"Mirror symmetry"의 두 판 사이의 차이

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5번째 줄: 5번째 줄:
 
* 1994 Kontsevich
 
* 1994 Kontsevich
 
* categorical equivalence of the following two categories
 
* 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
+
** derived category of bounded complexes of coherent sheaves on a smooth, complete, algebraic variety <math>X</math> over an algebraically closed field
−
** Fukaya category of the symplectic manifold $\tilde{X}$
+
** Fukaya category of the symplectic manifold <math>\tilde{X}</math>
 +
 
 +
 
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==elliptic curve case==
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* According to Kontsevich, the mirror partner of an algebraic manifold <math>M</math> should be a symplectic manifold <math>\tilde{M}</math> such that the derived category <math>D^b(M)</math> of bounded complexes of coherent sheaves on <math>M</math> is equivalent to a suitable version of Fukaya's category <math>F(\tilde{M})</math> of Lagrangian submanifolds of <math>M</math> equipped with a flat bundle. In this paper the authors verify this conjecture in the case when <math>M</math> is an elliptic curve <math>E_{\tau}=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})</math>, where <math>\tau=a+bi, b>0</math>. Here the mirror partner is a torus <math>\tilde{E}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z})</math> with Kähler metric <math>b(dx^2+dy^2)</math>. It is also equipped with a form <math>B=a dx\wedge dy</math>. The authors give a beautiful, very explicit description of the two categories involved in the mirror duality.
  
  
 
==related items==
 
==related items==
 
* [[Calabi-Yau threefolds]]
 
* [[Calabi-Yau threefolds]]
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 +
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==books==
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* [http://math.stanford.edu/~vakil/files/mirrorfinal.pdf Mirror Symmetry]
  
  
 
==exposition==
 
==exposition==
 +
* http://arxiv.org/abs/1506.07757
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* 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.
 
* 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.
 
* Chan, Kwokwai. “SYZ Mirror Symmetry for Toric Varieties.” arXiv:1412.7231 [math-Ph], December 22, 2014. http://arxiv.org/abs/1412.7231.
21번째 줄: 31번째 줄:
 
* Ueda, Kazushi. “Mirror Symmetry and K3 Surfaces.” arXiv:1407.1566 [math], July 6, 2014. http://arxiv.org/abs/1407.1566.
 
* 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
 
* http://www.kias.re.kr/file/NewsletterNo37.pdf
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* Lectures on Mirror Symmetry, Derived Categories, and D-branes<br> Authors: Anton Kapustin, Dmitri Orlov http://arxiv.org/abs/math/0308173
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* 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.
 
* Yau, Shing-Tung, ed. 1992. Essays on Mirror Manifolds. Hong Kong: International Press. http://www.ams.org/mathscinet-getitem?mr=1191418.
  
 
==articles==
 
==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.
 
* 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.
 
* Polishchuk, Alexander, and Eric Zaslow. 1998. “Categorical Mirror Symmetry: The Elliptic Curve.” Advances in Theoretical and Mathematical Physics 2 (2): 443–470.
30번째 줄: 44번째 줄:
 
* 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.
 
* 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.
 
* 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.
+
* Candelas, P., M. Lynker, and R. Schimmrigk. 1990. “Calabi-Yau Manifolds in Weighted <math>\bf P_4</math>.” Nuclear Physics. B 341 (2): 383–402. doi:10.1016/0550-3213(90)90185-G.
−
 
 
−
==books==
 
−
* [http://math.stanford.edu/~vakil/files/mirrorfinal.pdf Mirror Symmetry]
 
−
 
 
  
 
[[분류:개인노트]]
 
[[분류:개인노트]]
 
[[분류:physics]]
 
[[분류:physics]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
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[[분류:duality]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q5914418 Q5914418]
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===Spacy 패턴 목록===
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* [{'LOWER': 'mirror'}, {'LEMMA': 'symmetry'}]
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* [{'LOWER': 'mirror'}, {'LOWER': 'symmetry'}, {'OP': '*'}, {'LOWER': 'string'}, {'LOWER': 'theory'}, {'LEMMA': ')'}]

2021년 2월 17일 (수) 01:56 기준 최신판

introduction

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


books


exposition

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.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'mirror'}, {'LEMMA': 'symmetry'}]
  • [{'LOWER': 'mirror'}, {'LOWER': 'symmetry'}, {'OP': '*'}, {'LOWER': 'string'}, {'LOWER': 'theory'}, {'LEMMA': ')'}]
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