Editing Mirror symmetry (string theory) (section)

===Calabi–Yau manifolds===
{{main article|Calabi–Yau manifold}}
[[Image:Calabi yau.jpg|right|thumb|alt=Visualization of a complex mathematical surface with many convolutions and self intersections.|A cross section of a quintic [[Calabi–Yau manifold]] ]]
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]].<ref name=autogenerated1 /> A Calabi–Yau manifold is a special [[topological space|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]].<ref>{{harvnb|Yau|Nadis|2010|page=ix}}.</ref>

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]], and Nick Warner noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold.<ref>{{harvnb|Dixon|1988}}; {{harvnb|Lerche|Vafa|Warner|1989}}.</ref> Instead, two different versions of string theory called [[type IIA string theory]] and [[type IIB]] can be compactified on completely different Calabi–Yau manifolds giving rise to the same physics.{{efn|The shape of a Calabi–Yau manifold is described mathematically using an array of numbers called [[Hodge number]]s. The arrays corresponding to mirror Calabi–Yau manifolds are different in general, reflecting the different shapes of the manifolds, but they are related by a certain symmetry.<ref>For more information, see {{harvnb|Yau|Nadis|2010|pages=160–163}}.</ref>}} In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry.<ref>{{harvnb|Aspinwall et al.|2009|page=13}}.</ref>

The mirror symmetry relationship is a particular example of what physicists call a [[string duality|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.<ref>{{harvnb|Hori et al.|2003|page=xvi}}.</ref> Such dualities play an important role in modern physics, especially in string theory.{{efn|Other dualities that arise in string theory are [[S-duality]], [[T-duality]], and the [[AdS/CFT correspondence]].}}

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.<ref>{{harvnb|Zaslow|2008|page=523}}.</ref> The Calabi–Yau manifolds used in string theory are of interest in [[pure mathematics]], and mirror symmetry allows mathematicians to solve problems in [[enumerative geometry|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]]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.<ref>{{harvnb|Yau|Nadis|2010|page=168}}.</ref>

In physics, mirror symmetry is justified on physical grounds.<ref name=autogenerated10>{{harvnb|Hori|Vafa|2000}}.</ref> However, mathematicians generally require [[mathematical rigor|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]], a simplified version of string theory introduced by [[Edward Witten]],<ref name=autogenerated9>{{harvnb|Witten|1990}}.</ref> which has been rigorously proven by mathematicians.<ref>{{harvs|nb|last=Givental|year=1996|year2=1998}}; {{harvs|nb|last=Lian|last2=Liu||last3=Yau|year=1997|year2=1999a|year3=1999b|year4=2000}}.</ref> In the context of topological string theory, mirror symmetry states that two theories called the [[Topological A-model|A-model]] and [[Topological B-model|B-model]] are equivalent in the sense that there is a duality relating them.<ref name=autogenerated5>{{harvnb|Zaslow|2008|page=531}}.</ref> 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.<ref name=developments>{{harvnb|Hori et al.|2003|page=xix}}.</ref>