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

In the framework of the Chern-Simons gravity proposed by Witten, a transition amplitude of a torus universe in 2+1 dimensional quantum gravity is computed. This amplitude has the desired properties as a probability amplitude of the quantum mechanics of a torus universe, namely, it has a peak on the ”classical orbit” and it satisfies the Shrödinger equation of the 2+1 dimentional gravity. The discussion is given that the classical orbit dominance of the amplitude is not altered by taking the modular invariance into account and that this amplitude can serve as a covariant transition amplitude in a particular sense. A set of the modular covariant wavefunctions is also constructed and they are shown to be equivalent with the weight-1/2 Maass forms. e-mail address: [email protected]