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

TL;DR

This paper explores a (1+1)-dimensional quantum gravity model on a non-orientable M"obius band, analyzing its Pin structures, Dirac spectrum, and implications for black hole entropy, with detailed mathematical classifications and spectral computations.

Contribution

It introduces a detailed classification of Pin structures on the M"obius band in a JT gravity model and analyzes the Dirac operator's spectrum under twisted equivariance conditions.

Findings

Half-integer momentum quantization observed.

Spectral symmetry and vanishing mod-2 index established.

Non-perturbative saddle points double but do not change entropy.

Abstract

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.