Noncommutative Bloch analysis of Bochner Laplacians with nonvanishing gauge fields

Abstract

Given an invariant gauge potential and a periodic scalar potential V ̃ on a Riemannian manifold M ̃ with a discrete symmetry group Γ , consider a Γ -periodic quantum Hamiltonian H ̃ = − Δ ̃ B + V ̃ where Δ ̃ B is the Bochner Laplacian. Both the gauge group and the symmetry group Γ can be noncommutative, and the gauge field need not vanish. On the other hand, Γ is supposed to be of type I. With any unitary representation Λ of Γ one associates a Hamiltonian H Λ = − Δ B Λ + V on M = M ̃ / Γ where V is the projection of V ̃ onto M . We describe a construction of the Bloch decomposition of H ̃ into a direct integral whose components are H Λ , with Λ running over the dual space Γ ˆ . The evolution operator and the resolvent decompose correspondingly. Conversely, given Λ ∈ Γ ˆ , one can express the propagator K t Λ ( y 1 , y 2 ) (the kernel of exp ( − i t H Λ ) ) in terms of the propagator K ̃ t ( y 1 , y 2 ) (the kernel of exp ( − i t H ̃ ) ) as a weighted sum over Γ . Such a formula is known in theoretical physics for the case when the gauge field vanishes and M ̃ is a universal covering space of a multiply connected manifold M . We show that these constructions are mutually inverse. Analogous formulas exist for resolvents and their kernels (Green functions) as well.