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

Corollary: Conditional identification of fractional holonomy

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I-B-VP-0@1#cor:identification

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Fix $α, r$ , the orientation, and the state family of Definition def:state. The pair

(F_{β}, J_{φ, β})

determines $β \in [0, 1)$ . In contrast, $F_{β}$ and $⟨ T^{μ}_{ν} ⟩_{ren}$ together do not distinguish $β$ from $1 - β$ when these are distinct.

Proof. Equation (eq:wick) determines

u = β (1 - β) = \frac{1 - α ^{2} ( 1 + 48 π ^{2} r ^{2} F _{β} )}{6} .

For $0 < u < 1/4$ , the two possible values are

β_{\pm} = \frac{1 \pm 1 - 4 u}{2} .

The sign of $J_{φ}$ selects the upper or lower root by Corollary cor:parity. If $u = 0$ , the representative is $β = 0$ ; if $u = 1/4$ , it is $β = 1/2$ . The evenness of (eq:wick)–(eq:stress) proves the last assertion.

Displayed formulas in this passage

unlabeled · line 917
$(F_{β}, J_{φ, β})$
unlabeled · line 928
$u = β (1 - β) = \frac{1 - α ^{2} ( 1 + 48 π ^{2} r ^{2} F _{β} )}{6} .$
unlabeled · line 933
$β_{\pm} = \frac{1 \pm 1 - 4 u}{2} .$

Corollary: Conditional identification of fractional holonomy

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