine and orbifold constructions.

Conference paperA brief review is given of some of our recent work on Generalised Monstrous Moonshine using abelian orbifoldings of the Moonshine Module

Topics in perturbative quantum chromodynamics

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Estrategias de diseño instruccional en los MOOC : análisis de estudio de casos de la UNAM y la UNED

El presente estudio de caso exploratorio, tiene como objetivo identificar las estrategias instruccionales consideradas por los equipos pedagógicos de la UNAM y la UNED, en el diseño de siete cursos masivos abiertos en línea (MOOC, por sus siglas en inglés). Se identificaron cuatro aspectos, los cuales sirven de fundamento para el desarrolló la investigación: flexibilidad, conocimiento colectivo, emociones, evaluación-retroalimentación. La metodología empleada fue mixta, cualitativa-cuantitativa, utilizándose la entrevista y el cuestionario como instrumentos para obtener los datos. Los resultados mostraron que las estrategias relacionadas a la flexibilidad fueron parcialmente utilizadas privilegiando así la navegación lineal en los cursos. Por otro lado, la mayoría de los miembros de los equipos pedagógicos emplearon las estrategias de organización de comunidad de aprendizaje mientras que las de generación fueron poco utilizadas. El aspecto emocional fue tomado en cuenta, sin embargo se pudo observar que su empleo aún no se generaliza para ambos contextos universitarios. En lo que se refiere a las estrategias de evaluación y retroalimentación, se recomienda fortalecer el acompañamiento que se da a los participantes. En particular se identificó que en la UNAM existe libertad de cátedra, la cual se extiende al diseño de los MOOC, mientras que en la UNED, se trabaja con criterios y especificaciones metodológicas pre-establecidas. Dicha situación genera diferentes experiencias para los profesores y alumnos que participan en los cursos de ambas instituciones.La présente étude de cas exploratoire a pour objectif d’identifier les stratégies instructionnelles considérées par les équipes pédagogiques de l’UNAM et de l’UNED, qui ont participé à l’élaboration de sept cours en ligne, ouverts et massifs (CLOM ou MOOC en anglais). Nous avons décidé de focaliser notre recherche sur 4 aspects instructionnels à savoir, la flexibilité, les connaissances collectives, les émotions, l’évaluation et le feedback. La méthodologie employée a été à la fois qualitative et quantitative, au moyen d’une enquête et d’un questionnaire pour obtenir les données. Nous avons constaté que les stratégies liées à la flexibilité étaient partiellement utilisées au profit d’une navigation linéaire dans les cours. Par ailleurs, la plupart des membres de l’équipe pédagogique préfèrent se focaliser sur les stratégies d’organisation de communauté d'apprentissage plutôt que sur leur génération. L'aspect émotionnel a également été pris en compte mais à des degrés différents, selon le contexte universitaire. En ce qui concerne les stratégies d’évaluation et feedback, il conviendrait de renforcer l’accompagnement des participants afin qu’ils puissent mieux réaliser cette tâche. Nous avons observé qu’à l’UNAM, il existe une liberté de chaire qui s’étend jusqu’aux MOOC, tandis qu’à l’UNED, l’on travaille à partir de critères et de spécificités méthodologiques pré-établies. Cette situation permet aux professeurs et aux élèves qui participent aux cours dans les deux institutions, d’avoir des expériences différentes.Este estudo de caso exploratório tem como objetivo identificar as estratégias instruccionais consideradas pelas equipas pedagógica da UNAM e da UNED que participaram do design de sete cursos massivos abertos em linha (MOOC). Foram identificados quatro aspectos instruccionais em que a pesquisa foi desenvolvida: flexibilidade, conhecimento coletivo, emoções e retroalimentação-avaliação. A metodología usada foi mista qualitativa-quantitativa. A entrevista e o questionário foram utilizados como ferramentas para obtenção dos dados. Os resultados mostraram que as estratégias relacionadas com a flexibilidade foram parcialmente utilizadas pelas equipas pedagógicas, que privilegiaram os cursos de navegação linear. Por outro lado, ainda que a maioria dos professores utilizasse estratégias de organização de comunidade de aprendizagem, as estratégias didáticas de geração de conhecimento em fóruns foram pouco utilizadas. O aspecto emocional foi levado em conta, no entanto observou-se que seu emprego ainda não pode ser generalizado para ambos contextos universitários. Quanto às estratégias de avaliação e retroalimentação, será conveniente fortalecer o acompanhamento que foi dado aos alunos para que possam realizar melhor esta tarefa. Em particular, identificou-se que na UNAM existe liberdade de cátedra, o que é válido também para o desenho dos MOOC. Enquanto a UNED trabalha com critérios e especificidades metodológicas para todos os cursos. Esta diferença metodológica gera diferentes experiências para os professores e os alunos que participam dos cursos de cada uma das instituições.This exploratory case study aims to identify the instructional strategies considered by the teaching staff of the UNAM and UNED who participated in seven Massive Open Online Course (MOOC). We identified four instructional aspects on which we developed our research: flexibility, collective knowledge, emotions, evaluation and feedback. The methodology used was mixed, qualitative-quantitative, using interview and questionnaire as instruments to obtain data. It was found that the strategies related to flexibility were partially used privileging the linear navigation in the courses. On the other hand, most of the members of the teaching team prefer to focus on the organizational strategies of learning community than on their generation. The emotional aspect was also taken into account but at different levels according to the university. With regard to the strategies of assessment and feedback, it would be desirable to strengthen the accompaniment that is given to the participants so that they can better perform this task. In particular we identified that at UNAM, there is an academic freedom that extends to the design of the MOOC, while in UNED people work with some pre-established criteria and specifications. This situation generates different experiences for teachers and students who participate in these courses

Inclusive production of heavy quarkonia

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