Eigensections of flat unitary bundles

Question

I am a mathematician, and I have a question about the physics literature.

If we have a hyperbolic surface $X:=\Gamma\setminus\mathbb{H}$, then I can consider the Laplacian on it and study its eigenfunctions. These would correspond to eigenfunctions on the hyperbolic plane $\mathbb{H}$ that are invariant under the action of $\Gamma$. These naturally correspond to solving a Schrodinger equation on the $X$.

On the other hand, I could consider the more general setting of a flat unitary bundle $E$ over $X$ coming from surface representations $\Gamma=\pi_1(X)\rightarrow U(n)$. Then there is a Laplacian acting on sections of $E\rightarrow X$. The eigensections correspond to the eigenfunctions of the Laplacian acting on functions $\mathbb{H}\rightarrow\mathbb{C}^n$ that transform appropriately under the action of $\Gamma$.

My question is the following: are such eigensections physically meaningful?