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The local gauge symmetries of the Polyakov action are worldsheet diffeomorphisms and Weyl transformations. Under a reparametrization ${\sigma }^{\alpha }\mapsto {\sigma }^{\prime \alpha }$, the embedding coordinates $X^{\mu }$ transform as scalars and the worldsheet metric as a covariant tensor; the contracted integrand is a scalar density. Infinitesimally, with the active convention, $\delta X=-{\xi }^{\alpha }{\partial }_{\alpha }X$ and $\delta \gamma =-{\mathcal{L}}_{\xi }\gamma$. A Weyl transformation sends ${\gamma }_{\alpha \beta }\mapsto e^{2\omega }{\gamma }_{\alpha \beta }$ and leaves $X$ fixed. In two dimensions the factors from $\sqrt{-\gamma }$ and ${\gamma }^{\alpha \beta }$ cancel. Target-space Poincaré transformations are additional global symmetries, rather than worldsheet gauge symmetries.

These gauge freedoms put the metric locally in conformal gauge, ${\gamma }_{\alpha \beta }={\eta }_{\alpha \beta }$. Its variation must still be imposed: it gives the worldsheet stress tensor constraints $T_{++}=T_{--}=0$. The field equation is the wave equation, and periodicity of a closed string gives independent left- and right-moving oscillator sets. In a normalization with ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\prime }/2}{k}^{\mu }$, their commutators and normal-ordered generators arewith the same formulas for the tilded sector and zero mixed commutators.

The closed-string physical-state Virasoro conditions in the critical flat-space theory areThey act on the oscillator Fock space, and physical spurious (null) states are quotiented out. Equivalently physical states are the suitable BRST cohomology classes. Positive modes constrain kets; their adjoints constrain bras, so one does not impose every positive and negative matter mode on the same ket. Subtracting the two zero-mode equations gives closed-string level matching, $N=\tilde{N}$, and adding their mass-shell content gives $M^2=4(N-1)/{\alpha }^{\prime }$. Quantum consistency fixes the intercept to one and, for just these flat coordinate fields, $d=26$: the matter central charge $d$ cancels the reparametrization ghost central charge $-26$. Other dimensions require additional worldsheet degrees of freedom or a noncritical completion.

For the residual symmetry, take ${\sigma }^{\pm }=\tau \pm \sigma$. The conformal Killing equations leave independent transformations ${\sigma }^{+}\mapsto f({\sigma }^{+})$ and ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$, with a compensating Weyl rescaling restoring the flat metric. On the Euclidean plane the corresponding holomorphic vector fields are ${\ell }_n=-z^{n+1}{\partial }_z$ and their antiholomorphic partners. They satisfy the Witt algebra $[{\ell }_m,{\ell }_n]=(m-n){\ell }_{m+n}$. The conserved contour generators are the Virasoro modes $L_n=(2\pi i{)}^{-1}\oint z^{n+1}T(z)dz$. Their quantum Virasoro algebra iswith a second independent copy for the right movers. These are the generators of the residual conformal transformations, including their quantum central extension.

The two-dimensional quantum-gravity interpretation of strings follows from what is integrated, not from an analogy: at genus $h$, the Polyakov path integral integrates over both $X$ and the metric, divided by diffeomorphism and Weyl gauge volume. String perturbation theory sums these surface topologies, weighted by $g_s^{2h-2}$ before external-state coupling factors. Gauge fixing leaves worldsheet ghost determinants and integrals over worldsheet moduli. Thus it is two-dimensional quantum gravity coupled to the $d$ coordinate matter fields. A flat gauge is local; general higher-genus surfaces retain conformal-structure moduli and need not admit a globally flat representative.

Target-space gravity is a different conclusion. At $N=\tilde{N}=1$, the state ${ϵ}_{\mu \nu }{\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|k⟩$ is massless, transverse in both indices, and has longitudinal gauge identifications. Its symmetric traceless transverse polarization is a graviton; the antisymmetric and scalar polarizations give the Kalb–Ramond field and dilaton. The graviton gauge identification is precisely the linearized metric diffeomorphism. Moreover the leading string-frame massless effective action contains $\int \sqrt{-G}{e}^{-2\Phi }R$, and worldsheet Weyl invariance with constant dilaton and zero two-form requires $R_{\mu \nu }=0$ at leading order in ${\alpha }^{\prime }$. Thus both the propagating massless spin-two state and its low-energy interactions reproduce gravity. Closed strings contain the graviton and yield Einstein gravity in their low-energy massless sector, together with other fields and higher-derivative corrections; the bosonic tachyon instability remains.

The string nonlinear sigma model regards the target-space metric, the Kalb–Ramond field and the dilaton as couplings of the two-dimensional worldsheet theory. Schematically its Euclidean terms areQuantization produces running background couplings and a possible worldsheet Weyl anomaly. Gauge-fixed matter bosons contribute central charge $D$, while the reparameterization bc ghost system contributes $-26$ in each chirality. In the critical flat background cancellation therefore requires $D=26$. In a curved background, vanishing improved sigma-model beta functions further imposes field equations for $G,B,\Phi$. For example, the leading metric condition isThese are spacetime conditions because the couplings are functions of the embedding coordinates, not additional worldsheet coordinates.

The leading equations arise by varying the string-frame massless effective actionThe factor $e^{-2\Phi }$ is the sphere contribution in the string genus expansion. Higher sigma-model orders generate higher-derivative terms in the ${\alpha }^{\prime }$ expansion; higher genera give string-loop corrections. Thus conformal consistency of the quantum worldsheet determines the low-energy target equations and their action. This is a massless-sector truncation: the bosonic tachyon is not removed by writing an action only for $G,B,\Phi$.

For a brane embedding $X^{\mu }(\xi )$, the D-brane worldvolume pullbacks areThe dilaton is evaluated at the embedding. When $T_p$ denotes the physical tension in a background with constant mode ${\Phi }_0$, the convention is $\tilde{\Phi }=\Phi (X)-{\Phi }_0$, so $e^{-\tilde{\Phi }}=1$ in that vacuum. Equivalently one can use $\tilde{\Phi }=\Phi (X)$ and a coefficient with the constant $e^{-{\Phi }_0}$ not already included. The physics is the same; the two conventions must not count that constant factor twice.

Several features motivate the Dirac-Born-Infeld action. Without gauge or two-form fields, it is the relativistic induced-volume action for a tensionful extended object. An open-string boundary couples to its gauge potential $A$, so a bulk transformation $B\mapsto B+d\Lambda$ is canceled by $A\mapsto A-P[\Lambda ]/(2\pi {\alpha }^{\prime })$. Thus only $P[B]+2\pi {\alpha }^{\prime }F$ can enter. For slowly varying constant field strength, the open-string disk partition function resums its dependence into the determinant; expansion gives the Maxwell kinetic term. The disk has Euler characteristic one, giving $e^{-\Phi }$ rather than the closed-string sphere's $e^{-2\Phi }$. Worldvolume covariance and the induced metric then combine these ingredients into the determinant expression.

Explicitly, in flat background and static gauge $X^a={\xi }^a$ with transverse fluctuations $X^I$, the weak-field expansion starts asThe linear term in the antisymmetric matrix $F$ vanishes; the quadratic term follows from expanding $\frac{1}{2}\mathrm{Tr}\log (1+2\pi {\alpha }^{\prime }{\eta }^{-1}F)$. Thus the DBI action contains both the scalar and gauge kinetic energies, while higher powers and derivatives are string-scale corrections.

For two coincident D-branes, an oriented open string has independent labels $r,s\in \{1,2\}$ at its ends. Its Chan-Paton factors are the matrix units $E_{rs}$. The four massless vector modes span the Lie algebra of $U(2)$: diagonal strings supply the two Abelian fields, and oppositely oriented interbrane strings supply the off-diagonal modes. Joining strings multiplies these matrices, producing the non-Abelian commutators. Transverse vector polarizations become adjoint scalar fields. This is Coincident-D-brane gauge enhancement.

The massless gauge/scalar dynamics reduce to dimensionally reduced Yang-Mills theory in a flat background with constant dilaton and vanishing pulled-back $B$, at energies and field variations well below the string scale. More precisely, derivative corrections satisfy ${\alpha }^{\prime }{E}^2\ll 1$, and one expands in small dimensionless field strengths and scalar gradients rather than retaining the full nonlinear determinant. Weak string coupling suppresses loop corrections. After subtracting the constant brane vacuum energy and using the scalar normalization fixed below, the coefficient isA bosonic brane additionally has an open-string tachyon; the displayed action governs its massless sector and is not a stable complete truncation. For supersymmetric branes the corresponding leading action also has the massless fermions.

There is a sign qualification in the printed scalar potential. With the stated $D_a\varphi ={\partial }_a\varphi +i[A_a,\varphi ]$, take $A_a,{\varphi }^I$ Hermitian and $[{\varphi }^I,{\varphi }^J{]}^2$ to mean the literal matrix product. Then dimensional reduction hasConsequently $F_{IJ}{F}^{IJ}=-[{\varphi }^I,{\varphi }^J{]}^2$. The consistent low-energy action isThis is the Hermitian commutator potential in D-brane Yang-Mills theory. Its potential energy is $-\sum \mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2/(4g_{\mathrm{YM}}^2)\ge 0$, because a Hermitian-matrix commutator is anti-Hermitian. For a concrete check, take ${\varphi }^1=v{\sigma }_1$, ${\varphi }^2=v{\sigma }_2$ with Pauli matrices. Then $[{\varphi }^1,{\varphi }^2{]}^2=-4v^{4}I$, so the literal printed plus sign would make the static energy negative and unbounded below. The printed plus sign is consistent only if its square denotes the positive norm $[{\varphi }^I,{\varphi }^J{]}^{†}[{\varphi }^I,{\varphi }^J]$, rather than a literal square. The gauge-boson mass calculation below uses the unambiguous kinetic terms.

In an ordinary separated-brane vacuum the scalar expectation values commute and can be simultaneously diagonalized:Their eigenvalue vectors specify the transverse positions, up to the normalization to be determined. The trace encodes the center of mass; their difference encodes separation. When the two vectors coincide, every $U(2)$ transformation commutes with the vacuum. When they are distinct, only independent diagonal phases do, so the unbroken group is $U(1)\times U(1)$. The off-diagonal vectors acquire mass by the Higgs mechanism.

Write the off-diagonal entries of $A_a$ as $W_a$ and $W_a^{\ast }$ and put $\Delta v^I=v_1^I-v_2^I$. At a constant diagonal vacuum, $D_a{\varphi }^I=i[A_a,⟨{\varphi }^I⟩]$, soThe quadratic off-diagonal gauge kinetic term is $-f_{ab}^{\ast }{f}^{ab}/(2g_{\mathrm{YM}}^2)$ with $f_{ab}={\partial }_a{W}_b-{\partial }_b{W}_a$, and the scalar kinetic term supplies $-{\sum }_I(\Delta v^I{)}^2{W}_a^{\ast }{W}^a/g_{\mathrm{YM}}^2$. Comparing these with the complex massive-vector action givesHere the term W-boson means this broken-generator vector, not specifically an electroweak particle. The normalization places the gauge coupling outside the action, so no extra $g_{\mathrm{YM}}$ multiplies these $v_r^I$; rescaling to canonically normalized scalar fields moves that coupling into the usual Higgs mass formula.

The corresponding open-string vector oscillator stretched over distance $L=|X_1-X_2|$ has classical stretching mass $TL=L/(2\pi {\alpha }^{\prime })$. Its unstretched mass contribution vanishes because it is the vector level, rather than the bosonic tachyon level. Equating it with the gauge-theory mass givesThe component relation follows by rotational covariance and a common choice of transverse origin; a uniform translation adds the same constant to both brane coordinates. This is the D-brane scalar-position normalization. Together with the string tension determined in Question 1, it fixes the exact normalization of scalar eigenvalues as brane positions.

Use the free-boson operator product expansionWrite $E_p=:e^{ip\cdot X}:$ and $ϵ=z-w$. The needed Wick contractions are $\partial X^{\mu }(z)\partial X^{\nu }(w)\sim -{\alpha }^{\prime }{\eta }^{\mu \nu }/(2{ϵ}^2)$ and $\partial X^{\mu }(z)E_p(w)\sim -i{\alpha }^{\prime }{p}^{\mu }{E}_p(w)/(2ϵ)$. For $V_{\zeta }={\zeta }_{\mu }:\partial X^{\mu }{E}_p:$, the cross double contraction in the worldsheet stress tensor has a cubic pole: either of its two derivatives can contract with the derivative in $V_{\zeta }$, while the other contracts with $E_p$. Including the factor $-1/{\alpha }^{\prime }$ givesA primary operator has no cubic pole. Hence the first vertex is primary precisely whenThe antiholomorphic conformal weight is $\bar{h}={\alpha }^{\prime }{p}^2/4$. Primarity alone puts no mass-shell condition on $p$.

For $W_{\zeta }={\zeta }_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{E}_p:$, the holomorphic cubic pole is $-i{\alpha }^{\prime }{p}^{\mu }{\zeta }_{\mu \nu }:\bar{\partial }{X}^{\nu }{E}_p:/(2{ϵ}^3)$. The antiholomorphic operator product expansion gives the analogous pole involving $p^{\nu }{\zeta }_{\mu \nu }$. Thus the transversality of derivative-exponential primary vertices requiresThere is no trace-free requirement for primarity. A physical integrated string vertex operator must additionally have conformal weights $(1,1)$; for this vertex that gives $p^2=0$.

Now keep the fermionic signs in the real chiral Majorana fermion calculation. Contracting the external $\psi (w)$ through $:\psi (z)\partial \psi (z):$ givesMultiplying by $-1/2$ and Taylor expanding $\psi (z)$ at $w$ producesThis proves that $\psi$ is a primary operator with weight $h_{\psi }=1/2$. To find the central charge, in $T(z)T(w)$ the two double-contraction pairings contribute, before the overall factor $1/4$,The single Wick contractions give $2T(w)/{ϵ}^2+\partial T(w)/ϵ$, soComparing the leading term with $c/(2{ϵ}^4)$ proves $c_{\psi }=1/2$, as in the free chiral Majorana fermion conformal field theory.

With four embedding bosons and the ordinary reparameterization bc ghost system, cancellation of the worldsheet Weyl anomaly in each chirality requiresThus there are 44 real left-moving and 44 real right-moving fermions, equivalently 44 two-dimensional nonchiral Majorana fermions. These are internal free fermions in a four-dimensional bosonic string. Adding internal matter to the bosonic string does not introduce local worldsheet supersymmetry or its superghosts.

For the requested polynomial string vertex operators, work in the unprojected Neveu–Schwarz sector. A full model also chooses spin structures and a consistent projection, which the central-charge condition alone does not specify. If the holomorphic and antiholomorphic oscillator contributions to the conformal weights are ${\ell }_L,{\ell }_R$, physicality requiresTherefore only equal levels at most one can contribute to the requested nonpositive bosonic string mass spectrum.

With zero fermion insertions, level zero gives the tachyon vertex operator $E_p$, a scalar with $M^2=-4/{\alpha }^{\prime }$. Level one gives the massless closed-string vertex operator ${\zeta }_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{E}_p:$, with the transverse conditions above and the string-state gauge redundancy that removes longitudinal polarizations. The physical transverse polarization tensor decomposes into a symmetric trace-free graviton, an antisymmetric Kalb–Ramond field, and a scalar dilaton. In four spacetime dimensions these have respectively two, one, and one physical polarizations; the two-form can be dualized to a scalar.

With two fermion insertions there are two possibilities. One fermion in each chirality givesThese are $44^2=1936$ scalar tachyons before any projection. There is no spacetime derivative or spacetime polarization to impose a vector transversality condition.

Two different fermions in the same chirality form an antisymmetric fermion bilinear, a weight-one current $J^{ij}=:{\psi }^i{\psi }^j:$; equal-species undifferentiated bilinears vanish. Closed-string level matching then requires a weight-one bosonic derivative in the opposite chirality. The resulting string vertex operators areThey have $M^2=0$ and are gauge vectors, with $p\cdot A^{ij}=0$ and $A_{\mu }^{ij}\sim A_{\mu }^{ij}+p_{\mu }{\lambda }^{ij}$, and likewise for $\tilde{A}$. Each chirality supplies $\left(\genfrac{}{}{0ex}{}{44}{2}\right)=946$ gauge vectors. The fermion bilinears generate the two $\mathfrak{so}(44)$ current algebras associated with the Special orthogonal Lie algebra, giving the unprojected gauge group $SO(44{)}_L\times SO(44{)}_R$. The tempting vertex $:{\psi }^i{\psi }^j{E}_p:$ alone fails closed-string level matching because its weights are $(1,0)$ before the common momentum contribution. Additional derivatives raise the levels above the nonpositive-mass range; a product of a left and a right bilinear has four fermion insertions and is outside this requested sector. This completes the zero- and two-fermion vertices of an internally fermionized bosonic string; a specified global projection can remove some of the listed states.

Choose a Euclidean string worldsheet and the usual normalization of the Polyakov action. For a closed oriented worldsheet $\Sigma$, the string nonlinear sigma model isHere ${\epsilon }^{12}=1$ is the alternating density, not the tensor with components $1/\sqrt{g}$. The imaginary coefficient in the Kalb–Ramond field term is the Euclidean continuation; in Lorentzian signature this term is real. $G$ is the spacetime metric and $\Phi$ the dilaton.

All three terms have worldsheet diffeomorphism invariance, and are covariant under spacetime coordinate transformations when $X$, $G$, $B$ and $\Phi$ are transformed together. Under a Weyl transformation $g_{ab}\mapsto e^{2\omega }{g}_{ab}$, $\sqrt{g}{g}^{ab}$ is invariant in two dimensions, so the metric coupling has classical Weyl invariance. The antisymmetric coupling is independent of the worldsheet metric, so it too has classical Weyl invariance.

The curvature coupling requires care. An infinitesimal Weyl transformation gives ${\delta }_{\omega }(\sqrt{g}{R}^{(2)})=-2\sqrt{g}{\Delta }_g\omega$. Integration by parts on the closed worldsheet therefore gives the Weyl variation of a nonconstant dilaton coupling:A constant dilaton coupling is topological and Weyl invariant; a general varying dilaton coupling is not separately classically Weyl invariant. Its classical variation participates in canceling the quantum worldsheet Weyl anomaly, producing the improved background equations below.

To relate these background couplings to the flat-space bosonic string mass spectrum, expand $G_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }(X)$ and express $h$ and $B$ as superpositions of plane waves. The perturbation of the action is a sum of integrated string vertex operators of the formThis is the vertex of ${\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|p⟩$ in the massless first closed-string level. Its conformal weights are $(1+{\alpha }^{\prime }{p}^2/4,1+{\alpha }^{\prime }{p}^2/4)$, so marginality gives $p^2=0$. The primary operator conditions give transversality, and the string-state gauge redundancy removes longitudinal components. The symmetric trace-free polarization tensor is the graviton and perturbs $G$; its antisymmetric part is the Kalb–Ramond field and perturbs $B$. The scalar physical polarization is the dilaton. At the level of background fields its vertex involves the metric trace and the curvature improvement, so an arbitrary off-shell metric trace should not simply be identified with the physical dilaton without this distinction.

The spacetime gauge invariance of the antisymmetric coupling isIndeed its action is $S_B=i(2\pi {\alpha }^{\prime }{)}^{-1}{\int }_{\Sigma }{X}^{\ast }B$. Its variation isfor a closed worldsheet, by Stokes theorem. In components this is the total derivative ${\partial }_a({\epsilon }^{ab}{\Lambda }_{\nu }(X){\partial }_b{X}^{\nu })$: the term containing ${\partial }_a{\partial }_b{X}^{\nu }$ vanishes by antisymmetry. Thus only the Kalb-Ramond field strength $H$ can enter the local spacetime equations. With boundaries, a compensating boundary gauge-field transformation is needed.

Separate the dilaton into a constant ${\Phi }_0$ and a varying part. The Gauss-Bonnet theorem givesfor a connected closed orientable genus-$g$ worldsheet. Its Euclidean path-integral weight containsConsequently the sum over worldsheet topologies is the string genus expansion ${\sum }_{g\ge 0}{g}_s^{2g-2}{\mathcal{A}}_g$, with each coefficient integrating over that genus's worldsheet moduli. Each extra handle costs $g_s^2$, which is the loop-counting factor of string perturbation theory. With the usual normalization of $n$ external closed-string vertices, the connected amplitude carries $g_s^{2g-2+n}$. This dilaton Euler-characteristic weighting is distinct from the expansion in ${\alpha }^{\prime }$ of a fixed-worldsheet string nonlinear sigma model.

Finally quantize that two-dimensional theory. The background fields are its couplings. The background field expansion of a string sigma model, $X=x+\xi$ in target-space normal coordinates, contains curvature vertices quadratic in the fluctuation $\xi$. Contracting their two fluctuation indices produces the Ricci tensor multiplying $\partial x^{\mu }\bar{\partial }{x}^{\nu }$. Logarithmic short-distance contractions renormalize $G$, $B$, and $\Phi$, giving functional sigma-model beta functions. Gauge independence requires vanishing of the worldsheet Weyl anomaly, including the classical improvement from the varying dilaton, not merely an arbitrary choice of renormalization scale.

For clarity, in conventional leading-order normalization the improved coefficients areHere $H^2=H_{\mu \nu \rho }{H}^{\mu \nu \rho }$. The constant term in the dilaton coefficient is the matter-plus-ghost central charge deficit. At $D=26$, setting these coefficients to zero gives the leading spacetime field equationsFor the last equation, take the trace of the first, $R-H^2/4+2{\nabla }^2\Phi =0$, and combine it with ${\overline{\beta }}^{\Phi }=0$; this verifies the coefficients and signs. These are equivalently the equations from the leading string-frame massless effective actionHigher worldsheet orders give higher-derivative ${\alpha }^{\prime }$ corrections, whereas additional handles give string-loop corrections controlled by $g_s$. Thus quantum consistency of a two-dimensional string nonlinear sigma model determines the dynamics of its spacetime background fields. These leading coefficient conventions agree with the primary string-theory lectures' background-field calculation.

Actions and equations. Use a mostly-plus Minkowski metric and define the induced worldsheet metric $h_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. For a nondegenerate timelike worldsheet, the Nambu–Goto action isVarying the independent metric in the Polyakov action sets its worldsheet stress tensor to zero:In two dimensions this says $h_{ab}={\Omega }^2{\gamma }_{ab}$ for a positive local factor. The factor drops out of $\sqrt{-\gamma }{\gamma }^{ab}$, and substitution gives the Nambu–Goto action. Conversely, any nondegenerate induced metric solves the auxiliary-metric equation up to a Weyl transformation. This classical equivalence of Polyakov and Nambu–Goto actions is a statement about classical embeddings; quantum equivalence additionally requires treatment of the metric measure and anomaly.

Variation of $X^m$ in the metric action, followed by elimination of $\gamma$, gives the Nambu–Goto equations of motionA closed string has periodic $X$ and no spatial endpoint variation.

For a background metric $G$, Kalb–Ramond field $b$, and dilaton $\Phi$, one consistent Lorentzian convention isThe orientation fixes the two-form sign; here the Lorentzian curvature convention is chosen so Wick rotation gives the positive Euclidean dilaton term $S_{E,\Phi }=\int \sqrt{\gamma }\Phi R^{(2)}/(4\pi )$. This avoids hiding the convention in the topology argument. For constant vacuum value ${\Phi }_0$, the Gauss-Bonnet theorem gives $S_{E,\Phi }={\Phi }_0\chi$. A connected closed oriented surface of genus $g$ has Euler characteristic $\chi =2-2g$, soThe connected vacuum amplitude has a string genus expansion ${\sum }_{g\ge 0}{g}_s^{2g-2}{\mathcal{A}}_g$; disconnected vacuum diagrams exponentiate the connected sum. Each additional handle supplies a factor $g_s^2$. This is dilaton Euler-characteristic weighting.

The rotating circle. For the specified embedding, the squared spatial speed and tangent length are both $1/2$, and their spatial inner product is zero. Including $X^0=t$ therefore givesThe metric is constant. Its equation reduces to $\ddot{X}-X^{\prime \prime }=0$, satisfied by the left- and right-moving trigonometric components and the linear time component. On a constant-time slice with $0\le \sigma <2\pi$, the proper length of a string isIt is time-independent.

Let $R(\theta )$ be an ordinary two-dimensional rotation matrix. Rotate the first coordinate pair by $R(t)$ and the second by $R(-t)$. In these time-dependent Cartesian coordinates both pairs become $(\cos \sigma ,\sin \sigma )/2$. A further fixed orthogonal change of basis givesThus every spatial slice is a planar circle of radius $1/\sqrt{2}$; its plane rotates in the ambient four-space. This is a rigid circular string in four spatial dimensions, not a pulsating circle. The rotating axes establish its spatial shape, not a transformation to an inertial spacetime frame.

The momentum density obtained from the Nambu–Goto action is $P_m=-T\sqrt{-h}{h}^{ta}{\partial }_a{X}_m$. Here $P_0=-T$, hence the conserved target-space energy isThe material velocity is transverse to the tangent and has magnitude $1/\sqrt{2}$. Its Lorentz factor is $\sqrt{2}$, so the excess over $TL$ is kinetic energy. The spatial circle being stationary in rotating axes does not eliminate this energy.

Constraints and endpoints. Varying the multipliers in the Nambu-Goto phase-space action imposesThese are first-class constraints generating normal and tangential worldsheet diffeomorphisms. They remove the two longitudinal embedding degrees of freedom; they are not extra physical force laws. Hamilton's equations areFor an open string, integration by parts leaves the spatial boundary variationAllowed variational open-string boundary conditions must make this vanish for every permitted endpoint displacement, and be preserved by the evolution. The bracketed expression is the open-string endpoint momentum flux. For free displacements, set that flux to zero. In a boundary-preserving gauge with $u=0$ and finite nonzero $e$, this is the Neumann boundary condition $X^{\prime m}=0$ at each end. This shows that free-end string boundary conditions are consistent.

At such an end, $C=0$ gives $P^2=0$, and $\dot{X}=eP$ gives ${\dot{X}}^2=0$. For a nontrivial endpoint trajectory with nonzero time velocity, its spatial speed is therefore one. This null motion of a free string endpoint follows from the boundary condition together with the constraints, not merely from the bulk wave equation. The induced metric may degenerate right at a free end; the result is understood as the endpoint limit or in the Polyakov description.

For a fixed spatial $p$-plane, split coordinates into $X^A$ along its $(p+1)$-dimensional worldvolume, including time, and $X^i$ transverse to it. Set $X^{\prime A}=0$ and $X^i=y^i$ at the ends. Tangential variations are free and their momentum flux vanishes; normal variations vanish, so their flux need not vanish. Evolution preserves the fixed normal values with $P^i=0$ at the boundary in the same gauge. These mixed Neumann boundary conditions and Dirichlet boundary conditions consistently restrict the endpoint worldlines to the plane.

The massless open-string excitations then separate into a gauge vector along the worldvolume and transverse scalar fields. A scalar displacement changes the corresponding $y^i$; its interpretation is a fluctuation of the plane's embedding. These worldvolume fields from open-string massless states support the interpretation as a dynamical planar D-brane. In the bosonic theory the still lower ground state is a tachyon, indicating an unstable brane; the interpretation does not require pretending that this tachyon is a stable massless field. Appropriate superstring sectors can remove that instability.

Use $x^{\pm }=(x^0\pm x^{D-1})/\sqrt{2}$, transverse coordinates $x^i$, $i=1,\dots ,d=D-2$, and the Minkowski metricThe relativistic particle phase-space action becomesIn the light cone gauge $x^{+}=t$, solve the mass-shell condition for $p_{+}$, assuming $p_{-}\ne 0$. The reduced phase-space action is $\int dt({\dot{x}}^{-}{p}_{-}+{\dot{x}}^i{p}_i-H)$, withThe last equality selects the future-directed momentum sector and makes positivity transparent. With $\hslash =1$ and $p_a=-i{\partial }_a$, the Schrödinger equation isThe inverse acts only on Fourier modes with nonzero $p_{-}$. Multiplication by $2p_{-}$ gives $2{\partial }_{-}{\partial }_{+}\Psi =({\Delta }_{\perp }-{\mu }^2)\Psi$. ThereforeThe light-cone Hamiltonian thus gives the same Klein-Gordon equation as covariant quantization.

For the massive two-form field, take $\mu \ne 0$. Apply ${\partial }^n$ to its field equation. Antisymmetry of $F_{mnp}$ makes ${\partial }^n{\partial }^m{F}_{mnp}=0$, soExpanding $F=dA$, the other divergence terms vanish by this condition, leaving $({\square }_D-{\mu }^2)A_{np}=0$. The light-cone decomposition of a massive two-form makes its dependent components explicit. The divergence equation isTaking $n=-$ and $n=i$, respectively, givesThe $n=+$ equation follows from these expressions: the two terms containing ${\partial }_{+}{\partial }_i{A}_{-i}$ cancel and ${\partial }_i{\partial }_j{A}_{ji}=0$. Consequently $A_{-i}$ and $A_{ij}$ are independent, each satisfying the Klein-Gordon equation with mass $\mu$. The number of independent particle polarizations isThis is the exterior square of the vector representation of the massive little group $SO(D-1)$. In the analogous Proca equation, ${\partial }^m{A}_m=0$ determines $A_{+}$ from $A_{-}$ and $A_i$, leaving $D-1$ components. A massive field has no gauge freedom that would justify setting these longitudinal components to zero. If $\mu =0$, instead use the two-form gauge field symmetry $A\mapsto A+d\Lambda$: the light-cone gauge for a two-form removes $A_{-m}$, leaving $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$ transverse particle polarizations. The massive and massless counts are different.

In the closed-string mode expansion, $x^m,p_m$ are center-of-mass canonical variables, while ${\alpha }_k^i,{\tilde{\alpha }}_k^i$ are independent left- and right-moving transverse string oscillators. Their complex conjugates are ${\alpha }_{-k}^i,{\tilde{\alpha }}_{-k}^i$. The two zero-mode Lagrange multipliers impose the remaining mass-shell condition and closed-string level matching. The string level operators areTheir quantum definitions use normal ordering. The symplectic terms in the phase-space action givewith all brackets between distinct sectors zero. The nonzero-index string oscillators obey ${\alpha }_n^{i†}={\alpha }_{-n}^i$ and similarly for the right-moving sector. Define the momentum-labelled oscillator vacuum byFor $k>0$, $a_k^i={\alpha }_k^i/\sqrt{k}$ has $[a_k^i,a_l^{j†}]={\delta }_{kl}{\delta }^{ij}$. HenceStarting with $N|0;p⟩=0$, a finite product with $r_{ki}$ creation operators of mode $k$ has eigenvalue ${\sum }_{k,i}k{r}_{ki}$. The Fock space is generated by these products; both level operators have nonnegative integer eigenvalues. This establishes the integer string oscillator level property. Subtracting their physical zero-mode constraints enforces $N=\tilde{N}$.