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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.

Use ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$ and keep each constant odd supersymmetry parameter on the left. The transformation of a fermion may be written $\delta {\psi }_1^{\mu }={ϵ}_1{\partial }_{+}{X}^{\mu }$ or ${\partial }_{+}{X}^{\mu }{ϵ}_1$, since the bosonic derivative commutes with the parameter. The variation $\delta$ itself is even, so it obeys the ordinary product rule; the signs below arise when moving an odd parameter past an odd field.

First consider the ${ϵ}_1$ transformation. Suppress target indices and put $\mathcal{L}=2X_{+}\cdot X_{-}+i{\psi }_1\cdot {\partial }_{-}{\psi }_1+i{\psi }_2\cdot {\partial }_{+}{\psi }_2$. Its bosonic part varies asFor the first fermion, moving ${ϵ}_1$ to the left givesThe second fermion does not vary under ${ϵ}_1$. Since ${\partial }_{-}{X}_{+}={\partial }_{+}{X}_{-}$, these terms combine to ${\delta }_1\mathcal{L}=-i{ϵ}_1{\partial }_{+}({\psi }_1\cdot X_{-})$. The other parameter gives the same cancellation with $+\leftrightarrow -$ and $1\leftrightarrow 2$. ThereforeThis proves worldsheet supersymmetry invariance up to the rigid worldsheet supersymmetry boundary term, without using field equations.

For an open string, the boundary term is part of the invariance statement. For example, a Neumann endpoint has ${\partial }_{+}X={\partial }_{-}X$ and compatible fermion gluing ${\psi }_1=\kappa {\psi }_2$, $\kappa =\pm 1$. The spatial boundary flux vanishes when ${ϵ}_2=\kappa {ϵ}_1$. Both endpoints must admit the same preserved parameter. Opposite gluing signs, giving the Neveu–Schwarz sector, do not preserve a nonzero constant parameter at both ends. Thus the local bulk identity and the existence of a global rigid charge on a particular open-string sector are distinct. The local superconformal symmetry remains the basis of the string constraints; this is consistent with the spin-structure obstruction to constant worldsheet supersymmetry.

Now write $A_{\pm }^{\mu }={\partial }_{\pm }{X}^{\mu }$ and $J_{+}^1={\psi }_1\cdot A_{+}$, $J_{-}^2={\psi }_2\cdot A_{-}$. The other two components vanish. The action implies ${\partial }_{-}{\psi }_1=0$, ${\partial }_{+}{\psi }_2=0$ and ${\partial }_{+}{\partial }_{-}X=0$, so these supercurrents are chiral and conserved locally. At equal time, the given canonical commutation relations implyWith $q_{ϵ}={\pi }^{-1}\int d\sigma ({ϵ}_1{\psi }_1\cdot A_{+}+{ϵ}_2{\psi }_2\cdot A_{-})$, contraction with this commutator givesThe parameter-weighted charge is even. Since the parameters anticommute with the fermions,Using the PDF's normalization $\{{\psi }_i^{\mu }(\sigma ),{\psi }_j^{\nu }({\sigma }^{\prime })\}=\pi {\eta }^{\mu \nu }{\delta }_{ij}\delta (\sigma -{\sigma }^{\prime })$ yieldsThus the charge generates all the specified supersymmetry transformations. Dropping the factor $\pi$ in the fermion anticommutator, as happens in the converted TeX, would fail this check. The original PDF has the correct factor.

To calculate the supercurrent bracket, put $A={\partial }_{+}X$ and $J={\psi }_1\cdot A$. The equal-time bosonic relations giveBosonic and fermionic fields commute. Expanding the two current products, commuting the bosonic factors once and using the fermion anticommutator gives the noncentral partFor an ordinary smooth bilocal coefficient, the distributional derivative identity isFor the fermion bilinear, its coincident classical value, or its coincident normal-ordered value, is zero by antisymmetry against the symmetric target metric. The second term is ${\psi }_1\cdot {\partial }_{\sigma }{\psi }_1$. Since ${\partial }_{-}{\psi }_1=0$, we have ${\partial }_{\sigma }{\psi }_1={\partial }_{+}{\psi }_1$. It follows that the noncentral current bracket isThis is the canonical supercurrent bracket of free RNS matter. The other chirality similarly gives ${\theta }_{--}=({\partial }_{-}X{)}^2+(i/2){\psi }_2\cdot {\partial }_{-}{\psi }_2$ and ${\theta }_{+-}=0$ on shell.

The printed current identity is the classical, or noncentral, part of the algebra. Fully quantized coincident products require normal ordering; the matter algebra then also has a central contact term proportional to ${\delta }^{\prime \prime }$ in the current anticommutator. The 10 bosons and 10 real fermions contribute matter central charge $c=10+10/2=15$. The central term cannot be discarded in a literal quantum operator identity, although it is irrelevant to identifying the local noncentral stress tensor above.

The other brackets close on these same currents. The stress-tensor commutator contains $\theta {\delta }^{\prime }$ and ${\theta }^{\prime }\delta$ and, quantum mechanically, a central ${\delta }^{\prime \prime \prime }$ term. The stress-tensor–supercurrent commutator contains $J{\delta }^{\prime }$ and $J^{\prime }\delta$, expressing the conformal weight $3/2$ of $J$. The supercurrent anticommutator contains $\theta \delta$ and its central ${\delta }^{\prime \prime }$ term. In a conventional normalization of Fourier modes these form the N=1 super-Virasoro algebra:Here $C_{mn}$ and $D_{rs}$ are central terms supported at zero total mode number; their detailed coefficients are not needed for this request. The indices $r$ are half-integers in the Neveu–Schwarz sector and integers in the Ramond sector. Opposite chiralities commute or anticommute as appropriate.

For the closed RNS string, impose a copy of the constraints in each chirality. In the Neveu–Schwarz sector, the physical conditions areIn the Ramond sector, use vacuum-normal-ordered generators with zero Ramond intercept. ThenThe Ramond zero-mode equation is essential: its square gives the mass-shell condition in this convention, $G_0^2=L_0$, and on the oscillator ground sector it is the spacetime Dirac constraint. In a convention whose unshifted plane generator has $G_0^2=L_0-c/24$, the zero-mode equation instead sets that shifted combination to zero; these are the same condition, not competing intercepts.

For either choice, the corresponding tilded constraints hold on the other side. Both chiral mass-shell conditions and closed-string level matching are required:The state sectors are NS–NS, NS–R, R–NS and R–R. Quotienting physical null states removes unphysical longitudinal polarizations; equivalently, use the appropriate BRST cohomology including the superconformal ghosts. A consistent superstring also specifies the GSO projection. It is an additional sector selection, not a replacement for these local super-Virasoro constraints. Physical states satisfy both chiral positive-mode and zero-mode constraints, level matching, and the null-state quotient; Ramond states additionally obey the fermionic zero-mode constraint.

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.