I was wondering if anybody could clarify what the difference between the conformal scaling dimension $\Delta$ and the conformal weight $h$ is?
Is it correctly understood that $\Delta$ is related to the transformation properties of a field and $h$ is an eigenvalue of the Virasoro operator $L_0$? Or is it that $\Delta$ is for general dimensions, while $h$ is used in 2 dimensions?
I seem to have confused myself while reading Francesco et. al.'s Conformal Field
Theory.
Best Answer
Two dimensional CFTs separate into a left-moving sector and a right-moving sectors. The Virasoro generators $L_n$ act on the left-moving sector and ${\tilde L}_n$ act on the right-moving ones. Operators (or states due to the state-operator map) are labelled independently by representations of the left- and right-moving Virasoro. In particular, $h$ and ${\tilde h}$ are the eigenvalues of $L_0$ and ${\tilde L}_0$. These are called the conformal weights. $\Delta$ is the eigenvalue of the dilatation operator which is $D = L_0+ {\tilde L}_0$. In other words, $\Delta = h + {\tilde h}$.
In higher dimensions, there is no separation into left- or right- moving sectors. There is no analogue of $L_n$ and ${\tilde L}_n$. However, the dilatation operator $D$ still exists and representations are labelled by its eigenvalues, $\Delta$.