= Solution
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 is
$$
S=-\frac1{4\pi\alpha'}\int d^2\sigma\,
\left(\partial_aX^\mu\partial^aX_\mu-i\bar\psi^\mu\rho^a\partial_a\psi_\mu\right),
$$
where $\rho^a$ 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> $11$. Thus
$$
c_{\mathrm{tot}}=\frac32D-26+11=0
\quad\Longrightarrow\quad
\boxed{D=10}.
$$
In <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 are
$$
\psi(\sigma+2\pi)=+\psi(\sigma)\quad\text{(Ramond)},\qquad
\psi(\sigma+2\pi)=-\psi(\sigma)\quad\text{(Neveu--Schwarz)}.
$$
They 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,
$$
\{\psi_r^\mu,\psi_s^\nu\}=\eta^{\mu\nu}\delta_{r+s,0}.
$$
A <closed string> has independent left- and right-moving periodicities, giving four sector choices rather than two.
The <Ramond zero-mode Clifford algebra> is
$$
\{d_0^\mu,d_0^\nu\}=\eta^{\mu\nu},\qquad
\Gamma^\mu=\sqrt2\,d_0^\mu,\qquad
\{\Gamma^\mu,\Gamma^\nu\}=2\eta^{\mu\nu}.
$$
Hence 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 $\frac12\zeta(-1)=-1/24$. Per fermion it is the negative of the corresponding half-sum. Since $\zeta(-1,1/2)=1/24$, the eight transverse pairs give
$$
E_0^{\mathrm{NS}}=8\left(-\frac1{24}-\frac1{48}\right)=-\frac12,
\qquad
E_0^{\mathrm R}=8\left(-\frac1{24}+\frac1{24}\right)=0.
$$
Thus $a_{\mathrm{NS}}=1/2$, $a_{\mathrm R}=0$, and the <open string> <mass> formula is $\alpha'M^2=N-a$. This distinguishes the lowest states explicitly:
* In the <NS sector>, the level-zero scalar is a <tachyon> with $M^2=-1/(2\alpha')$. At level $1/2$, $\epsilon_\mu b_{-1/2}^\mu|0;p\rangle$ is a massless vector. The supercurrent and <Virasoro constraints> give $p\cdot\epsilon=0$ and the equivalence $\epsilon\sim\epsilon+\lambda p$. There are eight physical vector polarizations.
* In the <R sector>, the level-zero states are massless spacetime <spinors> satisfying the <Dirac equation>. Before the <GSO projection>, the two chiralities together give 16 on-shell polarizations; they split into the two eight-dimensional transverse spinor representations.
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 is
$$
P_{\mathrm{NS}}=\frac12\left(1-(-1)^{F_{\mathrm{osc}}}\right).
$$
It removes the scalar <tachyon> and keeps the massless vector. In the <R sector>, fermion parity contains the zero-mode chirality operator, so one may write
$$
P_{\mathrm R}=\frac12\left(1+\eta\Gamma_{11}(-1)^{F_{\mathrm{osc}}}\right),\qquad \eta=\pm1.
$$
Here $F_{\mathrm{osc}}$ 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>. \b[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:
$$
\boxed{M^2=\frac4{\alpha'}(N_L-a_L)=\frac4{\alpha'}(N_R-a_R)},\qquad
\boxed{N_L-a_L=N_R-a_R}.
$$
This shifted <closed-string level matching> is particularly important in the mixed sectors. The lowest states and their spacetime nature are as follows.
* The <NS-NS sector> is bosonic. Before projection, $(N_L,N_R)=(0,0)$ is a scalar <tachyon> with $M^2=-2/\alpha'$. The <GSO projection> on each side removes it. The surviving lowest level is $(1/2,1/2)$, with transverse states $b_{-1/2}^i\widetilde b_{-1/2}^j|0;p\rangle$. Decomposing their tensor product into symmetric traceless, antisymmetric and trace parts gives the <graviton>, <Kalb–Ramond field> and <dilaton>, with $35+28+1=64$ physical polarizations.
* The <RR sector> is bosonic, since it combines two spacetime <spinors>. Its lowest states have $(N_L,N_R)=(0,0)$ and are massless. The two <Dirac equations> on the bispinor polarizations identify them with on-shell differential-form field strengths. After projection they supply 64 bosonic polarizations. The actual form degrees depend on the two chosen <Ramond sector> chiralities.
* The <NS-R sector> and <R-NS sector> are fermionic. Their respective lowest level-matched pairs are $(1/2,0)$ and $(0,1/2)$, not two oscillator vacua. A vector tensored with a chiral spinor decomposes into its gamma-traceless vector-spinor and gamma trace. These give a massless <gravitino> and <dilatino>, with $56+8=64$ polarizations in each mixed sector. The vacuum pair would fail the shifted <closed-string level matching>, so it cannot be counted as a mixed-sector tachyon.
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 \b[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.
Back to article page
