# Mirror symmetry (string theory)

In mathematics 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 hidden dimensions of string theory. In this case, the manifolds are said to be "mirror" to one another.

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 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 nonrigorous ideas from theoretical physics, mathematicians have gone on to rigorously prove some of the mathematical predictions of mirror symmetry.

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.[1] Mirror symmetry is also a fundamental tool for doing calculations in string theory. Major approaches to mirror symmetry include homological mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.

## Overview

### Idea

Mirror symmetry is a particular example of what physicists call a duality. In physics, the term duality refers to a situation where two seemingly different physical systems 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.[2]

Like many of the dualities studied in theoretical physics, mirror symmetry was discovered in the context of string theory.[3] In string theory, particles are modeled not as zero-dimensional points but as one-dimensional extended objects called strings. One of the peculiar features of string theory is that it requires extra dimensions of spacetime for its mathematical consistency. In everyday life, there are three familiar dimensions of space (up/down, left/right, and forward/backward), and there is one dimension of time (later/earlier). Thus, in the language of modern physics, one says that spacetime is four-dimensional. In superstring theory, the version of the theory that incorporates supersymmetry, there are six extra dimensions of spacetime in addition to the four that are familiar from everyday experience.[4]

A three-dimensional projection of a Calabi–Yau manifold

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. 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.[5] A standard analogy for this is to consider multidimensional object such as a garden hose. 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.[6]

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.[7] 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,[8] and this version has been rigorously proven by mathematicians.[9] 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.[10]

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.[11] 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 algebraic geometry by solving equivalent problems for the mirror Calabi-Yau.[12] 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.[13]

### Complex geometry

To understand the kind of geometry that appears on one side of the mirror duality, consider the 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, first choose a pair of complex numbers $\omega_1$ and $\omega_2$ such that the quotient $\omega_1/\omega_2$ is not real. This last condition ensures that these points are not collinear. Then the chosen points determine a parallelogram whose other vertices are 0 and $\omega_1+\omega_2$. By identifying opposite sides of this parallelogram, one obtains a torus.

A torus can be constructed by identifying opposite sides of a parallelogram in the complex plane. This torus inherits a complex structure which, roughly speaking, describes the "shape" of the torus.

The tori obtained in this way are all equivalent in the sense that one torus can be continuously deformed into another.[14] On the other hand, they have an additional structure which makes it possible to distinguish them. Namely, the tori constructed in this way have a complex structure, meaning that a neighborhood of any point on such a torus looks just like a region in the complex plane.[10]

If in the construction of a torus one uses instead a pair of complex numbers $\omega_1'$ and $\omega_2'$ related to the original pair by rescaling by a common factor (that is, $\omega_1'=\lambda\omega_1$ and $\omega_2'=\lambda\omega_2$ for some complex number $\lambda$), then the result is an equivalent torus. It is therefore more convenient to parametrize the collection of tori by the ratio $\tau=\omega_1/\omega_2$, which does not change upon rescaling the $\omega_i$. One can assume, without loss of generality, that this parameter $\tau$ has positive imaginary part so that $\tau$ takes values in the upper half plane. One can also show that the parameters $\tau$, $\tau+1$, and $-1/\tau$ correspond to the same torus.

If two tori correspond to genuinely different values of $\tau$, then they have inequivalent complex structures.[15] One can think of the parameter $\tau$ 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, in the special case where "spacetime" is a torus, the theory depends continuously only on the parameter $\tau$.[10]

### Symplectic geometry

Another aspect of the geometry of a torus is the size of the torus. More precisely, one can 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 $\rho dxdy$ on this square. By integrating this area element over the unit square, one obtains the area $\rho$ 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.[10]

In mirror symmetry, the A-model of topological string theory is a theory that depends only on the symplectic geometry of spacetime. Thus, in the special case where "spacetime" is a torus, the A-model depends continuously only on the parameter $\rho$.[10]

### T-duality

A torus is the product of two circles. In this case the red circle is swept around axis defining the pink circle. $R_1$ is the radius of the red circle, $R_2$ is the radius of the pink one.

The above discussion shows that a torus can be studied as the space obtained by identifying opposite sides of a parallelogram in the complex plane. A particularly simple example is where the complex numbers $\omega_1$ and $\omega_2$ lie on the real and imaginary axes, respectively. In this case, one can write $\omega_1=R_1$ and $\omega_2=iR_2$ where $R_1$ and $R_2$ are real numbers. The complex structure on the torus obtained in this way is characterized by the number $\tau=iR_2/R_1$.

In the case of a two-dimensional torus, the symplectic structure is determined by an area element. One can choose coordinates $x$ and $y$ on the parallelogram so that each side of the parallelogram spanned by the chosen complex numbers has length 1. Then the area element of the torus is $R_1R_2dxdy$ which integrates to $R_1R_2$ on the unit square. Define the symplectic parameter $\rho$ to be the product $iR_1R_2$.

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).

Imagine now that the torus represents the "spacetime" for a physical theory. The fundamental objects of this theory will be strings propagating through the spacetime according to the rules of quantum mechanics. One of the basic dualities of string theory is T-duality, which states that a string propagating around a circle of radius $R$ is equivalent to a string propagating around a circle of radius $1/R$ in the sense that all observable quantities in one description are identified with quantities in the dual description.[16] For example, the momentum of a string in one description takes discrete values and is equal to the number of times the string winds around the circle in the dual description.[16] Applying T-duality to the longitudinal circle on the torus, one finds that there is an equivalent description in which spacetime is represented by a different torus. The duality changes $R_1$ to $1/R_1$, and thus it interchanges the complex and symplectic parameters: $\tau\leftrightarrow\rho$.

### The general case

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 above is the only compact Calabi-Yau manifold of (real) dimension two and therefore provides the simplest example of mirror symmetry.[17] 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.[5]

As in the above example, these Calabi-Yau manifolds may have very different shapes. The shape of a six-dimensional Calabi-Yau manifold is described mathematically using certain numerical invariants (numbers associated to the manifolds so that geometrically similar manifolds have the same associated numbers). For example, the shape of a Calabi-Yau manifold is roughly described by a number called the Euler characteristic, and Calabi-Yau mirror pairs can have Euler characteristics of opposite sign.[18] Many different-looking shapes have the same Euler characteristic, so this invariant provides only a crude description of the shape of a Calabi-Yau manifold. This crudeness can be refined, however, by breaking the Euler characteristic into a sum of numbers called Betti numbers.[19] A further refinement is provided by invariants called Hodge numbers, which exhibit interesting symmetries for mirror Calabi-Yau manifolds.[20]

In general in mirror symmetry, one is interested in computing physical quantities called correlation functions.[21] One example of a correlation function is the string partition function. In the A-model, the expression representing the string partition function involves infinitely many numbers called Gromov-Witten invariants, which can be difficult to compute. However, mirror symmetry relates the A-model partition function to the B-model partition function, which depends only on the classical complex geometry of the Calabi-Yau manifold and is easier to compute. This fact is what led mathematicians to become interested in mirror symmetry at its inception.[22]

## Applications

### Enumerative geometry

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, who asked how many circles in the plane are tangent to three given circles. In general, the solution to the problem of Apollonius is that there are eight such circles.[23] The picture on the right shows an example where the three given circles are illustrated in black.

Enumerative problems in mathematics often concern a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic is the surface illustrated on the left which is defined using a certain polynomial of 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.[24]

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.[23]

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."[25] 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. Candelas and his collaborators found that these six-dimensional Calabi-Yau manifolds can contain exactly 317,206,375 curves of degree three.[25]

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.[26] Although the methods used in this work were based on nonrigorous ideas from theoretical physics, mathematicians have gone on to rigorously prove some of the predictions of mirror symmetry. In particular, the enumerative predictions of mirror symmetry have now been rigorously proven.[27]

### Theoretical physics

In addition to its applications in 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.[28]

Outside of string theory, mirror symmetry is used to understand aspects of quantum field theory, the formalism that physicists use to describe elementary particles. For example, mirror symmetry can be used to understand the dynamics of gauge theories (a class of highly symmetric physical theories appearing, for example, in the standard model of particle physics). Such theories arise from strings propagating on a nearly singular background, and mirror symmetry is a useful tool for doing computations in these theories.[29] There is also a generalization of mirror symmetry called 3D mirror symmetry which relates pairs of quantum field theories in three spacetime dimensions.[30]

## Approaches

### Homological mirror symmetry

Open strings attached to a pair of D-branes

In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. 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.[31]

In string theory, a string may be open (forming a segment with two endpoints) or closed (forming a closed loop). D-branes are an important class of branes that arise when one considers open strings. 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 one imposes a certain mathematical condition on the system known as the Dirichlet boundary condition.[32]

Mathematically, branes can be described using the notion of a category.[33] 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, sets, vector spaces, or topological spaces) and the morphisms are given by functions between these structures.[34] One can also consider categories where the objects are D-branes and the morphisms between two branes $\alpha$ and $\beta$ are wavefunctions of open strings stretched between $\alpha$ and $\beta$.[35]

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.[36] The homological mirror symmetry conjecture of Maxim Kontsevich states that these two categories of branes are equivalent in a certain sense.[37]

### Strominger-Yau-Zaslow conjecture

Another approach to understanding mirror symmetry was suggested by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow in 1996.[38] According to their conjecture, which is now known as 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.[39]

Recall that the Overview section considered the case of a torus, which was viewed as the cartesian product of two circles. In other words, the torus was decomposed as the 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 $R_1$ to $1/R_1$.[40]

The SYZ conjecture generalizes this idea to the more complicated case of six-dimensional Calabi-Yau manifolds. As in the case of a torus, one can divide a six-dimensional Calabi-Yau into simpler pieces, which in this case are 3-tori (three-dimensional objects which generalize the notion of a torus) parametrized by a 3-sphere (a three-dimensional generalization of a sphere).[41] 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. In this way, the SYZ conjecture provides a geometric picture of how mirror symmetry acts on a Calabi-Yau manifold.[42]

## History and development

### Discovery

The idea of mirror symmetry can be traced back to the mid-1980s when it was noticed that a string propagating on a circle of radius $R$ is physically equivalent to a string propagating on a circle of radius $1/R$ in appropriate units.[43] This phenomenon is now known as T-duality and is understood to be closely related to mirror symmetry.[38]

In a paper from 1985, Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten showed that by compactifying string theory on a Calabi-Yau manifold, one obtains a theory roughly similar to the standard model of particle physics.[44] 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.[45]

By studying the relationship between Calabi-Yau manifolds and certain conformal field theories, Brian Greene and Ronen Plesser found nontrivial examples of the mirror relationship.[46] Further evidence for this relationship came from the work of Philip Candelas and two of his students, who surveyed a large number of Calabi-Yau manifolds by computer and found that they came in pairs as predicted by mirror symmetry.[47]

### Mirror symmetry finds applications

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,[48] some of which had resisted solution for decades or more.[49] 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 curves disagreed with the number obtained by Norwegian mathematicians Geir Ellingsrud and Stein Arild Strømme using ostensibly more rigorous techniques.[50] 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.[51]

### Mirror symmetry proved

In 1990, Edward Witten introduced topological string theory,[8] a simplified version of string theory, and physicists showed that there is a version of mirror symmetry for topological string theory.[52] This statement about topological string theory is usually taken as the definition of mirror symmetry in the mathematical literature.[53] 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 sheaves on a Calabi-Yau manifold and the Fukaya category of its mirror.[54]

Also around 1995, Kontsevich analyzed the results of Candelas, which gave a general formula for the problem of counting rational curves on a quintic threefold, and he reformulated these results as a precise mathematical conjecture.[55] In 1996, Alexander Givental posted a paper which claimed to prove this conjecture of Kontsevich.[56] 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.[57] Despite some controversy over who had published the first proof, these papers are now collectively seen as providing a mathematical proof of the results originally obtained by physicists using mirror symmetry.[27] In 2000, Kentaro Hori and Cumrun Vafa gave another physical proof of mirror symmetry based on T-duality.[58]

## Notes

1. ^ Hori et al. 2003; Aspinwall et al. 2009
2. ^ Hori et al. 2003, p. xvi
3. ^ Other dualities that arise in string theory are S-duality, T-duality, U-duality, and the AdS/CFT correspondence.
4. ^ Zwiebach 2009, p. 8
5. ^ a b Yau and Nadis 2010, Ch. 6
6. ^ This analogy is used for example in Greene 2000, p. 186
7. ^ Dixon 1988; Lerche, Vafa, and Warner 1989
8. ^ a b Witten 1990
9. ^ Givental 1996, 1998; Lian, Liu, Yau 1997, 1999, 2000
10. Zaslow 2008, p. 531
11. ^ Zaslow 2008, p. 523
12. ^ Yau and Nadis 2010, p. 168
13. ^ Hori et al. 2003, p. xix
14. ^ Zaslow 2008, p. 530
15. ^ More precisely, tori are parametrized by the fundamental domain for the modular group.
16. ^ a b Zaslow 2008, p. 532
17. ^ Zaslow 2008, p. 533
18. ^ Yau and Nadis 2010, p. 160
19. ^ Yau and Nadis 2010, p. 161
20. ^ Yau and Nadis 2010, p. 163
21. ^ Zaslow 2008, p. 529
22. ^ Zaslow 2008, p. 534
23. ^ a b Yau and Nadis 2010, p. 166
24. ^ Yau and Nadis 2010, p. 167
25. ^ a b Yau and Nadis 2010, p. 169
26. ^ Yau and Nadis 2010, p. 171
27. ^ a b Yau and Nadis 2010, p. 172
28. ^ Zaslow 2008, sec. 10
29. ^ Hori et al. 2003, p. 677
30. ^ Intriligator and Seiberg 1996
31. ^ Moore 2005, p. 214
32. ^ Moore 2005, p. 215
33. ^ Aspinwall et al. 2009
34. ^ A basic reference on category theory is Mac Lane 1998.
35. ^ Zaslow 2008, p. 536
36. ^ Aspinwall et al. 2009, p. 575
37. ^ Aspinwall et al. 2009, p. 616
38. ^ a b Strominger, Yau, and Zaslow 1996
39. ^ Yau and Nadis 2010, p. 174
40. ^ Yau and Nadis 2010, p. 175–8
41. ^ 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.
42. ^ Yau and Nadis 2010, p. 178
43. ^ This was first observed in Kikkawa and Yamasaki 1984 and Sakai and Senda 1986.
44. ^ Candelas et al. 1985
45. ^ This was observed in Dixon 1988 and Lerche, Vafa, and Warner 1989.
46. ^ Green and Plesser 1990; Yau and Nadis 2010, p. 158
47. ^ Candelas, Lynker, and Schimmrigk 1990; Yau and Nadis 2010, p. 163
48. ^ Candelas et al. 1991
49. ^ Yau and Nadis 2010, p. 165
50. ^ Yau and Nadis 2010, pp. 169–170
51. ^ Yau and Nadis 2010, p. 170
52. ^ Vafa 1992; Witten 1992
53. ^ Hori et al. 2003, p. xviii
54. ^ Kontsevich 1995a
55. ^ Kontsevich 1995b
56. ^ Givental 1996, 1998
57. ^ Lian, Liu, Yau 1997, 1999a, 1999b, 2000
58. ^ Hori and Vafa 2000

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