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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.22417 |
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Table of Contents:
- In this manuscript, we formulate a (1+1)-dimensional Jackiw-Teitelboim gravity toy model whose Euclidean spacetime manifold is the Möbius 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 Möbius band, which resolves the Möbius 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 \[ ψ(x+1, w) = γ^w ψ(x, -w). \] Half-integer momentum quantization, spectral symmetry, vanishing mod-2 index, and $η_D(0) = 0$ 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.