Tutorial on Scale and Conformal Symmetries in Diverse Dimensions

TL;DR

The paper clarifies how scale and conformal symmetries relate in diverse dimensions, showing that conformal invariance requires both Poincaré invariance and scale invariance with the field virial $V^\alpha$ being a total divergence, enabling an improved traceless energy-momentum tensor $\theta^{\mu\nu}_{CCJ}$ and conserved currents $J^\mu_f = \theta^{\mu\nu}_{CCJ} f_\nu$. It provides explicit examples of scale-invariant but non-conformal models—such as a scalar theory with $\mathcal{L}=\mathcal{L}(z)\varphi^{\frac{2D}{D-2}}$ and Maxwell theory in $D\neq4$—and discusses a special $D=3$ Maxwell formulation where a dual scalar yields a conformal structure. The paper then demonstrates a universal dimensional reduction of the conformal group from $D>2$ to a non-relativistic conformal group in one time dimension, yielding a velocity-independent conformal mechanics in $N$ spatial dimensions with transformations $\delta_S$ and $\delta_C$. Together, these results clarify when scale invariance implies conformal invariance and reveal hidden conformal structures arising from dimensional reduction, with implications for constructing conformally invariant theories across dimensions.

Abstract

We review the relation between scale and conformal symmetries in various models and dimensions. We present a dimensional reduction from relativistic to non-relativistic conformal dynamics.