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Many basic results and techniques in complex analysis play fundamental
roles in the representation theory of infinite-dimensional Lie
algebras and vertex operator algebras and in its applications
in mathematics and physics. The complex analysis method should be
mastered by every student interested in this representation theory
and its applications. I will discuss the following topics in this course:
- The complex analytic and geometric formulations of vertex operator
algebras, modules, intertwining operators, chiral and full conformal
field theories.
- Some results in complex analysis that play important roles in
the representation theory of infinite-dimensional Lie algebras and
vertex operator algebras.
- Some theorems in the representation theory whose formulations
and proofs need the formulations presented in Topic 1 and /or those
results discussed in Topic 2 above. This part of the material will
be chosen based on the interests of the students. Besides some basic
results, possible topics to be discussed include wave functions for
quantum Hall systems and intertwining operators, central charges and
the determinant line bundle, open-string vertex operator algebras and
D-branes, modular functors and full field algebras, open-closed
conformal field theories.
Prerequisites: First year graduate courses in algebra and analysis.
Text: There is no single text for this course. The material will
be from various research monographs and papers.
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