Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

Host, bundle, field normalization, and signature

Host, bundle, field normalization, and signature · lines 386–407

Let

N = (0, \infty) \times S^{1} \times R, q = d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, d V = α r d r d φ d z .

For $β \in [0, 1)$ , the flat Hermitian line bundle $L_{β}$ is represented on the angular cover by

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

In a lifted parallel frame its connection is ordinary differentiation. Put

q_{n} = n + β, ν_{n} = \frac{∣ q _{n} ∣}{α}, ψ_{n} (φ) = \frac{e ^{i q_{n} φ}}{2 π α} .

The spatial Hilbert space is $H_{β} = L^{2}_{β} (N, d V)$ .

Displayed formulas in this passage

unlabeled · line 389
$N = (0, \infty) \times S^{1} \times R, q = d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, d V = α r d r d φ d z .$
unlabeled · line 396
$u (r, φ + 2 π, z) = e^{2 π i β} u (r, φ, z) .$
unlabeled · line 401
$q_{n} = n + β, ν_{n} = \frac{∣ q _{n} ∣}{α}, ψ_{n} (φ) = \frac{e ^{i q_{n} φ}}{2 π α} .$