Third Order Fermionic and Fourth Order Bosonic Operators

TL;DR

This paper develops and analyzes third order fermionic and fourth order bosonic conformally invariant operators in higher spin spaces, extending previous work on powers of the Dirac operator with explicit solutions and intertwining operators.

Contribution

It introduces explicit constructions of third and fourth order conformally invariant operators in higher spin spaces, including their fundamental solutions and intertwining operators.

Findings

Constructed third order fermionic operators.

Constructed fourth order bosonic operators.

Provided fundamental solutions and intertwining operators.

Abstract

This paper continues the work of our previous paper [8], where we generalize kth-powers of the Euclidean Dirac operator D_x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we reconsider the generalizations of D_x^3 and D_x^4 to higher spin spaces in the case the target space is a degree k homogeneous polynomial space. Constructions of 3rd and 4th order conformally invariant operators in higher spin spaces are given; these are the 3rd order fermionic and 4th order bosonic operators. Fundamental solutions and intertwining operators of both operators are also presented here. These results can be easily generalized to cylinders and Hopf manifolds as in [7].