This is a follow-up to a recent question of mine Witten's proof of Wick Formula of QFT. The background of this question can be found there if needed, but I feel my question is simple enough regardless. When these notes write
and then takes contractions along edges using the form B^{-1}
which tensors are being contracted and how? And how is the Feynman amplitude of a graph $F_\Gamma$ being created from this process? I loosely see how $B^{-1}$ is the propagator but I cannot piece this together. From just the slots available, it should make sense that we use $B$ for these contractions, no? (considering the first two steps of the process)
Update: I believe my question will be answered by explaining the meaning of contraction in the following:
Assign $Q_k \in Sym^k (V ^∗ )$ to each internal vertex v of valence k, associating each copy of $V^*$ with (an end of) an edge. Also, to the i-th leg (external vertex) of $\Gamma$, i = 1, . . . , m assign some $f_i \in V^*$ . Now, for each edge contract two copies of $V^*$ associated to its ends using $a \in sym^2 (V )$. After all copies of $V^*$ get contracted, we obtain a number $F_\Gamma(f_1, . . . , f_m) \in \mathbb{R}$.
I think the confusion comes from terminology. In physics, the term 'contraction' means something else than contraction in mathematics. In proposition 1.1. it is given how to calculate a vacuum expectation value (thats the quantity in brackets with the subscript 0) in terms of sums of pairings. In physics, this process of pairing up is called contracting two elements. Hence summing over all pairings and inserting the pairs into $ B^{-1}(\cdot, \cdot) $ is what physicists mean by taking contractions.
The idea of Feynman diagrams is that you can visualise all different contractions (= pairings) by drawing them as graphs. A specific graph then already corresponds to a specific pairing. You then fill in the correct values into to the different $ B^{-1}(\cdot, \cdot)$ and multiply them all by potential constants $ g_m$ to get $ F_\Gamma$. Hence, summing over all diagrams gives you the sums over all pairings, which gives the you vacuum expectation value for a theory with interactions.
I couldn't answer the part after the update, because you did not cite the context of that piece. It appears not to be part of the book.
I must say, I think that it would be difficult to understand Feynman diagrams without first encountering them in a physics class, so I understand that you are having difficulties.