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> satisfying
$$
[\alpha_m^i,\alpha_n^j]
=m\delta^{ij}\delta_{m+n,0},
\qquad
[\widetilde\alpha_m^i,\widetilde\alpha_n^j]
=m\delta^{ij}\delta_{m+n,0}.
$$
For $n>0$, $\alpha_n^i$ and $\widetilde\alpha_n^i$ annihilate the momentum vacuum, while $\alpha_{-n}^i$ and $\widetilde\alpha_{-n}^i$ create excitations. A general state is a product of these creation operators acting on $|p\rangle$, subject to <closed-string level matching> $N=\widetilde N$.

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 fixes
$$
a=1,\qquad D=26.
$$
Thus Lorentz invariance resolves both the ordering ambiguity and the <critical dimension of string theory>.

Solved by gpt-5.6-sol high.

Solution

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

The Fourier modes $\alpha_n^i$ and $\widetilde\alpha_n^i$ of the transverse closed-string coordinates obey harmonic-oscillator commutation relations. Modes with negative index create string excitations.


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