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| {{String theory}}
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| In [[mathematics]] and [[theoretical physics]], '''mirror symmetry''' is a relationship between [[geometry|geometric]] objects called [[Calabi-Yau manifold]]s. It can happen, for two such six-dimensional [[manifolds]], that their shapes look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of [[string theory]].<ref>Yau and Nadis 2010</ref> In this case, the six-dimensional manifolds are said to be "mirror" to one another.
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| Mirror symmetry was originally discovered by physicists. Mathematicians became interested in mirror symmetry around 1990 when [[Philip Candelas]], Xenia de la Ossa, Paul Green, and Linda Parks showed that mirror symmetry could be used to count [[rational curve]]s on a Calabi-Yau manifold, thus solving a longstanding problem.<ref>Yau and Nadis 2010</ref> Although the original approach to mirror symmetry was based on [[mathematical rigor|nonrigorous]] ideas from theoretical physics, mathematicians have gone on to [[mathematical proof|rigorously prove]] some of the mathematical predictions of mirror symmetry.<ref>Givental 1996, 1998; Lian, Liu, Yau 1997, 1999, 2000</ref>
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| Today mirror symmetry is a major research topic in [[pure mathematics]], and mathematicians are working to develop a mathematical understanding of mirror symmetry based on physicists' intuition.<ref>Hori et al. 2003; Aspinwall et al. 2009</ref> Mirror symmetry is also a fundamental tool for doing calculations in string theory.<ref>Zaslow 2008</ref> Major approaches to mirror symmetry include [[homological mirror symmetry]] program of [[Maxim Kontsevich]]<ref>Kontsevich 2005</ref> and the [[SYZ conjecture]] of [[Andrew Strominger]], [[Shing-Tung Yau]], and [[Eric Zaslow]].<ref>Strominger, Yau, and Zaslow 1996</ref>
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| ==Overview==
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| ===The idea of mirror symmetry===
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| Mirror symmetry is a particular example of what physicists call a [[duality (physics)|duality]].<ref>Hori et al. 2003, p. xiv</ref> In physics, the term ''duality'' refers to a situation where two seemingly different [[physical system]]s turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be ''dual'' to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.
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| Like many of the dualities studied in [[theoretical physics]], mirror symmetry was discovered in the context of [[string theory]].<ref>Other dualities that arise in string theory are [[S-duality]], [[T-duality]], [[U-duality]], and the [[AdS/CFT correspondence]].</ref> In string theory, particles are modeled not as zero-dimensional points but as one-dimensional extended objects called [[string (physics)|strings]]. One of the peculiar features of string theory is that it requires [[extra dimensions]] of [[spacetime]] for its mathematical consistency. In [[superstring theory]], the version of the theory that incorporates [[supersymmetry|worldsheet supersymmetry]], there are six extra dimensions of [[spacetime]] in addition to the four that are familiar from everyday experience.<ref>Zwiebach 2009, p.8</ref>
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| [[Image:Calabi yau.jpg|left|thumb|200px|A three-dimensional projection of a [[Calabi–Yau manifold]]. ]]
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| In most realistic models of physics based on string theory, the extra dimensions are eliminated from the theory at low energies by a process known as [[compactification (physics)|compactification]]. This produces a theory in which spacetime has effectively a lower number of dimensions and the extra dimensions are "curled up" into complex shapes called [[Calabi-Yau manifolds]].<ref>Yau and Nadis 2010, Ch.6</ref> A standard analogy for this is to consider multidimensional object such as a garden hose.<ref>This analogy is used for example in Greene 2000, p.186</ref> If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions. In certain models based on string theory, the Calabi-Yau manifolds play a role analogous to the circumference of the hose.
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| In the late 1980s, it was noticed that given such a compactification of string theory, it is not possible to uniquely reconstruct a corresponding Calabi-Yau manifold. Instead, one finds that there are ''two'' Calabi-Yau manifolds that give rise to the same physics.<ref>Dixon 1988; Lerche, Vafa, and Warner 1989</ref> These manifolds are said to be "mirror" to one another. Although the full duality is still only a conjecture, there is a version of mirror symmetry in the context of [[topological string theory]], a simplified version of string theory introduced by [[Edward Witten]],<ref>Witten 1990</ref> and this version has been [[mathematical rigor|rigorously proven]] by mathematicians.<ref>Givental 1996, 1998; Lian, Liu, Yau 1997, 1999, 2000</ref> In the context of topological string theory, mirror symmetry states that two theories, the A-model and B-model, are equivalent in a certain precise sense.<ref>Zaslow 2008, p.531</ref>
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| Regardless of whether these Calabi-Yau compactifications of string theory provide a correct description of nature, the existence of the mirror symmetry relationship between different Calabi-Yau manifolds has significant mathematical consequences.<ref>Zaslow 2008, p.523</ref> The Calabi-Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve many problems in [[enumerative geometry|enumerative algebraic geometry]] by solving equivalent problems for the mirror Calabi-Yau.<ref>Yau and Nadis 2010, p.168</ref> Today mirror symmetry is an active area of research in mathematics, and mathematicians are still working to develop a mathematical understanding of mirror symmetry based on physicists' intuition.<ref>Hori et al. 2003, p. xix</ref>
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| ===Complex geometry===
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| {{main|Complex geometry}}
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| [[File:Torus.png|left|thumb|A torus]]
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| To understand the kind of geometry that appears on one side of the mirror duality, we will consider here a construction of a [[torus]], a closed surface with a single hole like a donut, by identifying points of the [[complex plane]]. To construct this torus, we first choose a pair of complex numbers <math>\omega_1</math> and <math>\omega_2</math> such that the quotient <math>\omega_1/\omega_2</math> is not [[real number|real]]. This last condition ensures that these points are not [[collinear]]. Then the chosen points determine a [[parallelogram]] whose other [[vertex (geometry)|vertices]] are 0 and <math>\omega_1+\omega_2</math>. By identifying opposite sides of this parallelogram, we obtain the desired torus.
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| [[Image:Lattice torsion points.svg|right|thumb|300px| A [[torus]] can be constructed by identifying opposite sides of a [[parallelogram]] in the [[complex plane]]. This torus inherits a [[complex manifold|complex structure]] which, roughly speaking, describes the "shape" of the torus. ]]
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| The tori obtained in this way are all equivalent in the sense that we can [[homotopy|continuously deform]] one torus into another.<ref>Zaslow 2008, p.530</ref> On the other hand, they have an additional structure which allows us to distinguish them.<ref>Zaslow 2008, p.531</ref> Namely, the tori constructed in this way have a [[complex manifold|complex structure]], meaning that a neighborhood of any point on such a torus looks just like a [[domain (mathematical analysis)|region]] in the complex plane.
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| If in the construction of a torus we had instead used a pair of complex numbers <math>\omega_1'</math> and <math>\omega_2'</math> related to the original pair by rescaling by a common factor (that is, <math>\omega_1'=\lambda\omega_1</math> and <math>\omega_2'=\lambda\omega_2</math> for some complex number <math>\lambda</math>), then we would get an equivalent torus. It is therefore more convenient to parametrize the collection of tori by the ''ratio'' <math>\tau=\omega_1/\omega_2</math>, which does not change when we rescale the <math>\omega_i</math>. One can assume, [[without loss of generality]], that this parameter <math>\tau</math> has positive [[imaginary part]] so that <math>\tau</math> takes values in the [[upper half plane]]. One can also show that the parameters <math>\tau</math>, <math>\tau+1</math>, and <math>-1/\tau</math> correspond to the same torus.
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| If two tori correspond to genuinely different values of <math>\tau</math>, then they have inequivalent complex structures.<ref>More precisely, tori are parametrized by the [[fundamental domain]] for the [[modular group]].</ref> One can think of the parameter <math>\tau</math> as describing the "shape" of a torus constructed by identifying opposite sides of a parallelogram. As explained above, mirror symmetry relates two physical theories, the A- and B-models of [[topological string theory]]. In this duality, the topological B-model depends only on the complex structure of [[spacetime]]. Thus, if we consider the theory in which "spacetime" is a torus, the theory will depend continuously only on the parameter <math>\tau</math>.<ref>Zaslow 2008, p.531</ref>
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| ===Symplectic geometry===
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| {{main|Symplectic geometry}}
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| Another aspect of the geometry of a torus is the size of the torus. More precisely, we can talk view a torus as the surface obtained by identifying opposite sides of a unit square, and the area of the torus is specified by an [[area element]] <math>\rho dxdy</math> on this square. By [[integral|integrating]] this area element over the [[unit square]], we obtain the [[area]] <math>\rho</math> of the corresponding torus. These concepts can be generalized to higher dimensions, and the area element is generalized by the notion of a [[symplectic form]]. The study of spaces equipped with a symplectic form is called [[symplectic geometry]].
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| In mirror symmetry, the A-model of [[topological string theory]] is a theory that depends only on the symplectic geometry of [[spacetime]]. Thus, if we consider the theory in which "spacetime" is a torus, then the A-model depends continuously only on the parameter <math>\rho</math>.<ref>Zaslow 2008, p.531</ref>
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| ===T-duality===
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| [[File:torus cycles.png|thumb|right|A torus is the product of two circles, in this case the red circle is swept around axis defining the pink circle. <math>R_1</math> is the radius of the red circle, <math>R_2</math> is the radius of the pink one.]]
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| {{main|T-duality}}
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| We have now seen how a torus can be obtained by identifying opposite sides of a parallelogram in the complex plane. A particularly simple example is where the complex numbers <math>\omega_1</math> and <math>\omega_2</math> lie on the real and imaginary axes, respectively. In this case, we can write <math>\omega_1=R_1</math> and <math>\omega_2=iR_2</math> where <math>R_1</math> and <math>R_2</math> are real numbers. As discussed above, the complex structure on the torus obtained in this way is characterized by the number <math>\tau=iR_2/R_1</math>.
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| We have explained how the symplectic structure of the torus is determined by an area element. We can choose coordinates <math>x</math> and <math>y</math> on our parallelogram so that each side of the parallelogram spanned by the chosen complex numbers has length 1. Then the area element of our torus is <math>R_1R_2dxdy</math> which integrates to <math>R_1R_2</math> on the unit square. We define the symplectic parameter <math>\rho</math> to be the product <math>iR_1R_2</math>.
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| Note that the torus can also be considered as a [[cartesian product]] of two circles. This means that at each point of an equatorial circle on the torus (illustrated in pink), there is a longitudinal circle (illustrated in red).
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| Let us now imagine that the torus represents the "spacetime" for a [[physical theory]]. The fundamental objects of this theory will be [[string (physics)|strings]] propagating through the spacetime according to the rules of [[quantum mechanics]]. One of the basic [[duality (physics)|dualities]] of [[string theory]] is [[T-duality]], which states that a string propagating around a circle of radius <math>R</math> is equivalent to a string propagating around a circle of radius <math>1/R</math> in the sense that all observable quantities in one description are identified with quantities in the dual description.<ref>Zaslow 2008, p.532</ref> For example, the [[momentum]] of a string in one description takes discrete values and is equal to the number of times the string [[winding number|winds]] around the circle in the dual description.<ref>Zaslow 2008, p.532</ref> Applying T-duality to the longitudinal circle on the torus, we find that there is an equivalent description in which spacetime is represented by a different torus. The duality changes <math>R_1</math> to <math>1/R_1</math>, and thus it interchanges the complex and symplectic parameters:
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| : <math>\tau\leftrightarrow\rho</math>.
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| In general, mirror symmetry is an equivalence of two physical theories that translates problems in [[complex geometry]] into problems in [[symplectic geometry]]. The torus considered here is the only [[compact topological space|compact]] Calabi-Yau manifold of (real) dimension two and is therefore the simplest example of mirror symmetry.<ref>Zaslow 2008, p.533</ref> In applications to string theory, one usually considers a six-dimensional Calabi-Yau manifold, where the six dimensions correspond to the six unobserved dimensions of spacetime.
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| ==Applications of mirror symmetry==
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| ===Enumerative geometry===
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| {{main|Enumerative geometry}}
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| [[File:Apollonius8ColorMultiplyV2.svg|thumb|right|[[Problem of Apollonius|Circles of Apollonius]].]]
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| Many of the important mathematical applications of mirror symmetry belong to the branch of mathematics called [[enumerative geometry]]. In enumerative geometry, one is interested in counting the number of solutions to geometric questions, typically using the techniques of [[algebraic geometry]]. One of the earliest problems of enumerative geometry was posed around the year 200 [[BCE]] by the ancient Greek mathematician [[Apollonius of Perga|Apollonius]], who asked how many circles in the plane are tangent to three given circles.<ref>Yau and Nadis 2010, p.166</ref> In general, the solution to the [[problem of Apollonius]] is that there are eight such circles. The picture on the right shows an example where the three given circles are illustrated in black.
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| [[File:Clebsch Cublic.png|thumb|left|The [[Clebsch cubic]].]]
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| Enumerative problems in mathematics often concern a class of geometric objects called [[algebraic varieties]] which are defined by the vanishing of [[polynomial]]s. For example, the [[Clebsch cubic]] is the surface illustrated on the left which is defined using a certain polynomial of [[degree of a polynomial|degree]] three in four variables. A celebrated result of nineteenth-century mathematicians [[Arthur Cayley]] and [[George Salmon]] states that there are exactly 27 straight lines that lie entirely on such a surface.<ref>Yau and Nadis 2010, p.167</ref>
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| Generalizing this problem, one can ask how many lines can be drawn on a quintic such as the Calabi-Yau illustrated above. This problem was solved by the nineteenth-century German mathematician [[Hermann Schubert]], who found that there are exactly 2,875 such lines. In 1986, geometer [[Sheldon Katz]] proved that the number of curves of degree two (such as circles) which lie entirely in the quintic is 609,250.<ref>Yau and Nadis 2010, p.166</ref>
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| By the year 1991, most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish. According to mathematician [[Mark Gross]], "As the old problems had been solved, people went back to check Schubert's numbers with modern techniques, but that was getting pretty stale."<ref>Yau and Nadis 2010, p.169</ref> The field was reinvigorated in May 1991 when physicists [[Philip Candelas]], Xenia de la Ossa, Paul Green, and Linda Parks showed that mirror symmetry could be used to count the number of degree three curves lie on a quintic Calabi-Yau. Roughly speaking, one can think of such a degree three curve as a sphere that lies entirely inside the Calabi-Yau.<ref>Yau and Nadis 2010, p.168</ref> Candelas and his collaborators found that such these six-dimensional Calabi-Yau manifolds can contain exactly 317,206,375 curves of degree three.<ref>Yau and Nadis 2010, p.169</ref>
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| In addition to counting degree-three curves on a quintic three-fold, Candelas and his collaborators obtained a number of more general results for counting rational curves which went far beyond the results obtained by mathematicians.<ref>Yau and Nadis 2010, p.171</ref> Although the methods used in this work were based on [[mathematical rigor|nonrigorous]] ideas from [[theoretical physics]], mathematicians have gone on to [[mathematical proof|rigorously prove]] some of the predictions of mirror symmetry. In particular, the enumerative predictions of mirror symmetry have now been rigorously proven.<ref>Yau and Nadis 2010, p.172</ref>
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| ===String theory===
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| In addition to its applications to enumerative geometry, mirror symmetry is a fundamental tool for doing calculations in [[string theory]]. When a calculation in string theory is impossible using the techniques of [[perturbation theory]], theorists can apply dualities such as mirror symmetry to translate the calculation into an equivalent calculation in a ''different'' theory. By outsourcing calculations to different theories in this way, theorists can calculate many quantities that are impossible to calculate without the use of dualities.<ref>Zaslow 2008, sec. 10</ref>
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| ==Approaches to mirror symmetry==
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| ===Homological mirror symmetry===
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| {{main|Homological mirror symmetry}}
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| [[File:D3-brane et D2-brane.PNG|thumb|right|350px|Open strings attached to a pair of [[D-brane]]s.]]
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| In [[string theory]] and related theories such as [[Supergravity|supergravity theories]], a ''brane'' is a physical object that generalizes the notion of a point particle to higher dimensions.<ref>Moore 2005, p.214</ref> For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher dimensional branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.
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| In string theory, a [[string (physics)|string]] may be open (forming a segment with two endpoints) or closed (forming a closed loop). [[D-brane]]s are an important class of branes that arise when one considers open strings.<ref>Moore 2005, p.215</ref> As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to the fact that we impose a certain mathematical condition on the system known as the [[Dirichlet boundary condition]].
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| Mathematically, branes can be described using the notion of a [[category (mathematics)|category]].<ref>Aspinwall et al. 2009</ref> This is a mathematical structure consisting of ''objects'', and for any pair of objects, a set of ''morphisms'' between them. In most examples, the objects are mathematical structures of some kind (for example, [[set (mathematics)|set]]s, [[vector spaces]], or [[topological spaces]]) and the morphisms are given by [[function (mathematics)|functions]] between these structures.<ref>A basic reference on category theory is Mac Lane 1998.</ref> We can also consider categories where the objects are D-branes and the morphisms between two branes <math>\alpha</math> and <math>\beta</math> are [[wavefunction]]s of open strings stretched between <math>\alpha</math> and <math>\beta</math>.<ref>Zaslow 2008, p.536</ref>
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| In the B-model of [[topological string theory]], the category of D-branes is constructed from the [[complex geometry]] of the Calabi-Yau manifold on which the strings propagate. In mathematical language, it is known as the [[derived category]] of [[coherent sheaves]] on the Calabi-Yau. On the other hand, the category of D-branes in the A-model is constructed from the [[symplectic geometry]] of the mirror Calabi-Yau. It is known in mathematics as the [[Fukaya category]].<ref>Aspinwall et al. 2009, p.575</ref> The [[homological mirror symmetry]] conjecture of [[Maxim Kontsevich]] states that these two categories of branes are equivalent in a certain sense.<ref>Aspinwall et al. 2009, p.616</ref>
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| ===Strominger-Yau-Zaslow conjecture===
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| {{main|SYZ conjecture}}
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| Another approach to understanding mirror symmetry was suggested by [[Andrew Strominger]], [[Shing-Tung Yau]], and [[Eric Zaslow]] in 1996.<ref>Strominger, Yau, and Zaslow 1996</ref> According to the SYZ conjecture, mirror symmetry can be understood by dividing a complicated Calabi-Yau manifold into simpler pieces and considering the effects of [[T-duality]] on these pieces.<ref>Yau and Nadis 2010, p.174</ref>
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| Recall that in the Overview section we considered the case of a [[torus]], and we viewed this torus as the [[cartesian product|product]] of two circles. This means that the torus can be viewed as the [[union (set theory)|union]] of a collection of longitudinal circles (such as the red circle in the image). There is an auxiliary space which says how these circles are organized, and this space is itself a circle (the pink circle). This space is said to ''parametrize'' the longitudinal circles on the torus. As explained above, mirror symmetry is equivalent to T-duality acting on the longitudinal circles, changing their radii from <math>R_1</math> to <math>1/R_1</math>.
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| The SYZ conjecture generalizes this idea to the more complicated case of six-dimensional Calabi-Yau manifolds. As in the case of a torus, we can divide a six-dimensional Calabi-Yau into simpler pieces, which in this case are [[3-torus|3-tori]] (three-dimensional objects which generalize the notion of a torus) parametrized by a [[3-sphere]] (a three-dimensional generalization of a sphere).<ref>More precisely, there is a 3-torus associated to every point on the three-sphere except at certain bad points, which correspond to singular tori. See Yau and Nadis 2010, pp.176--7.</ref> T-duality can be extended from circles to the three-dimensional tori appearing in this decomposition, and the SYZ conjecture states that mirror symmetry is equivalent to the simultaneous application of T-duality to these three-dimensional tori.<ref>Yau and Nadis 2010, p.178</ref> In this way, the SYZ conjecture provides a geometric picture of how mirror symmetry acts on a Calabi-Yau manifold.
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| ==History==
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| The idea of mirror symmetry can be traced back to the mid-1980s when it was noticed that a [[string (physics)|string]] propagating on a circle of radius <math>R</math> is physically equivalent to a string propagating on a circle of radius <math>1/R</math> in appropriate [[units of measurements|units]].<ref>This was first observed in Kikkawa and Yamasaki 1984 and Sakai and Senda 1986.</ref> This phenomenon is now known as [[T-duality]] and is understood to be closely related to mirror symmetry.
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| In a paper from 1985, [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], and [[Edward Witten]] showed that by [[compactification (physics)|compactifying]] string theory on a [[Calabi-Yau manifold]], one obtains a theory roughly similar to the [[standard model of particle physics]].<ref>Candelas et al. 1985</ref> Following this development, many physicists began studying Calabi-Yau compactifications, hoping to construct realistic models of particle physics based on string theory. It was noticed that given such a physical model, it is not possible to uniquely reconstruct a corresponding Calabi-Yau manifold. Instead, one finds that there are ''two'' Calabi-Yau manifolds that give rise to the same physics.<ref>This was observed in Dixon 1988 and Lerche, Vafa, and Warner 1989.</ref>
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| Mathematicians became interested in mirror symmetry around 1990 when physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parks showed that mirror symmetry could be used to solve certain problems in [[enumerative geometry]], some of which had resisted solution for decades or more.<ref>Yau and Nadis 2010, p.165</ref> These results were presented to mathematicians at a conference at the [[Mathematical Sciences Research Institute]] (MSRI) in [[Berkeley, California]] in May 1991. During this conference, it was noticed that one of the numbers Candelas had computed for the problem of counting [[rational curve]]s disagreed with the number obtained by [[Norwegians|Norwegian]] mathematicians [[Geir Ellingsrud]] and [[Stein Arild Strømme]] using ostensibly more rigorous techniques.<ref>Yau and Nadis 2010, p.169--170</ref> Many mathematicians at the conference assumed that Candelas's work contained a mistake since it was not based on rigorous mathematical arguments. However, after examining their solution, Ellingsrud and Strømme discovered an error in their computer code and, upon fixing the code, they got an answer that agreed with the one obtained by Candelas and his collaborators.<ref>Yau and Nadis 2010, p.170</ref>
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| In 1990, Edward Witten introduced [[topological string theory]],<ref>Witten 1990</ref> a simplified version of string theory, and physicists showed that there is a version of mirror symmetry for topological string theory.<ref>Vafa 1992; Witten 1992</ref> This statement about topological string theory is usually taken as the definition of mirror symmetry in the mathematical literature.<ref>Hori et al. 2003, p. xviii</ref> In an address at the [[International Congress of Mathematicians]] in 1994, mathematician [[Maxim Kontsevich]] presented a new mathematical conjecture based on the physical idea of mirror symmetry in topological string theory. Known as [[homological mirror symmetry]], this conjecture formalizes mirror symmetry as an equivalence of two mathematical structures: the [[derived category]] of [[coherent sheaf|coherent sheaves]] on a Calabi-Yau manifold and the [[Fukaya category]] of its mirror.<ref>Kontsevich 1995</ref>
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| Also around 1995, Kontsevich analyzed the results of Candelas, which gave a general formula for the problem of counting [[rational curve]]s on a [[quintic threefold]], and he reformulated these results as a precise mathematical conjecture. In 1996, [[Alexander Givental]] posted a paper which claimed to prove this conjecture of Kontsevich.<ref>Givental 1996, 1998</ref> Initially, many mathematicians found this paper hard to understand, and so there were doubts about the correctness of Givental's proof. In the years following this, [[Bong Lian]], [[Kefeng Liu]], and [[Shing-Tung Yau]] published an independent proof in a series of papers.<ref>Lian, Liu, Yau 1997, 1999, 2000</ref> Despite some controversy over who had published the first proof, these papers are now seen as providing a mathematical proof of the results originally obtained by physicists using mirror symmetry.<ref>Yau and Nadis 2010, p.172</ref>
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| ==See also==
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| * [[3D mirror symmetry]]
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| * [[Chern-Simons theory]]
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| * [[Donaldson-Thomas theory]]
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| * [[Monstrous moonshine]]
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| * [[Seiberg-Witten invariant]]
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| * [[Vertex operator algebra]]
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| * [[Wall-crossing]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| {{refbegin|2}}
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| * {{cite book |editor1-first=Paul |editor1-last=Aspinwall |editor2-first=Tom |editor2-last=Bridgeland |editor3-first=Alastair |editor3-last=Craw |editor4-first=Michael |editor4-last=Douglas |editor5-first=Mark |editor5-last=Gross |editor6-first=Anton |editor6-last=Kapustin |editor7-first=Gregory |editor7-last=Moore |editor8-first=Graeme |editor8-last=Segal |editor9-first=Balázs |editor9-last=Szendröi |editor10-first=P.M.H. |editor10-last=Wilson |title=Dirichlet Branes and Mirror Symmetry |year=2009 |publisher=American Mathematical Society}}
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| * {{cite journal |last1=Candelas |first1=Philip |last2=Horowitz |first2=Gary |last3=Strominger |first3= Andrew |last4=Witten |first4=Edward |year=1985 |title=Vacuum configurations for superstrings |journal=Nuclear Physics B |volume=258 |issue= |pages=46–74|bibcode = 1985NuPhB.258...46C |doi = 10.1016/0550-3213(85)90602-9 }}
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| * {{cite journal |last1=Dixon |first1=Lance |year=1988 |title=Some world-sheet properties of superstring compactifications, on orbifolds and otherwise |journal=ICTP Ser. Theoret. Phys. |volume=4 |pages=67–126}}
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| * {{cite journal |last1=Givental |first1=Alexander |year=1996 |title=Equivariant Gromov-Witten invariants |journal=International Mathematics Research Notices |volume=1996 |issue=13 |pages=613–663}}
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| * {{cite journal |last1=Givental |first1=Alexander |year=1998 |title=A mirror theorem for toric complete intersections |journal=Topological field theory, primitive forms and related topics |pages=141–175}}
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| * {{cite book |last1=Greene |first1=Brian |title=The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory |year=2000 |publisher=Random House }}
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| * {{cite book |editor1-first=Kentaro |editor1-last=Hori |editor2-first=Sheldon |editor2-last=Katz |editor3-first=Albrecht |editor3-last=Klemm |editor4-first=Rahul |editor4-last=Pandharipande |editor5-first=Richard |editor5-last=Thomas |editor6-first=Cumrun |editor6-last=Vafa |editor7-first=Ravi |editor7-last=Vakil |editor8-first=Eric |editor8-last= Zaslow|title=Mirror Symmetry |year=2003 |publisher=American Mathematical Society|url=http://math.stanford.edu/~vakil/files/mirrorfinal.pdf|isbn=0821829556}}
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| * {{cite journal |last1=Kikkawa |first1=Keiji |last2=Yamasaki |first2=Masami |year=1984 |title=Casimir effects in superstring theories |journal=Physics Letters B |volume=149 |issue=4 |pages=357–360|bibcode = 1984PhLB..149..357K |doi = 10.1016/0370-2693(84)90423-4 }}
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| * {{cite journal |last1=Kontsevich |first1=Maxim |year=1995 |title=Homological algebra of mirror symmetry |journal=Proceedings of the International Congress of Mathematicians |pages=120–139}}
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| * {{cite journal |last1=Lerche |first1=Wolfgang |last2=Vafa |first2=Cumrun |last3=Warner |first3=Nicholas |year=1989 |title=Chiral rings in <math>N = 2</math> superconformal theories |journal=Nuclear Physics B |volume=324 |issue=2 |pages=427–474|bibcode = 1989NuPhB.324..427L |doi = 10.1016/0550-3213(89)90474-4 }}
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| * {{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=1997 |title=Mirror principle, I |journal=Asian Journal of Math |volume=1 |pages=729–763}}
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| * {{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=1999 |title=Mirror principle, II |journal=Asian Journal of Math |volume=3 |pages=109–146}}
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| * {{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=1999 |title=Mirror principle, III |journal=Asian Journal of Math |volume=3 |pages=771–800}}
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| * {{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=2000 |title=Mirror principle, IV |journal=Surveys in Differential Geometry |pages=475–496}}
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| * {{cite book |last1=Mac Lane |first1=Saunders |title=Categories for the Working Mathematician |year=1998}}
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| * {{cite journal| author=Moore, Gregory | title=What is... a Brane?| journal=Notices of the AMS| year=2005 | url=http://www.ams.org/notices/200502/what-is.pdf |format=PDF| accessdate=June 2013 |page=214| volume=52}}
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| * {{cite journal |last1=Sakai |first1=Norisuke |last2=Senda |first2=Ikuo |year=1986 |title=Vacuum energies of string compactified on torus |journal=Progress of Theoretical Physics |volume=75 |issue=3 |pages=692–705|bibcode = 1986PThPh..75..692S |doi = 10.1143/PTP.75.692 }}
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| * {{cite journal |last1=Strominger |first1=Andrew |last2=Yau |first2=Shing-Tung |last3=Zaslow |first3=Eric |year=1996 |title=Mirror symmetry is T-duality |journal=Nuclear Physics B |volume=479 |issue=1 |pages=243–259|arxiv = hep-th/9606040 |bibcode = 1996NuPhB.479..243S |doi = 10.1016/0550-3213(96)00434-8 }}
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| * {{cite journal |last1=Vafa |first1=Cumrun |year=1992 |title=Topological mirrors and quantum rings |journal=Essays on mirror manifolds |pages=96–119}}
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| * {{cite journal |last1=Witten |first1=Edward |year=1990 |title=On the structure of the topological phase of two-dimensional gravity |journal=Nuclear Physics B |volume=340 |pages=281–332|bibcode = 1990NuPhB.340..281W |doi = 10.1016/0550-3213(90)90449-N }}
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| * {{cite journal |last1=Witten |first1=Edward |year=1992 |title=Mirror manifolds and topological field theory |journal=Essays on mirror manifolds |pages=121–160}}
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| * {{Cite book| first1 = Shing-Tung | last1 = Yau | first2 = Steve | last2 = Nadis | year = 2010 | title = The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions | publisher = Basic Books | isbn = 978-0-465-02023-2 }}
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| * {{Citation | last1=Zaslow | first1=Eric | contribution=Mirror Symmetry | year=2008 | title=The Princeton Companion to Mathematics | editor-last=Gowers | editor-first=Timothy}}
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| * {{cite book |last1=Zwiebach |first1=Barton |title=A First Course in String Theory |year=2009 |publisher=Cambridge University Press}}
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| {{refend}}
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| ==Further reading==
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| ===Popularizations===
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| * {{Cite book| first1 = Shing-Tung | last1 = Yau | first2 = Steve | last2 = Nadis | year = 2010 | title = The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions | publisher = Basic Books | isbn = 978-0-465-02023-2 }}
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| * {{cite web |url=http://www.claymath.org/library/senior_scholars/zaslow_physmatics.pdf |title=Physmatics |last1=Zaslow |first1=Eric |year=2005 |accessdate=September 11, 2013}}
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| * {{Citation | last1=Zaslow | first1=Eric | contribution=Mirror Symmetry | year=2008 | title=The Princeton Companion to Mathematics | editor-last=Gowers | editor-first=Timothy}}
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| ===Textbooks===
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| * {{cite book |editor1-first=Paul |editor1-last=Aspinwall |editor2-first=Tom |editor2-last=Bridgeland |editor3-first=Alastair |editor3-last=Craw |editor4-first=Michael |editor4-last=Douglas |editor5-first=Mark |editor5-last=Gross |editor6-first=Anton |editor6-last=Kapustin |editor7-first=Gregory |editor7-last=Moore |editor8-first=Graeme |editor8-last=Segal |editor9-first=Balázs |editor9-last=Szendröi |editor10-first=P.M.H. |editor10-last=Wilson |title=Dirichlet Branes and Mirror Symmetry |year=2009 |publisher=American Mathematical Society}}
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| * {{cite book |editor1-first=Kentaro |editor1-last=Hori |editor2-first=Sheldon |editor2-last=Katz |editor3-first=Albrecht |editor3-last=Klemm |editor4-first=Rahul |editor4-last=Pandharipande |editor5-first=Richard |editor5-last=Thomas |editor6-first=Cumrun |editor6-last=Vafa |editor7-first=Ravi |editor7-last=Vakil |editor8-first=Eric |editor8-last= Zaslow|title=Mirror Symmetry |year=2003 |publisher=American Mathematical Society|url=http://math.stanford.edu/~vakil/files/mirrorfinal.pdf|isbn=0821829556}}
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| {{Good article}}
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| {{DEFAULTSORT:Mirror Symmetry (String Theory)}}
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| [[Category:Algebraic geometry]]
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| [[Category:String theory]]
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| [[es:Simetría especular]]
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| [[sv:Spegelsymmetri]]
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| [[zh:镜像对称]]
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