Mirror symmetry (string theory)

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History Glossary
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In physics,

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 is four-dimensional.^[2] 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 a theoretical idea called supersymmetry, there are six extra dimensions of spacetime in addition to the four that are familiar from everyday experience.^[3]

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, in which the extra dimensions are assumed to "close up" on themselves to form circles.^[4] 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.^[5]

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.^[4] A Calabi–Yau manifold is a special space which is typically taken to be six-dimensional in applications to string theory. It is named after mathematicians Eugenio Calabi and Shing-Tung Yau.^[6]

After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions, many physicists began studying these manifolds. In the late 1980s,

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.^[10] Such dualities play an important role in modern physics, especially in string theory.^[b]

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.

By studying the relationship between Calabi–Yau manifolds and certain conformal field theories called Gepner models, Brian Greene and Ronen Plesser found nontrivial examples of the mirror relationship.^[22] 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.^[23]

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

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.^[31] In 1996, Alexander Givental posted a paper that claimed to prove this conjecture of Kontsevich.^[32] Initially, many mathematicians found this paper hard to understand, so there were doubts about its correctness. Subsequently, Bong Lian, Kefeng Liu, and Shing-Tung Yau published an independent proof in a series of papers.^[33] 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.^[34] In 2000, Kentaro Hori and Cumrun Vafa gave another physical proof of mirror symmetry based on T-duality.^[13]

Work on mirror symmetry continues today with major developments in the context of strings on

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

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, 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.^[37]

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."^[39] 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.^[39]

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.^[40] 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.^[34]

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 invariants, which are extremely difficult to compute. In the B-model, the calculations can be reduced to classical integrals and are much easier.^[41] 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.^[42]

Outside of string theory, mirror symmetry is used to understand aspects of

In string theory and related theories in physics, 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.^[46]

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 a condition that it satisfies, the Dirichlet boundary condition.^[47]

Mathematically, branes can be described using the notion of a

One can also consider categories where the objects are D-branes and the morphisms between two branes ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are

states

of open strings stretched between ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.

In the B-model of topological string theory, the D-branes are

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.[54] 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.^[55]

Strominger–Yau–Zaslow conjecture

Another approach to understanding mirror symmetry was suggested by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow in 1996.^[19] 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.^[56]

The simplest example of a Calabi–Yau manifold is a two-dimensional torus or donut shape.^[57] 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 of such circles.^[58]

One can choose an auxiliary circle ${\displaystyle B}$ (the pink circle in the figure) such that each of the infinitely many circles decomposing the torus passes through a point of ${\displaystyle B}$. This auxiliary circle is said to parametrize the circles of the decomposition, meaning there is a correspondence between them and points of ${\displaystyle B}$. The circle ${\displaystyle B}$ 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.^[52]

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. 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 ${\displaystyle B}$ is an ordinary

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 ${\displaystyle B}$ (a three-dimensional generalization of a sphere). Each point of ${\displaystyle B}$ 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.^[59]

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

. One of the basic dualities of string theory is T-duality, which states that a string propagating around a circle of radius ${\displaystyle R}$ is equivalent to a string propagating around a circle of radius ${\displaystyle 1/R}$ in the sense that all observable quantities in one description are identified with quantities in the dual description.

${\displaystyle p}$

${\displaystyle n}$

${\displaystyle n}$

${\displaystyle p}$

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 ${\displaystyle B}$ provides a kind of blueprint that describes how these tori are assembled into a Calabi–Yau manifold.^[62]

See also

• Donaldson–Thomas theory
• Wall-crossing

Notes