Function Theory for Higher Spin Equations

This text examines functions on Rn (rather than spinor-valued functions) with values in the Clifford algebra in higher dimensions. There is a close connection between the higher dimensional analogues of the Dolbeault complex and properties of solutions of higher spin analogues of the Rarita-Schwinger equations. An examination of a number of related questions that are now well understood forms the main topic of this book. Two different methods are presented in parallel for describing function theory for higher spin equations. One is based on results and language developed over many decades in the Clifford analysis setting; the other on differential geometry, in particular, from recent research concerning invariant differential operators on manifolds with a given parabolic structure. The reader requires only a standard knowledge of real and complex analysis, along with the basics of analysis on manifolds. Facts needed from the classification of invariant first-order systems on conformal manifolds and from the representation theory of the orthogonal groups are summarized in two appendices. The material will be of interest to graduate students and researchers in analysis, geometry, PDEs, and mathematical physics (electrodynamics, higher spin physics, and string theory).