Math 290 - Seminar on Mirror Symmetry - Spring 2011

D. Auroux - Thu., 5-6:30pm, Room 891 Evans

What's happening

• 1/20: Introduction: the SYZ and HMS conjectures (D. Auroux)
• 1/27: no meeting
• 2/3: Lagrangian Floer homology (R. Vianna) [Notes]
• 2/10: Product structure on Floer homology; twisted coefficients (R. Vianna) [Notes]
• 2/17: Special Lagrangians and their deformation theory; affine structure on the moduli space (Z. Sylvan) [Notes]
• 2/24: The complexified moduli space of special Lagrangians (uncorrected SYZ mirror); example: T^2 (Z. Sylvan) [Notes]
• 3/3: Homological mirror symmetry: the example of T^2 (after Polishchuk-Zaslow) (D. Pomerleano) [Notes]
• 3/10: Holomorphic discs and obstruction in Floer theory (V. Gripp) [Notes]
• 3/17: Obstruction in Floer theory continued: the example of S^2 (V. Gripp) [Notes]
• 3/31: The mirror of CP^1 (K. Lin)
• 4/7: SYZ for toric Fano manifolds (D. Auroux)
• 4/14: no meeting
• 4/21: A nontoric example: SYZ with corrections (K. Choi)
• 4/28: Introduction to Fukaya-Seidel categories (D. Auroux)

Outline

This seminar will present various geometric aspects of mirror symmetry. The main focus will be on the Strominger-Yau-Zaslow (SYZ) conjecture, according to which mirror symmetry is a duality between Lagrangian torus fibrations, and on Kontsevich's homological mirror symmetry (HMS), which predicts an equivalence between the Fukaya category of a symplectic manifold and the derived category of coherent sheaves of its mirror.

Main topics:

• Introduction: statement of the SYZ and HMS conjectures.
• (Special) Lagrangian torus fibrations and their geometry; moduli spaces of special Lagrangians and "uncorrected" SYZ mirror symmetry.
• Lagrangian Floer homology and the Fukaya category; HMS for the elliptic curve.
• Holomorphic discs and obstruction; the example of P^1.
• Landau-Ginzburg models, superpotentials.
• Mirror symmetry for toric Fano varieties.
• Wall-crossing and instanton corrections: a simple non-toric example.
• Corrected SYZ mirror symmetry; HMS for pairs.
• If time allows: mirror symmetry for pairs of pants and higher genus curves (SYZ and HMS).

Prerequisites: some prior knowledge of symplectic and complex geometry will be helpful.

References

The (handwritten) notes from my Fall '09 topics course will come in handy (especially the second half of the course). Other references:

• R. P. Thomas, The geometry of mirror symmetry, Encyclopedia of Mathematical Physics, Elsevier, 2006, pp. 439-448; arXiv:math.AG/0512412
• A. Polishchuk, E. Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2 (1998), 443-470; arXiv:math.AG/9801119
• C. H. Cho, Y. G. Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), 773-814; arXiv:math.SG/0308225
• D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. 1 (2007), 51-91; arXiv:math.SG/0706.3207
• D. Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, Surveys in Differential Geometry 13 (2009), 1-47; arXiv:math.SG/0902.1595
• M. Gross, D. Huybrechts, D. Joyce, Calabi-Yau manifolds and related geometries, Lectures from the Summer School held in Nordfjordeid, June 2001, Universitext, Springer, 2003.