Calabi-Yau & Mirror Symmetry Seminar

One should never try to prove anything that is not almost obvious.
                         -- Alexander Grothendieck

Abstract and Introduction

• Basically this seminar goes 2 ways at the beginning:
• Mathematically, we come from the definition of Calabi-Yau manifolds and K3 surfaces, then go to moduli spaces of K3, last arrive at moduli space of sheaves on K3
• Physically, we start from the introduction to QFT, including SUSY and renormalization group flow, then come to BPS solitons and fundamental string theory and introduce D-branes.
• Then the 2 ways compose to the proof of mirror symmetry. The final goal of this seminar is homoloical mirror symmetry. Last we seek to learn some application of mirror symmetry, especially topological string theory.

References

• For Mathematics side:
• Daniel Huybrechts, Lectures on K3 surfaces. (PDF)
• Daniel Huybrechts, The Geometry of Moduli Spaces of Sheaves. (PDF)
• For Physics side:
• Clay Institute, Mirror Symmetry. (PDF)

Contributor

Rourou Ma, Yu Wu, Hefeng Xu, Yushu Zhu

Timetable

Basically we hold the seminar online via Tencent Meeting every Sunday 2000 CST/ 1400 CET
• Here is the lecture note which will be updated every week: NOTE (PDF)
• Week 1 (Mar.5, 2023)
  • Yushu Zhu
  • Introduction to C-Y manifolds and K3 surface.
  Refenrence
  • Arnaud Beauville, Complex Algebraic Surfaces (PDF)
• Week 2 (Mar.19, 2023) Postponed due to the speaker
  • Hefeng Xu
  • Introduction to QFT (I).
    Note
• Week 3 (Mar.29, 2023)
  • Yushu Zhu
  • K3 surface (I): up to constructing Picard group.
  Refenrence
  • Arnaud Beauville, Complex Algebraic Surfaces (PDF)
• Week 4 (Apr.9, 2023)
  • Hefeng Xu
  • Introduction to QFT (II): Asymptotic expansion and construction of Feynman diagrams.
  Refenrence
  Note
• Week 5 (Apr.30, 2023)
  • Yushu Zhu
  • K3 surface (II): More on line bundle and Picard group, up to Riemann-Roch Theorem
  Refenrence
  • A.Rosenschon, Lecture Note on Algebraic Surfaces (PDF)
  • Devlin Mallory, Complex Algebraic Surfaces (PDF)
  • Beltram, Note on Hilbert Scheme (PDF)
• Week 6 (May 1, 2023)
  • Hefeng Xu
  • Introduction to QFT (III): Stokes Phenomenon and Determinants.
  Refenrence
  E. Witten, arXiv: 1001.2933. Analytic Continuation Of Chern-Simons Theory
• Week 7 (May 14, 2023)
  • Yushu Zhu
  • Riemann-Roch Theorem: complete proof on K3 surfaces.
  Refenrence
  • A.Rosenschon, Lecture Note on Algebraic Surfaces (PDF)
  • Devlin Mallory, Complex Algebraic Surfaces (PDF)
  • Beltram, Note on Hilbert Scheme (PDF)
• Week 8 (May 28, 2023)
  • Hefeng Xu
  • Introduction to QFT (IV): Determinants in infinite dim theory and BRST introduction I : DGLA.
  Refenrence
  • S. Li, K. Zeng, Homotopy Algebra in Higher Spin Theory (PDF)
• Week 9 (May 31, 2023)
  • Hefeng Xu
  • Complementary for Introduction to QFT (III): Stokes Phenomenon and Determinants.
  Refenrence
  E. Witten, arXiv: 1001.2933. Analytic Continuation Of Chern-Simons Theory
• Week 10 (June 21, 2023)
  • Yushu Zhu
  • Riemann-Roch Theorem on K3 surfaces and Hodge Index Theorm.
  Refenrence
  • A.Rosenschon, Lecture Note on Algebraic Surfaces (PDF)
• Week 11 (Aug 19, 2023)
  • Hefeng Xu
  • DGLA and BRST algebra.
• Week 12 (Aug 27, 2023)
  • Hefeng Xu
  • BRST algebra w/ Feynman diagram representation. Introduction to Effective BV formalism.
  Refenrence
  • A.Rosenschon, Lecture Note on Algebraic Surfaces (PDF)

Useful Links

Here are some useful links to some other institutes, personnels and information. Hope these can help you out.

• Links (Click to expand)

  • BIMSA lecture: Calabi-Yau Manifolds in Algebraic Geometry
  • Prof. Si Li, YMSC, homepage
  • Mirror Symmetry on nlab
  • Lectures on Homological Mirror Symmetry in University of Edinburgh
  • Homological Mirror Symmetry Program HP of IAS
  • Syzygies and mirror symmetry at AMS
  • Note "Mathematical Quantum Field Theory" by U.Schreiber