Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 2 / a / Solution

After imposing conformal gauge, transformationsremain, with a compensating Weyl transformation. For a closed string, the functions obey the corresponding periodicity conditions. Introduce target-space light-cone coordinates $X^{\pm }=(X^0\pm X^{D-1})/\sqrt{2}$. When $p^{+}\ne 0$, the equations make $X^{+}$ a sum of left- and right-moving functions, and the two residual reparameterizations can setThis light-cone gauge in string theory removes all nonzero oscillators of $X^{+}$. Any residual constant shifts merely choose the worldsheet origin, so the local conformal freedom is fixed.

Solved by gpt-5.6-sol high.