MAT 1739F
Instructor: K. Hori

Mirror symmetry plays a central role in the study of geometry of string theory.
In mathematics, it reveals a surprising connection between symplectic geometry and algebraic geometry.
In physics, it provides a conceptual guide as well as powerful computational tools, especially in compactifications to four-dimensions.

Outline:
1. Background:
• Supersymmetry and homological algebra
• Non-linear sigma models (NLSM)
• Landau-Ginzburg models
• topological field theory and topological strings
2. Linear sigma models, moduli space of theories
3. Mirror Symmetry
4. Mirror Symmetry involving D-branes
* B-branes in NLSM - holomorphic bundles, coherent sheaves
* B-branes in LG models - level sets, matrix factorizations
* A-branes in NLSM - Lagrangian submanifolds, Floer homology
* A-branes in LG models - vanishing cycles and Picard-Lefschetz monodromy

References:
The course does not follow a textbook but the following may be useful.
1. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil and E. Zaslow, ``Mirror Symmetry'' Clay Mathematics Monographs Vol 1 (AMS, 2003).
2. P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison, E. Witten, ``Quentum Fields and Strings: A Course for Mathematicians'' (AMS 1999).