Editing Mirror symmetry (string theory) (section)

==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 <math>R</math> is physically equivalent to a string propagating on a circle of radius <math>1/R</math> in appropriate [[units of measurements|units]].<ref>This was first observed in {{harvnb|Kikkawa|Yamasaki|1984}} and {{harvnb|Sakai|Senda|1986}}.</ref> This phenomenon is now known as [[T-duality]] and is understood to be closely related to mirror symmetry.<ref name=autogenerated3>{{harvnb|Strominger|Yau|Zaslow|1996}}.</ref> In a paper from 1985, [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], 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]] that also consistently incorporates an idea called supersymmetry.<ref>{{harvnb|Candelas|Horowitz|Strominger|Witten|1985}}.</ref> 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.<ref>This was observed in {{harvnb|Dixon|1988}} and {{harvnb|Lerche|Vafa|Warner|1989}}.</ref>

By studying the relationship between Calabi–Yau manifolds and certain [[conformal field theory|conformal field theories]] called Gepner models, [[Brian Greene]] and Ronen Plesser found nontrivial examples of the mirror relationship.<ref>{{harvnb|Greene|Plesser|1990}}; {{harvnb|Yau|Nadis|2010|page=158}}.</ref> 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.<ref>{{harvnb|Candelas|Lynker|Schimmrigk|1990}}; {{harvnb|Yau|Nadis|2010|page=163}}.</ref>

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<ref>{{harvnb|Candelas|de la Ossa|Green|Parkes|1991}}.</ref> that had resisted solution for decades or more.<ref name=autogenerated7>{{harvnb|Yau|Nadis|2010|page=165}}.</ref> These results were presented to mathematicians at a conference at the [[Mathematical Sciences Research Institute]] (MSRI) in [[Berkeley, California]] 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 [[Norwegians|Norwegian]] mathematicians [[Geir Ellingsrud]] and Stein Arild Strømme using ostensibly more rigorous techniques.<ref>{{harvnb|Yau|Nadis|2010|pages=169–170}}.</ref> 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.<ref>{{harvnb|Yau|Nadis|2010|page=170}}</ref>

In 1990, Edward Witten introduced topological string theory,<ref name=autogenerated9 /> a simplified version of string theory, and physicists showed that there is a version of mirror symmetry for topological string theory.<ref>{{harvnb|Vafa|1992}}; {{harvnb|Witten|1992}}.</ref> This statement about topological string theory is usually taken as the definition of mirror symmetry in the mathematical literature.<ref>{{harvnb|Hori et al.|2003|page=xviii}}.</ref> In an address at the [[International Congress of Mathematicians]] in 1994, mathematician [[Maxim Kontsevich]] presented a new mathematical conjecture based on the physical idea of mirror symmetry in topological string theory. Known as [[homological mirror symmetry]], this conjecture formalizes mirror symmetry as an equivalence of two mathematical structures: the [[derived category]] of [[coherent sheaves]] on a Calabi–Yau manifold and the [[Fukaya category]] of its mirror.<ref>{{harvnb|Kontsevich|1995b}}.</ref>

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.<ref>{{harvnb|Kontsevich|1995a}}.</ref> In 1996, [[Alexander Givental]] posted a paper that claimed to prove this conjecture of Kontsevich.<ref>{{harvs|nb|last=Givental|year=1996|year2=1998}}</ref> 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.<ref>{{harvs|nb|last=Lian|last2=Liu||last3=Yau|year=1997|year2=1999a|year3=1999b|year4=2000}}.</ref> 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.<ref name=autogenerated13>{{harvnb|Yau|Nadis|2010|page=172}}.</ref> In 2000, Kentaro Hori and Cumrun Vafa gave another physical proof of mirror symmetry based on T-duality.<ref name=autogenerated10 />

Work on mirror symmetry continues today with major developments in the context of strings on [[Riemann surface|surfaces]] with boundaries.<ref name=developments /> In addition, mirror symmetry has been related to many active areas of mathematics research, such as the [[McKay correspondence]], [[topological quantum field theory]], and the theory of [[wall-crossing|stability conditions]].<ref>{{harvnb|Aspinwall et al.|2009|p=vii}}.</ref> 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.<ref>{{harvnb|Zaslow|2008|page=537}}.</ref>