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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.22417 |
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| _version_ | 1866915452004859904 |
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| author | Chakraborty, Anik |
| author_facet | Chakraborty, Anik |
| 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse Mobius Spacetime in 1+1D Quantum Gravity: Functional Analytic Structures, Dirac Spectrum, and Pin Geometry Chakraborty, Anik General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry Functional Analysis 57R15, 81T50, 83C45, 46A03, 58J50, 53C05, 55R10 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. |
| title | Inverse Mobius Spacetime in 1+1D Quantum Gravity: Functional Analytic Structures, Dirac Spectrum, and Pin Geometry |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry Functional Analysis 57R15, 81T50, 83C45, 46A03, 58J50, 53C05, 55R10 |
| url | https://arxiv.org/abs/2507.22417 |