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×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.