Seiberg–Witten theory and monstrous moonshine

Abstract

Abstract We study the relation between the instanton expansion of the Seiberg–Witten (SW) prepotential for D = 4, ${\cal N}=2$SU(2) SUSY gauge theory for Nf = 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 = e2πiτ in terms of $A^2=\frac{\Lambda ^2}{16 a^2}$ (Nf = 0) or $\frac{\Lambda ^2}{32a^2}$ (Nf = 1) are all integer-coefficient polynomials of the moonshine coefficients of the modular j-function. A relationship between the Alday–Gaiotto–Tachikawa (AGT) c = 25 Liouville conformal field theory (CFT) and the c = 24 vertex operator algebra CFT of the moonshine module is also suggested.