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With string tension $T=1/(2\pi {\alpha }^{\prime })$, the free-ended open string $X^0=a\tau$, $X^1+i{X}^2=a\cos \sigma e^{i\tau }$, $0\le \sigma \le \pi$, is straight at every target time. Its Nambu–Goto equations of motion and Virasoro constraints hold, and its ends move at light speed. Integrating the target-space Noether charges of a string gives $M=\pi Ta$ and $J=\pi T{a}^2/2$. Thus $J={\alpha }^{\prime }{M}^2$. Extending the same map to $0\le \sigma \le 2\pi$ describes a double-covered folded closed string, with twice the mass and twice the angular momentum, hence $J={\alpha }^{\prime }{M}^2/2$. The different slopes record the different boundary topology.

Free endpoint variations set the open string flux $e{T}^2{X}_m^{\prime }+u{P}_m$ to zero. In a boundary-adapted temporal gauge, freedom to vary time forces $u=0$ at the end. Nonzero lapse then gives $X^{\prime m}=0$, a Neumann boundary condition.

The symmetry used to classify an open string must preserve its endpoint boundary background as well as its momentum. A Neumann-Dirichlet open-string boundary condition breaks target Lorentz transformations mixing ND and common directions with Neumann boundary conditions. For a D1–D25 pair with 24 ND directions, the common worldvolume has Lorentz group $SO(1,1)$, whose connected massive little group is trivial. The transverse $SO(24)$ remains as an internal rotation group. Thus a massive 24-component oscillator multiplet is not an $SO(25)$ vector of the full 26-dimensional massive little group; a vector of that full group would have 25 components.

A transverse open string coordinate has an ND boundary condition when one endpoint obeys a Neumann boundary condition and the other a Dirichlet boundary condition. On a strip of width $\pi$, its nonconstant spatial eigenfunctions are $\cos ((n+1/2)\sigma )$ when the free endpoint is at zero. They solve the endpoint equations because their derivatives vanish at zero and their values vanish at $\pi$. Reversing the endpoints gives sine eigenfunctions with the same frequencies. The fixed endpoint removes the dynamical worldsheet zero mode.

In a boundary-adapted temporal gauge, a free open string endpoint has $X^{\prime m}=0$. The Nambu–Goto phase-space constraints then imply $P^2=0$, while $\dot{X}=eP$ implies ${\dot{X}}^2=0$. For $X^0=t$, its spatial speed is one. This null boundary is compatible with a timelike interior worldsheet.

The string level operator for a free-ended open string is $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$. The quantum condition is ${\alpha }^{\prime }{M}^2=N-a$, where $a$ is the normal-ordering constant of a string. Lorentz-consistent bosonic string theory has $a=1$.

For an ordinary critical open string with all Neumann boundary conditions, there are 24 independent transverse creation oscillators at each positive mode number $n$. Summing their nonnegative occupation numbers gives one factor $(1-w^n{)}^{-24}$. The coefficient $d_p=[w^p]F(w)$ counts oscillator polarizations at level $p$, excluding continuous center-of-mass momentum and additional endpoint multiplicities. Cauchy coefficient formula gives $d_p=(2\pi i{)}^{-1}\oint F(w)w^{-p-1}dw$.

Tree-level scattering of four ordered open string tachyons gives the Veneziano amplitude. Its poles factorize through intermediate open-string states.

Use Minkowski spacetime with signature $(-,+,\dots ,+)$, and write the string tension as $T=1/(2\pi {\alpha }^{\prime })$. If $X^{\mu }(\tau ,\sigma )$ is the string embedding map, its induced worldsheet metric is ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. The Nambu–Goto action isThe first variation uses $\delta {\gamma }_{ab}=2{\partial }_{(a}{X}_{\mu }{\partial }_{b)}\delta X^{\mu }$ and therefore givesAn integration by parts yields the bulk Nambu–Goto equations of motion and the endpoint term:For a closed string, periodicity cancels the endpoint term. For a freely moving open string, arbitrary endpoint variations require zero momentum flux, which is the free-end string boundary condition.

Choose orthogonal conformal coordinates on the nondegenerate part of the worldsheet, so that ${\gamma }_{\tau \sigma }=0$ and ${\gamma }_{\tau \tau }+{\gamma }_{\sigma \sigma }=0$. The Nambu–Goto equations of motion then become a wave equation, with the accompanying Virasoro constraints:Here a dot and a prime mean ${\partial }_{\tau }$ and ${\partial }_{\sigma }$. The last two equations are essential: a solution of the wave equation alone need not be a relativistic fundamental string motion.

For a free-ended open string, take $0\le \sigma \le \pi$ and $a>0$, and setwith the other coordinates constant. Every component satisfies the wave equation, andMoreover $X^{\prime }=0$ at both endpoints, giving the Neumann boundary condition. At fixed target time $t=a\tau$, the spatial image is the straight segmentThus its angular velocity is $1/a$. The speed at coordinate $r$ is $|r|/a$, so the endpoints have the null motion of a free string endpoint. The induced worldsheet metric degenerates there; the conserved densities below have finite limits, so the endpoint degeneracy causes no divergent charge.

In these coordinates ${\Pi }^{\tau \mu }=T{\dot{X}}^{\mu }$. The target-space Noether charges of a string give the energy, momentum and angular momentum:Since this is the centre-of-momentum frame, the rest mass is $M=E$. Eliminating $a$ proves the free-ended straight-string Regge relation:A spacetime Lorentz transformation of the solution changes its frame, not this rest-frame relation.

There is also a folded closed string version: use the same formula over $0\le \sigma \le 2\pi$. The spatial segment is then covered twice, with folds at $\sigma =0,\pi$, and the closed-string mode expansion is periodic. Each charge is twice its open string value, soThe folds also have degenerate induced metric and are understood through the finite canonical densities or a limiting smooth motion. The two classical Regge trajectories have different slopes because the closed string has two coincident branches. No quantum string intercept enters this classical calculation.

The Nambu–Goto action is invariant under worldsheet diffeomorphisms. Its canonical Nambu–Goto phase-space constraints areIn conformal gauge, $\Pi =T\dot{X}$, so these become $(\dot{X}\pm X^{\prime }{)}^2=0$. Their smeared generators implement the residual conformal transformations rather than independent propagating degrees of freedom. Classically their Fourier modes form the classical Virasoro constraint algebra. Quantization promotes them to the Virasoro algebra, and its constraints select the physical string states from the indefinite covariant Fock space.

For clarity, start with a Neumann boundary condition at both ends of an open string, $0\le \sigma \le \pi$. Its open-string mode expansion isCanonical quantization gives the string oscillator relationsWith ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$, the worldsheet stress-energy tensor obeysSubstituting the open-string mode expansion into this quadratic expression derives the Virasoro generators:The colons mean normal ordering, placing the negative-mode creation operators to the left. The string level operator $N$ increases by $r$ when an oscillator ${\alpha }_{-r}$ is added.

Using the commutator once givesApplying this to the quadratic expression for $L_n$ produces $(m-n)L_{m+n}$, together with the double-contraction Virasoro central extension. To determine its coefficient, take $m>0$ and evaluate $[L_m,L_{-m}]$ on a zero-momentum oscillator vacuum. Only the creation pairs with $1\le r\le m-1$ contribute, givingHere $D$ is the target spacetime dimension; the timelike free boson also contributes one to the central charge, since a double contraction contains two metric factors. ThusThis derives both the Virasoro generators and their quantum anomaly.

In old covariant string quantization, the physical-state Virasoro conditions for an open string arewhere $a$ is the normal-ordering constant of a string. Requiring all positive and negative modes to annihilate a state would contradict the nonzero Virasoro central extension; the positive-mode prescription is the appropriate covariant subsidiary condition. The zero-mode equation gives the open bosonic string mass spectrum,For a closed string, there are two commuting Virasoro algebras. With uncompactified zero modes, $L_0={\alpha }^{\prime }{p}^2/4+N_R$ and ${\tilde{L}}_0={\alpha }^{\prime }{p}^2/4+N_L$. Imposing both positive-mode conditions and both zero-mode conditions gives the closed-string level matching $N_L=N_R$ and $M^2=4(N_L-a)/{\alpha }^{\prime }$.

Covariance initially retains the timelike string oscillators. For example, ${\alpha }_{-1}^0|0;p⟩$ has negative norm because ${\eta }^{00}=-1$. Its existence in the unconstrained space is not an inconsistency if it is absent from the physical quotient. The no-ghost theorem for the critical bosonic string states that, at $D=26$ and $a=1$, the inner product on physical string states at ordinary nonzero momentum is positive semidefinite. After quotienting physical null string states, this physical space is represented by the 24 positive-norm transverse string oscillators.

A spurious string state has the form ${\sum }_{n>0}{L}_{-n}|{\chi }_n⟩$. Its inner product with every physical string state vanishes, by $L_n|\psi ⟩=0$ and $L_n^{†}=L_{-n}$. Whenever such a state is itself physical, it is a null string state and represents a gauge redundancy, not an extra polarization. Consequently the physical quotient has no negative-norm propagating states, and agrees with light-cone gauge in string theory. This is the significance of the theorem; a proof is not needed here. It does not eliminate the bosonic tachyon, whose negative mass squared is distinct from negative norm, nor does it remove the auxiliary worldsheet ghost fields used in gauge fixing.

The Polyakov action introduces an independent worldsheet metric $h_{ab}$:Its redundancies are worldsheet diffeomorphisms and Weyl transformations. A Polyakov path integral over all metrics without dividing by this gauge volume would count physically equivalent configurations repeatedly. Fixing conformal gauge therefore needs the Faddeev-Popov determinant, not merely the substitution of a flat metric into the action.

After using the Weyl transformation to remove the trace of a metric variation, the infinitesimal worldsheet diffeomorphism ghost operator isThe Faddeev-Popov determinant of this operator is represented by anticommuting fields: a vector Faddeev-Popov ghost $c^a$ and a symmetric traceless Faddeev-Popov antighost field $b^{ab}$. In Euclidean coordinates their worldsheet ghost action is, up to a consistent overall normalization,The two chiral copies of this bc system have conformal weights $(2,-1)$ and anticommutators $\{b_m,c_n\}={\delta }_{m+n,0}$. They are auxiliary fields for the gauge determinant, not additional spacetime particles. Their use is different from the unwanted timelike negative-norm string states of covariant matter quantization.

The chiral holomorphic stress-energy tensor of the bc system isDouble contractions in its operator product expansion give the central charge of reparameterization ghosts $c_{\mathrm{gh}}=-26$. Equivalently, a fermionic bc system with antighost weight $\lambda$ has $c=1-3(2\lambda -1{)}^2$, and $\lambda =2$ gives $-26$. The free coordinate matter contributes $c_{\mathrm{m}}=D$. Thus the total worldsheet Weyl anomaly vanishes for the flat critical bosonic string theory precisely whenThis statement assumes no additional matter or compensating Liouville field theory.

BRST symmetry expresses the gauge symmetry after gauge fixing by replacing its infinitesimal parameter with the Grassmann variable $c$. In one chiral conformal patch, its graded action has the formthere is an analogous barred copy for a closed string. The Grassmann parity makes $s^{2}X=0$: the term from $sc$ cancels the term from applying $s$ to $\partial X$, while the term with $c^2$ vanishes. The gauge-fixed chiral transformations are on shell when their conservation equations are used. At the quantum level, BRST nilpotence also tests the anomaly.

For example, an open string BRST charge can be written using the matter Virasoro generators asUse the ghost oscillator vacuum $b_{n\ge 0}|0{⟩}_{mathrmgh}=c_{n>0}|0{⟩}_{mathrmgh}=0$ and the corresponding oscillator normal ordering. In this convention the last term is $-c_0$ in the critical theory. With a differently shifted ghost $L_0$, the displayed intercept must be shifted consistently; one must not count the same shift twice. Squaring $Q_B$ shows why both critical conditions are needed. Its anomalous terms, in this convention, have coefficientsThe independent cubic and linear terms vanish at $D=26$, $a=1$. The equivalent chiral BRST current is $j_B=c(T_{mathrmm}+T_{mathrmgh}/2)+(3/2){\partial }^{2}c$, with its contour integral defining the charge.

The graded identity $\{Q_B,b_n\}=L_n^{\mathrm{tot}}$ recovers the gauge constraints. In particular, on $|\psi ⟩\otimes |0{⟩}_{mathrmgh}$, BRST-closed imposes $L_{n>0}^{\mathrm{m}}|\psi ⟩=0$ and $(L_0^{\mathrm{m}}-1)|\psi ⟩=0$. Physical states are classes in BRST cohomology,at the appropriate ghost number: one for the usual open-string vertex and two for the closed-string vertex, with the closed-string zero-mode and closed-string level matching conditions also imposed. Since $Q_B^2=0$, an exact shift preserves closure. BRST Ward identities make exact insertions decouple from physical amplitudes and establish gauge independence. Together with the no-ghost theorem for the critical bosonic string, this explains how the auxiliary ghosts enforce gauge invariance while the physical spectrum retains positive norm.

The RNS string adds worldsheet Majorana fermions ${\psi }^{\mu }$ to the coordinate fields $X^{\mu }$. The new fields are spinors on the worldsheet but vectors under the target Lorentz group; this distinction matters when interpreting their oscillator states. In a flat target, a convenient Lorentzian convention for the free gauge-fixed action iswhere ${\rho }^a$ are two-dimensional gamma matrices. The bosonic and fermionic kinetic terms are related by worldsheet supersymmetry. Before fixing local worldsheet supersymmetry, one introduces a worldsheet gravitino along with the worldsheet metric. After superconformal gauge fixing, the residual matter constraints are the Virasoro constraints and the constraints generated by the superconformal current $G\propto \psi \cdot \partial X$.

Each real worldsheet Majorana fermion contributes $1/2$ to the chiral central charge, while each coordinate contributes one. The reparameterization bc system contributes $-26$, and fixing local worldsheet supersymmetry also produces commuting superconformal ghosts of weights $(3/2,-1/2)$, with central charge $11$. ThusIn light-cone gauge in string theory there are eight physical transverse coordinate fields and eight transverse worldsheet Majorana fermions.

On the doubled spatial interval of an open string, the two allowed fermionic periodicities areThey arise from the relative signs of the left-right fermion gluing conditions at the two ends. Thus the Ramond sector has integer-mode oscillators $d_n$, including $d_0$, and the Neveu–Schwarz sector has half-integer oscillators $b_r$, $r\in \mathbb{Z}+1/2$, without zero modes. In either sector,A closed string has independent left- and right-moving periodicities, giving four sector choices rather than two.

The Ramond zero-mode Clifford algebra isHence its ground states carry a Spinor representation of the Lorentz group. The zero-mode supercurrent constraint reduces on a ground state to the Dirac equation $p_{\mu }{\Gamma }^{\mu }u=0$. In the Neveu–Schwarz sector the oscillator vacuum instead carries a scalar representation. Consequently the open string NS sector describes spacetime bosons, while the R sector describes spacetime fermions. Exciting a target-vector oscillator does not change this statistics assignment.

The string intercepts follow from the transverse zero-point energy. Per boson the integer-mode contribution is $\frac{1}{2}\zeta (-1)=-1/24$. Per fermion it is the negative of the corresponding half-sum. Since $\zeta (-1,1/2)=1/24$, the eight transverse pairs giveThus $a_{\mathrm{NS}}=1/2$, $a_{\mathrm{R}}=0$, and the open string mass formula is ${\alpha }^{\prime }{M}^2=N-a$. This distinguishes the lowest states explicitly:

The supersymmetric GSO projection keeps odd oscillator fermion number in the NS sector. Defining $F_{\mathrm{osc}}$ so that the NS vacuum has $F_{\mathrm{osc}}=0$, its projector isIt removes the scalar tachyon and keeps the massless vector. In the R sector, fermion parity contains the zero-mode chirality operator, so one may writeHere $F_{\mathrm{osc}}$ counts nonzero-mode fermionic excitations. On the ground states, this keeps one Majorana-Weyl spinor chirality: 16 real covariant components, or eight on-shell polarizations after the Dirac equation. The projected massless open-string states are an eight-polarization gauge vector and an eight-polarization gaugino. The compatible projected tower exhibits spacetime supersymmetry; unprojected worldsheet supersymmetry by itself did not eliminate the tachyon or equalize these massless counts.

For a closed string, both chiral zero-mode constraints must hold:This shifted closed-string level matching is particularly important in the mixed sectors. The lowest states and their spacetime nature are as follows.

For type IIA superstring theory, the two R sector ground-state chiralities are opposite. The Ramond–Ramond potentials can be represented by $C_1,C_3$, with transverse counts $8+56=64$. The two mixed-sector gravitinos have opposite chiralities, and the ten-dimensional theory is nonchiral. For type IIB superstring theory, the two R sector chiralities agree; the Ramond–Ramond potentials are $C_0,C_2,C_4$, with self-dual five-form field strength, giving $1+28+35=64$. Its two gravitinos have the same chirality. Thus both projected type II theories have 128 massless bosonic and 128 massless fermionic polarizations and no tachyon, while their chiralities and form spectra differ.

If unoriented open strings are included in the complete theory, the supersymmetric example is type I string theory. Worldsheet orientation reversal projects the closed type IIB superstring theory spectrum and identifies the two mixed sectors; the surviving closed fields include the graviton, dilaton, a Ramond–Ramond potential $C_2$, and one gravitino and dilatino. The open states form vector and gaugino multiplets with Chan-Paton factors; tadpole and anomaly cancellation select $SO(32)$. This is an additional orientation projection, not part of the GSO projection itself. The heterotic string instead pairs one supersymmetric chiral side with a bosonic chiral side carrying an internal gauge sector; it does not have a Ramond choice on both sides.

Use a mostly-plus Minkowski metric and string tension $T=(2\pi {\alpha }^{\prime }{)}^{-1}$. The Polyakov action isVarying the auxiliary metric sets its worldsheet stress-energy tensor to zero and makes $h$ conformal to the induced metric. Substitution gives the equivalent Nambu–Goto action $-T\int \sqrt{-\det ({\partial }_{a}X\cdot {\partial }_{b}X)}$. The string coupling is separate from $T$. A constant dilaton contributes $S_{\Phi }={\Phi }_0\chi (\Sigma )$ to the Euclidean action, by the Gauss-Bonnet theorem, including its boundary term when necessary. With $g_s=e^{{\Phi }_0}$, a connected worldsheet has weight $g_s^{-\chi }$. For an oriented closed surface of genus $g$, this is $g_s^{2g-2}$; with $b$ boundaries it is $g_s^{2g+b-2}$. Normalizing each external closed string vertex with a factor $g_s$ gives $g_s^{2g+M-2}$ for $M$ external closed strings. This is the string genus expansion.

The local gauge symmetries are worldsheet diffeomorphisms, under which $X$ is a scalar field, and Weyl transformations $h_{ab}\mapsto e^{2\omega (\sigma )}{h}_{ab}$. Two coordinate functions and one Weyl function locally fix the three components of the two-dimensional metric, giving conformal gauge $h_{ab}=e^{2\omega }{\eta }_{ab}$ and then $\omega =0$. Globally one instead chooses a fiducial metric and integrates over the worldsheet moduli; conformal Killing transformations can remain. Target-space Poincaré transformations are global symmetries, not additional local gauge freedoms.

For an infinitesimal coordinate change $v$, the trace-free metric variation isAfter the Weyl trace is removed, gauge fixing produces the Faddeev-Popov determinant of this operator. An anticommuting vector $c^a$ and symmetric traceless tensor $b_{ab}$ represent that determinant through a Grassmann Gaussian integral. The resulting worldsheet ghost action is, up to field-normalization conventions,In complex coordinates this is a sum of chiral $b\bar{\partial }c$ and antichiral $\tilde{b}\partial \tilde{c}$ terms. The worldsheet ghost fields have conformal weights two and minus one. Their central charge is $-26$, while the $d$ free embedding bosons contribute $d$. In the critical dimension the total anomaly cancels, consistently with the stated nilpotence of the BRST charge.

Physical states are BRST cohomology classes at the appropriate ghost number: $\ker Q/\mathrm{im}Q$, with $Q|\varphi ⟩=0$ and $|\varphi ⟩\sim |\varphi ⟩+Q|\chi ⟩$. Nilpotence ensures that an exact state is closed and that the equivalence is well-defined. For the usual open string representatives the physical ghost number is one. BRST-exact excitations describe gauge or null directions rather than additional physical particles.

To calculate the ghost oscillator Virasoro generators, use the elementary fermionic identitieswhen $A,B,C$ are odd. In the first commutator only $\{c_{-k},b_n\}={\delta }_{k,n}$ survives, givingIn the second only $\{b_{m+k},c_n\}={\delta }_{m+k+n,0}$ survives, givingNormal-ordering constants commute with the string oscillators and do not change these results.

For the last request choose the ghost oscillator vacuum $|\downarrow ⟩$ with $b_{m>0}|\downarrow ⟩=c_{m>0}|\downarrow ⟩=b_0|\downarrow ⟩=0$. The given conditions restrict a general state to $|\varphi ⟩=|\Psi ⟩\otimes |\downarrow ⟩$ with no nonzero ghost excitations. Indeed, the positive modes annihilate both members of each creation–annihilation pair, and $b_0$ selects one member of the ghost zero-mode doublet. Use the ghost Virasoro zero-mode convention compatible with the displayed explicit $-c_0$ term. Reordering the charge givesThe normal-ordered cubic term annihilates $|\downarrow ⟩$. If both $c$ modes are creators, $m,n\ge 0$, the $b_{m+n}$ mode is an annihilator, including $b_0$. If either $c$ is an annihilator, normal ordering puts an annihilator on the right. ConsequentlyThese ghost states are independent. Acting with $b_0$ isolates $(L_0^{\alpha }-1)|\Psi ⟩$, while $b_m$, $m>0$, isolates $L_m^{\alpha }|\Psi ⟩$. Thus closure is equivalent toConversely these conditions make every term in $Q|\varphi ⟩$ vanish. They are exactly the physical-state Virasoro conditions for an open string; the cohomology quotient additionally identifies physically equivalent representatives.

Write the integrand, without its common overall factor, as $\mathcal{L}={\partial }_{+}X\cdot {\partial }_{-}X-{\psi }_1\cdot {\partial }_{-}{\psi }_1-{\psi }_2\cdot {\partial }_{+}{\psi }_2$. Keep each constant Grassmann variable on the left. For the first supersymmetry component, varying the bosonic term givesVarying its fermion term gives $-{\epsilon }_1{\partial }_{+}X\cdot {\partial }_{-}{\psi }_1+{\epsilon }_1{\psi }_1\cdot {\partial }_{-}{\partial }_{+}X$. The plus sign in the second term comes from moving the odd parameter past ${\psi }_1$. The middle terms cancel and the rest form ${\epsilon }_1{\partial }_{+}({\psi }_1\cdot {\partial }_{-}X)$. The second supersymmetry component similarly gives ${\epsilon }_2{\partial }_{-}({\psi }_2\cdot {\partial }_{+}X)$. Hence, off shell,This proves the local total-derivative identity for rigid worldsheet supersymmetry; the action is invariant when the corresponding boundary flux vanishes and the transformation preserves the allowed fields. A constant parameter preserves a periodic boson and a periodic fermion. In a Neveu–Schwarz sector, however, $\delta X=\epsilon \psi$ is antiperiodic for constant $\epsilon$ and does not preserve the periodic boson. Thus the bulk identity does not supply a global constant supercharge in every spin sector. This is the spin-structure obstruction to constant worldsheet supersymmetry; the local superconformal formulation handles the different sectors consistently.

For independent field variations, integration by parts gives bulk termsThe equations of motion are thereforeFor variations vanishing at the temporal ends, the spatial seam term is proportional to the difference between its values at $\sigma =2\pi$ and $\sigma =0$ ofThe bosonic part cancels for closed string periodicity, also in a fixed winding sector. For a real fermion impose ${\psi }_a(2\pi )=s_a{\psi }_a(0)$ and the same relation on its variation. Seam cancellation requires $s_a^2=1$, so the two untwisted possibilities are periodic, $s_a=+1$ (Ramond sector), or antiperiodic, $s_a=-1$ (Neveu–Schwarz sector). The two chiralities of a closed string can choose these signs independently, giving R–R, NS–NS, R–NS and NS–R sectors. This discussion assumes the ordinary Lorentz-invariant untwisted boundary conditions of the flat theory.

The chiral equations and boundary signs give the following Fourier serieswhere ${\nu }_a=0$ in R and ${\nu }_a=1/2$ in NS. Thus integer or half-integer modes follow directly from the Fourier series monodromy. With the conventional string oscillator normalization their anticommutation relations and reality conditions areFor an open string the endpoint gluing identifies the two sets of coefficients, leaving one R or NS tower.

In ten-dimensional light-cone gauge in string theory, the eight transverse bosonic modes have positive integer level and each NS fermionic creator has a positive half-integer level. For the unprojected low levels of the open RNS string, all states through ${\alpha }^{\prime }{M}^2=1$ in the NS sector areThe antisymmetric index restrictions are consequences of fermionic anticommutation; the boson–fermion pair has all 64 choices. The displayed partitions exhaust levels at most $3/2$. Here the NS normal-ordering constant of a string is $a=1/2$.

In the R sector $a=0$, and the zero modes satisfy $\{{\psi }_0^i,{\psi }_0^j\}={\delta }^{ij}$. Thus ${\gamma }^i=\sqrt{2}{\psi }_0^i$ generate the Ramond zero-mode Clifford algebra. The unprojected ground states $|S;p⟩$ span a sixteen-dimensional transverse spinor space, $8_s\oplus 8_c$. All states in the requested range areThere is no intervening half-level, since all nonzero R modes have integer level. Acting with the zero modes only moves within the ground spinor space and does not generate further independent copies.

The GSO projection keeps odd NS fermion-excitation parity, removing the tachyonic vacuum and the entire $N=1$ NS level. In R it keeps one eigenvalue of total fermion parity, whose zero-mode part is spinor chirality. It leaves eight massless spinor states. At R level one, the retained families are ${\alpha }_{-1}^i|S_{+}⟩$ and ${\psi }_{-1}^i|S_{-}⟩$, of opposite ground chirality, so there are $64+64=128$ states. The retained NS states are spacetime bosons and the R states spacetime fermions. Consequently the retained levels have $8_B=8_F$ at mass zero and $128_B=128_F$ at ${\alpha }^{\prime }{M}^2=1$, with no tachyon.

Equality at every higher mass level follows from the standard Jacobi abstruse identity, or equivalently the equality of the transverse generating functionsThe left side counts the projected NS bosons, the right side the projected R fermions. Their common coefficients start $8+128q+\cdots$. This consistent truncation yields spacetime supersymmetry; the bulk worldsheet symmetry alone would not remove the tachyon or impose the projection.

The first massless level of a closed string contains a spin-two field, an antisymmetric two-form and a scalar: the graviton, Kalb–Ramond field and dilaton. The polarization redundancies found in Question 2 become, in position space, the linearized transformations $\delta G_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$ and $\delta B_{\mu \nu }={\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }$. Thus the massless spin-two field already carries the gauge symmetry of linearized general relativity.

To describe slowly varying backgrounds, promote these polarizations to fields of the embedding coordinates. The Euclidean string nonlinear sigma model isHere ${ϵ}^{ab}$ is the antisymmetric coordinate density. Quantum consistency requires worldsheet Weyl anomaly cancellation, not merely the classical background equations. At first order in ${\alpha }^{\prime }$ the bosonic sigma-model Weyl anomaly coefficients givewhere $H=dB$ is the Kalb-Ramond field strength. The scalar anomaly condition, combined with the trace of the metric condition in the critical dimension, givesThese are the Euler–Lagrange equations of the leading string-frame massless effective actionFor example, variation of $B$ gives the divergence equation above. Varying $\Phi$ and integrating its kinetic variation by parts gives $R+4{\nabla }^2\Phi -4(\nabla \Phi {)}^2-H^2/12=0$. Combining this with the metric variation yields the Ricci equation. Thus the string's quantum conformal consistency reproduces target-space gravity coupled to matter. Equivalently, expanding string scattering amplitudes at small momenta gives the same action: massless exchange poles are reproduced by these fields, while analytic terms from massive exchange become local higher-derivative interactions.

The apparent dilaton-dependent coefficient of $R$ is a frame choice. Put $\varphi =\Phi -{\Phi }_0$ and, in dimension $D>2$, define the Einstein-frame metricTo see the cancellation explicitly, write $G=e^{2\omega }{g}^{(E)}$, $\omega =2\varphi /(D-2)$. ThenThe Laplacian term is a boundary term. Combining the last term with the original positive dilaton kinetic term produces $-4({\nabla }_E\varphi {)}^2/(D-2)$. The three inverse metrics in $H^2$ give its remaining exponential factor. HenceFor the uncompactified bosonic theory $D=26$, these coefficients are $-1/6$ and $e^{-\varphi /3}$. The Ricci term is now the Einstein-Hilbert action; its metric variation gives the Einstein field equations with stress-energy tensor from the scalar and two-form. The leading low-energy theory is general relativity coupled to the massless string fields.

The approximation requires characteristic momenta $E\sqrt{{\alpha }^{\prime }}\ll 1$ and small curvature in string units. Massive string modes have masses of order $1/\sqrt{{\alpha }^{\prime }}$ and can be integrated out, generating corrections with extra powers of ${\alpha }^{\prime }{E}^2$ or ${\alpha }^{\prime }R$. The string genus expansion supplies independent loop corrections controlled by $g_s=e^{{\Phi }_0}$; at weak coupling tree-level gravity dominates. Dimensional analysis gives ${\kappa }_{26}^2\propto g_s^2({\alpha }^{\prime }{)}^{12}$.