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

Abstract

The critical curve $\mathcal{C}$ on which $\mathrm{Im}\hat{\tau }=0$, $\hat{\tau }=a_D/a$, determines hyperbolic domains whose Poincaré metric is constructed in terms of $a_D$ and $a$. We describe $\mathcal{C}$ 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 ${\partial }_{\hat{\tau }}⟨\mathrm{tr}{\varphi }^2{⟩}_{\hat{\tau }}=\frac{1}{8\pi i{b}_1}⟨\varphi {⟩}_{\hat{\tau }}^2$, where $b_1$ is the one-loop coefficient of the beta function.

11 pages, LaTex file, Expanded version: new results, technical details explained, misprints corrected and references added

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• Modular Invariant Regularization of String Determinants and the Serre GAGA Principle
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