= Solution
Use the mostly-positive target <Minkowski spacetime> metric and a <closed string> coordinate $0\leq\sigma<2\pi$. In <conformal gauge>, variation of the <Polyakov action> gives $(\partial_\tau^2-\partial_\sigma^2)X^\mu=0$. The compact coordinate, denoted $X^{25}$, can change by $2\pi Rw$ around the <closed string>, with $w\in\mathbb Z$. Write $w^\mu=w\delta^{\mu,25}$. Integrating the two chiral wave solutions and expanding their periodic parts gives the <closed-string mode expansion>
$$
X^\mu=x^\mu+\alpha'p^\mu\tau+Rw^\mu\sigma
+i\sqrt{\frac{\alpha'}2}\sum_{n\ne0}\frac1n
\left(\alpha_n^\mu e^{-in(\tau-\sigma)}+
\widetilde\alpha_n^\mu e^{-in(\tau+\sigma)}\right).
$$
Reality imposes $\alpha_{-n}=(\alpha_n)^*$ and the analogous condition in the other <chirality>. The classical compact momentum is continuous; quantization of a wavefunction on the circle subsequently gives $p^{25}=n/R$ with $n\in\mathbb Z$. The <momentum and winding modes> are most conveniently packaged as
$$
p_R=\frac nR-\frac{wR}{\alpha'},\qquad
p_L=\frac nR+\frac{wR}{\alpha'},\qquad
\alpha_0^{25}=\sqrt{\frac{\alpha'}2}p_R,\quad
\widetilde\alpha_0^{25}=\sqrt{\frac{\alpha'}2}p_L.
$$
For noncompact directions both zero modes are $\sqrt{\alpha'/2}\,p^\mu$. Thus the formula includes both independent <string oscillator> families and all compact topological sectors.
Define the <worldsheet stress-energy tensor> by $\theta_{ab}=-2(-\gamma)^{-1/2}\delta I/\delta\gamma^{ab}$. Its normalization and components are
$$
\theta_{ab}=\frac1{2\pi\alpha'}
\left(\partial_aX\cdot\partial_bX-
\frac12\gamma_{ab}\gamma^{cd}\partial_cX\cdot\partial_dX\right),
\qquad
\theta_{\tau\tau}=\theta_{\sigma\sigma}
=\frac{\dot X^2+X'^2}{4\pi\alpha'},\quad
\theta_{\tau\sigma}=\frac{\dot X\cdot X'}{2\pi\alpha'}.
$$
An overall rescaling of the named <worldsheet stress tensor> does not alter its vanishing. Variation of the independent <worldsheet metric> imposes $\theta_{ab}=0$, equivalently $(\dot X+X')^2=(\dot X-X')^2=0$. These are the <Virasoro constraints> that restrict the otherwise general wave solution.
The conjugate momentum density is $P_\mu=\dot X_\mu/(2\pi\alpha')$. The gauge-fixed <Hamiltonian> generating $\tau$ translation is therefore
$$
H=\int_0^{2\pi}\frac{\dot X^2+X'^2}{4\pi\alpha'}d\sigma
=L_0+\widetilde L_0,
\qquad
L_m=\frac12\sum_{k\in\mathbb Z}\alpha_{m-k}\cdot\alpha_k,
$$
with the analogous tilded generators. In particular, before imposing the <Virasoro constraints>,
$$
H=\frac{\alpha'}2p_{\rm nc}^2+
\frac{\alpha'}2\left(\frac nR\right)^2+
\frac{w^2R^2}{2\alpha'}+N+\widetilde N,
\qquad
N=\sum_{k>0}\alpha_{-k}\cdot\alpha_k.
$$
The target momentum includes its timelike component, so this is not a positive particle-energy formula. On the classical constraint surface the reparameterization <Hamiltonian> vanishes; target energy is the separate charge $p^0$. The difference $L_0-\widetilde L_0=0$ imposes the residual spatial constraint.
In the quantum theory the same quadratic expressions require <normal ordering>: $[\alpha_m^\mu,\alpha_n^\nu]=m\eta^{\mu\nu}\delta_{m+n,0}$ makes different orderings differ by divergent constants. Their regularized finite remainder is the <normal-ordering constant of a string>, or <string intercept>. The <Virasoro central extension> also appears. In the critical 26-dimensional <bosonic string theory>, the 24 transverse <string oscillators> give one-unit <string intercept> in each <chirality>. For example their regulated chiral vacuum energy is $24\sum_{k>0}k/2=-1$ using <zeta function regularization>. The physical zero-mode constraints are $(L_0-1)|\Psi\rangle=(\widetilde L_0-1)|\Psi\rangle=0$.
Let $M$ be the mass seen in the remaining 25 noncompact dimensions. Substituting the compact zero modes in these two constraints gives
$$
M^2=p_R^2+\frac4{\alpha'}(N-1)
=p_L^2+\frac4{\alpha'}(\widetilde N-1).
$$
Taking their mean and difference yields
$$
\boxed{M^2=\frac{n^2}{R^2}+\frac{w^2R^2}{\alpha'^2}
+\frac2{\alpha'}(N+\widetilde N-2),\qquad
\widetilde N-N+nw=0.}
$$
This sign of <compact-circle closed-string level matching> follows from assigning tildes to the left-moving sector. Under <T-duality>, set $R'=\alpha'/R$, $n'=w$, $w'=n$. Then $p_L'=p_L$ and $p_R'=-p_R$, so reversing the compact right-moving <string oscillators> implements the transformation while leaving $N,\widetilde N$ unchanged. Both the mass equation and <closed-string level matching> are invariant. This proves invariance of the full <closed string> spectrum, including its <string oscillator> degeneracies, rather than only the zero-mode contribution.
A <D-brane> with $p$ spatial <worldvolume> directions, called a D$p$-brane, is a $(p+1)$-dimensional timelike submanifold on which <open string> endpoints have <Neumann boundary conditions> tangentially and <Dirichlet boundary conditions> transversely. Apply <T-duality> to the compact coordinate by $\widetilde X=X_L-X_R$. The differential relations
$$
\partial_\tau\widetilde X=\partial_\sigma X,\qquad
\partial_\sigma\widetilde X=\partial_\tau X
$$
interchange the two endpoint conditions. Consequently a D$p$-brane wrapping the original circle becomes a D$(p-1)$-brane localized on the dual circle. Its compact <Wilson line> specifies that position. This is <T-duality of D-brane boundary conditions>.
\b[The separated-brane configuration must instead have the circle transverse to the <D-branes>.] Two parallel <D-branes> wrapping the same circle cannot be separated along that <worldvolume> direction. Thus the latter part describes a new orientation, with both endpoints subject to compact <Dirichlet boundary conditions>. If their positions differ by $L$, an oriented <Open string stretched between D-branes> has displacement
$$
L_w=L+2\pi Rw,\qquad w\in\mathbb Z.
$$
On $0\leq\sigma\leq\pi$, its compact classical solution has a linear part $y_1+L_w\sigma/\pi$, plus sine modes vanishing at the endpoints. This adds $L_w^2/(4\pi^2\alpha')$ to the <Virasoro generator> zero mode. Thus, for momentum $\mathbf p$ along the <D-branes>,
$$
\boxed{E_w^2=\mathbf p^2+\left(\frac{L_w}{2\pi\alpha'}\right)^2+
\frac{N-1}{\alpha'},\qquad
E_{0,w}^2=\left(\frac{L_w}{2\pi\alpha'}\right)^2-\frac1{\alpha'}.}
$$
The second equation is the quantum rest-energy squared of the <string oscillator> ground state; the smallest one uses the shortest allowed compact separation $|L_w|$. Classically the straight string energy is $T|L_w|=|L_w|/(2\pi\alpha')$. The quantum ground state is a <tachyon> if $|L_w|<2\pi\sqrt{\alpha'}$, so in that regime it is an instability rather than a state with real rest energy. Dropping the <string intercept> would describe the classical stretched string, not its quantum ground state.
The transverse-circle <T-duality> changes these two <D-branes> to D$(p+1)$-branes wrapping the dual circle. Their positions become the eigenvalues $A_i=y_i/(2\pi\alpha')$ of a constant gauge connection. The <Wilson loop> phases are $e^{i2\pi R'A_i}=e^{iy_i/R}$. A string connecting <D-brane> $1$ to <D-brane> $2$ has dual compact momentum
$$
p'_{25}=\frac w{R'}+A_2-A_1
=\frac{L+2\pi Rw}{2\pi\alpha'}.
$$
Its momentum contribution to $E^2$ is exactly the former stretching term, including every winding sector. Qualitatively the two geometrically coincident wrapped <D-branes> are distinguished by their <Wilson lines>, and their connecting strings have shifted momentum quantization. This is how <brane separation becomes a Wilson line>.
Back to article page
