Topology-changing amplitudes in 2 + 1 dimensional gravity

Nontrivial spatial or spacetime topologies in 2+1 gravity have moduli that can become dynamical.

“One can consider nontrivial spatial or spacetime topologies [277, 279]. Such “cosmological” solutions have moduli — a finite number of parameters that distinguish among geometrically inequivalent constant curvature manifolds — and these can become dynamical.”

The geometric structure description is closely related to the first-order Chern-Simons formalism, which connects back to the standard metric formalism.

“Fortunately, the description is closely related to the more familiar first-order Chern-Simons formalism [108, 277, 279, 2], which, in turn, can connect us back to the standard metric formalism.”

The first-order action (14) is equivalent to that of a Chern-Simons theory.

“As first noted by Achúcarro and Townsend [2] and further developed by Witten [277, 279], the first-order action (14) is equivalent to that of a Chern-Simons theory.”

The path integral for the covariant first-order action (14) reduces to a ratio of determinants that has a topological interpretation as the analytic or Ray-Singer torsion.

“The path integral for this action was first fully analyzed in two seminal papers by Witten [277, 279], who showed that it reduced to a ratio of determinants that has an elegant topological interpretation as the analytic or Ray-Singer torsion [230].”

The integral over the moduli space of flat connections sometimes diverges.

“This integral sometimes diverges [279]; the significance of that divergence is not understood.”

If g > 1 then qρ : π1S → SO(2,1)0 is injective and qρ(π1S) is a discrete cocompact subgroup of SO(2,1)0.

“Proposition 1. If g > 1 then qρ : π 1 S → SO (2 , 1) 0 is injective and qρ ( π 1 S ) is a discrete cocompact subgroup of SO (2 , 1) 0 .”

Conversely, if |e(ρ)| = 2g - 2 then ρ is discrete and faithful.

“Conversely Goldman shows in [6] that if | e ( ρ ) | = 2 g -2 then ρ is discrete and faithful”