Decomposition of the polynomial kernel of arbitrary higher spin Dirac operators

Abstract

In a series of recent papers, we have introduced higher spin Dirac operators, which are generalisations of the classical Dirac operator. Whereas the latter acts on spinorvalued functions, the former act on functions taking values in arbitrary irreducible half-integer highest weight representations for the spin group. In this paper, we describe how the polynomial kernel spaces of such operators decompose in irreducible representations of the spin group. We will hereby make use of results from representation theory.

The higher spin Laplace operator in several vector variables

“…We explain how Branson's method, based on generalised gradients [32], can be formulated on the level of a transvector algebra, as this allows us to also obtain results on the space of solutions. Despite being a partial result only, the solutions obtained in this paper (the so-called solutions of type A) are important for two reasons: first of all, our experience from the first-order case has taught us that the 'missing' solutions can be obtained from the ones described in the paper using twistor operators (see [17]). Secondly, the solutions of type A are exactly the solutions satisfying the socalled tranversality conditions which arise in the framework of the celebrated ambient method (the mathematical framework underlying both the AdS/CFT correspondence and the tractor calculus).…”

“…Whereas this theory originally focused most of its attention on the Dirac operator, see for instance [12,23], the area of interest has substantially grown once it became clear that also higher spin Dirac operators and super versions thereof were elegantly described using similar techniques. We refer for instance to [5,6,11,16,17] for the case of first-order operators. As for the case of second-order operators, for which we refer to e.g.…”

“…Next, we will also focus on a special subspace of ker(D λ ), referred to as solutions of type A (a notion introduced in the framework of Rarita-Schwinger operators, see [5]). These are important for 2 reasons: first of all, our experience from the first-order case has taught us that full kernel (polynomial solutions) can be obtained from the type A solutions in terms of a 'translation principle' using twistor operators (see [18]). Secondly, in the case of symmetric tensor fields defined on a space of signature (1, m + 1), the solutions of type A are exactly the solutions satisfying the so-called transversality conditions arising in the ambient method (this is the mathematical framework underlying the AdS/CFT correspondence and the tractor calculus).…”

Spectra of the Rarita-Schwinger operator on some symmetric spaces

“…For example, polynomial solutions, fundamental solution and Clifford-Cauchy kernel were studied in [7], [8]. Furthermore, the generalization to the higher spin operators is in progress, [11], etc. On non-Euclidean spaces, the harmonic analysis of the Dirac operator has been studied a lot.…”

General Massless Field Equations for Higher Spin in Dimension 4

“…Hence it was a surprise that this is not true for the Rarita-Schwinger case (see [5]). In general, it was shown that the structure of the space of homogeneous solutions for higher spin became quite complicated (see [12,11]). On the other hand, the space of homogeneous solutions of massless field equations was shown to be an irreducible module in many cases (see [1,2,17]).…”