Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 1 / ii / 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 isAn additive constant is physically irrelevant because it can be absorbed into the zero mode.