Several Dirac Operator in parabolic geometry

TL;DR

This thesis constructs and analyzes sequences of invariant differential operators starting from the Dirac operator in parabolic geometry, extending known results in Clifford analysis to higher dimensions and multiple variables.

Contribution

It introduces a new framework using parabolic geometry to describe and analyze sequences of Dirac-type operators in multiple variables, including explicit formulas for specific cases.

Findings

Sequences of invariant differential operators are constructed starting with the Dirac operator.

The structure of these sequences is characterized for odd dimensions and conjectured for even dimensions.

Explicit formulas are derived for the case of two variables (k=2).

Abstract

In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,\S)\to C^\infty((\R^n)^k,\S)$ ($\S$ is the spinor module). This operator is the Cauchy-Riemann operator for n=2 and its resolution is the Dolbeault complex. For higher n, the resolution of D is not known in general. While this problem was treated many times in the language of Clifford analysis and some partial results are known, we give a description of this operator in Parabolic geometry, which is a special type of Cartan geometry modeled on $G/P$, where P is a Parabolic subgroup of G. We construct sequences of invariant differential operators starting with the Dirac operator in several variables and assume that these sequences coinside in some cases with the resolution. We describe the structure of these sequences precisely in case the dimension $n$ is odd and give a conjecture that these sequences have similar structure for n even, $k\leq n/2$ (the s.c. {\it stable range}). We also give some information about these sequences in case n even, k>n/2. In the last chapter, explicite formulas for the operators are derived for the case k=2.