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Causalité des théories de supergravité. (Causality of supergravity theories). (French) Zbl 0604.53047
Élie Cartan et les mathématiques d’aujourd’hui, The mathematical heritage of Elie Cartan, Sémin. Lyon 1984, Astérisque, No.Hors Sér. 1985, 79-93 (1985).
[For the entire collection see Zbl 0573.00010.]
This paper is an interesting review of the status of supergravity theories with respect to coherence and causality: coherence means the complete integrability of the system of partial differential equations of the gauge invariant theory considered, and causality is preserved when the characteristic cone is on or inside the metrical cone determined by the metric.
In Simple Supergravity coupling the Einstein-Cartan metric with a spinor field \(\psi_{\rho}\) having their values in a \({\mathbb{Z}}_ 2\) graded commutative algebra \({\mathcal A}\), coherence can be established with a convenient choice of gauge conditions. Existence theorems are obtained when \({\mathcal A}\) is restricted to a Grassmann algebra \({\mathcal G}\) and causality follows when \({\mathcal G}\) has a finite number of generators.
For Kaluza-Klein type theories of \(d=4+n\) dimensions to be coherent, one requires the introduction of additional fields. In the case of Cremmer- Julia-Scherk theory \((d=11)\), coherence is obtained by the addition of a 3-form taking values in \({\mathcal A}^+\) and causality may be restored with a convenient choice of gauge conditions.
Reviewer: S.Kichenassamy
MSC:
53C80 Applications of global differential geometry to the sciences
83E50 Supergravity
58J45 Hyperbolic equations on manifolds