Inverse Mobius Spacetime in 1+1D Quantum Gravity: Functional Analytic Structures, Dirac Spectrum, and Pin Geometry

In this manuscript, we formulate a (1+1)-dimensional Jackiw-Teitelboim gravity toy model whose Euclidean spacetime manifold is the M\"obius band $M$. Since $M$ is non-orientable, the relevant spin-statistics structure is Pin rather than Spin. To emphasize the role of orientation reversal, we refer to the universal orientable cover $\widetilde{M}$ as the inverse M\"obius band, which resolves the M\"obius twist into an infinite ribbon equipped with a $\mathbb{Z}$ deck action. We compute the Stiefel-Whitney classes $w_1$, $w_2$, classify all $\mathrm{Pin}^\pm$ structures, construct the associated pinor bundles, and analyze the Dirac operator under the twisted equivariance condition $$\psi(x+1, w) = \gamma^w \psi(x, -w).$$ Half-integer momentum quantization, spectral symmetry, vanishing mod-2 index, and $\eta_D(0) = 0$ follow. In JT gravity, the two inequivalent Pin lifts in each parity double the non-perturbative saddle-point sum, yet leave the leading Bekenstein-Hawking entropy unchanged. Full proofs and heat-kernel calculations are provided for completeness.