The Green function for the normalization in the question is
$$
\langle X(z,\bar z)X(w,\bar w)\rangle=-\frac12\log|z-w|^2+	ext{constant}.
$$
Differentiating at distinct points gives the holomorphic two-point function
$$
\boxed{\langle\partial_zX(z)\partial_wX(w)\rangle
=-\frac1{2(z-w)^2}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-306/2/a/solution