The higher spin Dirac operators on 3-dimensional manifolds. (English) Zbl 1021.53026

In physics the classical Dirac operator is used to describe particles with spin \(1/2\). For describing particles of higher spin one needs higher spin Dirac operators. In order to define these spin Dirac operators, it makes sense to restrict to the case of \(3\)-dimensional manifolds. One reason for restricting to dimension \(n=3\) is that the representation theory for \(Spin(3)=SU(2)\) is easier than for \(Spin(n)\), \(n>3\). To any non-negative integer \(m\) there is unique \(m+1\)-dimensional irreducible complex representation \(V_m\) of \(Spin(3)\). The corresponding associated bundle is the bundle \(S_m\) of spinors of spin \(m/2\). The higher spin Dirac operators \(D_m^0\) resp. \(D_m^\pm\) are certain first order differential operators mapping section of \(S_m\) to sections of \(S_m\) resp. \(S_{m\pm 1}\). For example \(D_1^0\) is the classical Dirac operator.
The article by Homma gives a well written definition and presentation of the higher spin Dirac operators. The main result of the article are Bochner type identities for such higher spin Dirac operators. As a corollary one obtains eigenvalue estimates for certain Laplace type operators. In the last sections the spectrum of the higher spin Dirac operators \(D_m^0\) on the 3-dimensional sphere with constant curvature are calculated, using representation theory. A variation also yields the spectra of the Laplace-type operators mentionned above.


53C27 Spin and Spin\({}^c\) geometry
53C80 Applications of global differential geometry to the sciences
58J90 Applications of PDEs on manifolds
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