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Nonlinear sigma-models are a class of two-dimensional quantum
field theories from which physicists have obtained many deep
conjectures in topology and geometry. Unfortunately nonlinear
sigma-models are still not constructed mathematically. In this
course, I will cover the following topics:
- Nonlinear sigma-models
from a mathematical point of view; conjectures obtained by
physicists, including, in particular, those on Calabi-Yau manifolds
and elliptic genera.
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The mathematical construction of linear
sigma-models, that is, sigma-models with flat target spaces.
- My recent results in a program of constructing nonlinear
sigma-models; Laplacians and sheaves of meromorphic open-string
vertex algebras and modules on Riemannian manifolds.
- My recent results on Witten's Dirac-like operator on a loop
space and their connection with Witten genus.
Prerequisites: Algebra and analysis at the level of
first year graduate courses. Exposure to the theory of
vertex operator algebras and Riemnnian geometry will be
helpful but is not required.
Text: Some papers to be distributed in the classes, including, in particular:
- E. Witten, Elliptic genera and quantum field theory, Comm. Math.
Phys. 109 (1987), 525-536.
- B. Greene, String Theory on Calabi-Yau Manifolds, hep-th/9702155.
- Yi-Zhi Huang, Meromorphic open-string vertex algebras, arXiv:1204.1774.
- Yi-Zhi Huang, Meromorphic open-string vertex algebras and
Riemannian manifolds,arXiv:1205.2977.
- Some other papers.
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