= Solution
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 matrices>
$$
\gamma^0=\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\qquad
\gamma^1=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad
\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}.
$$
Take $\bar\psi=\psi^TC$ with $C=\gamma^0$. Define the <chirality> matrix with the chosen worldsheet orientation by
$$
\gamma_*=-\gamma^0\gamma^1=\begin{pmatrix}1&0\\0&-1\end{pmatrix},
\qquad P_\pm=\frac{1\pm\gamma_*}{2}.
$$
It 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> $\gamma^\mu\partial_\mu\psi=0$ becomes
$$
(\partial_\tau+\partial_\sigma)\psi_+=0,\qquad
(\partial_\tau-\partial_\sigma)\psi_-=0.
$$
Hence
$$
\boxed{\psi_+=F(\tau-\sigma)\text{ is right-moving},\qquad
\psi_-=G(\tau+\sigma)\text{ is left-moving}.}
$$
The 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 $\epsilon$ be a constant Grassmann-odd <Majorana spinor>. With the displayed conventions,
$$
\delta\bar\psi^a=-\bar\epsilon\gamma^\mu\partial_\mu X^a.
$$
The minus sign follows from $(\gamma^\mu)^TC=-C\gamma^\mu$. A second identity from the <Majorana Grassmann bilinear interchange> is $\bar\chi\gamma^\nu\gamma^\mu\epsilon=\bar\epsilon\gamma^\mu\gamma^\nu\chi$. Keeping these Grassmann signs is essential.
The bosonic kinetic variation is $2i\partial_\mu X_a\,\bar\epsilon\,\partial^\mu\psi^a$. The two fermionic kinetic variations, using the two identities above, give the full result
$$
\begin{aligned}
\delta\mathcal L={}&
2i\partial_\mu X_a\,\bar\epsilon\,\partial^\mu\psi^a
-i\bar\epsilon\gamma^\mu\gamma^\nu\partial_\mu X_a\,\partial_\nu\psi^a
+i\bar\epsilon\psi_a\,\partial_\mu\partial^\mu X^a\\
={}&\partial_\nu\!\left(i\bar\epsilon\gamma^\nu\gamma^\mu
\psi_a\,\partial_\mu X^a\right).
\end{aligned}
$$
For 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 \b[variation is an off-shell total <derivative>]:
$$
\boxed{\delta I=\int_{\partial\Sigma}d\Sigma_\nu\,
i\bar\epsilon\gamma^\nu\gamma^\mu\psi_a\partial_\mu X^a.}
$$
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 $a_{\rm NS}=1/2$. The massless level is $N=1/2$ and has states
$$
b_{-1/2}^{\,i}|0;k\rangle_{\rm NS},\qquad i=1,\ldots,8.
$$
The <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 $a_{\rm R}=0$ and the ground states are massless. The eight transverse fermion zero modes obey
$$
\{d_0^i,d_0^j\}=\delta^{ij},\qquad
\Gamma^i=\sqrt2d_0^i,\qquad
\{\Gamma^i,\Gamma^j\}=2\delta^{ij}.
$$
This <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 $\operatorname{SO}(8)$, 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> satisfies
$$
\boxed{N_{\rm bosonic}=8=N_{\rm fermionic}.}
$$
The <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.
Back to article page
