Pin Equivariance and Local Dirac Boundary Problems on the Flat Mobius Band

Abstract

We study complex rank-two pinor fields on the flat Möbius band and the relation between their deck equivariance and Dirac boundary realizations. With explicit Clifford conventions, the two lifts for each Pin type are represented by $R=c{\sigma }_1$, where $c=\pm 1$ for Pin+ and $c=\pm i$ for Pin-. The relevant longitudinal monodromy is $cJ$, with $J\varphi (w)={\sigma }_1\varphi (-w)$, rather than the fiber matrix alone. Its eigenspaces give shifts $0,1/2$ in the positive case and $1/4,3/4$ in the negative case. We identify the resulting geometric Sobolev spaces and construct self-adjoint elliptic realizations of $D=-i({\sigma }_1{\partial }_x+{\sigma }_2{\partial }_w)$ with a local boundary condition defined by the longitudinal tangent field. An exact graph estimate gives compact resolvent and Fredholm mappings on nonnegative Sobolev levels. The spectrum consists of a single unpaired transverse branch and paired branches whose longitudinal shifts alternate with transverse parity. Consequently, the kernel and spectral symmetry depend on the lift and boundary condition; the two negative Pin lifts have eta values $1/2$ and $-1/2$ in the stated convention. We also give a secular equation and an exact kernel criterion for the full translation-invariant family of local self-adjoint elliptic boundary lines.

arXiv comment15 Pages, No Figure