YITP-95-10 Back-Reaction on the Topological Degrees of Freedom in (2+1)-Dimensional Spacetime

We investigate the back-reaction e ect of the quantum eld on the topological degrees of freedom in (2+1)-dimensional toroidal universe, M ' T 2 R. Constructing a homogeneous model of the toroidal universe, we examine explicitly the back-reaction e ect of the Casimir energy of a massless, conformally coupled scalar eld, with a conformal vacuum. The back-reaction causes an instability of the universe: The torus becomes thinner and thinner as it evolves, while its total 2-volume (area) becomes smaller and smaller. The back-reaction caused by the Casimir energy can be compared with the in uence of the negative cosmological constant: Both of them make the system unstable and the torus becomes thinner and thinner in shape. On the other hand, the Casimir energy is a complicated function of the Teichm uller parameters ( ; ) causing highly non-trivial dynamical evolutions, while the cosmological constant is simply a constant. Since the spatial section is a 2-torus, we shall write down the partition function of this system, xing the path-integral measure for gravity modes, with the help of the techniques developed in string theories. We show explicitly that the partition function expressed in terms of the canonical variables corresponding to the (redundantly large) original phase space, is reduced to the partition function de ned in terms of the physical-phase-space variables with a standard Liouville measure. This result is compatible with the general theory of the path integral for the 1st-class constrained systems.