Manuscript XVI-I@1 · XVI-I.tex · definition · a7046a8d

Definition: Reduced open-index blocks

The reduced graph and its combinatorial normalization › Open tensor legs and source-contracted reduction · lines 984–1012

XVI-I@1#def:blocks, XVI-I@1#eq:Vblock, XVI-I@1#eq:U

For $∣ α ∣ + ∣ β ∣ \leq 2$ , define on $Ω$

V^{α β}_{I J} [h] (y, w) = (- 1)^{∣ α ∣ + ∣ β ∣} χ_{I} (w) K, L, a, b \sum L_{K L; ab} [h] (w) \times (\partial_{y}^{α} \partial_{w}^{a} D_{F}^{I K} (y, w)) (\partial_{y}^{β} \partial_{w}^{b} D_{F}^{J L} (y, w)) .

The source-contracted reduced kernel is

U_{A}^{\circ} [h] (y, w) = I, J, α, β \sum c^{A}_{I J; α β} (y) V^{α β}_{I J} [h] (y, w) .

It still has the source/root index $A$ . The scalar insertion slot is not contracted.

Displayed formulas in this passage

XVI-I@1#eq:Vblock · line 987
$V^{α β}_{I J} [h] (y, w) = (- 1)^{∣ α ∣ + ∣ β ∣} χ_{I} (w) K, L, a, b \sum L_{K L; ab} [h] (w) \times (\partial_{y}^{α} \partial_{w}^{a} D_{F}^{I K} (y, w)) (\partial_{y}^{β} \partial_{w}^{b} D_{F}^{J L} (y, w)) .$
XVI-I@1#eq:U · line 1002
$U_{A}^{\circ} [h] (y, w) = I, J, α, β \sum c^{A}_{I J; α β} (y) V^{α β}_{I J} [h] (y, w) .$