鏡像對稱 (弦理論)

在代數幾何和理論物理中，鏡像對稱是指卡拉比-丘流形之間的一種特殊關係，即兩種卡丘流形雖然在幾何上差別很大，但是作為弦理論的額外維度時卻是等價的。這樣的一對流形被稱為鏡像流形。

鏡像對稱最早是由物理學家發現的。1990年左右，菲利普·坎德拉斯（英語：Philip Candelas）、齊妮婭·德·拉·奧薩（Xenia de la Ossa）、保羅·格林（Paul Green）和琳達·帕克斯（Linda Parks）發現它可以用於枚舉幾何，因此激發了數學家對此的興趣。枚舉幾何是研究幾何問題解的數量的數學分支。坎德拉斯和他的合作者證明了鏡像對稱可用於計算卡丘流形上有理曲線的數目，從而解決了一個長期的難題。儘管鏡像對稱最初的方法是從物理出發的，數學上並不嚴格，它的許多數學預測已經被嚴格證明了。

目前，鏡像對稱是純數學中的熱門話題，數學家正在物理直覺的基礎上探索鏡像對稱的嚴格數學化表述。鏡像對稱也是進行弦論和量子場論計算的重要工具，這兩者都是物理學家用來描述基本粒子的理論。鏡像對稱的數學表述主要有馬克西姆·孔采維奇的同調鏡像對稱，以及安德魯·施特羅明格、丘成桐和埃里克·扎斯洛（英語：Eric Zaslow）的SYZ猜想（英語：SYZ conjecture）。

參見

唐納森–托馬斯理論（英語：Donaldson–Thomas theory）
Wall-crossing

註釋

參考文獻

Aspinwall, Paul; Bridgeland, Tom; Craw, Alastair; Douglas, Michael; Gross, Mark; Kapustin, Anton; Moore, Gregory; Segal, Graeme; Szendröi, Balázs; Wilson, P.M.H. (編). Dirichlet Branes and Mirror Symmetry. American Mathematical Society. 2009. ISBN 978-0-8218-3848-8.
Candelas, Philip; de la Ossa, Xenia; Green, Paul; Parks, Linda. A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory. Nuclear Physics B. 1991, 359 (1): 21–74. Bibcode:1991NuPhB.359...21C. doi:10.1016/0550-3213(91)90292-6.
Candelas, Philip; Horowitz, Gary; Strominger, Andrew; Witten, Edward. Vacuum configurations for superstrings. Nuclear Physics B. 1985, 258: 46–74. Bibcode:1985NuPhB.258...46C. doi:10.1016/0550-3213(85)90602-9.
Candelas, Philip; Lynker, Monika; Schimmrigk, Rolf. Calabi–Yau manifolds in weighted $\mathbb {P} _{4}$ . Nuclear Physics B. 1990, 341 (1): 383–402. Bibcode:1990NuPhB.341..383C. doi:10.1016/0550-3213(90)90185-G.
Dixon, Lance. Some world-sheet properties of superstring compactifications, on orbifolds and otherwise. ICTP Ser. Theoret. Phys. 1988, 4: 67–126. ISBN 978-9971-5-0452-6.
Givental, Alexander. Equivariant Gromov-Witten invariants. International Mathematics Research Notices. 1996, 1996 (13): 613–663. doi:10.1155/S1073792896000414.
Givental, Alexander. A mirror theorem for toric complete intersections. Topological field theory, primitive forms and related topics. 1998: 141–175. ISBN 978-1-4612-6874-1. doi:10.1007/978-1-4612-0705-4_5.
Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. Random House. 2000. ISBN 978-0-9650888-0-0.
Greene, Brian; Plesser, Ronen. Duality in Calabi–Yau moduli space. Nuclear Physics B. 1990, 338 (1): 15–37. Bibcode:1990NuPhB.338...15G. doi:10.1016/0550-3213(90)90622-K.
Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Vakil, Ravi; Zaslow, Eric (編). Mirror Symmetry (PDF). American Mathematical Society. 2003. ISBN 0-8218-2955-6. （原始內容 (PDF)存檔於2006-09-19）.
Hori, Kentaro; Vafa, Cumrun. Mirror Symmetry. 2000. arXiv:hep-th/0002222  .
Intriligator, Kenneth; Seiberg, Nathan. Mirror symmetry in three-dimensional gauge theories. Physics Letters B. 1996, 387 (3): 513–519. Bibcode:1996PhLB..387..513I. arXiv:hep-th/9607207  . doi:10.1016/0370-2693(96)01088-X.
Kikkawa, Keiji; Yamasaki, Masami. Casimir effects in superstring theories. Physics Letters B. 1984, 149 (4): 357–360. Bibcode:1984PhLB..149..357K. doi:10.1016/0370-2693(84)90423-4.
Kontsevich, Maxim. The Moduli Space of Curves. Birkhäuser: 335. 1995a. ISBN 978-1-4612-8714-8. doi:10.1007/978-1-4612-4264-2_12. |chapter=被忽略 (幫助)
Kontsevich, Maxim. Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians. 1995b: 120–139. Bibcode:1994alg.geom.11018K. arXiv:alg-geom/9411018  .
Lerche, Wolfgang; Vafa, Cumrun; Warner, Nicholas. Chiral rings in ${\mathcal {N}}=2$ superconformal theories. Nuclear Physics B. 1989, 324 (2): 427–474. Bibcode:1989NuPhB.324..427L. doi:10.1016/0550-3213(89)90474-4.
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung. Mirror principle, I. Asian Journal of Math. 1997, 1: 729–763. Bibcode:1997alg.geom.12011L. arXiv:alg-geom/9712011  .
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung. Mirror principle, II. Asian Journal of Math. 1999a, 3: 109–146. Bibcode:1999math......5006L. arXiv:math/9905006  .
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung. Mirror principle, III. Asian Journal of Math. 1999b, 3: 771–800. Bibcode:1999math.....12038L. arXiv:math/9912038  .
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung. Mirror principle, IV. Surveys in Differential Geometry. 2000: 475–496. Bibcode:2000math......7104L. arXiv:math/0007104  .
Mac Lane, Saunders. Categories for the Working Mathematician. 1998. ISBN 978-0-387-98403-2.
Moore, Gregory. What is ... a Brane? (PDF). Notices of the AMS. 2005, 52: 214 [June 2013].
Sakai, Norisuke; Senda, Ikuo. Vacuum energies of string compactified on torus. Progress of Theoretical Physics. 1986, 75 (3): 692–705. Bibcode:1986PThPh..75..692S. doi:10.1143/PTP.75.692.
Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric. Mirror symmetry is T-duality. Nuclear Physics B. 1996, 479 (1): 243–259. Bibcode:1996NuPhB.479..243S. arXiv:hep-th/9606040  . doi:10.1016/0550-3213(96)00434-8.
Vafa, Cumrun. Topological mirrors and quantum rings. Essays on mirror manifolds. 1992: 96–119. Bibcode:1991hep.th...11017V. ISBN 978-962-7670-01-8. arXiv:hep-th/9111017  .
Wald, Robert. General Relativity. University of Chicago Press. 1984. ISBN 978-0-226-87033-5.
Witten, Edward. On the structure of the topological phase of two-dimensional gravity. Nuclear Physics B. 1990, 340 (2–3): 281–332. Bibcode:1990NuPhB.340..281W. doi:10.1016/0550-3213(90)90449-N.
Witten, Edward. Mirror manifolds and topological field theory. Essays on mirror manifolds. 1992: 121–160. ISBN 978-962-7670-01-8.
Yau, Shing-Tung; Nadis, Steve. The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. 2010. ISBN 978-0-465-02023-2.
Zaslow, Eric. Mirror Symmetry. Gowers, Timothy (編). The Princeton Companion to Mathematics. 2008. ISBN 978-0-691-11880-2.
Zwiebach, Barton. A First Course in String Theory. Cambridge University Press. 2009. ISBN 978-0-521-88032-9.

擴展閱讀

科普

Yau, Shing-Tung; Nadis, Steve. The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. 2010. ISBN 978-0-465-02023-2.
Zaslow, Eric. Physmatics. 2005. arXiv:physics/0506153  .
Zaslow, Eric. Mirror Symmetry. Gowers, Timothy (編). The Princeton Companion to Mathematics. 2008. ISBN 978-0-691-11880-2.

教材

Aspinwall, Paul; Bridgeland, Tom; Craw, Alastair; Douglas, Michael; Gross, Mark; Kapustin, Anton; Moore, Gregory; Segal, Graeme; Szendröi, Balázs; Wilson, P.M.H. (編). Dirichlet Branes and Mirror Symmetry. American Mathematical Society. 2009. ISBN 978-0-8218-3848-8.
Cox, David; Katz, Sheldon. Mirror symmetry and algebraic geometry. American Mathematical Society. 1999. ISBN 978-0-8218-2127-5.
Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Vakil, Ravi; Zaslow, Eric (編). Mirror Symmetry (PDF). American Mathematical Society. 2003. ISBN 0-8218-2955-6. （原始內容 (PDF)存檔於2006-09-19）.