Chapter 15 — Applications

Open source document →

1. #1

   For a -connection on a finite graph , define the discrete Chern--Simons-type invariant:

   $$\mathrm{CS}(A):=\sum _{△\in \mathcal{T}}{d}_G({\Omega }_{△}(A),e{)}^2.$$
2. #2

   Define the topological signature of a fruit with doors :

   $$\mathrm{Sig}(F):=(\{{\beta }_k(K_F){\}}_k,\{{\beta }_k(K_F,L){\}}_k,\mathrm{Barcode}(\Sigma ))$$
3. #3

   Two graph connections on the same graph are gauge-equivalent iff they produce the same holonomies on all fundamental loops (Proposition 3.10). The number of gauge-inequivalent flat connections is:

   $$|\mathrm{Hom}({\pi }_1(\mathcal{G}),G)/G|$$
4. #4

   Bottleneck analysis. For fruits , the effective coupling is:

   $$J(F_1,F_2):=\sum _{i\in F_1}\sum _{j\in F_2}{W}_t(i,j).$$