Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · section · f4bd2b11

Geometry, bundle, and non-global-hyperbolicity

The regular host and the spectral covariance › Geometry, bundle, and non-global-hyperbolicity · lines 254–295

I-B-VP-2@1#eq:host, I-B-VP-2@1#eq:character

Fix

0 < α < 1, 0 \leq β < 1.

The spacetime and metric are

M = R_{t} \times (0, \infty)_{r} \times (R /2 π Z)_{φ} \times R_{z}, g = - d t^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2} .

The future orientation is that of $\partial_{t}$ . The completion adds the worldsheet

W = R_{t} \times {o} \times R_{z} .

It is not part of $M$ .

A lifted parallel representation of the charged line bundle is fixed by

Φ (φ + 2 π) = e^{2 π i β} Φ (φ) .

Changing a local parallel frame multiplies the covariance by a smooth nowhere-zero phase, and therefore does not change its wavefront set.

Write

q_{N} = d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, d V = α r d r d φ d z .

Let $B_{β}$ be the nonnegative Friedrichs extension of $- Δ_{q_{N}}$ on the twisted spatial Hilbert space. The extension is an input, as in I-B-G CheungIBG,KayStuder. No distributional curvature-coupling interaction at $W$ is introduced.

Displayed formulas in this passage

I-B-VP-2@1#eq:host · line 261
$M = R_{t} \times (0, \infty)_{r} \times (R /2 π Z)_{φ} \times R_{z}, g = - d t^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2} .$
I-B-VP-2@1#eq:character · line 276
$Φ (φ + 2 π) = e^{2 π i β} Φ (φ) .$
unlabeled · line 257
$0 < α < 1, 0 \leq β < 1.$
unlabeled · line 270
$W = R_{t} \times {o} \times R_{z} .$
unlabeled · line 285
$q_{N} = d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, d V = α r d r d φ d z .$

Geometry, bundle, and non-global-hyperbolicity

Displayed formulas in this passage

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