Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 3 / d / Solution

The free-boson worldsheet theory factorizes into independent holomorphic and antiholomorphic sectors. The factors $\partial X^{\mu }$ contract only in the holomorphic sector and produce one tensor $T^{\mu \kappa \rho }$, while the $\bar{\partial }{X}^{\nu }$ factors independently produce $T^{\nu \lambda \sigma }$. Their product is the closed-string version of left-right factorization, often summarized at tree level as closed-string kinematics being a square of open-string kinematics.