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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.

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.

A Majorana spinor equals its charge conjugation, ${\psi }^c=C{\bar{\psi }}^T=\psi$, with conventional phase choices absorbed into $C$. The two-component worldsheet Majorana fermion therefore has no independent complex conjugate components; in a Majorana representation its components can be real Grassmann variables. The same condition applies to the constant supersymmetry parameter. Grassmann statistics matter in the variation: replacing the spinors by commuting numerical vectors would give the wrong bilinear interchange signs.

The displayed rigid transformation is understood in flat conformal gauge. Take $h_{\mu \nu }={\eta }_{\mu \nu }=\mathrm{diag}(-1,1)$, $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$, and chooseHere $C={\gamma }^0$ is a valid invariant charge-conjugation form: $C^T=-C$ and $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$. These concrete matrices make all signs checkable; equivalent Majorana conventions give the same result with their consistently transformed bars.

Let $A=\sqrt{{\alpha }^{\prime }/2}$ and $B=1/\sqrt{2{\alpha }^{\prime }}$, so ${\alpha }^{\prime }B=A$. Suppressing the common factor $-1/(4\pi {\alpha }^{\prime })$ in the action, the flat kinetic density isThe rigid variations have $\delta X^a=iA\overline{ϵ}{\psi }^a$ and $\delta {\psi }^a=B{\gamma }^{\nu }{\partial }_{\nu }{X}^aϵ$. The supersymmetry variation is even, so its product rule has no extra graded sign. From the Majorana Grassmann bilinear interchange identities,For example the last identity follows by moving the Grassmann-odd parameter through $\psi$, then using $C^T=-C$ and the transpose relations twice. No equation of motion has been used.

The two kinetic variations areThe Clifford algebra turns the first fermionic term plus the bosonic variation into $iA{\partial }_{\nu }{X}_a\overline{ϵ}{\gamma }^{\mu }{\gamma }^{\nu }{\partial }_{\mu }{\psi }^a$. In the second term, commute the partial derivatives and use the same algebra: the antisymmetric gamma product drops out, leaving $iA\square X_a\overline{ϵ}{\psi }^a$. Together they are precisely the rigid worldsheet supersymmetry boundary term:Therefore $\delta S=-(4\pi {\alpha }^{\prime }{)}^{-1}{\int }_{\partial \Sigma }d{\Sigma }_{\mu }{J}^{\mu }$. It vanishes on a closed or periodic worldsheet, for compactly supported changes, or under compatible supersymmetric endpoint conditions. The local total-derivative identity is off shell; the boundary assumptions are needed to call the integrated action invariant.

The flat-gauge qualification is substantive. On an arbitrary curved worldsheet, spinors need a zweibein and spin connection and a constant spinor parameter need not exist. A fully covariant locally supersymmetric action also involves the worldsheet gravitino; the isolated ordinary-derivative expression does not prove rigid invariance for arbitrary $h_{\mu \nu }$. We have proved the intended rigid symmetry of its flat gauge-fixed action, with the fermionic term inside the same integral and overall normalization. If the displayed last term were read as outside the integral, it would not even define an action.

For phenomenology, bosonic string theory has no spacetime fermions and its usual vacuum contains a tachyon. The spinning string has a Ramond sector, whose fermionic zero modes form a spacetime Clifford algebra and give spacetime spinor states, as well as a Neveu–Schwarz sector. In suitable consistent theories the GSO projection removes the tachyonic NS ground state and keeps the appropriate Ramond chirality. The critical dimension of the RNS superstring is 10 rather than 26: $D$ bosons and $D$ real fermions have matter central charge $3D/2$ per chiral sector, while the reparameterization and superconformal ghosts contribute $-26+11=-15$, so the total anomaly cancels at $D=10$.

Suitable GSO-projected superstrings admit spacetime fermions and a tachyon-free spectrum, making them more promising for particle physics than the bosonic string. Rigid worldsheet supersymmetry alone does not establish a tachyon-free spacetime theory or realistic phenomenology. The projection, consistent sectors and compactification in string theory are additional input; spacetime supersymmetry and a realistic four-dimensional spectrum are not automatic consequences of this classical action.