More on Seiberg-Witten theory and Monstrous moonshine: A new simple method of calculating the prepotential

Abstract

We continue the study of a relationship between the instanton expansion of the Seiberg-Witten (SW) prepotential of D=4, N=2SU(2) SUSY gauge theory and the Monstrous moonshine. As was done in [10], we expand the inverse of the modular j-function in u−1 around u=∞, where u is the familiar u parameter for the respective SW curves, and compute the complex gauge coupling τ as a function of the scalar vev a by using the Fourier expansion of j(τ) and the relation between u and a obtained by the Picard-Fuchs equation. In this way, the instanton expansion of the prepotential is related to the dimensions of representations of the Monster group. We show that, for the cases of Nf=2 and 3, q again has an expansion whose coefficients are all integer-coefficient polynomials of the moonshine coefficients if the expansion variables are appropriately chosen. This hints some unknown relation between the Liouville CFT and the vertex operator algebra CFT with different central charges. We also demonstrate that this new method of calculating the SW prepotential developed here is useful by performing some explicit computations.