Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

Formal duality and the two defects

Formal duality and the two defects · lines 1252–1273

X-C-ST@1#eq:pullbacks

The variation is calculated before imposing the plateau. Let $ξ, ζ$ now be arbitrary smooth compact interior vector fields, and let $q^{'} = F^{*} q$ have the full law (eq:metriclaw).

Write

F^{*} F^{♯} = 1 + κ X_{ξ} + κ^{2} (X_{ζ} + \frac{1}{2} X_{ξ}^{2}), = 1 + κ D_{ξ} + κ^{2} (D_{ζ} + \frac{1}{2} D_{ξ}^{2}), D_{ξ} a = \partial_{μ} (ξ^{μ} a) .

The first acts on scalar functions. The second is density pullback, including the Jacobian.

Displayed formulas in this passage

X-C-ST@1#eq:pullbacks · line 1261
$F^{*} F^{♯} = 1 + κ X_{ξ} + κ^{2} (X_{ζ} + \frac{1}{2} X_{ξ}^{2}), = 1 + κ D_{ξ} + κ^{2} (D_{ζ} + \frac{1}{2} D_{ξ}^{2}), D_{ξ} a = \partial_{μ} (ξ^{μ} a) .$

Nearby in this section

lemma: Lemma: Formal scalar–density duality
definition: Definition: Coordinate and switch defects