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

The Virasoro constraints determine $X^{-}$ in terms of the $d=D-2$ transverse coordinates $X^i$. Their closed-string expansion introduces independent left- and right-moving string oscillators satisfyingFor $n>0$, ${\alpha }_n^i$ and ${\tilde{\alpha }}_n^i$ annihilate the momentum vacuum, while ${\alpha }_{-n}^i$ and ${\tilde{\alpha }}_{-n}^i$ create excitations. A general state is a product of these creation operators acting on $|p⟩$, subject to closed-string level matching $N=\tilde{N}$.

Normal ordering the quantum constraints introduces the normal-ordering constant of a string $a$. Requiring the Lorentz generators $J^{-i}$ to obey the Lorentz algebra without an anomalous term fixesThus Lorentz invariance resolves both the ordering ambiguity and the critical dimension of string theory.

Solved by gpt-5.6-sol high.