Seiberg-Witten Theory and Monstrous Moonshine

We study the relation between the instanton expansion of the Seiberg-Witten prepotential for $D=4$, $\mathcal{N}=2$ $SU(2)$ 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 $A^2=\frac{{\mathrm{\Lambda }}^2}{16a^2}$ ($N_f=0$) or $\frac{{\mathrm{\Lambda }}^2}{16\sqrt{2}{a}^2}$ ($N_f=1$) 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.