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The bosons are the metric, two-form, dilaton, RR one-form and RR three-form, with physical dimensions $35+28+1+8+56=128$. Two opposite-chirality gravitinos and two dilatinos supply $2(56+8)=128$ fermionic degrees of freedom. They form the ten-dimensional type IIA supergravity multiplet. Counts refer to on-shell polarizations, not raw field components.

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.

Complexifying the Clifford algebra removes the signature dependence of its representation dimension. Pair the $2k$ gamma matrices into operators $a_i,a_i^{†}$ with $\{a_i,a_j^{†}\}={\delta }_{ij}$ and $\{a_i,a_j\}=0$. The $k$ commuting occupation operators have eigenvalues zero or one. Starting from a Clifford vacuum, applying every subset of the $k$ fermionic creation operators produces $2^k$ independent states. Conversely, these operators construct matrices on a $2^k$-dimensional space, and their products span its full matrix algebra. Thus an irreducible complex Clifford algebra module in $D=2k$ hasA general Clifford module is a direct sum of these; the statement concerns the minimal spinor.

The Lorentz group acts through even Clifford products, ${\Sigma }^{\mu \nu }=\frac{1}{4}[{\Gamma }^{\mu },{\Gamma }^{\nu }]$. More precisely this is a representation of its double cover, the Spin group. The normalized product ${\Gamma }_{\ast }$ of all $2k$ matrices squares to one and anticommutes with each ${\Gamma }^{\mu }$, while commuting with every ${\Sigma }^{\mu \nu }$. Its eigenspaces are the two Weyl spinor representations, so the Dirac spinor is reducible under the connected spin group:Each summand is irreducible. The Clifford module itself remains irreducible because an individual ${\Gamma }^{\mu }$ interchanges the summands. In $D=2k+1$ one can use the old ${\Gamma }_{\ast }$ as the additional matrix, with an appropriate phase for its signature. No independent chirality matrix remains: the product of all $2k+1$ matrices is central and fixed to a scalar on an irreducible module. There are two choices of this central sign for the complex Clifford algebra, but each restricts to the same irreducible $2^k$-dimensional spinor representation of the connected spin group. An odd-dimensional spinor has no Weyl splitting. Majorana spinor reality conditions depend additionally on dimension and signature; they must be imposed separately from this complex dimension count.

For the massless field counts, use the rotational part $SO(D-2)$ of the little group and real physical polarizations. A massless p-form gauge field has only transverse, antisymmetric components after its gauge redundancy and equations of motion are imposed, giving $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$ polarizations. A graviton has a symmetric traceless transverse tensor, giving $(D-2)(D-1)/2-1=D(D-3)/2$. A massless fermion obeys a Dirac equation which halves its off-shell spinor components. In ten dimensions a Majorana-Weyl spinor has sixteen real components off shell and eight physical components; in nine dimensions a Majorana spinor has sixteen off shell and eight physical components; in eleven dimensions a Majorana spinor has thirty-two off shell and sixteen physical components. A gravitino is a transverse vector-spinor with its gamma trace removed, leaving $(D-3)$ times the corresponding physical spinor dimension. Thus the ten-dimensional Majorana-Weyl gravitino has $7\times 8=56$ polarizations, the nine-dimensional Majorana gravitino has $6\times 8=48$, and the eleven-dimensional Majorana gravitino has $8\times 16=128$.

The common Neveu-Schwarz sector of type IIA supergravity and type IIB supergravity consists of a metric tensor, the Kalb–Ramond field $B_2$, and the dilaton $\varphi$. Their counts are respectively $35$, $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$, and $1$. The Ramond-Ramond sectors and fermions distinguish the theories. In type IIA supergravity the independent bosonic potentials and counts areThere are two Majorana-Weyl gravitini of opposite chirality, giving $56+56=112$, and two Majorana-Weyl dilatini of opposite chirality, giving $8+8=16$. Their total is $128$. With gravitini labelled ${\psi }^{+}$ and ${\psi }^{-}$, the corresponding dilatini have signs $-$ and $+$ respectively. Type IIA is nonchiral.

In type IIB supergravity, the independent bosonic potentials and counts areThe five-form field strength associated with $C_4$ is a self-dual differential form, so only half the seventy transverse four-form components are independent. In Lorentzian ten dimensions the Hodge star operator on five-forms squares to one, making a real self-duality condition possible. The two Majorana-Weyl gravitini have the same chirality and contribute $112$ states. Both Majorana-Weyl dilatini have the opposite chirality to those gravitini and contribute $16$ states. Type IIB is chiral, with $128+128$ bosonic and fermionic polarizations.

For dimensional reduction, retain the massless zero modes on a flat circle, with no flux, gauging or fermion twist. A metric splits into a lower-dimensional metric, one vector and one scalar. An ordinary $p$-form splits into a $p$-form and a $(p-1)$-form, according as it has no compact index or one. This preserves counts by Pascal's identity, $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)=\left(\genfrac{}{}{0ex}{}{D-3}{p}\right)+\left(\genfrac{}{}{0ex}{}{D-3}{p-1}\right)$.

Reducing type IIA supergravity to nine dimensions gives the following decompositions; each number is a count of physical states:Thus the bosonic maximal nine-dimensional supergravity multiplet contains one graviton, three massless vectors, two massless two-forms, one massless three-form and three real scalars, withEach ten-dimensional Majorana-Weyl gravitino becomes one nine-dimensional Majorana gravitino and one Majorana spin-one-half field, giving $56=48+8$. Each ten-dimensional dilatino gives one more Majorana spin-one-half field. The fermionic multiplet therefore has two gravitini and four spin-one-half fields, with $2(48)+4(8)=128$. The fields belong to one maximal gravity supermultiplet, not separate interacting matter multiplets.

Reducing type IIB supergravity gives $g_{\mu \nu },A_{\mu }^{\mathrm{metric}},\rho$ from the metric; one two-form and one vector each from $B_2$ and $C_2$; and the two scalars $\varphi ,C_0$. The four-form potential yields a nine-dimensional four-form and three-form, but the ten-dimensional self-dual differential form condition relates their field strengths. Retaining either one gives $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=\left(\genfrac{}{}{0ex}{}{7}{4}\right)=35$ independent polarizations, not seventy. The same bosonic count, three scalars, three vectors, two two-forms and one three-form, follows. The fermionic reduction likewise gives two gravitini and four spin-one-half fields. These are the same maximal nine-dimensional supergravity spectrum, as expected from T-duality. The spectrum is nonchiral in nine dimensions, since the odd-dimensional spinor representation has no independent Weyl chirality; the ten-dimensional distinction is lost on restriction to nine-dimensional Lorentz symmetry.

Eleven-dimensional supergravity has a metric with $11(8)/2=44$ states, a three-form with $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$, and a Majorana gravitino with $128$. For direct reduction on a flat two-torus, write the internal indices as $i=1,2$. The metric gives $g_{\mu \nu }$, two vectors $g_{\mu i}$ and three symmetric components $g_{ij}$; the three-form gives $A_{\mu \nu \rho }$, two two-forms $A_{\mu \nu i}$ and one vector $A_{\mu 12}$. HenceThe eleven-dimensional Majorana spinor restricts to two nine-dimensional Majorana spinors. The vector-spinor consequently yields two gravitini and four spin-one-half fields, giving $128=2(48)+4(8)$. This is again the same $128+128$ maximal nine-dimensional multiplet.

For reduction of massless type IIA supergravity to four dimensions, use a flat six-torus and retain all zero modes, without flux or projections. Internal indices $i=1,\dots ,6$ giveThe multiplicities $21,15,20$ come respectively from a symmetric pair of six internal indices, $\left(\genfrac{}{}{0ex}{}{6}{2}\right)$, and $\left(\genfrac{}{}{0ex}{}{6}{3}\right)$. Before dualizing, this gives one graviton, $28$ vectors, $63$ scalars, seven two-forms and one three-form. The seven two-forms provide seven scalar polarizations, whereas the three-form has no local propagating polarization. Thus the final bosonic spectrum is one graviton, $28$ vectors and $70$ scalars, with $2+56+70=128$ states.

A ten-dimensional Majorana-Weyl spinor decomposes into four four-dimensional Majorana spinors. Each ten-dimensional gravitino supplies four four-dimensional gravitini and twenty-four spin-one-half fields, with $56=4(2)+24(2)$. Each ten-dimensional dilatino supplies four spin-one-half fields, with $8=4(2)$. Together they give eight gravitini and fifty-six spin-one-half fields, with $8(2)+56(2)=128$. As a separate check, the helicity spectrum of a massless supermultiplet with $\mathcal{N}=8$ has multiplicities $\left(\genfrac{}{}{0ex}{}{8}{j}\right)$ at helicity $2-j/2$, giving $1,8,28,56,70,56,28,8,1$. The reduced theory has the field content of four-dimensional N=8 supergravity, with $128+128$ physical states, exactly as in ten and eleven dimensions. Other compact manifolds or projections can reduce the number of preserved supercharges; the flat-torus assumption is essential to this spectrum.

Here is the local two-form scalar duality including its coupling dependence. For this calculation use signature $(-+++)$, $H_{\mu \nu \rho }=3{\partial }_{[\mu }{B}_{\nu \rho ]}$, and the contravariant volume tensor ${ϵ}^{0123}=+1$. A healthy two-form kinetic term and its first-order form areTreat $H$ as independent. Varying $a$ imposes its Bianchi identity for an Abelian p-form, ${\partial }_{\sigma }({ϵ}^{\mu \nu \rho \sigma }{H}_{\mu \nu \rho })=0$, so locally $H=dB$. Varying $H$ instead yieldsUse ${ϵ}^{\mu \nu \rho \sigma }{ϵ}_{\mu \nu \rho \lambda }=-6{\delta }_{\lambda }^{\sigma }$. The original kinetic term becomes $+(g_B^2/2)(\partial a{)}^2$, and the multiplier term becomes $-g_B^2(\partial a{)}^2$. Both terms must be substituted; replacing $H$ only in the original kinetic term would produce the wrong sign. ThusAt fixed normalization of the Bianchi multiplier, the dual kinetic coefficient is the inverse of the original coefficient: if the scalar convention is $-\frac{1}{2g_a^2}(\partial a{)}^2$, then $g_a=1/g_B$. A canonical scalar is $g_{B}a$ for constant $g_B$, but that rescaling hides the formal coupling inversion and changes any assigned scalar periodicity. For a scalar-dependent positive kinetic matrix ${\mathcal{G}}_{IJ}$ of several two-forms, the same calculation gives $-\frac{1}{2}({\mathcal{G}}^{-1}{)}^{IJ}\partial a_I\partial a_J$. Additional topological couplings modify the multiplier's derivative terms but do not change this basic inversion of the two-form kinetic matrix. The Hodge star operator exchanges the two-form Euler-Lagrange equations and Bianchi identities for an Abelian p-form with those of the dual scalars; the duality is local and global flux sectors require separate treatment.

A four-dimensional three-form instead has $F_4=d{C}_3=f{\mathrm{vol}}_4$. Its equation of motion, $d(g_3^{-2}\ast F_4)=0$, makes $f/g_3^2$ a spacetime constant. For constant coupling the first-order expression in this convention can be writtenHere the first-order construction includes a multiplier $q(F_4-d{C}_3)$: varying $C_3$ imposes $dq=0$, and the displayed density retains the term fixing the flux normalization. Eliminating $f$ gives $f=g_3^{2}q$ and ${\mathcal{L}}_{\mathrm{eff}}=-g_3^2{q}^2/2$. Thus it can encode a constant-flux contribution to the vacuum energy, but no local massless particle or scalar wave. On the zero-flux perturbative vacuum it contributes no state. The seven two-forms are scalar duals; the three-form is nondynamical, which completes the four-dimensional field count without discarding a possible global flux parameter.