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An oscillator vacuum is annihilated by all positive-frequency oscillator annihilation operators. For a Ramond sector string this leaves a degenerate space on which the fermionic zero modes act through the Ramond zero-mode Clifford algebra. Vacuum annihilation therefore does not select a unique spinor state.

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, $(L_0-a)|\mathrm{phys}⟩=0$, and the R sector also has the zero-mode condition $F_0|\mathrm{phys}⟩=0$. 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 requiresThis 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 ${\varphi }_{\mathrm{gh}}$ the bosonized superghost scalar. The spacetime supercharges are generated byand 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 $\mathrm{Spin}(8)$. Choose the left Ramond ground representation $8_s$ and the right one $8_c$. ThenThe 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 $\left(\genfrac{}{}{0ex}{}{8}{1}\right)=8$ and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$ 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 hasThese fields form the massless type IIA supergravity multiplet.

For the first massive level of a chiral RNS sector, restore ${\alpha }^{\prime }$ throughThe first positive value is one. The NS states therefore have level $N_{\mathrm{NS}}=3/2$. A complete transverse basis after GSO projection isAll contain odd fermion-excitation parity. Their respective dimensions are $8$, $8^2=64$ and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$, so 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 $\mathrm{Spin}(9)$ representations $44+84$: a symmetric traceless rank-two tensor and a three-form, since $44\downarrow \mathrm{Spin}(8)=35_v+8_v+1$ and $84\downarrow \mathrm{Spin}(8)=56_v+28$.

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 areThe 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 $8\times 8=64$ states. HenceThese first massive GSO-projected Ramond states form the gamma-traceless vector-spinor representation of $\mathrm{Spin}(9)$, of dimension $9\cdot 16-16=128$. 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. ConsequentlyTheir 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.

The PDF's displayed zero-mode term has no derivative. Read literally, $d_0\cdot d_0$ vanishes for classical Grassmann variables and supplies no zero-mode symplectic structure. The subsequent canonical-algebra requests therefore require the standard kinetic term $\frac{i}{2}{d}_0\cdot {\dot{d}}_0$. We use that intended correction explicitly; the rest of the displayed action fixes the nonzero-mode normalization.

The Ramond level operator, with vacuum-annihilating normal ordering, isThe canonical oscillator relations, for transverse indices $I,J$, areThe hermiticity convention is ${\alpha }_k^{†}={\alpha }_{-k}$ and $d_k^{†}=d_{-k}$. The commuting bosonic zero mode is supplied by the center-of-mass momentum. For $k>0$, define $a_k^I={\alpha }_k^I/\sqrt{k}$ and $f_k^I=d_k^I$. Their bosonic occupation numbers are $0,1,2,\dots$ and their fermionic occupation numbers are $0,1$. Therefore, on the Fock space generated from an oscillator vacuum,Equivalently, creation operators raise the level by $k$, since $[N_R,{\alpha }_{-k}^I]=k{\alpha }_{-k}^I$ and $[N_R,d_{-k}^I]=k{d}_{-k}^I$. The zero modes commute with $N_R$ and do not change the level. The multiplier imposesso the $N=0$ states are massless. In the Ramond sector the bosonic and fermionic oscillator zero-point contributions cancel, consistently with the stated zero intercept. The massless ground states are spacetime spinors, as the Ramond zero-mode Clifford algebra now shows.

Normalize ${\gamma }^I=\sqrt{2}{\hat{d}}_0^I$. ThenLet $v=|0⟩\ne 0$. Each positive-frequency bosonic annihilator commutes with ${\gamma }^I$, while each fermionic annihilator anticommutes with it. Applying either annihilator to ${\gamma }^{I}v$ therefore gives zero. Thus all eight ${\gamma }^{I}v$ are oscillator vacua. For real $a_I$, the hermitian operator $A={\sum }_I{a}_I{\gamma }^I$ satisfies $A^2=({\sum }_I{a}_I^2)1$, soThis proves the real independence of Clifford-generated vectors, and hence their linear independence over $\mathbb{R}$.

The same argument applies to the nonzero vacuum ${\gamma }^{1}v$, because $({\gamma }^1{)}^2=1$. It gives eight real-linearly independent oscillator vacua ${\gamma }^I{\gamma }^{1}v$. They include $v$ itself, at $I=1$. Products of two zero modes preserve vacuum annihilation just as products of one do.

For the chirality matrix ${\gamma }_9={\gamma }^1{\gamma }^2\cdots {\gamma }^8$, reversing eight anticommuting factors introduces $(-1{)}^{8\cdot 7/2}=1$. HenceMoving any ${\gamma }^I$ through the other seven factors also gives ${\gamma }_9{\gamma }^I=-{\gamma }^I{\gamma }_9$. If ${\gamma }_{9}v=v$, thenThe first collection has negative chirality; the second has positive chirality. If $u_{+}$ and $u_{-}$ have these respective chiralities, hermiticity gives $⟨u_{+},u_{-}⟩=⟨{\gamma }_9{u}_{+},u_{-}⟩=⟨u_{+},{\gamma }_9{u}_{-}⟩=-⟨u_{+},u_{-}⟩$, so they are orthogonal. Combining the two real-independent collections therefore gives at least sixteen real-linearly independent oscillator vacua, eight in each chirality.

The real qualification in the question matters: the particular eight vectors generated from an arbitrary complex $v$ need not be independent over $\mathbb{C}$. Nevertheless the dimension bound from paired Clifford involutions also follows from the full Clifford algebra. Define four commuting hermitian involutions $J_j=i{\gamma }^{2j-1}{\gamma }^{2j}$, $j=1,\dots ,4$. Their joint spectral projections preserve the vacuum space, so it contains a nonzero common eigenvector $w$. Multiplication by ${\gamma }^{2j-1}$ flips the eigenvalue of $J_j$ and leaves the other three eigenvalues unchanged. The sixteen products obtained by independently choosing whether to apply these four odd-indexed Gamma matrices to $w$ consequently have distinct joint eigenvalue quadruples. They are nonzero, mutually orthogonal oscillator vacua. Thus the unprojected vacuum space also has complex dimension at least sixteen, with eight states of each chirality in the minimal representation. A further chiral projection is an additional physical restriction, not part of the oscillator-vacuum conditions here.