A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Abstract

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.where ρ := ρ r,s : Spin r,s → GL(Σ r,s ) is a complex spinor representation, Θ g : Spin g M → SO g M is a metric spin structure and Spin r,s ⊂ GL + m is the spin group. Main Theorem 1 (universal spinor bundle, cf. Theorem 2.25). There exists a finite dimensional vector bundleπ Σ SM :ΣM → J 1 π r,s such that for each metric g ∈ S r,s (M ), there existsῑ g such thatcommutes. Here, J 1 π r,s denotes the first jet bundle of π r,s . Moreover,π Σ SM carries a connection, a metric and a Clifford multiplication such thatῑ g is a morphism of (generalized) Dirac bundles (see Definition 2.27). In addition,π Σ SM is natural with respect to spin diffeomorphisms. ♦The claims of the last assertions mean that not only the vector bundle structure of any metric spinor bundle π g M can be recovered fromπ Σ SM , but also its spinorial connection, its metric and its Clifford multiplication, see Theorem 2.25 for the precise meaning and a proof. We would like to emphasize that the morphism j 1 (g) in (1.2) is the 1-jet of the metric g and that the naturality assertion is formulated with respect to diffeomorphisms and not just isometries, see Section 2.6 for details. arXiv:1504.01034v3 [math.DG] 6 Oct 2015 ∼ = 9 9 We setκ V := κ V • θ. ♦ Diagram (2.1) easily extends to spin manifolds as follows: Let {ρ r,s : Spin r,s → GL(Σ r,s )} r,s∈N be a fixed choice of Spin r,s -representations. The construction of κ V andκ V induces bundle maps κ M : GL + M → S r,s M,κ M : GL + M → S r,s M, (2.2) by setting κ M | GL + x M := κ TxM andκ M | GL + x M :=κ TxM for any x ∈ M . Definition 2.3 (universal spinor bundle). The map π Σ SM : ΣM → S r,s M [b, σ] →κ M (b), where ΣM := GL + M × ρr,s Σ r,s is called universal spinor (vector) bundle. We obtain the following diagram ΣM π Σ SM / / π Σ M 9 9 S r,s M π r,s / / M, (2.3)where π Σ M := π r,s • π Σ SM . The map π Σ M is called universal spinor (fiber) bundle of M . Its sections are called universal spinor fields.♦

Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

“…Such a relation between families of metrics with special holonomy and solutions of the constraint equations was already conjectured by Leistner and Lischewski, see [29]. We essentially show that the conditions in [29,Table 1] is satisfied if and only if the divergence condition (31) in our Appendix D is satisfied.…”

“…However we were told that there was also work by Bismut. The concepts were later properly formalized under the name 'universal spinor bundle' in [4] and [31], where a finite-dimensional fiber bundle with a partial connection is constructed whose sections correspond to the elements of F such that the parallel transport corresponds to the BBGM parallel transport. The connection is given in terms of horizontal spaces H (g,ϕ) , i.e.…”

“…The lemma follows from the construction of the universal spinor bundle given in [31]. The strategy in that paper is as follows: Let π S : SM → M be the bundle of positive definite symmetric bilinear forms on T M. A complex vector bundle π univ : M → SM is constructed, called the universal spinor bundle, which carries a scalar product, a Clifford multiplication with vectors in T M and partial connection on M with respect to π S : SM → M. The Clifford multiplication is given by a bilinear map cl g :…”

Pin groups in general relativity

“…It is based on the elementary fact that the twofold spin cover of the orthonormal frame bundle O g (M ) has to be compatible with a twofold cover of the full frame bundle F (M ). Although we derived this fact from the setting outlined in §4 (going back to [1] in the Riemannian and [9,11,15] in the Lorentzian case), the use of double covers of the full frame bundle -and hence our conclusion that only two Pin groups are admissible -is common to many other approaches, such as the more 'global' formalism developped in [4,5,6]. This is not a mere coincidence.…”

Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors

“…However we were told that there was also work by Bismut. The concepts were later properly formalized under the name 'universal spinor bundle' in [3] and [29], where a finite-dimensional fiber bundle with a partial connection is constructed whose sections correspond to the elements of F such that the parallel transport corresponds to the BBGM parallel transport. The connection is given in terms of horizontal spaces H (g,ϕ) , i.e.…”