Monstrous moonshine

The mathematical study of symmetry, is called group theory. The symmetries themselves, are the various operations that can be performed, on a geometric shape, without changing the shape itself. Mathematicians define a collection of symmetries as a "group". A finite group, is a group that describes only a finite amount of symmetries. A classification of a mathematical group, would be a list, of all of it's objects. The consensus, among mathematicians, was that finite groups are to diverse to be sufficiently classified. However, what was done, was the classification of all simple finite groups, which could lead to the construction of a sufficient classification of finite groups. This classification would be a list of all possible simple finite groups. The construction itself, contained several infinite families of groups. These were the simple finite groups that fell into natural categories. The classification, however, also contained 26 groups that don't fit into any kind of family: sporadic groups. These sporadic groups, could not be categorized. The monster group, is the largest sporadic group, and has 10^53 elements. That number is more than 1,000 times the number of atoms contained in the Earth. The monster group is a massive algebraic entity that contained sufficient information to capture this new kind of symmetry. 

Modular functions, are a certain class of functions, that, when graphed, form a kind of repeating pattern. Belonging to this special class of functions is the j-function, one of the most fundamental objects in number theory. The graph of the j-function ha a repeating pattern. John McKay and John Thompson, in the summer of 1978, are going to propose that the j-function of number theory and the theory of finite groups, are more related that one would initially anticipate. At first, it appeared like an odd coincidence. What McKay and Thompson noticed is that numbers that arose in the monster group's analysis, were very closely related to numbers in the j-function's formula. What McKay noticed was that the first significant coefficient for the j-function was the number: 196,884. John McKay made the connection: this is the sum of the first two special dimensions of the monster group (1 + 196,883). These were the first two dimensions that acted in special ways for the monster group. This was quite a peculiar discovery, since, there was really no reason for physicists to believe that the monster group and the j-function were related in any way. However, John Thompson, is going to be interested in this peculiar connection between modular functions and simple finite groups. What Thompson discovered, was that, the 2nd coefficient of the j-function: 21,493,760, is the sum of the monster group’s first 3 special dimensions (1 + 196,883 + 21,296,876). It was apparent that the j-function and the monster group were more intimately related than what was once anticipated. 

In 1979, John Horton Conway and Simon Norton are going to name this relationship:  "monstrous moonshine". The reason they named it thus, is, since it seemed to be such a far-fetched idea. Monstrous moonshine, thus, was this unexpected relationship between the modular functions (the j-function in particular) with the monster group. 

Richard Borcherds proved that the bridge between the monster group and the j-function, was, indeed: string theory. It was a particular version of string theory that existed at the time. It was 24 dimensional, and, physicists were not particularly enamored with it. However, it led to some alarming mathematical results. The idea was that the j-function could show the different energy levels that strings could oscillate at, while, the monster group would capture the information for the symmetry of those energy levels. It was useful, since, the massive monster group, could now be studied, by means of the j-function, where it is much easier to make calculations.

However, string theory's connection to monstrous moonshine did not end here. Yuji Tachikawa, Hirosi Ooguri and Tohru Eguchi, in 2010, proved connections between another sporadic group (the Mathieu group M24) with string theory. The idea began with K3 surfaces, which are 4-dimensional objects that appear in string theory. K3 surfaces, were not understood well enough at the time, to explain how strings could oscillate in these dimensions. However, a function was constructed, that counted the physical states that appear in these K3 surfaces. What these string theorists realized is that, this function, could be written in such a fashion, that, they were the same as the M24 group. The M24 group has about 250 elements. Physicists as well as mathematicians were all very excited over the discovery of a new moonshine. 

Another proposal will be Umbral moonshine, this was proposed by Jeffrey A. Harvey, John Duncan and Miranda Cheng, a generalization moonshine, that is later going to be mathematically proven by Duncan himself, Ken Ono, and Michael Griffin. Umbral moonshine was conjectured in 2012. The Umbral moonshine conjecture proposes that, in addition to monstrous moonshine, there are 23 other moonshines. Each of these moonshines is a correspondence between a particular symmetric group with the coefficients of a special function. These other moonshines, also, appear to have connections to string theory, and with it’s 4-dimensional K3 surfaces.

Lastly, Edward Witten, in 2007, had speculated that the string theory that appears in monstrous moonshine, could have connections to a particular model of 3-spacetime dimensional gravitation.