Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · assumption · 8a9d0682

Assumption: Fixed background and charged field

Prescribed geometry, bundle, and spectral state · lines 202–247

I-B-VP-0@1#ass:inputs, I-B-VP-0@1#eq:metric, I-B-VP-0@1#eq:equivariance

Fix

0 < μ < 2 π, α = 1 - \frac{μ}{2 π} \in (0, 1) .

The regular spacetime is

M = R_{t} \times (0, \infty)_{r} \times (R /2 π Z)_{φ} \times R_{z}

with

g = - d t^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2} .

The completed apex worldsheet $W = {r = 0}$ is not part of $M$ . We use $ℏ = c = 1$ .

For $β \in [0, 1)$ , the charged complex scalar is represented in a lifted parallel frame by

Φ (r, φ + 2 π, z) = e^{2 π i β} Φ (r, φ, z) .

This equation fixes the meridian and character conventions. The massless equation is

(- D^{μ} D_{μ} + ξ R) Φ = 0.

On $M$ , the metric and bundle curvatures vanish, so in a parallel frame the equation is $□_{g} Φ = 0$ . No distributional $ξ R$ interaction at $W$ is modeled.

The spatial operator is the Friedrichs extension

B_{β} = A_{F, β} \geq 0

of the covariant expression $- Δ_{q}$ , where

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

Displayed formulas in this passage

I-B-VP-0@1#eq:metric · line 215
$g = - d t^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2} .$
I-B-VP-0@1#eq:equivariance · line 224
$Φ (r, φ + 2 π, z) = e^{2 π i β} Φ (r, φ, z) .$
unlabeled · line 205
$0 < μ < 2 π, α = 1 - \frac{μ}{2 π} \in (0, 1) .$
unlabeled · line 210
$M = R_{t} \times (0, \infty)_{r} \times (R /2 π Z)_{φ} \times R_{z}$
unlabeled · line 231
$(- D^{μ} D_{μ} + ξ R) Φ = 0.$
unlabeled · line 239
$B_{β} = A_{F, β} \geq 0$
unlabeled · line 243
$q = d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, d V = α r d r d φ d z .$

Nearby in this section

section: Prescribed geometry, bundle, and spectral state