Milnor's isospectral tori and harmonic maps

A well-known question asks whether the spectrum of the Laplacian on a Riemannian manifold \((M,g)\) determines the Riemannian metric \(g\) up to isometry. A similar question is whether the energy spectrum of all harmonic maps from a given Riemannian manifold \((\Sigma,h)\) to \(M\) determines the Riemannian metric on the target space. We consider this question in the case of harmonic maps between flat tori. In particular, we show that the two isospectral, non-isometric \(16\)-dimensional flat tori found by Milnor cannot be distinguished by the energy spectrum of harmonic maps from \(d\)-dimensional flat tori for \(d\leq 3\), but can be distinguished by certain flat tori for \(d\geq 4\). This is related to a property of the Siegel theta series in degree \(d\) associated to the \(16\)-dimensional lattices in Milnor's example.