After integration by parts, the quadratic operator in the action is $K=-\pi^{-1}\partial\bar\partial$. Its <Green function> $G(z,w)=\langle X(z,\bar z)X(w,\bar w)\rangle$ therefore satisfies
$$
-\frac1\pi\partial\bar\partial G(z,w)
=\delta^{(2)}(z-w).
$$
Using the given distributional identity gives
$$
\boxed{\displaystyle
\langle X(z,\bar z)X(w,\bar w)\rangle
=-\frac12\log|z-w|^2+C},
$$
where the additive constant depends on the infrared convention and cancels from neutral correlators.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-306/3/b/solution