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

The Darboux-Treibich-Verdier (DTV) potential ∑ₖ₌₀³nₖ(nₖ+1)(z+ ωₖ2;τ) is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. { 68} (1992), 217-236). In this paper, we study the generalized Lam\'{e} equation with the DTV potential {equation*} y(z)=[ ∑ₖ₌₀³nₖ(nₖ+1)(z+{ ωₖ}{2};τ)+B] y(z), nₖ∈ N {equation*} from the monodromy aspect. We prove that the map from (τ, B) to the monodromy data $(r,s)$ satisfies a surprising universal law dτ dB≡8π² dr 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.