Conformal invariance from scale invariance in nonlinear sigma models

Yu Nakayama

doi:10.1103/physrevd.102.065018

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 $2n + \epsilon$ dimensions, but the naive extension has a small loophole, which indeed shows an obstruction in non-linear sigma models in $2+\epsilon$ dimensions. Even though it could have failed due to the loophole, we show that scale invariance does imply conformal invariance of non-linear sigma models in $2+\epsilon$ dimension from the seminal work by Perelman on the Ricci flow.

Highlights

The advent of conformal bootstrap approaches to critical phenomena (e.g., [1] for a review) raises a renewed interest in understanding about under which conditions the conformal symmetry emerges
Since we have seen that the fixed points of φ4 theories are conformal invariant in the d 1⁄4 4 þ ε dimensions, we expect that the fixed points of the nonlinear sigma models in d 1⁄4 2 þ ε dimensions are conformal invariant
The direct renormalization group interpretation of Perelman’s entropy in nonlinear sigma models in two dimension was not obvious, but we find that it has a direct connection with conformal invariance of nonlinear sigma models in d 1⁄4 2 þ ε dimensions

Summary

INTRODUCTION

The advent of conformal bootstrap approaches to critical phenomena (e.g., [1] for a review) raises a renewed interest in understanding about under which conditions the conformal symmetry emerges. It is typically the case that scale invariance, Poincareinvariance (Euclidean invariance), and unitarity (reflection positivity) give rise to the enhanced conformal symmetry Some arguments supporting this empirical fact exist in even space-time dimensions, in particular two [2] and four dimensions [3,4,5,6], but we do not have general arguments in odd dimensions, say, in three dimensions.. Gradient flow of the renormalization group beta function in d 1⁄4 2n þ ε dimensions once we know that it is a gradient flow in d 1⁄4 2n dimensions This typically implies conformal invariance in (perturbative) scale invariant fixed point in d 1⁄4 2n þ ε dimensions if any. This, on the other hand, suggests that a general argument without a loophole would be quite nontrivial: at least it should directly imply Perelman’s theorem on the Ricci flow

A SIMPLE ARGUMENT AND POSSIBLE LOOPHOLES

B I ð6Þ where aðgÞ ε 2

A direct approach

A gradient approach 1

A gradient approach 2

DISCUSSION