Eleven-Dimensional Spin Structure and Majorana Bundle

Let $(M,g)$ be as in Eleven-Dimensional Oriented Lorentzian Geometry, and supply a time orientation $\tau$, meaning a continuous choice ${\tau }_x$ of one component of $\{v\in T_{x}M:g_x(v,v)<0\}$ at each $x\in M$. Put $V={\mathbb{R}}^{11}$ with $\eta =\mathrm{diag}(-1,1,\dots ,1)$ and standard basis $(E_0,\dots ,E_{10})$, and let ${\mathcal{C}}^{+}\subset V$ be the component of $\{v:\eta (v,v)<0\}$ containing $E_0$. Let $F_{g,\tau }^{+}M$ be the bundle of oriented linear isometries $e:V\to T_{x}M$ that send ${\mathcal{C}}^{+}$ to ${\tau }_x$. First define the group $SO(V,\eta ):=\{A\in GL(V):A^T\eta A=\eta ,\det A=1\},$ with the smooth submanifold structure inherited from $GL(V)$. Indeed, the derivative at an isometry $A$ of $B\mapsto B^T\eta B$ is $H\mapsto H^T\eta A+A^T\eta H$, which is onto the symmetric matrices: for symmetric $S$, take $H=\frac{1}{2}A{\eta }^{-1}S$. Thus the isometry set is a smooth submanifold, and its determinant-one part is a union of components; multiplication and inversion are smooth matrix maps, so $SO(V,\eta )$ is a finite-dimensional Lie group. Define the orthochronous subgroup $S{O}^{+}(V,\eta ):=\{A\in SO(V,\eta ):A({\mathcal{C}}^{+})={\mathcal{C}}^{+}\}\mathrm{.}$ It is the right structure group of $F_{g,\tau }^{+}M$ and is the identity component of $SO(V,\eta )$ for Lorentz signature. Put $\mathfrak{s}\mathfrak{o}(V,\eta ):=T_{I}SO(V,\eta )$.

Form $\mathrm{C}\mathrm{l}(V,\eta )$ with the plus-quadratic-form convention $v^2=\eta (v,v)1$ (Universal Clifford algebra of a quadratic space). Let ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(10,1):={\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(V,\eta )$. Its canonical action is $\lambda (s)v=\alpha (s)v{s}^{-1}$, using the grading involution $\alpha$ from Real Clifford group, Pin, and the two Spin subgroups; it is a two-sheeted topological covering of $SO(V,\eta )$ by Pin and full-even Spin are topological double covers. Equip ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(10,1)$ with the lifted smooth atlas: whenever $U\subset SO(V,\eta )$ is a smooth coordinate neighborhood evenly covered by $\lambda$ and $W$ is one sheet over $U$, use $\lambda {\mathrm{\mid }}_W:W\to U$ as a smooth chart. These charts have smooth transition maps because their transitions are the corresponding matrix-group coordinate changes; $\lambda$ is a local diffeomorphism. Since $\lambda$ is a group homomorphism, multiplication and inversion are smooth in this atlas. Define ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1):={\lambda }^{-1}(S{O}^{+}(V,\eta ))$ and let ${\lambda }^{+}$ be the restriction of $\lambda$; it is a two-sheeted smooth covering of $S{O}^{+}(V,\eta )$ with the restricted smooth structure. Write ${\mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}}^{+}(10,1):=T_1{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1)$ and $d{\lambda }^{+}:{\mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}}^{+}(10,1)\to \mathfrak{s}\mathfrak{o}(V,\eta )$ for the induced Lie-algebra isomorphism.

A spin structure on the time-oriented Lorentzian manifold $(M,g,\tau )$ is a smooth right principal ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1)$-bundle ${\pi }_P:P\to M$ and a smooth two-sheeted covering $\mathrm{\Lambda }:P\to F_{g,\tau }^{+}M$ which is a map over ${\mathrm{id}}_M$ and is ${\lambda }^{+}$-equivariant. Explicitly, if ${\pi }_F:F_{g,\tau }^{+}M\to M$ is the frame projection, then ${\pi }_F\circ \mathrm{\Lambda }={\pi }_P$ and $\mathrm{\Lambda }(ps)=\mathrm{\Lambda }(p){\lambda }^{+}(s)$ for every $p\in P$ and $s\in {\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1)$. In particular, each map $P_x\to (F_{g,\tau }^{+}M{)}_x$ between fibers over $x\in M$ is a two-sheeted cover. This is supplied structure data; no existence claim is made for an arbitrary $(M,g,\tau )$.

For the CJS convention, also supply a real vector space $\mathrm{\Delta }$ of dimension $32$ and a unital representation $\rho :\mathrm{C}\mathrm{l}(V,\eta )\to {\mathrm{End}}_{\mathbb{R}}(\mathrm{\Delta })$. Its restriction to ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1)$ is required to be smooth for the lifted Lie-group structure and defines the associated real vector bundle

$S:=P{\times }_{\rho {\mathrm{\mid }}_{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}}}\mathrm{\Delta }⟶M,$

with equivalence $(ps,v)\sim (p,\rho (s)v)$. A Majorana spinor is a smooth real section of $S$; on an open set $U$, “nonzero” means not identically zero on $U$. The quotient has its smooth rank-$32$ vector-bundle structure induced from principal-bundle sections and the smooth representation (Associated bundles, Associated vector bundles are well-defined). Requiring $(P,\mathrm{\Lambda },\mathrm{\Delta },\rho )$ as supplied data defines the structure and does not assert a Clifford-classification theorem or the existence of such data for every Lorentzian eleven-manifold. The selected dimension $32$ matches the source-reported CJS Majorana convention (Hamilton, §3.6; Gauntlett–Pakis, Appendix A); this mathematical definition only uses the supplied representation.

The Levi-Civita connection determines the spinor-bundle connection as follows. On an open set $U\subseteq M$, let $e=(e_0,\dots ,e_{10}):U\to F_{g,\tau }^{+}M$ be a smooth oriented, time-oriented orthonormal frame and define ${\omega }^e\in {\mathrm{\Omega }}^1(U;\mathfrak{s}\mathfrak{o}(V,\eta ))$ by ${\mathrm{\nabla }}_X^{LC}{e}_B={\sum }_A{e}_A{\omega }^e{}^A{}_B(X)\mathrm{.}$ Metric compatibility makes each matrix ${\omega }^e(X)$ $\eta$-skew. After shrinking $U$, choose a smooth local lift $s:U\to P$ with $\mathrm{\Lambda }\circ s=e$ and set ${\omega }^s:=(d{\lambda }^{+}{)}^{-1}{\omega }^e\in {\mathrm{\Omega }}^1(U;{\mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}}^{+}(10,1))$. If $ϵ{\mathrm{\mid }}_U=[s,\psi ]$ in the associated-bundle chart, define ${\mathrm{\nabla }}_X^Sϵ{\mathrm{\mid }}_U:=[s,X(\psi )+{\rho }_{\ast }({\omega }^s(X))\psi ],$ where ${\rho }_{\ast }:=d(\rho {\mathrm{\mid }}_{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}})$ is the differentiated representation. On an overlap with $s^{\mathrm{\prime }}=sh$, the frame and spin connection forms transform by $e^{\mathrm{\prime }}=e\lambda (h)$ and ${\omega }^{s^{\mathrm{\prime }}}={\mathrm{Ad}}_{h^{-1}}{\omega }^s+h^{-1}dh$, while the components satisfy ${\psi }^{\mathrm{\prime }}=\rho (h{)}^{-1}\psi$. These transformation laws make the displayed local operators agree, so they define a smooth vector-bundle connection in the sense of Connection on a smooth vector bundle. This is the connection induced from the Levi-Civita connection through the spin lift, not an additional independent choice.

For a $p$-form $\omega$, Clifford multiplication uses

$\omega \cdot ϵ:=\frac{1}{p!}{\omega }_{A_1\cdots A_p}{\gamma }^{A_1\cdots A_p}ϵ,$

in an oriented orthonormal frame. Let $i_{\eta }:V\to \mathrm{C}\mathrm{l}(V,\eta )$ be the canonical generator map of the Clifford-algebra definition. In a local spin frame $s$ above $e$, write $e_A(x):=e(x)E_A$ and ${\gamma }_A:=\rho (i_{\eta }(E_A))=\rho (i_{\eta }(e(x{)}^{-1}{e}_A(x)))$, with ${\gamma }^A:={\eta }^{AB}{\gamma }_B$. The displayed action on $[s,\psi ]$ means $[s,(1\mathrm{/}p!){\omega }_{A_1\cdots A_p}{\gamma }^{A_1\cdots A_p}\psi ]$. The product in this formula is the normalized antisymmetrization

${\gamma }^{A_1\cdots A_p}:=\frac{1}{p!}{\sum }_{\sigma \in S_p}\mathrm{sgn}(\sigma ){\gamma }^{A_{\sigma (1)}}\cdots {\gamma }^{A_{\sigma (p)}},$

where $S_p$ is the permutation group of $\{1,\dots ,p\}$.

Thus ${\gamma }_A{\gamma }_B+{\gamma }_B{\gamma }_A=2{\eta }_{AB}I$. Equivariance makes this action independent of the chosen spin frame. This is the convention used in Hamilton, §3.1 and §3.6, and Gauntlett–Pakis, Appendix A.

The lift map is displayed by the commutative square $\begin{array}{ccc}P& \stackrel{\mathrm{\Lambda }}{\to }& F_{g,\tau }^{+}M\\ \downarrow & & \downarrow \\ M& \stackrel{{\mathrm{i}\mathrm{d}}_M}{\to }& M\end{array}$. Its TikZ source is

The square commutes because $\mathrm{\Lambda }$ is a bundle map over the identity of $M$; its fibers are the two-element fibers of the covering ${\lambda }^{+}$. The diagram records the definition only and is not used as a proof of existence.