Editing String theory (section)

== Connections to mathematics ==

In addition to influencing research in [[theoretical physics]], string theory has stimulated a number of major developments in [[pure mathematics]]. Like many developing ideas in theoretical physics, string theory does not at present have a [[mathematical rigor|mathematically rigorous]] formulation in which all of its concepts can be defined precisely. As a result, physicists who study string theory are often guided by physical intuition to conjecture relationships between the seemingly different mathematical structures that are used to formalize different parts of the theory. These conjectures are later proved by mathematicians, and in this way, string theory serves as a source of new ideas in pure mathematics.<ref name=Deligne/>

=== 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/>

=== Monstrous moonshine ===
{{main|Monstrous moonshine}}

[[File:Labeled Triangle Reflections.svg|left|thumb|upright=1|alt=An equilateral triangle with a line joining each vertex to the midpoint of the opposite side|An equilateral triangle can be rotated through 120°, 240°, or 360°, or reflected in any of the three lines pictured without changing its shape.]]

[[Group theory]] is the branch of mathematics that studies the concept of [[symmetry]]. For example, one can consider a geometric shape such as an equilateral triangle. There are various operations that one can perform on this triangle without changing its shape. One can rotate it through 120°, 240°, or 360°, or one can reflect in any of the lines labeled {{math|''S''<sub>0</sub>}}, {{math|''S''<sub>1</sub>}}, or {{math|''S''<sub>2</sub>}} in the picture. Each of these operations is called a ''symmetry'', and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a [[group (mathematics)|group]]. In this particular example, the group is known as the [[dihedral group]] of [[order (group theory)|order]] 6 because it has six elements. A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a [[finite group]].<ref name=Dummit/>

Mathematicians often strive for a [[classification theorems|classification]] (or list) of all mathematical objects of a given type. It is generally believed that finite groups are too diverse to admit a useful classification. A more modest but still challenging problem is to classify all finite ''simple'' groups. These are finite groups that may be used as building blocks for constructing arbitrary finite groups in the same way that [[prime number]]s can be used to construct arbitrary [[Integer|whole number]]s by taking products.{{efn|More precisely, a nontrivial group is called ''[[simple group|simple]]'' if its only [[normal subgroup]]s are the [[trivial group]] and the group itself. The [[Jordan–Hölder theorem]] exhibits finite simple groups as the building blocks for all finite groups.}} One of the major achievements of contemporary group theory is the [[classification of finite simple groups]], a mathematical theorem that provides a list of all possible finite simple groups.<ref name=Dummit/>

This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. The latter groups are called the "sporadic" groups, and each one owes its existence to a remarkable combination of circumstances. The largest sporadic group, the so-called [[monster group]], has over {{math|10<sup>53</sup>}} elements, more than a thousand times the number of atoms in the Earth.<ref name="Klarreich 2015"/>

[[Image:KleinInvariantJ.jpg|thumb|upright=1.4|A graph of the [[j-invariant|{{math|''j''}}-function]] in the complex plane]]

A seemingly unrelated construction is the [[j-invariant|{{math|''j''}}-function]] of [[number theory]]. This object belongs to a special class of functions called [[modular function]]s, whose graphs form a certain kind of repeating pattern.<ref>[[#Gannon|Gannon]], p. 2</ref> Although this function appears in a branch of mathematics that seems very different from the theory of finite groups, the two subjects turn out to be intimately related. In the late 1970s, mathematicians [[John McKay (mathematician)|John McKay]] and [[John G. Thompson|John Thompson]] noticed that certain numbers arising in the analysis of the monster group (namely, the dimensions of its [[irreducible representation]]s) are related to numbers that appear in a formula for the {{math|''j''}}-function (namely, the coefficients of its [[Fourier series]]).<ref>[[#Gannon|Gannon]], p. 4</ref> This relationship was further developed by [[John Horton Conway]] and [[Simon P. Norton|Simon Norton]]<ref name=Conway/> who called it [[monstrous moonshine]] because it seemed so far fetched.<ref>[[#Gannon|Gannon]], p. 5</ref>

In 1992, [[Richard Borcherds]] constructed a bridge between the theory of modular functions and finite groups and, in the process, explained the observations of McKay and Thompson.<ref>[[#Gannon|Gannon]], p. 8</ref><ref name=Borcherds/> Borcherds' work used ideas from string theory in an essential way, extending earlier results of [[Igor Frenkel]], [[James Lepowsky]], and [[Arne Meurman]], who had realized the monster group as the symmetries of a particular{{which|date=February 2016}} version of string theory.<ref name=FLM/> In 1998, Borcherds was awarded the [[Fields medal]] for his work.<ref>[[#Gannon|Gannon]], p. 11</ref>

Since the 1990s, the connection between string theory and moonshine has led to further results in mathematics and physics.<ref name="Klarreich 2015"/> In 2010, physicists [[Tohru Eguchi]], [[Hirosi Ooguri]], and Yuji Tachikawa discovered connections between a different sporadic group, the [[Mathieu group M24|Mathieu group {{math|''M''<sub>24</sub>}}]], and a certain version{{which|date=November 2016}} of string theory.<ref name=EOT/> [[Miranda Cheng]], John Duncan, and [[Jeffrey A. Harvey]] proposed a generalization of this moonshine phenomenon called [[umbral moonshine]],<ref name=CDH/> and their conjecture was proved mathematically by Duncan, Michael Griffin, and [[Ken Ono]].<ref name=DGO/> Witten has also speculated that the version of string theory appearing in monstrous moonshine might be related to a certain simplified model of gravity in three spacetime dimensions.<ref name=Witten2007/>