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

Abstract Instanton properties of the characteristic connection $${\nabla }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> on an integrable $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> manifold as well as instanton condition of the torsion connection $${\nabla }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> on a Spin (7) manifold are investigated. It is shown that for an integrable $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> manifold with $${\nabla }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> -parallel Lee form the curvature of the characteristic connection is a $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> instanton exactly when the torsion 3-form is $${\nabla }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> -parallel. It is observed that on a compact Spin (7) manifold with $${\nabla }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> 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.