Conformal Ward identity minus sign error

Question

I'm trying to track down what seems like a fairly crucial minus-sign error in Di Francesco et al's conformal field theory book. The minus sign has to do with the Ward identity for Lorentz transformations (ie, rotations in Euclidean signature). On page 107, Eq. (4.66) states $$ \langle (T^{\rho \nu} - T^{\nu \rho}) X \rangle = -i \sum_i \delta(x-x_i) S^{\nu \rho}_i \langle X \rangle.\tag{4.66} $$ Upon specializing to $d = 2$ dimensions in chapter 5, they have the alternate equation $$ \varepsilon_{\mu \nu} \langle T^{\mu \nu} X \rangle = -i \sum_i \delta(x-x_i) s_i \langle X \rangle $$ Here is the issue: if $S^{\nu \rho}_i = \varepsilon^{\nu \rho} s_i$, then these two equations are inconsistent by a minus sign. This is easy to see, for example, upon letting $\rho = 0$ and $\nu = 1$ in the first equation. On the other hand, if $S^{\nu \rho}_i = -\varepsilon^{\nu \rho} s_i$, then this is inconsistent with the previous definitions of both $S^{\nu \rho}$ and $s$: under a Lorentz transformation in $d$ dimensions, the fields are defined to transform as (c.f. Eq. (2.132) on page 40): $$ \phi(x) \mapsto \phi'(x') = \phi(x) - \frac{i}{2} \omega_{\alpha \beta} S^{\alpha \beta} \phi + \ldots $$ where $\omega_{\alpha \beta} = -\omega_{\beta \alpha}$ and $S^{\alpha \beta} = -S^{\beta \alpha}$ are bot antisymmetric. In $d = 2$, we can write $\frac{1}{2} \omega_{\alpha \beta} S^{\alpha \beta} = \omega_{01} S^{01} \equiv \omega S$. On the other hand, quasi-primary fields are defined to transform under rotations in two dimensions as (c.f. Eq. (5.22) on page 116, with $w = e^{i \theta}z$): $$ \phi'(w, \bar{w}) = \left( \frac{dw}{dz} \right)^{-h} \left( \frac{d\bar{w}}{d\bar{z}} \right)^{-\bar{h}} \phi (z,\bar{z}) = e^{-i(h - \bar{h})\theta} \phi(z,\bar{z}) - e^{-i s \theta} \phi(z,\bar{z}) $$ So it seems that we really ought to take $S^{\nu \rho}_i = \varepsilon^{\nu \rho} s_i$, and not with a minus sign.

As far as I can tell, the very first equation seems to be consistent with all previous notation and definitions in defining Lorentz transformations, the stress-energy tensor, etc; while the sign in the second equation seems crucial for obtaining the correct form of the conformal Ward identity in Eq. (5.39) on page 120. Am I making a simple mistake somewhere by believing these equations are inconsistent?

Answer 1

I managed to answer my own question. The parameters $\omega_{\alpha \beta}$ in the Lorentz transform are defined by $$ x^{\mu} \mapsto x'^{\mu} = x^{\mu} + \eta^{\mu \alpha} \omega_{\alpha \beta} x^{\beta} + \ldots $$ In particular, in $d = 2$ we have $$ x^0 \mapsto x^0 + \omega_{01} x^1 + \ldots, x^1 \mapsto x^1 - \omega_{01} x^0 + \ldots $$ Notice that for small positive $\omega_{01}$, we have a clockwise rotation of the complex plane. We should therefore identify $\omega_{01}$ with $-\theta$ in the original question: ie, if we want a counterclockwise rotation $z \mapsto e^{i \theta} z$ to correspond to a field transformation $\phi \mapsto e^{-i s \theta} \phi$, we need to take $S^{\nu \rho} = - s \varepsilon^{\nu \rho}$ with a minus sign.