Construction of Arbitrary Order Conformally Invariant Operators in Higher Spin Spaces

TL;DR

This paper completes the explicit construction of conformally invariant differential operators of arbitrary order in higher spin spaces, using convolution operators and representation theory, extending previous classifications.

Contribution

It provides explicit formulas for higher order conformally invariant operators in higher spin theory, including convolution type operators and their inverses, expanding the understanding of conformal invariance.

Findings

Constructed explicit higher order conformally invariant differential operators.

Introduced convolution type operators as inverses of differential operators.

Connected convolution operators with Rarita-Schwinger and Spin group representations.

Abstract

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slov\'{a}k has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for arbitrary order conformally invariant differential operators, where by conformally invariant we mean equivariant with respect to the conformal group of $\mathbb{S}^m$ acting in Euclidean space $\mathbb{R}^m$. We name these the fermionic operators when the order is odd and bosonic operators when the order is even. Our approach explicitly uses convolution type operators to construct conformally invariant differential operators. These convolution type operators are examples of Knapp-Stein operators and they can be considered as the inverses of the corresponding differential operators. Intertwining operators of these convolution type operators are provided and intertwining operators of differential operators follow immediately. This reveals that our convolution type operators and differential operators are all conformally invariant. This also gives us a class of conformally invariant convolution type operators in higher spin spaces. Their inverses, when they exist, are conformally invariant pseudo-differential operators. Further we use Stein Weiss gradient operators and representation theory for the Spin group to naturally motivate the Rarita-Schwinger operators.