Chapter 15 — Applications

Open source document →

1. #1

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

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

   Define the topological signature of a fruit with doors :

   $$\mathrm{Sig}(F):=\bigl(\{\beta_k(K_F)\}_k,\;\{\beta_k(K_F,L)\}_k,\;\mathrm{Barcode}(\Sigma)\bigr)$$
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:

   $$J(F_1,F_2):=\sum_{i\in F_1}\sum_{j\in F_2}W_t(i,j).$$