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<p>A <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{Z}_{2}$</EquationSource> </InlineEquation>-harmonic spinor on a 3-manifold <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>Y</mi> </math></EquationSource> <EquationSource Format="TEX">$Y$</EquationSource> </InlineEquation> is a solution of the Dirac equation on a bundle that is twisted around a submanifold <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Z</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{Z}$</EquationSource> </InlineEquation> of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{Z}_{2}$</EquationSource> </InlineEquation>-harmonic spinors over the space of parameters <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(g,B)$</EquationSource> </InlineEquation> consisting of a metric and perturbation to the spin connection. The main result states that near a <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{Z}_{2}$</EquationSource> </InlineEquation>-harmonic spinor with <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Z</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{Z}$</EquationSource> </InlineEquation> smooth, the universal moduli space projects to a codimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Z</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{Z}$</EquationSource> </InlineEquation>, necessitating the use of the Nash-Moser Implicit Function Theorem.</p>

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Deformations of \(\mathbb{Z}_{2}\)-Harmonic Spinors on 3-Manifolds

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Gregory J. Parker

摘要

A Z 2 $\mathbb{Z}_{2}$ -harmonic spinor on a 3-manifold Y $Y$ is a solution of the Dirac equation on a bundle that is twisted around a submanifold Z $\mathcal{Z}$ of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of Z 2 $\mathbb{Z}_{2}$ -harmonic spinors over the space of parameters ( g , B ) $(g,B)$ consisting of a metric and perturbation to the spin connection. The main result states that near a Z 2 $\mathbb{Z}_{2}$ -harmonic spinor with Z $\mathcal{Z}$ smooth, the universal moduli space projects to a codimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set Z $\mathcal{Z}$ , necessitating the use of the Nash-Moser Implicit Function Theorem.