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課程名稱 |
幾何與拓樸場論一
Geometry and Topological Field Theory (Ⅰ) |
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開課學期 |
110-1 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
王金龍 |
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課號 |
MATH5266 |
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課程識別碼 |
221 U9120 |
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班次 |
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學分 |
4.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一3,4(10:20~12:10)星期四3,4(10:20~12:10) |
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上課地點 |
天數101天數101 |
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備註 |
初選不開放。
總人數上限：20人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1101MATH5266_GTFT1 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
In this course, we will give the early details of “mirror symmetry” from both the mathematical and physical theories. More advanced topics are left to the second semester.
本課程採實體上課. 欲選修同學 (須符合課程要求) 請於 9/23 第一堂課當天找我領取選課授權碼. |
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課程目標 |
During the last 30 years, the “mirror symmetry” discovered by string theory proved to be one of the most fundamental breakthrough in the study of Calabi-Yau manifolds and 3-dimensional algebraic geometry. Many difficult problems are solved with the help of it, e.g. the counting curves problem. However, the full strength of it has not yet been fully understood. The major reason is that the full correspondence between physics and mathematics requires many new notions to be rigorously developed. The Clay Lecture by Vafa et al. was a serious attempt toward this goal. The main focus of this course is to go through the contour of it in detail from BOTH the Mathematical and Physical point of view. |
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課程要求 |
Prerequisites: Algebraic geometry (at least Riemann surfaces), differential geometry (tensors, differential forms, Riemannian geometry, curvature etc.), basics in calculus of variations (Lagrangian and Hamiltonian formalisms), general intersect in theoretic physics. |
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預期每週課後學習時數 |
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Office Hours |
備註： Welcome for more discussions. Fixed office hour Monday 2:00-3:30. You may also check other time with me if needed. |
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指定閱讀 |
MIRROR SYMMETRY, Vafa et al. Clay Mathematical Monographs, Vol. 1, 2003.
Note: Most of the contents can be made more transparent and rigorous with 參考書目. In particular the mathematical details can be found in the mathematics literature (with *). |
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參考書目 |
*1. Deligne et. al. eds, Quantum Fields and Strings Vol. I, II. AMS and IAS 1999
*2. D. Cox and S. Katz, Mirror symmetry and algebraic geometry, AMS 1999
*3. Y. Manin, Gauge field theory and complex geometry, Springer-Verlag 1988
4. S.-T. Yau ed., Essays on mirror manifolds 1991 (revised Mirror Symmetry I, Int. Press 1998)
5. R.P. Feymann and A.R. Hibbs, Quantum mechanics and path integrals, 1965
6. M. Green, J. Schwarz and E. Witten, Superstring theory, 1987
7. Polchinski, String theory I, II, Cambridge University Press 1998
8. A. Zee, Quantum field theory in a nutshell
Note: 1-3 are mathematics literature, 4-8 are physics literature. 8 is more elementary for people with not much physics background. |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Homework |
50% |
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2. |
Reports |
50% |
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週次 |
日期 |
單元主題 |
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第1週 |
9/23, 9/27 |
QFT in d = 0 (Ch.8 and 9) |
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第2週 |
9/30, 10/4 |
QFT in d = 1, quantum mechanics with SUSY (10.1-10.3) |
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第3週 |
10/7 |
QFT in d = 1, sigma models and instantons (10.4-10.5) |
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第4週 |
10/14, 10/18 |
QFT in 1 + 1 dim, free Bosonic scalar theory, sigma model (11.1-11.2) |
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第5週 |
10/21, 10/25 |
QFT in 1 + 1 dim, T-duality and Dirac Fermion (11.2-11.3) |
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第6週 |
10/28, 11/1 |
N = (2, 2) SUSY (Ch.12) and non-linear sigma models (13.1-13.2) |
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第7週 |
11/4, 11/8 |
RG flow (Ch.14) |
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第8週 |
11/11, 11/15 |
Chiral rings and TFT (Ch.16) |
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第9週 |
11/18, 11/22 |
Linear sigma models (Ch.15) |
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第10週 |
11/25, 11/29 |
tt*, BPS solitons, and D branes (Ch.17-19) |
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第11週 |
12/2 |
Introduction to physics "proof" of mirror symmetry (Ch.20) |
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第12週 |
12/6, 12/9 |
Moduli of stable maps and GW invariants (Ch.21-26) |
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第13週 |
12/13, 12/16 |
Localization on moduli space and quantum DE (Ch.27-28) |
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第14週 |
12/20, 12/23 |
Proof of the classical mirror conjecture (Ch.29-30) |
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第15週 |
12/30 |
Final reports I |
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第16週 |
1/6 |
Final reports II |
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第17週 |
1/13 |
Final reports III |
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