Four-loop-order β-function of nonlinear σ-models in symmetric spaces

The beta-function for the grassmannian nonlinear σ-model of symmetry U(N)U(p)∗U(N−p) has been calculated directly in four-loop order in d=2+ε dimensions. Using isomorphisms and information from 1N expansions I obtain the four-loop β-function for a large class of manifolds. Consequences are: (i) the degeneracy of the exponent ν for chiral models on the group manifolds SU(N) and SO(N) in three-loop order is lifted in four-loop order; (ii) the conductivity exponent at the mobility edge for the orthogonal case acquires a negative correction; (iii) the β-function bends over in the symplectic (i.e. spin-orbit coupling) case which suggests a nontrivial mobility edge fixed-point in d = 2 dimensions.