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

Parallel and periodic gauges

Noether normalization and renormalized current › Parallel and periodic gauges · lines 637–670

I-B-VP-0@1#eq:gaugecheck

Write the connection as $D = \partial + i a$ . In the lifted parallel representation $a = 0$ . The periodic field

v = e^{- i β φ} Φ

has

D_{φ} v = (\partial_{φ} + i β) v, H_{per} (X, X^{'}) = e^{- i β (φ - φ^{'})} H_{par} (X, X^{'}) .

Thus

J_{φ} = i [(\partial_{φ} - \partial_{φ^{'}} + 2 i β) H_{per}] = i [(\partial_{φ} - \partial_{φ^{'}}) H_{par}] .

The endpoint phase must also be included in the subtracted parametrix in periodic gauge. Using ordinary derivatives on $H_{per}$ alone would produce the spurious shift $2 β [H_{par}]$ .

Changing angular lifts multiplies a bundle kernel by the corresponding character. The local fiber identification in the coincidence operation cancels that change. The final current is a global ordinary one-form.

Displayed formulas in this passage

I-B-VP-0@1#eq:gaugecheck · line 651
$J_{φ} = i [(\partial_{φ} - \partial_{φ^{'}} + 2 i β) H_{per}] = i [(\partial_{φ} - \partial_{φ^{'}}) H_{par}] .$
unlabeled · line 641
$v = e^{- i β φ} Φ$
unlabeled · line 645
$D_{φ} v = (\partial_{φ} + i β) v, H_{per} (X, X^{'}) = e^{- i β (φ - φ^{'})} H_{par} (X, X^{'}) .$