# Large-N critical NLSM (equation 13.115 of Peskin and Schroeder)

Any opinions if the equation 13.115 of Peskin and Schroeder is true on arbitrary manifolds in arbitrary dimensions for the same Lagrangian? I a priori see no problem.

The point I also want to ask is - if one evaluates that equation for spacetime >2 then one will miss the symmetry broken phase of the theory - right? - BUT isn't the saddle point obtained from it still the critical point of the corresponding theory on whatever is the (>2)-manifold?

Is this in general true that the large-N critical NLSM on a manifold $M$ is the same as the theory of $N$ non-interacting conformally coupled scalars on $M$?

Or in otherwords - will the Lagrange multiplier field in NLSM have to have at the large-N saddle point an expectation value which is the same as the conformally coupled mass for scalars on that manifold?