Manuscript Em01-B@3 · Em01-B-v3.tex · section · 69189992

Finite vertex index set

Complete branch–vertex inventory › Finite vertex index set · lines 1991–2053

Let $I, J$ be the fixed incoming and residual tensor-component sets, and let $S = {0, 1, 2, 3}$ . The finite expansion of $F_{δ}$ is the union of three types:

D_{kin} = {(I, J, μ, ν) : I \in I, J \in J, μ, ν \in S},

D_{tr} = {(I, J, μ, λ, ν) : I \in I, J \in J, μ, λ, ν \in S},

D_{mass} = {(I, J, μ, ν) : I \in I, J \in J, μ, ν \in S} .

Terms with zero metric coefficient simply contribute zero.

For a given internal row $K_{b}$ , the operations are:

Type	Crossed entry and operator $D$	Output $O$	Coefficient inside $O$
Kinetic	$\partial_{2, μ} C_{I}^{μν}$	$\partial_{y^{ν}}$	$i g (y) K_{b}^{I, J}$ .
Trace kinetic	$\partial_{2}^{ν} C_{I}^{μ λ}$	$\partial_{y^{ν}}$	$- \frac{i}{2} g (y) η_{μ λ} K_{b}^{I, J}$ .
Mass	$C_{I}^{μν}$ , $D = id$	$id$	$- \frac{i m ^{2}}{2} g (y) η_{μν} K_{b}^{I, J}$ .

In every row, pull back the crossed entry by $f$ , multiply by the displayed coefficient, and apply $O$ . The fibre contraction is the basis–dual-basis contraction of the repeated $I$ , leaving $J$ . Summing this explicitly defined finite index set gives $i V (K_{b}, C_{δ})$ .

The output derivative is applied after multiplication. When expanded, it differentiates the switch, internal kernel, and both relative-variable dependences of the pulled-back crossed entry. In contrast, $D$ differentiates only the crossed scalar slot before pullback.

Displayed formulas in this passage

unlabeled · line 1998
$D_{kin} = {(I, J, μ, ν) : I \in I, J \in J, μ, ν \in S},$
unlabeled · line 2003
$D_{tr} = {(I, J, μ, λ, ν) : I \in I, J \in J, μ, λ, ν \in S},$
unlabeled · line 2009
$D_{mass} = {(I, J, μ, ν) : I \in I, J \in J, μ, ν \in S} .$