The zero-mode <Virasoro constraint> is
$$
-\alpha'M^2+\frac{|x_2-x_1|^2}{4\pi^2\alpha'}+N-1=0,
$$
where the <string level operator> is
$$
\boxed{N=\sum_{n=1}^\infty\alpha_{-n}\mathbin\cdot\alpha_n}.
$$
Since the <string tension> is
$$
\boxed{T=\frac1{2\pi\alpha'}},
$$
the mass formula is
$$
\boxed{M^2=T^2|x_2-x_1|^2+\frac1{\alpha'}(N-1)}.
$$

For a string beginning and ending on the same brane, the $N=1$ states are massless. Oscillators polarized tangentially give a gauge field on the $(p+1)$-dimensional worldvolume, while transverse polarizations give scalar fields describing fluctuations of the brane position. Strings joining two separated branes have the additional stretching mass $T|x_2-x_1|$ and are charged under the two endpoint gauge groups. When the two branes coincide, these off-diagonal states also become massless and <Coincident-D-brane gauge enhancement> enlarges $U(1)\times U(1)$ to $U(2)$. The bosonic open string also retains its $N=0$ tachyon.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 4 / 3 / Solution

Let $M$ be the mass measured in the uncompactified 25-dimensional spacetime. The quantum zero-mode constraints are
$$
0=L_0-a=\frac{\alpha'}4(-M^2+p_L^2)+N-a,
\qquad
0=\widetilde L_0-a=\frac{\alpha'}4(-M^2+p_R^2)+\widetilde N-a.
$$
The <string level operators>
$$
N=\sum_{r>0}\alpha_{-r}\mathbin\cdot\alpha_r,
\qquad
\widetilde N=\sum_{r>0}\widetilde\alpha_{-r}\mathbin\cdot\widetilde\alpha_r
$$
have nonnegative integer eigenvalues. Adding the constraints and using $(p_L^2+p_R^2)/2=n^2/R^2+w^2R^2/\alpha'^2$ gives
$$
\boxed{M^2=\frac{n^2}{R^2}+\frac{w^2R^2}{\alpha'^2}
+\frac2{\alpha'}(N+\widetilde N-2a)}.
$$
Their difference gives <closed-string level matching>, $N-\widetilde N+nw=0$ with the present left-right convention. The <normal-ordering constant of a string> $a$ is the regularized zero-point energy generated when oscillator products are normal ordered.

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 1 / d / Solution

{title2=$N$}

The string level operator is the weighted oscillator number $N=\sum_{n>0}\alpha_{-n}\mathbin\cdot\alpha_n$. A closed string has independent operators $N$ and $\widetilde N$ for its two chiral sectors.


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