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

Abstract

The critical curve ${\cal C}$ on which ${\rm Im}\,\hatτ=0$, $\hatτ=a_D/a$, determines hyperbolic domains whose Poincaré 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 $\partial_{\hatτ} \langle {\rm tr}\,ϕ^2\rangle_{\hat τ}={1\over 8πi b_1}\langle ϕ\rangle_{\hatτ}^2$, where $b_1$ is the one-loop coefficient of the beta function.