Koebe 1/4-Theorem and Inequalities in N=2 Super-QCD

TL;DR

This paper explores inequalities and relations in N=2 Super-QCD using complex analysis tools like the Koebe 1/4-theorem, deriving new bounds and a closed-form prepotential related to the renormalization group and gauge theory parameters.

Contribution

It introduces new inequalities and relations for N=2 Super SU(2) Yang-Mills theory based on complex analysis and Schwarzian equations, providing a novel geometric perspective.

Findings

Derived inequalities involving $u$, $a_D$, and $a$ using the Koebe 1/4-theorem.

Obtained a closed-form expression for the prepotential as a function of $a$.

Established a relation between derivatives of $raket{ ext{tr}\,^2}$ and $raket{}$ involving the beta function coefficient.

Abstract

The critical curve ${\cal C}$ on which ${\rm Im}\,\hat\tau =0$, $\hat\tau=a_D/a$, determines hyperbolic domains whose Poincar\'e metric is constructed in terms of $a_D$ and $a$. We describe ${\cal C}$ in a parametric form related to a Schwarzian equation and prove new relations for $N=2$ Super $SU(2)$ Yang-Mills. In particular, using the Koebe 1/4-theorem and Schwarz's lemma, we obtain inequalities involving $u$, $a_D$ and $a$, which seem related to the Renormalization Group. Furthermore, we obtain a closed form for the prepotential as function of $a$. Finally, we show that $\partial_{\hat\tau} \langle {\rm tr}\,\phi^2\rangle_{\hat \tau}={1\over 8\pi i b_1}\langle \phi\rangle_{\hat\tau}^2$, where $b_1$ is the one-loop coefficient of the beta function.