Some observations on deformed Donaldson–Thomas connections

Let $X^{7}$ be a $7$-manifold with a $\rm G_{2}$-structure $\varphi\in\Omega^{3}(X)$. For the definition of $G_{2}$-structures, see for example [kawai2021mirror, Section 2.2]. We use the same sign convention there. Denote by $g$, $\mathrm{vol}$ and $\ast$ the induced Riemannian metric, volume form and Hodge star operator, respectively. Let $(L,h)\to X$ be a smooth complex Hermitian line bundle over $X$. We denote by $\mathcal{A}_{0}$ the affine space of Hermitian connections on $(L,h)$. Given $\nabla\in\mathcal{A}_{0}$, we regard its curvature $F_{\nabla}$ as a $\sqrt{-1}\mathbb{R}$-valued closed $2$-form on $X$.

DDT connections appeared in the context of mirror symmetry. They were introduced in [lee2009geometric] as “mirrors” of calibrated (associative) submanifolds. Historically, deformed Hermitian Yang–Mills (dHYM) connections were introduced first in [lyz2000FM] as “mirrors” of special Lagrangian submanifolds. There is also a similar notion of dDT connections for a manifold with a ${\rm Spin}(7)$-structure ([lee2009geometric, kawai2021FM]). As the names indicate, dDT connections can also be considered as analogues of Donaldson–Thomas connections ($G_{2}$-instantons).

Thus it is natural to expect that dDT connections would have similar properties to associative submanifolds and $G_{2}$-instantons. We show that it is indeed the case in [kawai2020deformation, kawai2021mirror]. For example, the moduli space of dDT connections is $b^{1}$-dimensional and canonically orientable if we perturb the $G_{2}$-structure. Any dDT connection on a compact $G_{2}$-manifold is a global minimizer of the “mirror volume” and its value is topological by the “mirror” of associator equality. We could also prove similar statements in the ${\rm Spin}(7)$ case in [kawai2021deformationSpin(7), kawai2021mirror]. Moreover, dDT connections are given by critical points of the Chern-Simons type functional in [karigiannis2009hodge, Theorem 5.13]. The variational characterization is known only for the $G_{2}$ case, and no such characterization is known for the ${\rm Spin}(7)$ case.

This paper is organized as follows. In Section 2, we study the existence of a dDT connection. Known examples of dDT connections are either trivial or constructed in [lotay2020examples, fowdar2022examples], and are very few in number. So it would be important to consider the existence problem. We first see that the formal “large radius limit” of the defining equation of dDT connections is that of $G_{2}$-instantons. Thus it is natural to expect that dDT connections for a “sufficiently large” $G_{2}$-structure will behave like $G_{2}$-instantons. Moreover, it is known that any complex Hermitian line bundle admits a $G_{2}$-instanton on a compact holonomy $G_{2}$-manifold. Then we show the following from these facts.

In Section 3, we formulate the dDT equation in terms of a multi-moment map. The multi-moment map is a generalization of the moment map introduced in [Madsen2012multi, Madsen2013closed]. The dHYM equation is described as the zero of a certain moment map on a infinite dimensional symplectic manifold ([collins2021moment, Section 2], [collins2021survey, Section 2.1]). Analogously, we show that the dDT equation is described as the zero of a certain multi-moment map.

In Section 4, we study the gradient flow of the Karigiannis-Leung functional introduced in [karigiannis2009hodge] whose critical points are dDT connections. It is known that the gradient flow equation of the Chern-Simons functional on an oriented 3-manifold $X^{3}$ agrees with the ASD equation on $\mathbb{R}\times X^{3}$. This is an important observation in instanton Floer homology for 3-manifolds. We show that there is an analogous relation between dDT equations for $G_{2}$- and ${\rm Spin}(7)$-manifolds using the Karigiannis-Leung functional. This will establish a new link between 3, 4-manifold theory and $G_{2}$-, ${\rm Spin}(7)$-geometry, and we might define analogues of instanton Floer homology using dDT connections.

In this section, we show the existence of a dDT connection if a 7-manifold has full holonomy $G_{2}$ and the $G_{2}$-structure is “sufficiently large”.

Suppose that $(X,\varphi)$ is a compact holonomy $G_{2}$-manifold. Let $(L,h)\to X$ be a smooth complex Hermitian line bundle over $X$. Set

where $\nabla_{0}\in\mathcal{A}_{0}$ is any fixed connection and $\Omega^{1}$ is the space of 1-forms on $X$. Denote by $\mathcal{G}_{U}$ the group of unitary gauge transformations of $(L,h)$, which acts on $\mathcal{A}_{0}$. Explicitly,

where $\Omega^{0}_{\mathbb{C}}$ is the space of $\mathbb{C}$-valued smooth functions, and the action $\mathcal{G}_{U}\times\mathcal{A}_{0}\rightarrow\mathcal{A}_{0}$ is defined by $(\lambda,\nabla)\mapsto\lambda^{*}\nabla:=\lambda^{-1}\circ\nabla\circ\lambda$. When $\lambda=f\cdot\mathrm{id}_{L}$ for $f\in C^{\infty}(X,S^{1})$, we have

Thus the $\mathcal{G}_{U}$-orbit through ${\nabla}\in\mathcal{A}_{0}$ is given by ${\nabla}+\mathcal{K}_{U}\cdot\mathrm{id}_{L}$, where

Note that the curvature 2-form $F_{\nabla}$ is invariant under the action of $\mathcal{G}_{U}$.

Consider the family of $G_{2}$-structures

all of which induce holonomy $G_{2}$ metrics. The defining equation of dDT connections with respect to $\varphi_{r}$ is given by

Thus, formally taking the ”large radius limit”, which means the leading behaviour of $\mathcal{F}_{r}({\nabla})$ as $r\to\infty$, we obtain

This is exactly the defining equation of $G_{2}$-instantons. Thus it is natural to expect that dDT connections for a sufficiently large $G_{2}$-structure will behave like $G_{2}$-instantons. The following would be well-known for $G_{2}$-instantons on a smooth complex Hermitian line bundle, but we give the proof for completeness.

Using this, we can show the following.

It is known that there is a moment map picture in the dHYM case. In particular, the dHYM equation is described as the zero of a certain moment map on a infinite dimensional symplectic manifold. See for example [collins2021moment, Section 2] or the survey article [collins2021survey, Section 2.1]. Analogously, we show that the dDT equation is described as the zero of a certain multi-moment map. First, recall the definition of the multi-moment map in [Madsen2012multi, Madsen2013closed].

Let $X$ be a compact 7-manifold with a coclosed $G_{2}$-structure $\varphi$ ($d*\varphi=0$) and $(L,h)\to X$ be a smooth complex Hermitian line bundle over $X$. Let $\mathcal{A}_{0}$ be the space of Hermitian connections of $(L,h)$. Define a map $\mathcal{F}_{G_{2}}:\mathcal{A}_{0}\to{\sqrt{-1}}\Omega^{6}$ by

Then the space of dDT connections is given by $\mathcal{F}_{G_{2}}^{-1}(0)$. Denote by $\mathcal{G}_{U}$ the group of unitary gauge transformations of $(L,h)$ acting $\mathcal{A}_{0}$ canonically as in \tagform@2.1. Since $\mathcal{G}_{U}=C^{\infty}(X,S^{1})$, the Lie algebra ${\mathfrak{g}}_{U}$ of $\mathcal{G}_{U}$ is identified with the space ${\sqrt{-1}}\Omega^{0}$ of ${\sqrt{-1}}\mathbb{R}$-valued functions on $X$. Note that $\mathcal{P}_{\mathfrak{g}}=\Lambda^{2}{\mathfrak{g}}$ since $\mathcal{G}_{U}$ is abelian. Define a 3-form $\Theta\in\Omega^{3}(\mathcal{A}_{0})$ on $\mathcal{A}_{0}$ by

where ${\nabla}\in\mathcal{A}_{0}$ and $\alpha_{1},\alpha_{2},\alpha_{3}\in{\sqrt{-1}}\Omega^{1}=T_{\nabla}\mathcal{A}_{0}$. We first show the following as required in Definition 3.1.

We also need the following lemma.

Denote by $Z^{6}$ the space of closed 6-forms on $X$. Define a map $\iota_{Z^{6}}:Z^{6}\to\Lambda^{2}{\mathfrak{g}}_{U}^{*}$ by

for $\xi\in Z^{6}$ and $f_{1},f_{2}\in{\sqrt{-1}}\Omega^{0}={\mathfrak{g}}_{U}$.

Since we assume that $d*\varphi=0$, we see that ${\sqrt{-1}}\mathcal{F}_{G_{2}}({\nabla})\in Z^{6}$ for any ${\nabla}\in\mathcal{A}_{0}$. By Lemma 3.3, $\iota_{Z^{6}}$ is injective. Hence we have $\nu^{-1}(0)=\mathcal{F}_{G_{2}}^{-1}(0)$. In this sense, we can regard the dDT equation as the zero of a multi-moment map.

In the dHYM case, the “$\mathcal{J}$ functional” defined in [collins2021moment, Remark 2.15] or [collins2021survey, Lemma 2.6 (ii)] is convex along geodesics and the critical points are solutions of the dHYM equation. Hence it plays an important role in the existence problem.

In the dDT case, there is a functional whose critical points are dDT connections. See Section 4.2. However, no metric has yet been found that makes the functional convex along geodesics. Since no such results have been found for associative submanifolds, it might be difficult to relate the functional to the existence problem.

However, as we see in the next section, we have an observation as in the case of instanton Floer homology for 3-manifolds by using the functional in Section 4.2. We might develop the theory like instanton Floer homology using dDT connections.

It is known that the gradient flow equation of the Chern-Simons functional on an oriented 3-manifold $X^{3}$ agrees with the ASD equation on $\mathbb{R}\times X^{3}$. See for example [Donaldson2002, Section 2.5.3]. This is an important observation in instanton Floer homology for 3-manifolds. We show that there is an analogous relation between dDT equations for $G_{2}$- and ${\rm Spin}(7)$-manifolds.

Let $X^{7}$ be a 7-manifold with a $G_{2}$-structure $\varphi$ and $(L,h)\to X^{7}$ be a smooth complex Hermitian line bundle over $X^{7}$. Let $\{{\nabla}_{t}\}_{t\in\mathbb{R}}$ be a family of Hermitian connections of $(L,h)\to X^{7}$. We identify this with a connection $\widetilde{\nabla}$ of $\pi^{*}L\to\mathbb{R}\times X^{7}$, where $\pi:\mathbb{R}\times X^{7}\to X^{7}$ is the projection. If we set

where $a_{t}\in\Omega^{1}(X^{7})$, we have $\widetilde{\nabla}=\pi^{*}{\nabla}_{0}+{\sqrt{-1}}\pi^{*}a_{t}\cdot\mathrm{id}_{\pi^{*}L}$ and the curvature $F_{\widetilde{\nabla}}$ of $\widetilde{\nabla}$ is given by