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

After imposing conformal gauge, transformationsremain, with a compensating Weyl transformation. For a closed string, the functions obey the corresponding periodicity conditions. Introduce target-space light-cone coordinates $X^{\pm }=(X^0\pm X^{D-1})/\sqrt{2}$. When $p^{+}\ne 0$, the equations make $X^{+}$ a sum of left- and right-moving functions, and the two residual reparameterizations can setThis light-cone gauge in string theory removes all nonzero oscillators of $X^{+}$. Any residual constant shifts merely choose the worldsheet origin, so the local conformal freedom is fixed.

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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.

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The bosonic string mass spectrum isWith Lorentz invariance, $a=1$ and $D=26$, so $N=0$ is tachyonic, $N=1$ is massless, and higher levels are massive. To display the degeneracies in general notation, put $d=D-2$. The numbers of states in one chiral sector at levels $N=0,1,2,3$ areLevel matching pairs any left state with any right state at the same level, so the total closed-string degeneracies areAt $D=26$ these are $1$, $576$, $104976$, and $10240000$. If “first four levels” is taken to exclude the ground state, the next chiral degeneracy iswhich is $25650$ at $d=24$, giving total degeneracy $25650^2=657922500$.

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