= Solution
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, $(L_0-a)|\mathrm{phys}\rangle=0$, and the <R sector> also has the zero-mode condition $F_0|\mathrm{phys}\rangle=0$. 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 requires
$$
\boxed{\frac{3D}{2}-26+11=0\quad\Longrightarrow\quad D=10.}
$$
This 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 $\phi_{\rm gh}$ the bosonized superghost scalar. The spacetime supercharges are generated by
$$
Q_A=\oint dz\,e^{-\phi_{\rm gh}/2}S_A(z),
$$
and 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> $\operatorname{Spin}(8)$. Choose the left Ramond ground representation $8_s$ and the right one $8_c$. Then
$$
\begin{aligned}
\mathrm{NS\! -\! NS}:&\quad 8_v\otimes8_v=35_v\oplus28\oplus1,\\
\mathrm{R\! -\! R}:&\quad 8_s\otimes8_c=8_v\oplus56_v,\\
\mathrm{NS\! -\! R}:&\quad 8_v\otimes8_c=8_s\oplus56_s,\\
\mathrm{R\! -\! NS}:&\quad 8_s\otimes8_v=8_c\oplus56_c.
\end{aligned}
$$
The <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 $\binom81=8$ and $\binom83=56$ 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> has
$$
\boxed{n_B=35+28+1+8+56=128,\qquad n_F=2(56+8)=128.}
$$
These fields form the massless <type IIA supergravity> multiplet.
For the <first massive level of a chiral RNS sector>, restore $\alpha'$ through
$$
\frac{\alpha'M^2}{4}=N_{\mathrm{NS}}-\frac12=N_R.
$$
The first positive value is one. The NS states therefore have level $N_{\mathrm{NS}}=3/2$. A complete transverse basis after <GSO projection> is
$$
b_{-3/2}^i|0\rangle,\qquad
\alpha_{-1}^i b_{-1/2}^j|0\rangle,\qquad
b_{-1/2}^i b_{-1/2}^j b_{-1/2}^k|0\rangle\quad(i<j<k).
$$
All contain odd fermion-excitation parity. Their respective dimensions are $8$, $8^2=64$ and $\binom83=56$, so \b[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 $\operatorname{Spin}(9)$ representations $44+84$: a symmetric traceless rank-two tensor and a three-form, since $44\downarrow\operatorname{Spin}(8)=35_v+8_v+1$ and $84\downarrow\operatorname{Spin}(8)=56_v+28$.
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 are
$$
\alpha_{-1}^i|s_c\rangle,\qquad d_{-1}^i|s_s\rangle.
$$
The 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 $8\times8=64$ states. Hence
$$
\boxed{n_R=64+64=128=n_{\mathrm{NS}},\qquad M^2=4/\alpha'.}
$$
These <first massive GSO-projected Ramond states> form the gamma-traceless vector-spinor representation of $\operatorname{Spin}(9)$, of dimension $9\cdot16-16=128$. 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. Consequently
$$
\boxed{n_B=2(128)^2=32768,\qquad n_F=2(128)^2=32768.}
$$
Their 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.
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