Nonlinear models in 2 + ε dimensions

Abstract

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 coefficients 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 φ on Euclidean 2 + ε space whose values φ( x) lie in a finite dimensional differentiable manifold M. The action is S(φ) = Λ ε f dx 1 2 T −1g ij(φ(x)) ∂ μ φ i ( x) ∂ μ φ j ( x), where Λ −1 is the short distance cutoff and T −1 g ij 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 ij 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 the diffeomorphisms (i.e., the reparametrizations of transformations) of M. The techniques used are the standard power counting arguments combined with generalizations of the BRS transformation and the method of quadratic identities. The second part of the renormalization is crucial for renormalizing the standard models, since it implies the renormalization of internal symmetry. It is carried out to the point of identifying the finite dimensional cohomology spaces containing possible obstructions to the renormalization of the transformation laws, and of noting the absence of obstructions when M has finite fundarmental group and nonabelian semisimple isometry group. The renormalization group equation for the metric coupling is Λ −1( ∂ ∂ Λ −1)g ij = −β ij(g) , β ij(T −1g) = −εT −1g ij + R ij + 1 2 TR ipqrR jpqr + O(T 2) . R ipqr is the curvature tensor and R ij = R ipjp the Ricci tensor of the metric g ij . The β-function β ij ( g) is a vector field on the infinite dimensional space of Riemannian metrics on M. Two results on global properties of β are obtained. When M is a homogeneous space G H , the β-function is shown to be a gradient on the finite dimensional space of G-invariant metric couplings on M. And, when M is a two dimensional compact manifold, the β-function is shown to be a gradient on the infinite dimensional space of metrics on M. The rest of the results are concerned with fixed points. The fixed points are shown to correspond to the metrics satisfying a generalized Einstein equation, R ij − ag ij = ∇ i v j + ∇ j v i , a = ±1 or 0, for v i some vector field on M. Known solutions to these equations are discussed and some of their general properties described. In particular, it is shown that infrared instability occurs in at most a finite number of directions in the infinite dimensional space of metric couplings.