Editing String theory (section)

=== 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/>