1. Kontsevich-Segal formulation of two-dimensional conformal field theory.

2. Modular functors and weakly conformal field theories.

3. Vertex operator algebras and their geometry.

4. Operator product expansion and Intertwining operator algebras.

5. Modular invariance.

6. Verlinde formula and modular tensor categories.

7. Genus-zero and genus-one full conformal field theories.

8. Open-closed conformal field theories.

9. Main unsolved problems.

Prerequisites: First year graduate courses in algebra and analysis.

Text: No text book. Lectures are based on many papers that are available online. I will give links below to the papers relevant to the lectures. I also plan to write a book on two-dimensional conformal field theory based on the notes for this course. Here is a tentative table of contents of the book:

Two-dimensional conformal field theory

Papers relevant to the lectures:

1. G. Segal, The definition of conformal field theory, in Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, ed. U. Tillmann, London Mathematical Society Lecture Note Series, Vol. 308, Cambridge University Press, 2004.

   pdf file at University of Oxford

2. Y.-Z. Huang, Riemann surfaces with boundaries and the theory of vertex operator algebras, in: Vertex Operator Algebras in Mathematics and Physics, ed. S. Berman, Y. Billig, Y.-Z. Huang and J. Lepowsky, Fields Institute Communications, Vol. 39, Amer. Math. Soc., Providence, 2003, 109--125.

   arXiv:math/0212308

   pdf file

3. G. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Comm. Math. Phys. 123 (1989), 177-254.

   Scanned pdf file (KEK scanned document)

4. G. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Comm. Math. Phys. 123 (1989), 177-254.

   Scanned pdf file (KEK scanned document)

5. Y.-Z. Huang, Geometric interpretation of vertex operator algebras, Proc. Natl. Acad. Sci. USA 88 (1991), 9964--9968.

   pdf file at the journal web site

6. Y.-Z. Huang, Generalized rationality and a Jacobi identity for intertwining operator algebras, Selecta Math. 6 (2000), 225--267.

   arXiv:q-alg/9704008

7. Y.-Z. Huang, Differential equations and intertwining operators, Comm. Contemp. Math. 7 (2005), 375--400.

   arXiv:math/0206206

   pdf file

8. Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I, Selecta Mathematica 1 (1995), 699-756.

   arXiv:hep-th/9309076

9. Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, III, J. Pure Appl. Alg. 100 (1995), 141--171.

   arXiv:q-alg/9505018

10. Y.-Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Alg. 100 (1995), 173--216.

    arXiv:q-alg/9505019

11. Y.-Z. Huang, Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories, in: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov, Contemporary Math., Vol. 202, Amer. Math. Soc., Providence, 1997, 335--355.

    arXiv:q-alg/9512024

12. Y.-Z. Huang, Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649--706.

    arXiv:math/0303049

    pdf file

13. Y.-Z. Huang, Vertex operator algebras and the Verlinde conjecture, Comm. Contemp. Math. 10 (2008), 103--154.

    arXiv:math/0406291

    pdf file

14. Y.-Z. Huang, Rigidity and modularity of vertex tensor categories, Comm. Contemp. Math. 10 (2008), 871--911.

    arXiv:math/0502533

    pdf file

15. Y.-Z. Huang and L. Kong, Full field algebras, Comm. Math. Phys. 272 (2007), 345--396.

    arXiv:math/0511328

    pdf file

16. Y.-Z. Huang and L. Kong, Modular invariance for conformal full field algebras, Trans. Amer. Math. Soc. 362 (2010), 3027--3067.

    arXiv:math/0609570

    pdf file

17. Y.-Z. Huang and L. Kong, Open-string vertex algebras, , tensor categories and operadsComm. Math. Phys. 250 (2004), 433--471.

    arXiv:math/0308248

    pdf file