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Take $0\le \sigma \le \pi$ and split the coordinates into $a=0,\dots ,p$ along the branes and $I=p+1,\dots ,25$ transverse to them. In conformal gauge the wave equation permits a separated mode $e^{-i\omega \tau }f(\sigma )$ with $f^{\prime \prime }+{\omega }^{2}f=0$. For Neumann boundary conditions at both ends, $f^{\prime }(0)=f^{\prime }(\pi )=0$ selects $\cos m\sigma$ and integer $m$. The zero-frequency solution is independent of $\sigma$, with a constant position and a term linear in time. Therefore the open-string mode expansion isThe canonical momentum density integrated over this interval is $k^a$.

For Dirichlet boundary conditions, subtract the static interpolation between fixed endpoint positions $y_0^I$ and $y_{\pi }^I$. The remaining field vanishes at both ends, selecting $\sin m\sigma$ with integer $m$ and excluding a momentum worldsheet zero mode. The expansion isHere ${\alpha }_m^{I†}={\alpha }_{-m}^I$. There is no factor $i$ in the sine series with this reality convention: changing $m$ to $-m$ reverses both the sine and the denominator. These are the integer-mode expansion with parallel D-brane endpoints; they are not the half-integer modes of a coordinate with a Neumann condition at one endpoint and a Dirichlet condition at the other.

The endpoints can move on $(p+1)$-dimensional worldvolumes but are fixed in the transverse directions. Thus these are open strings attached to D-branes, or stretched between parallel D-branes at $y_0$ and $y_{\pi }$. The tangential momenta describe propagation on the brane; the transverse positions describe its embedding. Let $L=|y_{\pi }-y_0|$. The linear spatial worldsheet zero mode contributes $L^2/(4{\pi }^2{\alpha }^{\prime })$ to $L_0$. Quantization with string intercept one givesThe first term is also the square of the classical stretched-string energy $TL$, since the string tension is $T=1/(2\pi {\alpha }^{\prime })$.

For two coincident D3-branes, label the ends of an oriented open string by $i,j\in \{1,2\}$. The four endpoint combinations give Chan-Paton factors ${\lambda }_{ij}$, which form the full algebra of $2\times 2$ matrices. Under a change of brane basis these transform as $\lambda \mapsto U\lambda U^{-1}$, the adjoint action of $U(2)$. At $L=0$, the first oscillator level is massless. Its tangential polarization supplies a four-dimensional vector $A_a$, while the 22 transverse polarizations supply real adjoint scalar fields ${\Phi }^I$. This is the bosonic D3-brane low-energy field content.

The endpoint matrices multiply in their boundary order in disk amplitudes. The difference between the two orders in the three-vector interaction gives the matrix commutator; the kinematic factor is the one-derivative Yang-Mills theory cubic vertex. Factorization and the vector gauge redundancy then supply the quartic interaction with the same commutator. Equivalently, the leading low-energy action is the dimensional reduction of 26-dimensional Yang-Mills theory, withFor Hermitian scalar fields the last displayed term corresponds to a nonnegative potential, because their commutators are anti-Hermitian. Higher-derivative terms are suppressed at energies well below $1/\sqrt{{\alpha }^{\prime }}$. Ignoring the tachyon, the gauge sector is four-dimensional $U(2)$ Yang-Mills theory, accompanied by 22 adjoint scalar fields. It is not pure Yang-Mills theory, nor the supersymmetric six-scalar D3-brane theory of a ten-dimensional superstring.

Separate the two branes and use the D-brane scalar-position normalizationA Yang-Mills gauge transformation preserves this expectation value only when it is diagonal if $y_1\ne y_2$. Thus the unbroken group is $U(1)\times U(1)$. The scalar field kinetic term supplies a mass for the off-diagonal vectors, sinceSumming over $I$ gives $m_{12}^2=L^2/(4{\pi }^2{\alpha }^{\prime 2})$, exactly the mass of the stretched $N=1$ string. The two diagonal gauge vectors remain massless. This identifies brane separation with an adjoint Higgs mechanism, rather than an explicit deletion of the off-diagonal strings.

For the ground state of an open string joining the two different branes, $N=0$, so the bosonic stretched-string tachyon threshold isAt equality it is massless, and above it the stretched ground state has positive mass squared. The same-brane ground strings still have $M_0^2=-1/{\alpha }^{\prime }$; this calculation removes only the stretched-string instability, not the original instability of the bosonic branes.

The Polyakov action has independent worldsheet metric ${\gamma }_{ab}$ and embedding fields $X^{\mu }$. Varying the metric before imposing a gauge givesWorldsheet diffeomorphisms and Weyl transformations allow conformal gauge ${\gamma }_{ab}=e^{2\omega }{\eta }_{ab}$ locally. The Weyl factor then cancels out of the classical action, leavingThese are free two-dimensional scalar fields, but the surviving metric equations requireEquivalently, ${\dot{X}}^2+X^{\prime 2}=0$ and $\dot{X}\cdot X^{\prime }=0$. Gauge fixing does not license dropping these Virasoro constraints.

For an ordinary open string, the boundary conditions relate the two chiral sectors. With ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{k}^{\mu }$, their independent Fourier generators areThe classical oscillator Poisson brackets $\{{\alpha }_m^{\mu },{\alpha }_n^{\nu }\}=-im{\eta }^{\mu \nu }{\delta }_{m+n,0}$ give the classical Virasoro constraint algebraThus they are first-class constraints. In the quantum theory one applies normal ordering to the product and uses $[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\eta }^{\mu \nu }{\delta }_{m+n,0}$. The oscillator commutator givesFor example, for $m>0$ the extra vacuum term is $(D/2){\sum }_{r=1}^{m-1}r(m-r)=D(m^3-m)/12$. This exhibits the Virasoro central extension rather than concealing it in a classical constraint equation. The matter central charge is $D$; the worldsheet ghost fields have charge $-26$. Quantum consistency sets $D=26$ and the string intercept to one.

In old covariant string quantization the physical-state Virasoro conditions for an open string arePhysical string states are then identified modulo physical null string states. One does not impose both positive and negative generators as annihilation conditions: their central extension would obstruct that prescription. The resulting null-state quotient of a string agrees with the transverse light-cone spectrum at nonzero momentum.

At $N=0$, the oscillator vacuum $|k⟩$ is a single spacetime scalar field with ${\alpha }^{\prime }{M}^2=-1$: the bosonic tachyon. At $N=1$, write $|\Psi ⟩={ϵ}_{\mu }{\alpha }_{-1}^{\mu }|k⟩$. The zero-mode equation gives $k^2=0$, and $L_1$ gives $k\cdot ϵ=0$. The null descendant $L_{-1}|k⟩=\sqrt{2{\alpha }^{\prime }}k\cdot {\alpha }_{-1}|k⟩$ identifies $ϵ\sim ϵ+\lambda k$. This leaves 24 physical polarizations of a massless gauge vector.

At $N=2$, use the level-two open-string polarization decompositionThe mass is $M^2=1/{\alpha }^{\prime }$. The oscillator commutators give all the nontrivial positive-mode conditions:Higher positive modes annihilate this level. To see the quotient explicitly, the 25 transverse-parameter null states $L_{-1}(\xi \cdot {\alpha }_{-1}|k⟩)$, with $k\cdot \xi =0$, shiftThe additional level-two scalar Virasoro null state $(L_{-2}+\frac{3}{2}{L}_{-1}^2)|k⟩$ shiftsSubstitution into the two physical equations verifies the null-state conditions; the second residual is $(D-26)/2$. Because $k$ is timelike, these null shifts span all components of $b$, so choose a representative with $b=0$. The physical conditions then say $k^{\mu }{h}_{\mu \nu }=0$ and $h^{\mu }{}_{\mu }=0$. In the rest frame this is a traceless second-rank tensor in 25 spatial dimensions, with $25\cdot 26/2-1=324$ polarizations. The light-cone basis independently gives $24$ states ${\alpha }_{-2}^i|k⟩$ and $24\cdot 25/2=300$ symmetric states ${\alpha }_{-1}^i{\alpha }_{-1}^j|k⟩$. Hence

For the fermionic extension, an explicit convention prevents ambiguity in the signs. Keep worldsheet signature $(-,+)$ and useTake $\psi =({\psi }_1,{\psi }_2{)}^T$ and the constant parameter $ϵ=({ϵ}_1,{ϵ}_2{)}^T$ to be real Grassmann-odd Majorana components, with $\bar{\psi }={\psi }^T{\rho }^0$. Put ${\partial }_{\pm }={\partial }_{\tau }\pm {\partial }_{\sigma }$. Apart from the common factor $1/(4\pi {\alpha }^{\prime })$, the combined Lagrangian isThe rigid transformations can be writtenor, in components,For the ${ϵ}_2$ part, the bosonic variation is $i{ϵ}_2({\partial }_{+}{\psi }_1\cdot {\partial }_{-}X+{\partial }_{+}X\cdot {\partial }_{-}{\psi }_1)$, while the fermionic variation is $-i{ϵ}_2{\partial }_{-}X\cdot {\partial }_{+}{\psi }_1+i{ϵ}_2{\psi }_1\cdot {\partial }_{+}{\partial }_{-}X$. Anticommuting the constant parameter past the fermion is essential in the second term. The first terms cancel. The other chiral component works in the same way, giving the chiral proof of rigid worldsheet supersymmetryThus the action is invariant off shell up to a boundary term. The transformation is a genuine supersymmetry: on $X$ its commutator for parameters $ϵ,\zeta$ is the translationOn $\psi$ the same translation follows using ${\partial }_{+}{\psi }_1=0$ and ${\partial }_{-}{\psi }_2=0$; the remaining cross terms are proportional to these Dirac equations. This is on-shell closure of rigid worldsheet supersymmetry. On a surface with boundary, compatible endpoint conditions must make the boundary flux vanish. On a compact worldsheet, a constant parameter must preserve the spin structure; the local identity alone does not make a constant transformation compatible with every antiperiodic sector.

The fermions contribute to the worldsheet stress-energy tensor and produce a fermionic superconformal current. In holomorphic normalization with ${\psi }^{\mu }(z){\psi }^{\nu }(w)\sim {\eta }^{\mu \nu }/(z-w)$,The free-field contractions give $G(z)G(w)\sim (2c/3)/(z-w{)}^3+2T(w)/(z-w)$ and conformal weight $3/2$ for $G$, where $c=D+D/2=3D/2$. Consequently the modes obey the N=1 super-Virasoro algebraDropping the central terms gives the classical graded constraint algebra. Integer $r$ gives the Ramond sector; half-integer $r$ gives the Neveu–Schwarz sector. In the displayed conformal-plane convention $G_0^2=L_0-c/24$ in the Ramond sector. Shifting to the Ramond vacuum-normal-ordered zero mode gives the usual Ramond supercurrent zero-mode square $G_0^2=L_0^{(R)}$.

Rigid symmetry of matter already supplies these currents, but promoting their vanishing to new gauge constraints requires coupling to worldsheet supergravity and varying its worldsheet gravitino. In superconformal gauge one retains $T=G=0$. The fermionic current constraints and their physical-state conditions extend the bosonic Virasoro conditions. The ghost charges are $-26+11=-15$, so cancellation with $3D/2$ gives the critical dimension of the RNS superstring $D=10$; the NS and Ramond intercepts are respectively $1/2$ and zero. This distinguishes the global supersymmetry established above from its locally gauged string completion.

Use the critical ten-dimensional RNS string and light-cone gauge in string theory, leaving eight transverse bosons and eight transverse fermions. The bosonic oscillators ${\alpha }_{-m}^i$ have integer $m>0$. In the Neveu–Schwarz sector the fermionic oscillators $b_{-r}^i$ have positive half-integer $r$, while in the Ramond sector the fermionic oscillators $d_{-m}^i$ have integer $m$, including zero modes. Their anticommutators are $\{b_r^i,b_s^j\}={\delta }^{ij}{\delta }_{r+s,0}$ and $\{d_m^i,d_n^j\}={\delta }^{ij}{\delta }_{m+n,0}$. The level and mass formulas areBefore projection the NS ground state is a tachyon of mass squared $-1/(2{\alpha }^{\prime })$. Its oscillator tower describes spacetime bosons. The Ramond zero-mode Clifford algebra $\{\sqrt{2}{d}_0^i,\sqrt{2}{d}_0^j\}=2{\delta }^{ij}$ makes the Ramond vacuum a 16-dimensional spinor of $SO(8)$, split into the two chiral eight-dimensional spinors. The Ramond tower describes spacetime fermions; there is no Ramond tachyon.

Choose the supersymmetric GSO projection. In the NS sector it keeps states with an odd number of fermionic creation operators, eliminating the zero-oscillator tachyon. In the Ramond sector it fixes the combined parity of the zero-mode chirality and the nonzero-mode fermion number. Thus the Ramond vacuum retains one chirality; adding an odd number of nonzero-mode fermions requires the opposite zero-mode chirality. The projected towers both have ${\alpha }^{\prime }{M}^2=0,1,2,\dots$. Here the counts refer to oscillator polarizations for a single endpoint multiplicity; any common Chan-Paton multiplicity multiplies both counts.

At the first retained mass level, $M^2=0$, the NS states are $b_{-1/2}^i|0;k⟩$ for $i=1,\dots ,8$, furnishing the eight polarizations of the ten-dimensional gauge vector. The Ramond ground state is one chiral spinor with eight physical polarizations. HenceThese form the massless supersymmetric vector multiplet.

At the next mass level, ${\alpha }^{\prime }{M}^2=1$, the NS level is $3/2$. The odd-fermion condition leaves exactlyTheir counts are $8$, $8\cdot 8=64$ and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$. Anticommutation requires distinct indices in the third family. There are no other partitions of $3/2$ with odd fermion number, so these are all 128 bosons. They assemble into the massive little-group $SO(9)$ representations of dimensions 44 and 84: a traceless symmetric two-tensor and a three-form.

At Ramond level one, the only possibilities are ${\alpha }_{-1}^i|s⟩$ and $d_{-1}^i|s^{\prime }⟩$. The spinors $|s⟩$ and $|s^{\prime }⟩$ each have eight components and have opposite zero-mode chirality because the fermionic oscillator reverses the GSO parity. Consequently there are $64+64=128$ fermions. They form a gamma-traceless vector-spinor of the massive little group: a vector-spinor has $9\cdot 16$ components and its gamma trace removes 16. Therefore the first massive level of the open RNS string hasThis is the open-string mass normalization; the same chiral oscillator count used in a closed string would correspond to ${\alpha }^{\prime }{M}^2=4$.

Count oscillator polarizations at fixed center-of-mass momentum. In a critical closed bosonic string, each chiral sector has 24 transverse oscillator species. A species at frequency $m$ can be occupied $0,1,2,\dots$ times, so its contribution is $(1-q^m{)}^{-1}$. The full chiral generating function is thereforeThe two sectors are independent, while closed-string level matching restricts their oscillator numbers to the same $N$. Thus the number of states at the closed-string mass level is $g_N=d_N^2$, not $d_{2N}$.

Put $q=e^{-\beta }$, with $\beta >0$. The Dedekind eta function obeysThe transformed argument $2\pi i/\beta$ has large imaginary part, so its product approaches one and $\eta (2\pi i/\beta )\sim e^{-{\pi }^2/(6\beta )}$. HenceThis establishes the singular exponential as well as its power prefactor. The corrections to the transformed eta product are exponentially small in $1/\beta$.

The Cauchy coefficient formula, followed by $q=e^{-\beta }$, gives a steepest-descent integral with leading formLet $n=N-1$. The saddle of $S(\beta )=n\beta +4{\pi }^2/\beta$ isAlong the vertical steepest-descent direction $\beta ={\beta }_{\ast }+it$, the quadratic part is $-S^{\prime \prime }({\beta }_{\ast })t^2/2$. Thus the saddle-point approximation gives the Gaussian prefactorThe point $q=1$ supplies the largest exponential contribution. At another fixed root of unity of order $k>1$, the corresponding modular singularity gives a smaller exponential $e^{4\pi \sqrt{n}/k}$, so it does not change this leading asymptotic. Squaring the chiral result gives the large-level degeneracy of a closed bosonic stringReplacing $N-1$ by $N$ gives the equivalent usual leading asymptotic. The power $-27/2$ matters: the two-sector degeneracy is not just an unspecified exponential.

Since $M=2\sqrt{(N-1)/{\alpha }^{\prime }}$, the exponential is $e^{{\beta }_{H}M}$ withHere Boltzmann's constant is one. Coarse-graining the levels into a density per unit rest mass introduces the Jacobian $dN/dM={\alpha }^{\prime }M/2$. ConsequentlyThis is distinct from the degeneracy per level, whose power in $M$ is $-27$. The internal-state contribution to a canonical partition function behaves at large mass asIt converges for $\beta >{\beta }_H$ and diverges for $\beta <{\beta }_H$. At $\beta ={\beta }_H$, this particular internal rest-mass integral converges because of the power $M^{-26}$; one should not discard the prefactor and claim divergence there without specifying the full thermodynamic ensemble. For example, including continuous momenta in 25 spatial dimensions multiplies the high-mass integrand by a factor proportional to $M^{25/2}$, still leaving convergence of this ideal one-string integral at the endpoint, although sufficiently high derivatives are singular. Multi-string effects, volume and interactions affect the detailed limiting thermodynamics.

The leading microcanonical entropy grows as $S(E)\sim {\beta }_{H}E$, with logarithmic corrections, so increasingly large energies are stored in increasingly excited, long strings rather than in an arbitrarily hot ordinary gas. The Hagedorn temperature is the boundary of canonical convergence from above in inverse temperature; extrapolating the free-string canonical description to $T>T_H$ fails. This counting does not settle the interacting phase above that scale. Also, the bosonic ground-state tachyon is a separate vacuum instability; the high-level asymptotic is meaningful as a formal spectrum calculation without asserting a stable bosonic thermal vacuum.

The starting point is a physical massless closed-string stateIts symmetric traceless part is a spacetime spin-two field, the graviton; the antisymmetric part is the Kalb–Ramond field, and the scalar field part gives the dilaton. The physical-state conditions impose transverse polarizations, and longitudinal null states identifyfor the symmetric component. In position space this is the linearized diffeomorphism $\delta h_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$. Thus the massless spin-two state has the gauge redundancy required of a gravitational field, rather than an arbitrary massive tensor interaction.

Closed-string amplitudes factorize on this graviton pole. Its leading soft coupling is universal and couples to the energy-momentum tensor of the other string states. Decoupling longitudinal polarizations enforces the gravitational Ward identities. The two-derivative cubic graviton vertex extracted from the low-momentum three-point amplitude is the cubic vertex of the Einstein-Hilbert action; continuing these Ward identities requires the nonlinear diffeomorphism-invariant completion. A complementary way to derive the same dynamics is to demand quantum consistency of propagation in a curved background.

Replace the flat target metric in the worldsheet action by $G_{\mu \nu }(X)$, and allow the two-form and dilaton backgrounds. After Euclidean continuation, their couplings have the schematic normalizationThese backgrounds are couplings of a string nonlinear sigma model. Quantum Weyl invariance requires their Weyl anomaly coefficients to vanish. To first order in ${\alpha }^{\prime }$, the metric equation isThe dilaton and two-form equations accompany it. In a critical-dimensional background with $H=0$ and constant $\Phi$, the leading metric equation is $R_{\mu \nu }=0$, precisely the vacuum Einstein field equations. Matter and other massless string fields supply the corresponding stress-energy when retained. This argument also explains why arbitrary curved target metrics are not consistent string backgrounds.

The equations arise from the leading string-frame massless effective actionA constant-dilaton expansion of its curvature term directly gives the Einstein-Hilbert kinetic and self-interaction terms. For a varying dilaton, introduce the Einstein-frame metric $G_{\mu \nu }^E=e^{-4\Phi /(D-2)}{G}_{\mu \nu }$. The Weyl rescaling of the metric and the corresponding transformation of the Ricci scalar and an integration by parts giveThis displays ordinary Einstein gravity coupled to the dilaton and two-form. Gravity need not be the only low-energy field: those fields, compactification moduli and gauge fields have to be retained or stabilized as appropriate. Four-dimensional gravity additionally requires compactification from the critical target dimension, with its Newton constant depending on the compactification volume.

The approximation is controlled when external energies obey $E\sqrt{{\alpha }^{\prime }}\ll 1$ and curvatures obey ${\alpha }^{\prime }|R_{\mu \nu \rho \sigma }|\ll 1$. Massive string modes then produce suppressed higher-derivative interactions, such as higher-curvature corrections. Small string coupling $g_s=e^{{\Phi }_0}$ also suppresses higher-genus quantum corrections; dimensionally $G_D$ scales as $g_s^2{\alpha }^{\prime (D-2)/2}$. Thus Einstein gravity is the universal leading two-derivative dynamics of the massless closed-string spin-two field, with additional massless matter, string-length corrections and quantum corrections systematically controlled.

For the bosonic theory this is the massless-sector effective description in 26 dimensions; the tachyon means that it is not by itself a proof of a stable vacuum. Tachyon-free supersymmetric string theories have the same massless graviton mechanism in ten dimensions before compactification. Removing the tachyon and obtaining a realistic four-dimensional model require extra physical input beyond the emergence of the Einstein action.

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.

Use the mostly-positive target Minkowski spacetime metric and a closed string coordinate $0\le \sigma <2\pi$. 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 expansionReality imposes ${\alpha }_{-n}=({\alpha }_n{)}^{\ast }$ 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 asFor noncompact directions both zero modes are $\sqrt{{\alpha }^{\prime }/2}{p}^{\mu }$. 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 areAn overall rescaling of the named worldsheet stress tensor does not alter its vanishing. Variation of the independent worldsheet metric imposes ${\theta }_{ab}=0$, equivalently $(\dot{X}+X^{\prime }{)}^2=(\dot{X}-X^{\prime }{)}^2=0$. These are the Virasoro constraints that restrict the otherwise general wave solution.

The conjugate momentum density is $P_{\mu }={\dot{X}}_{\mu }/(2\pi {\alpha }^{\prime })$. The gauge-fixed Hamiltonian generating $\tau$ translation is thereforewith the analogous tilded generators. In particular, before imposing the Virasoro constraints,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 $L_0-{\tilde{L}}_0=0$ imposes the residual spatial constraint.