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

Theorem: CP 1 family

Scale-invariant irregular extensions › The half-holonomy line family · lines 2083–2160 · proof included or linked

I-B-VP@3#thm:line

Let $β = 1/2$ , $α > 1/2$ , and $ν = 1/ (2 α)$ . For

v = (cos θ, sin θ e^{i χ}),

take $E_{0}$ on $v$ and Friedrichs on $v^{⊥}$ . Then $C = c_{0} v v^{†}$ , and

δ G = \frac{sinh ( η / ( 2 α ))}{4 π ^{2} α r r ^{'} sinh η} w (φ) w (φ^{'}),

w (φ) = cos θ e^{i φ /2} + sin θ e^{i χ} e^{- i φ /2} .

Put

u = sin 2 θ cos (φ - χ), ϖ = sin 2 θ sin (φ - χ), k_{ξ} = 6 ξ - 1.

For one complex field,

δ F = \frac{1 + u}{8 π ^{2} α ^{2} r ^{2}}, δ j_{φ} = - \frac{e cos 2 θ}{8 π ^{2} α ^{2} r ^{2}}, δ j_{r} = δ j_{t} = δ j_{z} = 0,

δ T^{t}_{t} = δ T^{z}_{z} δ T^{r}_{r} δ T^{φ}_{φ} δ T^{r}_{φ} = \frac{8 α ^{2} k _{ξ} ( 1 + u ) - 2 k _{ξ} u - 1}{48 π ^{2} α ^{4} r ^{4}}, = \frac{- 4 α ^{2} k _{ξ} ( 1 + u ) - 2 k _{ξ} u - 1}{48 π ^{2} α ^{4} r ^{4}}, = \frac{4 α ^{2} k _{ξ} ( 1 + u ) + 1}{16 π ^{2} α ^{4} r ^{4}}, = α^{2} r^{2} δ T^{φ}_{r} = - \frac{k _{ξ} ϖ}{8 π ^{2} α ^{2} r ^{3}} .

They satisfy the local conservation and trace identities. At conformal coupling the correction is independent of $(θ, χ)$ and equals the one-mode conformal correction.

Proof. In (eq:CW), take $V = v$ , $H_{W} = 0$ , and use equal orders. This gives $C = c_{0} v v^{†}$ . The kernel is therefore (eq:E0kernel) with its angular mode replaced by $w$ .

The necessary angular identities are

∣ w ∣^{2} = 1 + u, ∣ w^{'} ∣^{2} = \frac{1 - u}{4}, w^{''} = - \frac{1}{4} w, w^{'} w ˉ - w w ˉ^{'} = i cos 2 θ .

Together with the radial second jet in the proof of Theorem thm:shift, they give the displayed coincidence derivatives. At $ξ = 1/6$ , substitution in (eq:pointsplit) gives $- [48 π^{2} α^{4} r^{4}]^{- 1} diag (1, 1, - 3, 1)$ . For general $ξ$ , apply twice the improvement operator to $δ F$ . In particular

\nabla_{r} \nabla_{φ} (r^{- 2} (1 + u)) = 3 r^{- 3} ϖ,

which gives the off-diagonal entry. The Ward identities also follow from the smooth bisolution property, independently of these component formulas.

Displayed formulas in this passage

unlabeled · line 2115
$δ T^{t}_{t} = δ T^{z}_{z} δ T^{r}_{r} δ T^{φ}_{φ} δ T^{r}_{φ} = \frac{8 α ^{2} k _{ξ} ( 1 + u ) - 2 k _{ξ} u - 1}{48 π ^{2} α ^{4} r ^{4}}, = \frac{- 4 α ^{2} k _{ξ} ( 1 + u ) - 2 k _{ξ} u - 1}{48 π ^{2} α ^{4} r ^{4}}, = \frac{4 α ^{2} k _{ξ} ( 1 + u ) + 1}{16 π ^{2} α ^{4} r ^{4}}, = α^{2} r^{2} δ T^{φ}_{r} = - \frac{k _{ξ} ϖ}{8 π ^{2} α ^{2} r ^{3}} .$
unlabeled · line 2088
$v = (cos θ, sin θ e^{i χ}),$
unlabeled · line 2093
$δ G = \frac{sinh ( η / ( 2 α ))}{4 π ^{2} α r r ^{'} sinh η} w (φ) w (φ^{'}),$
unlabeled · line 2099
$w (φ) = cos θ e^{i φ /2} + sin θ e^{i χ} e^{- i φ /2} .$
unlabeled · line 2104
$u = sin 2 θ cos (φ - χ), ϖ = sin 2 θ sin (φ - χ), k_{ξ} = 6 ξ - 1.$
unlabeled · line 2110
$δ F = \frac{1 + u}{8 π ^{2} α ^{2} r ^{2}}, δ j_{φ} = - \frac{e cos 2 θ}{8 π ^{2} α ^{2} r ^{2}}, δ j_{r} = δ j_{t} = δ j_{z} = 0,$
unlabeled · line 2141
$∣ w ∣^{2} = 1 + u, ∣ w^{'} ∣^{2} = \frac{1 - u}{4}, w^{''} = - \frac{1}{4} w, w^{'} w ˉ - w w ˉ^{'} = i cos 2 θ .$
unlabeled · line 2153
$\nabla_{r} \nabla_{φ} (r^{- 2} (1 + u)) = 3 r^{- 3} ϖ,$