A Three-Dimensional Conformal Field Theory

Abstract

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 $Σ\times R$ ($Σ=S^1 \times S^1, S^2, H^2$). 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 $Σ$ or its Riemannian curvature, but the conformal class of the metric.

7 pages, TeX, UR-1368/ER-40685-818 (Talk presented by S.G. at the 16-th Annual Montreal-Rochester-Syracuse- Toronto (MRST) Meeting:``What Next? Exploring the Future of High-Energy Physics'', held at McGill University, Montreal, Canada, 11--13 May 1994. To appear in Proceedings published by World Scientific.)