A Three-Dimensional Conformal Field Theory

TL;DR

This paper investigates three-dimensional conformal field theories by analyzing the large N limit of the O(N) non-linear sigma model at its fixed point across various curved manifolds, emphasizing the role of conformal class over size or curvature.

Contribution

It provides a detailed study of a specific 3D conformal field theory on different manifolds, highlighting the importance of conformal class in quantum phase transitions.

Findings

The critical behavior depends on the conformal class of the metric.

The size or curvature of the manifold does not determine the phase transition.

The analysis uses zeta function regularization on manifolds like S^1×S^1, S^2, and H^2.

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 $\Sigma \times R$ ($\Sigma=S^1 \times S^1, S^2, H^2$). 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.