Chapter 3: manifestly rational universal quantization (k)
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(k)
Let us abbreviate θ1...j;n = δ1...j;n ◦ ηw , and θ(1) = ∆ ◦ ηw : ...
...
... ...
θ
δ
=
ηw ...
...
Then the diagrammatics continue to allow to pull down vertices: ...
...
...
θ
θ
=
...
...
...
With these we have a nice description of the ?-product ?w ˜w ◦ ◦ u,v = τ R abbreviations, −1 (ϕu ⊗ ϕw ) = ◦(id −θ) ◦ ◦ (ϕu ⊗ ϕw ). It is implemented by a sum of all diagrams of the following shape: ...
...
...
P
?
=
...
...
...
...
...
ways to ... attach Θ1...j some number of θs ... ×~#∆s u
u ...
v
v ...
We have written u for ϕu : X → C0 (R) ⊗ X, and the power on ~ counts the number of ∆ s in the fully expanded-out diagram. What’s important to emphasize is that each θ vertex occurs at its own level, and at each level we sum over all ways to attach it to the diagram below it. We now sort the sum by the degree in ~, and by the data of which incoming strands participate and which pass through to the top without interacting with some θ vertex. Fixing a power in ~ and some collection of incoming strands, we can lump into a single vertex Θ the sum of all diagrams that contribute to that power in ~ and in which the fixed