I am a mathematician, and I have a question about the physics literature.
If we have a hyperbolic surface X:=Γ∖H, then I can consider the Laplacian on it and study its eigenfunctions. These would correspond to eigenfunctions on the hyperbolic plane H that are invariant under the action of Γ. 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 Γ=π1(X)→U(n). Then there is a Laplacian acting on sections of E→X. The eigensections correspond to the eigenfunctions of the Laplacian acting on functions H→Cn that transform appropriately under the action of Γ.
My question is the following: are such eigensections physically meaningful?