Seiberg-Witten Theory and Monstrous Moonshine

TL;DR

This paper reveals a deep connection between the instanton expansion coefficients of Seiberg-Witten theory for certain supersymmetric gauge theories and the coefficients of the monstrous moonshine modular functions, suggesting a link between gauge theory and moonshine phenomena.

Contribution

It introduces a simple method to compute the Seiberg-Witten prepotential and demonstrates that its expansion coefficients relate to moonshine coefficients, bridging gauge theory and moonshine.

Findings

Coefficients are integer polynomials of moonshine coefficients.

Established a link between Seiberg-Witten theory and monstrous moonshine.

Suggested a relationship between Liouville CFT and moonshine module CFT.

Abstract

We study the relation between the instanton expansion of the Seiberg-Witten prepotential for $D=4$, ${\cal N}=2$ $SU(2)$ SUSY gauge theory for $N_f=0$ and $1$ and the monstrous moonshine. By utilizing a newly developed simple method to obtain the SW prepotential, it is shown that the coefficients of the expansion of $q=e^{2\pi \tau}$ in terms of $A^2=\frac{\Lambda^2}{16 a^2}$ ($N_f=0$) or $\frac{\Lambda^2}{16 \sqrt{2}a^2}$ ($N_f=1$) are all integer coefficient polynomials of the moonshine coefficients of the modular $j$-function. A relationship between the AGT $c = 25$ Liouville CFT and the $c = 24$ vertex operator algebra CFT of the moonshine module is also suggested.