O(N) SIGMA-MODEL AS A 3-DIMENSIONAL CONFORMAL FIELD-THEORY

We study a three-dimensional conformal field theory in terms of its pa rtition function on arbitrary curved spaces. The large N limit of the non-linear sigma model at the non-trivial fixed point is shown to be a n example of a conformal field theory, using zeta function regularizat ion. We compute the critical properties of this model in various space s of constant curvature (R(2) X S-1, S-1 X S-1 X R, S-2 X R, H-2 X R, S-1 X S-1 X S-1 and S-2 X S-1) and we argue that what distinguishes th e different cases is not the Riemann curvature but the conformal class of the metric. In the case H-2 X R (constant negative curvature), the O(N) symmetry is spontaneously broken at the critical point. In the c ase S-2 X R (constant positive curvature) we find that the free energy vanishes, consistent with conformal equivalence of this manifold to R (3), although the correlation length is finite. In the zero-curvature cases, the correlation length is finite due to finite size effects. Th ese results describe two-dimensional quantum phase transitions or thre e-dimensional classical ones.