Transition Amplitude in 2+1 dimensional Chern-Simons Gravity on a Torus

TL;DR

The paper computes a transition amplitude for the evolution of torus moduli in 2+1D gravity by blending Chern-Simons gravity with the ADM formalism. The amplitude concentrates on the classical orbit and satisfies the ADM Schrödinger equation, enabling a probabilistic interpretation of York-time evolution. By enforcing modular covariance through a sum over SL(2, Z) images, the authors relate covariant wavefunctions to weight-1/2 Maass forms and connect the CS–ADM pictures within a unified framework. These results illuminate time and modular invariance in quantum gravity on fixed spatial topology and point toward extensions to higher-genus cases and loop-quantized formalisms.

Abstract

The discussions on the modular invariance in section 5 are refined.