We will start with the worldsheet action under massless background fields - the graviton $G_{\mu\nu}$ and Kalb-Ramond field $B_{\mu\nu}$ (we choose to exclude the dilaton $\Phi$ that also appears in the theory in general) given in the standard coordinates $(\tau,\sigma)$ as:
$$S=-\frac{1}{4\pi\alpha'}\int d^2\sigma\sqrt{g}\left[ G_{\mu\nu}g^{\alpha\beta}+iB_{\mu\nu}\epsilon^{\alpha\beta }\right]\partial_{\alpha}X^{\mu}\partial_{\beta}X^\nu \tag{1}$$
Using Weyl symmetry we will reduce to a simpler action:
$$S=-\frac{1}{4\pi\alpha'}\int d^2\sigma\left[ G_{\mu\nu}\eta^{\alpha\beta}+iB_{\mu\nu}\epsilon^{\alpha\beta }\right]\partial_{\alpha}X^{\mu}\partial_{\beta}X^\nu \tag{2}$$
we can write this action explicitly as:
$$ S=-\frac{1}{4\pi\alpha'}\int d^{2}\sigma\left[G_{\mu\nu}\left(\partial_{\sigma}X^{\mu}\partial_{\sigma}X^{\nu}-\partial_{\tau}X^{\mu}\partial_{\tau}X^{\nu}\right)+iB_{\mu\nu}\left(\partial_{\sigma}X^{\mu}\partial_{\tau}X^{\nu}-\partial_{\tau}X^{\mu}\partial_{\sigma}X^{\nu}\right)\right] \tag{3} $$
Now, moving to the complex set of coordinates given by: $$z=\sigma-\tau,\quad \bar{z}=\sigma+\tau \tag{4}$$ and derivatives: $$\partial_z\equiv \partial = \frac{1}{2}(\partial_\sigma+\partial_\tau),\quad \partial_{\bar{z}}\equiv \bar{\partial} = \frac{1}{2}(\partial_\sigma+\partial_\tau) \tag{5}$$ we will substitute eq.(4-5) in eq.(3) and get:
$$S = -\frac{1}{8\pi\alpha'}\int\left(d\bar{z}^{2}-dz^{2}\right)\left[\left(G_{\mu\nu}-iB_{\mu\nu}\right)\partial X^{\mu}\bar{\partial}X^{\nu}+\left(G_{\mu\nu}+iB_{\mu\nu}\right)\bar{\partial}X^{\mu}\partial X^{\nu}\right] \tag{6} $$ Where the measure using eq.(3-4) is given by: $$d^2\sigma=d\sigma d\tau=\frac{1}{4}(d\bar{z}^2-dz^2) \tag{7}$$ which seems like strange object to be integrated over.
In many sources the measure have a special form such that: $$d^{2}\sigma=-\frac{i}{2}dzd\bar{z}\equiv-\frac{i}{2}d^{2}z \tag{8}$$
which yield to the action:
$$ S = \frac{1}{4\pi\alpha'}\int d^{2}z\left[\left(iG_{\mu\nu}+B_{\mu\nu}\right)\partial X^{\mu}\bar{\partial}X^{\nu}+\left(iG_{\mu\nu}-B_{\mu\nu}\right)\bar{\partial}X^{\mu}\partial X^{\nu}\right] \tag{9} $$
So does eq.(6) or eq.(9) is the right one? or maybe there is different way to show the worldsheet action in the complex coordinates $z,\bar{z}$? (Unfortunately I didn't find any source expressing this action in terms of $z,\bar{z}$ coordinates so I tried derive it myself).
