Solution
= Solution
The quadratic worldsheet action makes each $X^\mu$ a free boson. Its <Green function> in the complex plane inverts the operator in the action and uses $\partial\bar\partial\log|z-w|^2=2\pi\delta^{(2)}(z-w)$ with the corresponding convention. The result is
$$
\boxed{\left\langle X^\mu(z,\bar z)X^\nu(w,\bar w)\right\rangle
=-\frac{\alpha'}2\eta^{\mu\nu}\log|z-w|^2}.
$$
An additive constant is physically irrelevant because it can be absorbed into the zero mode.