Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law

Abstract

The Darboux-Treibich-Verdier (DTV) potential $\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ \omega_{k}}{2};\tau)$ is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. {\bf 68} (1992), 217-236). In this paper, we study the generalized Lam\'{e} equation with the DTV potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ \omega_{k}}{2};\tau)+B\bigg] y(z),\quad n_{k}\in \mathbb{N} \end{equation*} from the monodromy aspect. We prove that the map from $(\tau, B)$ to the monodromy data $(r,s)$ satisfies a surprising universal law $d\tau\wedge dB\equiv8\pi^2 dr\wedge ds.$ Our proof applies Panlev\'{e} VI equation and modular forms. We also give applications to the algebraic multiplicity of (anti)periodic eigenvalues for the associated Hill operator.