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Each of $D$ embedding bosons contributes one to the chiral central charge, and each real worldsheet Majorana fermion contributes one half. The reparameterization bc system contributes $-26$, and the superconformal ghosts contribute $+11$. Vanishing total worldsheet Weyl anomaly therefore requires $3D/2-26+11=0$, or $D=10$. This differs from the critical dimension 26 of the bosonic string, which has no worldsheet fermions or superconformal ghosts.

Introduce anticommuting matter fields ${\psi }^{\mu }$ as worldsheet Majorana fermions, transforming as target-spacetime vectors and two-dimensional spinors. The RNS string pairs them with $X^{\mu }$ through worldsheet supersymmetry. In flat superconformal gauge the matter action is, up to the conventional fermion normalization,and the field equations split $\psi$ into left- and right-moving components. The supersymmetry constraints include both the stress tensor and the supercurrent, so unphysical polarizations are removed rather than interpreting every fermionic worldsheet component as a physical particle.

The possible spin structures produce two sectors. On the closed chiral circle, or the doubled open string, the Neveu–Schwarz sector has $\psi (\sigma +2\pi )=-\psi (\sigma )$ and half-integer modes $b_r$. The Ramond sector has periodic $\psi$ and integer modes $d_n$, including $d_0$. Canonical quantization replaces the classical Grassmann variables by operators satisfyingIn particular ${\Gamma }^{\mu }=\sqrt{2}{d}_0^{\mu }$ obeys $\{{\Gamma }^{\mu },{\Gamma }^{\nu }\}=2{\eta }^{\mu \nu }$. This is the Ramond zero-mode Clifford algebra in covariant form. Its ground-state space must carry a representation of the Clifford algebra of the target metric, and hence transforms as a spinor, not a scalar vacuum. Indeed the zero-mode Lorentz generators proportional to $[{\Gamma }^{\mu },{\Gamma }^{\nu }]/4$ act on it as the spin representation. This is the essential mechanism converting anticommuting worldsheet matter into spacetime spinorial states.

The zero-mode supercurrent constraint on a Ramond ground state is proportional to $k\cdot d_0$. It becomesthe momentum-space Dirac equation. The Ramond level operator has zero intercept, so the ground state is massless. Exciting bosonic or nonzero fermionic oscillators gives tensor-spinor excitations; the Clifford ground-state factor is retained. In the Neveu–Schwarz sector there is no fermionic zero-mode degeneracy. Its ground state is a scalar and its excited states carry integer-spin tensor representations, despite the anticommuting oscillator construction. Thus worldsheet fermionic oscillators do not by themselves make every target-space state a fermion.

Consistent projection also matters. The GSO projection removes the NS tachyon and selects a definite Ramond chirality in the standard supersymmetric spectrum. At the critical dimension ten, the projected massless NS vector has eight transverse polarizations, while the projected Ramond ground state has eight physical spinor polarizations after the massless Dirac constraint and the Majorana-Weyl spinor conditions. The projection permits their organization into spacetime supersymmetry multiplets. The assignment of spacetime fermionic statistics is compatible with spinorial transformation, vertex-operator locality and the Spin-statistics theorem; it is not just a relabeling of the Grassmann variables in the classical action.

For a closed string the left and right sectors can be chosen independently subject to level matching and the consistent spin-structure sum. The NS-NS sector and Ramond–Ramond sector have bosonic spacetime representations; a pair of Ramond spinors produces integer-spin fields. The mixed NS-R sector and R-NS sector instead carry one spinor factor and yield spacetime fermions, including gravitino and dilatino states. Opposite versus equal chiralities of the two Ramond sectors distinguish the type IIA superstring theory and type IIB superstring theory choices.

Quantum consistency of this RNS matter-plus-gauge system requires $d=10$: its matter central charge is $d+d/2$, the reparameterization ghosts contribute $-26$ and the commuting superconformal ghosts contribute $11$, so the total is $3d/2-15$. Spacetime fermions arise through Ramond matter zero modes, their Clifford/spinor representations and the physical-state/projection conditions. The anticommuting $b,c$ gauge-fixing ghosts already present in the bosonic path integral are not such matter fields and do not create physical spacetime fermions.

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.

Take ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$ on the strip $0\le \sigma \le \pi$. Varying the Grassmann-odd worldsheet Majorana fermions gives ${\partial }_{-}{\psi }_1=0$, ${\partial }_{+}{\psi }_2=0$ and, up to the orientation of the strip boundary, the surface termAt each endpoint it vanishes if ${\psi }_1=\eta {\psi }_2$ with $\eta =\pm 1$, with the same relation on variations. A common sign can be absorbed by redefining one chiral field, so set ${\eta }_0=+1$. Since ${\psi }_1=f(\tau +\sigma )$ and ${\psi }_2=f(\tau -\sigma )$, the second endpoint imposes $f(u+2\pi )={\eta }_{\pi }f(u)$. Equal endpoint signs give the periodic Ramond sector, and opposite endpoint signs give the antiperiodic Neveu–Schwarz sector:These gluing choices preserve the Neumann-endpoint worldsheet supersymmetry conditions and give integer or half-integer worldsheet Majorana fermion modes, respectively.

First fix a normalization issue in the original PDF. With its stated canonical string oscillator algebra, the standard superconformal current mode iswithout the printed factor $1/2$. To check the Ramond supercurrent zero-mode square directly, pair positive and negative integer modes:The zero-mode Clifford relations give $({\alpha }_0\cdot {\psi }_0{)}^2={\alpha }_0^2/2$. Different $Q_m$ anticommute, and so do $Q_m$ with the zero-mode term. The two same-sign terms in $Q_m^2$ vanish by antisymmetry of the fermions and symmetry of the bosonic product. The cross terms giveThe second term comes from commuting ${\alpha }_m$ through ${\alpha }_{-m}$, so the frequency factor $m$ is essential. ThereforeHere $L_0^R$ uses vacuum-annihilating Ramond normal ordering; its string intercept is zero. A plane-to-cylinder central shift must be included consistently if a different zero-mode convention is used.

For the mode $F_r=G_r/2$ actually printed, the algebra instead gives $\{F_0,F_0\}=L_0^R/2$. Already the oscillator-free term gives $\{F_0,F_0\}={\alpha }_0^2/4$, whereas $2L_0^R={\alpha }_0^2$. Thus the requested identity is not valid with both that prefactor and the supplied string oscillator algebra. Removing the prefactor gives the identity just derived. Renaming $L_0^R/4$ as $L_0$ would fix this isolated equality but would spoil the usual Virasoro normalization and the subsequent mass formula. We use the standard normalized modes $G_r$ in the remaining physical constraints; their vanishing is unaffected by a common nonzero rescaling.

The RNS string physical-state conditions in the two sectors areThe last condition is the fermionic zero-mode constraint; its square implies the Ramond mass-shell condition. Physical null string states are quotiented. Matter central charge is $3d/2$, while the reparameterization worldsheet ghost fields and superconformal ghosts contribute $-26+11=-15$. Thus $d=10$ is the critical dimension of the RNS superstring, with $a_{NS}=1/2$ and $a_R=0$.

To solve the constraints, choose $X^{\pm }=(X^0\pm X^9)/\sqrt{2}$, so ${\eta }_{+-}=-1$, and work on a patch with ${\alpha }_0^{+}=\sqrt{2{\alpha }^{\prime }}{p}^{+}\ne 0$. In light-cone gauge in string theory fix all nonzero ${\alpha }_n^{+}$ and all ${\psi }_r^{+}$ to zero. For the Ramond sector, the printed restriction to nonzero fermion modes is not by itself complete: one must also reduce the longitudinal fermionic zero modes using the zero-mode supersymmetry constraint. The usual reduced light-cone description sets the remaining ${\psi }_0^{+}$ to zero as well and retains the transverse Clifford algebra. Covariantly this reduction is the Dirac equation on the ground-state particle polarization; it is not an extra unconstrained canonical string oscillator.

With this reduction the string oscillator generators have the formRepeated $i$ runs over the eight transverse directions. The coefficient in $L_n^{\perp }$ comes from the fermion worldsheet stress tensor $-\psi \partial \psi /2$; for $n=0$, positive and negative $r$ combine to give the occupation energy $r{\psi }_{-r}{\psi }_r$. Thus the light-cone super-Virasoro constraints are explicitly solved byHere $a$ is $1/2$ in the NS sector and zero in the R sector. The zero-mode equation says $2p^{+}{p}^{-}=p_i^2+(N-a)/{\alpha }^{\prime }$. All longitudinal excitations are consequently determined by transverse bilinears. The critical dimension and string intercepts are necessary for the resulting quantum theory to preserve the target Lorentz algebra.

The frequency-weighted Neveu–Schwarz level operator isThe PDF's displayed number operator omits the fermion frequency weights and the $r=1/2$ term. With the stated anticommutators that cannot be a level operator: it would assign no level to the first fermionic excitation. The bosonic term already includes its frequency through $[{\alpha }_n,{\alpha }_{-n}]=n$, whereas the fermionic term must include its frequency explicitly.

One independent way to obtain the string intercept is the transverse zero-point energy. Each periodic boson contributes $\frac{1}{2}{\sum }_{n\ge 1}n=-1/24$. Each antiperiodic real fermion contributes $-\frac{1}{2}{\sum }_{r\ge 1/2}r=-1/48$, with the minus sign due to fermionic normal ordering. For example a common exponential cutoff givesSubtracting the local divergent terms leaves $8(-1/24-1/48)=-1/2$, so the NS string intercept is $a=1/2$. In the Ramond sector the integer fermions instead contribute $+1/24$ per component and cancel the bosonic contribution, giving $a_R=0$.

The unprojected NS sector oscillator vacuum has $N=0$ andIt is a spacetime scalar tachyon. The first excitation ${\psi }_{-1/2}^i|0;p⟩$ has $N=1/2$ and is massless, with eight vector particle polarizations. Covariantly its particle polarization obeys $p\cdot ϵ=0$ modulo the longitudinal null particle polarization, leaving the same eight degrees of freedom.

The R sector oscillator vacuum is also massless, but it is degenerate because $\{{\psi }_0^{\mu },{\psi }_0^{\nu }\}={\eta }^{\mu \nu }$. The operators ${\Gamma }^{\mu }=\sqrt{2}{\psi }_0^{\mu }$ give the target Clifford algebra, and $G_0|u;p⟩=0$ becomes $p_{\mu }{\Gamma }^{\mu }u=0$. At nonzero null momentum this Dirac equation leaves sixteen unprojected on-shell spinor states. Equivalently the eight transverse zero modes give the sixteen-dimensional spin representation $8_s\oplus 8_c$ of $SO(8)$. Thus the Ramond ground state describes spacetime fermions, not a single scalar oscillator vacuum.

To define the GSO projection explicitly, let $f_{NS}$ count excited NS fermions and give its vacuum even parity. ThenIt retains odd fermion number, removes the tachyon, and retains the eight massless vector particle polarizations. In the Ramond sector let ${\Gamma }_{\ast }$ be transverse chirality and $f_R$ count nonzero-mode fermions. Choose one consistent chirality sign $s=\pm 1$ and useEach operator squares to itself. At the ground level $P_R$ retains one eight-dimensional spinor chirality, so the retained spectrum has eight massless bosonic and eight massless fermionic states. The parity convention can instead assign the NS vacuum odd parity and write a plus projector; the retained states are identical.

Equality continues at every mass level. Let $q$ count ${\alpha }^{\prime }{M}^2$, suppressing common momentum and endpoint-label factors, and set $D(q)={\prod }_{n\ge 1}(1-q^n{)}^8$. The projected string oscillator generating functions areThe Ramond factor eight follows because either chirality projector selects eight ground states for every nonzero-oscillator parity. The Jacobi abstruse identity makes the bracket equal to $16q^{1/2}{\prod }_{n\ge 1}(1+q^n{)}^8$, soThis proves level-by-level equality of bosonic and fermionic multiplicities. The appropriate spacetime supercharge from an RNS spin field relates the two sectors; the projection ensures that it acts within the retained state space. These sector and chirality conventions agree with the primary superstring lecture notes. Equal total counts alone are not the definition of the projection; they follow from the stated parity selection and the string oscillator identity.

Take ${\eta }^{\alpha \beta }=\mathrm{diag}(-1,1)$ with coordinate zero equal to $\tau$ and coordinate one to $\sigma$. In the supplied representation setThey satisfy the Clifford algebra $\{{\rho }^{\alpha },{\rho }^{\beta }\}=2{\eta }^{\alpha \beta }1$. For anticommuting spinor components and an ordinary matrix $B$, relabelling the two summed indices givesNow $C^T=-C$, whereas $C{\rho }^0=-1$ and $C{\rho }^1=\mathrm{diag}(-1,1)$ are symmetric. Therefore the Majorana Grassmann bilinear interchange yieldsThese signs require fermionic, Grassmann-odd fields; commuting spinors would have the opposite interchange signs.

In flat conformal gauge, suppress the common factor $(4\pi {\alpha }^{\prime }{)}^{-1}$ and write the combined density as $\mathcal{L}=-{\partial }_{\alpha }X\cdot {\partial }^{\alpha }X+\bar{\psi }\cdot {\rho }^{\alpha }{\partial }_{\alpha }\psi$. The supersymmetry variation including its odd parameter is an even derivation. From $({\rho }^{\alpha }{)}^{T}C=-C{\rho }^{\alpha }$ we obtainAlso $(C{\rho }^{\alpha }{\rho }^{\beta }{)}^T=-C{\rho }^{\beta }{\rho }^{\alpha }$, so interchanging odd spinors gives $\bar{\chi }{\rho }^{\alpha }{\rho }^{\beta }ϵ=\overline{ϵ}{\rho }^{\beta }{\rho }^{\alpha }\chi$. The two density variations, with target contractions understood, are thereforeIn the second term the symmetric second derivative annihilates the antisymmetric part of ${\rho }^{\beta }{\rho }^{\alpha }$. Using $2{\eta }^{\alpha \beta }-{\rho }^{\beta }{\rho }^{\alpha }={\rho }^{\alpha }{\rho }^{\beta }$ now gives the exact off-shell identityThus the action is invariant when this boundary flux vanishes and the transformation preserves the field boundary conditions, for example with suitable falloff or compatible supersymmetric endpoint conditions. On a spatial circle, periodic fields admit the constant parameter; antiperiodic fermions with periodic bosons require the local superconformal treatment, since a nonzero constant parameter would make the boson variation antiperiodic. The constant parameter and ordinary derivatives describe rigid supersymmetry in flat gauge; a locally supersymmetric curved-worldsheet formulation includes the worldsheet gravitino. The result is the rigid worldsheet supersymmetry boundary term without using field equations.

For the commutator on $X$, keep the order of the odd parameters:For the fermion, the order is equally important. Moving ${ϵ}_2$ through both odd factors in $({\overline{ϵ}}_1{\partial }_{\alpha }\psi ){ϵ}_2$ incurs two minus signs. ConsequentlyTo check the spinor matrix explicitly, set ${ϵ}_1=(a,b{)}^T$, ${ϵ}_2=(c,d{)}^T$, with all four components anticommuting and monomials ordered $a,b,c,d$. ThenDirect multiplication gives ${\rho }^{\alpha }M+M{\rho }^{\alpha }=2V^{\alpha }1$. HenceVariation of the fermion action gives the Dirac equation ${\rho }^{\alpha }{\partial }_{\alpha }\psi =0$. On this equation the second term vanishes, provingThis is on-shell closure of rigid worldsheet supersymmetry; the action's invariance was off shell, but its fermionic transformation algebra closes in this field content only on shell.

The associated superconformal current is proportional to $\psi \cdot \partial X$, of conformal weight $3/2$. Its fermionic modes $G_r$ extend the stress-tensor generators to the N=1 super-Virasoro algebra:The modes are half-integral in the Neveu–Schwarz sector and integral in the Ramond sector. Their anticommutator giving a Virasoro generator is the conformal-mode counterpart of the translation commutator just derived. Ten coordinate bosons and ten Majorana fermions give matter central charge $c=10+10/2=15$, canceled by the usual reparametrization and superconformal ghosts in the critical RNS string.

A Majorana spinor equals its transform under charge conjugation, so its spinor components are not independent of their conjugates. In a real two-dimensional gamma matrix representation it can be taken real, with anticommuting Grassmann variables as components. Each target index labels a separate worldsheet Majorana fermion; it is not itself a target-space spinor.

Use worldsheet signature $(-,+)$ and the real gamma matricesTake $\bar{\psi }={\psi }^{T}C$ with $C={\gamma }^0$. Define the chirality matrix with the chosen worldsheet orientation byIt squares to one and anticommutes with each gamma matrix. Chirality is its eigenvalue, selected by the chiral projectors. Its overall sign is conventional.

Writing $\psi =({\psi }_{+},{\psi }_{-}{)}^T$, the fermion equation of motion ${\gamma }^{\mu }{\partial }_{\mu }\psi =0$ becomesHenceThe first profile travels toward increasing $\sigma$, and the second toward decreasing $\sigma$. Thus the choice of chirality orientation realizes the requested worldsheet chirality and propagation direction correspondence; reversing the orientation interchanges the labels.

For the rigid worldsheet supersymmetry variation, let $ϵ$ be a constant Grassmann-odd Majorana spinor. With the displayed conventions,The minus sign follows from $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$. A second identity from the Majorana Grassmann bilinear interchange is $\bar{\chi }{\gamma }^{\nu }{\gamma }^{\mu }ϵ=\overline{ϵ}{\gamma }^{\mu }{\gamma }^{\nu }\chi$. Keeping these Grassmann signs is essential.

The bosonic kinetic variation is $2i{\partial }_{\mu }{X}_a\overline{ϵ}{\partial }^{\mu }{\psi }^a$. The two fermionic kinetic variations, using the two identities above, give the full resultFor the second equality use ${\gamma }^{\nu }{\gamma }^{\mu }=2{\eta }^{\mu \nu }-{\gamma }^{\mu }{\gamma }^{\nu }$ and symmetry of the second embedding derivative. No field equation has been used. Thus the variation is an off-shell total derivative:This is the rigid worldsheet supersymmetry boundary term. It vanishes when the boundary flux vanishes, for example on a closed worldsheet with compatible fields and variations. On a spatial circle, a constant supersymmetry parameter must also respect the chosen spin structure. Periodic fermions allow this rigid transformation; antiperiodic fermions and a constant parameter would give an antiperiodic $\delta X$ and fail to preserve a periodic bosonic embedding. The local action identity remains valid, but global symmetry requires compatible boundary data. This is the spin-structure obstruction to constant worldsheet supersymmetry.

For the spectrum, assume the usual critical RNS string, physical-state constraints, a nonzero null target momentum, and the supersymmetric GSO projection. Analyze just one chiral sector. Its matter central charge is $d+d/2$; the bc system and superconformal ghosts contribute $-26+11$. Therefore $3d/2-15=0$ gives the critical dimension of the RNS superstring $d=10$. Light-cone gauge in string theory leaves eight transverse bosonic and eight transverse fermionic string oscillator directions.

In the Neveu–Schwarz sector, the normal-ordering intercept is $a_{\mathrm{NS}}=1/2$. The massless level is $N=1/2$ and has statesThe GSO projection retains these eight transverse vector polarizations while removing the tachyonic ground state. They are spacetime bosons, although the creating string oscillator is a worldsheet fermion. In covariant language transversality and the longitudinal null-state quotient leave $10-2=8$ polarizations.

In the Ramond sector, bosonic and fermionic zero-point energies cancel, so $a_{\mathrm{R}}=0$ and the ground states are massless. The eight transverse fermion zero modes obeyThis Ramond zero-mode Clifford algebra acts on a sixteen-dimensional ground-state space. Its two chirality subspaces each have dimension eight. The GSO projection keeps one, an $8_s$ or $8_c$ spinor of $\mathrm{SO}(8)$, the rotation subgroup of the massless little group. These are spacetime fermions. Equivalently a ten-dimensional Majorana-Weyl spinor has sixteen real components before the massless Dirac equation reduces the physical polarization count to eight.

Consequently the massless chiral RNS spectrum after GSO projection satisfiesThe GSO projection is essential: without it, the transverse Ramond sector would retain both eight-dimensional spinor chiralities. These are counts for one chiral sector, not the tensor-product counts of the full closed-string spectrum.

The fields constitute an RNS string coupled to two-dimensional worldsheet supersymmetry. The zweibein and worldsheet gravitino impose constraints; they do not supply extra propagating string polarizations. Worldsheet diffeomorphisms, local frame rotations and Weyl transformations put the metric locally in conformal gauge, $h_{\mu \nu }=e^{2\omega }{\eta }_{\mu \nu }$. Local supersymmetry together with super-Weyl symmetry removes the worldsheet gravitino, giving superconformal gauge ${\chi }_{\mu }=0$. On a general closed worldsheet, moduli and spin structures remain and must still be summed or integrated over; gauge fixing is not a declaration that every surface is globally a flat cylinder.

Variation with respect to the metric and worldsheet gravitino before gauge fixing gives the stress-tensor and supercurrent constraints $T_{++}=T_{--}=0$ and $G_{+}=G_{-}=0$. The remaining matter consists of free $X^a$ and their worldsheet fermions. In covariant quantization, these become the N=1 super-Virasoro algebra physical-state conditions: positive Virasoro and supercurrent modes annihilate a physical state, $(L_0-a)|\mathrm{phys}⟩=0$, and the R sector also has the zero-mode condition $F_0|\mathrm{phys}⟩=0$. Null gauge states are quotiented out. Equivalently, physical states are BRST cohomology classes, so BRST-exact operators do not represent additional physical states.

The diffeomorphism $bc$ ghosts have central charge $-26$, while the bosonic superconformal ghosts $(\beta ,\gamma )$ have charge $11$. Matter contributes $D+D/2$. Therefore quantum gauge consistency requiresThis is the critical dimension of the RNS superstring. In light-cone gauge in string theory, choose nonzero $p^{+}$, make $X^{+}$ linear in worldsheet time and set ${\psi }^{+}=0$. The stress-tensor and supercurrent constraints solve for the longitudinal $X^{-}$ and ${\psi }^{-}$ oscillators. Only eight transverse bosons and eight transverse fermions remain, with positive norm. This explicitly eliminates the time-like and longitudinal unphysical states.

In a chiral sector the fermions have half-integral modes in the NS sector and integral modes in the R sector. The normal-ordering constant of a string is $a_{\mathrm{NS}}=1/2$ and $a_R=0$. The GSO projection retains odd fermion-excitation parity in the NS sector, removing its tachyonic vacuum. In the R sector it keeps one chirality of the zero-mode spinor, with oscillator parity included in the projection. The Ramond zero-mode Clifford algebra then leaves eight ground-state polarizations in light-cone gauge in string theory.

Worldsheet supersymmetry alone does not imply spacetime supersymmetry. The GSO projection makes the spinorial worldsheet currents mutually local with the retained vertex operators and pairs their NS and R states. More explicitly, let $S_A$ be an RNS spin field and ${\varphi }_{\mathrm{gh}}$ the bosonized superghost scalar. The spacetime supercharges are generated byand similarly in the other chiral sector. The spin field has conformal weight $10/16=5/8$, and its superghost factor has weight $3/8$, so the current has weight one. Its operator products generate the spacetime translation operator; schematically $\{Q_A,Q_B\}\propto (C{\Gamma }^a{)}_{AB}{P}_a$. This is the spacetime supercharge from an RNS spin field. For type IIA superstring theory, the left and right Ramond projections select opposite ten-dimensional Majorana-Weyl chiralities, giving 32 real supercharges and a nonchiral spacetime theory.

The massless physical states are best counted with the transverse little group $\mathrm{Spin}(8)$. Choose the left Ramond ground representation $8_s$ and the right one $8_c$. ThenThe NS-NS sector supplies the graviton, Kalb–Ramond field and dilaton, with $35$, $28$ and $1$ polarizations. The RR sector supplies a one-form potential and a three-form potential, with $\left(\genfrac{}{}{0ex}{}{8}{1}\right)=8$ and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$ polarizations. The mixed sectors supply two gravitinos of opposite chirality, each with $56$ polarizations, and two dilatinos, each with $8$. Covariantly the dilatino has chirality opposite to its corresponding supersymmetry parameter; the two chirality sets are both present. Thus the massless type IIA spectrum hasThese fields form the massless type IIA supergravity multiplet.

For the first massive level of a chiral RNS sector, restore ${\alpha }^{\prime }$ throughThe first positive value is one. The NS states therefore have level $N_{\mathrm{NS}}=3/2$. A complete transverse basis after GSO projection isAll contain odd fermion-excitation parity. Their respective dimensions are $8$, $8^2=64$ and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$, so the NS count is $128$. These are the first massive GSO-projected Neveu–Schwarz states. The middle family decomposes into $35_v+28+1$. Together the families assemble into the massive $\mathrm{Spin}(9)$ representations $44+84$: a symmetric traceless rank-two tensor and a three-form, since $44\downarrow \mathrm{Spin}(8)=35_v+8_v+1$ and $84\downarrow \mathrm{Spin}(8)=56_v+28$.

In the R sector, the first massive level is $N_R=1$. If the retained right-moving ground spinor is $8_c$, its two families areThe Ramond spinors have opposite zero-mode chiralities in these two families: inserting one fermionic oscillator reverses the oscillator contribution to the GSO condition, so the second family must use the opposite zero-mode chirality. Each family has $8\times 8=64$ states. HenceThese first massive GSO-projected Ramond states form the gamma-traceless vector-spinor representation of $\mathrm{Spin}(9)$, of dimension $9\cdot 16-16=128$. Its transverse branching is $8_s+56_s+8_c+56_c$. The right-moving sector therefore contains $128$ bosons and $128$ fermions, a massive chiral superstring supersymmetry multiplet.

For the closed string, closed-string level matching requires the left and right shifted levels to agree. Both chiral sectors at this first massive mass have $128$ NS states and $128$ R states. The bosons lie in the NS-NS sector and RR sector, while the mixed sectors are fermionic. ConsequentlyTheir total is $65536=2^{16}$. A massive ten-dimensional state with 32 real supercharges and no central charges has sixteen fermionic creation operators in its rest-frame supersymmetry algebra, giving a long multiplet of this size. Thus the chiral equality, the four closed sectors and level matching are consistent with the first massive type IIA long supermultiplet, rather than merely matching an isolated number of right-moving states.

Use the classically equivalent Polyakov action, fix worldsheet diffeomorphisms and Weyl invariance, and perform a Wick rotation of the string worldsheet. In complex coordinates, the worldsheet ghost action for the closed string isThe worldsheet ghost fields are anticommuting. Their conformal weights are $b:(2,0)$, $c:(-1,0)$ and $\bar{b}:(0,2)$, $\bar{c}:(0,-1)$. The two chiral sectors are counted separately.

In either chiral sector, each embedding coordinate is a free boson conformal field theory with central charge one. For the bc system, putting $J=2$ in $c_{bc}=-2(6J^2-6J+1)$ gives $c_{bc}=-26$. Cancellation of the worldsheet Weyl anomaly requiresThe antiholomorphic sector gives the same equation, rather than doubling the required critical dimension of string theory. This dimension count assumes flat target spacetime with just the embedding fields; more general internal conformal field theories change the matter accounting.

For the spinning string, the worldsheet Majorana fermions have conformal weight $1/2$. At $J=1/2$, the complex anticommuting bc system has $c=1$; a real Majorana spinor contributes half of this. Hence $D$ real worldsheet Majorana fermions contribute $D/2$ per chiral sector. Ramond sector versus Neveu–Schwarz sector changes their mode numbers and vacuum energies, not this local central charge.

The additional local worldsheet supersymmetry has a fermionic gauge parameter, so its superconformal ghosts are commuting fields forming a beta-gamma system, with conformal weights $\beta :3/2$ and $\gamma :-1/2$ in the holomorphic sector. Commuting statistics reverse the sign of the bc system formula, givingTogether with the reparameterization worldsheet ghost fields, their total central charge is $-26+11=-15$. ThusThese cancellations make the string worldsheet gauge symmetries compatible with quantization; in the BRST symmetry formulation they enter the nilpotence condition for the BRST operator.

Local supersymmetry on a string worldsheet relates embedding coordinates to their worldsheet Majorana fermion partners. Fixing this fermionic gauge symmetry introduces commuting superconformal ghosts.