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

Definition: Complete vertex operation

Frozen expression and complete contraction › Explicit vertex and its arbitrary-test action · lines 623–645

Em01-B@3#def:vertex, Em01-B@3#eq:vertex, Em01-B@3#eq:vertex-test

For $K = (K^{I, J}) \in D^{'} (W; V_{int})$ and smooth $C$ , define

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

For every compact joint dual test $Φ = (Φ_{J})$ , this means

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

Displayed formulas in this passage

Em01-B@3#eq:vertex · line 627
$V (K, C)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (C)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (C) .$
Em01-B@3#eq:vertex-test · line 636
$⟨ V (K, C), Φ ⟩ = - I, J \sum ⟨ K^{I, J}, g (y) [Z_{I}^{ν} (C) \partial_{y^{ν}} Φ_{J} + \frac{m ^{2}}{2} M_{I} (C) Φ_{J}] ⟩ .$