Lorentzian spin-lift interface

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $(M,g,N,o)$ be an oriented and time-oriented smooth Lorentzian four-manifold, and let $P_g^{+}(M)$ be its smooth oriented time-oriented orthonormal frame bundle from the oriented-frame-bundle proposition. Set ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(1,3):=SL(2,\mathbb{C})$ and let $\rho :SL(2,\mathbb{C})\to S{O}^{+}(1,3)$ be the Hermitian-matrix covering homomorphism of the Hermitian-matrix double-cover proposition. A Lorentzian spin structure is a smooth right principal $SL(2,\mathbb{C})$-bundle $\tilde{P}\to M$ together with a smooth map $\lambda :\tilde{P}\to P_g^{+}(M)$ over the identity of $M$ such that

$\lambda (ua)=\lambda (u)\rho (a)(u\in \tilde{P},a\in SL(2,\mathbb{C})),$

and the restriction of $\lambda$ to each fiber is a surjective two-sheeted covering map. The map is over the identity, meaning ${\pi }^{+}\circ \lambda =\tilde{\pi }$; the square below displays this base condition and is not used as a proof.

A smooth finite-dimensional complex representation here consists of a finite-dimensional complex vector space $V_{\sigma }$ and a smooth group homomorphism $\sigma :SL(2,\mathbb{C})\to GL(V_{\sigma })$, where $GL(V_{\sigma })$ is the group of invertible complex-linear maps on $V_{\sigma }$, with the smooth structure inherited as an open subset of the real vector space of complex-linear endomorphisms of $V_{\sigma }$. Its associated vector bundle is the quotient $\tilde{P}{\times }_{\sigma }{V}_{\sigma }:=(\tilde{P}\times V_{\sigma })\mathrm{/}\sim ,$ where $\sim$ is the equivalence relation generated by $(ua,v)\sim (u,\sigma (a)v)$ for $u\in \tilde{P}$, $a\in SL(2,\mathbb{C})$ and $v\in V_{\sigma }$; equivalently $(u,v)\sim (ua,\sigma (a{)}^{-1}v)$, using the right-principal associated-bundle convention. Write $[u,v]$ for the class of $(u,v)$. Since every $\sigma (a)$ is complex-linear, the fiberwise vector-space operations $[u,v]+[u,w]=[u,v+w]$ and $z[u,v]=[u,zv]$ for $z\in \mathbb{C}$ are independent of the representative. The associated-bundle local-triviality proposition gives local triviality; on an overlap with local sections satisfying $s_j(x)=s_i(x)a_{ij}(x)$, the smooth principal charts give transition maps $(x,v)\mapsto (x,\sigma (a_{ij}(x))v)$. These are smooth because $\sigma$ is smooth and define the smooth vector-bundle structure. The defining spinor representation here is $V_{\sigma }={\mathbb{C}}^2$ with the action $\sigma (a)v=av$ for $a\in SL(2,\mathbb{C})$ and column vectors $v\in {\mathbb{C}}^2$. It is a smooth group homomorphism because matrix multiplication is smooth and $(ab)v=a(bv)$. Its associated bundle is called the rank-two complex spinor bundle in this convention; a spinor field is a smooth section of that bundle.

To define a lift of a Lorentz connection, first supply a principal connection $H^{+}\subset T{P}_g^{+}(M)$ on the frame bundle: it is a smooth vector subbundle complementary to $V{P}^{+}:=\ker (d{\pi }^{+})$ and invariant under the right $S{O}^{+}(1,3)$ action, as in the principal-connection definition. This $H^{+}$ is additional global data; the local affine connection and local matrix $\omega$ in the local tetrad-connection definition do not determine it unless they are specified as restrictions of a chosen global metric-compatible affine connection whose induced principal connection is the supplied $H^{+}$. A lift is a supplied principal connection $\tilde{H}\subset T\tilde{P}$, likewise a smooth right-invariant horizontal complement to $V\tilde{P}:=\ker (d\tilde{\pi })$, such that $d{\lambda }_u({\tilde{H}}_u)=H_{\lambda (u)}^{+}$ at each $u\in \tilde{P}$. It induces a covariant derivative on the associated vector bundle by horizontal differentiation: a section is represented by a smooth equivariant map $f:\tilde{P}\to V_{\sigma }$ with $f(ua)=\sigma (a{)}^{-1}f(u)$, and its derivative along a smooth base vector field $X$ at $x$ is ${\mathrm{\nabla }}_{X}s(x):=[u,d{f}_u({\tilde{X}}_u^H)]$ for $u\in {\tilde{P}}_x$. Here $s(x)=[u,f(u)]$, and the horizontal lift is the smooth field ${\tilde{X}}_u^H:=(d{\tilde{\pi }}_u{\mathrm{\mid }}_{{\tilde{H}}_u}{)}^{-1}{X}_x$; the restriction is a linear isomorphism because $\tilde{H}$ complements the vertical kernel. Right invariance and uniqueness give ${\tilde{X}}_{ua}^H=d{R}_a({\tilde{X}}_u^H)$. Differentiating $f\circ R_a=\sigma (a{)}^{-1}f$ therefore gives $d{f}_{ua}({\tilde{X}}_{ua}^H)=\sigma (a{)}^{-1}d{f}_u({\tilde{X}}_u^H)$, and the associated-bundle relation makes $[u,d{f}_u({\tilde{X}}_u^H)]$ independent of $u$. In principal charts this formula is smooth, is real $C^{\mathrm{\infty }}(M)$-linear in $X$ and complex-linear in $s$. For $h\in C^{\mathrm{\infty }}(M,\mathbb{C})$, the product rule applied to $(h\circ \tilde{\pi })f$ gives ${\mathrm{\nabla }}_X(hs)=X(h)s+h{\mathrm{\nabla }}_{X}s$. These are the covariant-derivative properties used here. The local Clifford-compatibility calculation is a separate proposition about supplied gamma matrices and does not follow from the existence of $\sigma$ alone. A spin structure, a global spinor field, and a lifted connection are extra data: this definition asserts neither existence nor uniqueness of a spin lift, global tetrad, or connection lift. The matrix group and covering homomorphism are the Lorentzian ones specified above.

Local Lorentz tetrad and connection

Oriented time-oriented orthonormal frame bundle

The Lorentzian spin double cover from Hermitian matrices

Frame bundles and associated vector bundles

Principal g bundle and associated fiber bundle

Associated bundle is locally trivial and functorial under pullback

Smooth principal $G$-bundle

Principal connections and vertical curvature

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)