The DT-instanton equation on almost Hermitian $6$-manifolds

The notions of holomorphic bundles and Hermitian Yang Mills connections have proven to be very fruitful in complex geometry. When considering Hermitian vector bundles, the Hitchin-Kobayashi correspondence [D, UY, LT] yields a relation between the algebro-geometric notion of a stable holomorphic vector bundle and the more differential geometric one of a Hermitian-Yang-Mills (HYM) connection. The goal of the present paper is to study some natural generalizations of these objects in almost complex geometry. The most well known of such generalizations are pseudo-holomorphic and pseudo-Hermitian-Yang-Mills (pHYM) connections. In specific situations, these have been studied by several authors, see [Charbonneau2016] and [Bryant2006] for example. The major goal of the current paper is to study a system of partial differential equations whose solutions give a further generalization of the notion of a HYM connection on real $6$-dimensional almost Hermitian manifolds. To the authors’ knowledge such equations first appeared in Richard Thomas’s thesis [Thomas1997] (page 29). These equations have also independently appeared in the physics literature, for instance in [Baulieu1998, Baulieu1998b, Iqbal2008] and references therein. More recently, the same equations were studied by Yuuji Tanaka ([Tanaka2008], [Tanaka2013], [Tanaka2014]) who constructed the only known (nontrivial) examples of solutions in [Tanaka2008]. These rely on a very general version of the Hitchin-Kobayashi correspondence and require the underlying almost Hermitian manifold to actually be Kähler. In that direction, our results give the first nontrivial solutions to these equations on non-Kähler almost Hermitian manifolds. For instance, one of the examples explored in this paper focuses on $\mathbb{F}_{2}$, the manifold of full flags in $\mathbb{C}^{3}$. In that example, we give nontrivial solutions to these equations for several almost Hermitian structures compatible with the nearly Kähler almost complex structure.

Let $(X,g,J)$ be an almost Hermitian manifold, ${\rm G}$ a compact semisimple¹¹1Similar equations to those considered here can be written if ${\rm G}$ not semisimple. However, for the sake of simplifying some statements we shall restrict to this case. Lie group, and $P\rightarrow X$ a principal ${\rm G}$-bundle. A connection $A$ on $P$ is called pseudo-holomorphic if its curvature $F_{A}$ is of type $(1,1)$ and pHYM if we further have

where $\Lambda F_{A}=\ast(F_{A}\wedge\omega^{2})$ denotes contraction with respect to the associated $2$-form $\omega(\cdot,\cdot)=g(J\cdot,\cdot)$. These notions are word by word adaptations of the respective notions in the case where $(X,g,J)$ is Hermitian. However, for the general almost Hermitian structure there may not exist (even locally) solutions to these equations, see [Bryant2006]. Similarly, in the non-integrable case there is no analogue of the function theory relating the existence of a (pseudo)-holomorphic connection with local holomorphic framings. When $X$ is real $6$-dimensional, there is a further generalization of the HYM equations which has a better general existence theory. This is an equation for a pair $(A,u)$, consisting of a connection $A$ on $P$ and a Higgs field $u\in\Omega^{0,3}(X,\mathfrak{g}_{P}^{\mathbb{C}})$, required to satisfy

where $\overline{\partial}_{A}^{*}=-\ast\partial_{A}\ast$ and $\ast$ denotes the $\mathbb{C}$-linear extension of the Hodge-$\ast$ operator associated with $g$. In this paper we shall refer to these as the DT-instanton equations, and call a pair $(A,u)$ solving them a DT-instanton. The reason for this nomenclature is the point of view adopted in Richard Thomas and Yuuji Tanaka’s work, where these equations are regarded as the basis for a possible differential geometric approach to the Donaldson-Thomas invariants, constructed by Thomas for Calabi-Yau $3$-folds using algebraic geometry, see [Thomas1997] and [Thomas2001]. Such a program is still far from being completed, but its success would extend the theory of DT-invariants to symplectic (or even almost Hermitian) real $6$-dimensional manifolds.

As it will become clear in the course of the article, a particular case for these equations happens when $(J,g)$ admits a certain compatible ${\rm SU}(3)$-structure. In that situation, the equations can be rewritten as in proposition 3, leading to a simplification of the analysis which can be carried out in several particular cases of interest. For instance, when the ${\rm SU}(3)$-structure is Calabi-Yau or nearly Kähler, the DT-instanton equations actually reduce to the pHYM ones, see proposition 5. These vanishing theorems further motivate the DT-instanton equations as being a natural generalization of the HYM ones which, for the generic almost Hermitian structure, we expect to have more solutions then the pHYM equations. To test these ideas, in this paper we solve these equations in very specific examples, namely for invariant almost Hermitian structures on the manifold of full flags in $\mathbb{C}^{3}$. In particular, we obtain the first examples of DT-instantons on a compact manifold with $\overline{\partial}_{A}^{*}u\neq 0$, and these are also the first non-trivial examples on non-Kähler manifolds. For future reference, DT-instantons $(A,u)$ with $A$ an irreducible connection and $\overline{\partial}_{A}^{*}u\neq 0$ will be called irreducible.

In fact, of these two almost complex structures, $J^{ni}$ is the only one with topologically trivial canonical bundle. As a consequence of this observation, for the almost Hermitian structures in $\mathcal{C}^{ni}$ we will be able to simplify the search for solutions to the DT-instanton equations 2.4–2.5 by making use of proposition 3. Indeed, in this case we are able to classify ${\rm SU}(3)$-invariant DT-instantons with gauge group ${\rm SO}(3)$. To state this classification we start with some preparation. Let $r\in(\mathfrak{t}^{2})^{*}$ be an integral weight of ${\rm SU}(3)$ and $e^{r}:T^{2}\rightarrow{\rm U}(1)$ the induced group homomorphism. Then, denote by $\lambda_{r}:T^{2}\rightarrow{\rm SO}(3)$ the homomorphism obtained by composing $e^{r}$ with the degree one embedding of ${\rm U}(1)$ as the maximal torus of ${\rm SO}(3)$. This can be used to construct the ${\rm SU}(3)$-homogeneous ${\rm SO}(3)$-bundles

All the ${\rm SU}(3)$-homogeneous ${\rm SO}(3)$-bundles on $\mathbb{F}_{2}$ are of this form, and in our first main result we classify ${\rm SU}(3)$-invariant DT-instantons on such bundles. As way of preparing for the statement it is important to note that the bundles $P_{r}$ are all reducible to the ${\rm U}(1)$-bundles $L_{r}$ associated with the homomorphism $e^{r}$. The Hitchin-Kobayashi correspondence, and also our analysis, yields that for $(g,J^{i})\in\mathcal{C}^{i}$ an irreducible HYM connection on the bundle $P_{r}$ exists if and only if $\deg(L_{r})<0$. In other words, for invariant Hermitian structures $(g,J^{i})\in\mathcal{C}^{i}$, the quantity $\deg(L_{r})$ controls the existence of HYM connections on the bundle $P_{r}$. Here, by $\deg(L_{\beta})$, i.e. the degree of $L_{\beta}$, we mean the value of $c_{1}(L_{\beta})\cup[\omega^{2}]$ evaluated against the fundamental class of $(\mathbb{F}_{2},J)$, which we regard as a real valued function on both $\mathcal{C}^{i}$ and $\mathcal{C}^{ni}$. Our main result, stated below, shows that, for invariant almost Hermitian structures $(g,J^{ni})\in\mathcal{C}^{ni}$, the same quantity controls the existence of invariant DT-instantons on $P_{r}$.

The previous result, i.e. Theorem 1, is a summarized version of our main result stated as Theorem 3 which constructs these DT-instantons and the functions $\deg(L_{r})$ explicitly. In particular, this gives the first existence theorem for solutions of the equations 2.4–2.5 with $\overline{\partial}_{A}^{*}u\neq 0$, which are also the first nontrivial examples outside the Kähler world.

We must further mention on the appearance of the restriction that the integral weights $r$ are roots of ${\rm SU}(3)$ follows in our case from the imposition that the DT-instantons $(A,u)$ be ${\rm SU}(3)$-invariant. In fact, we do not expect that this condition must hold for general DT-instantons, but of course we do not know if these exist on other $P_{r}$.

An immediate consequence of the explicit nature of our formulas for the degrees $\deg(L_{r})$ as functions on $\mathcal{C}^{ni}$ is that it is very easy to check that they cannot all be simultaneously negative. Furthermore, they all vanish if and only if $g=g^{nk}$ is the metric compatible with the nearly Kähler structure. This gives the result below, which will be presented in a more detailed manner as corollary 5.

In fact, one can explicitly pick a $1$-parameter family of compatible $2$-forms $\{\omega_{s}\}_{s\in I\subset\mathbb{R}}$ so that for some root $\deg(L_{r})(\omega_{s})$ crosses zero. Then, one can check that as $\deg(L_{r})(\omega_{s})\searrow 0$, the DT-instanton $(A,u)$ constructed by theorem 1 become obstructed and reducible. See examples 4–5 and the accompanying figures for illustrations of this phenomena. From a Physics point of view, this phenomenon can be interpreted as analogous to crossing a wall from the supersymmetric region in $\mathcal{C}^{ni}$ to a non-supersymmetric one. Namely, it can be thought of as analogous to supersymmetric breaking by internal gauge fields. See [Anderson2009] for the description of this phenomenon in the Kähler case.

In section 6 we turn to the problem of classifying the ${\rm SU}(3)$-invariant pHYM connections with gauge group ${\rm SO}(3)$. In theorem 2 we prove that the existence of such irreducible pHYM connections requires the almost complex structure to be integrable and we write down the resulting HYM connections explicitly.

An almost Hermitian $6$-manifold is a triple $(X^{6},J,g)$, where $X$ is real $6$-dimensional smooth manifold equipped with an almost complex structure $J$ and a compatible Riemannian metric $g$, i.e $g(J\cdot,J\cdot)=g(\cdot,\cdot)$. In this situation, the associated $(1,1)$-form is given by $\omega(\cdot,\cdot)=g(J\cdot,\cdot)$. The pair $(J,g)$ determines a reduction of the structure group of the frame bundle of $X$ from $\mathrm{GL}(6,\mathbb{R})$ to ${\rm U}(3)$. Thus, when convenient, we refer to the pair $(J,g)$ as an ${\rm U}(3)$-structure. In this manner, a ${\rm SU}(3)$-structure compatible with a given $(J,g)$ consists of the extra data of a real $3$-form $\Omega_{1}$ satisfying

where $\Omega_{2}=J\Omega_{1}=\ast\Omega_{1}$. In particular, the complex valued $3$-form $\Omega=\Omega_{1}+i\Omega_{2}$ is of type $(3,0)$ with respect to $J$. Indeed, such a triple $(J,g,\Omega_{1})$ determines a reduction of the structure group of the frame bundle to ${\rm SU}(3)$. In order to settle on the nomenclature we now recall various different kinds of ${\rm SU}(3)$-structures we will be considering

Notice that a Calabi-Yau structure is also nearly Calabi-Yau and half-flat, while a nearly Kähler structure is also half-flat.

Throughout this paper $G$ will be a compact Lie group, with Lie algebra $\mathfrak{g}$, and $P$ a principal $G$-bundle over $X$. The adjoint bundle of $P$ will be denoted by $\mathfrak{g}_{P}$. Recall, for example from [Bryant2005], that

When $(X,J,g)$ is an Hermitian manifold, the notions of pseudo-holomorphic and pHYM connections obviously coincide with the usual ones of holomorphic and HYM connections respectively. This motivates the definitions above in the setting of general almost Hermitian structures where much less is known. See for example [Charbonneau2016, Bryant2005], and references therein, for the case of nearly Kähler manifolds.

The conclusion of remark 2 is a shadow of the first of the following two interesting facts about nearly Kähler structures, see [Bryant2005, Foscolo2016, Verbitsky2011]:

• (a)

  All closed $2$-forms of type $(1,1)$ are primitive, i.e satisfy $F\wedge\omega^{2}=0$. This follows from the computation in remark 2 by replacing the curvature by any such $2$-form.

• (b)

  All closed, $2$-forms of type $(1,1)$ are harmonic. This follows easily from the previous bullet and the fact that any primitive $(1,1)$-form satisfies $\ast F=-F\wedge\omega$, hence

  $$d\ast F=-F\wedge d\omega=-3F\wedge\Omega_{1}=0,$$

  as $F$ is of type $(1,1)$.

Robert Bryant identified in [Bryant2005] a class of Hermitian-structures on a $6$-manifold for which closed $2$-forms of type $(1,1)$ are primitive. Bearing in mind the previous remark, it would not be surprising if they had a good local existence theory for pseudo-holomorphic and pHYM connections. Indeed, these structures are called quasi-integrable and were shown in [Bryant2005] to locally admit as many pseudo-holomorphic and pHYM connections as the integrable ones.

However, for the general almost Hermitian structure, the pHYM equations are overdetermined modulo gauge. Motivated by this we shall now introduce an elliptic equation (modulo gauge) whose solutions we regard as generalizing the notion of pHYM equation to the generic almost Hermitian structure.

To the authors’ knowledge, these equations first appeared in [Thomas1997] and were considered in this set-up already in [Donaldson2009]. They have been studied by Donaldson-Segal [Donaldson2009] and Yuuji Tanaka [Tanaka2008], [Tanaka2013], [Tanaka2014] who suggest these equations as a possible analytic approach to DT-invariants. On noncompact Calabi-Yau manifolds they were studied by the second named author in [Oliveira2014] and [Oliveira2016]. The same equations have also been studied by physicists such as in [Baulieu1998] and [Iqbal2008].

If $X$ is compact and $J$ integrable, then any DT-instanton induces a holomorphic structure on any associated complex vector bundle. Indeed, it follows from the fact that $u$ is of type $(0,3)$ and the Bianchi identity that

Taking the inner product with $u$ and integrating by parts we get that $\|\overline{\partial}_{A}^{*}u\|^{2}_{L^{2}}=0$. Thus $F_{A}^{0,2}=0$ and so $(A,u)$ solves

In particular, this proves the following.

In fact, notice that as $u$ is of type $(0,3)$ we have that $\overline{\partial}_{A}^{*}u=-\ast\partial_{A}\ast u=-i\ast\partial_{A}u$, so that $\overline{u}\in H^{0}(X,K_{X}\otimes\mathfrak{g}^{\mathbb{C}}_{P})$, where $\mathfrak{g}^{\mathbb{C}}_{P}=\mathfrak{g}_{P}\otimes_{\mathbb{R}}\mathbb{C}$.

For Kähler manifolds we can prove that if the scalar curvature is positive then $u=0$ and the equations 3.1 coincide with the HYM equations. We state this as follows.

One other interesting case when the equations simplify is when the almost Hermitian structure admits a compatible ${\rm SU}(3)$-structure. Of course, this is a very restrictive condition. Indeed, an almost Hermitian structure admits a compatible ${\rm SU}(3)$-structure if and only if the canonical bundle $K_{X}:=\Lambda^{3,0}_{\mathbb{C}}X$ is (topologically) trivial. When this is the case we shall say that a compatible ${\rm SU}(3)$-structure is pseudo-holomorphcially trivial if there is a $\Omega$ so that $\overline{\partial}\Omega=0$. Under this further restriction we can rewrite the equations as follows.

In several cases the DT-instanton equations 2.4–2.5 reduce to the pHYM equations, which further motivates studying the DT-instantons as an extension of the pHYM connections for the general almost Hermitian structure. In this direction, we start by proving that if there is a compatible half-flat ${\rm SU}(3)$-structure, then the DT instanton equations 3.2–3.3 reduce to a simpler equation with only one Higgs field.

For a compact Calabi-Yau manifold the DT-instanton equation further reduces to the HYM equations. Indeed, if the ${\rm SU}(3)$-structure is Calabi-Yau it also is half-flat and the DT instanton equations can be written as 3.6–3.6, with $\Phi_{1}=0$. Moreover, from the Bianchi identity and $d\Omega_{2}=0$ it follows that

Thus $\Delta|\Phi_{2}|^{2}=-2|\nabla_{A}\Phi_{2}|^{2}\leq 0$, and again, if $X$ is compact, $G$ semisimple and $A$ is irreducible must have $\Phi_{2}=0$. Hence, equations yields 3.6–3.6, so that $A$ is actually HYM. In the case of noncompact Calabi-Yau manifolds, irreducible DT-instantons with $\overline{\partial}^{*}u\neq 0$ for semisimple Lie groups do exist, and are expected to be related to special Lagrangian submanifolds, see [Oliveira2014] for more on this. On nearly Kähler manifolds, a similar vanishing result holds true, as we now state.