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by Liliana Usvat
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.
In string theory, the different types of observed elementary particles arise from the different quantum states of these strings. In addition to the types of particles postulated by the standard model of particle physics, string theory naturally incorporates gravity, and is therefore a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter.
Aside from this hypothesized role in particle physics, string theory is now widely used as a theoretical tool in physics, and it has shed light on many aspects of quantum field theory and quantum gravity.
In mathematics and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi-Yau manifolds. It can happen, for two such six-dimensional manifolds, that their shapes look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of string theory. In this case, the six-dimensional 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 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. 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.
Like many of the dualities studied in theoretical physics, mirror symmetry was discovered in the context of string theory. 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 superstring theory, the version of the theory that incorporates worldsheet supersymmetry, there are six extra dimensions of spacetime in addition to the four that are familiar from everyday experience.
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.
"Chance favors the prepared mind." - Louis Pasteur