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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, mirror symmetry is a relationship between geometric objects called
Calabi–Yau manifold In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
s. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as
extra dimension In physics, extra dimensions are proposed additional space or time dimensions beyond the (3 + 1) typical of observed spacetime, such as the first attempts based on the Kaluza–Klein theory. Among theories proposing extra dimensions are: ...
s of
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
. Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when
Philip Candelas Philip Candelas, (born 24 October 1951, London, UK) is a British physicist and mathematician. After 20 years at the University of Texas at Austin, he served as Rouse Ball Professor of Mathematics at the University of Oxford until 2020 and is a Fe ...
,
Xenia de la Ossa Xenia de la Ossa Osegueda (born 30 June 1958, San José, Costa Rica) is a theoretical physicist whose research focuses on mathematical structures that arise in string theory. She is a professor at Oxford's Mathematical Institute. Academic car ...
, Paul Green, and Linda Parkes showed that it could be used as a tool in
enumerative geometry In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory. History The problem of Apollonius is one of the earliest examp ...
, 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 curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously. Today, mirror symmetry is a major research topic in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
, and mathematicians are working to develop a mathematical understanding of the relationship based on physicists' intuition. Mirror symmetry is also a fundamental tool for doing calculations in string theory, and it has been used to understand aspects of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the formalism that physicists use to describe
elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, anti ...
. Major approaches to mirror symmetry include the
homological mirror symmetry Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address t ...
program of
Maxim Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
and the
SYZ conjecture The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled "Mirror Symmetry is ''T''- ...
of
Andrew Strominger Andrew Eben Strominger (; born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his w ...
,
Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
, and
Eric Zaslow Eric Zaslow is an American mathematical physicist at Northwestern University. Biography Zaslow attended Harvard University, earning his Ph.D. in physics in 1995, with thesis "Kinks, twists, and folds : exploring the geometric musculature of qua ...
.


Overview


Strings and compactification

In physics,
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
is a theoretical framework in which the point-like particles of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
are replaced by one-dimensional objects called
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
. These strings look like small segments or loops of ordinary string. String theory describes how strings propagate through space and interact with each other. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
,
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
, and other properties determined by the vibrational state of the string. Splitting and recombination of strings correspond to particle emission and absorption, giving rise to the interactions between particles. There are notable differences between the world described by string theory and the everyday world. 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
is four-dimensional. One of the peculiar features of string theory is that it requires
extra dimensions In physics, extra dimensions are proposed additional space or time dimensions beyond the (3 + 1) typical of observed spacetime, such as the first attempts based on the Kaluza–Klein theory. Among theories proposing extra dimensions are: ...
of spacetime for its mathematical consistency. In
superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string theor ...
, the version of the theory that incorporates a theoretical idea called
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
, there are six extra dimensions of spacetime in addition to the four that are familiar from everyday experience. One of the goals of current research in string theory is to develop models in which the strings represent particles observed in high energy physics experiments. For such a model to be consistent with observations, its spacetime must be four-dimensional at the relevant distance scales, so one must look for ways to restrict the extra dimensions to smaller scales. In most realistic models of physics based on string theory, this is accomplished by a process called
compactification Compactification may refer to: * Compactification (mathematics), making a topological space compact * Compactification (physics), the "curling up" of extra dimensions in string theory See also * Compaction (disambiguation) Compaction may refer t ...
, in which the extra dimensions are assumed to "close up" on themselves to form circles.Yau and Nadis 2010, Ch. 6 In the limit where these curled up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a 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 on the surface of the hose would move in two dimensions.


Calabi–Yau manifolds

Compactification can be used to construct models in which spacetime is effectively four-dimensional. However, not every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In a viable model of particle physics, the compact extra dimensions must be shaped like a
Calabi–Yau manifold In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
. A Calabi–Yau manifold is a special
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
which is typically taken to be six-dimensional in applications to string theory. It is named after mathematicians
Eugenio Calabi Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...
and
Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
. After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions, many physicists began studying these manifolds. In the late 1980s, Lance Dixon, Wolfgang Lerche,
Cumrun Vafa Cumrun Vafa ( fa, کامران وفا ; born 1 August 1960) is an Iranian-American theoretical physicist and the Hollis Professor of Mathematics and Natural Philosophy at Harvard University. Early life and education Cumrun Vafa was born in Tehran ...
, and Nick Warner noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold. Instead, two different versions of string theory called type IIA string theory and
type IIB In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theory, superstring theories in ten dimens ...
can be compactified on completely different Calabi–Yau manifolds giving rise to the same physics. In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry. The mirror symmetry relationship is a particular example of what physicists call a physical duality. In general, the term ''physical duality'' refers to a situation where two seemingly different physical theories turn out to be equivalent in a nontrivial way. If one theory can be transformed so it looks just like another theory, the two are said to be dual under that transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena. Such dualities play an important role in modern physics, especially in string theory. Regardless of whether Calabi–Yau compactifications of string theory provide a correct description of nature, the existence of the mirror duality between different string theories has significant mathematical consequences. The Calabi–Yau manifolds used in string theory are of interest in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
, and mirror symmetry allows mathematicians to solve problems in enumerative algebraic geometry, a branch of mathematics concerned with counting the numbers of solutions to geometric questions. A classical problem of enumerative geometry is to enumerate the
rational curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s on a Calabi–Yau manifold such as the one illustrated above. By applying mirror symmetry, mathematicians have translated this problem into an equivalent problem for the mirror Calabi–Yau, which turns out to be easier to solve. In physics, mirror symmetry is justified on physical grounds.Hori and Vafa 2000 However, mathematicians generally require rigorous proofs that do not require an appeal to physical intuition. From a mathematical point of view, the version of mirror symmetry described above is still only a conjecture, but there is another version of mirror symmetry in the context of
topological string theory In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological ...
, a simplified version of string theory introduced by
Edward Witten Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
,Witten 1990 which has been rigorously proven by mathematicians. In the context of topological string theory, mirror symmetry states that two theories called the A-model and B-model are equivalent in the sense that there is a duality relating them.Zaslow 2008, p. 531 Today mirror symmetry is an active area of research in mathematics, and mathematicians are working to develop a more complete mathematical understanding of mirror symmetry based on physicists' intuition.Hori et al. 2003, p. xix


History

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 Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
. This phenomenon is now known as
T-duality In theoretical physics, T-duality (short for target-space duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories descr ...
and is understood to be closely related to mirror symmetry.Strominger, Yau, and Zaslow 1996 In a paper from 1985,
Philip Candelas Philip Candelas, (born 24 October 1951, London, UK) is a British physicist and mathematician. After 20 years at the University of Texas at Austin, he served as Rouse Ball Professor of Mathematics at the University of Oxford until 2020 and is a Fe ...
,
Gary Horowitz Gary T. Horowitz (born April 14, 1955 in Washington, D.C.) is an American theoretical physicist who works on string theory and quantum gravity. Biography Horowitz studied at Princeton University (Bachelor 1976) and obtained his Ph.D. in 1979 at t ...
,
Andrew Strominger Andrew Eben Strominger (; born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his w ...
, 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 The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It wa ...
that also consistently incorporates an idea called supersymmetry. Following this development, many physicists began studying Calabi–Yau compactifications, hoping to construct realistic models of particle physics based on string theory. Cumrun Vafa and others noticed that given such a physical model, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold. Instead, there are two Calabi–Yau manifolds that give rise to the same physics. By studying the relationship between Calabi–Yau manifolds and certain
conformal field theories A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
called Gepner models,
Brian Greene Brian Randolph Greene (born February 9, 1963) is a American theoretical physicist, mathematician, and string theorist. Greene was a physics professor at Cornell University from 19901995, and has been a professor at Columbia University since 1 ...
and Ronen Plesser found nontrivial examples of the mirror relationship. Further evidence for this relationship came from the work of Philip Candelas, Monika Lynker, and Rolf Schimmrigk, who surveyed a large number of Calabi–Yau manifolds by computer and found that they came in mirror pairs. Mathematicians became interested in mirror symmetry around 1990 when physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that mirror symmetry could be used to solve problems in enumerative geometry that had resisted solution for decades or more.Yau and Nadis 2010, p. 165 These results were presented to mathematicians at a conference at the
Mathematical Sciences Research Institute The Simons Laufer Mathematical Sciences Institute (SLMath), formerly the Mathematical Sciences Research Institute (MSRI), is an independent nonprofit mathematical research institution on the University of California campus in Berkeley, Califo ...
(MSRI) in
Berkeley, California Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Irish bishop and philosopher George Berkeley. It borders the cities of Oakland and Emer ...
in May 1991. During this conference, it was noticed that one of the numbers Candelas had computed for the counting of rational curves disagreed with the number obtained by
Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe * Norwegians, both a nation and an ethnic group native to Norway * Demographics of Norway *The Norwegian language, including ...
mathematicians Geir Ellingsrud and Stein Arild Strømme using ostensibly more rigorous techniques. 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. In 1990, Edward Witten introduced topological string theory, a simplified version of string theory, and physicists showed that there is a version of mirror symmetry for topological string theory. This statement about topological string theory is usually taken as the definition of mirror symmetry in the mathematical literature. In an address at the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
in 1994, mathematician
Maxim Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
presented a new mathematical conjecture based on the physical idea of mirror symmetry in topological string theory. Known as
homological mirror symmetry Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address t ...
, this conjecture formalizes mirror symmetry as an equivalence of two mathematical structures: the
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on a Calabi–Yau manifold and the Fukaya category of its mirror. 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 In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds. The Hodge diamond of a non-singular quintic 3-fold is Mathem ...
, and he reformulated these results as a precise mathematical conjecture. In 1996,
Alexander Givental Alexander Givental (russian: Александр Борисович Гивенталь) is a Russian-American mathematician working in symplectic topology and singularity theory, as well as their relation to topological string theories. He graduat ...
posted a paper that claimed to prove this conjecture of Kontsevich. Initially, many mathematicians found this paper hard to understand, so there were doubts about its correctness. Subsequently, Bong Lian,
Kefeng Liu Kefeng Liu ( Chinese: 刘克峰; born 12 December 1965), is a Chinese-American mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Cala ...
, and Shing-Tung Yau published an independent proof in a series of papers. Despite 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.Yau and Nadis 2010, p. 172 In 2000, Kentaro Hori and Cumrun Vafa gave another physical proof of mirror symmetry based on T-duality. Work on mirror symmetry continues today with major developments in the context of strings on
surfaces A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat * Sur ...
with boundaries. In addition, mirror symmetry has been related to many active areas of mathematics research, such as the
McKay correspondence In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of . Each node represents an irreducible representation of . If are irreducibl ...
,
topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathe ...
, and the theory of stability conditions. At the same time, basic questions continue to vex. For example, mathematicians still lack an understanding of how to construct examples of mirror Calabi–Yau pairs though there has been progress in understanding this issue.


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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. One of the earliest problems of enumerative geometry was posed around the year 200
BCE Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
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 In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies ...
is that there are eight such circles.Yau and Nadis 2010, p. 166 Enumerative problems in mathematics often concern a class of geometric objects called
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
which are defined by the vanishing of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s. For example, the
Clebsch cubic In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surf ...
(see the illustration) is defined using a certain polynomial of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
three in four variables. A celebrated result of nineteenth-century mathematicians
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, C ...
and
George Salmon George Salmon FBA FRS FRSE (25 September 1819 – 22 January 1904) was a distinguished and influential Irish mathematician and Anglican theologian. After working in algebraic geometry for two decades, Salmon devoted the last forty years of his ...
states that there are exactly 27 straight lines that lie entirely on such a surface. Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi–Yau manifold, such as the one illustrated above, which is defined by a polynomial of degree five. This problem was solved by the nineteenth-century German mathematician
Hermann Schubert __NOTOC__ Hermann Cäsar Hannibal Schubert (22 May 1848 – 20 July 1911) was a German mathematician. Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite n ...
, who found that there are exactly 2,875 such lines. In 1986, geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is 609,250. 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."Yau and Nadis 2010, p. 169 The field was reinvigorated in May 1991 when physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that mirror symmetry could be used to count the number of degree three curves 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. 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. Although the methods used in this work were based on physical intuition, mathematicians have gone on to prove rigorously some of the predictions of mirror symmetry. In particular, the enumerative predictions of mirror symmetry have now been rigorously proven.


Theoretical physics

In addition to its applications in enumerative geometry, mirror symmetry is a fundamental tool for doing calculations in string theory. In the A-model of topological string theory, physically interesting quantities are expressed in terms of infinitely many numbers called
Gromov–Witten invariant In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic man ...
s, which are extremely difficult to compute. In the B-model, the calculations can be reduced to classical
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s and are much easier. By applying mirror symmetry, theorists can translate difficult calculations in the A-model into equivalent but technically easier calculations in the B-model. These calculations are then used to determine the probabilities of various physical processes in string theory. Mirror symmetry can be combined with other dualities to translate calculations in one theory into equivalent calculations in a different theory. By outsourcing calculations to different theories in this way, theorists can calculate quantities that are impossible to calculate without the use of dualities. Outside of string theory, mirror symmetry is used to understand aspects of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the formalism that physicists use to describe
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
s. For example,
gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
are a class of highly symmetric physical theories appearing in the standard model of particle physics and other parts of theoretical physics. Some gauge theories which are not part of the standard model, but which are nevertheless important for theoretical reasons, arise from strings propagating on a nearly singular background. For such theories, mirror symmetry is a useful computational tool. Indeed, mirror symmetry can be used to perform calculations in an important gauge theory in four spacetime dimensions that was studied by
Nathan Seiberg Nathan "Nati" Seiberg (; born September 22, 1956) is an Israeli American theoretical physicist who works on quantum field theory and string theory. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, United ...
and Edward Witten and is also familiar in mathematics in the context of
Donaldson invariant In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting ...
s. There is also a generalization of mirror symmetry called 3D mirror symmetry which relates pairs of quantum field theories in three spacetime dimensions.


Approaches


Homological mirror symmetry

In string theory and related theories in physics, a ''
brane 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. Branes are dynamical objects which can propagate through spacetime accordin ...
'' 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. In string theory, a string may be open (forming a segment with two endpoints) or closed (forming a closed loop).
D-brane In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
s 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 a condition that it satisfies, the
Dirichlet boundary condition In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
. Mathematically, branes can be described using the notion of a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. This is a mathematical structure consisting of ''objects'', and for any pair of objects, a set of ''
morphisms In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
'' between them. In most examples, the objects are mathematical structures (such as sets,
vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, or
topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
) and the morphisms are functions between these structures. One can also consider categories where the objects are D-branes and the morphisms between two branes \alpha and \beta are states of open strings stretched between \alpha and \beta.Zaslow 2008, p. 536 In the B-model of topological string theory, the D-branes are complex submanifolds of a Calabi–Yau together with additional data that arise physically from having charges at the endpoints of strings. Intuitively, one can think of a submanifold as a surface embedded inside the Calabi–Yau, although submanifolds can also exist in dimensions different from two. In mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi–Yau.Aspinwal et al. 2009, p. 575 In the A-model, the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold. Roughly speaking, they are what mathematicians call special Lagrangian submanifolds. This means among other things that they have half the dimension of the space in which they sit, and they are length-, area-, or volume-minimizing.Yau and Nadis 2010, p. 175 The category having these branes as its objects is called the Fukaya category. The derived category of coherent sheaves is constructed using tools from
complex geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, a branch of mathematics that describes geometric curves in algebraic terms and solves geometric problems using
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s. On the other hand, the Fukaya category is constructed using
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
, a branch of mathematics that arose from studies of
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
. Symplectic geometry studies spaces equipped with a
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...
, a mathematical tool that can be used to compute
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
in two-dimensional examples. The homological mirror symmetry conjecture of Maxim Kontsevich states that the derived category of coherent sheaves on one Calabi–Yau manifold is equivalent in a certain sense to the Fukaya category of its mirror. This equivalence provides a precise mathematical formulation of mirror symmetry in topological string theory. In addition, it provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.


Strominger–Yau–Zaslow conjecture

Another approach to understanding mirror symmetry was suggested by Andrew Strominger, Shing-Tung Yau, and
Eric Zaslow Eric Zaslow is an American mathematical physicist at Northwestern University. Biography Zaslow attended Harvard University, earning his Ph.D. in physics in 1995, with thesis "Kinks, twists, and folds : exploring the geometric musculature of qua ...
in 1996. According to their conjecture, now known as the SYZ conjecture, mirror symmetry can be understood by dividing a Calabi–Yau manifold into simpler pieces and then transforming them to get the mirror Calabi–Yau. The simplest example of a Calabi–Yau manifold is a two-dimensional
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
or donut shape. Consider a circle on this surface that goes once through the hole of the donut. An example is the red circle in the figure. There are infinitely many circles like it on a torus; in fact, the entire surface is a
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of such circles. One can choose an auxiliary circle B (the pink circle in the figure) such that each of the infinitely many circles decomposing the torus passes through a point of B. This auxiliary circle is said to ''parametrize'' the circles of the decomposition, meaning there is a correspondence between them and points of B. The circle B is more than just a list, however, because it also determines how these circles are arranged on the torus. This auxiliary space plays an important role in the SYZ conjecture. The idea of dividing a torus into pieces parametrized by an auxiliary space can be generalized. Increasing the dimension from two to four real dimensions, the Calabi–Yau becomes a
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...
. Just as the torus was decomposed into circles, a four-dimensional K3 surface can be decomposed into two-dimensional tori. In this case the space B is an ordinary
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. Each point on the sphere corresponds to one of the two-dimensional tori, except for twenty-four "bad" points corresponding to "pinched" or
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
tori. The Calabi–Yau manifolds of primary interest in string theory have six dimensions. One can divide such a manifold into 3-tori (three-dimensional objects that generalize the notion of a torus) parametrized by a 3-sphere B (a three-dimensional generalization of a sphere). Each point of B corresponds to a 3-torus, except for infinitely many "bad" points which form a grid-like pattern of segments on the Calabi–Yau and correspond to singular tori. Once the Calabi–Yau manifold has been decomposed into simpler parts, mirror symmetry can be understood in an intuitive geometric way. As an example, consider the torus described above. Imagine that this torus represents the "spacetime" for a
physical theory Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimen ...
. The fundamental objects of this theory will be strings propagating through the spacetime according to the rules of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. 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.Zaslow 2008, p. 532 For example, a string has
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the
winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...
. If a string has momentum p and winding number n in one description, it will have momentum n and winding number p in the dual description. By applying T-duality simultaneously to all of the circles that decompose the torus, the radii of these circles become inverted, and one is left with a new torus which is "fatter" or "skinnier" than the original. This torus is the mirror of the original Calabi–Yau. T-duality can be extended from circles to the two-dimensional tori appearing in the decomposition of a K3 surface or to the three-dimensional tori appearing in the decomposition of a six-dimensional Calabi–Yau manifold. In general, the SYZ conjecture states that mirror symmetry is equivalent to the simultaneous application of T-duality to these tori. In each case, the space B provides a kind of blueprint that describes how these tori are assembled into a Calabi–Yau manifold.Yau and Nadis 2010, p. 178–9


See also

*
Donaldson–Thomas theory In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual num ...
*
Wall-crossing In algebraic geometry and string theory, the phenomenon of wall-crossing describes the discontinuous change of a certain quantity, such as an integer geometric invariant, an winding number, index or a space of BPS state, across a codimension-one w ...


Notes


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* * * {{DEFAULTSORT:Mirror Symmetry (String Theory) Algebraic geometry Symplectic geometry Mathematical physics String theory