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 $0\leq\sigma\leq\pi$, the conserved open-string charges are
$$
P^\mu=T\int_0^\pi d\sigma\,\dot X^\mu,
\qquad
J^{\mu\nu}=T\int_0^\pi d\sigma\,(X^\mu\dot X^\nu-X^\nu\dot X^\mu).
$$
The endpoint fluxes vanish for the allowed boundary conditions.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-306/1/b/solution