镜像对称 (弦理论)

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. It can happen that two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent if they are employed as extra dimensions of string theory. In this case, they are called mirror manifolds.

Mirror symmetry was originally discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parks showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously.

Today mirror symmetry is a major research topic in pure mathematics, and mathematicians are working to develop a mathematical understanding of the relationship based on physicists' intuition. Mirror symmetry is also a fundamental tool for doing calculations in string theory, and it has been used to understand aspects of quantum field theory, the formalism that physicists use to describe elementary particles. Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.

参见

• Donaldson–Thomas theory
• Wall-crossing

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. 
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• 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. 
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• 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. 
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• 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). Mirror Symmetry (PDF). American Mathematical Society. 2003. ISBN 0-8218-2955-6.