46 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Figure 3. The renormalization group differential equation. Next consider γ2, given by Since there are no external edges there is no dependence on a ∈ C∞(M), and we find wγ 2 (P (ε, L),I) = l1,l2,l3∈[ε,L] x,y∈M Kl 1 (x, y)Kl 2 (x, y)Kl 3 (x, y)d VolM×M . Using the fact that Kl(x, y) l− dim M/2 e−d(x−y)2/l + higher order terms for l small, we can see that the limit of wγ 2 (P (ε, L),I) as ε → 0 is singular. Let γ3 be the graph

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