A New Class of Conformal Field Theories with Anomalous Dimensions

The Wilsonian renormalization group (WRG) equation is used to derive a newclass of scale invariant field theories with nonvanishing anomalous dimensionsin 2-dimensional ${\cal N}=2$ supersymmetric nonlinear sigma models. When thecoordinates of the target manifolds have nontrivial anomalous dimensions,vanishing of the $\beta$ function suggest the existence of novel conformalfield theories whose target space is not Ricci flat. We construct suchconformal field theories with ${\bf U}(N)$ symmetry. The theory has one freeparameter a corresponding to the anomalous dimension of the scalar fields. Thenew conformal field theories are well behaved for positive a and have thecentral charge 3N, while they have curvature singularities at the boundary fora<0. When the target space is of complex 1-dimension, we obtain the explicitform of the Lagrangian, which reduces to two different kinds of free fieldtheories in weak and in strong coupling limit. As a consistency test, theanomalous dimensions are reproduced in these two limits. The target space inthis case looks like a semi-infinite cigar with one-dimension compactified to acircle.Comment: 14 pages, 3 figures; comments and references adde