ANNALS OF PHYSICS 163, 318419 (1985) Nonlinear Models in 2 + E Dimensions*

The general nonlinear scalar model is studied at asymptotically low temperature near two dimensions. The low temperature expansion is renormalized and effective algorithms are derived for calculation to all orders in the renormalized expansion. The renormalization group coefftcients are calculated in the two loop approximation and topological properties of the renormalization group equations are investigated. Special attention is paid to the infrared instabilities of the fixed points, since they provide the continuum limits of the model. The model consists of a scalar field # on Euclidean 2 + E space whose values d(x) lie in a fmite dimensional differentiable manifold M. The action is S(d) = Ae J dx tT-‘g&d(x)) a,@(x)a,@(x), where A- ’ is the short distance cutoff and T-‘g, is a (positive definite) Riemannian metric on M, called the metric coupling. The standard nonlinear models are the special cases in which M is a homogeneous space (the quotient G/H of a Lie group G by a compact subgroup H) and g, is some G-invariant Riemannian metric on M. G acts as a global internal symmetry group. The renormalization of the model is divided into two parts: showing that the action retains its form under renormalization and showing that renormalization respects the action of th