Manuscript En03-A@1 · En03-A.tex · section · 5398614e

Cone charts and overlap components

External conical geometry and certified local moves › Cone charts and overlap components · lines 1210–1274

En03-A@1#eq:cone-metric, En03-A@1#eq:rotation, En03-A@1#eq:developing, En03-A@1#eq:seam-transitions, En03-A@1#eq:particle-word

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Use

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

On the punctured cone,

d s^{2} = - d T^{2} + d r^{2} + α^{2} r^{2} d θ^{2}, r > 0,

with $θ$ periodic modulo $2 π$ . Its completion has a timelike conical axis. The spinless point-particle interpretation is standard DeserJackiwHooft1984; no field equation is derived.

On the universal cover,

D (T, r, θ) = (T, r cos (α θ), r sin (α θ)) .

For the deck transformation $T (T, r, θ) = (T, r, θ + 2 π)$ , define

R (μ) = 100 0 cos μ sin μ 0 - sin μ cos μ .

Then

D \circ T^{- 1} = R (μ) \circ D .

Choose $0 < ϵ < π /4$ , with angular lifts

U_{1} = (- ϵ, π + ϵ), U_{2} = (π - ϵ, 2 π + ϵ),

restricted where necessary to injective developing charts. Their two overlap components have transitions

Λ_{2 \leftarrow 1}^{π} = I, Λ_{1 \leftarrow 2}^{0} = R (μ) .

The positively oriented particle word is

(Λ_{1 \leftarrow 2}^{0}, 0) (Λ_{2 \leftarrow 1}^{π}, 0) = (R (μ), 0) .

Overlap-component indexing prevents suppression of the nontrivial seam. No smooth flat chart contains the axis.

Displayed formulas in this passage

En03-A@1#eq:cone-metric · line 1219
$d s^{2} = - d T^{2} + d r^{2} + α^{2} r^{2} d θ^{2}, r > 0,$
En03-A@1#eq:rotation · line 1237
$R (μ) = 100 0 cos μ sin μ 0 - sin μ cos μ .$
En03-A@1#eq:developing · line 1247
$D \circ T^{- 1} = R (μ) \circ D .$
En03-A@1#eq:seam-transitions · line 1259
$Λ_{2 \leftarrow 1}^{π} = I, Λ_{1 \leftarrow 2}^{0} = R (μ) .$
En03-A@1#eq:particle-word · line 1266
$(Λ_{1 \leftarrow 2}^{0}, 0) (Λ_{2 \leftarrow 1}^{π}, 0) = (R (μ), 0) .$
unlabeled · line 1213
$η = diag (- 1, 1, 1), 0 < μ < 2 π, α = 1 - \frac{μ}{2 π} .$
unlabeled · line 1229
$D (T, r, θ) = (T, r cos (α θ), r sin (α θ)) .$
unlabeled · line 1253
$U_{1} = (- ϵ, π + ϵ), U_{2} = (π - ϵ, 2 π + ϵ),$