Nonlinear Models in2+εDimensions

A generalization of the nonlinear $\sigma$ model is considered. The field takes values in a compact manifold $M$ and the coupling is determined by a Riemannian metric on $M$. The model is renormalizable in $2+\epsilon$ dimensions, the renormalization group acting on the infinite-dimensional space of Riemannian metrics. Topological properties of the $\beta$ function and solutions of the fixed-point equation $R_{\mathrm{ij}}-\alpha g_{\mathrm{ij}}={\nabla }_i{v}_j+{\nabla }_j{v}_i$, $\alpha =\pm 1 or 0$, are discussed.