Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

The indexed vertex

Fixed package, tests, conventions, and smoothing › The indexed vertex · lines 430–480

Em01-E@2#eq:vertex, Em01-E@2#eq:vertextest

Let $I$ run over a fixed finite internal component set and let

K^{I, J} \in D^{'} (U) .

For a smooth symmetric array $H_{I}^{μν}$ on $R^{8}$ , define

f (w) = (y - a, y - b),

Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μ λ} \partial_{2}^{ν} H_{I}^{μ λ}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .

Fix a smooth observation $g$ and mass $m \geq 0$ . Define

V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H) .

Its test action is

⟨ i V (K, H), Φ ⟩ = - i I, J \sum ⟨ K^{I, J}, g (y) [Z_{I}^{ν} (H) \partial_{y^{ν}} Φ_{J} + \frac{m ^{2}}{2} M_{I} (H) Φ_{J}] ⟩ .

The three vertex types are kinetic, trace kinetic, and mass. The internal row $K_{9, mass}$ and the crossed-vertex mass term are different entries.

Displayed formulas in this passage

Em01-E@2#eq:vertex · line 453
$V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H) .$
Em01-E@2#eq:vertextest · line 464
$⟨ i V (K, H), Φ ⟩ = - i I, J \sum ⟨ K^{I, J}, g (y) [Z_{I}^{ν} (H) \partial_{y^{ν}} Φ_{J} + \frac{m ^{2}}{2} M_{I} (H) Φ_{J}] ⟩ .$
unlabeled · line 433
$K^{I, J} \in D^{'} (U) .$
unlabeled · line 437
$f (w) = (y - a, y - b),$
unlabeled · line 440
$Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μ λ} \partial_{2}^{ν} H_{I}^{μ λ}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .$