O( N) sigma model as a three-dimensional conformal field theory

Abstract

We study a three-dimensional conformal field theory in terms of its partition 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 an example of a conformal field theory, using zeta function regularization. We compute the critical properties of this model in various spaces of constant curvature (R 2 × S 1, S 1 × S 1 × R, S 2 × R, H 2 × R, S 1 × S 1 × S 1 and S 2 × S 1) and we argue that what distinguishes the different cases is not the Riemann curvature but the conformal class of the metric. In the case HZ X R (constant negative curvature), the O( N) symmetry is spontaneously broken at the critical point. In the case S 2 × 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. These results describe two-dimensional quantum phase transitions or three-dimensional classical ones.