Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

Twisted sections and deficiency indices

Flat bundles and the restricted extension ledger › Twisted sections and deficiency indices · lines 448–476

I-B-G@2#eq:twist, I-B-G@2#eq:radial

For $β \in [0, 1)$ , represent a section of a flat complex line bundle by a function on the angular cover satisfying

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

Its norm is taken over one angular period. Denote the resulting Hilbert space by $H_{β} = L^{2}_{β} (N, d V)$ . In this lifted parallel representation the flat covariant derivative is the ordinary derivative.

Let $A_{0, β} = - Δ_{q}$ on compactly supported smooth sections. Angular separation gives

u = n \in Z \sum \frac{e ^{i (n + β) φ}}{2 π α} u_{n} (r, z), ν_{n} = \frac{∣ n + β ∣}{α} .

The radial expression on $L^{2} ((0, \infty), r d r)$ is

ℓ_{ν} = - \partial_{r}^{2} - \frac{1}{r} \partial_{r} + \frac{ν ^{2}}{r ^{2}} .

Displayed formulas in this passage

I-B-G@2#eq:twist · line 452
$u (r, φ + 2 π, z) = e^{2 π i β} u (r, φ, z) .$
I-B-G@2#eq:radial · line 471
$ℓ_{ν} = - \partial_{r}^{2} - \frac{1}{r} \partial_{r} + \frac{ν ^{2}}{r ^{2}} .$
unlabeled · line 463
$u = n \in Z \sum \frac{e ^{i (n + β) φ}}{2 π α} u_{n} (r, z), ν_{n} = \frac{∣ n + β ∣}{α} .$

Twisted sections and deficiency indices

Displayed formulas in this passage

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