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Orbifold conformal field theories play an particularly important role in conformal field theory and in the applications of conformal field theory. The moonshine module vertex operator algebra constructed by Frenkel-Lepowsky-Meurman is the first example of orbifold conformal field theories. The recent results on the construction of self-dual (or holomorphic) vertex operator algebras of central charge 24 and on the generalized moonshine depend heavily on (abelian) orbifold conformal field theory.
This is an introductory course on orbifold conformal field theory.
1. Vertex operator algebras and examples.
2. Modules for vertex operator algebras and examples.
3. Twisted modules for vertex operator algebras and examples.
4. Intertwining operators and twisted intertwining operators.
5. The orbifold theory conjecture and the existing results.
6. The moonshine module as an orbifold theory.
7. G-crossed tensor categories.
Prerequisites: Basic courses in algebra and analysis. I will start from the very beginning of the theory of vertex operator algebras.
Text: No textbook. Papers and chapters in books will be discussed in the classes.
Papers relevant to the lectures:
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I. B. Frenkel, J. Lepowsky and A. Meurman,
Vertex Operator Algebras and the Monster,
Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988.
- Yi-Zhi Huang,
A nonmeromorphic extension of the moonshine module vertex operator algebra, in: Moonshine, the Monster and related topics, Proc. Joint Summer Research Conference, Mount Holyoke, 1994, ed. C. Dong and G. Mason, Contemporary Math., Vol. 193, Amer. Math. Soc., Providence, 1996, 123--148.
arXiv:hep-th/9406190
- Yi-Zhi Huang,
Generalized twisted modules associated to general automorphisms of a vertex operator algebra, Comm. Math. Phys. 298 (2010), 265--292.
arXiv:0905.0514
- Yi-Zhi Huang, Two constructions of grading-restricted vertex (super)algebras, J. Pure
Appl. Alg. 220 (2016), 3628-3649.
arXiv:1507.06098
- Yi-Zhi Huang,
Some open probelms in mathematical two-dimensional conformal field theory, in: Proceedings of the Conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, held at University of Notre Dame, Notre Dame, Indiana, August 14-18, 2015, ed. K. Barron, E. Jurisich, H. Li, A. Milas, K. C. Misra, Contemp. Math, Vol. 695, American Mathematical Society, Providence, RI, 2017, 123--138.
arXiv:1606.04493
- Yi-Zhi Huang, Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms, J. Alg. 493 (2018), 346--380.
arXiv:1702.05845
- Yi-Zhi Huang, Twist vertex operators for twisted modules, to appear.
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- Yi-Zhi Huang, A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra, to appear.
pdf file
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