Mirror symmetry (disambiguation)

Last updated August 20, 2020

Mirror symmetry may refer to:

Mirror symmetry (string theory), a relation between two Calabi–Yau manifolds in string theory
Homological mirror symmetry, a mathematical conjecture about Calabi–Yau manifolds made by Maxim Kontsevich
Mirror symmetry conjecture, a mathematical conjecture about mirror symmetry
P-symmetry, symmetry under parity inversion
Reflection symmetry, which mirror symmetry is a synonym for
Mirror-symmetry breaking

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{{About|physics|string algorithms|atical theory|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 interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.

Calabi–Yau manifold Riemannian manifold with SU(n) holonomy

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 theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.

Shing-Tung Yau is an American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University.

In theoretical physics, T-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 describes strings propagating in an imaginary spacetime shaped like a circle of some radius $, while the other theory describes strings propagating on a spacetime shaped like a circle of radius proportional to . The idea of T-duality was first noted by Bala Sathiapalan in an obscure paper in 1987. The two T-dual theories are equivalent in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, momentum in one description takes discrete values and is equal to the number of times the string winds around the circle in the dual description.$

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional spaces.

Andrew Eben Strominger 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 work on Calabi–Yau compactification and topology change in string theory, and on the stringy origin of black hole entropy. He is a senior fellow at the Society of Fellows, and is the Gwill E. York Professor of Physics.

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.

In theoretical physics, topological string theory is a version of string theory. Topological string theory was first thought of and is studied by physicists such as Edward Witten and Cumrun Vafa.

Alexander Givental is a Russian American mathematician working in the area of symplectic topology, singularity theory and their relations to topological string theories. He earned his Ph.D. under the supervision of V. I. Arnold, in 1987. He provided the first proof of the mirror conjecture for Calabi–Yau manifolds that are complete intersections in toric ambient spaces, in particular for quintic hypersurfaces in P⁴. He is now Professor of Mathematics at the University of California, Berkeley.

Fedor Alekseyevich Bogomolov is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at the Steklov Institute in Moscow before he became a professor at the Courant Institute in New York. He is most famous for his pioneering work on hyperkähler manifolds.

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-duality".

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 according to the rules of quantum mechanics. They have mass and can have other attributes such as charge.

Richard Paul Winsley Thomas FRS is a British mathematician working in several areas of geometry. He is a professor at Imperial College London. He studies moduli problems in algebraic geometry, and ‘mirror symmetry’—a phenomenon in pure mathematics predicted by string theory in theoretical physics.

In theoretical physics, particularly string theory and M-theory, the notion of a flop-transition is basically the shrinking of a sphere in a Calabi-Yau space to the point of tearing. Based on typical spacetime topology, this is not possible due to mathematical technicalities. On the other hand, Mirror symmetry allows for the mathematical similarity between two distinct Calabi-Yau manifolds. If one undergoes a flop-transition, the mirror of it should result in identical mathematical properties, which it does.

A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for $-dimensional convex polyhedra. The most famous examples of the polar duality provide Platonic solids: e.g., the cube is dual to octahedron, the dodecahedron is dual to icosahedron. There is a natural bijection between the -dimensional faces of a -dimensional convex polyhedron and -dimensional faces of the dual polyhedron and one has . In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special -dimensional convex lattice polytopes which are called reflexive polytopes.$

Mark William Gross is an American mathematician, specializing in differential geometry, algebraic geometry, and mirror symmetry.

Paul Stephen Aspinwall is a British theoretical physicist and mathematician, who works on string theory and also algebraic geometry.

Xenia de la Ossa Osegueda is a theoretical physicist whose research focuses on mathematical structures that arise in string theory. She is a professor at Oxford's Mathematical Institute.

In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold to integrals from a family of varieties. In short, this means there is a relation between the number of genus $algebraic curves of degree on a Calabi-Yau variety and integrals on a dual variety . These relations were original discovered by Candelas, De la Ossa, Green, and Parkes in a paper studying a generic quintic threefold in as the variety and a construction from the quintic Dwork family giving . Shortly after, Sheldon Katz wrote a summary paper outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.$

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