Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 1 / a / Solution

The Euler-Lagrange equation of the gauge-fixed Polyakov action is the two-dimensional wave equationIts general solution is a sum of left- and right-moving functions. Periodicity in $\sigma$ and a Fourier expansion give the closed-string mode expansionThe zero modes satisfy ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\prime }/2},p^{\mu }$. Here $x^{\mu }$ and $p^{\mu }$ are the center-of-mass position and momentum, while the nonzero string oscillators describe shape excitations.