Learning Seminar

This learning seminar will mainly explore the physical origin of mirror symmetry (MS) and Strominger-Yau-Zaslow (SYZ) construction of mirror Calabi-Yau 3-folds. There will be roughly 10-12 sessions in Winter term II. 

Schedule (Meeting @ Friday 13:05-13:55 MATH126)

1. Jan. 09 - Organization meeting

2. Jan. 16 - Peilin - [Gross01] Classical Geometry of Calabi-Yau Manifolds Part I 

Calabi-Yau n-folds, Complex structure, Universal deformation, Infinitesimal deformation, Integrability condition, Maurer-Cartan equation.

3. Jan. 23 - Peilin - [Gross01] Classical Geometry of Calabi-Yau Manifolds Part II

Tangent space of Def(X), Bogomolov-Tian-Todorov theorem. Kodaira-Spencer map.

4. Jan. 30 - Peilin -  [Gross01] Classical Geometry of Calabi-Yau Manifolds Part III (Written Note)

K\"ahler cone, 1st Chern class, Bogomolov decomposition theorem, Complexified Kahler moduli, Outline of Physical story.

5. Feb. 06 - Peilin - [HKKPTVVZ] Physical Origin of MS Part I

Path-Integral, Correlation functions, N=(2,2) supersymmetry, Chiral and twisted chiral fields.

6. Feb. 13 - Peilin - [HKKPTVVZ] Physical Origin of MS Part II

Axial anomaly, N=(2,2) SUSY-invariant theories and algebra, Statement of MS, (Twisted-)Chiral operators, Q_A/Q_B-cohomology.

7. Feb. 20 - BREAK - NO TALKS :)

8. Feb. 27 - Peilin - [HKKPTVVZ] Physical Origin of MS Part III (Written Note)

A-/B-Twist,  A-model/B-model, Localization principle, Holomorphic maps in A-model, Constant maps in B-model.

9. Mar. 06 - (Different room @ MATH 105) - Peilin - [Gross01] J-holomorphic curves and GW-invariants Part I

Motivation, J-holomorphic curves, Moduli as zero locus of section \Bar{\partial}_{J}, Simple curves, Regularity/Transversality.

10. Mar. 13 - Peilin - [Gross01] J-holomorphic curves and GW-invariants Part II (Written Note)

Upcoming talks

1. [Gross01] Variation and Degeneration of Hodge Structures 

2. [Gross01] A Mirror Conjecture and Quintic Hypersurface Example [CdGP]

3. [SYZ] and Classical Results on Semi-Flat Case

4. PDE and Analysis (TBD)

Tentative list of papers and materials (To be modified)

1. [CdGP] Candelas, P., Xenia, C., Green, P.S. and Parkes, L., 1991. A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Physics B, 359(1), pp.21-74. 

2. [SYZ] Strominger, A., Yau, S.T. and Zaslow, E., 1996. Mirror symmetry is T-duality. Nuclear Physics B, 479(1-2), pp.243-259. 

3. [Gross01] Gross, M., 2001. Calabi-Yau Manifolds and Mirror Symmetry. Part II in Calabi-Yau Manifolds and Related Geometries, Lectures at a Summer School in Nordfjordeid, Norway.  

[Gross12] Gross, M., 2012. Mirror symmetry and the Strominger-Yau-Zaslow conjecture. arXiv preprint arXiv:1212.4220.

4. [Joyce03]  Joyce, D. 2003. Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications. 

5. [GW00] Gross, M. and Wilson, P.M.H. 2000. Large Complex Structure Limits of K3 Surfaces.

More analysis/PDE papers:

6. [CJL12] Collins-Jacob-Lin: Non-compact SYZ (https://arxiv.org/abs/2012.05416)

7. [Li19] Yang Li: SYZ for Fermat family (https://arxiv.org/abs/1912.02360)

Tentative list of topics (To be modified)

1. Physical origin of mirror symmetry from string theory and D-branes.

2. [CdGP] paper on Quintic CY3s.

3. [SYZ] paper on Conjectured geometric construction of mirror CY3s.

4. Gross's survey on Mirror symmetry [Gross01]  and SYZ conjecture [Gross12] .

Tentative list of books (To be modified)

1. [HKKPTVVZ] Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R. and Zaslow, E. 2003. Mirror Symmetry. Clay Mathematics monographs, Volume 1.

2. [ABCDGKSSW] Aspinwall, P.S., Bridgeland, T., Craw, A., Douglas, M.R., Gross, M., Kapustin, A., Moore, G.W., Segal, G., Szendr\"oi, B. and Wilson, P.M.H. 2009. Dirichlet Branes and Mirror Symmetry. Clay Mathematics monographs, Volume 4.

3. [CK] Cox, D.A., and Katz, S. Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, Volume 68.

Further plan (To be modified)

1. SYZ conjecture: Semi-flat case and Classical results (1-2 Session)

Special Lagrangians [McLean], Legendre transformation [Hitchin], Fourier-Mukai transformation (HMS). 

2. Introduction to toric geometry (1 Session)

Cones and fans, Polytopes, K\"ahler cones, Symplectic geometry, Fano varieties, Reflexive polytopes, Automorphism.

3. SYZ on toric CY3s (3-4 Session)

Batyrev construction, Quintic example, Toric complete intersections, Voisin-Borcea construction.