Course Descriptions

16:640:554 - Selected Topics in Algebra

James Lepowsky

Subtitle:

Monstrous Moonshine and Generalized Moonshine

Text: 

I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988.

Prerequisites: 

Interest in the themes mentioned in the description.

Description: 

In the late 1970s, conjectures called "Monstrous Moonshine" remarkably connected the Monster finite simple group, which at the time had not yet been proved to exist, to certain modular functions in number theory. A direct mathematical connection was made, in the listed text (a research monograph), by means of the construction of what in retrospect came to be understood as an "orbifold string theory." This will be a survey course, with no specific prerequisites, in which we will selectively discuss a wide range of aspects of this rich field, which deeply involves many fields of mathematics as well as string theory and conformal field theory in physics. We will outline the features and the structure of much of the theory, illustrating many features with concrete examples involving both mathematics and physics. Many important current research problems will be highlighted, and many references will be mentioned and discussed, for students interested in more details. Much of what we specifically focus on will depend on the interests of the students. 

Some specific topics will be:

Brief introduction to vertex operator algebra theory, with examples.

Sketch of "twisting" in vertex operator algebra theory and string theory.

Sketch of Borcherds's proof of the Conway-Norton conjectures for the Moonshine Module vertex operator algebra and relations with string theory.

Sketch of Carnahan's work on "Generalized Moonshine," which expands the connection with modular function theory.

Relations of all of the above with problems in theoretical physics.

Schedule of Sections: