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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

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    $\begingroup$ 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 $-d$. $\endgroup$
    – BRayhaun
    Commented Sep 10, 2021 at 4:58
  • $\begingroup$ @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 ? $\endgroup$
    – alpha
    Commented Sep 10, 2021 at 5:25
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    $\begingroup$ 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. $\endgroup$
    – BRayhaun
    Commented Sep 10, 2021 at 5:49

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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.

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  • $\begingroup$ What if the conserved current is not primary ? I think the argument doesn't work then although i am not sure if non-primary spin 1 conserved currents can exist. $\endgroup$
    – Bronsteinx
    Commented Jan 1, 2023 at 5:40
  • $\begingroup$ Well then it can have a different dimension but its multiplet will have stuff below the unitarity bound. $\endgroup$ Commented May 19 at 10:43

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