DescriptionA vertex operator algebra (VOA) is an infinite-dimensional vector space V with infinitely many heavily constrained bilinear products. It is an algebra of vertex operators, which are formal differential operators used by physicists as quantum fields that describe the creation and propagation of strings. In mathematics, vertex operators provide realisations of affine Kac-Moody algebras, among other things. The construction of some vertex operators is associated with lattices, and therefore a connection with modular functions is to be expected.A famous VOA is the Monster VOA, which has the Monster group (largest sporadic finite simple group) as its group of symmetries. It is also called the Moonshine module, as it is central in the mathematical framework that explains an observation made by the mathematician John McKay in 1978, namely that the coefficients of one of the best studied function in number theory (the j-function, which is modular for the group SL(2,Z)) may be decomposed in dimensions of irreducible representations of the Monster group. This apparent connection between Number Theory and Group Theory - later coined `Monstrous Moonshine - was totally unexpected, yet it led to a Fields Medal being awarded to the mathematician Richard Borcherds in 1998 for his use of VOAs in his proof of the Conway-Norton Moonshine conjecture. You can listen to one of his expository talks on Monstrous Moonshine to get some idea. In 2010, another Moonshine phenomenon was observed by three Japanese mathematical physicists (Eguchi, Ooguri and Tachikawa), this time relating a string theory involving the well-known 4-dimensional K3 surface and the sporadic group `Mathieu 24'. Intriguingly, and despite the relentless efforts of a small but dedicated community of Moonshine mathematicians and physicists, the construction of the VOA for Mathieu Moonshine is still missing. Since 2010, other instances of Moonshines and generalised Moonshines have been uncovered, but their significance and relevance in theoretical physics remain elusive. Here is an outreach article on new Moonshines . In this project, you will be expected to learn the basics of VOAs and how they relate to Monstrous Moonshine and Mathieu Moonshine. Subsequently, you will be given the opportunity to expand on particular aspects of VOAs and/or Moonshine according to your interests.
PrerequisitesAlgebra II and Complex Analysis II are prerequisites for this project. In particular, you should have a good understanding of the basics of group theory taught in the second term of Algebra II. Representation Theory IV and Topics in Algebra and Geometry IV as co-requisites might enable you to push the project in more advanced directions.Resources- T. Gannon, Monstrous Moonshine: the first 25 years.- T. Gannon, Moonshine beyond the monster, Cambridge University Press(2006); available online through DU Library. - T. Gannon, Monstrous Moonshine and the classification of CFT. - M. Gaberdiel and R. Volpato, Mathieu Moonshine and orbifold K3s. Further resources will be made available to those choosing the project. |
email: Anne Taormina