52 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Figure 4. The weight Cγ(I[ε],f1,f2,f3) attached to a graph with 3 external vertices. These correlation functions will be defined as a sum over graphs. Let Γn denote the set of graphs γ with n univalent vertices, which are labelled as v1,...,vn. These vertices will be referred to as the external vertices. The remaining vertices will be called the internal vertices. The internal vertices of a graph γ ∈ Γn can be of any valency, and are labelled by a genus g(v) ∈ Z≥0. The internal vertices of genus 0 must be at least trivalent. For a graph γ ∈ Γn, and smooth functions f1,...,fn ∈ C∞(M), we will define Cγ(I[L])(f1,...,fn) ∈ R by contracting certain tensors attached to the edges and the vertices. We will label each internal vertex v of genus i and valency k by Ii,k[L] : C∞(M)⊗H(v) → R. Let f1,...,fn ∈ C∞(M) be smooth functions on M. The external vertex vi of γ will be labelled by the distribution C∞(M) → R φ → M fφ. Any edge e joining two internal vertices will be labelled by P (L, ∞) ∈ C∞(M 2 ). The remaining edges, which join two external vertices or join an external and an internal vertex, will be labelled by P (0, ∞), which is a distribution on M 2 . As usual, we can contract all these tensors to define an element Cγ(I[L])(f1,...,fn) ∈ R. One may worry that because some of the edge are labelled by the distribution P (0, ∞), this expression is not well defined. However, because the external edges are labelled by smooth functions fi, there are no problems. Figure 4 describes Cγ(I[ε],f1,f2,f3) for a particular graph γ.

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