SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry

Understand

We study the expectation value of a nonplanar Wilson graph operator in SL(2,C) Chern-Simons theory on $S^3$.

• In particular we analyze its asymptotic behaviour in the double-scaling limit in which both the representation labels and the Chern-Simons coupling are taken to be large, but with fixed ratio.
• When the Wilson graph operator has a specific form, motivated by loop quantum gravity, the critical point equations obtained in this double-scaling limit describe a very specific class of flat connection on the graph complement manifold.
• We find that flat connections in this class are in correspondence with the geometries of constant curvature 4-simplices.