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. Many results on the construction and classification 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. The nonabelian orbifold conformal field theory still needs to be developed.
This is an introductory course on the representation theory of vertex operator algebras and orbifold conformal field theory. The topics covered in the course include:
- Vertex operator algebras and examples.
- Modules for vertex operator algebras and examples.
- Twisted modules for vertex operator algebras and examples.
- Construction of modules and twisted modules.
- Intertwining operators and twisted intertwining operators.
- The orbifold theory conjectures.
- The construction of tensor product bifunctors.
- The moonshine module as an orbifold theory.
Text: No textbook. Papers and chapters in books will be discussed in the classes.
Papers, books and lecture notes to be used in the lectures:
- Y.-Z. Huang, Lecture notes on vertex algebras and quantum vertex algebras, 96 pages on April 26, 2020, for the graduate course "Math 555: Selected Topics in Algebra: Vertex algebras and quantum vertex algebras," Spring, 2020.
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- Y.-Z. Huang,
Generalized twisted modules associated to general automorphisms of a vertex operator algebra, Comm. Math. Phys. 298 (2010), 265--292.
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- Y.-Z. Huang, Twist vertex operators for twisted modules, J. Alg., 539 (2019), 53--83.
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- Y.-Z. Huang, A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra, 39 pages, Comm. Math. Phys. 377 (2020), 909-945.
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- Generators, spanning sets and existence of
twisted modules for a grading-restricted vertex
(super)algebra, 41 pages, Selecta Math. 26 (2020), Paper no. 62.
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- Lower-bounded and grading-restricted twisted modules
for affine vertex (operator) algebras, 36 pages, J. Pure Appl. Alg. 225 (2021), Paper no. 106618.
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- Intertwining operators among twisted modules
associated to
not-necessarily-commuting automorphisms, J. Alg. 493 (2018), 346--380.
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- Representation theory of vertex operator algebras and orbifold conformal field theory, in: Lie groups, number
theory, and vertex algebras, ed. by D. Adamovic,
A. Dujella, A. Milas and P. Pandzic,
Contemp. Math., Vol. 768, Amer. Math. Soc., Providence, RI, 2021, 221–252.
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- 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.
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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, Two constructions of grading-restricted vertex (super)algebras, J. Pure
Appl. Alg. 220 (2016), 3628-3649.
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