Mirror symmetry

homological mirror symmetry

• 1994 Kontsevich
• categorical equivalence of the following two categories
  • derived category of bounded complexes of coherent sheaves on a smooth, complete, algebraic variety \(X\) over an algebraically closed field
  • Fukaya category of the symplectic manifold \(\tilde{X}\)

elliptic curve case

• According to Kontsevich, the mirror partner of an algebraic manifold \(M\) should be a symplectic manifold \(\tilde{M}\) such that the derived category \(D^b(M)\) of bounded complexes of coherent sheaves on \(M\) is equivalent to a suitable version of Fukaya's category \(F(\tilde{M})\) of Lagrangian submanifolds of \(M\) equipped with a flat bundle. In this paper the authors verify this conjecture in the case when \(M\) is an elliptic curve \(E_{\tau}=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})\), where \(\tau=a+bi, b>0\). Here the mirror partner is a torus \(\tilde{E}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z})\) with Kähler metric \(b(dx^2+dy^2)\). It is also equipped with a form \(B=a dx\wedge dy\). The authors give a beautiful, very explicit description of the two categories involved in the mirror duality.

related items

• Calabi-Yau threefolds

books

• Mirror Symmetry

exposition

• http://arxiv.org/abs/1506.07757
• Port, Andrew. “An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves.” arXiv:1501.00730 [math], January 4, 2015. http://arxiv.org/abs/1501.00730.
• Quigley, Callum. “Mirror Symmetry in Physics: The Basics.” arXiv:1412.8180 [hep-Th], December 28, 2014. http://arxiv.org/abs/1412.8180.
• Chan, Kwokwai. “SYZ Mirror Symmetry for Toric Varieties.” arXiv:1412.7231 [math-Ph], December 22, 2014. http://arxiv.org/abs/1412.7231.
• Clader, Emily, and Yongbin Ruan. “Mirror Symmetry Constructions.” arXiv:1412.1268 [hep-Th, Physics:math-Ph], December 3, 2014. http://arxiv.org/abs/1412.1268.
• Hosono, Shinobu, and Hiromichi Takagi. “Mirror Symmetry and Projective Geometry of Fourier-Mukai Partners.” arXiv:1410.1254 [math], October 6, 2014. http://arxiv.org/abs/1410.1254.
• Chan, Kwokwai. “The Strominger-Yau-Zaslow Conjecture and Its Impact.” arXiv:1408.6062 [math], August 26, 2014. http://arxiv.org/abs/1408.6062.
• Ueda, Kazushi. “Mirror Symmetry and K3 Surfaces.” arXiv:1407.1566 [math], July 6, 2014. http://arxiv.org/abs/1407.1566.
• http://www.kias.re.kr/file/NewsletterNo37.pdf
• Lectures on Mirror Symmetry, Derived Categories, and D-branes Authors: Anton Kapustin, Dmitri Orlov http://arxiv.org/abs/math/0308173
• Yau, Shing-Tung, ed. 1992. Essays on Mirror Manifolds. Hong Kong: International Press. http://www.ams.org/mathscinet-getitem?mr=1191418.

articles

• Schaug, Andrew. “Quantum Mirror Symmetry for Borcea-Voisin Threefolds.” arXiv:1510.08333 [math-Ph], October 28, 2015. http://arxiv.org/abs/1510.08333.
• Cao, Yalong, and Naichung Conan Leung. “Remarks on Mirror Symmetry of Donaldson-Thomas Theory for Calabi-Yau 4-Folds.” arXiv:1506.04218 [math-Ph], June 12, 2015. http://arxiv.org/abs/1506.04218.
• Kanazawa, Atsushi, and Siu-Cheong Lau. “Geometric Transitions and SYZ Mirror Symmetry.” arXiv:1503.03829 [math], March 12, 2015. http://arxiv.org/abs/1503.03829.
• Sheridan, Nicholas. “Homological Mirror Symmetry for Calabi-Yau Hypersurfaces in Projective Space.” arXiv:1111.0632 [math], November 2, 2011. http://arxiv.org/abs/1111.0632.
• Hiep, Dang Tuan. “Rational Curves on Calabi-Yau Threefolds: Verifying Mirror Symmetry Predictions.” arXiv:1409.3712 [math], September 12, 2014. http://arxiv.org/abs/1409.3712.
• Polishchuk, Alexander, and Eric Zaslow. 1998. “Categorical Mirror Symmetry: The Elliptic Curve.” Advances in Theoretical and Mathematical Physics 2 (2): 443–470.
• Kontsevich, Maxim. 1995. “Homological Algebra of Mirror Symmetry.” In Proceedings of the International Congress of Mathematicians, Vol.\ 1, 2 (Zürich, 1994), 120–139. Basel: Birkhäuser. http://www.ams.org/mathscinet-getitem?mr=1403918.
• Candelas, Philip, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes. 1991. “A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory.” Nuclear Physics. B 359 (1): 21–74. doi:10.1016/0550-3213(91)90292-6.
• Greene, B. R., and M. R. Plesser. 1990. “Duality in Calabi-Yau Moduli Space.” Nuclear Physics. B 338 (1): 15–37. doi:10.1016/0550-3213(90)90622-K.
• Candelas, P., M. Lynker, and R. Schimmrigk. 1990. “Calabi-Yau Manifolds in Weighted \(\bf P_4\).” Nuclear Physics. B 341 (2): 383–402. doi:10.1016/0550-3213(90)90185-G.

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