Manuscript XIII-A@1 · XIII-A.tex · section · f56e91c7

The independent auxiliary smearing

A3: the coefficientwise restriction kernel › The independent auxiliary smearing · lines 965–1000

XIII-A@1#eq:linear-tests, XIII-A@1#eq:uq, XIII-A@1#eq:vh

Let

q \in C_{c}^{\infty} (W; Sym^{2} T M), h \in C_{c}^{\infty} (M_{T}^{\circ}; R),

and define

L_{q} = \int q^{μν} (D^{*} b)_{μν} d^{4} x, G_{h} = \int h η^{α β} γ_{α β} d^{4} x .

Both are even, ghost-number-zero regular linear functionals. Every $L_{q}$ belongs to $N_{W}$ .

Compact integration by parts gives

L_{q} = \int u_{q, λ} b^{λ} d^{4} x, u_{q, λ} = η_{λ ν} \partial_{μ} q^{μν} - \frac{1}{2} \partial_{λ} (η_{μν} q^{μν}) .

The metric dual smearing is

v_{h}^{α β} = η^{α β} h .

These are the actual off-shell derivative kernels. In particular, $L_{q}$ has no metric derivative kernel off shell. Replacing its argument by $D γ$ before computing the product would change the lift and obscure which block performs the contraction.

Displayed formulas in this passage

XIII-A@1#eq:linear-tests · line 973
$L_{q} = \int q^{μν} (D^{*} b)_{μν} d^{4} x, G_{h} = \int h η^{α β} γ_{α β} d^{4} x .$
XIII-A@1#eq:uq · line 983
$L_{q} = \int u_{q, λ} b^{λ} d^{4} x, u_{q, λ} = η_{λ ν} \partial_{μ} q^{μν} - \frac{1}{2} \partial_{λ} (η_{μν} q^{μν}) .$
XIII-A@1#eq:vh · line 992
$v_{h}^{α β} = η^{α β} h .$
unlabeled · line 968
$q \in C_{c}^{\infty} (W; Sym^{2} T M), h \in C_{c}^{\infty} (M_{T}^{\circ}; R),$

The independent auxiliary smearing

Displayed formulas in this passage

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