Math 263A: Affine Lie Algebras and Modular Forms (Fall 2020)

This class will meet Tuesdays and Thursdays from 12:30-1:50

About half the course will be devoted to general Kac-Moody Lie algebras and their representation theory. These are infinite-dimensional Lie algebras that were discovered independently in the 1970's by Victor Kac and Robert Moody. A highlight of their theory is the generalized Weyl Character formula.

After the general theory is developed we will turn to affine Lie algebras and modular forms.

After the finite-dimensional simple Lie algebras, the most important class is the affine Lie algebras. These arose in mathematical physics, particularly conformal field theory. Remarkably, the characters, and partial characters called string functions are modular forms.

Given an affine Lie algebra and a level, a finite set of theta functions is singled out, corresponding to the fields of a WZW conformal field theory. Thus there is an operation on these theta functions called fusion and remarkably the modular transformation $\tau\mapsto -1/\tau$ as it effects these theta functions gives the "S-matrix" that encodes the deeper aspects of the fusion ring.

A primary reference will be Kac's book Infinite-dimensional Lie algebras. The material on modular forms is in Chapter 13, which is closely related to the 1984 paper of Kac and Peterson. Other useful material may be found in the book Conformal Field Theory by Di Francesco, Mathieu and Senechal, Chapters 10 and 14-17.

Zoom and Canvas

Use Canvas to obtain a Zoom link to the lectures.

Lecture Notes

Lecture notes will be posted here starting September 15, 2020.

• Lecture 0: Modular Forms. Lecture of September 15, 2020. Modular forms, Theta Functions, Comparison with the Weyl Denominator.
• Lecture 1: Lie algebras with a triangular decomposition. Lecture of September 17. Triangular decompositions, Verma modules, Irreducible highest weight modules, review of finite-dimensional semisimple Lie algebras.
• Lecture 2: Finite-Dimensional Simple Lie Algebras. Lecture of September 22. Finite-dimensional simple Lie algebras, the affine Weyl group, central extensions.
• Lecture 3: Affine Lie Algebras. Lecture of September 24. Untwisted affine Lie algebras, the Weyl group, level.
• Lecture 4: Kac-Moody Lie Algebras. Lecture of September 29. Free Lie algebras; Kac-Moody Lie algebras
• Lecture of October 5, 2020: characters of Verma and integrable highest weight representations, BGG resolution.
• Lecture 5: Integrable representations. Lecture of October 7. The Serre relations; integrable representations; BGG Category $\mathcal{O}$.
• Lecture 6: The Casimir Operator. finite dimensional case; the Kac-Moody inner product; the Kac-Moody Casimir operator.
• Lecture of October 13. Invariant bilinear form; Casimir element.
• Lecture of October 15. Casimir element (continued).
• Lecture 7: The Kac-Weyl character formula.
• Lecture of October 20 Kac-Weyl formula I.
• Lecture of October 22 Kac-Weyl formula II.
• Lecture of October 27 Kac-Weyl formula concluded. Affine Lie algebras.
• Lecture of October 29. Integrable affine representations.
• Lecture of November 3. Affine Weyl group identities.
• Lecture of November 5. Macdonald Identities.
• Lecture of November 10. Modular Forms, Theta Functions.
• Lecture of November 12. Characters and string functions as theta functions.
• Lecture of November 17. Characters and string functions as modular forms (continued). The S-matrix
• Lecture of November 19. The Fusion Category as a modular tensor category.