Lorentzian spin structure

Definition

Let $(M,g)$ be an oriented and time-oriented Lorentzian $n$ -manifold of signature $(1,n-1)$ , with negative directions listed first, and let $P_{\mathrm{SO}^+(1,n-1)}(M)$ be its bundle of oriented, time-oriented pseudo-orthonormal frames. Write $\mathrm{Spin}^+(1,n-1)$ for the full preimage of $\mathrm{SO}^+(1,n-1)$ under the spin covering, as in the restricted spin group. A Lorentzian spin structure is a principal $\mathrm{Spin}^+(1,n-1)$ -bundle $P_{\mathrm{Spin}}\to M$ and an equivariant double covering

\Phi:P_{\mathrm{Spin}}\longrightarrow P_{\mathrm{SO}^+(1,n-1)}(M)

whose restriction to each fiber is induced by the spin double covering $\mathrm{Spin}^+(1,n-1)\to\mathrm{SO}^+(1,n-1)$ . Defining $\mathrm{Spin}^+$ by this full preimage keeps the fiber map two-to-one also in the low-dimensional signature $(1,1)$ .

Relation to the Riemannian definition

This is the signature- $(1,n-1)$ analogue of a Riemannian spin structure. The reduction to $\mathrm{SO}^+(1,n-1)$ records both space orientation and time orientation; without those choices the appropriate structure group is larger and the lifting problem changes.

For an oriented, time-oriented Lorentzian manifold, existence is again controlled by

w_2(TM)=0.

When structures exist, their isomorphism classes form a torsor for $H^1(M;\mathbb Z/2)$ , subject to the usual hypotheses on the manifold.

Associated geometry

The Lorentzian Levi–Civita connection lifts uniquely to the spin principal bundle. Choosing a real or complex representation of the signature-dependent spin group produces a Lorentzian spinor bundle and its spin connection. Clifford contraction of that connection gives a Lorentzian Dirac operator; on flat Minkowski spacetime this specializes to the Minkowski Dirac operator.

Convention warning

Labels such as $\mathrm{Spin}(1,n-1)$ and $\mathrm{Spin}(n-1,1)$ are not interchangeable until the sign convention for the Clifford algebra has been stated. Here $g$ has one negative direction and Clifford multiplication satisfies $c(v)^2=-g(v,v)$ .

References

Helga Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, Teubner, 1981. Bibliographic record. Relevant: Chapters 1–2.
Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §1.3.