Mirror symmetry - 편집 역사

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메타데이터: 새 문단

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imported>Pythagoras0: /* articles */

2015-10-29T06:59:38Z

articles

imported>Pythagoras0: /* exposition */

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exposition

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articles

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@@ 52번째 줄: / 52번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q5914418 Q5914418]

← 이전 판		2020년 12월 28일 (월) 08:00 판
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	[[분류:duality]]		[[분류:duality]]
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		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q5914418 Q5914418]

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 * 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
+** 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>
 ==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.
+* 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.
@@ 31번째 줄: / 31번째 줄: @@
 * 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<br> Authors: Anton Kapustin, Dmitri Orlov http://arxiv.org/abs/math/0308173
+* 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.
@@ 44번째 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.
 * 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.
 [[분류:개인노트]]

← 이전 판		2020년 11월 13일 (금) 11:02 판
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	[[분류:duality]]		[[분류:duality]]
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← 이전 판		2016년 9월 21일 (수) 07:25 판
7번째 줄:		7번째 줄:
	** 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 $X$ over an algebraically closed field
	** Fukaya category of the symplectic manifold $\tilde{X}$		** 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.

@@ 31번째 줄: / 31번째 줄: @@
 ==articles==
 * 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.

@@ 18번째 줄: / 18번째 줄: @@
 ==exposition==
 * 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.

@@ 11번째 줄: / 11번째 줄: @@
 ==related items==
 * [[Calabi-Yau threefolds]]
@@ 26번째 줄: / 30번째 줄: @@
 ==articles==
 * 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.
@@ 33번째 줄: / 38번째 줄: @@
 * 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.
 [[분류:개인노트]]

← 이전 판		2015년 3월 5일 (목) 07:18 판
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