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Symmetric analogues of Rarita-Schwinger equations. (English) Zbl 1025.58013

The Dirac operator and twisted Dirac operators have been studied extensively during the last decades and were used in many other branches of mathematics and mathematical physics. Rarta-Schwinger equation is the spin analogue of Dirac equation in dimension 4.
In the paper under review, a generalization of the classical Rarita-Schwinger equations for spin 3/2 fields to the case of spin fields with values in irreducible representation spaces with weight \(k+1/2\) is given. First, a suitable definition of the Rarita-Schwinger operator on any Riemannian manifold is settled and its properties with respect to the conformal group action are studied. The corresponding first order systems are elliptic and their fundamental solution is constructed. The representation character of polynomial solutions of these equations on a flat space and their relationships are described in details.

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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