Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

The actual regulated problem

Cauchy operators and the regulated coordinate problem › The actual regulated problem · lines 880–920

X-A-conv@1#eq:regcoord, X-A-conv@1#eq:coorddata, X-A-conv@1#eq:Xexp, X-A-conv@1#eq:C

The coordinates are scalar functions with an external label $μ$ . Define them by

\partial_{α} (A (q_{g, ε})^{α β} \partial_{β} X^{μ}_{g, ε}) = 0 (mod κ^{3}),

with

X^{μ}_{g, ε} ∣_{Σ} = x^{μ} ∣_{Σ}, \partial_{t} X_{g, ε}^{μ} ∣_{Σ} = δ^{μ}_{0} .

Equivalently, (eq:regcoord) is $□_{q_{g, ε}} X^{μ}_{g, ε} = 0$ formally. The auxiliary metric is exactly (eq:auxmetric).

Write

X^{μ}_{g, ε} = x^{μ} + κ w^{μ} + κ^{2} v^{μ} (mod κ^{3}) .

Define

C^{μ} (a) = \partial_{ν} a^{ν μ} - \frac{1}{2} \partial^{μ} tr a = - \partial_{ν} a_{1} (a)^{ν μ} .

Displayed formulas in this passage

X-A-conv@1#eq:regcoord · line 884
$\partial_{α} (A (q_{g, ε})^{α β} \partial_{β} X^{μ}_{g, ε}) = 0 (mod κ^{3}),$
X-A-conv@1#eq:coorddata · line 893
$X^{μ}_{g, ε} ∣_{Σ} = x^{μ} ∣_{Σ}, \partial_{t} X_{g, ε}^{μ} ∣_{Σ} = δ^{μ}_{0} .$
X-A-conv@1#eq:Xexp · line 904
$X^{μ}_{g, ε} = x^{μ} + κ w^{μ} + κ^{2} v^{μ} (mod κ^{3}) .$
X-A-conv@1#eq:C · line 911
$C^{μ} (a) = \partial_{ν} a^{ν μ} - \frac{1}{2} \partial^{μ} tr a = - \partial_{ν} a_{1} (a)^{ν μ} .$