Global Symmetries, Volume Independence, and Continuity in Quantum Field Theories

We discuss quantum field theories with global $SU(N)$ and $O(N)$ symmetries for which temporal direction is compactified on a circle of size $L$ with periodicity of fields up to a global symmetry transformation, i.e., twisted boundary conditions. Such boundary conditions correspond to an insertion of the global symmetry operator in the partition function. We argue in general and prove in particular for $CP(N-1)$ and $O(N)$ nonlinear sigma models that large-$N$ volume independence holds. Further we show that the $CP(N-1)$ theory is free from the Affleck phase transition confirming the Ünsal-Dunne continuity conjecture.