A Three-Dimensional Conformal Field Theory

This talk is based on a recent paper$^{1}$ of ours. In an attempt to understand three-dimensional conformal field theories, we study in detail one such example --the large $N$ limit of the $O(N)$ non-linear sigma model at its non-trivial fixed point -- in the zeta function regularization. We study this on various three-dimensional manifolds of constant curvature of the kind $\Sigma \times R$ ($\Sigma=S^1 \times S^1, S^2, H^2$). This describes a quantum phase transition at zero temperature. We illustrate that the factor that determines whether $m=0$ or not at the critical point in the different cases is not the `size' of $\Sigma$ or its Riemannian curvature, but the conformal class of the metric.