Conformal invariance from scale invariance in nonlinear sigma models

Abstract

There exists a certain argument that in even dimensions, scale invariant quantum field theories are conformal invariant. We may try to extend the argument in 2n+ϵ dimensions, but the naive extension has a small loophole, which indeed shows an obstruction in nonlinear sigma models in 2+ϵ dimensions. Even though it could have failed due to the loophole, we show that scale invariance does imply conformal invariance of nonlinear sigma models in 2+ϵ dimension from the seminal work by Perelman on the Ricci flow.