More on Seiberg-Witten Theory and Monstrous Moonshine

TL;DR

This paper explores the connection between Seiberg-Witten theory and monstrous moonshine, showing that the instanton expansion coefficients relate to moonshine coefficients for $N_f=2$ and 3, and introduces a new computational method.

Contribution

It extends previous work by demonstrating the moonshine connection for $N_f=2$ and 3 and develops a new method for calculating the SW prepotential.

Findings

Coefficients are integer polynomials of moonshine coefficients.

The expansion variable $q$ relates to the modular $j$-function.

New computational approach is effective for explicit calculations.

Abstract

We continue the study of a relationship between the instanton expansion of the Seiberg-Witten (SW) prepotential of $D = 4$, ${\cal N }= 2$ $SU(2)$ SUSY gauge theory and the monstrous moonshine. Extending the previous results, we show for the cases of $N_f=2$ and $3$ that $q=e^{2\pi i\tau}$, where $\tau$ is the complex gauge coupling, again has an expansion whose coefficients are all integer-coefficient polynomials of the moonshine coefficients of the modular $j$-function in terms of an appropriate expansion variable. We also demonstrate that the new method of calculating the SW prepotential developed here is useful by performing some explicit computations.