Holonomy Asymptotics along Quartic Differential Rays

Abstract

Let \(X\) be a closed Riemann surface and let \(q\in H^0(X,K^4)\) be a nonzero holomorphic quartic differential on \(X\). For \(t >0\), the ray \(tq\) determines a family of Hitchin representations in the \(\operatorname{PSp}(4,\mathbb R)\)-Hitchin component. We study, as \(t\to+\infty\), the asymptotic behavior of their holonomy along closed curves. We obtain explicit asymptotic formulas for all singular values and for the absolute values of all eigenvalues of the holonomy. Their logarithmic growth rates are given by integrating the local fourth roots of \(q\) along the saddle connections forming the geodesic representative of the curve with respect to the singular flat metric \(\lvert q\rvert^{1/2}\). No restriction is imposed on the orders of the zeros of \(q\).