Spin(8) vector and half-spin modules

Let $V={\mathbb{R}}^8$ with its standard oriented orthonormal basis $e_1,\dots ,e_8$, Euclidean inner product $⟨,⟩$, and quadratic form $Q(v)=⟨v,v⟩$. Use the plus-$Q$ Clifford algebra $\mathrm{C}\mathrm{l}(V,Q)$, generated by $V$ with relations $v^2=Q(v)1$. Write $G={\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(V,Q)$ for the even subgroup generated by unit vectors in this algebra. Its vector action is ${\rho }_x(v)=\alpha (x)v{x}^{-1}$, where $\alpha$ is the algebra grading involution that sends each vector to its negative. Write $\mathrm{Spin}(8)=G$.

Let $W={\mathbb{C}}^4$ with ordered basis $u_1,\dots ,u_4$ and let $\mathrm{\Delta }={\mathrm{\Lambda }}^{\bullet }W$. Give its ordered wedge basis the Hermitian inner product making it orthonormal. For $1\le j\le 4$, let ${ϵ}_j$ be exterior multiplication by $u_j$ and let ${\iota }_j$ delete $u_j$ from an ordered basis wedge, with the sign determined by its position, and be zero when $u_j$ is absent. Define $C_{2j-1}={ϵ}_j+{\iota }_j$ and $C_{2j}=i({ϵ}_j-{\iota }_j)$. These operators satisfy the plus-$Q$ Clifford relations for the complexification of $V$; the resulting $\mathrm{C}\mathrm{l}(V,Q){\otimes }_{\mathbb{R}}\mathbb{C}$ action on $\mathrm{\Delta }$ is the complex spin module. Write $c(v)$ for the Clifford action of $v\in V$ on $\mathrm{\Delta }$, obtained by extending $e_i\mapsto C_i$ linearly. Its oriented volume operator is $\omega =C_1{C}_2\cdots C_8$, and ${\mathrm{\Delta }}^{+}$ and ${\mathrm{\Delta }}^{-}$ are its $+1$ and $-1$ eigenspaces.

A real structure on a complex representation is a conjugate-linear involution commuting with the group action; its fixed subspace is a real representation whose complexification is the original complex representation. Let $K$ denote coefficient conjugation on $\mathrm{\Delta }$, put $A=C_2{C}_4{C}_6{C}_8$, and define the real structure $J=AK$. Give $V$ its standard real inner product and give each fixed real space $({\mathrm{\Delta }}^{\pm }{)}^J$ the restriction of the real part of the Hermitian inner product. Set $8_v=V$ with the vector action $\rho$, $8_s=({\mathrm{\Delta }}^{+}{)}^J$, and $8_c=({\mathrm{\Delta }}^{-}{)}^J$, each with the Spin(8) action induced by Clifford multiplication.

For this choice of orientation and chirality, $8_s$ and $8_c$ are the positive and negative real half-spin representations. Reversing the chirality convention exchanges their labels. The fixed dimensions, invariance, and irreducibility asserted by this construction are established in the following well-definedness lemma.

Universal Clifford algebra of a quadratic space

Real Clifford group, Pin, and the two Spin subgroups

Exterior Algebra Of A Finite Free Module

Exterior Algebra Basis Monomials

PBW basis and exterior associated graded for a Clifford algebra

Universal property of the Clifford algebra

Real Pin and full-even Spin images, kernels, and definite compactness

The Axiom of Choice

Spin(8) Clifford modules, real forms, and D4 weights

These are representations of the compact Euclidean group Spin(8). Their construction does not establish the existence of Lorentzian ten-dimensional Spin(1,9) spinor bundles or identify those bundles with Euclidean half-spin modules.

For comparison of names only, adopt the light-cone convention reported in Weigand, §§6.5–6.7. There $\mathbf{16}$ and ${\mathbf{16}}^{\mathrm{\prime }}$ denote the positive- and negative-chirality ten-dimensional Majorana–Weyl representations; $\downarrow$ denotes restriction to $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(1,1)\times \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(8)$, with the first entry the $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(1,1)$ weight and the second a transverse $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(8)$ module: $\mathbf{16}\downarrow =(+\frac{1}{2},8_s)\oplus (-\frac{1}{2},8_c),{\mathbf{16}}^{\mathrm{\prime }}\downarrow =(+\frac{1}{2},8_c)\oplus (-\frac{1}{2},8_s)\mathrm{.}$ The source's open-string $R^{+}$ and $R^{-}$ denote the $+1$ and $-1$ eigenspaces of its R-sector G-parity operator $G=\mathrm{\Gamma }(-1{)}^F$, where $\mathrm{\Gamma }$ is ten-dimensional chirality and $F$ is the source's R-sector fermion-number operator; they transform as $8_s$ and $8_c$, respectively. Its Type IIB closed-string GSO choice pairs equal transverse chiralities and Type IIA pairs opposite chiralities. These are reported light-cone naming conventions, not a derivation of the GSO projection or a Lorentzian spinor-bundle construction. Matching the source's positive transverse label $8_s$ to $({\mathrm{\Delta }}^{+}{)}^J$ is the convention adopted here.