Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 4 / c / Solution

Integrating by parts makes the $Z$ dependence $-i\int Z{\partial }_{\alpha }{p}^{\alpha }$. Its functional integral imposes the constraint ${\partial }_{\alpha }{p}^{\alpha }=0$. On the simply connected plane, the Poincare lemma therefore permits the local and global parametrizationSubstitution into $S_2$ givesThis is the Buscher procedure for the translation isometry in $z$, and its T-duality replaces $G_{zz}=V$ by ${\tilde{G}}_{\tilde{z}\tilde{z}}=V^{-1}$.