Chapter 13 — Yang–Mills Flow

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1. #1

   Let be a graph with triangle set (all 3-cliques). For a -valued connection , the holonomy of triangle is:

   $$\Omega_\triangle(A):=g_{ij}\cdot g_{jk}\cdot g_{ki}\in G.$$
2. #2

   Let be a graph with triangle set (all 3-cliques). For a -valued connection , the holonomy of triangle is: The discrete Yang--Mills energy:

   $$\mathcal{E}(A):=\sum_{\triangle\in\mathcal{T}}d_G(\Omega_\triangle(A),e)^2.$$
3. #3

   On with the product Riemannian structure :

   $$\frac{dA_s}{ds}=-\mathrm{grad}\,\mathcal{E}(A_s),$$
4. #4

   On with the product Riemannian structure : i.e., for each edge :

   $$\frac{dg_{ij}(s)}{ds}=-\frac{\partial\mathcal{E}}{\partial g_{ij}}\bigg|_{\mathfrak{g}}.$$