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The Superstring Origin of Moonshine

Jan 20 2016

One of the most powerful mathematical principles guiding our current understanding of quantum physics is symmetry. We encounter many symmetric objects in our daily life—a square can be rotated about its center in increments of 90 degrees and look the same, while a circle can be rotated any angle. These transformations can be combined (i.e. performed one after another) to form an abstract mathematical structure known as a group. If a physical theory or system possesses symmetries—or more formally speaking, transforms under the action of a group--this often puts very useful constraints on its dynamics and properties. On the other hand, new insights may proceed in the opposite direction: the formal properties of an abstract structure, such as a group, are better understood when they are realized in a concrete physical system. The mathematical subject of moonshine, described below, is a prominent example of this relationship. The authors describe how the seemingly miraculous mathematical properties of certain groups and functions find a natural explanation — and maybe some unexpected generalizations —   when interpreted in terms of physical models in string theory.

Monstrous moonshine refers to the surprising relationship, discovered in the 1970's, between a certain function called the J-function and the largest finite simple group (of order ~10^54), called the Monster group. The former is a modular function: a function defined on the upper half of the complex plane that is invariant under the group SL(2, Z) which, roughly speaking, is the group of discrete symmetries of a torus. One of the key observations of Monstrous moonshine is the fact that the Fourier coefficients of the J-function decompose naturally into dimensions of representations of the Monster group. It was soon realized that the J-function is just the simplest example of a set of 170 modular functions, the McKay-Thompson series, invariant under some very special subgroups of SL(2,R). All of these subgroups are "genus-zero". This means that the quotient of the upper half plane by the relevant subgroup is homeomorphic to a sphere. These Monstrous moonshine observations connected the fields of number theory and group theory in an unforeseen way. 

Later, part of the Monstrous moonshine observation was explained via the construction, due to Frenkel, Lepowsky, and Meurman (FLM), of a certain two-dimensional conformal field theory (or vertex operator algebra). This special vertex operator algebra has a Monster group symmetry, and its partition function is equal to the J-function. In this language, the Monstrous moonshine observation is the natural statement that the states of the theory are organized in representations of the discrete symmetry group, and the McKay-Thompson series are partition functions enriched by an additional insertion of a Monster group element. The Monstrous moonshine conjectures were proven by Borcherds in the 1990's, who constructed and utilized a generalized Kac-Moody algebra called the Monster Lie Algebra, but there has been no satisfactory explanation of the genus zero property and no physical interpretation (not even at the level of the FLM conformal field theory!) of the Monster Lie Algebra.

These longstanding questions have been addressed by the authors in their new preprint. They construct a class of heterotic string compactifications related to the FLM Monster theory, and compute indices counting supersymmetric (BPS) states in these theories. The supersymmetric indices turn out to be closely related to the McKay-Thompson series. By studying certain limits of the indices, they are able to prove that the McKay-Thompson series enjoy the modular properties delineated by the Monstrous moonshine conjectures. The authors further show that the genus zero groups are nothing but groups of T-dualities in their heterotic models. The Monster Lie Algebra and natural generalizations thereof play a central role in the construction, arising as algebras of spontaneously broken gauge symmetries. The authors show that the BPS states in these theories form representations of these algebras. It has long been suspected, starting with the work of Harvey and Moore, that algebras of BPS states comprise generalized Kac-Moody algebras and play a central role in string compactifications. In this paper, the authors realize a variation on this idea and show that the BPS states in their theories actually form a module over such an algebra.