$N=2$ Super Yang-Mills and Subgroups of $SL(2,Z)$

TL;DR

This paper explores specific subgroups of SL(2,Z) relevant to N=2 Super Yang-Mills theories, constructing hyperelliptic curves with modular form coefficients, and deriving exact beta-functions for certain gauge theories.

Contribution

It identifies appropriate SL(2,Z) subgroups for N=2 Super Yang-Mills with various flavors and constructs associated hyperelliptic curves, providing new insights into their modular properties and beta-functions.

Findings

Constructed SU(3) curve up to two constants using uniqueness arguments.

Derived closed-form beta-functions for SU(2) and SU(3) theories.

Analyzed the modular structure of hyperelliptic curves for different gauge groups.

Abstract

We discuss $SL(2,Z)$ subgroups appropriate for the study of $N=2$ Super Yang-Mills with $N_f=2n$ flavors. Hyperelliptic curves describing such theories should have coefficients that are modular forms of these subgroups. In particular, uniqueness arguments are sufficient to construct the $SU(3)$ curve, up to two numerical constants, which can be fixed by making some assumptions about strong coupling behavior. We also discuss the situation for higher groups. We also include a derivation of the closed form $\beta$-function for the $SU(2)$ and $SU(3)$ theories without matter, and the massless theories with $N_f=n$.