= Solution
Distinguish target-space light-cone coordinates from the worldsheet $\pm$ labels. In target space take $X^\pm=(X^0\pm X^{25})/\sqrt2$, so the <Minkowski metric> is $ds^2=-2dX^+dX^-+\sum_i(dX^i)^2$. After <conformal gauge> has been chosen, the residual transformations of $\tau+\sigma$ and $\tau-\sigma$ may be used, on a patch with $p^+\ne0$, to make
$$
X^+=x^++\kappa\tau.
$$
For an open string on $0\le\sigma\le\pi$, the conventional normalization is $\kappa=2\alpha'p^+$. For the closed-string convention of the preceding solution, $\kappa=\alpha'p^+$. In either case, the two <Virasoro constraints> read
$$
-2\partial_\pm X^+\partial_\pm X^-+\sum_i(\partial_\pm X^i)^2=0.
$$
Since $\partial_\pm X^+=\kappa/2$, they determine the longitudinal coordinate:
$$
\boxed{\partial_\pm X^-=\frac1\kappa\sum_i(\partial_\pm X^i)^2.}
$$
Its oscillator content is fixed by the transverse coordinates, leaving $d-2$ independent fields. This is the <light-cone gauge in string theory>; it solves the constraints rather than imposing them afresh on each transverse state.
For <Neumann boundary conditions> at both endpoints, the open-string expansion is
$$
X^\mu=x^\mu+2\alpha'p^\mu\tau+i\sqrt{2\alpha'}\sum_{n\ne0}\frac{\alpha_n^\mu}{n}e^{-in\tau}\cos(n\sigma).
$$
The transverse <string oscillators> satisfy $[\alpha_m^i,\alpha_n^j]=m\delta^{ij}\delta_{m+n,0}$, $i,j=1,\ldots,24$, and $\alpha_{n>0}^i|0,p\rangle=0$. Define $L_n^\perp=\tfrac12\sum_m:\!\alpha_{n-m}^i\alpha_m^i\!:$ and $\alpha_0^+=\sqrt{2\alpha'}p^+$. The solved longitudinal modes, including the zero-mode constraint, are
$$
\alpha_n^-=\frac{L_n^\perp-a\delta_{n0}}{\alpha_0^+}.
$$
The critical <bosonic string theory> has 24 transverse bosons. Their regulated oscillator vacuum energy is $24\cdot\tfrac12\sum_{n\ge1}n=-1$, since the finite term of $\sum_{n\ge1}ne^{-\varepsilon n}=\varepsilon^{-2}-1/12+O(\varepsilon^2)$ is $-1/12$. Thus the <normal-ordering constant of a string> is $a=1$. A normalized transverse <Fock space> basis for general excited <physical string states> is
$$
\boxed{|\{k_{ni}\};p\rangle=\prod_{n\ge1}\prod_{i=1}^{24}\frac{(\alpha_{-n}^i/\sqrt n)^{k_{ni}}}{\sqrt{k_{ni}!}}|0,p\rangle,
\qquad k_{ni}\in\{0,1,2,\ldots\},}
$$
with finitely many nonzero occupation numbers. Arbitrary physical states are superpositions of this basis, with each momentum on its corresponding <mass shell>. The <string level operator> and <open bosonic string mass spectrum> give
$$
N=\sum_{n,i}n k_{ni},\qquad
\boxed{M^2=\frac{N-1}{\alpha'},\qquad p^-=\frac{p_i p_i+(N-1)/\alpha'}{2p^+}.}
$$
All spatial directions have free endpoints, so \b[this string ends on a space-filling D25-brane]. More generally, a <D-brane> has <Neumann boundary conditions> along its worldvolume and <Dirichlet boundary conditions> in its transverse directions; a D$p$-brane has $p$ spatial worldvolume directions. A stack may additionally supply <Chan-Paton factors>, without changing the oscillator construction above.
To see <T-duality of D-brane boundary conditions> directly, decompose the compact coordinate as $X= X_L(\tau+\sigma)+X_R(\tau-\sigma)$ and define its dual by $\widetilde X=X_L-X_R$. Differentiation gives
$$
\partial_\tau\widetilde X=\partial_\sigma X,\qquad
\partial_\sigma\widetilde X=\partial_\tau X.
$$
Therefore the original <Neumann boundary condition> $\partial_\sigma X=0$ becomes $\partial_\tau\widetilde X=0$ at each endpoint. The dual endpoint coordinate is time-independent: \b[the dual condition is Dirichlet]. Conversely a fixed endpoint in $X$ becomes Neumann in the dual coordinate. For a circle, the <T-duality> radius is $\widetilde R=\alpha'/R$. A wrapped Neumann brane loses one spatial dimension in the dual picture, for example D25 becomes D24. The two dual endpoint constants need not be identical; their positions encode the original Wilson-line data.
For the separated-brane problem, use two static parallel D$p$-branes and no relative angle or background field. Let their displacement in transverse directions be $d^I$, with $\sum_I(d^I)^2=d^2$. A Dirichlet coordinate contains the classical term
$$
X^I(\sigma,\tau)=y_1^I+\frac{d^I}{\pi}\sigma+\text{integer-moded sine oscillators}.
$$
The spatial gradient contributes to the worldsheet Hamiltonian
$$
\frac1{4\pi\alpha'}\int_0^\pi d\sigma\sum_I(X'^I)^2=\frac{d^2}{4\pi^2\alpha'}.
$$
Parallel branes have integer modes in their NN and DD directions, so their oscillator vacuum energy is still $-1$. The zero-mode <Virasoro constraint> is consequently
$$
\alpha'(-E^2+k_\parallel^2)+\frac{d^2}{4\pi^2\alpha'}+N-1=0.
$$
With <string tension> $T=1/(2\pi\alpha')$, the <Open string stretched between D-branes> has
$$
\boxed{M_N^2=(Td)^2+\frac{N-1}{\alpha'}.}
$$
The classical stretching energy is $Td$, while the quantum ground-state energy, when real, is
$$
\boxed{E_0(k_\parallel)=\sqrt{k_\parallel^2+\frac{d^2}{4\pi^2\alpha'^2}-\frac1{\alpha'}},\qquad
E_0(0)=\sqrt{(Td)^2-\frac1{\alpha'}}.}
$$
The vacuum correction is an additive term in energy squared, not a constant subtraction from the classical stretching energy. It follows that the <bosonic stretched-string tachyon threshold> is
$$
\boxed{d_0=2\pi\sqrt{\alpha'},\qquad M_0^2<0\ \Longleftrightarrow\ d<d_0.}
$$
Below this threshold the lowest state is a <tachyon>: small-momentum modes have imaginary frequency and the background is unstable, rather than having an ordinary real negative ground-state energy. It is massless at the threshold. The assumption of parallel branes is needed for the integer moding used here; relative angles change the vacuum contribution.
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