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For simple roots $(1,0)$ and $(-1,1)$, coroot integrality gives the weight lattice $P={\mathbb{Z}}^2\cup (\mathbb{Z}+1/2{)}^2$. This is the lattice for the Spin group $\mathrm{Spin}(5)$ and its Lie algebra. Representations descending to $SO(5)$ use the integer coset, because a $2\pi$ rotation in either coordinate plane must act trivially.

After complexifying a nondegenerate metric in $D=2k$ dimensions, pair gamma matrices into $k$ fermionic creation and annihilation operators. Their binary occupation states span $2^k$ dimensions, and the gamma matrices generate all endomorphisms of this space. The Dirac module is therefore irreducible as a Clifford module. The even algebra preserves the chirality grading, splitting its Lorentz-spinor representation into two irreducible Weyl modules of dimension $2^{k-1}$. In $D=2k+1$, adjoining the old chirality matrix as the extra gamma gives an irreducible $2^k$-dimensional Spin group representation with no Weyl splitting. The two possible signs of that extra gamma distinguish Clifford modules but give equivalent Spin group modules.

At NS level $3/2$, the light-cone families are $b_{-3/2}^i|0⟩$, ${\alpha }_{-1}^i{b}_{-1/2}^j|0⟩$, and three distinct $b_{-1/2}$ oscillators. Their dimensions are $8+64+\left(\genfrac{}{}{0ex}{}{8}{3}\right)=128$. All have odd fermion-excitation parity. They assemble into the massive Spin group representations $44+84$, a symmetric traceless tensor and a three-form of Spin(9).

Give the real root span an inner product with ${\epsilon }_1,{\epsilon }_2$ orthonormal. The B2 root system consists of four short roots on the coordinate axes and four long roots on the diagonals. Choose the long-first simple rootsThe positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2={\epsilon }_1$, and ${\alpha }_1+2{\alpha }_2={\epsilon }_1+{\epsilon }_2$. The corresponding simple reflections areThey generate every signed permutation of the two coordinates. Thus the Weyl group has order eight and is the symmetry group of a square; every element sends $(x,y)$ to $(\pm x,\pm y)$ or $(\pm y,\pm x)$, with independent signs.

Since ${\alpha }_1,{\alpha }_2$ form an integral basis of ${\mathbb{Z}}^2$, the root lattice is $Q=\mathbb{Z}{\epsilon }_1\oplus \mathbb{Z}{\epsilon }_2$. The simple coroots are ${\alpha }_1^{\vee }={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2^{\vee }=2{\epsilon }_2$. Therefore the weight lattice isSolving $⟨{\Lambda }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$ gives the fundamental weightsThe cone of dominant weights isIndeed $a=x-y$ and $b=2y$ are exactly the two simple-coroot pairings. This describes representations of the Lie algebra, or of its simply connected Spin group; half-integral weights are allowed. For representations descending to $\mathrm{SO}(5)$ itself, the weight must lie in ${\mathbb{Z}}^2$, equivalently $b$ must be even.

Identify ${\mathbb{R}}^3$ with the traceless Hermitian matrices $X=x\cdot \sigma$, where ${\sigma }_i$ are the Pauli matrices. Then $\det X=-|x{|}^2$. For $U\in SU(2)$ the action $X\mapsto UX{U}^{†}$ preserves trace, Hermiticity and determinant, so gives an orthogonal transformation of ${\mathbb{R}}^3$. Since $SU(2)$ is connected, the determinant of this transformation is $+1$.

An element in the kernel commutes with every ${\sigma }_i$, hence is scalar. Unitarity and determinant one restrict it to $\pm I$. Every three-dimensional rotation is a rotation through some angle $\theta$ around a unit axis $n$. It is obtained byIndeed multiplication with ${\sigma }_i{\sigma }_j={\delta }_{ij}I+i{ϵ}_{ijk}{\sigma }_k$ gives the Rodrigues rotation formula for $UX{U}^{†}$. This proves surjectivity, not just equality of Lie-algebra dimensions. ThereforeA full $2\pi$ rotation lifts from $I$ to $-I$, explaining the double cover and the Spin group interpretation.

Start with the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$. For the dimension count, complexify and rescale the timelike matrix so that the generators obey the Euclidean complex relation $\{{\gamma }_a,{\gamma }_b\}=2{\delta }_{ab}$. This changes neither the complex representation dimension nor its decomposition under the complexified even algebra. In $D=2k$, pair the generators intoDirect substitution gives $\{a_i,a_j\}=\{b_i,b_j\}=0$ and $\{a_i,b_j\}={\delta }_{ij}$. The commuting operators $n_j=b_j{a}_j$ satisfy $n_j^2=n_j$, so their occupation eigenvalues are zero or one. An irreducible module contains an all-zero occupation state $|0⟩$, obtained by lowering the occupied entries of a simultaneous eigenvector. The statesare nonzero and independent by their distinct joint occupation eigenvalues. They span an invariant module; irreducibility makes this the full module. Conversely this Fock construction supplies a representation on exactly these $2^k$ states. Products of occupation projectors and raising/lowering operators give its matrix units, so the generated algebra is the full matrix algebra. Therefore the minimal complex Dirac spinor has

This module is irreducible under the full Clifford algebra, but the Spinor representation of the Lorentz group is generated by the even bilinears ${\Sigma }^{\mu \nu }=\frac{1}{4}[{\gamma }^{\mu },{\gamma }^{\nu }]$. The chirality matrix in the Euclideanized basis isIts two eigenspaces therefore remain invariant under Lorentz transformations. Each has half the Fock states, since any gamma matrix bijects them. The even algebra connects every pair of occupation states of the same parity and has no further invariant subspaces within either one. HenceThese are the irreducible Weyl spinors. Globally they are ordinary representations of $\mathrm{Spin}(D-1,1)$, the double cover; calling them representations of $SO(D-1,1)$ uses the conventional projective spinor terminology.

For $D=2k+1$, adjoin ${\gamma }_{2k+1}=\pm {\gamma }_{\ast }$. It anticommutes with the old matrices and gives a Clifford module of the same dimension $2^k$. The new even generators $[{\gamma }_a,{\gamma }_{2k+1}]$ interchange the two old chirality eigenspaces, so the odd-dimensional Lorentz spinor is irreducible of complex dimension $2^k$, with no Weyl splitting. The product of all odd-dimensional gamma matrices is central and acts as a scalar; it cannot be a new anticommuting chirality operator. Its two signs distinguish the two complex odd-Clifford modules, whose restrictions to Spin group are equivalent. Majorana spinor reality conditions, when the signature permits them, are additional conditions and should not be confused with complex chirality.

For the reduction, take the massless zero modes on a flat seven-torus with no flux or projections. The eleven-dimensional fields are the metric $G_{MN}$, a three-form $A_{MNP}$ and a Majorana gravitino ${\Psi }_M$. Split $M=(\mu ,i)$ with $\mu =0,\dots ,3$ and $i=1,\dots ,7$. The metric supplies $G_{\mu \nu }$, seven vectors $G_{\mu i}$ and 28 scalar fields $G_{ij}=G_{ji}$. The three-form supplies a four-dimensional three-form $A_{\mu \nu \rho }$, seven two-forms $A_{\mu \nu i}$, $\left(\genfrac{}{}{0ex}{}{7}{2}\right)=21$ vectors $A_{\mu ij}$ and $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$ scalars $A_{ijk}$.

A massless two-form in four dimensions has one physical particle polarization and can be dualized to a scalar by $H_3=\ast d\sigma$; the seven two-forms thus supply seven more scalars. The four-dimensional three-form has no local particle polarization, since its four-form field strength is locally a constant times the volume form. The total bosonic content is consequentlyThe eleven-dimensional Majorana spinor has 32 real components, decomposing into eight four-dimensional Majorana spinors. Accordingly ${\Psi }_{\mu }$ supplies eight gravitini, while ${\Psi }_i$ supplies $7\times 8=56$ spin-$1/2$ fields after diagonalizing the gravitino/matter kinetic terms. The resulting theory has 32 real supercharges, hence is four-dimensional N=8 supergravity, withThe on-shell counts check the reduction: the metric has $11(11-3)/2=44$ polarizations and the three-form has $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$, totaling 128. In four dimensions $2+28\cdot 2+70=128$ bosonic polarizations match $8\cdot 2+56\cdot 2=128$ fermionic ones. Higher Kaluza–Klein levels, fluxes or projections would be additional choices beyond this massless toroidal reduction.

There are two necessary qualifications to the literal formulation. The whole group $G$ is itself a closed normal subgroup, so the claim about containment in the centre must concern proper closed normal subgroups. Also, the homomorphism conclusion needs a nontrivial target $H$: for example, $SU(2)\to \{1\}$ has surjective differential onto the zero Lie algebra, but its group kernel is all of $SU(2)$ and no quotient by a finite subgroup is a one-point group. We prove the precise assertions, including these exceptional cases.

Write $\mathfrak{g}=\mathrm{Lie}(G)$. The centre $\mathfrak{z}(\mathfrak{g})$ is invariant under the Adjoint representation, so irreducibility makes it either zero or all of $\mathfrak{g}$. In the latter case $\mathfrak{g}$ is Abelian. For connected $G$, the adjoint action is then trivial: the group is generated by exponentials, and ${\mathrm{Ad}}_{\exp X}=e^{\mathrm{ad}X}=I$. A nonzero trivial representation is irreducible only in dimension one. A connected compact one-dimensional Lie group is a circle, so $G\cong U(1)$.

For every other $G$, the Lie algebra of its centre is zero. The centre is a closed compact Lie subgroup; a zero-dimensional compact Lie group is finite. Let $N$ be a closed normal subgroup. The closed-subgroup theorem identifies $\mathfrak{n}=\mathrm{Lie}(N)$ as an adjoint-invariant subspace of $\mathfrak{g}$, hence $\mathfrak{n}=0$ or $\mathfrak{n}=\mathfrak{g}$. If the latter holds, $N$ contains an identity neighbourhood and is open, so connectedness forces $N=G$. If $N$ is proper, it therefore has zero Lie algebra and is finite. For any $a\in N$, the continuous map $g\mapsto ga{g}^{-1}$ takes connected $G$ into the finite set $N$, so it is constant and equal to $a$. ThusProper closed subgroups of the circle are finite as well, and of course central.

Now let $d\Phi$ be surjective. The submersion theorem makes $\Phi (G)$ contain an identity neighbourhood in $H$, so it is an open subgroup. Since $H$ is connected, $\Phi (G)=H$. Its group kernel $F$ is closed and normal. If $H$ is nontrivial, $F$ is proper and the preceding argument gives a finite central group kernel; this also holds for $G\cong U(1)$, whose proper closed subgroups are finite. The Lie-group first isomorphism theorem givesIf $H$ is trivial, the correct statement is instead $H=G/G$.

For the Spin groups, use the real Clifford algebra with $v^2=-\Vert v{\Vert }^2$. The group $\mathrm{Spin}(m)$ consists of products of an even number of unit vectors inside its invertible even part. Conjugation on the embedded ${\mathbb{R}}^m$ defines the standard double covering $\mathrm{Spin}(m)\to SO(m)$ with group kernel $\{\pm 1\}$. The Clifford relations show that a unit-vector reflection, with the usual twisted conjugation, is a hyperplane reflection; the even products give orientation-preserving transformations. The Cartan–Dieudonné theorem states that every orthogonal transformation is a product of reflections, proving surjectivity. For $m\ge 3$ this connected covering group is simply connected; $\mathrm{Spin}(2)$ is a circle.

Here is an explicit exterior-square construction of the requested six-dimensional map. Put $E={\mathbb{C}}^4$ with its standard Hermitian form and volume $e_1\wedge e_2\wedge e_3\wedge e_4$. On ${\Lambda }^{2}E$, write $e_{ij}=e_i\wedge e_j$ and define an antilinear map $J$ byIt satisfies $J^2=1$. Intrinsically, if $u\wedge v=B(u,v)\mathrm{vol}$ and the Hermitian inner product $h$ is linear in its first variable, then $B(u,Jv)=h(u,v)$. Because $SU(4)$ preserves both $B$ and $h$, it commutes with $J$. This is the real structure of the SU4 exterior square.

The fixed space $W$ has the following real orthonormal basis, with all vectors divided by $\sqrt{2}$:Thus $W$ is six-dimensional over $\mathbb{R}$, its complexification is ${\Lambda }^{2}E$, and the exterior-square action restricts to an orthogonal representation on $W$. Its determinant is one because $SU(4)$ is connected. We obtain $R:SU(4)\to SO(W)\cong SO(6)$.

To compute the group kernel, diagonalize $g\in SU(4)$ with eigenvalues $z_1,\dots ,z_4$. If $R(g)=I$, its complexified exterior-square action is the identity, hence $z_i{z}_j=1$ for every $i<j$. Comparing pairs shows that all $z_i$ are equal, and their common value has square one. A unitary matrix with that single eigenvalue is scalar, so $\ker R=\{\pm I_4\}$. Conversely, those two scalar matrices plainly act trivially on the exterior square.

The differential of $R$ is injective because its group kernel is finite. Both Lie algebras have dimension $15$, so the differential is an isomorphism. Its image is consequently open in connected $SO(6)$, hence all of $SO(6)$. ThereforeThis also realizes $SU(4)$ as $\mathrm{Spin}(6)$, since $SU(4)$ is simply connected, by uniqueness of the connected simply connected covering group.

The circle group is $U(1)=\{z\in \mathbb{C}:|z|=1\}$ under multiplication. For continuous finite-dimensional complex representations, its irreducibles are the one-dimensional charactersA group representation is a continuous homomorphism to the invertible linear maps; it is irreducible when it has no proper nonzero invariant subspace. Compactness permits an invariant Hermitian inner product by averaging. Since the resulting unitary operators commute for $U(1)$, simultaneous diagonalization reduces an irreducible to a character, and the continuous characters of the circle have the stated integer winding number.

The group $SU(2)$ consists of unitary two-by-two complex matrices of determinant one. Each has the unique formidentifying it homeomorphically with the unit 3-sphere in ${\mathbb{R}}^4$. This is the Spin group $\mathrm{Spin}(3)$. A unitary matrix is conjugate to $\mathrm{diag}(e^{i\theta },e^{-i\theta })$, and its conjugacy class is determined by $0\le \theta \le \pi$, or equivalently its trace $2\cos \theta$.

The irreducible representations of SU2 are $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$ for integers $n\ge 0$, with dimension $n+1$. The symmetric power can be realized as homogeneous degree-$n$ polynomials in two variables, with the induced linear action. On the diagonal torus its eigenvalues are $e^{i(n-2j)\theta }$ for $j=0,\dots ,n$, so its character, the trace of the representing operator, isAt the endpoints use the continuous values ${\chi }_n(0)=n+1$ and ${\chi }_n(\pi )=(-1{)}^n(n+1)$. The highest-weight classification says this list is complete.

For the final action, multiplication of block matrices gives $(AB{)}_1=A_1{B}_1$, so conjugation is a linear group representation. Decompose the underlying three-dimensional space as $V_1\oplus V_0$. Its endomorphism space isThe two off-diagonal blocks are each $V_1$, since the invariant alternating form identifies $V_1^{\ast }$ with $V_1$. The scalar part of $\mathrm{End}(V_1)$ is $V_0$ and its traceless part is $V_2$: the same alternating form identifies traceless endomorphisms with symmetric tensors of degree two. The lower scalar block is a second $V_0$. ThusThe dimensions add to $2+4+3=9$, and its character is $2{\chi }_0+2{\chi }_1+{\chi }_2$.

The special orthogonal group in five dimensions isIts Lie algebra consists of real skew-symmetric matrices. There are $5\cdot 4/2=10$ independent entries above the diagonal, so $\dim SO(5)=10$. Equivalently, the orthogonality equations impose fifteen independent constraints on twenty-five matrix entries; the determinant condition chooses a component without changing the dimension.

Fix the fifth coordinate. The matrices $\mathrm{diag}(S,1)$, $S\in SO(4)$, form an explicit subgroup. Under this subgroup the defining vector space splits as ${\mathbb{R}}^4\oplus \mathbb{R}{\mathbf{e}}_5$, so its branching rule is $5\downarrow SO(4)=4\oplus 1$. An element of the so5 Lie algebra can be written uniquely asConjugation by $\mathrm{diag}(S,1)$ sends $B$ to $SB{S}^{-1}$ and $w$ to $Sw$. The first summand is the six-dimensional Adjoint representation of a Lie algebra of $\mathfrak{so}(4)$, and the second is its four-dimensional vector representation. Hence the SO5 to SO4 branching gives

For the left SU(2) subgroup of SO(4), identify ${\mathbb{R}}^4$ with the quaternions. Left multiplication by a unit quaternion is a real orthogonal transformation and gives an embedded SU(2) group. More generally, $q\mapsto aq{b}^{-1}$ gives the double cover $SU(2{)}_L\times SU(2{)}_R\to SO(4)$ with kernel $\{(1,1),(-1,-1)\}$. After complexifying, the vector representation is $(2,2)$ and the Adjoint representation is $(3,1)\oplus (1,3)$. Restricting to the left factor turns the right factor into a multiplicity space. ThereforeThese are decompositions into complex irreducible representations; the real $4$ is the underlying real representation of a quaternionic doublet. Combining the branching rules gives

Write a weight as $\lambda =(x,y)$. The integrality conditions for the B2 root system give $2x\in \mathbb{Z}$ and $y-x\in \mathbb{Z}$. Hence $x=m/2$, $y=m/2+n$, with $m,n\in \mathbb{Z}$. ThusThis is the B2 weight lattice, with an integer square lattice and a second square lattice shifted by $(1/2,1/2)$. The eight roots of a root system areThe short roots lie on the coordinate axes and the long roots on the diagonals. The positive roots for the given simple-root choice are $\alpha$, $\beta$, $\alpha +\beta$, $2\alpha +\beta$.

The integrality conditions determine the weight lattice of the Lie algebra, equivalently of the simply connected Spin group $\mathrm{Spin}(5)$. For the global special orthogonal group $SO(5)$, a $2\pi$ rotation in either coordinate plane is the identity, so a genuine group representation requires integer $x,y$. Thus the half-integer coset contains spin representations that do not descend to $SO(5)$. Both representations requested here have integer weights, so their diagrams are unaffected by this distinction.

The root system of the displayed $SO(4)$ subgroup is $\{\pm (1,1),\pm (1,-1)\}$. The two orthogonal pairs give its two commuting $\mathfrak{su}(2)$ factors. Choose the left factor to have root $\delta =(1,1)$; exchanging the two diagonal pairs exchanges left and right. Its coroot pairs with a weight asThis demonstrates the diagonal-root SU(2) embedding in SO(5) directly. The short-axis root $\alpha$ would instead give $2x$, and therefore a different subgroup: on the vector representation it would produce a triplet and two singlets rather than two doublets and a singlet.

For the vector representation, simultaneously rotate the first and second coordinate planes. Over $\mathbb{C}$, the two planes give opposite pairs of weights, while the fifth coordinate is fixed. Hence its weight diagram iswith every weight multiplicity equal to one. Evaluating $x+y$ gives $+1$ twice, $-1$ twice and $0$ once, exactly two SU(2) representations of dimension two and one singlet.

For the Adjoint representation, the root-space decomposition has one one-dimensional space for each of the eight roots and a two-dimensional zero-weight Cartan subalgebra. ThusThe coroot values $x+y$ have multiplicities $1,2,4,2,1$ at $-2,-1,0,1,2$. One zero-weight state joins the $\pm 2$ states to make a triplet; the $\pm 1$ states form two doublets, leaving three zero-weight singlets. This verifies the earlier branching rule and accounts for all ten dimensions.

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.

For the stated matrix exponential parametrization in the connected proper orthochronous Lorentz group, use the corresponding Spinor representation of the Lorentz group:The spacetime argument is inverse-transformed because this is an active transformation of the field at a fixed coordinate $x$. Equivalently, with $x^{\prime }=\Lambda x$, ${\psi }^{\prime }(x^{\prime })=D(\Lambda )\psi (x)$. The identity $[S^{\mu \nu },{\gamma }^{\rho }]={\eta }^{\nu \rho }{\gamma }^{\mu }-{\eta }^{\mu \rho }{\gamma }^{\nu }$ integrates to $D^{-1}{\gamma }^{\mu }D={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$, which ensures covariance of the Dirac equation.

Strictly, $D$ depends on a lift to the Spin group, not just on the final Lorentz transformation: two lifts differ by sign. The specified generator exponential chooses a lift along its continuous path from the identity. This qualification is essential for the full-rotation comparison in part (g).

Insert ${\Omega }_{jk}=-{ϵ}_{jkl}{\varphi }^l$ and the result for $S^{jk}$ into the Spinor representation of the Lorentz group. The contraction of two Levi-Civita symbols ${\sum }_{j,k}{ϵ}_{jkl}{ϵ}^{jkm}=2{\delta }_l{}^m$ givesThe matrix exponential of this block diagonal matrix is the stated $O$, so ${\psi }^{\prime }(x)=O\psi ({\Lambda }^{-1}x)$. The plus sign in its exponent follows from both minus signs above; it is tied to the rotation-parameter convention in the PDF.

For a rotation through $2\pi$ about $z$, the Pauli matrix has eigenvalues $\pm 1$, hence $e^{i\pi {\sigma }^3}=-I_2$. The spacetime rotation is the identity, whileThis is the spinor sign under a full spatial rotation. A vector returns to itself already at $2\pi$. The comparison concerns the exponentiated Lorentz group representations: a Lie algebra by itself has no distinguished element called a $2\pi$ rotation. The sign records the nontrivial lift of this rotation loop to the Spin group.

The other element is $\boxed{-M}$. Conjugation by $M$ and $-M$ is identical, so the homomorphism $SU(2)\to SO(3)$ has kernel $\{\pm I\}$ and is the Spin group double cover.

The displayed $R(\phi )$ rotates the coordinate components in the $xy$ plane by angle $-\phi$. Sinceits spin representative isAt a full turn,Thus a $2\pi$ spatial rotation changes the sign of a spin-one-half state, while a $4\pi$ rotation returns it to itself. This realizes the fact that the Spin group is a double cover of the proper orthochronous Lorentz group; observable spinor bilinears are unchanged by the sign.

An irreducible complex Clifford algebra module in even spacetime dimension $2k$ has dimension $2^k$ and splits under the connected Spin group into two Weyl spinor modules of dimension $2^{k-1}$. In odd dimension $2k+1$, an irreducible spinor still has dimension $2^k$ but has no Weyl splitting. Reality conditions depend on signature and are additional to these complex counts.

For a Dirac spinor a spatial rotation acts in each chiral block by a matrix exponential $e^{\pm i\theta \mathbf{n}\cdot \mathbf{\sigma }/2}$, depending on angle orientation. Since $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I_2$ for a unit axis, the Pauli matrix multiplication law gives this exponential as $\cos (\theta /2)I_2\pm i\sin (\theta /2)\mathbf{n}\cdot \mathbf{\sigma }$. At $\theta =2\pi$ it is $-I_2$, and at $4\pi$ it is $I_2$. A vector returns already at $2\pi$. This describes the lift of the rotation loop to the Spin group, rather than a single-valued representation of the ordinary rotation group on spinors.

A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.