Conformal Field Theory and Representations

This class meets Tuesdays and Thursdays 10:30-11:50 in 384H

Conformal Field Theory (CFT) is a branch of physics with origins in solvable lattice models and string theory. But the mathematics that it has inspired has applications in pure mathematics in modular forms, representation theories of various infinite-dimensional Lie algebras and vertex algebras, Monstrous Moonshine, geometric Langlands theory, knot theory and topological quantum computation. (Of course we will not cover all these applications.)

This is a large subject and we will have to be very selective. Our guiding principle will be to develop enough to see how the representation theories of two classes of infinite-dimensional Lie algebras applies. These are the Virasoro and affine Lie algebras.

We will discuss unitary representations of the Virasoro algebra and integrable highest-weight representations of affine Lie algebra, and relate these back to CFT.

Some references for the course are:

• Schottenloher, A mathematical introduction to Conformal Field Theory
• [DMS] Di Francesco, Mathieu and Senechal, Conformal Field Theory
• Kac and Raina, Bombay Lectures on Infinite-Dimensional Lie Algebras
• [FBZ] Frenkel and Ben-Zvi, Vertex Algebras and Algebraic Curves
• [BPZ] Belavin, Polyakov and Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984).
• A useful summary article "Introduction to conformal field theory and infinite-dimensional algebras" by David Olive may be found in the book Physics, Geometry and Topology, H.C. Lee (ed.), 1990.

All of these texts are available on-line through the Stanford libraries. Unfortunately Kac's book Vertex Algebras for Beginners, which we follow in Lecture 12 does not seem to be available on-line. But you can get the first few pages, where the relationship with CFT is explained, through Google Books.

I also recommend the lectures on YouTube by Tobias Osborne. These are mainly based on the Notes of Paul Ginsparg.

For vertex algebras, there is a good book by Frenkel and Ben-Zvi called Vertex Algebras and Algebraic Curves, also available on-line through the Stanford Libraries.

Lecture Notes

• Lecture 1: Review of Quantum Mechanics. Operators on Hilbert Space; Quantum Mechanics; The Path Integral.
• Lecture 2: Conformal mappings. Orthogonal Groups; Conformal Completion and Global Conformal mappings; Local conformal transformations and the Witt Lie algebra.
• Lecture 3: Quantum Field Theory. Central Extensions and Projective Representations; Wightman Axioms of CFT.
• Lecture 4: Representation Theory. Highest weight representations; Unitary Representations of the Virasoro Algebra.
• Lecture 5: The Free Boson. Review: the Wightman axioms; The Free Boson.
• Lecture 6: Affine Lie Algebras Kac-Moody Lie Algebras; (Untwisted) Affine Kac-Moody Lie Algebras; Representations and modular forms; Sage methods.
• Lecture 7: Correlation Functions and Wick Rotation Tube domains; Correlation Functions.
• Lecture 8: Vertex Algebras Locality; Vertex Algebras; The Heisenberg Vertex Algebra.
• Lecture 9: More about Vertex Algebras. Review of Vertex Algebras; Normal ordering; the Heisenberg VA continued.
• Lecture 10: Operator Product Expansions and Associativity. Euclidean CFT; OPE in Vertex Algebras.
• Lecture 11: OPE and Associativity for Vertex Algebras. Overview; Preliminaries; Proof of Associativity.
• Lecture 12: From CFT to VA. Overview; Conformal Field Theories; Chiral Algebras.
• Lecture 13: Primary Fields and the Virasoro Algebra. Overview; Quasi-Primary Fields; Primary Fields and $\mathbf{Vir}$; The Energy-Momentum tensor.
• Lecture 14: Virasoro Vertex Algebras. Conformal Vertex Algebras, Reconstruction Theorem for Vertex Algebra, Verma Modules and Virasoro VA.
• Lecture 15: Virasoro Discrete Series. Primitive vectors; The Kac Determinant Formula.
• Lecture 16: Minimal Models I. Singular vectors; Minimal Models I.
• Lecture 17: Fusion for Minimal Models. Differential equations for correlation functions; Singular conformal families; Fusion.
• Lecture 18: Minimal Models II. Fusion of degenerate fields; The simplest minimal model $M(4,3)$; General minimal models; The Ising Model.
• Lecture 19: Modularity and the Partition Function. Rational CFT; Theta functions; Automorphicity of the Partition function.
• Lecture 20: WZW CFT.

I will not assume much physics background. Some knowledge of Lie algebras will be helpful.

Uses of CFT

In physics CFT and vertex operators arise in different contexts:

• String Theory (2d Lorentzian theories)
• Statistical Mechanics (2d Euclidean theories)
• Quantum Hall Effect, Soliton theory etc.

In mathematics the avatar of CFT is the vertex algebra. This is an algebraic structure that might have remained hidden were it not for physical motivation. Vertex operators and vertex algebras have many applications.

• Monstrous Moonshine
• Geometric Langlands Theory
• Modular forms
• Representation theory, algebraic combinatorics, etc.

Conformal field theory led to Kac and I.Frenkel's study of the basic representation of an affine Lie algebra, which appeared in string theory. This representation, and more generally the highest weight integrable representations of affine Lie algebras, appeared in unexpected places. For example, thanks to Lascoux, Leclerc and Thibon, Misra and Miwa, Kleshchev, etc. its crystal basis is now understood to be closely related to the modular representation of the symmetric group. On the other hand, Kac and Peterson proved that the characters of these representations are modular forms, a fact discovered in connection with conformal field theory, where the primary fields of Wess-Zumino-Witten CFT are modular forms. These same fields have a composition law coming from the operator product expansions of CFT giving rise to a modular tensor category with applications to knot theory and deep connections to quantum groups. These modular tensor categories are also important in one approach to topological quantum computing.

Returning to the Kac-Frenkel description of the basic representation of affine Lie algebras by vertex operators, this description led to Borcherd's axiomatic description of vertex algebras leading to his solution of the Monstrous Moonshine conjectures. Vertex algebras continue to be important, occurring for example in one approach to the geometric Langlands theory.

In summary, conformal field theory has stimulated a remarkable amount of important mathematics, much of it very surprising and unexpected.