= Solution
Complexifying the <Clifford algebra> removes the signature dependence of its representation dimension. Pair the $2k$ <gamma matrices> into operators $a_i,a_i^\dagger$ with $\{a_i,a_j^\dagger\}=\delta_{ij}$ 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 $D=2k$ has
$$
\boxed{\dim_{\mathbb C}S_{\rm Dirac}=2^k.}
$$
A general Clifford module is a direct sum of these; the statement concerns the minimal spinor.
The <Lorentz group> acts through even Clifford products, $\Sigma^{\mu\nu}=\tfrac14[\Gamma^\mu,\Gamma^\nu]$. More precisely this is a representation of its double cover, the <Spin group>. The normalized product $\Gamma_*$ 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 \b[the Dirac spinor is reducible under the connected spin group]:
$$
\boxed{S_{\rm Dirac}=S_+\oplus S_-,\qquad\dim_{\mathbb C}S_\pm=2^{k-1}.}
$$
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 $\Gamma_*$ 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. \b[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 $\binom{D-2}{p}$ 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 $7\times8=56$ polarizations, the nine-dimensional Majorana <gravitino> has $6\times8=48$, and the eleven-dimensional Majorana <gravitino> has $8\times16=128$.
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> $\phi$. Their counts are respectively $35$, $\binom82=28$, and $1$. The Ramond-Ramond sectors and fermions distinguish the theories. In <type IIA supergravity> the independent bosonic potentials and counts are
$$
\begin{array}{c|ccccc|c}
\text{field}&g_{MN}&B_2&\phi&C_1&C_3&\text{total}\\\hline
\text{polarizations}&35&28&1&\binom81=8&\binom83=56&128
\end{array}
$$
There 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. \b[Type IIA is nonchiral.]
In <type IIB supergravity>, the independent bosonic potentials and counts are
$$
\begin{array}{c|cccccc|c}
\text{field}&g_{MN}&B_2&\phi&C_0&C_2&C_4&\text{total}\\\hline
\text{polarizations}&35&28&1&1&\binom82=28&\tfrac12\binom84=35&128
\end{array}
$$
The 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. \b[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>, $\binom{D-2}{p}=\binom{D-3}{p}+\binom{D-3}{p-1}$.
Reducing <type IIA supergravity> to nine dimensions gives the following decompositions; each number is a count of physical states:
$$
\begin{array}{c|l|r}
\text{ten-dimensional field}&\text{nine-dimensional fields}&\text{counts}\\\hline
g_{MN}&g_{\mu\nu},\ A_\mu^{\rm metric},\ \rho&27+7+1\\
B_2&B_2,\ A_\mu^B&21+7\\
\phi&\phi&1\\
C_1&A_\mu^C,\ C_y&7+1\\
C_3&C_3,\ C_{\mu\nu y}&35+21
\end{array}
$$
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, with
$$
\boxed{27+3(7)+2(21)+35+3=128.}
$$
Each 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 $g_{\mu\nu},A_\mu^{\rm metric},\rho$ from the metric; one two-form and one vector each from $B_2$ and $C_2$; and the two scalars $\phi,C_0$. 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 $\binom73=\binom74=35$ 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 \b[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 $\binom93=84$, 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 $A_{\mu12}$. Hence
$$
44=27+2(7)+3,\qquad84=35+2(21)+7.
$$
The 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 \b[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 $i=1,\ldots,6$ give
$$
\begin{array}{c|l|r}
\text{field}&\text{four-dimensional fields}&\text{polarizations}\\\hline
g_{MN}&g_{\mu\nu},\ 6g_{\mu i},\ 21g_{ij}&2+6(2)+21=35\\
B_2&B_{\mu\nu},\ 6B_{\mu i},\ 15B_{ij}&1+6(2)+15=28\\
\phi&\phi&1\\
C_1&C_\mu,\ 6C_i&2+6=8\\
C_3&C_{\mu\nu\rho},\ 6C_{\mu\nu i},\ 15C_{\mu ij},\ 20C_{ijk}&0+6(1)+15(2)+20=56
\end{array}
$$
The multiplicities $21,15,20$ come respectively from a symmetric pair of six internal indices, $\binom62$, and $\binom63$. 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 $\binom8j$ at helicity $2-j/2$, giving $1,8,28,56,70,56,28,8,1$. \b[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 $\epsilon^{0123}=+1$. A healthy two-form kinetic term and its first-order form are
$$
\mathcal L_B=-\frac1{12g_B^2}H_{\mu\nu\rho}H^{\mu\nu\rho},\qquad
\mathcal L_1=-\frac1{12g_B^2}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac16\epsilon^{\mu\nu\rho\sigma}H_{\mu\nu\rho}\partial_\sigma a.
$$
Treat $H$ as independent. Varying $a$ imposes its <Bianchi identity for an Abelian p-form>, $\partial_\sigma(\epsilon^{\mu\nu\rho\sigma}H_{\mu\nu\rho})=0$, so locally $H=dB$. Varying $H$ instead yields
$$
H^{\mu\nu\rho}=g_B^2\epsilon^{\mu\nu\rho\sigma}\partial_\sigma a.
$$
Use $\epsilon^{\mu\nu\rho\sigma}\epsilon_{\mu\nu\rho\lambda}=-6\delta^\sigma_\lambda$. 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. Thus
$$
\boxed{\mathcal L_{\rm dual}=-\frac{g_B^2}{2}\partial_\mu a\,\partial^\mu a.}
$$
At fixed normalization of the Bianchi multiplier, the dual kinetic coefficient is the inverse of the original coefficient: if the scalar convention is $-\tfrac1{2g_a^2}(\partial a)^2$, then $g_a=1/g_B$. A canonical scalar is $g_Ba$ 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 $-\tfrac12(\mathcal G^{-1})^{IJ}\partial a_I\partial a_J$. 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 $F_4=dC_3=f\,\operatorname{vol}_4$. Its equation of motion, $d(g_3^{-2}*F_4)=0$, makes $f/g_3^2$ a spacetime constant. For constant coupling the first-order expression in this convention can be written
$$
\mathcal L_1=\frac{f^2}{2g_3^2}-qf,\qquad \partial_\mu q=0.
$$
Here the first-order construction includes a multiplier $q(F_4-dC_3)$: varying $C_3$ imposes $dq=0$, and the displayed density retains the term fixing the flux normalization. Eliminating $f$ gives $f=g_3^2q$ and $\mathcal L_{\rm eff}=-g_3^2q^2/2$. 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. \b[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.
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