# Conformal dimension of conserved current and Current Algebra in CFT

In the 2019 ESI lecture notes on WZW model, right before equation (3.1) Lorenz Eberhardt claims that any conserved current of a CFT has conformal weight (1,0) or (0,1). Can someone please explain why this is true?

Further, I am not sure if the following is directly related to the above question or not, while studying chiral fields we define chiral current to be a quasi-primary chiral field which has conformal dimension $$h = 1$$ $$^{[1]}$$. Why are we exclusively selecting chiral fields with $$h = 1$$?

[1] Introduction to Conformal Field Theory: With Applications to String Theory by Plauschinn and Ralph Blumenhagen, Page 37 Section 2.7

• This is actually a pretty general statement. In a $(d+1)$ dimensional CFT, the conserved current of a continuous symmetry is a dimension $d$ operator. One way to see this is to note that $Q \sim \int d^dx j^0$ is the associated charge of the symmetry, and by definition of being a symmetry, should commute with the stress tensor. Commuting with the stress tensor essentially means that $Q$ has dimension $0$. But then this in turn means that $j^0$ has dimension $d$, since $d^dx$ has dimension $-d$. Commented Sep 10, 2021 at 4:58
• @BRayhaun this might be very elementary, but I want to understand this a little better. d (here = 1) - the dimension that you calculated is the length scale dimension right, why should it become the conformal dimension as well ? Commented Sep 10, 2021 at 5:25
• When I say dimension here, I'm abbreviating scaling dimension'' which refers to how objects behave under transformations of the form $x\to \lambda x$. So we say that spacetime coordinates have scaling dimension $-1$. By the way, in 2d CFT all kinds of special stuff happens: for comparison, scaling dimension refers to $h+\bar h$. In general dimension, there is no notion of holomorphic vs anti-holomorphic dimension. Commented Sep 10, 2021 at 5:49

Except in special cases, a conserved current is always understood to be a spin one conserved current. The fact that these have dimension $$d -1$$ in $$CFT_d$$ follows from the conformal algebra. Given a primary operator of spin one, we can create a state by acting on the vacuum. One of its descendants will be $$P_\mu \left | J^\mu \right > = \partial_\mu J^\mu(0) \left | 0 \right >$$ since momentum generates a spatial translation.
We can now act with the special conformal generator to get \begin{align} K_\nu P_\mu \left | J^\mu \right > &= [K_\nu, P_\mu] \left | J^\mu \right > \\ &= 2(D \delta_{\mu\nu} - M_{\nu\mu}) \left | J^\mu \right > \\ &= 2(\Delta - d + 1) \left | J^\nu \right >. \end{align} This vanishes if and only if $$\Delta = d - 1$$. The generalization of this to higher spin is $$\Delta = d + \ell - 2$$. So in 2d, the scaling dimension of a conserved current is the same as its spin. This is why generators of a chiral symmetry always have one conformal weight equal to zero and the other equal to an integer or half integer.