Instanton properties of the characteristic connection <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\nabla }$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation> on an integrable <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> manifold as well as instanton condition of the torsion connection <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\nabla }$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation> on a Spin(7) manifold are investigated. It is shown that for an integrable <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> manifold with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\nabla }$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>-parallel Lee form the curvature of the characteristic connection is a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> instanton exactly when the torsion 3-form is <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\nabla }$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>-parallel. It is observed that on a compact Spin(7) manifold with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">${\nabla }$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation> closed torsion 3-form the torsion connection is a Spin(7) instanton if and only if the torsion 3-form is parallel with respect to the torsion connection.

Parallel torsion and $G_2, Spin(7)$ instantons

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2026年6月29日

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Stefan Ivanov,
Alexander Petkov,
Luis Ugarte

摘要

Instanton properties of the characteristic connection ${\nabla }$ $\nabla$ on an integrable $G_2$ $G_{2}$ manifold as well as instanton condition of the torsion connection ${\nabla }$ $\nabla$ on a Spin(7) manifold are investigated. It is shown that for an integrable $G_2$ $G_{2}$ manifold with ${\nabla }$ $\nabla$ -parallel Lee form the curvature of the characteristic connection is a $G_2$ $G_{2}$ instanton exactly when the torsion 3-form is ${\nabla }$ $\nabla$ -parallel. It is observed that on a compact Spin(7) manifold with ${\nabla }$ $\nabla$ closed torsion 3-form the torsion connection is a Spin(7) instanton if and only if the torsion 3-form is parallel with respect to the torsion connection.

Parallel torsion and \(G_2, Spin(7)\) instantons

摘要