Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 1 / b / Solution

Translations $\delta X^{\mu }=a^{\mu }$ and Lorentz transformations $\delta X^{\mu }={\omega }^{\mu }{}_{\nu }{X}^{\nu }$ leave the action invariant because it depends only on derivatives and Lorentz contractions. Noether theorem gives the stated currents, whose equations are ${\partial }_{\alpha }{P}^{\alpha \mu }=0$ and ${\partial }_{\alpha }{J}^{\alpha \mu \nu }=0$. For $0\le \sigma \le \pi$, the conserved open-string charges areThe endpoint fluxes vanish for the allowed boundary conditions.