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镜像对称 (弦理论)

行進 29, 2023

在代数几何和理论物理中，镜像对称是指卡拉比-丘流形之间的一种特殊关系，即两种卡丘流形虽然在几何上差别很大，但是作为弦理论的额外维度时却是等价的。这样的一对流形被称为镜像流形。

镜像对称最早是由物理学家发现的。1990年左右，菲利普·坎德拉斯（英语：Philip Candelas）、齐妮娅·德·拉·奥萨（Xenia de la Ossa）、保罗·格林（Paul Green）和琳达·帕克斯（Linda Parks）发现它可以用于枚举几何（英语：enumerative geometry），因此激发了数学家对此的兴趣。枚举几何是研究几何问题解的数量的数学分支。坎德拉斯和他的合作者证明了镜像对称可用于计算卡丘流形上有理曲线的数目，从而解决了一个长期的难题。尽管镜像对称最初的方法是从物理出发的，数学上并不严格，它的许多数学预测已经被严格证明了。

目前，镜像对称是纯数学中的热门话题，数学家正在物理直觉的基础上探索镜像对称的严格数学化表述。镜像对称也是进行弦论和量子场论计算的重要工具，这两者都是物理学家用来描述基本粒子的理论。镜像对称的数学表述主要有马克西姆·孔采维奇的同调镜像对称（英语：Homological mirror symmetry），以及安德鲁·施特罗明格、丘成桐和埃里克·扎斯洛（英语：Eric Zaslow）的SYZ猜想（英语：SYZ conjecture）。

参见

  • 唐纳森–托马斯理论（英语：Donaldson–Thomas theory）
  • Wall-crossing

注释

参考文献

  • Aspinwall, Paul; Bridgeland, Tom; Craw, Alastair; Douglas, Michael; Gross, Mark; Kapustin, Anton; Moore, Gregory; Segal, Graeme; Szendröi, Balázs; Wilson, P.M.H. (编). 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  . 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 (编). (PDF). American Mathematical Society. 2003. ISBN 0-8218-2955-6. （原始内容 (PDF)存档于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 (帮助)
  • 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   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 (编). 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. 

扩展阅读

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 (编). 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. (编). 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 (编). (PDF). American Mathematical Society. 2003. ISBN 0-8218-2955-6. （原始内容 (PDF)存档于2006-09-19）. 

行進 29, 2023
镜像对称, 弦理论, 在代数几何和理论物理中, 镜像对称是指卡拉比, 丘流形之间的一种特殊关系, 即两种卡丘流形虽然在几何上差别很大, 但是作为弦理论的额外维度时却是等价的, 这样的一对流形被称为镜像流形, 镜像对称最早是由物理学家发现的, 1990年左右, 菲利普, 坎德拉斯, 英语, philip, candelas, 齐妮娅, 奥萨, xenia, ossa, 保罗, 格林, paul, green, 和琳达, 帕克斯, linda, parks, 发现它可以用于枚举几何, 英语, enumerative, . 在代数几何和理论物理中 镜像对称是指卡拉比 丘流形之间的一种特殊关系 即两种卡丘流形虽然在几何上差别很大 但是作为弦理论的额外维度时却是等价的 这样的一对流形被称为镜像流形 镜像对称最早是由物理学家发现的 1990年左右 菲利普 坎德拉斯 英语 Philip Candelas 齐妮娅 德 拉 奥萨 Xenia de la Ossa 保罗 格林 Paul Green 和琳达 帕克斯 Linda Parks 发现它可以用于枚举几何 英语 enumerative geometry 因此激发了数学家对此的兴趣 枚举几何是研究几何问题解的数量的数学分支 坎德拉斯和他的合作者证明了镜像对称可用于计算卡丘流形上有理曲线的数目 从而解决了一个长期的难题 尽管镜像对称最初的方法是从物理出发的 数学上并不严格 它的许多数学预测已经被严格证明了 目前 镜像对称是纯数学中的热门话题 数学家正在物理直觉的基础上探索镜像对称的严格数学化表述 镜像对称也是进行弦论和量子场论计算的重要工具 这两者都是物理学家用来描述基本粒子的理论 镜像对称的数学表述主要有马克西姆 孔采维奇的同调镜像对称 英语 Homological mirror symmetry 以及安德鲁 施特罗明格 丘成桐和埃里克 扎斯洛 英语 Eric Zaslow 的SYZ猜想 英语 SYZ conjecture 目录 1 参见 2 注释 3 参考文献 4 扩展阅读 4 1 科普 4 2 教材参见 编辑唐纳森 托马斯理论 英语 Donaldson Thomas theory Wall crossing注释 编辑参考文献 编辑Aspinwall Paul Bridgeland Tom Craw Alastair Douglas Michael Gross Mark Kapustin Anton Moore Gregory Segal Graeme Szendroi Balazs Wilson P M H 编 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 P 4 displaystyle 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 编 Mirror Symmetry PDF American Mathematical Society 2003 ISBN 0 8218 2955 6 原始内容 PDF 存档于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 Birkhauser 335 1995a ISBN 978 1 4612 8714 8 doi 10 1007 978 1 4612 4264 2 12 chapter 被忽略 帮助 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 N 2 displaystyle 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 编 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 扩展阅读 编辑科普 编辑 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 编 The Princeton Companion to Mathematics 2008 ISBN 978 0 691 11880 2 教材 编辑 Aspinwall Paul Bridgeland Tom Craw Alastair Douglas Michael Gross Mark Kapustin Anton Moore Gregory Segal Graeme Szendroi Balazs Wilson P M H 编 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 编 Mirror Symmetry PDF American Mathematical Society 2003 ISBN 0 8218 2955 6 原始内容 PDF 存档于2006 09 19 取自 https zh wikipedia org w index php title 镜像对称 弦理论 amp oldid 67597725, 维基百科，wiki，书籍，书籍，图书馆，

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