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 Σ × R (Σ = S 1 × 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 Σ or its Riemannian curvature, but the conformal class of the metric.