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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 1 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-306/1/ii/solution