Some observations on deformed Donaldson-Thomas connections

TL;DR

This paper explores properties of deformed Donaldson-Thomas (dDT) connections on $G_2$-manifolds, revealing conditions for their existence, their description via multi-moment maps, and their relation to gradient flows and Floer homology analogues.

Contribution

It identifies key geometric and analytical features of dDT connections, including existence criteria, their characterization as zeroes of multi-moment maps, and links to gradient flow equations.

Findings

dDT connections exist on $G_2$-manifolds with full holonomy and large $G_2$-structure

dDT equations are zeros of a multi-moment map

Gradient flow of a Chern-Simons type functional aligns with ${ m Spin}(7)$ dDT equations

Abstract

A deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ 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 $G_2$-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 $G_2$ and the $G_2$-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 ${\rm Spin}(7)$ 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.