What's the conserved quantities correspond to the generator of conformal transformation

Question

What's the conserved quantity corresponding to the generator of conformal transformations?

Answer 1

The conformal charge in CFTs is a special case of any other standard treatment of classical field theory (e.g. chapter 1 of Peskin & Schroeder).

In flat space with $(1, n)$ signature and spacetime $(t,x_1, ..x_n)$ coordinates, we consider a conformal killing vector field with components $\epsilon^\mu$. We assume our field theory yields some energy-momentum tensor with components $T_{\mu\nu}$.

The current $ j_\mu = T_{\mu\nu} \epsilon^\mu $ satisfies $\partial^\mu j_\mu = 0$

As usual the conserved charge is $ Q = \int dx^n j_0 $ is the conserved charge, where the integral is taken over the spatial dimensions.

In CFTs we often choose different coordinates in which these equations will not take these exact forms. Discussion of conformal charges are found in standard CFT books, for example Blumenhagen gives a concise and to the point discussion.