I have a question about the expressions for free field Green's functions in conformal field theory. It comes from three origins
1) In Polchinski's string theory volume I p36, it is given
$$ \frac{1}{\pi \alpha'} \partial_z \partial_{\bar{z}} X^{\mu}(z,\bar{z}) X^{\nu} (z',\bar{z}') = -\eta^{\mu \nu} \delta^2 (z-\bar{z}', \bar{z}-\bar{z}') \tag{2.1.20} $$
2) In David Tong's string theory lecture note, p 186 (in the number at the bottom) or p 193 (counting by acrobat reader),
$$\langle Y^a(z,\bar{z}) Y^b(\omega,\bar{\omega}) \rangle = G^{ab} (z,\bar{z}; \omega,\bar{\omega} )$$ $$ \partial \bar{\partial } G^{ab} (z,\bar{z}) = - 2 \pi \delta^{ab} \delta (z,\bar{z}) \tag{7.33} $$
here $\alpha'$ has been scaled by $X^a = \bar{x}^a + \sqrt{\alpha'} Y^a$ in p 185/192. Why there is a factor of 2 difference in the RHS of (2.1.20) (Polchinski) and (7.33) (Tong)?
In page 77, it is given
$$ \langle \partial^2 X(\sigma) X(\sigma') \rangle = - 2\pi \alpha' \delta(\sigma-\sigma') \tag{4.20}$$
I am not sure what is the definition of $\partial^2$. Since $\partial \bar{\partial}=\frac{1}{4} (\partial_1^2 + \partial_2^2) $ and $\delta^2(z,\bar{z})=\frac{1}{2} \delta(\sigma^1) \delta(\sigma^2)$ (Polchinski p33),the factor at RHS of Eq. (4.20) in David Tong's lecture note seems to agree with Polchinski (2.1.20), but not Tong's (7.33)
In addition, David Tong's lecture is related to
We'll be fairly explicit here, but if you want to see more details then the best place to look is the original paper by Abouelsaood, Callan, Nappi and Yost, \Open Strings in Background Gauge Fields", Nucl. Phys. B280 (1987) 599
3) In the original paper, it is given
$$\frac{1}{2\pi\alpha'} \square G(z,z')= - \delta(z-z') \tag{2.7}$$
where $\square=\partial_{\tau}^2 + \partial_{\sigma}^2$.
Again, a factor of two difference between (2.7) and (2.1.20) of Polchinski. In Eq. (2.1) of that paper states
$$ S= \frac{1}{2\pi \alpha'} \left[ \frac{1}{2} \int_{M^2} d^2 z \partial^a X_{\mu} \partial_a X^{\mu} + i \int_{\partial M} d \tau A_{\mu} \partial_{\tau} X^{\nu} \right] $$
In Polchinski's string theory volume I, p32, it is given
$$ S = \frac{1}{4\pi \alpha'} \int d^2 \sigma \left( \partial_1 X^{\mu} \partial_1 X_{\mu} + \partial_2 X^{\mu} \partial_2 X_{\mu} \right) \tag{2.1.1}$$
Since $d^2 z = 2 d \sigma^1 d \sigma^2 \tag{2.1.7}$ , is that the origin of factor of 2 difference? Because combininig Eqs.(2.1.1) and (2.1.7) leads to $$ S = \frac{1}{2\pi \alpha'} \int d^2 z \left( \partial_1 X^{\mu} \partial_1 X_{\mu} + \partial_2 X^{\mu} \partial_2 X_{\mu} \right) $$