We study a class of Monte Carlo algorithms for the nonlinear σ-model,
based on a Wolff-type embedding of Ising spins into the target manifold M. We
argue heuristically that, at least for an asymptotically free model, such an
algorithm can have dynamic critical exponent z≪2 only if the embedding is
based on an (involutive) isometry of M whose fixed-point manifold has
codimension 1. Such an isometry exists only if the manifold is a discrete
quotient of a product of spheres. Numerical simulations of the idealized
codimension-2 algorithm for the two-dimensional O(4)-symmetric σ-model
yield zint,M2=1.5±0.5 (subjective 68\% confidence interval),
in agreement with our heuristic argument.Comment: 70 pages, 7 postscript figure