Torsion-induced gauge structure in curved quantum waveguides
We investigate the effective dynamics of a particle confined near a space curve. In the strict thin-layer reduction of the nondegenerate transverse ground state, torsion does not enter the local effective Hamiltonian, which contains only the curvature-induced scalar geometric potential. In contrast, for a thin guide with finite transverse width, a leading-order adiabatic projection onto the twofold-degenerate first-excited transverse band renders the rotation of the Frenet normal frame dynamically relevant and generates a matrix-valued Abelian gauge potential. Using a projection-based derivation in a co-rotating Frenet-frame basis, we show that this effective gauge potential is directly determined by the local torsion of the curve. The resulting effective Hamiltonian takes a gauge-covariant form and produces two transverse-mode branches whose parabolic dispersions are shifted in opposite directions in momentum space. For closed curves, the associated holonomy is controlled by the integrated torsion and leads to geometric interference. These results provide a direct realization of a Wilczek--Zee-type connection induced purely by spatial geometry in curved quantum waveguides. We further construct a classical-wave analogue using the degenerate bending modes of an isotropic elastic rod, demonstrating that the same torsion-induced gauge structure appears in continuum wave physics.