<p>Conformal manifolds <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> are open subsets of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> endowed with the metric <Equation ID="Equ3"> <EquationSource Format="TEX">$\begin{aligned} g_\lambda =\frac{dx_1^2+\ldots +dx_n^2}{\lambda ^2} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>g</mi> <mi>λ</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> </mrow> <msup> <mi>λ</mi> <mn>2</mn> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is called the conformal function. We show that there exists the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Dirac operator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$D_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, acting on functions valued by the Clifford algebra on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$M_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation>. The operator behaves similarly to the usual Euclidean Dirac operator. We develop <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-dependent potential theory for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Delta _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> on conformal manifolds, prove refined Poincaré lemmata, and establish Helmholtz-type decompositions for multivector fields.</p>

Dirac Operators on Conformal Manifolds

期刊论文
发表日期：

2025年11月14日

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H. Orelma,
N. Vieira

摘要

Conformal manifolds $M_\lambda $ $M_{λ}$ are open subsets of $\mathbb {R}^n$ $R^{n}$ endowed with the metric $\begin{aligned} g_\lambda =\frac{dx_1^2+\ldots +dx_n^2}{\lambda ^2} \end{aligned}$ $\begin{matrix} g_{λ} = \frac{d x_{1}^{2} + \dots + d x_{n}^{2}}{λ^{2}} \end{matrix}$ where $\lambda $ $λ$ is called the conformal function. We show that there exists the $\alpha $ $α$ -Dirac operator $D_\alpha $ $D_{α}$ , with $\alpha \in \mathbb {R}$ $α \in R$ , acting on functions valued by the Clifford algebra on $M_\lambda $ $M_{λ}$ . The operator behaves similarly to the usual Euclidean Dirac operator. We develop $\alpha $ $α$ -dependent potential theory for $\Delta _\alpha $ $Δ_{α}$ on conformal manifolds, prove refined Poincaré lemmata, and establish Helmholtz-type decompositions for multivector fields.