Some observations on deformed Donaldson–Thomas connections

A deformed Donaldson–Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a <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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. It can also be considered as an analogue of a <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. In this paper, we see that some important observations that appear in other geometric problems are also found in the dDT case as follows. (1) A dDT connection exists if a 7-manifold has full holonomy <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> and the <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> -structure is “sufficiently large”. (2) The dDT equation is described as the zero of a certain multi-moment map. (3) The gradient flow equation of a Chern–Simons type functional of Karigiannis and Leung, whose critical points are dDT connections, agrees with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Spin</mml:mo> <mml:mo>(</mml:mo> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> version of the dDT equation on a cylinder with respect to a certain metric on a certain space. This can be considered as an analogue of the observation in instanton Floer homology for 3-manifolds.