Math 581E
Geometric Introduction to Mirror Symmetry

Charles Doran and Amer Iqbal

Autumn 2006, Monday/Wednesday/Friday, 12:30-1:20
The study of Mirror Symmetry represents perhaps the most active and broad-reaching interaction between pure mathematics and theoretical physics in recent years. Mirror symmetry, a set of evolving conectures coming from string theory, suggests above all a surprising connection between symplectic and algebraic geometry. Varius questions about the geometry of Calabi-Yau manifolds (and even other complex manifolds and algebraic varieties) can be answered by studying their "mirror partners". These conjectures are at once concrete -- involving very explicit and approachable computations which require relatively little background to appreciate -- and very deep.

The aim of this course is to provide a comprehensive introduction to both physical and mathematical mirror symmetry from a geometric point of view. Throughout the course we will not only try to develop the physical intuition behind the approach of Hori-Vafa, but will alo provide a rigorous mathematical treatment of the Batyrev-Borisov description of mathematical mirror pairs. One important aspect of the course will be the many explicit examples which will be worked out to explain the concepts. In this regard the "Superseminar" (a seminar organized by some of the graduate sutdents in the department) will be used on a parallel track to cover some of these examples in more detail.

Topics:

  1. Review of Toric geometry
  2. Reflexive polytopes
  3. Two dimensional conformal field theory
  4. Compact Calabi-Yau hypersurfaces
  5. Sigma models with Calabi-Yau target spaces
  6. Batyrev-Borisov proposal for mirror pairs
  7. Topological twisting and A/B models
  8. Hori-Vafa proposal for mirror symmetry
  9. Periods for non-compact Calabi-yau manifolds
  10. Periods for compact Calabi-yau manifolds
  11. Curve counting (Target space/D-brane point of view)
  12. K-theoretic mirror symmetry conjecture

Prerequisites: Some basic knowledge of algebraic geometry (at the level of the two-quarters introductory course) OR familiarity with string theory is sufficient. This course is designed to be accessible to both mathematics and physics graduate students.