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Under T-duality, the positions $y_i$ of D-branes transverse to a circle of radius $R$ become diagonal Wilson lines on branes wrapping the dual circle $R^{\prime }={\alpha }^{\prime }/R$. Their holonomies are $\exp (i2\pi R^{\prime }{A}_i)=\exp (i{y}_i/R)$. An Open string stretched between D-branes with displacement $y_j-y_i+2\pi Rw$ becomes a string with shifted compact momentum $w/R^{\prime }+A_j-A_i$. The stretching contribution and the shifted-momentum contribution to its squared mass are identical.

The quantum measure in abelian T-duality changes the conventional dilaton by ${\phi }^{\prime }=\phi -\frac{1}{2}\log g_{yy}$ in alpha-prime-one units for a spacelike isometry. The regulated Gaussian integration over the auxiliary gauge field produces this curvature coupling. It is consistent with invariance of the string-frame density $e^{-2\phi }\sqrt{\det g}$ under the local Buscher rules.

Consider the closed bosonic string on a rectangular product of spatial circles with no background mixing between them. Let $Y^j$ have period $2\pi R_j$, take $0\le \sigma <2\pi$, and choose $Y_L(\tau +\sigma )$ to be the tilded oscillator sector. Closed-string topology permitsSingle-valued center-of-mass wavefunctions give quantized momentum $p^j=n_j/R_j$, $n_j\in \mathbb{Z}$. These are the momentum and winding modes. The oscillator modes remain integer moded; the zero modes are split asThe two intercept-one Virasoro constraints, with noncompact $k^2=-M^2$, giveThus the mass formula and compact-circle closed-string level matching condition areHere $N$ labels right movers and $\tilde{N}$ left movers; exchanging the labels changes the displayed sign convention, not the spectrum. Noncompact oscillator polarizations are still restricted by positive-mode Virasoro conditions and null-state identifications. T-duality exchanges $n_j,w_j$ and $R_j,{\alpha }^{\prime }/R_j$, preserving masses and the physical state conditions.

At generic radius, neutral $(N,\tilde{N})=(1,1)$ states with one noncompact oscillator and one compact oscillator give two massless vectors per circle. They are the two Abelian gauge fields associated with the compact metric and two-form components. Additional charged vectors can occur at oscillator level sum one. For one circle, set all other compact charges to zero. Take $n=w=\pm 1$ and $(N,\tilde{N})=(1,0)$, with a right-moving noncompact oscillator ${\alpha }_{-1}^{\mu }$; or take $n=-w=\pm 1$ and $(N,\tilde{N})=(0,1)$, with a left-moving noncompact oscillator ${\tilde{\alpha }}_{-1}^{\mu }$. These obey level matching. Their common squared mass isTherefore four extra charged vector particles become massless at $R=\sqrt{{\alpha }^{\prime }}$, the self-dual circle. Away from that radius they are massive. They are genuine vector states: the noncompact level-one oscillator carries a vector polarization, and the Virasoro constraint plus its null identification impose transversality and remove the gauge polarization at zero mass.

The resulting self-dual circle current algebra makes the gauge enhancement explicit. At the special radius, the left-moving charges above have $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$, $p_R=0$. With $Y_L(z)Y_L(w)\sim -{\alpha }^{\prime }\log (z-w)/2$, the currentshave conformal weight one. The operator products give $J^3{J}^3\sim 1/[2(z-w{)}^2]$, $J^3{J}^{\pm }\sim \pm J^{\pm }/(z-w)$ and $J^{+}{J}^{-}\sim 1/(z-w{)}^2+2J^3/(z-w)$. They form a level-one SU(2) current algebra; the right-moving copy supplies another SU(2). Pairing each chiral current with the opposite-sector noncompact derivative yields a weight-$(1,1)$ vector vertex. Thus the generic $U(1{)}_L\times U(1{)}_R$ becomes $SU(2{)}_L\times SU(2{)}_R$, with four charged generators in addition to the two Cartan vectors. Several self-dual circles give the corresponding product enhancement.

Compactification also creates scalar states and moduli; not every additional massless state is a vector. The mechanism just shown uses the intercept-one bosonic spectrum. A superstring projection changes the allowed oscillator states, so the same mass formula cannot simply be applied to a GSO-projected superstring without modification.

A $(p+1)$-form gauge potential $A_{p+1}$ couples electrically to a spatial $p$-brane through the Wess-Zumino brane couplingThe worldvolume dimension supplies exactly the degree required for integration. Under $A\mapsto A+d\Lambda$, the change is $e{\int }_{\partial \Sigma }{X}^{\ast }\Lambda$, so a closed worldvolume is gauge invariant; an open one needs suitable endpoint couplings. If $F_{p+2}=d{A}_{p+1}$ has an ordinary dual potential, $\ast F$ has degree $d-p-2$ and the dual potential degree $d-p-3$. Thus the electric and magnetic dimensions of a form-coupled brane areElectric charge is measured by ${\int }_{S^{d-p-2}}\ast F$, magnetic charge by ${\int }_{S^{p+2}}F$, on spheres linking the corresponding objects. Flux normalization, and possible Chern-Simons modifications of charge definitions, must be fixed in interacting backgrounds; this is the basic linear form-field dictionary.

In ten dimensions the common NS-NS fields are the graviton $g_{\mu \nu }$, Kalb–Ramond field $B_2$, and dilaton $\Phi$. Their on-shell bosonic polarizations are $35,28,1$, respectively. The graviton count is $10(10-3)/2=35$, and a massless $k$-form has $\left(\genfrac{}{}{0ex}{}{8}{k}\right)$ transverse polarizations. The RR fields distinguish the two theories:

Each gravitino has $56$ physical polarizations, and each dilatino $8$, giving $128$ fermions in either theory. In IIA the two supersymmetry sectors have opposite ten-dimensional chiralities, and the dilatino in each sector has chirality opposite to its gravitino. In IIB both gravitini can be designated positive-chirality and both dilatini negative-chirality. The self-duality of the IIB five-form halves the unconstrained four-form count $\left(\genfrac{}{}{0ex}{}{8}{4}\right)=70$ to $35$. These are the massless type IIA spectrum and massless type IIB spectrum, not raw component counts.

The Neveu–Schwarz two-form couples electrically to the fundamental string and magnetically to the NS5-brane in both theories. IIA $C_1$ couples electrically to a D0-brane and magnetically to a D6-brane; $C_3$ couples electrically to a D2-brane and magnetically to a D4-brane. The magnetic descriptions use dual seven-form and five-form potentials, not extra independent polarizations. In IIB, $C_2$ gives D1-brane and D5-brane charges, while self-dual $C_4$ gives the self-dual D3-brane. The scalar $C_0$ couples to a Euclidean D-instanton, whose magnetic partner is the D7-brane. Momentum and, after circle compactification, Kaluza-Klein monopole charge come from the metric sector.

The full brane inventory also involves nonpropagating top-form potentials. A D8-brane is an IIA domain-wall source for Romans mass $F_0$, described dually by a nine-form potential; it is not generated by the propagating $C_1,C_3$ massless spectrum alone. A space-filling IIB D9-brane has a ten-form coupling and requires appropriate global RR-charge consistency. Such fields add no ordinary massless polarizations. This distinction avoids confusing the list of propagating particles with all possible charged extended objects.

For T-duality, compactify a spatial coordinate with $X^9\sim X^9+2\pi R$. Closed strings have integer momentum $n$ and winding $w$, withThe mapleaves $p_L$ unchanged and reverses $p_R$. Equivalently, it reverses $X_R^9$ while keeping $X_L^9$, and reverses the corresponding right-moving worldsheet fermion to preserve worldsheet supersymmetry. Multiplication of the right Ramond ground-state spinor by ${\Gamma }^9$ reverses its ten-dimensional chirality. Left and right Ramond projections therefore change from opposite chirality to equal chirality, or conversely:The zero-mode mass contribution $(n/R{)}^2+(wR/{\alpha }^{\prime }{)}^2$ and the oscillator spectrum are unchanged. The Buscher dilaton shift gives $g_s^{\prime }=g_s\sqrt{{\alpha }^{\prime }}/R$, which preserves the nine-dimensional gravitational coupling proportional to $g_s^2/R$. The mixed metric and two-form components along the circle are exchanged; RR potentials with a circle leg map to one lower degree and those without it to one higher degree, in the corresponding gauge-covariant description.

For open strings, the derivative relations ${\partial }_{\tau }\tilde{X}={\partial }_{\sigma }X$ and ${\partial }_{\sigma }\tilde{X}={\partial }_{\tau }X$ exchange Neumann boundary conditions with Dirichlet boundary conditions. Consequently a wrapped D$p$-brane becomes a D$(p-1)$-brane, while a transverse one becomes a wrapped D$(p+1)$-brane. The even-dimensional IIA branes and odd-dimensional IIB branes therefore transform consistently with the closed-string chirality flip; transverse positions and compact gauge Wilson lines are exchanged.

Let $Y$ be the compact coordinate, identified modulo $2\pi R$. The closed string may wrap the circle:Single-valued momentum eigenfunctions $e^{ipY}$ require $p=n/R$, $n\in \mathbb{Z}$. These are the momentum and winding modes. Write the chiral zero-mode momenta asThe left-moving string oscillators will be tilded: $N_L=\tilde{N}$, $N_R=N$. This fixes the sign in compact-circle closed-string level matching.

Define the lower-dimensional mass by $M^2=-p_{\mathrm{noncompact}}^2$. The two closed-string physical-state Virasoro conditions, with bosonic string intercept one, areTheir average and difference derive the complete free bosonic string mass spectrum on the circle:The allowed states have nonnegative integer oscillator levels and obey the Virasoro constraints within each chiral sector. Compactification has not removed oscillator polarizations; it has added quantized compact momentum and allowed winding while changing the lower-dimensional interpretation of those polarizations.

At large $R$, the momentum and winding modes with $w=0$ have closely spaced momentum energies $|n|/R$, whereas nonzero winding costs $|w|R/{\alpha }^{\prime }$. At small $R$ their roles reverse. The precise equality is T-duality:It leaves $p_L$ unchanged and reverses $p_R$, and leaves the mass formula and closed-string level matching unchanged. On the string embedding map it reverses the right-moving part of $Y$. Therefore the small-radius spectrum is the large-radius spectrum of the dual theory, rather than a spectrum in which all charged states simply become heavy.

At a generic radius, the neutral level $(N_L,N_R)=(1,1)$ contains the lower-dimensional graviton, Kalb–Ramond field and dilaton, two gauge bosons from the mixed components of the metric and two-form, and the radius modulus of a string compactification. The two vector charges are the Cartan charges of $U(1{)}_L\times U(1{)}_R$.

Consider instead the following charged states:Every state satisfies compact-circle closed-string level matching. The single noncompact string oscillator supplies a vector polarization. At the massless point, the level-one Virasoro constraint imposes transverse polarization and its physical null string state identifies polarizations differing by the noncompact momentum. Thus these are genuine lower-dimensional vector states. Their squared mass isThus four additional charged massless spin-one particles appear atFor $(N_L,N_R)=(0,1)$ the charges have $p_R=0$, $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$; for $(1,0)$ they have $p_L=0$, $p_R=\pm 2/\sqrt{{\alpha }^{\prime }}$. This identifies the two pairs of charged roots.

The enhanced gauge group can be derived rather than just inferred from counting. At the self-dual circle, the chiral compact boson satisfies $Y_L(z)Y_L(u)\sim -({\alpha }^{\prime }/2)\log (z-u)$. The self-dual circle current algebra has currentsThe exponentials have conformal weight ${\alpha }^{\prime }{p}_L^2/4=1$. Free-boson contractions give the operator product expansionsThey form the level-one SU(2) Lie algebra, and there is an independent right-moving copy. The string vertex operators $J_L^a\bar{\partial }{X}^{\mu }$ and $\partial X^{\mu }{J}_R^a$ give the six massless vectors. ConsequentlyChanging $R$ gives the four charged vectors the mass found above, consistent with the radius deformation as a Higgs mechanism.

For completeness, the enhanced spectrum also has scalar fields. Using a compact rather than noncompact oscillator in the four singly excited charged families gives four scalars. At $(N_L,N_R)=(0,0)$, closed-string level matching requires $nw=0$; the mass equation then gives the four charges $(\pm 2,0),(0,\pm 2)$. These exceptional massless ground states of a bosonic circle give another four scalars. Together with the radius scalar they form nine scalars in the $(3,3)$ representation, in addition to the neutral dilaton. The ever-present neutral bosonic tachyon still has $M^2=-4/{\alpha }^{\prime }$. The enhancement establishes the additional massless vectors, not stability of the entire bosonic vacuum.

Take $0\le \sigma <2\pi$. Variation of the conformal-gauge action, with periodic variations, gives $({\partial }_{\tau }^2-{\partial }_{\sigma }^2)X^{\mu }=0$. Hence $X^{\mu }=X_R^{\mu }(\tau -\sigma )+X_L^{\mu }(\tau +\sigma )$. In an uncompactified direction, periodicity allows equal linear slopes and independent periodic parts. Expanding the periodic parts in a Fourier series yields the closed-string mode expansionHere ${\alpha }_m$ labels the right-moving sector and ${\tilde{\alpha }}_m$ the left-moving sector; exchanging the names is harmless if done consistently. Integration of the canonical momentum density $(2\pi {\alpha }^{\prime }{)}^{-1}{\dot{X}}^{\mu }$ over the string gives $p^{\mu }$, fixing the zero-mode coefficient. Reality gives ${\alpha }_m^{†}={\alpha }_{-m}$ and the analogous relation for tildes, with ${\alpha }_0={\tilde{\alpha }}_0=\sqrt{{\alpha }^{\prime }/2}p$.

The closed-string physical-state Virasoro conditions arewhere $L_0={\alpha }^{\prime }{p}^2/4+N$ and ${\tilde{L}}_0={\alpha }^{\prime }{p}^2/4+\tilde{N}$. Thus closed-string level matching gives $N=\tilde{N}$, and $M^2=-p^2=4(N-1)/{\alpha }^{\prime }$. The massless level is $N=\tilde{N}=1$. Its general state isThe string oscillator commutators give $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$: the two terms obtained by commuting through the quadratic generator coincide and cancel its factor $1/2$. Therefore $L_1$ replaces ${\alpha }_{-1}^{\mu }$ by ${\alpha }_0^{\mu }$, while $L_{m\ge 2}$ produces positive string oscillators that annihilate this level. The same holds independently for tildes. The complete polarization conditions areNull string states identify ${\zeta }_{\mu \nu }$ with ${\zeta }_{\mu \nu }+p_{\mu }{\lambda }_{\nu }+{\tilde{\lambda }}_{\mu }{p}_{\nu }$, with transverse $\lambda ,\tilde{\lambda }$. At nonzero momentum this leaves $24\cdot 24=576$ physical polarizations. Their symmetric traceless, antisymmetric and transverse-trace parts are the graviton, Kalb–Ramond field and dilaton.

The original PDF's condition ${\alpha }^{\prime }{p}^2=4$ is inconsistent with calling the displayed excited states massless. It is the on-shell condition of the string oscillator vacuum as a physical tachyon. For the level-one state, that value would give $L_0={\tilde{L}}_0=2$, violating the physical conditions. The ket $|0;p⟩$ used as the basis for a massless excitation must instead carry the excitation's momentum $p^2=0$; by itself it is then not an on-shell physical vacuum state.

For the compact coordinate, $X^1(\tau ,\sigma +2\pi )=X^1(\tau ,\sigma )+2\pi wR$, $w\in \mathbb{Z}$. Its momentum is $n/R$, $n\in \mathbb{Z}$, since the centre-of-mass wavefunction is single-valued on the circle. Thus its zero modes becomewithThese are momentum and winding modes. Let $k$ be the 25-dimensional noncompact momentum and $M^2=-k^2$. The two zero-mode constraints now giveTheir difference and average give, respectively,This also displays T-duality $R\leftrightarrow {\alpha }^{\prime }/R$, $n\leftrightarrow w$, with one chiral momentum reversed.

At a generic radius, the massless solutions are only $n=w=0$, $N=\tilde{N}=1$. Their fields in 25 dimensions are a graviton (275 polarizations), antisymmetric two-form (253), dilaton (one), two abelian vectors from $G_{a1}$ and $B_{a1}$ (23 each), and the radius scalar (one). Hence the generic massless spectrum has $275+253+1+46+1=576$ polarizations, with gauge group $U(1{)}_L\times U(1{)}_R$. Here generic excludes isolated radii where exceptional massless ground states of a bosonic circle occur. Indeed $N=\tilde{N}=0$ forces $nw=0$, and becomes massless at $R=|n|\sqrt{{\alpha }^{\prime }}/2$ or $R=2\sqrt{{\alpha }^{\prime }}/|w|$. These additional thresholds matter in the bosonic theory.

At the self-dual circle $R=\sqrt{{\alpha }^{\prime }}$, masslessness and level matching readSince $N,\tilde{N}\ge 0$, their sum is at most two. For sum two, $n=w=0$ and level matching gives the ordinary $(1,1)$ family. For sum one the four possibilities areThey have one right-moving ${\alpha }_{-1}$ or one left-moving ${\tilde{\alpha }}_{-1}$ excitation, as their levels specify. Each family has 24 physical polarizations: a 25-dimensional vector with 23 polarizations and one compact-oscillator scalar. Thus these families add 96 states.

For sum zero, level matching forces $nw=0$, while the mass equation requires $n^2+w^2=4$. It gives four additional scalar states with charges $(\pm 2,0)$ and $(0,\pm 2)$, without any string oscillators. They must not be omitted. The massless spectrum at the bosonic self-dual circle therefore has 100 additional polarizations, 676 in total. The four new vector fields enhance the gauge group to $SU(2{)}_L\times SU(2{)}_R$. Including the original radius scalar, the nine non-dilaton scalars form its $(3,3)$ representation. Equivalently, the full count is $275+253+1+6\cdot 23+9=676$.

Use target signature $(-,+,\dots ,+)$ and a closed-string spatial coordinate $0\le \sigma \le 2\pi$. Write $Y$ for the compact coordinate and $X^A$ for the noncompact coordinates. In conformal gauge, the wave equation factorizes, so every coordinate is a sum of a function of $\tau +\sigma$ and a function of $\tau -\sigma$. The closed-string boundary conditions areThus the derivatives of each chiral part are periodic and have integer Fourier modes; only their worldsheet zero modes can produce the winding displacement. Integrating these Fourier series gives the closed-string mode expansionThe oscillator reality conditions are ${\alpha }_m^{†}={\alpha }_{-m}$ and similarly for the tilded modes. The coefficient of $\tau$ follows by integrating the canonical momentum density $\dot{Y}/(2\pi {\alpha }^{\prime })$. Single-valued wavefunctions of the center coordinate $y\sim y+2\pi R$ have momentum $n/R$, with $n\in \mathbb{Z}$. Equivalently, the compact chiral worldsheet zero modes are ${\alpha }^{\prime }{p}_L(\tau +\sigma )/2$ and ${\alpha }^{\prime }{p}_R(\tau -\sigma )/2$, whereHere tildes denote the left-moving sector. This convention will also fix the sign in compact-circle closed-string level matching.

Choose a noncompact light-cone pair and impose light-cone gauge in string theory $X^{+}=x^{+}+{\alpha }^{\prime }{p}^{+}\tau$, with $p^{+}\ne 0$. The Virasoro constraints $({\partial }_{\pm }X{)}^2=0$ then solve for ${\partial }_{\pm }{X}^{-}$ in terms of the transverse coordinates. For example, with ${\partial }_{\pm }={\partial }_{\tau }\pm {\partial }_{\sigma }$, one has ${\partial }_{\pm }{X}^{-}=({\partial }_{\pm }{X}^i{)}^2/(2{\alpha }^{\prime }{p}^{+})$. Thus there are two independent transverse oscillator Fock spaces, including $Y$ among the transverse coordinates. Quantization gives $[{\alpha }_m^i,{\alpha }_k^j]=m{\delta }^{ij}{\delta }_{m+k,0}$, and the same relation for the tilded oscillators, with the two sectors commuting. Lorentz consistency gives the critical dimension of the bosonic string $D=26$ and string intercept one in each sector. Accordinglycount 24 transverse oscillator species. The two zero-mode constraints give the lower-dimensional bosonic string mass spectrumTaking their difference and average yieldsEvery choice of transverse string oscillators and integer charges satisfying this closed-string level matching is a physical string state in light-cone gauge. The eliminated longitudinal oscillators do not supply additional polarizations.

To classify all massless states, positivity of the two compact terms implies $N+\tilde{N}\le 2$. When the sum is two, masslessness forces $n=w=0$, and closed-string level matching then forces $N=\tilde{N}=1$. The states ${\alpha }_{-1}^i{\tilde{\alpha }}_{-1}^j|0⟩$ have $24^2=576$ polarizations. In 25 noncompact dimensions, their decomposition gives a graviton, a Kalb–Ramond field, a dilaton, two Abelian gauge vectors from the metric and two-form with one compact index, and the radius scalar field. Their physical polarization counts areThe first two numbers are the symmetric traceless second-rank tensor and antisymmetric second-rank tensor of the massless little group $SO(23)$.

When $N+\tilde{N}=1$, closed-string level matching requires $|nw|=1$. The compact contribution obeysand equality occurs precisely at $R=\sqrt{{\alpha }^{\prime }}$. The charges $n=w=\pm 1$ have $(N,\tilde{N})=(1,0)$; the charges $n=-w=\pm 1$ have $(N,\tilde{N})=(0,1)$. Each charge has 24 polarizations supplied by its single excited sector. At a general radius these have

Finally, $N=\tilde{N}=0$ requires $nw=0$. The exceptional massless ground states of a bosonic circle occur atEach charge is a scalar field. These special radii must be included if “general radius” is to mean a complete classification, rather than only a generic radius. At a generic radius the massless spectrum is the neutral 576-state spectrum above; at exceptional radii one adds exactly the states just listed. The neutral ground state remains tachyonic.

The mapsends $p_L$ to $p_L$ and $p_R$ to $-p_R$. Reversing all right-moving compact string oscillators, ${\alpha }_m^Y\mapsto -{\alpha }_m^Y$, preserves their commutators and oscillator number; the left-moving oscillators are unchanged. Both the mass formula and compact-circle closed-string level matching are therefore unchanged, with a one-to-one map of the complete oscillator basis. This proves T-duality of the full spectrum, including winding, massive and tachyonic states.

At the self-dual circle the four single-oscillator charge families give $4\cdot 24=96$ new massless states. Each family decomposes into a 25-dimensional gauge vector with 23 polarizations and one scalar field. There are also four massless ground states, with charges $(\pm 2,0)$ and $(0,\pm 2)$. Thus the massless spectrum at the bosonic self-dual circle hasThe gauge interpretation follows directly from the chiral compact-boson operators. With $⟨Y_L(z)Y_L(0)⟩=-{\alpha }^{\prime }\log z/2$, the exponential $e^{ik{Y}_L}$ has conformal weight ${\alpha }^{\prime }{k}^2/4$. At the self-dual radius,have weight one and satisfy $J^3(z)J^3(0)\sim 1/(2z^2)$, $J^3(z)J^{\pm }(0)\sim \pm J^{\pm }(0)/z$, and $J^{+}(z)J^{-}(0)\sim z^{-2}+2J^3(0)/z$. These are the level-one $SU(2)$ current relations; the right sector supplies another copy. Multiplying these currents by a noncompact oscillator produces six gauge vectors in total. The nine scalar string vertex operators $J_L^a{J}_R^b$ form the $(3,3)$ representation, accounting for the radius scalar field and eight new scalar fields. The remaining neutral fields are the graviton, two-form and dilaton.

A radius deformation as a Higgs mechanism is the expectation value of the Cartan–Cartan scalar string vertex operator $J_L^3{J}_R^3\propto \partial Y_L\bar{\partial }{Y}_R$. Its stabilizer is $U(1{)}_L\times U(1{)}_R$, soThe four compact single-oscillator scalar field polarizations become longitudinal polarizations of the four massive vectors, exactly as in the Higgs mechanism. The other four extra scalar fields have masses $4/R^2-4/{\alpha }^{\prime }$ or $4R^2/{\alpha }^{\prime 2}-4/{\alpha }^{\prime }$ and need not remain massless; one pair becomes tachyonic on either side of the self-dual point. Together with the original bosonic tachyon, this means that the gauge-symmetry interpretation is formal and does not assert a stable bosonic-string vacuum.

Distinguish target-space light-cone coordinates from the worldsheet $\pm$ labels. In target space take $X^{\pm }=(X^0\pm X^{25})/\sqrt{2}$, so the Minkowski metric is $d{s}^2=-2d{X}^{+}d{X}^{-}+{\sum }_i(d{X}^i{)}^2$. After conformal gauge has been chosen, the residual transformations of $\tau +\sigma$ and $\tau -\sigma$ may be used, on a patch with $p^{+}\ne 0$, to makeFor an open string on $0\le \sigma \le \pi$, the conventional normalization is $\kappa =2{\alpha }^{\prime }{p}^{+}$. For the closed-string convention of the preceding solution, $\kappa ={\alpha }^{\prime }{p}^{+}$. In either case, the two Virasoro constraints readSince ${\partial }_{\pm }{X}^{+}=\kappa /2$, they determine the longitudinal coordinate: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 isThe transverse string oscillators satisfy $[{\alpha }_m^i,{\alpha }_n^j]=m{\delta }^{ij}{\delta }_{m+n,0}$, $i,j=1,\dots ,24$, and ${\alpha }_{n>0}^i|0,p⟩=0$. Define $L_n^{\perp }=\frac{1}{2}{\sum }_m:{\alpha }_{n-m}^i{\alpha }_m^i:$ and ${\alpha }_0^{+}=\sqrt{2{\alpha }^{\prime }}{p}^{+}$. The solved longitudinal modes, including the zero-mode constraint, areThe critical bosonic string theory has 24 transverse bosons. Their regulated oscillator vacuum energy is $24\cdot \frac{1}{2}{\sum }_{n\ge 1}n=-1$, since the finite term of ${\sum }_{n\ge 1}n{e}^{-\epsilon n}={\epsilon }^{-2}-1/12+O({\epsilon }^2)$ 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 iswith 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 giveAll spatial directions have free endpoints, so 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 $\tilde{X}=X_L-X_R$. Differentiation givesTherefore the original Neumann boundary condition ${\partial }_{\sigma }X=0$ becomes ${\partial }_{\tau }\tilde{X}=0$ at each endpoint. The dual endpoint coordinate is time-independent: 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 $\tilde{R}={\alpha }^{\prime }/R$. 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 termThe spatial gradient contributes to the worldsheet HamiltonianParallel branes have integer modes in their NN and DD directions, so their oscillator vacuum energy is still $-1$. The zero-mode Virasoro constraint is consequentlyWith string tension $T=1/(2\pi {\alpha }^{\prime })$, the Open string stretched between D-branes hasThe classical stretching energy is $Td$, while the quantum ground-state energy, when real, isThe 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 isBelow 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.

The three identifications areThey match the same lower-dimensional theory, including its moduli of a string compactification, charged states and extended branes. Different weakly coupled descriptions occupy different limits of its moduli space.