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

Varying the Polyakov action with respect to the worldsheet metric givesIn conformal gauge, variation of $X$ gives the wave equationwhile the Virasoro constraints become $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. The boundary variation is $-T\int d\tau ,X^{\prime }\cdot \delta X$ at each endpoint. It vanishes through a Neumann boundary condition $X^{\prime \mu }=0$, a Dirichlet boundary condition $\delta X^{\mu }=0$, or a direction-by-direction mixture of the two.

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.

Orthogonality of the cosine modes gives ${\int }_0^{\pi }\cos (n\sigma )d\sigma =0$ and ${\int }_0^{\pi }\cos (n\sigma )\cos (m\sigma )d\sigma =\pi {\delta }_{nm}/2$. Since the supplied expansion uses ${\alpha }^{\prime }{p}^{\mu }\tau$ rather than the more usual $2{\alpha }^{\prime }{P}^{\mu }\tau$,With ${\alpha }_{-n}={\alpha }_n^{\ast }$ for a real embedding,The first term is orbital angular momentum and the second is the contribution of the string oscillators.

A rigidly rotating stretched solution iswith all other coordinates constant. Each coordinate obeys the wave equation and $X^{\prime }=0$ at $\sigma =0,\pi$. Directly, $\dot{X}\cdot X^{\prime }=0$ andso both Virasoro constraints hold. Its energy and planar angular momentum areConsequently $J=M^2/(2\pi T)={\alpha }^{\prime }{M}^2$, the classical leading open-string Regge trajectory.