Lorentzian Type IIB Spinor Bundle Data

Let $M$ be a smooth $10$-manifold with a smooth Lorentzian metric $g$ of inertia $(9,1,0)$, an orientation, and a time orientation, the latter being a smooth choice of one of the two connected components of the timelike cone in each tangent space. Write $V={\mathbb{R}}^{10}$ and $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$. Define $Q$ to be the set of linear isometries $u:(V,\eta )\to (T_{x}M,g_x)$ over points $x\in M$ that preserve orientation and the chosen future cone, with projection ${\pi }_Q(u)=x$. Its right action is $u\cdot A=u\circ A$ for $A\in {\mathrm{S}\mathrm{O}}^{+}(9,1)$, the subgroup of $O(V,\eta )$ preserving orientation and time orientation.

Let $\mathrm{Cl}(V,\eta )$ be the unital real associative algebra generated by $V$ with relations $vw+wv=2\eta (v,w)1$, and write ${\mathrm{Cl}}^0(V,\eta )$ for its even subalgebra, spanned by products of an even number of generators. For an even product $s=v_1\cdots v_{2k}$ of vectors with $\eta (v_i,v_i)=\pm 1$, let $\lambda (s)v:=sv{s}^{-1}$. The subgroup of such invertible even Clifford elements whose conjugation maps lie in ${\mathrm{S}\mathrm{O}}^{+}(9,1)$ is ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(9,1)\subset {\mathrm{Cl}}^0(V,\eta )$; $\lambda$ is its connected two-sheeted covering homomorphism with kernel $\{1,-1\}$.

The covering, smoothness and connectedness assertions here have the following explicit justification in these Clifford conventions. The group of even normalized-vector products is the full-even Spin group of Pin and full-even Spin are topological double covers; for even elements its twisted adjoint is precisely $v\mapsto sv{s}^{-1}$. Thus the map to full $SO(9,1)$ is onto with central kernel $\{\pm 1\}$ and is a topological double cover. Restricting that actual cover to the open subgroup $S{O}^{+}(9,1)$ gives the asserted two sheets over this subgroup; connectedness still requires a separate argument below.

For the smooth assertion, write ${\epsilon }_a=\eta (e_a,e_a)$ and ${\mathfrak{o}}_{\eta }=\{L:L^T\eta +\eta L=0\}$. The convergent matrix logarithm near the identity maps $S{O}^{+}(9,1)$ into ${\mathfrak{o}}_{\eta }$, and matrix exponential maps a sufficiently small neighborhood of zero in ${\mathfrak{o}}_{\eta }$ back into this group; their absolutely convergent power-series identities make them inverse local charts. The same series may be differentiated to every order on smaller norm balls: the differentiated tails are bounded by convergent scalar geometric-series derivatives. Hence these are smooth charts, with smooth matrix multiplication and inversion. For $L\in {\mathfrak{o}}_{\eta }$ put $b(L)={\sum }_{a<b}\frac{L_{ab}}{2{\epsilon }_b}{e}_a{e}_b\mathrm{.}$ The PBW basis in PBW basis and exterior associated graded for a Clifford algebra makes the bivectors independent. The Clifford relation gives $[e_a{e}_b,e_c]=2({\eta }_{bc}{e}_a-{\eta }_{ac}{e}_b)$, so $[b(L),v]=Lv$; this is a linear isomorphism from ${\mathfrak{o}}_{\eta }$ to the bivector space. The explicit section of Local Spin lift from the matrix logarithm is $s(A)={\exp }_{\mathrm{C}\mathrm{l}}(b(\log A))$. That supplier proves it belongs to the full-even Spin group and satisfies $s(A)vs(A{)}^{-1}=Av$. The preceding convergent-series argument proves that this section is smooth, strengthening its stated continuity here by an explicit calculation. The two cover charts are exactly $s(A)$ and $-s(A)$; translating them by any fixed lift gives all charts. Their transition maps are smooth because Clifford multiplication and inversion are smooth on invertible elements. Equivalently these are compatible embedded charts in the finite-dimensional Clifford algebra, with inverse the smooth adjoint matrix map. Thus $\lambda$ is a smooth local diffeomorphism, not only a continuous covering. Its derivative is $d\lambda (b(L))=L,$ obtained by differentiating $\lambda ({\exp }_{\mathrm{C}\mathrm{l}}(tb(L)))=\exp (tL)$, and is a Lie-algebra isomorphism with the displayed inverse $b$.

The set $Q$ has the smooth principal-bundle structure used below. At any $x\in M$, choose a positively oriented, future-directed $g_x$-orthonormal basis and extend it to a smooth local frame $V_0,\dots ,V_9$ on an open neighborhood $U$ of $x$. After shrinking $U$, define $u_0=\frac{V_0}{\sqrt{-g(V_0,V_0)}},W_a=V_a-{\sum }_{b<a}{\eta }_{bb}g(V_a,u_b)u_b,u_a=\frac{W_a}{\sqrt{g(W_a,W_a)}}(1\le a\le 9)\mathrm{.}$ The denominators remain positive because $V_0$ is timelike and the successive orthogonalized spatial vectors are spacelike at $x$; shrinking also keeps $u_0$ future-directed and the frame orientation positive. Thus $u=(u_0,\dots ,u_9)$ is a smooth local section of $Q$. The map $(y,A)\mapsto u(y)\circ A$ identifies $U\times {\mathrm{S}\mathrm{O}}^{+}(9,1)$ with $Q{\mathrm{\mid }}_U$, with inverse $q\mapsto ({\pi }_Q(q),u({\pi }_Q(q){)}^{-1}\circ q)$. On overlaps, two such sections differ by a smooth map into ${\mathrm{S}\mathrm{O}}^{+}(9,1)$, so the transition maps are smooth left multiplications on the group coordinate and commute with its right action. These charts make $Q$ a smooth right principal ${\mathrm{S}\mathrm{O}}^{+}(9,1)$-bundle.

To prove connectedness specifically in signature $(9,1)$, apply Polar decomposition of a real indefinite orthogonal matrix after ordering the positive and negative axes. Its path joins any $A\in S{O}^{+}(9,1)$ to a block matrix $\mathrm{diag}(1,R)$ with $R\in SO(9)$. The timelike sign remains positive along that path, since the time component of a timelike unit vector cannot cross zero. The spatial group $SO(9)$ is path connected: rotate $R{e}_9$ to $e_9$ in their plane (use a half-turn in a plane through $e_9$ for its antipode), then repeat on the orthogonal complement; the final one-dimensional determinant-one block is the identity. Thus $S{O}^{+}(9,1)$ is path connected. Lift any such path from $I$ to $A$ starting at $1$, using finitely many consecutive evenly covered neighborhoods along its compact parameter interval. This reaches one of the two lifts of $A$. The two lifts are in the same path component: for two spacelike orthonormal vectors $e_1,e_2$, the curve $q(\theta )=\cos \theta +\sin \theta e_1{e}_2=e_1(\cos \theta e_1+\sin \theta e_2),0\le \theta \le \pi ,$ is a normalized even product, begins at $1$, ends at $-1$, and projects to spatial rotations, hence stays in the restricted preimage. Multiply this curve by the reached lift to reach the other lift. Every point of the restricted preimage is therefore joined to $1$; it is connected. This argument uses the actual spacelike two-plane and makes no connectedness claim for the exceptional signature $(1,1)$.

A spin structure is a smooth right principal ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(9,1)$-bundle $P\to M$ and a smooth map $\tilde{\lambda }:P\to Q$ over ${\mathrm{id}}_M$ satisfying $\tilde{\lambda }(ps)=\tilde{\lambda }(p)\lambda (s)$ and restricting on every fiber to the two-sheeted covering $\lambda$.

Supply a real $32$-dimensional Dirac Clifford module, meaning a unital algebra representation $c:\mathrm{Cl}(V,\eta )\to {\mathrm{End}}_{\mathbb{R}}({\mathrm{\Delta }}_{9,1})$, with $c(v)c(w)+c(w)c(v)=2\eta (v,w)I$. Restricting this representation to ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(9,1)\subset {\mathrm{Cl}}^0(V,\eta )$ defines the smooth group representation $\rho (s):=c(s)$; it is smooth because $c$ is linear on the finite-dimensional Clifford algebra, and its inverse is $\rho (s^{-1})$. In a positively oriented orthonormal basis $(e_0,\dots ,e_9)$ of $V$, define the chirality operator ${\mathrm{\Gamma }}_{11}:=c(e_0)\cdots c(e_9)$. It satisfies ${\mathrm{\Gamma }}_{11}^2=I$ and is independent of the oriented orthonormal basis because the Spin action commutes with the even volume element. Put ${\mathrm{\Delta }}_{9,1}^{\pm }:=\ker ({\mathrm{\Gamma }}_{11}\mp I)$; an invertible vector generator anticommutes with ${\mathrm{\Gamma }}_{11}$ and maps these eigenspaces isomorphically, so each has real dimension $16$. Since Spin elements are even, $\rho$ preserves each summand; write ${\rho }_{\pm }=\rho {\mathrm{\mid }}_{{\mathrm{\Delta }}_{9,1}^{\pm }}$.

Using the right-principal associated-bundle convention $[ps,\psi ]=[p,\rho (s)\psi ]$, define the full spinor bundle $S:=P{\times }_{\rho }{\mathrm{\Delta }}_{9,1}$ and its chiral subbundles $S_{\pm }:=P{\times }_{{\rho }_{\pm }}{\mathrm{\Delta }}_{9,1}^{\pm }$. A section is a smooth map $ϵ:M\to S$ with bundle projection ${\pi }_S\circ ϵ={\mathrm{id}}_M$; local sections are defined over open subsets in the same way. If $p\in P_x$ projects to the orthonormal frame $\tilde{\lambda }(p)=(e_a)$ and $Y=Y^a{e}_a\in T_{x}M$, define Clifford multiplication by $Y$ by $Y\cdot [p,\psi ]=[p,Y^{a}c(e_a)\psi ]$. It is independent of $p$, since $\rho (s)c(v)\rho (s{)}^{-1}=c(\lambda (s)v)$. For a covector $\alpha$, set $\alpha \cdot ϵ:={\alpha }^{\mathrm{♯}}\cdot ϵ$ using ${\alpha }^{\mathrm{♯}}=g^{-1}(\alpha ,\cdot )$. For $F\in {\mathrm{\Omega }}^p(M)$ define $F\cdot ϵ=(1\mathrm{/}p!)F_{a_1\dots a_p}{\mathrm{\Gamma }}^{a_1\dots a_p}ϵ$, where ${\mathrm{\Gamma }}^{a_1\dots a_p}$ is the unit-weight antisymmetrized product of the raised orthonormal generators.

The smooth structure on $S$ is the associated vector-bundle atlas: over a cover with smooth local spin frames $s_i$, use $(x,v)\mapsto [s_i(x),v]$. If $s_j=s_i{a}_{ij}$ on an overlap, the coordinate transition from $s_i$ to $s_j$ is $(x,v_i)\mapsto (x,\rho (a_{ij}(x){)}^{-1}{v}_i)$, which is smooth because $a_{ij}$ and $\rho$ are smooth. The restricted transition maps likewise give the smooth subbundle structures on $S_{\pm }$.

For completeness, the Levi-Civita connection used below exists uniquely for this smooth nondegenerate metric. For smooth local vector fields define $K(X,Y,Z):=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)+g([X,Y],Z)+g([Z,X],Y)-g([Y,Z],X)\mathrm{.}$ Nondegeneracy of $g$ determines ${\mathrm{\nabla }}_X^{\mathrm{L}\mathrm{C}}Y$ by $2g({\mathrm{\nabla }}_X^{\mathrm{L}\mathrm{C}}Y,Z)=K(X,Y,Z)$ for every $Z$; in a chart its coefficients are ${\mathrm{\nabla }}_{{\mathrm{\partial }}_M}^{\mathrm{L}\mathrm{C}}{\mathrm{\partial }}_N={\mathrm{\Gamma }}^P{}_{MN}{\mathrm{\partial }}_P,{\mathrm{\Gamma }}^P{}_{MN}=\frac{1}{2}{g}^{PQ}({\mathrm{\partial }}_M{g}_{NQ}+{\mathrm{\partial }}_N{g}_{MQ}-{\mathrm{\partial }}_Q{g}_{MN})\mathrm{.}$ The inverse metric is smooth by Matrix inversion preserves $C^k$ regularity where the determinant is nonzero, so these local coefficients are smooth. The Lie-bracket product rules and symmetry of $g$ give $K(fX,Y,Z)=fK(X,Y,Z)$, $K(X,Y,fZ)=fK(X,Y,Z)$, and $K(X,fY,Z)=fK(X,Y,Z)+2X(f)g(Y,Z)$, so the construction satisfies the affine-connection rules of Affine connection on a smooth manifold. They also give $K(X,Y,Z)-K(Y,X,Z)=2g([X,Y],Z)$ and $K(X,Y,Z)+K(X,Z,Y)=2Xg(Y,Z)$, proving respectively zero torsion and metric compatibility. The same identities force this formula for any torsion-free metric-compatible connection, proving uniqueness.

In a local oriented, time-oriented orthonormal frame $(e_a)$ and a local spin lift $s:U\to P$, define the Levi-Civita connection one-form ${\omega }^a{}_b$ by ${\mathrm{\nabla }}_X^{\mathrm{L}\mathrm{C}}{e}_b={\omega }^a{}_b(X)e_a$; metric compatibility gives ${\omega }_{ab}:={\eta }_{ac}{\omega }^c{}_b=-{\omega }_{ba}$. The induced spin connection one-form is $A_s:=(d\lambda {)}^{-1}\omega$, where $d\lambda :\mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}(9,1)\to \mathfrak{s}\mathfrak{o}(9,1)$ is the Lie-algebra isomorphism of the double cover. If a spinor section is represented locally as $ϵ=[s,\psi ]$, define ${\mathrm{\nabla }}_X^Sϵ:=[s,X(\psi )+d\rho (A_s(X))\psi ]\mathrm{.}$ Under a change $s^{\mathrm{\prime }}=sa$, put $l=\lambda (a)$. Changing orthonormal frames in ${\mathrm{\nabla }}^{\mathrm{L}\mathrm{C}}e=e\omega$ and differentiating $e^{\mathrm{\prime }}=el$ gives ${\omega }^{\mathrm{\prime }}=l^{-1}\omega l+l^{-1}dl$. The proved Lie-algebra isomorphism satisfies $d\lambda (\mathrm{Ad}(a^{-1})B)=l^{-1}d\lambda (B)l$ by differentiating conjugation, and $d\lambda (a^{-1}da)=l^{-1}dl$ by differentiating the group homomorphism. Its injectivity therefore proves $A_{s^{\mathrm{\prime }}}=\mathrm{Ad}(a^{-1})A_s+a^{-1}da$. The local component is ${\psi }^{\mathrm{\prime }}=\rho (a{)}^{-1}\psi$. Differentiating this expression and the smooth representation gives $d{\psi }^{\mathrm{\prime }}+d\rho (A_{s^{\mathrm{\prime }}}){\psi }^{\mathrm{\prime }}=\rho (a{)}^{-1}(d\psi +d\rho (A_s)\psi );$ the derivative of $\rho (a{)}^{-1}$ cancels the $a^{-1}da$ term. Thus the local covariant derivatives glue to a smooth connection on the associated bundle. This proves the connection construction used here rather than presuming a smooth Lie covering from topological double-cover exactness alone. Differentiating $\rho (s)c(v)\rho (s{)}^{-1}=c(\lambda (s)v)$ shows that this connection is Clifford-compatible: ${\mathrm{\nabla }}_X^S(Y\cdot ϵ)=({\mathrm{\nabla }}_X^{\mathrm{L}\mathrm{C}}Y)\cdot ϵ+Y\cdot {\mathrm{\nabla }}_X^Sϵ\mathrm{.}$ The even Spin action and connection preserve $S_{\pm }$, while Clifford multiplication by a vector or one-form interchanges them. A homogeneous $p$-form preserves chirality for even $p$ and interchanges it for odd $p$, since ${\mathrm{\Gamma }}_{11}$ anticommutes with every one-form generator. This definition records supplied spin structure and Clifford-module data; it asserts neither existence of those data nor existence of a nonzero parallel spinor.

Smooth manifolds and their smooth charts

A smooth tensor field

Oriented smooth manifolds and oriented charts

Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms

Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form

Principal g bundle and associated fiber bundle

Associated bundles

Smooth sections, local sections, and support

Affine connection on a smooth manifold

Christoffel symbols of an affine connection

The Lie bracket of smooth vector fields

Leibniz rules for the Lie bracket with function multiples

Matrix inversion preserves $C^k$ regularity where the determinant is nonzero

Pin and full-even Spin are topological double covers

Local Spin lift from the matrix logarithm

Polar decomposition of a real indefinite orthogonal matrix

PBW basis and exterior associated graded for a Clifford algebra