Editing String theory (section)

=== Mirror symmetry ===
{{main|Mirror symmetry (string theory)}}

[[File:Clebsch Cubic.png|thumb|right|alt=A complex mathematical surface in three dimensions.|The [[Clebsch cubic]] is an example of a kind of geometric object called an [[algebraic variety]]. A classical result of [[enumerative geometry]] states that there are exactly 27 straight lines that lie entirely on this surface.]]

After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many physicists began studying these manifolds. In the late 1980s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold.<ref>[[#Hori|Hori]], p. xvii</ref> Instead, two different versions of string theory, type IIA and type IIB, can be compactified on completely different Calabi–Yau manifolds giving rise to the same physics. In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called [[mirror symmetry (string theory)|mirror symmetry]].<ref name="Aspinwall et al. 2009"/>

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. 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]], a branch of mathematics concerned with counting the numbers of solutions to geometric questions.<ref name="Aspinwall et al. 2009"/><ref>[[#Hori|Hori]]</ref>

Enumerative geometry studies a class of geometric objects called [[algebraic varieties]] which are defined by the vanishing of [[polynomial]]s. For example, the [[Clebsch cubic]] illustrated on the right is an algebraic variety defined using a certain polynomial of [[degree of a polynomial|degree]] three in four variables. A celebrated result of nineteenth-century mathematicians [[Arthur Cayley]] and [[George Salmon]] states that there are exactly 27 straight lines that lie entirely on such a surface.<ref>[[#Yau|Yau and Nadis]], p. 167</ref>

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.<ref>[[#Yau|Yau and Nadis]], p. 166</ref>

By the year 1991, most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish.<ref name="Yau and Nadis 2010, p. 169">[[#Yau|Yau and Nadis]], p. 169</ref> 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 translate difficult mathematical questions about one Calabi–Yau manifold into easier questions about its mirror.<ref name=Candelas1991/> In particular, they used mirror symmetry to show that a six-dimensional Calabi–Yau manifold can contain exactly 317,206,375 curves of degree three.<ref name="Yau and Nadis 2010, p. 169"/> In addition to counting degree-three curves, Candelas and his collaborators obtained a number of more general results for counting rational curves which went far beyond the results obtained by mathematicians.<ref>[[#Yau|Yau and Nadis]], p. 171</ref>

Originally, these results of Candelas were justified on physical grounds. However, mathematicians generally prefer rigorous proofs that do not require an appeal to physical intuition. Inspired by physicists' work on mirror symmetry, mathematicians have therefore constructed their own arguments proving the enumerative predictions of mirror symmetry.{{efn|Two independent mathematical proofs of mirror symmetry were given by Givental<ref name=Givental1996/><ref name=Givental1998/> and Lian et al.<ref name=Lian1997/><ref name=Lian1999/><ref name=Lian2000/>}} 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>[[#Hori|Hori]], p. xix</ref> Major approaches to mirror symmetry include the [[homological mirror symmetry]] program of [[Maxim Kontsevich]]<ref name="Kontsevich 1995"/> and the [[SYZ conjecture]] of Andrew Strominger, Shing-Tung Yau, and [[Eric Zaslow]].<ref name=SYZ/>