Instantons on Sasakian 7-manifolds

TL;DR

This paper investigates contact instantons on 7-dimensional Sasakian manifolds, establishing their moduli space's structure, smoothness conditions, and Kähler property, with special cases linking to Calabi-Yau geometry.

Contribution

It introduces a finite-dimensional local model for the moduli space of contact instantons, derives smoothness conditions, and shows the Kähler nature of the moduli space in Sasakian geometry.

Findings

Finite-dimensional local model for moduli space

Cohomological conditions for smoothness

Moduli space of selfdual contact instantons is Kähler

Abstract

We study a natural contact instanton (CI) equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related both to the transverse Hermitian Yang-Mills (tHYM) condition and the G_2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreducible solutions. We derive cohomological conditions for smoothness, and we express its dimension in terms of the index of a transverse elliptic operator. Finally we show that the moduli space of selfdual contact instantons (ASDI) is K\"ahler, in the Sasakian case. As an instance of concrete interest, we specialise to transversely holomorphic Sasakian bundles over contact Calabi-Yau 7-manifolds, and we show that, in this context, the notions of contact instanton, integrable G_2-instanton and HYM connection coincide.

Equations

$$\lambda F_A=\ast (\sigma \wedge F_A),$$

$$\pm F_A=\ast (\eta \wedge (d\eta {)}^{n-2}\wedge F_A).$$

$${\hat{F}}_A:=(F_A,\omega )=0\text{and}{F}_A^{0,2}=0.$$

$$\phi :=\eta \wedge d\eta +\mathrm{Im}(\epsilon ),$$

$$\sigma :=\eta \wedge d\eta .$$

$$L_{\sigma }:=\ast (\sigma \wedge \cdot ):{\Omega }^2({\mathfrak{g}}_E)\to {\Omega }^2({\mathfrak{g}}_E)$$

$${\Omega }^2({\mathfrak{g}}_E)=(\underset{{\Omega }_H^2}{\underbrace{{\Omega }_6^2\oplus {\Omega }_8^2\oplus {\Omega }_1^2}}\oplus {\Omega }_V^2)({\mathfrak{g}}_E)$$

$$p:{\Omega }_H^2({\mathfrak{g}}_E)\to {\Omega }_{6\oplus 1}^2({\mathfrak{g}}_E):=({\Omega }_6^2\oplus {\Omega }_1^2)({\mathfrak{g}}_E).$$

$$\mathcal{M}:=\{[A]\in \mathcal{B}|p(F_A)=0\},$$

$$d_7:=p\circ d_A:{\Omega }_H^1({\mathfrak{g}}_E)\to {\Omega }_{6\oplus 1}^2({\mathfrak{g}}_E).$$

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$${\mathrm{L}}^k:=\frac{{\Omega }^k({\mathfrak{g}}_E)}{I},k=0,1,2,3,\text{with}I:=⟨{\Omega }_8^2({\mathfrak{g}}_E)⟩\subset \left({\Omega }^{\bullet }({\mathfrak{g}}_E),\wedge \right).$$

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$$d_B:=d{|}_{{\Omega }_B^k(M)}:{\Omega }_B^k(M)\to {\Omega }_B^{k+1}(M),$$

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$${\mathrm{index}}_T(A)=\text{dim}(H_B^0)-\text{dim}(H_B^1)+\text{dim}(H_B^2).$$

$$F_A=\ast (\eta \wedge d\eta \wedge F_A).$$

$$\dim (T_{[A]}{\mathcal{M}}^{\ast })=\dim (H^1(\mathrm{C})).$$

$$\dim (T_{[A]}{\mathcal{M}}^{\ast })=\dim (H_B^2)-{\mathrm{index}}_T(A).$$

$$\ast (\xi ┘\ast \beta )={\xi }^{♭}\wedge ({\ast }^2\beta )=(-1{)}^{p(n-p)}\eta \wedge \beta ,$$

$$i_{\xi }(\ast \beta )=(-1{)}^p\ast (\eta \wedge \beta ),\forall \beta \in {\Omega }^p(M).$$

$$T:=\eta \wedge i_{\xi }(\cdot ):{\Omega }^p(M)\to {\Omega }^p(M),$$

$$T^2(\alpha )$$

$$=T(\alpha ).$$

$$\alpha =(1-\eta \wedge i_{\xi })\alpha +\eta \wedge i_{\xi }\alpha ={\alpha }_H+{\alpha }_V=\underset{{\alpha }_H}{\underbrace{i_{\xi }(\eta \wedge \alpha )}}+\underset{{\alpha }_V}{\underbrace{\eta \wedge i_{\xi }\alpha }},$$

$$i_{\xi }(\ast (d\eta \wedge \alpha ))=\ast \left(\sigma \wedge \alpha \right),\text{with}\sigma =\eta \wedge d\eta .$$

$$\ast F_A=\pm \sigma \wedge F_A.$$

$${\ast }_T(\beta )=(-1{)}^{m-1-k}\ast (\beta \wedge \eta ).$$

$$\ast \alpha ={\ast }_T\alpha \wedge \eta ,\forall \alpha \in {\Omega }_H^{\bullet }(M).$$

$${\ast }_T(\beta )=\frac{1}{2}J\beta \wedge {\omega }^2.$$

$$0$$

Full text

As an instance of concrete interest, we specialise to transversely holomorphic Sasakian bundles over contact Calabi-Yau $7$-manifolds, as studied by Calvo-Andrade, Rodríguez and Sá Earp [Calvo-Andrade2016], and we show that in this context the notions of contact instanton, integrable ${\rm G}_{2}$-instanton and HYM connection coincide.

1. 1 Introduction

2. 1.1 Instantons in $7$-dimensions: contact, Hermitian Yang-Mills and ${\rm G}_{2}$

3. 1.2 Local model of the moduli space

4. 1.3 Geometric structures on the moduli space

5. 2 Preliminaries on Sasakian geometry

6. 2.1 Horizontal forms and the contact instanton equation

7. 2.2 Splitting of complexified differential forms

8. 2.3 Proof of Proposition 1.1

9. 3 Gauge theory on $7$-dimensional Sasakian manifolds

10. 3.1 Infinitesimal deformations of contact instantons

11. 3.2 The extended complex of selfdual contact instantons

12. 3.3 ${\rm G}_{2}$-instantons on contact Calabi-Yau manifolds

13. 3.4 The Yang-Mills and Chern-Simons functionals

14. 4 The moduli space of contact instantons in $7$-dimensions

15. 4.1 The associated elliptic complex of a contact instanton

16. 4.2 Deformation theory of SDCI

17. 4.3 Obstruction and smoothness

18. 5 Geometric structures on the contact instanton moduli space $\mathcal{M}^{\ast}$

19. 5.1 Riemannian metric on $\mathcal{M}^{\ast}$

20. 5.2 Complex structure on $\mathcal{M}^{\ast}$

21. 5.3 Kähler structure on $\mathcal{M}^{\ast}$

22. A Sasakian vector bundles

23. A.1 Differential forms on Sasakian manifolds

24. A.2 Partial connections

25. A.3 Hermitian and holomorphic structures

1 Introduction

We describe the moduli space of solutions to a natural gauge-theoretic equation, on a suitable class of vector bundles over a Sasakian manifold $M$. We recall that on a $d$-dimensional manifolds M $d\geq 4$, if $F_{A}$ is the curvature of a connection $A$ the classical instanton equation $\pm F_{A}=\ast F_{A}$ [Donaldson1990], can be generalised relatively to an appropriate $(d-4)$-form $\sigma$ by [Donaldson1998, Tian2000]

[TABLE]

for eigenvalues $\lambda$ of the operator $\ast(\sigma\wedge\cdot):\Omega^{2}(M)\to\Omega^{2}(M)$. On a Sasakian $7$-manifolds, the contact structure $\eta$ induces a natural $3$-form $\sigma=\eta\wedge d\eta$ in which $\ast(\sigma\wedge\cdot)$ provides $\lambda\in\{\pm 1,-2\}$ (see §2.1.4). Indeed, in that context we argue that the meaningful condition is the selfdual contact instanton (SDCI) equation. We adopt the approach of Baraglia and Hekmati [Baraglia2016], who described the moduli space of contact instantons on contact metric $5$-manifolds, studying obstructions to smoothness and determining its expected dimension as the index of an elliptic operator transverse to the Reeb foliation. We will see that these results admit precise analogues in the appropriate $7$-dimensional setup, while some new distinct gauge-theoretic phenomena also occur.

1.1 Instantons in $7$-dimensions: contact, Hermitian Yang-Mills and ${\rm G}_{2}$

Let $(M^{2n+1},\eta,\xi)$ denote a contact manifold, with contact form $\eta$ and Reeb vector field $\xi$ [Boyer2008, blair2010contact]. Then the natural $(2n-3)$-form $\sigma=\eta\wedge(d\eta)^{n-2}$ provides an instance of (1.1):

[TABLE]

Solutions of (1.2) are said to be self dual contact instantons (SDCI) for $\lambda=1$ (respectively anti-selfdual contact instanton (ASDCI) for $\lambda=-1$).

When the contact manifold is endowed in addition with a Sasakian structure, namely an integrable transverse complex structure $\Phi$ and a compatible metric $g$, Biswas [Biswas2010] proposes a natural notion of Sasakian holomorphic structure for complex vector bundles $E\to M$ (see Appendix A). We recall that a connection $A$ on a complex vector bundle over a Kähler manifold is said to be Hermitian Yang-Mills (HYM) if

[TABLE]

This notion indeed extends to Sasakian bundles, by taking $\omega:=d\eta\in\Omega^{1,1}(M)$ as a ‘transverse Kähler form’, and defining HYM connections to be the solutions of (1.3) in that sense. The well-known concept of Chern connection also extends, namely as a connection mutually compatible with the holomorphic structure (integrable) and a given Hermitian bundle metric (unitary), see [Biswas2010]*§ 3.

An important class of Sasakian manifolds are those endowed with a contact Calabi-Yau (cCY) structure [Definition 3.8], the Riemannian metric of which have transverse holonomy ${\rm SU}(n)$, in the sense of foliations, corresponding to the existence of a global transverse holomorphic volume form $\varepsilon\in\Omega^{n,0}(M)$ [habib2015some]. Furthermore, when $n=3$, such cCY $7$-manifolds are naturally endowed with a ${\rm G}_{2}$-structure defined by the $3$-form

[TABLE]

which is cocalibrated, in the sense that its Hodge dual $\psi:=\ast_{g}\varphi$ is closed under the de Rham differential. When a $3$-form $\varphi$ on a $7$-manifold defines a ${\rm G}_{2}$-structure, the instanton condition (1.1) for $\sigma=\varphi$ and $\lambda=1$ is referred to as the ${\rm G}_{2}$-instanton equation. On holomorphic Sasakian bundles over closed cCY $7$-manifolds, it has the distinctive feature that integrable solutions are indeed Yang-Mills critical points, even though the ${\rm G}_{2}$-structure has torsion [Calvo-Andrade2016].

As a first point of interest, we will describe exactly how the three above concepts of instanton interrelate on suitable Sasakian $7$-manifolds. The contact instanton equation (1.2) in this case is determined by the natural $3$-form

[TABLE]

The operator $\ast(\sigma\wedge\cdot)$ splits the space of $2$-forms into $\{\pm 1,-2\}$-eigenspaces [§2.1.4]. However, the $(-2)$-eigenspace is $1$-dimensional, spanned by $d\eta$ and rather uninteresting, so we will focus on the $(\pm 1)$-eigenspaces, which in some sense still signify the instanton equation (1.1) as an (anti-) selfduality condition. The SDCI and HYM conditions are related as follows.

Proposition 1.1**.**

Let ${\bf E}\to M$ be a complex Sasakian bundle over $(M^{7},\eta,\xi,g,\Phi)$. If a connection $A\in\mathcal{A}({\bf E})$ is SDCI, then $A$ is a transverse HYM connection. Conversely, if moreover ${\bf E}$ underlies a holomorphic Sasakian bundle $\mathcal{E}$ [Definition A.5] and $A\in\mathcal{A}(\mathcal{E})$ is an integrable HYM connection, then $A$ is a SDCI [§2.3].

In particular, on a contact Calabi-Yau $7$-manifold, the three notions of instanton are related in the following ways:

Theorem 1.1**.**

Let $\mathcal{E}$ be a holomorphic Sasakian bundle over a contact Calabi-Yau manifold $(M^{7},\eta,\xi,g,\Phi)$ endowed with its natural ${\rm G}_{2}$-structure (1.4); then the following hold:

1. (i)

Every solution of the contact instanton equation $\pm F_{A}=\ast(\sigma\wedge F_{A})$ is also a solution of $\ast(\varphi\wedge F_{A})=\pm F_{A}$, i.e., every contact instanton is a ${\rm G}_{2}$-instanton [Proposition 3.11]. 2. (ii)

A Chern connection is a ${\rm G}_{2}$-instanton if, and only if, it is a contact instanton [Proposition 3.12]. 3. (iii)

*A Chern connection is HYM if, and only if, it is a ${\rm G}_{2}$-instanton **[Calvo-Andrade2016]**Lemma  21.

In particular, among Chern connections, the three notions are equivalent.

1.2 Local model of the moduli space

Our first main result is a complete description of the local deformation theory of $7$-dimensional Sasakian contact instantons. Let ${\bf E}\to M$ be a Sasakian vector bundle with compact, connected, semi-simple structure group ${\rm G}$, and denote by ${\mathfrak{g}}_{E}$ its adjoint bundle and by $\Omega^{k}({\mathfrak{g}}_{E})$ the ${\mathfrak{g}}_{E}$-valued $k$-forms on $M$. The operator

[TABLE]

induces the irreducible splitting [cf. (2.13)]

[TABLE]

where $\Omega_{6}^{2}({\mathfrak{g}}_{E})$, $\Omega_{8}^{2}({\mathfrak{g}}_{E})$ and $\Omega_{1}^{2}({\mathfrak{g}}_{E})$ are the eigenspaces associated to $-1$, $1$ and $-2$, respectively. In Proposition 3.1, we show that Chern connections with curvature in $\Omega^{2}_{6}({\mathfrak{g}}_{E})$, i.e. anti-selfdual contact instantons, are necessarily flat. Hence the meaningful notion of Sasakian instanton in this case is that of SDCI, with curvature in $\Omega^{2}_{8}({\mathfrak{g}}_{E})$, which is the kernel of the projection map

[TABLE]

The Fréchet Lie group $\mathcal{G}$ of smooth gauge transformations acts smoothly on the space $\mathcal{A}({\bf E})$ of connections on ${\bf E}$, and the topological quotient $\mathcal{B}:=\mathcal{A}/\mathcal{G}$ is a Hausdorff space. We denote by $\mathcal{B}^{\ast}\subset\mathcal{B}$ the open subspace of irreducible connections, by $\mathcal{M}\subset\mathcal{B}$ the set of gauge equivalence classes of solutions to the SDCI equation:

[TABLE]

and, accordingly, by $\mathcal{M}^{\ast}\subset\mathcal{M}$ its irreducible stratum. Linearising the SDCI condition, in terms of the projection (1.7), we introduce:

[TABLE]

In Proposition 3.4, we will see that a local model for the moduli space $\mathcal{M}^{\ast}$ of SDCI is given by the cohomology group $H^{1}({\rm C}):=\frac{\operatorname{ker}(d_{7})}{\mathop{\mathrm{im}}(d_{A})}$ of the deformation complex

[TABLE]

This complex, however, is not elliptic, and in order to compute the dimension of $H^{1}({\rm C})$ we resort to an auxiliary construction, studied in §4. We introduce the quotient spaces of $k$-forms modulo the Lie algebra ideal $I$ generated by $\Omega^{2}_{8}({\mathfrak{g}}_{E})$:

[TABLE]

For a natural choice of differentials $D_{k}$, the ${\rm L}^{k}$ spaces fit in a complex [Proposition 4.5]:

[TABLE]

We denote by $H^{\bullet}:=H^{\bullet}({\rm L})$ the cohomology of (1.11), and indeed the first cohomology group $H^{1}$ is isomorphic to the infinitesimal deformations of $[A]\in\mathcal{M}$ as a contact instanton.

Relatively to the Reeb orbits, a $S^{1}$-invariant differential form $\alpha\in\Omega^{\bullet}(M)$ is called basic. The graded ring $\Omega^{\bullet}_{B}(M)$ of basic forms inherits a natural basic de Rham differential

[TABLE]

and the cohomology of $d_{B}$ is referred to as the basic de Rham cohomology. Restricting the differentials $D_{k}$ in (1.11) to basic forms in ${\rm L}^{k}=\Omega^{k}(M)/\langle\Omega^{2}_{8}\rangle$, we obtain a basic complex [Proposition 4.5]:

[TABLE]

We denote by $H^{\bullet}_{B}:=H^{\bullet}({\rm L}_{B})$ the corresponding basic cohomology. If $A$ is a contact instanton, the transverse index of $A$ is defined as the index of the basic complex (1.12):

[TABLE]

In particular, when $A$ is irreducible, $\mathop{\mathrm{index}_{T}}(A)=\text{dim}(H^{2}_{B})-\text{dim}(H^{1}_{B})$. In summary, we formulate a local model for the moduli space $\mathcal{M}^{\ast}$ in terms of basic cohomology:

Theorem 1.2**.**

Let ${\bf E}\to M$ be a Sasakian $G$-bundle over a closed connected Sasakian manifold $(M^{7},\eta,\xi,g,\Phi)$, with adjoint bundle ${\mathfrak{g}}_{E}$, and denote by $\mathcal{M}^{\ast}$ the moduli space (1.8) of irreducible selfudal contact instantons, i.e. solutions of (1.1) for $\lambda=1$:

[TABLE]

Then, the following hold:

1. (i)

The tangent space of $\mathcal{M}^{\ast}$ at $[A]$, i.e. the space of infinitesimal deformations of $[A]$ as a contact instanton, is isomorphic to the finite-dimensional cohomology group $H^{1}({\rm C}):=\frac{\operatorname{ker}(d_{7})}{\mathop{\mathrm{im}}(d_{A})}$ of the complex (1.10), and

[TABLE] 2. (ii)

The dimension of $\mathcal{M}^{\ast}$ near $[A]$ can be computed from the cohomology of the basic complex (1.12), which is elliptic transversely to the Reeb foliation, namely there is an isomorphism $H^{1}\cong H^{1}_{B}$, where $H^{1}_{B}$ is the cohomology of (1.12):

[TABLE] 3. (iii)

The local model of $\mathcal{M}^{*}$ is cut out as the zero set of an obstruction map [Definition 4.31], which vanishes precisely when $H^{2}_{B}=0$ [Proposition 4.27]. Thus, for an irreducible selfdual contact instanton $A$ such that $H^{2}_{B}=0$, $\mathcal{M}^{\ast}$ is smooth near $A$, with finite dimension $\dim\mathcal{M}^{\ast}=-\mathop{\mathrm{index}_{T}}(A)$ [Corollary 4.26].

Remark 1.2*.*

Parts $(i)$ and $(ii)$ in Theorem 1.2 establish somewhat independently that the tangent space near an irreducible contact instanton is finite-dimensional, since it occurs as the first cohomology group in both complexes (1.11) and (1.12). However, in terms of the obstruction theory, we learn something finer from $(ii)$ and $(iii)$. In the context of $(i)$, the moduli space near an acyclic point, i.e. $h^{0}({\rm C})=h^{2}({\rm C})=0$ in (1.10), would be necessarily [math]-dimensional, whereas the complex (1.12) in terms of basic cohomology is merely transverse-elliptic, hence the moduli space near an acyclic smooth point, with $h^{0}_{B}=h^{2}_{B}=0$, can in principle have nonzero dimension $-\mathop{\mathrm{index}_{T}}(A)$.

Remark 1.3*.*

For most steps in our arguments, it suffices to assume $M$ compact and connected, with possibly nontrivial boundary. However, in Theorem 1.1–$(iii)$, taken from [Calvo-Andrade2016]*Lemma  21, and in Propositions 3.14 and 4.16, one actually needs $M$ to be closed.

Remark 1.4*.*

For a Sasakian $7$-manifold with positive transverse scalar curvature, the second basic cohomology group $H_{B}^{2}$ should vanish, and therefore $\mathcal{M}^{*}$ is smooth. This is announced here as Conjecture 5.13, to be expanded in subsequent work. In particular, $\mathop{\mathrm{index}_{T}}(A)=-\text{dim}(H^{1}_{B})$, by Theorem 1.2–(ii).

Regarding the various notions of instanton related by Theorem 1.1, Theorem 1.2 has the following significance:

Corollary 1.5**.**

Let $\mathcal{E}\to M$ be a holomorphic Sasakian bundle over a $7$-dimensional cCY manifold $(M^{7},\eta,\xi,g,\Phi)$, endowed with its natural ${\rm G}_{2}$-structure (1.4). Among Chern connections in $\mathcal{A}({\bf E})$, the three notions of instanton coincide: SDCI, HYM connections and ${\rm G}_{2}$-instantons. The complex (1.10) describes their local deformations, and Theorem 1.2 describes their moduli space.

1.3 Geometric structures on the moduli space

We are furthermore interested in geometric structures on the SDCI moduli space $\mathcal{M}^{\ast}$ [cf. (1.8)], in the hope that this might lead in the future to constructing new invariants of Sasakian manifolds.

By means of comparison, under suitable assumptions, the moduli space of ASD contact instantons on a Sasakian $5$-manifold $(M^{5},\eta,\xi,g,\Phi)$ is Kähler [Baraglia2016]*§ 4.3, and moreover hyper-Kähler in the transverse Calabi-Yau case. We will see that, even though this does not generalise verbatim to the $7$–dimensional contact Calabi-Yau case, $\mathcal{M}^{\ast}$ still inherits a natural Kähler structure:

Theorem 1.3**.**

In the situation of Theorem 1.2, the moduli space $\mathcal{M}^{\ast}$ of irreducible SDCI carries a natural Riemannian metric [cf. (5.2)], a complex structure $\mathcal{J}$ induced by the transverse complex structure $J:=\Phi|_{{\rm H}}$, and a symplectic $2$-form $\Omega$, such that $(\mathcal{M}^{*},\mathcal{J},\Omega)$ is a Kähler manifold.

Outline: In §2.1.4, we describe the local splitting (2.13) of $\Omega^{2}(M)$ under the contact structure and the operator $L_{\sigma}:=\ast(\sigma\wedge\cdot)$ from (1.6). Another natural decomposition of $\Omega^{2}({\mathfrak{g}}_{E})$ comes from the transverse complex structure induced by $\Phi\in{\mathrm{End}}(TM)$, and both are related by (2.16). Furthermore, the endomorphism $\Phi$ provides a notion of transverse holomorphicity for complex vector bundles over Sasakian manifolds [Appendix A], hence also notions of unitary and integrable connections. We show, in Proposition 3.1, that imposing these conditions on a connection forces its curvature component in $\Omega_{6}^{2}({\mathfrak{g}}_{E})$ to vanish, hence the only nontrivial theory in this context is the SDCI case.

Parts (i) and (ii) of Theorem 1.1 are proven respectively in Propositions 3.11 and 3.12. The proof of Theorem 1.2 is organised as follows: (i) is the content of Proposition 3.4, which uses an auxiliary elliptic complex [Proposition 3.7] to establish that this local model has finite dimension; (ii) is an immediate consequence [Corollary 4.18] of Proposition 4.17; and (iii) requires a thorough study of the moduli space of the obstruction theory of SDCI, under the $5$-dimensional paradigm from [Baraglia2016], culminating in Proposition 4.27. Finally, in §5 we study the geometry of the moduli space, showing that $\mathcal{M}^{\ast}$ inherits a complex structure [ Proposition 5.7] and a Kähler $2$-form [Proposition 5.10], thus proving Theorem 1.3.

Funding: This work was supported by the Higher Education Improvement Coordination - Brasil (CAPES) - Finance Code 001; the Brazilian National Council for Scientific and Technological Development (CNPq) [141215/2019-4 to L.P., 307217/2017-5 to H.S.E.]; and São Paulo Research Foundation (Fapesp) [2017/20007-0, 2018/21391-1 to H.S.E.].

2 Preliminaries on Sasakian geometry

We follow the standard references for Sasakian geometry [Boyer2008, blair2010contact]. A Sasakian structure on a smooth manifold $M^{2n+1}$ is a quadruple $(\eta,\xi,g,\Phi)$ such that $(M,g)$ is a Riemannian manifold, $(M,\eta)$ is a contact manifold with Reeb field $\xi$ and $\Phi\in{\mathrm{End}}(TM)$ is a transverse complex structure, satisfying the following compatibility relations:

1. (i)

$g(\xi,\xi)=1,$ 2. (ii)

$\Phi\circ\Phi=-{\rm I}_{TM}+\eta\otimes\xi,$ 3. (iii)

$g(\Phi X,\Phi Y)=g(X,Y)-\eta(X)\eta(Y),$ 4. (iv)

$\nabla^{g}_{X}\xi=-\Phi X,$ 5. (v)

$(\nabla^{g}_{X}\Phi)(Y)=g(X,Y)\xi-\eta(Y)X,$

where $X,Y$ are vector fields on $M$ and $\nabla^{g}$ is the Levi-Civita connection of $g$. In that case we say $(M,\eta,\xi,g,\Phi)$ is a Sasakian manifold, equivalently, $M$ is Sasakian manifold if and only if the metric cone $(\mathbb{R}_{+}\times M,dr^{2}+r^{2}\cdot g)$ is Kähler.

2.1 Horizontal forms and the contact instanton equation

2.1.1 Vertical and horizontal forms

Hereinafter we set $\sigma:=\eta\wedge d\eta$. For a $p$-form $\alpha$ and a vector field $v$ on $M$, the metric $g$ is compatible, in the sense that $\ast(v\lrcorner\alpha)=v^{\flat}\wedge\ast\alpha$, where $v^{\flat}:=g(v,\cdot)$ and $\ast$ is the Hodge operator of $g$. Notice that $\xi^{\flat}=\eta$, thus applying the above formula to $v=\xi$ and $\alpha=\ast\beta$ for a $p$-form $\beta$, we obtain

[TABLE]

or, equivalently,

[TABLE]

Furthermore, the contact structure induces a natural operator

[TABLE]

which is a projection:

[TABLE]

As such, it splits any $\alpha\in\Omega^{\bullet}(M)$ into horizontal and vertical components:

[TABLE]

this provides a splitting $\Omega^{\bullet}(M)=\Omega^{\bullet}_{H}(M)\oplus\Omega^{\bullet}_{V}(M)$, where $\Omega^{\bullet}_{H}(M)$ and $\Omega^{\bullet}_{V}(M)$ are the horizontal and vertical parts, respectively.

2.1.2 The contact instanton equation in $7$ dimensions

From now on, unless otherwise stated, we will fix $\dim M=7$. Equation (2.3) suggests a natural ‘instanton equation’ as follows: consider $\alpha\in\Omega^{2}(M)$, applying the contraction (2.1) to $d\eta\wedge\alpha\in\Omega^{4}(M)$, we obtain

[TABLE]

This motivates the introduction of operator $L_{\sigma}\colon\Omega^{2}_{H}(M)\to\Omega^{2}_{H}(M)$ in (1.6). In next section we show that $\pm 1$ are eigenvalues of $L_{\sigma}$ and that $L_{\sigma|_{\Omega^{2}_{V}(M)}}=0$, and this extends in the natural way to $\Omega^{2}({\mathfrak{g}}_{E})$. If $\alpha=F_{A}$ is the curvature of a connection $A$ on a suitable vector bundle ${\bf E}\to M$, a natural instance of the contact instanton equation is $\ast(\sigma\wedge F_{A})=i_{\xi}(\ast(d\eta\wedge F_{A}))=\pm F_{A},$ or, equivalently,

[TABLE]

Notice that $\eta\wedge(d\eta)^{3}\neq 0$ is a volume form on $M$, and $\ast(\sigma\wedge d\eta)=cd\eta$ for a constant $c\neq 1$. This contrasts with the $5$-dimensional case in [Baraglia2016] and classical $4$-dimensional gauge theory, in both of which $d\eta$ is selfdual.

2.1.3 Transverse Hodge star operator and Sasakian Kähler identities

Let us briefly describe the transverse complex geometry on a Sasakian manifold $(M,\eta,\xi,g,\Phi)$, referring to [Boyer2008] for a thorough treatment. Relatively to the Reeb foliation, the usual Hodge star induces a transverse Hodge star operator $\ast_{T}\colon\Omega_{H}^{k}(M)\to\Omega_{H}^{m-(k+1)}(M)$ by the formula [tondeur2012foliations]*§ 12

[TABLE]

Both operators are compatible, in the sense that

[TABLE]

The following result appears in the literature in various guises [ibid.], but since it will play a central role in §5.3, we include a full proof here.

Lemma 2.1**.**

Let $(M,\eta,\xi,g,\Phi)$ be a Sasakian manifold and denote by $J:=\Phi|_{{\rm H}}$ the restriction of $\Phi\in{\mathrm{End}}(TM)$ to the horizontal distribution, cf. (A.1). If $\beta\in\Omega^{1}_{H}({\mathfrak{g}})$ is a transverse $1$–form, then

[TABLE]

Proof.

By excess in degree, for any $\beta\in\Omega^{1}_{H}({\mathfrak{g}})$ the transverse $7$–form $\beta\wedge\omega^{3}$ is zero, so

[TABLE]

The conclusion now follows by a short computation:

[TABLE]

Now, acting on $(p,q)$-forms, we have well-defined operators

[TABLE]

which naturally extend to ${\mathfrak{g}}$–valued forms. In terms of the transverse Hodge star (2.5), the operators (2.7) have adjoints

[TABLE]

Let $\langle\cdot,\cdot\rangle_{\mathfrak{g}}$ be an invariant metric on ${\mathfrak{g}}$, an inner product on $\Omega^{\bullet}({\mathfrak{g}}_{E})$ is defined by:

[TABLE]

Note that from (2.6) the inner product in (2.9) can be rewritten as $(\alpha,\beta)_{M}=\displaystyle\int_{M}\langle\alpha\wedge\ast_{T}\beta\rangle\wedge\eta.$ Denoting the exterior product with $\omega:=d\eta$ by

[TABLE]

its adjoint with respect to the transverse Hodge star (2.5) is the transverse Lefschetz operator

[TABLE]

For later use in §5, we recall the Sasakian Kähler identities:

Lemma 2.2** ([Boyer2008]*Lemma 7.2.7).**

On a Sasakian manifold, the following properties hold:

1. (i)

$[\Lambda,\partial_{B}]=-{\textbf{i}}\bar{\partial}_{B}^{\ast}$** 2. (ii)

$[\Lambda,\bar{\partial}_{B}]={\textbf{i}}\partial_{B}^{\ast}$** 3. (iii)

$\partial_{B}\bar{\partial}_{B}^{\ast}+\bar{\partial}_{B}^{\ast}\partial_{B}=\partial_{B}^{\ast}\bar{\partial}_{B}+\bar{\partial}_{B}\partial_{B}^{\ast}=0$** 4. (iv)

$\partial_{B}^{\ast}\partial_{B}+\partial_{B}\partial_{B}^{\ast}=\bar{\partial}_{B}^{\ast}\bar{\partial}_{B}+\bar{\partial}_{B}\bar{\partial}_{B}^{\ast}$**

• (v)

Defining $\Delta_{\partial_{B}}:=\partial_{B}^{\ast}\partial_{B}+\partial_{B}\partial_{B}^{\ast}$ and $\Delta_{\bar{\partial}_{B}}:=\bar{\partial}_{B}^{\ast}\bar{\partial}_{B}+\bar{\partial}_{B}\bar{\partial}\textbf{}^{\ast}$, then:

[TABLE]

2.1.4 Eigenspaces of $L_{\sigma}=\ast(\sigma\wedge\cdot)$ from the contact structure

We will now examine how (2.4) splits into components according to the eigenspaces of $L_{\sigma}$ defined in (1.6). Let $(x_{1},\dots,x_{7})$ be Sasakian Darboux coordinates on $M$ [blair2010contact]*Theorem  3.1, such that the contact form $\eta$ is given by

[TABLE]

Let $X_{i}:=\frac{\partial}{\partial x_{i}}$ and $dx^{i_{1}\cdots i_{k}}:=dx^{i_{1}}\wedge\cdots\wedge dx^{i_{k}}$; in particular, $X_{7}=\xi$ is the Reeb vector field, and the transverse symplectic $2$-form is expressed by $\omega:=d\eta=dx^{14}+dx^{25}+dx^{36}$. In these coordinates, the projection $T$ defined in (2.2) acts as follows:

[TABLE]

Therefore, the decomposition (2.3) determines a $15$-dimensional horizontal space $\Omega^{2}_{H}$ and a $6$-dimensional vertical space $\Omega^{2}_{V}$:

[TABLE]

Moreover, $L_{\sigma}$ acts on the horizontal space

[TABLE]

as follows:

[TABLE]

For immediate convenience, let us fix the following notation:

[TABLE]

[TABLE]

[TABLE]

It is easy to check that

[TABLE]

Hence the operator $L_{\sigma}$ defined in (1.6) splits $\Omega^{2}_{H}(M)$ into eigenspaces associated to $\{-2,-1,1\}$, respectively:

[TABLE]

Therefore $2$-forms decompose irreducibly as

[TABLE]

Of course, this decomposition extends naturally to ${\mathfrak{g}}_{E}$-valued $2$-forms.

Lemma 2.3**.**

The decompositions (2.3) and (2.14) are orthogonal with respect to the inner product (2.9).

Proof.

For $\alpha\in\Omega^{2}_{H}(M)$ and $\beta\in\Omega^{2}_{V}(M)$ we have

[TABLE]

Moreover, the operator $\ast(\sigma\wedge(\cdot))\colon\Omega^{2}_{H}(M)\to\Omega^{2}_{H}(M)$ is self-adjoint: for $\alpha,\beta\in\Omega^{2}_{H}(M)$,

[TABLE]

hence its eigenspaces are orthogonal. ∎

2.2 Splitting of complexified differential forms

We establish some notation and elementary facts about the complexified tangent bundle, which are largely adapted from [Biswas2010] and reviewed in Appendix A. The contact structure splits the tangent bundle as $TM=H\oplus N_{\xi}$ (A.1), where $H=\operatorname{ker}(\eta)$ and $N_{\xi}$ is the real line bundle spanned by the Reeb field $\xi$. The transverse complex structure $\Phi$ satisfies $(\Phi|_{H})^{2}=-1$, so the eigenvalues of the complexified operator $\Phi^{\mathbb{C}}$ are $\pm{\bf i}$, with ${\bf i}:=\sqrt{-1}$. The complexification $H_{\mathbb{C}}:=H\otimes_{\mathbb{R}}\mathbb{C}$ splits as $H_{\mathbb{C}}=H^{1,0}\oplus H^{0,1}$ (A.3), so we obtain a decomposition of direct sum of vector bundles (A.4)

[TABLE]

This induces the decomposition of vector bundles (A.5)

[TABLE]

where $\Omega^{p,q}_{H}(M):=\Gamma(M,\Lambda^{p}(H_{\mathbb{C}})^{\ast}\otimes\Lambda^{q}(H_{\mathbb{C}})^{\ast})$. Now, let us study more closely the space of $2$-forms, from the ‘transverse complex’ point of view. Still in local Darboux coordinates $(x_{1},\cdots,x_{7})$, we denote the transverse complex coordinates by

[TABLE]

We will denote, as usual, $dz^{j}:=dx^{j}+{\textbf{i}}dx^{j+3}$ and $d\bar{z}^{j}:=dx^{j}-{\textbf{i}}dx^{j+3}$ for $j=1,2,3.$ In terms of the bases $\{v_{i}\}_{i=1}^{6}$ and $\{w_{i}\}_{i=1}^{8}$, from (2.13), the space $\Omega^{2,0}(M)$ is locally spanned by

[TABLE]

we also compute

[TABLE]

Since $\omega:=d\eta$ is nowhere-vanishing and has type $(1,1)$ [Biswas2010]*Corollary  3.1, it determines an orthogonal complement $\Omega^{1,1}_{\perp}(M)$ in

[TABLE]

which is expressed in local Darboux coordinates by

[TABLE]

Therefore we obtain two decompositions for the horizontal $2$-forms:

[TABLE]

Note that $\Omega^{0,2}(M)$ is spanned by

[TABLE]

consistently with the fact that $\Omega^{0,2}(M)\cong\overline{\Omega^{2,0}}(M)$. Still by inspection, we have:

[TABLE]

[TABLE]

[TABLE]

From the expressions (2.17) and (2.19), we obtain immediately:

Lemma 2.4**.**

Let $\Omega^{2}_{8}$ and $\Omega^{2}_{6}$ be the eigenspaces of the operator $L_{\sigma}$ from (1.6), associated to the eigenvalues $1$ and $-1$, respectively [cf. (2.13)]; then

1. (i)

A $2$-form $\alpha$ belongs to $\Omega^{2}_{8}$ if, and only if, $\alpha$ it is a real form of type $(1,1)$ and orthogonal to $\omega$. 2. (ii)

A $2$-form $\alpha$ belongs to $\Omega^{2}_{6}$ if, and only if, $\alpha=\beta+\overline{\beta}$, for some $\beta$ of type $(2,0)$.

2.3 Proof of Proposition 1.1

The proof is based on Lemma 2.4, which yields a characterisation curvature forms in $\Omega^{2}_{6}({\mathfrak{g}}_{E})$ and $\Omega^{2}_{8}({\mathfrak{g}}_{E})$. We denote by $\Lambda^{k}(T^{\ast}M)_{\mathbb{C}}$ the complexification of $\Lambda^{k}(T^{\ast}M)$. Let also:

[TABLE]

The conjugation on $\Lambda^{k}(T^{\ast}M)_{\mathbb{C}}$ naturally induce a conjugation on $\Omega^{k}(T^{\ast}M)_{\mathbb{C}}\otimes{\mathfrak{g}}_{E}$ such that $\overline{\Omega^{p,q}({\mathfrak{g}}_{E})}=\Omega^{q,p}({\mathfrak{g}}_{E})$. We have the following characterisation of SDCI solutions:

Proposition 2.5**.**

Let ${\bf E}\to M$ be a Sasakian $G$-bundle with adjoint bundle ${\mathfrak{g}}_{E}$. A connection $A\in\mathcal{A}$ satisfies the SDCI equation $\ast F_{A}=\sigma\wedge F_{A}$ if, and only if:

[TABLE]

In that case, moreover, $\overline{F}_{A}=F_{A}.$

Proof.

That $A$ is a SDCI if, and only if, (2.21) holds is an immediate consequence of Lemma 2.4–(i) and (2.16), since $\Omega^{1,1}_{\perp}({\mathfrak{g}}_{E})\cong\Omega^{2}_{8}({\mathfrak{g}}_{E})$. Now, assuming that is the case, the reality condition is manifest in local Darboux coordinates, cf. (2.15). Finally, we see in (2.17) that the generators $\{w_{i}\}_{i=1}^{8}$ of $\Omega^{2}_{8}$ are all real: $\overline{w_{i}}=w_{i}$. ∎

Proof of Proposition 1.1.

If $A$ is a SDCI on a Sasakian bundle ${\bf E}\to M$ [Definition A.4], it follows from (2.16) that $F_{A}\in\Omega^{1,1}_{8}(M)$, in particular $F_{A}^{0,2}=0$. Furthermore, Proposition 2.5 gives immediately $\hat{F}_{A}=0$, hence $A$ is HYM. Conversely, if moreover ${\bf E}\to M$ underlies a holomorphic bundle $\mathcal{E}\to M$ [Definition A.5] and $A$ is a an integrable HYM connection, then $\hat{F}_{A}=0$ and $F^{0,2}=0$. Since $A$ is also unitary, the curvature $F_{A}$ is of type $(1,1)$, cf. [Donaldson1990, Proposition 2.1.56], and we conclude form (2.16) that $A$ is a SDCI. ∎

3 Gauge theory on $7$-dimensional Sasakian manifolds

In $4$-dimensional gauge theory, reversing orientation of the base manifold interchanges SD and ASD connections, thus the two theories are equivalent. On the other hand, in the $5$-dimensional contact case, the SD and ASD equations are studied separately [Baraglia2016] and some differences appear; for instance, the coboundary map in the long exact sequence in [Baraglia2016, Proposition 3.3] is only zero in the selfdual case. On a merely contact manifold $(M^{7},\eta)$, one could a priori study connections with curvature in $\Omega^{2}_{6}$ or $\Omega^{2}_{8}$. However, on a Sasakian manifold $(M^{7},\eta,g,\Phi)$, there is a natural choice favouring the SDCI equation ($\lambda=1$) in (1.2):

[TABLE]

That the alternative theory is trivial follows immediately from Lemma 2.4–$(ii)$:

Proposition 3.1**.**

Let $\mathcal{E}\to M^{7}$ be a holomorphic Sasakian vector bundle [Definition A.5]. A Chern connection $A\in\mathcal{A}(\mathcal{E})$ such that $F_{A}\in\Omega^{2}_{6}({\mathfrak{g}}_{E})$ is necessarily flat.

3.1 Infinitesimal deformations of contact instantons

The main purpose this Section is to prove Theorem 1.2–$(i)$. Namely, the linearisation of the moduli space of SDCI $\mathcal{M}^{\ast}$ is given by the cohomology group $H^{1}({\rm C}):=\frac{\operatorname{ker}(d_{7})}{\mathop{\mathrm{im}}(d_{A})}$ of the deformation complex

[TABLE]

We will show that, even though the complex $C\bullet$ is manifestly not elliptic, one can establish that $H^{1}({\rm C})$ is finite-dimensional by means of an auxiliary extended complex. Finally, we will relate contact instantons and ${\rm G}_{2}$-instantons in the $7$-dimensional contact Calabi-Yau setting, as outlined in [Calvo-Andrade2016]. Under the hypotheses of Theorem 1.2, we describe the linearisation of the moduli space of SDCI over a Sasakian $7$-manifold, following the approach of [itoh1983moduli].

Proposition 3.2**.**

An element $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$ gives an infinitesimal SDCI deformation if, and only if, $\alpha\in\operatorname{ker}(d_{7})$.

Proof.

Given a connection $A$ on the $G$-bundle $E\to M$, the induced covariant exterior derivative $d_{A}\colon\Omega^{k}({\mathfrak{g}}_{E})\to\Omega^{k+1}({\mathfrak{g}}_{E})$ squares to an algebraic curvature operator:

[TABLE]

Furthermore, the transversal complex structure splits $d_{A}=\partial_{A}+\overline{\partial}_{A}$, acting on $\Omega^{p,q}_{H}({\mathfrak{g}}_{E})$ [cf. (2.20)] according to bi-degree:

[TABLE]

Let $\alpha\in\Omega^{p,q}({\mathfrak{g}}_{E})$, and note that

[TABLE]

If $A$ is SDCI, i.e., $F_{A}$ is of type $(1,1)$ and orthogonal to $\omega$ [cf. Proposition 2.5], replacing (3.1) in the above formula and comparing bi-degree, we obtain:

[TABLE]

On the other hand, let $\{A_{t}\}_{|t|<\varepsilon}$ be a path of SDCI connections with $A_{0}=A$. An infinitesimal deformation $\alpha:=\left.\frac{d}{dt}\right|_{t=0}(A_{t})\in\Omega^{1}({\mathfrak{g}}_{E})$ varies the curvature by $F_{A_{t}}=F_{A}+td_{A}\alpha+t^{2}\alpha\wedge\alpha.$ Decomposing $\alpha=\alpha^{(1,0)}+\alpha^{(0,1)}$, in $(1,0)$ and $(0,1)$ part respectively,

[TABLE]

so, by Lemma 2.4 item $(i)$, the infinitesimal deformation $\alpha$ satisfies:

[TABLE]

Using the definition of $d_{7}$ in (1.9), the above relations give

[TABLE]

The linearisation at $A\in\mathcal{A}$ of the $\mathcal{G}$-action is $-d_{A}\colon\Omega^{0}({\mathfrak{g}}_{E})\to\Omega^{1}({\mathfrak{g}}_{E})$, so a natural transverse ‘Coulomb’ slice is given by the orthogonal complement $\operatorname{ker}(d_{A}^{\ast})$ of $\mathop{\mathrm{im}}(d_{A})\subset\Omega^{1}({\mathfrak{g}}_{E})$ [Donaldson1990, p. 131]. For small $\varepsilon>0$, we set:

[TABLE]

where $\|\cdot\|$ is a suitable Sobolev norm. Following [Donaldson1990, (4.2.6)], a neighbourhood of $[A]\in\mathcal{B}$ is described by the quotient of $S_{A,\varepsilon}$ under $\pi\colon A\in\Omega^{1}({\mathfrak{g}}_{E})\mapsto[A]\in\mathcal{B}=\mathcal{A}/\mathcal{G}$, as in the following slice lemma (see also [Freed1991]):

Lemma 3.3**.**

For every $A\in\mathcal{M}^{\ast}$, there exists $\varepsilon=\varepsilon(A)>0$ such that the restriction $\pi|_{{S_{A,\varepsilon}}}$ is a homeomorphism onto an open set $\pi(S_{A,\varepsilon})\subset\mathcal{B}$.

In view of the slice condition in (3.3), we consider the restriction of (1.9) to $\operatorname{ker}(d^{\ast}_{A})$ as follows:

[TABLE]

Proposition 3.4**.**

In the context of Theorem 1.2, the infinitesimal deformations of an irreducible selfdual contact instanton $A$ are described by the complex

[TABLE]

The tangent space $T_{[A]}\mathcal{M}^{\ast}$ is isomorphic to the first cohomology group $H^{1}({\rm C}):=\frac{\operatorname{ker}(d_{7})}{\mathop{\mathrm{im}}(d_{A})}$.

Proof.

We know from Proposition 3.2 that infinitesimal SDCI deformations of $A$ lie in $\operatorname{ker}(d_{7})$. On the other hand, if $A$ is an irreducible SDCI connection, then $d_{7}\circ d_{A}=p\circ F_{A}=0$, so (3.4) is indeed a complex. Furthermore the derivative of the action of the gauge group $\mathcal{G}$ at $A$ is $-d_{A}\colon\Omega^{0}({\mathfrak{g}})\to\Omega^{1}({\mathfrak{g}})$, i.e., the tangent space to the $\mathcal{G}$-orbit is $\mathop{\mathrm{im}}(d_{A})$ [Donaldson1990, p. 37], hence $H^{1}({\rm C})$ represents the tangent space $T_{[A]}\mathcal{M}^{\ast}$ at $[A]$. ∎

Proposition 3.4 shows Theorem 1.2–(i). Observe that, in contrast to the $4$-dimensional case, the complex ${\rm C}^{\bullet}$ in (1.10) is not elliptic (eg. by comparing bundle ranks in (3.4)), so we do not know a priory whether $H^{1}({\rm C})$ is finite-dimensional. In the next section, we show in Proposition 3.7 that this is indeed the case, but we must resort to an auxiliary construction.

3.2 The extended complex of selfdual contact instantons

Let us prove that the first cohomology group $H^{1}({\rm C})$ in (1.10) is finite-dimensional.

Lemma 3.5**.**

Let $\sigma=\eta\wedge d\eta$ the natural $3$-form on the Sasakian manifold $(M^{7},\eta,\xi,g,\Phi)$, as in (1.5). The linear map

[TABLE]

satisfies

[TABLE]

Proof.

The transverse symplectic $2$-form $d\eta=:\omega$ is given in local Darboux coordinates [blair2010contact, Theorem. 3.1] by $d\eta=dx^{14}+dx^{25}+dx^{36}$, so $d\eta^{2}=-2(dx^{1245}+dx^{1346}+dx^{2356})$, and therefore $\ast\sigma$ is given by:

[TABLE]

Using the decomposition (2.14) of $\Omega^{2}(M)$ to express $L_{\sigma}$ in the bases $\{w_{i}\}_{i=1}^{8}$ and $\{v_{j}\}_{j=1}^{6}$ from (2.13), we obtain both claims by direct computation. ∎

In terms of the projected differential (1.10), the previous Lemma gives the identification

[TABLE]

since $L_{\ast\sigma}|_{\Omega^{2}_{1}\oplus\Omega^{2}_{V}}\colon\Omega^{2}_{1}\oplus\Omega^{2}_{V}\xrightarrow[]{\sim}\Omega^{6}(M)$ [cf. Lemma 3.5] and, in local Darboux coordinates [blair2010contact, Theorem. 3.1], we can define an isomorphism $T\colon\Omega_{V}^{2}\xrightarrow[]{\sim}\Omega^{2}_{6}$ by

[TABLE]

These identifications can be summarised in the following diagram:

[TABLE]

It is worth mentioning that a canonical isomorphism $\Omega_{V}^{2}\simeq\Omega_{6}^{2}$ is given by the global transverse holomorphic volume form, in the special case of contact Calabi-Yau $7$-manifolds, cf. Definition 3.8 and Lemma 3.10 below.

From the definition (1.8) of $\mathcal{M}$, we know that this space is described (modulo gauge) near an instanton $A$ as the zero locus of the map

[TABLE]

where the neighbourhood $S_{A,\varepsilon}$ from (3.3) is transversal to $\mathcal{G}$-orbits. Let us check that $\Psi$ is a Fredholm map, so that standard theory provides a finite-dimensional local model for $\mathcal{M}$. The linearisation of $\Psi$ at the origin is

[TABLE]

and $d_{7}|_{\operatorname{ker}(d^{\ast}_{A})}\colon\Omega^{1}({\mathfrak{g}}_{E})\to\Omega^{2}_{6\oplus 1}({\mathfrak{g}}_{E})$ is shown to be Fredholm via the ‘Euler characteristic’ map

[TABLE]

associated to the complex ${\rm C}^{\bullet}$ (1.10). Note that:

[TABLE]

By (3.7), we identify $d_{7}$ with $L_{\ast\sigma}\circ d_{A}\colon\Omega^{1}({\mathfrak{g}}_{E})\to\Omega^{6}({\mathfrak{g}}_{E})$, then we can consider the extended complex

[TABLE]

If (3.11) is elliptic, then $\mathbbm{D}_{A}$ is Fredholm and, in particular, $\operatorname{ker}(\mathbbm{D}_{A})$ is finite-dimensional. To see that explicitly, we will need the following elementary technical facts:

Lemma 3.6**.**

1. (i)

Let $L_{\ast\sigma}\colon\Omega^{2}(M)\to\Omega^{6}(M)$ the operator defined in (3.5), then $[L_{\ast\sigma},d_{A}]=0$. 2. (ii)

The formal adjoint of $d_{7}$ defined by (1.9) is given by $d_{7}^{\ast}=\ast d_{7}\ast\colon\Omega^{6}({\mathfrak{g}}_{E})\to\Omega^{1}({\mathfrak{g}}_{E})$.

Proof.

For $(i)$, note from (3.6) that $d(\ast\sigma)=d(d\eta^{2})=0$, so the assertion is straightforward:

[TABLE]

For (ii) we use (i) and the identification in (3.7); for any $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$ and $\beta\in\Omega^{6}({\mathfrak{g}}_{E})$,

[TABLE]

Proposition 3.7**.**

If $A$ is a connection with curvature $F_{A}\in\Omega^{2}_{8}$, the extended complex (3.11) is elliptic.

Proof.

We follow the argument of [SaEarp2009, Proposition 1.22]. Fix a non zero section $\varsigma$ of $\pi\colon T^{\ast}M\setminus\{0\}\to M$, so we have the symbol complex

[TABLE]

To see the exactness of (3.12) at the middle, take $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$ such that $\ast\sigma\wedge\varsigma\wedge\alpha=0$. We need to show that $\alpha$ lies in $(\Omega^{0}({\mathfrak{g}}_{E})).\varsigma$. Note that $\varsigma\wedge\alpha\in\operatorname{ker}(L_{\ast\sigma})=\Omega^{2}_{8}\oplus\Omega^{2}_{6}$ [cf. (3.5)], so, by definition of the eigenspaces $\Omega^{2}_{8}$ and $\Omega^{2}_{6}$ [cf. (2.13)], we obtain:

[TABLE]

We show that (3.13) implies that $\varsigma\wedge\alpha=0$ and this finish the proof. Let $\{e^{i}\}_{i=1}^{7}$ be a basis of $T^{\ast}_{x}M$ with $e^{1}=\varsigma$. Then $\mu\in\Lambda^{k}T_{x}^{\ast}M$, can be written

[TABLE]

where $\gamma$ and $\beta$ are products just involving $e^{2},\cdots,e^{7}$. Let $\alpha=\varsigma\wedge\gamma+\beta$, $\eta=\varsigma\wedge\gamma^{\prime}+\beta^{\prime}$ and $d\eta=\varsigma\wedge\gamma^{\prime\prime}+\beta^{\prime\prime}$ where $\gamma,\gamma^{\prime},\gamma^{\prime\prime}$ and $\beta,\beta^{\prime},\beta^{\prime\prime}$ are products just involving $e^{2},\cdots,e^{7}$, hence (3.13) becomes

[TABLE]

so $\beta^{\prime}\wedge\beta^{\prime\prime}\wedge\varsigma\wedge\beta=\ast(\varsigma\wedge\beta)$ the left-hand side of the above equality involves $\varsigma$, while the right-hand side does not, so $\varsigma\wedge\beta=0$. Hence

[TABLE]

as claimed, and so $\alpha=f\wedge\varsigma.$ ∎

Together, Propositions 3.4 and 3.7 prove Theorem 1.2–$(i)$, i.e., that the space of infinitesimal deformations of $A$ is finite-dimensional and isomorphic to the first cohomology $H^{1}({\rm C})$ of (1.10). In §4, we address such deformations from another perspective, showing that this local model is isomorphic to the first cohomology group of the complex (1.11) [cf. Theorem 1.2–$(ii)$]. As observed in Remark 1.2, to assume that $H^{2}({\rm C})$ vanishes would in general be much too strong, leading to a [math]-dimensional local model. Instead, we can show the smoothness of the moduli space under a weaker obstruction theory, in terms of $H^{2}_{B}$ [cf. Proposition 4.27].

3.3 ${\rm G}_{2}$-instantons on contact Calabi-Yau manifolds

We have seen that the Sasakian structure $(\eta,\xi,g,\Phi)$ on $M^{7}$ naturally induces a moduli space of SDCI. In the contact Calabi-Yau case, moreover, the Sasakian $7$-manifold carries in fact a transverse ${\rm SU}(3)$-structure, and hence a natural ${\rm G}_{2}$-structure [habib2015some]. As such, it may indeed be seen as somewhat of an interpolation between $CY^{3}$-geometry and ${\rm G}_{2}$-geometry. We explain the relationship between ${\rm G}_{2}$-instantons and contact instantons [see Proposition 3.12] in that context, following the approach of [Calvo-Andrade2016].

Definition 3.8**.**

A Sasakian manifold $(M^{2n+1},\eta,\xi,\Phi,\varepsilon)$ is said to be a contact Calabi-Yau manifold (cCY) if $\varepsilon$ is a nowhere-vanishing transverse form of horizontal type $(n,0)$ [cf. (A.1)] such that

[TABLE]

It is well-known that, for a Calabi-Yau 3-fold $({\rm Z},\omega,\varepsilon)$, the product ${\rm Z}\times{\rm S}^{1}$ has a natural torsion-free ${\rm G}_{2}$-structure defined by: $\varphi:=dt\wedge\omega+\mathop{\mathrm{Im}}(\varepsilon),$ where $t$ is the coordinate on ${\rm S}^{1}$. The Hodge dual of $\varphi$ is

[TABLE]

and the induced metric $g_{\varphi}=g_{\rm Z}+dt\otimes dt$ is the Riemannian product metric on ${\rm Z}\times S^{1}$ with holonomy $\mathrm{Hol}(g_{\varphi})=\text{SU}(3)\subset{\rm G}_{2}$. The contact Calabi-Yau structure essentially emulates all of these features, albeit its ${\rm G}_{2}$-structure has some symmetric torsion. Sasakian manifolds with transverse holonomy ${\rm SU}(n)$ are studied by Habib and Vezzoni; a number of facts from [habib2015some, § 6.2.1] can be summarised as follows:

Proposition 3.9**.**

Every cCY manifold $(M^{7},\eta,\xi,\Phi,\varepsilon)$ carries a cocalibrated ${\rm G}_{2}$-structure

[TABLE]

with torsion $d\varphi=\omega\wedge\omega$ and Hodge dual $4$-form $\psi=\ast\varphi=\frac{1}{2}\omega\wedge\omega+\eta\wedge\mathop{\mathrm{Re}}(\varepsilon)$. Here $\omega:=d\eta$ and $\sigma:=\eta\wedge\omega$, as in (1.5).

Lemma 3.10**.**

On a cCY manifold $(M^{7},\eta,\xi,\Phi,\varepsilon)$, the operator

[TABLE]

satisfies $L_{\varepsilon}|_{\Omega_{6}^{2}\oplus\Omega_{8}^{2}\oplus\Omega_{1}^{2}}\equiv 0$ and $L_{\varepsilon}|_{\Omega_{V}^{2}(M)}\colon\Omega_{V}^{2}(M)\xrightarrow[]{\sim}\Omega_{6}^{2}(M)$ is an isomorphism.

Proof.

In the local Darboux transverse complex coordinates $z_{i}$ defined in (2.15), we have $\varepsilon=dz^{1}\wedge dz^{2}\wedge dz^{3}$. In particular, $\mathop{\mathrm{Im}}(\varepsilon)=(dx^{234}+dx^{126})-(dx^{135}+dx^{456})$. Using the decomposition (2.14) of $2$-forms to compute $L_{\varepsilon}$ in the bases $\{w_{i}\}_{i=1,\dots,8}$ and $\{v_{j}\}_{j=1,\dots,6}$ of (2.13), we obtain immediately $L_{\varepsilon}|_{\Omega_{6}^{2}\oplus\Omega_{8}^{2}\oplus\Omega_{1}^{2}}\equiv 0$ and, by inspection,

[TABLE]

So $L_{\varepsilon}|_{\Omega_{V}^{2}(M)}\colon\Omega_{V}^{2}(M)\xrightarrow[]{\sim}\Omega_{6}^{2}(M)$ is an isomorphism and coincides with $T$ in (3.8). ∎

The ${\rm G}_{2}$-instanton equation on a cCY $7$-manifold reads:

[TABLE]

or, equivalently, $F_{A}\wedge\psi=0,$ where $\psi$ is the dual $4$-form (3.14).

Proposition 3.11**.**

On a cCY $7$-manifold, the SDCI equation (1.2) implies the ${\rm G}_{2}$-instanton equation (3.17).

Proof.

If $A$ is a SDCI, then $F_{A}=\ast(\sigma\wedge F_{A})\in\Omega^{2}_{8}({\mathfrak{g}}_{E})\subset\operatorname{ker}L_{\varepsilon}$, by Lemma 3.10. Therefore

[TABLE]

If the complex Sasakian bundle ${\bf E}\to M^{7}$ has a holomorphic structure, then, at least among Chern connections (mutually compatible with the holomorphic structure and a Hermitian metric), the sets of solutions of both equations actually coincide:

Proposition 3.12**.**

Let $\mathcal{E}\to M$ be a Sasakian holomorphic vector bundle [cf. Definition A.5] on a cCY $7$-manifold with its natural ${\rm G}_{2}$-structure (3.15). Then a Chern connection $A$ on $\mathcal{E}$ is a ${\rm G}_{2}$-instanton if, and only if, $A$ is a SDCI as in (2.4).

Proof.

If $A$ is the Chern connection, then $F_{A}\in\Omega^{1,1}(M)$ [cf. Proposition A.2], so taking account of the bi-degree of the transverse holomorphic volume form $\varepsilon$ [cf. Definition 3.8], it follows that $F_{A}\wedge\mathop{\mathrm{Im}}(\varepsilon)=0$. Therefore

[TABLE]

3.4 The Yang-Mills and Chern-Simons functionals

We will describe two natural gauge-theoretic action functionals on a Sasakian $G$-bundle ${\bf E}\to M^{7}$ [Definition A.4], adapting the approach of [sa2014generalised]. This section culminates at topological Yang-Mills energy bounds on holomrphic Sasakian bundles, in terms of integrable (anti-)selfdual contact instantons.

The Yang-Mills functional acts on the space $\mathcal{A}$ of connections on ${\bf E}$ and it is defined by

[TABLE]

The curvature $F_{A}$ splits orthogonally (2.3) into horizontal and vertical parts, respectively:

[TABLE]

so $\operatorname{YM}(A)$ has at least two independent components. Moreover, the horizontal part further splits orthogonally, by Lemma 2.3:

[TABLE]

hence

[TABLE]

Given $A\in\mathcal{A}$, the * charge* of $A$ is defined by

[TABLE]

Proposition 3.13**.**

Let $A\in\mathcal{A}$ be a connection on ${\bf E}$, the charge of A (3.20) is determined by the horizontal curvature:

[TABLE]

Furthermore,

[TABLE]

These bounds are saturated if and only if $A$ is ASDCI or SDCI.

Proof.

It suffices to show (3.21), because then (3.22) follows from (3.19). Using the definition of $\kappa(A)$ in (3.20) and the definition of $F_{H}^{+}$ given above, we compute:

[TABLE]

Analogously,

[TABLE]

Therefore, $\kappa(A)=\|F_{H}^{+}\|^{2}-\|F_{H}^{-}\|^{2}.$ Now using the inequality $|a^{2}-b^{2}|\leq a^{2}+b^{2}$, we obtain the following bounds:

[TABLE]

Under the hypothesis $F_{V}=0$, if $A$ is a SDCI, then $F^{-}_{H}=0$, and it follows from (3.21) that the latter bound is saturated. ∎

Now, fix a reference connection $A_{0}\in\mathcal{A}$ on ${\bf E}$. The Chern-Simons action is defined by: $\text{CS}(A_{0})=0$ and

[TABLE]

Lemma 3.14**.**

Let ${\bf E}\to M^{7}$ be a Sasakian $G$-bundle over a closed $7$-manifold. For any connection $A^{\prime}=A+\alpha\in\mathcal{A}$, for $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$, on the underlying complex vector bundle $E$, we have:

[TABLE]

Proof.

Let $A\in\mathcal{A}$ and let $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$ be a variation of $A$. From standard Chern-Weil theory, we know that:

[TABLE]

where $\delta=\delta(A,\alpha)\in\Omega^{1}({\mathfrak{g}}_{E})$ given by $\delta=F_{A}\wedge\alpha+\frac{1}{2}d_{A}\alpha\wedge\alpha+\frac{1}{3}\alpha\wedge\alpha\wedge\alpha.$ Since $M$ is closed, by Stokes’ theorem we obtain

[TABLE]

Now, we analyse the term $\int_{M}\mathop{\mathrm{tr}}\nolimits\delta\wedge d\sigma$ for a transverse $1$-form $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$:

[TABLE]

The last equality holds since the $7$-form $\mathop{\mathrm{tr}}\nolimits(\alpha\wedge\alpha\wedge\alpha)\wedge d\eta^{2}$ is basic; the same is true for $\mathop{\mathrm{tr}}\nolimits(d_{A}\alpha\wedge\alpha)\wedge d\eta^{2}$. ∎

Corollary 3.15**.**

Among Chern connections, the charge is independent of the Hermitian structure; it is a holomorphic Sasakian topological invariant, denoted by $\kappa(\mathcal{E})$.

Proof.

Fixing a reference Chern connection $A_{0}\in\mathcal{A}(\mathcal{E})$, we know from Proposition A.2 that $F_{A_{0}}\wedge d\eta^{2}$ has type $(3,3)$, so the defect term $\int_{M}\mathop{\mathrm{tr}}\nolimits(F_{A_{0}})\wedge d\eta^{2}\wedge\alpha$ vanishes, by excess in bi-degree, for any $\alpha\in\Omega^{1,0}({\mathfrak{g}}_{E})$. Therefore $\kappa(A_{0})=\kappa(A_{0}+\alpha)$. ∎

Now, let $A\in\mathcal{A}$ be a connection on ${\bf E}$. From (3.19) and (3.21) we have

[TABLE]

It follows that $\operatorname{YM}$ attains an absolute minimum either at selfdual contact instantons, i.e. when $F_{H}^{-}=0$ and $F_{V}=0$, or at anti-selfdual contact instantons, i.e. when $F_{H}^{+}=0$ and $F_{V}=0$. Furthermore, from Corollary 3.15, among Chern connections we can replace $\kappa(A)=\kappa(\mathcal{E})$ in the above equalities, thus the sign of $\kappa(\mathcal{E})$ obstructs the existence of one or the other type of solution.

4 The moduli space of contact instantons in $7$-dimensions

We showed in §3 that there exists a finite-dimensional local model for the moduli space near an irreducible SDCI. In this Section we will show part (ii) of Theorem 1.2, namely, that its dimension can be computed from an associated transverse elliptic complex. Our strategy is inspired by [Baraglia2016, § 3].

4.1 The associated elliptic complex of a contact instanton

Consider a Sasakian $G$-bundle ${\bf E}\to(M^{7},\eta,\xi,g,\Phi)$ and ${\mathfrak{g}}_{E}$ its adjoint bundle. Recalling that $\Omega^{k}_{H}({\mathfrak{g}}_{E})$ denotes the space of horizontal forms, i.e., $\alpha\in\Omega^{k}({\mathfrak{g}}_{E})$ for which $i_{\xi}\alpha=0$, we introduce the following operators:

[TABLE]

Moreover, $\alpha\in\Omega^{k}_{H}({\mathfrak{g}}_{E})$ is called basic if $i_{\xi}d_{A}\alpha=0$; we denote by $\Omega^{k}_{B}({\mathfrak{g}}_{E})$ the space of basic forms.

Remark 4.1*.*

The operators $d_{T}$ and $d_{V}$ are defined in [Baraglia2016, § 3.1] for a $5$-dimensional contact manifold. Note that the restriction of the Lie derivative along the Reeb field $\mathcal{L}_{\xi}|_{\Omega^{k}_{H}({\mathfrak{g}}_{E})}=d_{V}$, thus $\Omega^{k}_{B}({\mathfrak{g}}_{E})=\operatorname{ker}(d_{V})$. Moreover, we can write $d_{T}(\alpha)=i_{\xi}(\eta\wedge d_{A}\alpha)$, i.e., $d_{T}\alpha$ is just the horizontal part of $d_{A}\alpha$. In particular, $d_{T}$ coincides with the usual covariant exterior differential $d_{A}$ on basic forms.

Remark 4.2*.*

Even though almost all results in this section hold for $M^{7}$ just compact, in Proposition 4.16 we need it to be actually closed.

The graded ring $\left(\Omega^{\bullet}({\mathfrak{g}}_{E}),\wedge\right)$ has a natural graded Lie algebra structure, given in local coordinates by the bi-linear map

[TABLE]

The following properties are immediate to check, for $\Phi\in\Omega^{p}({\mathfrak{g}}_{E})$, $\Psi\in\Omega^{q}({\mathfrak{g}}_{E})$ and $\Theta\in\Omega^{p}({\mathfrak{g}}_{E})$:

[TABLE]

[TABLE]

The next result is a $7$-dimensional adaptation of [Baraglia2016, Lemma 3.1], under the assumption that $(M^{7},\eta,\xi,g)$ is a K-contact manifold, i.e., that the Reeb field $\xi$ is Killing. Every Sasakian manifold is $K-$contact but, at dimensions greater than $3$, the $K-$contact condition is strictly weaker. Yet, if the manifold is compact and Einstein, both notions again coincide [boyer2001einstein]:

Lemma 4.3**.**

Let ${\rm I}=\text{Span}\{\Omega^{2}_{8}\}\subset\Omega^{\bullet}({\mathfrak{g}}_{E})$ be the algebraic ideal of the graded Lie algebra $\left(\Omega^{\bullet}({\mathfrak{g}}_{E}),\wedge\right)$ generated by $\Omega^{2}_{8}({\mathfrak{g}}_{E})$; if the Reeb field $\xi$ is a Killing vector field, then $d_{A}({\rm I})\subset{\rm I}$.

Proof.

Since ${\rm I}$ lies in the image of the exterior product $\Omega^{2}_{8}\otimes\Omega^{\bullet}({\mathfrak{g}}_{E})\to\Omega^{\bullet}({\mathfrak{g}}_{E})$, by linearity it suffices to show that $d_{A}(\alpha\otimes\psi)\in{\rm I}$, where $\alpha\in\Omega^{2}_{8}$ in (2.13) and $\psi\in\Omega^{0}({\mathfrak{g}}_{E})$. Indeed,

[TABLE]

and the last two terms clearly lie in ${\rm I}$. In order to show that $\eta\wedge d_{V}\alpha\in{\rm I}$, we check that $d_{V}\alpha=i_{\xi}d_{A}\alpha=\mathcal{L}_{\xi}\alpha$ belongs to $\Omega^{2}_{8}$. By the eigenspace decomposition (2.13), if $\alpha\in\Omega^{2}_{8}$, then $\alpha=\ast(\sigma\wedge\alpha)$. Furthermore, since $\xi$ is Killing, the Lie derivative $\mathcal{L}_{\xi}$ commutes with the star Hodge operator, so

[TABLE]

We deduce that $\mathcal{L}_{\xi}\alpha$ satisfies the instanton equation (2.4) for $\lambda=1$, i.e., $\mathcal{L}_{\xi}\alpha\in\Omega^{2}_{8}$. ∎

Lemma 4.3 shows that $d_{A}$ descends to a derivation $D\colon{\rm L}^{k}\to{\rm L}^{k+1}$ on the quotient ${\rm L}^{\bullet}={\rm L}^{\bullet}({\mathfrak{g}}_{E}):=\Omega^{\bullet}({\mathfrak{g}}_{E})/{\rm I},$ which is indeed a complex, since $d_{A}^{2}=F_{A}\in\Omega^{2}_{8}\subset{\rm I}$. Therefore the quotient has the structure of a differential graded Lie algebra:

[TABLE]

Remark 4.4*.*

The statement of Lemma 4.3 follows analogously if one takes, instead, the ideal ${\rm I}=\text{Span}\{\Omega^{2}_{6}\}$, with ${\rm L}^{\bullet}=\Omega^{\bullet}({\mathfrak{g}}_{E})/{\rm I}$. This will have no further bearing in this article, since after all we know that the corresponding instantons in that case are trivial [Proposition 3.1], but, whatever the case, we have explicit characterisations for the spaces ${\rm L}^{k}$ as follows.

Proposition 4.5**.**

The spaces ${\rm L}^{k}$ of complex (4.3), for $k=0,\dots,3$, admit the following decompositions:

1. (i)

For ${\rm I}=\text{Span}\{\Omega^{2}_{8}({\mathfrak{g}}_{E})\}$,

[TABLE] 2. (ii)

For ${\rm I}=\text{Span}\{\Omega^{2}_{6}({\mathfrak{g}}_{E})\}$,

[TABLE]

In either case, ${\rm L}^{k}=0$ for $k\geq 4$.

Proof.

For $(i)$, the identifications are immediate for ${\rm L}^{0},{\rm L}^{1}$ and ${\rm L}^{2}$, combining $\Omega^{k}({\mathfrak{g}}_{E})=\Omega^{k}_{H}({\mathfrak{g}}_{E})\oplus\eta\wedge\Omega^{k-1}_{H}({\mathfrak{g}}_{E})$, as in (A.5), and the natural decomposition of ${\mathfrak{g}}_{E}$-valued $2$-forms $\Omega^{2}({\mathfrak{g}}_{E})$ induced by (2.14):

[TABLE]

To show that $\Omega^{3}_{H}({\mathfrak{g}}_{E})\subset{\rm I}$, let $\{z_{j}\}_{j=1}^{3}$ be the Darboux transverse complex coordinates as in (2.15). Following [bedulli2007ricci, § 2.1], we denote by

[TABLE]

the real and imaginary parts of the local transverse holomorphic volume form $dz^{1}\wedge dz^{2}\wedge dz^{3}$, respectively. For every point $x\in M$, $(H_{x},\omega=d\eta)$ is a transverse symplectic vector space, and $J:=\Phi|_{H}$ is a complex structure. The group ${\rm SU}(3)$ acts irreducibly on $\Lambda^{5}H^{\ast}$ and $H^{\ast}$, while $\Lambda^{2}H^{\ast}$ and $\Lambda^{3}H^{\ast}$ decompose as follows [bedulli2007ricci, § 2]:

[TABLE]

where

[TABLE]

and

[TABLE]

Our goal is to show that all subspaces in the decomposition of $\Lambda^{3}H^{\ast}$ lie in ${\rm I}$. The following exterior multiplication table (row $\wedge$ column) is easy to compute:

[TABLE]

From that we obtain a set of generators for $\Lambda^{3}_{12}H^{\ast}$:

[TABLE]

In terms of the $\{w_{i}\}_{i=1}^{8}$ of (2.13), this translates into

[TABLE]

To see that $\Lambda_{6}^{3}H^{\ast}\subset{\rm I}$, note that $\omega\wedge dx^{i}=dx^{14i}+dx^{25i}+dx^{36i}$. Equivalently, in terms of $w_{7}$ and $w_{8}$ in (2.13),

[TABLE]

Let us have a closer look at terms of the form $dx^{36i}$. For $i=3$ and $i=6$, clearly $dx^{36i}=0$; for $i=1$ and $i=4$, necessarily $dx^{36i}=\pm w_{7}\wedge dx^{i}$; for $i=2$ and $i=5$, one has $dx^{36i}=\pm w_{8}\wedge dx^{i}$, hence in all instances $\Lambda_{6}^{3}H^{\ast}=\{w_{7}\wedge dx^{i}+w_{8}\wedge dx^{i}+3dx^{36i}\}_{i=1}^{6}\subset{\rm I}$. Finally, we observe that:

[TABLE]

whereas

[TABLE]

i.e., $(X_{i}\lrcorner\Theta_{+})\in\Lambda^{2}_{8}H^{\ast}$, and thus $\Lambda_{\mathop{\mathrm{Re}}}^{3}H^{\ast}\subset{\rm I}$. That $\Lambda_{\mathop{\mathrm{Im}}}^{3}H^{\ast}\subset{\rm I}$ follows analogously.

For $(ii)$, it is still manifest that ${\rm L}^{0},{\rm L}^{1},{\rm L}^{2}\subset{\rm I}$. As to ${\rm L}^{3}$, we can inspect directly the basis elements $\{v_{i}\}_{i=1}^{6}$ for $\Omega^{2}_{6}$ introduced in (2.13). We claim that every element in the Darboux coordinate basis for $\Omega^{3}_{H}({\mathfrak{g}}_{E})$ is obtained from an element of $\Omega^{1}_{6}({\mathfrak{g}}_{E})$. Indeed,

[TABLE]

as well as

[TABLE]

Similarly, one can easily check that $\Omega^{4}_{H}({\mathfrak{g}}_{E}),\;\Omega^{5}_{H}({\mathfrak{g}}_{E})$ and $\Omega^{6}_{H}({\mathfrak{g}}_{E})$ lie in ${\rm I}$, thus ${\rm L}^{k}=0$ for $k\geq 4$. ∎

Lemma 4.6**.**

The followings maps preserve ${\rm I}$ (whether ${\rm I}$ be generated by $\Omega^{2}_{8}({\mathfrak{g}}_{E})$ or by $\Omega^{2}_{6}({\mathfrak{g}}_{E})$)

[TABLE]

so these maps descend to the quotient $L^{\bullet}$.

Proof.

If $\alpha\wedge\beta\in{\rm I}$, for some $\alpha\in\Omega^{\bullet}({\mathfrak{g}}_{E})$ and $\beta\in\Omega^{2}_{8}({\mathfrak{g}}_{E})$,

[TABLE]

Therefore the map $i_{\xi}$ is a well-defined contraction of bi-degree $(0,-1)$ on ${\rm L}^{\bullet}$. We also denote by $L_{H}^{k}$ the kernel of contraction inside of ${\rm L}^{k}$, so that ${\rm L}^{k,0}={\rm L}^{k}_{H}$. ∎

We denote the induced maps on the quotient ${\rm L}^{\bullet}$ as follows:

[TABLE]

In summary, by Lemma 4.5 we have the following

Proposition 4.7**.**

If a connection $A$ has curvature $F_{A}\in\Omega^{2}_{8}({\mathfrak{g}}_{E})$, the associated complex (1.11) is elliptic:

[TABLE]

Proof.

Denote by $\pi\colon T^{\ast}M\setminus\{0\}\to M$ the fibre bundle obtained by removal of the zero section in $T^{*}M$, and by $p:\Omega^{\bullet}({\mathfrak{g}}_{E})\to{\rm L}^{\bullet}$ the quotient projection, i.e., $D_{k}=p\circ d_{A}^{k}$. Given $\varsigma\in T^{\ast}M\setminus\{0\}$, the first order symbol functor $\sigma_{1}$ satisfies:

[TABLE]

The fibre $\pi^{\ast}\left({\rm L}^{k}\right)_{\varsigma}$ is isomorphic to $\Lambda^{k}T_{x}^{\ast}M/{\rm I}$, where ${\rm I}=\langle\Omega^{2}_{8}\rangle$, hence the associated $1$-symbol is

[TABLE]

and we assert that the associated symbol complex is exact:

[TABLE]

Using the identifications from Proposition 4.5, the above complex becomes

[TABLE]

Exactness at positions $k=0,1,2$ is shown by the same argument, so we prove explicitly for $k=2$ and $k=3$.

$k=2$: Let $\alpha\in(\Lambda^{2}_{6\oplus 1}\oplus\eta\wedge(\Lambda^{1}_{H}))(T_{x}^{\ast}M)$ such that $\varsigma\wedge\alpha\in\langle\Lambda^{2}_{8}\rangle$. Since $\varsigma$ is non zero, this forces to $\varsigma\wedge\alpha=0$, and we know that the de Rham complex is elliptic, so there exists $\beta\in\Lambda^{1}(T_{x}^{\ast}M)$ such that $\varsigma\wedge\beta=\alpha$. Hence we have exactness at $(\Lambda^{2}_{6\oplus 1}\oplus\eta\wedge(\Lambda^{1}_{H}))(T_{x}^{\ast}M)$.

$k=3$: This is similar to the $k=2$ case, but we need to show that $\beta\in(\Lambda^{2}_{6\oplus 1}\oplus\eta\wedge(\Lambda^{1}_{H}))(T_{x}^{\ast}M)$. A priori, $\beta=\beta_{1}+\beta_{2}$, with

[TABLE]

Since $\beta_{2}$ projects to zero under $p$, we have $\alpha=p(\varsigma\wedge\beta_{1})$, so (3.12) is exact. ∎

The above complex in Proposition 4.7 is referred to as the associated complex to the selfdual contact instanton $A$ (Proposition 4.7 can be shown in the same way if $F_{A}\in\Omega^{2}_{6}({\mathfrak{g}}_{E})$). We denote by $H^{k}:=H^{k}({\rm L})$ the cohomology groups of ${\rm L}^{\bullet}$ (1.11), $H^{1}$ can be interpreted as the space of infinitesimal deformations of the contact instanton $A$, so that

[TABLE]

represents the expected dimension of the moduli space.

4.2 Deformation theory of SDCI

The splitting (A.1) of the tangent space $TM=H\oplus N_{\xi}$ by the Reeb vector field defines a bi-grading on $\Omega^{\bullet}({\mathfrak{g}}_{E})$:

[TABLE]

In particular,

[TABLE]

Since the ideal ${\rm I}$ is bi-graded, the bi-grading descends to the quotients ${\rm L}^{\bullet}$ to define components ${\rm L}^{k,d}$. In our case of main interest, ${\rm I}=\langle\Omega^{2}_{8}({\mathfrak{g}}_{E})\rangle$ induces the complex (1.11):

[TABLE]

Define in the quotient the followings maps

[TABLE]

Lemma 4.8**.**

Let $D_{T}$ and $D_{V}$ be the operators defined in (4.10), then we have the following identities

[TABLE]

and

[TABLE]

Proof.

From $D^{2}=0$, we obtain

[TABLE]

so, $D_{T}^{2}+\omega\wedge D_{V}=0=D_{V}D_{T}-D_{T}D_{V}$, hence (4.11) and (4.12) follow. ∎

The basic forms in ${\rm L}^{k}$ are denoted by ${\rm L}^{k}_{B}=\operatorname{ker}(D_{V})\subset{\rm L}^{k}$ and, by Proposition 4.5, $\operatorname{ker}(D_{V})$ can be identified with $\Omega^{\bullet}_{B}({\mathfrak{g}}_{E})$. Since $D_{V}$ and $D_{T}$ commute (4.11),

[TABLE]

Moreover, $D_{T}$ restricts to $D_{B}\colon{\rm L}_{B}^{k}\to{\rm L}_{B}^{k+1}$, so $D_{B}$ defines a basic deformation complex. Considering ${\rm I}=\langle\Omega^{2}_{8}({\mathfrak{g}}_{E})\rangle$, the associated basic complex (1.12) becomes

[TABLE]

Denote by $H_{B}^{\bullet}:=H^{\bullet}({\rm L}_{B})$ the cohomology of the basic complex ${\rm L}^{\bullet}_{B}$. This is not an elliptic complex, yet it is elliptic transversely to the Reeb foliation. In particular, its cohomology $H_{B}^{\bullet}$ is finite-dimensional [kacimi1990operateurs, Theorem  3.2.5], and we will see that this complex computes the dimension of the moduli space of contact instantons.

Lemma 4.9**.**

Let $D_{T}$ and $D_{V}$ be the operators defined in (4.10), then the formal adjoints of $D_{V}$ and $D_{T}$ are given by:

[TABLE]

[TABLE]

Furthermore,

[TABLE]

Proof.

For (4.13) note that, since $\mathcal{L}_{\xi}$ coincides with $D_{V}$ on $\Omega_{H}^{\ast}({\mathfrak{g}}_{E})$, if $\xi$ is a Killing vector then $D_{V}$ and $\ast_{T}$ commute, and so

[TABLE]

Furthermore, since $D_{T}D_{V}=D_{V}D_{T}$ we obtain (4.15). To show (4.14), let $\alpha\in\Omega^{k-1}_{H}({\mathfrak{g}}_{E})$ and $\beta\in\Omega^{k}_{H}({\mathfrak{g}}_{E})$:

[TABLE]

and the assertion follows by observing that $i_{\xi}D\eta(\alpha)=i_{\xi}(D(\eta\wedge\alpha))=i_{\xi}(\omega\wedge\alpha-\eta\wedge D\alpha)=-D_{T}\alpha$. ∎

We now adapt a number of fundamental insights from [Baraglia2016, Proposition 3.3]. We begin by introducing the transverse Laplacian on the complex ${\rm L}^{\bullet}$:

Definition 4.10**.**

The Laplacian of $D$, with respect to the inner product defined in (2.9), is

[TABLE]

and the transverse Laplacian is defined by:

[TABLE]

Clearly $\Delta_{T}$ and $\Delta$ have the same symbol, so $\Delta_{T}$ is an elliptic operator. Let us denote the spaces of $\Delta-$harmonic and $\Delta_{T}-$harmonic forms, respectively, by

[TABLE]

Since $\langle\Delta_{T}\alpha,\alpha\rangle=\|D_{T}\alpha\|^{2}+\|D_{T}^{\ast}\alpha\|^{2}+\|D_{V}\alpha\|^{2}$,

[TABLE]

On basic forms, in particular, $\Delta_{T}$-harmonicity is equivalent to $\Delta$-harmonicity. Note that $\Delta_{T}$ respects the bi-grading defined in (4.8), hence we can split $\mathcal{H}^{k}_{T}$ into components $\mathcal{H}^{k,d}_{T}$. Since $L_{\eta}\colon{\rm L}^{k,0}\to{\rm L}^{k,1}$ (4.5) is an isomorphism, we know from the outset that

[TABLE]

Lemma 4.11**.**

There exists an isomorphism $\phi\colon\mathcal{H}_{T}^{k,0}\tilde{\to}H_{B}^{k}$, where $\mathcal{H}_{T}^{k,0}$ is just above defined and $H_{B}^{k}$ is the $k$-th cohomology group of the basic complex (1.12).

Proof.

Let the morphism $\phi\colon\mathcal{H}_{T}^{k,0}\to H_{B}^{k}$ send $\alpha\in\mathcal{H}_{T}^{k,0}$ to its equivalence class $[\alpha]\in H_{B}^{k}$. This is well-defined, by (4.18), since $\mathcal{H}^{k}_{T}\subset\ker(D_{V})=\Omega^{k}_{B}({\mathfrak{g}}_{E})$ and $D$ coincides with $D_{T}$ on basic forms, so indeed $\mathcal{H}^{k}_{T}\subset\operatorname{ker}(D_{B})$, therefore $\alpha$ defines a equivalence class in $H^{k}_{B}$.

We first check injectivity of $\phi$. If $\phi(\alpha)=[\alpha]=0\in H_{B}^{k}$, then

[TABLE]

$D_{V}\beta=0$ because $\beta$ is a basic form, hence

[TABLE]

Applying $D^{\ast}_{T}$ to $\alpha$ follows that $0=D^{\ast}_{T}\alpha=D^{\ast}_{T}D_{T}\beta$, now taking inner product with $\beta$ in the last equality we obtain that $D_{T}\beta=0=\alpha$.

To check surjectivity, let $\alpha$ be closed and basic, i.e, $\alpha\in{\rm L}^{k,0}$ and $D_{T}\alpha=0=D_{V}\alpha.$ Elliptic theory implies that $\alpha=\beta+\Delta_{T}\gamma$, where $\beta\in\operatorname{ker}(\Delta_{T})$. From (4.15), we have $D_{V}\Delta_{T}=\Delta_{T}D_{V}$, hence

[TABLE]

It follows that $D_{V}\gamma$ is $\Delta_{T}$-harmonic, in particular $D_{V}^{2}\gamma=0$, and so

[TABLE]

This implies

[TABLE]

so, $\alpha=\beta+D_{T}D_{T}^{\ast}\gamma$. Now, $D_{T}^{\ast}\gamma$ is a basic form, since $D_{V}D_{T}^{\ast}\gamma=D_{T}^{\ast}D_{V}\gamma=0$, so in fact $\alpha=\beta+D_{B}(D_{T}^{\ast}\gamma)$. This shows that every element in $H_{B}^{k}$ has a $\Delta_{T}$-harmonic representative, thus $\phi$ is surjective. ∎

Lemma 4.12**.**

Consider the vector spaces $A^{k}:=\mathcal{H}_{T}^{k,0}\oplus\mathcal{H}_{T}^{k-1,0}$, together with the differential $\hat{{\rm d}}$ defined by:

[TABLE]

The following hold:

1. (i)

$(A^{\bullet},\hat{{\rm d}})$* form a chain complex.* 2. (ii)

There is a chain map $j^{\bullet}\colon(A^{\bullet},\hat{{\rm d}})\to({\rm L}^{\bullet},D)$ into the complex (4.3), defined by

[TABLE] 3. (iii)

$\displaystyle\chi(A)=\sum_{k=0}^{3}(-1)^{k}\dim(H^{k}(A))=0.$**

Proof.

1. (i)

Clearly $\hat{{\rm d}}^{2}=0$, and $\hat{{\rm d}}$ is well-defined in view of the isomorphism $\mathcal{H}_{T}^{k,0}\cong H_{B}^{k}$ from Lemma 4.11:

[TABLE] 2. (ii)

The maps $j^{k}$ in (4.21) fit in the following diagram:

[TABLE]

To see that (4.22) is commutative, the only nontrivial step to check is $Dj^{1}=j^{2}\hat{{\rm d}}.$ Given $(\alpha,\beta)\in A^{1}$, by definition, $D\alpha=0=D_{V}\beta$, hence

[TABLE] 3. (iii)

We know, from Lemma 4.11, that each chain $A^{k}$ in (4.20) is finite-dimensional, for $k=0,1,2,3$, and

[TABLE]

The assertion now follows from the rank-nullity Theorem. ∎

Lemma 4.13**.**

Let ${\rm L}^{k}$, $k=1,2$ be the chain spaces defined in Proposition 4.5. For each $[\alpha]\in H^{k}$, its harmonic representative can be written as:

[TABLE]

such that $D_{V}\beta=D_{T}^{\ast}\beta=0$ and $D_{V}\gamma=D_{T}\gamma=0$.

Proof.

The harmonic representative of $[\alpha]\in H^{k}$ has the general form $\alpha=\beta+\eta\wedge\gamma$ by definition of ${\rm L}^{k}$ (see Proposition 4.5 item $(i)$). We know from (4.10) that $D\alpha=D_{T}\alpha+\eta\wedge D_{V}\alpha$, so

[TABLE]

and

[TABLE]

In summary, we obtain the following relations:

[TABLE]

Applying $D_{T}$ to $(iii)$ and $D_{V}$ to $(ii)$, and commuting by (4.11), we obtain $D_{T}D_{T}^{\ast}\beta-D_{V}^{2}\beta=0$. Taking the inner product with $\beta$, we have

[TABLE]

therefore $D_{V}\beta=D_{T}^{\ast}\beta=0$, and so $D_{V}\gamma=D_{T}\gamma=0$. ∎

Proposition 4.14**.**

The chain map $j^{\ast}\colon(A^{\bullet},\hat{{\rm d}})\to({\rm L}^{\bullet},D)$ defined in (4.21) is a quasi-isomorphism, i.e., the induced map in cohomology $j^{k}\colon H^{k}(A^{\bullet})\to H^{k}({\rm L}^{\bullet})$ is an isomorphism, for each $k=0,1,2,3$.

Proof.

There are four cases to consider:

$k=0$: This case is trivial, since $j^{0}$ is essentially the inclusion map.

$k=3$: We must show that $H^{3}\cong\{0\}\oplus\mathcal{H}^{2,0}_{T}$. From Hodge theory, we have isomorphisms $H^{k}\cong\mathcal{H}^{k}$, so we show that $\mathcal{H}^{3}\cong\mathcal{H}^{2,0}_{T}$. Indeed, $j^{3}$ is just exterior multiplication by $\eta$, so

[TABLE]

$k=1$: By (4.23), a class $[\alpha]\in H^{1}$ has a harmonic representative of the form,

[TABLE]

and we first need to be sure that $\beta,\gamma$ are $\Delta_{T}$-harmonic. By (4.18) and Lemma 4.13, it only remains to check that $D_{T}^{\ast}\gamma=0$ and $D_{T}\beta=0$. The former holds because $\gamma\in\Omega^{0}_{H}({\mathfrak{g}}_{E})\subset\ker(D_{T}^{\ast})$. For the latter, take the inner product with $\beta$ after the following computation:

[TABLE]

Now, let us check that $j^{1}\colon H^{1}(A^{\bullet})\to H^{1}({\rm L}^{\bullet})$ is an isomorphism. From relation $(i)$ in (4.24), we have $\omega\wedge\gamma=0$, and thus $(\beta,\gamma)\in\mathcal{H}^{1,0}_{T}\oplus\mathcal{H}^{0,0}_{T}$ and $\hat{{\rm d}}(\beta,\gamma)=(\omega\wedge\gamma,0)=(0,0)$, i.e., $j^{1}$ is surjective. For injectivity, suppose that $[j^{1}(\beta,\gamma)]=[\beta+\eta\wedge\gamma]=0\in H^{1}({\rm L}^{\bullet}),$ so $\beta+\eta\wedge\gamma=D\delta$, for some $\delta\in\Omega^{0}({\mathfrak{g}}_{E})$, and so

[TABLE]

Comparing types, we have $\beta=D_{T}\delta$ and $\gamma=D_{V}\gamma$, but $\beta$ and $\gamma$ are $\Delta_{T}$-harmonic, so indeed $(\beta,\gamma)=(0,0)$.

$k=2$: From Lemma 4.13, $\alpha=\beta+\eta\wedge\gamma\in\left(\Omega^{2}_{6}\oplus\Omega^{2}_{1}\right)({\mathfrak{g}}_{E})\oplus\eta\wedge\Omega^{1}_{H}({\mathfrak{g}}_{E})={\rm L}^{2},$ where $D_{V}\beta=D_{T}^{\ast}\beta=0$ and $D_{V}\gamma=D_{T}\gamma=0$. Note that $\omega\wedge\gamma=0$, since $\omega\wedge\gamma\in{\rm L}^{3}={\rm L}^{2,1}\cong\eta\wedge(\Omega^{2}_{6}\oplus\Omega^{2}_{1})({\mathfrak{g}}_{E})$, so relation $(i)$ in (4.24) implies that $D_{T}\beta=0$, i.e., $j^{2}\colon H^{2}(A^{\bullet})\to H^{2}({\rm L}^{\bullet})$ is surjective. For injectivity, recall that the complex $({\rm L}^{\bullet},D)$ in (4.3) is elliptic [Proposition 4.7] on a compact odd-dimensional manifold, so

[TABLE]

similarly, $0=\sum(-1)^{k}\dim(H^{k}(A^{\bullet}))$ (see item (iii) of Lemma 4.12). Since $j^{k}$ is an isomorphism for $k=0,1,3$, we conclude that $\dim(H^{2}({\rm L}^{\bullet}))=\dim(H^{2}(A^{\bullet}))$, hence $j^{2}$ is also an isomorphism. ∎

Remark 4.15*.*

Lemma 4.11 and Proposition 4.14 can be shown in the same fashion if we adopt the ideal ${\rm I}=\text{Span}\{\Omega^{2}_{6}\}$, instead of ${\rm I}=\text{Span}\{\Omega^{2}_{8}\}$.

Proposition 4.16**.**

Assume the Sasakian manifold $M^{7}$ is closed (see Remark 4.2). Then the map

[TABLE]

induced in cohomology by $L_{\omega}\colon{\rm L}^{k}\to{\rm L}^{k+2}$ [cf. Lemma 4.6], is injective.

Proof.

Since $D\omega=0$, the exterior product map for $\omega$ is a well-defined map in cohomology. Given $\alpha\in\ker(\omega\wedge\cdot)\subset H^{0}_{B}$, clearly $[\alpha\otimes\omega]=0\in H^{2}_{B}$, so there exists some $\beta\in\Omega^{1}_{B}$ such that $\alpha\otimes\omega=D_{B}\beta$. Then

[TABLE]

Therefore, it suffices to show that the integrand is exact. Indeed:

[TABLE]

At an instanton, the cohomologies $H^{\bullet}$ and $H^{\bullet}_{B}$ of the complexes (1.11) and (1.12), respectively, fit in a Gysin sequence analogous to [tondeur2012foliations, Theorem 10.13]:

Proposition 4.17**.**

At a SDCI, the complex (4.20) induces a long exact sequence in cohomology:

[TABLE]

Proof.

In order to show the exactness of (4.25), we use the isomorphisms $H^{\bullet}(A)\cong H^{\bullet}$ [Proposition 4.14] and $H^{k}_{B}\cong\mathcal{H}^{k,0}_{T}$ [Lemma 4.11]. We recall that the differential $\hat{{\rm d}}^{k}\colon A^{k}\to A^{k+1}$ from (4.20) maps $(\alpha,\beta)\in\mathcal{H}_{T}^{k,0}\oplus\mathcal{H}_{T}^{k-1,0}$ to $(\omega\wedge\beta,0)\in\mathcal{H}_{T}^{k+1,0}\oplus\mathcal{H}_{T}^{k,0}$.

• •

First note that $H^{0}(A)=\ker(\hat{{\rm d}}^{0})=\mathcal{H}^{0,0}_{T}\cong H^{0}_{B},$ hence we obtain exactness in

[TABLE]

Also note that $H^{3}(A)=\mathcal{H}^{2,0}_{T}\cong H^{2}_{B},$ hence we obtain exactness in

[TABLE]

It remains to show the exactness of

[TABLE]

We proceed from left to right.

• •

From Proposition 4.16, the map $L_{\omega}\colon H^{0}_{B}\to H^{2}_{B}$ is injective. Moreover,

[TABLE]

so mapping $H^{1}\to 0$ gives exactness at $H^{1}_{B}$, $H^{1}$ and $H^{0}_{B}$.

• •

The map $H^{2}_{B}\to H^{2}$ is induced by the inclusion ${\rm L}^{2}_{B}\to{\rm L}^{2}$, and its kernel consists of exact basic forms. By Proposition 4.14, these are identified with the image of $L_{\omega}\colon\mathcal{H}^{0}_{T}\to\mathcal{H}^{2,0}_{T}$, hence (4.25) is exact at $H^{2}_{B}$.

• •

We assert that the map

[TABLE]

induced in cohomology by $i_{\xi}\colon L^{2}\to L^{2}_{B}$, is surjective. Indeed, if $\alpha\in{\rm L}^{1}$ is basic and closed, the form $\eta\wedge\alpha$ belongs to ${\rm L}^{2}$, and $D(\eta\wedge\alpha)=\omega\wedge\alpha$ is a basic $3$-form, i.e., $\omega\wedge\alpha\in{\rm L}^{3,0}$[cf. (4.9)], so it is zero, and $(i_{\xi})_{\ast}[\eta\wedge\alpha]=[\alpha]$, as claimed.

• •

Finally, $H^{2}(A)=\frac{\mathcal{H}_{T}^{2,0}\oplus\mathcal{H}_{T}^{1,0}}{\mathop{\mathrm{im}}(\hat{{\rm d}}^{1})}$ and

[TABLE]

so, $H^{2}\cong\frac{H^{2}_{B}}{\omega\wedge(\mathcal{H}^{0,0}_{T})}\oplus H^{1}_{B}\cong\frac{H^{2}_{B}}{\omega\wedge(H^{0}_{B})}\oplus H^{1}_{B}$. ∎

The following immediate consequence of Proposition 4.17 proves part (ii) of Theorem 1.2:

Corollary 4.18**.**

At a SDCI, the inclusion ${\rm L}_{B}^{1}\to{\rm L}^{1}$ induces an isomorphism $H^{1}_{B}\cong H^{1}$, thus the expected dimension of the moduli space (1.8) is $h^{1}_{B}:=\dim H^{1}_{B}$ (1.12).

4.3 Obstruction and smoothness

In this section we will establish that, if the second cohomology group $H^{2}_{B}$ of the basic complex in (1.12) vanishes, then the moduli space $\mathcal{M}^{\ast}$ of irreducible SDCI defined in (1.8) has a local structure of a smooth manifold (Corollary 4.26). Our approach follows the general scheme of [Donaldson1990, § 4.2], adapting to $7$ dimensions some crucial insights from [Baraglia2016, §3.1 & §3.2].

4.3.1 The obstruction map

Lemma 4.19**.**

Let $A$ be a SDCI and $\alpha\in{\rm L}^{1}\cong\Omega^{1}({\mathfrak{g}}_{E})$ [cf. Proposition 4.5]; the connection $A+\alpha$ remains a SDCI if, and only if, $\alpha$ is a Maurer-Cartan element of $({\rm L}^{\bullet},D)$ [cf. (4.2) and (4.3)], i.e.,

[TABLE]

Proof.

Given $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$, the curvature of the connection $A+\alpha$ is

[TABLE]

Hence $A+\alpha$ still has curvature in $\Omega^{2}_{8}({\mathfrak{g}}_{E})$, if, and only if, the Maurer-Cartan torsion $D\alpha+\frac{1}{2}[\alpha,\alpha]$ lies in ${\rm I}=\langle\Omega^{2}_{8}({\mathfrak{g}}_{E})\rangle$, and therefore it vanishes in the quotient. ∎

Notation 4.20**.**

We denote by $\mathcal{A}^{+}$ the elements in ${\rm L}^{1}$ satisfying the Maurer-Cartan condition.

We introduce the ${\rm W}^{2}_{m}$-Sobolev norms on smooth sections $\Gamma({\mathfrak{g}}_{E})$:

[TABLE]

where $\nabla^{j}s\in\Gamma({\mathfrak{g}}_{E}\otimes T^{0,j}(M))$, and $|\nabla^{j}s|$ is determined by the metric on $M$ and the fibrewise inner product (2.9) on ${\mathfrak{g}}_{E}$.

Let $\mathcal{A}_{m}$ denote the space of ${\rm W}_{m}^{2}$-connections on $E$, and define $\mathcal{G}_{m+1}$ as the topological group of ${\rm W}_{m+1}^{2}$-gauge transformations. For $m>3$, Sobolev multiplication holds [Donaldson1990, Appendix II], hence $\mathcal{G}_{m+1}$ has the structure of an infinite-dimensional Lie group modeled on a Hilbert space; its Lie algebra is the space of ${\rm W}_{m+1}^{2}$ sections of ${\mathrm{End}}({\mathfrak{g}}_{E})$. Moreover, the gauge group $\mathcal{G}_{m+1}$ acts smoothly on $\mathcal{A}_{m}$, so we denote by

[TABLE]

the Hausdorff orbit space with the quotient topology [Donaldson1990, Lemma 4.2.4]. Let $\mathcal{M}_{m}\subset\mathcal{B}_{m}$ denote the set of moduli of solutions to the selfdual ($\lambda=1$) contact instanton equation (1.1) and, accordingly, denote by $\mathcal{M}^{\ast}_{m}\subset\mathcal{M}_{m}$ the stratum of irreducible elements.

We recall that a contact instanton $A\in\mathcal{A}$ defines a complex $({\rm L}^{\bullet},D)$ [cf. (4.3)]. The ${\rm L}^{k}$ spaces consist of smooth sections of vector bundles on $M$ (Proposition 4.5), and we denote their ${\rm W}^{2}_{m}$-completions by ${\rm L}^{\bullet}_{m}$. From the formal adjoint of $D$,

[TABLE]

we define the Laplacian $\Delta\colon{\rm L}^{k}_{m}\to{\rm L}^{k}_{m-2}$ as in (4.16). Denote by $G$ its Green operator, i.e., the inverse of $\Delta$ on the orthogonal complement $\ker(\Delta)^{\perp}$, the orthogonal projection onto harmonic sections is denote by

[TABLE]

finally set

[TABLE]

Definition 4.21**.**

The Kuranishi map is defined by

[TABLE]

Lemma 4.22**.**

The Kuranishi map in Definition 4.21 is invertible near $0\in{\rm L}^{1}_{m}$ (4.2).

Proof.

For $m>3$, $\mathscr{F}$ is a smooth map from the Hilbert space ${\rm L}^{1}$ to itself. The linearisation of $\mathscr{F}$ at $0\in{\rm L}^{1}_{m}$ is the identity:

[TABLE]

hence there exists a smooth inverse $\mathscr{F}^{-1}$ near $0\in{\rm L}^{1}_{m}$. Take $c>0$ sufficiently small, such that

[TABLE]

For $\beta\in U_{\varepsilon}$ and $\alpha:=\mathscr{F}^{-1}(\beta)$, i.e., $\beta=\alpha+\frac{1}{2}\delta[\alpha,\alpha]$,

[TABLE]

so, $\alpha$ is a smooth point, by elliptic regularity. ∎

We define the obstruction map of the deformation complex $({\rm L}^{\bullet},D)$ [cf. (4.3)] by:

[TABLE]

where $U_{\varepsilon}$ is defined in (4.30) and $\mathcal{H}^{2}=\operatorname{ker}(\Delta)\subset{\rm L}^{2}$.

Lemma 4.23**.**

For $m>3$, let $U_{\varepsilon}$ be a neighbourhood of $0\in{\rm L}^{1}_{m}$ as in (4.30), on which the inverse $\mathscr{F}^{-1}$ of the map (4.29) is defined. Then $\mathscr{F}^{-1}$ maps $\Psi^{-1}(\{0\})$ diffeomorphically to a neighbourhood $W$ of the set

[TABLE]

where $\mathcal{A}^{+}$ is defined in Notation 4.20.

Proof.

Given $\alpha\in{\rm L}^{k}_{m}$,

[TABLE]

i.e.,

[TABLE]

Now, we show that $\mathscr{F}^{-1}(\Psi^{-1}(0))\subset S$. If $\beta\in\Psi^{-1}(0)$, set $\alpha=\mathscr{F}^{-1}(\beta)$, and apply $D^{\ast}$ to $\beta=\alpha+\frac{1}{2}\delta[\alpha,\alpha]$:

[TABLE]

On the other hand, applying $D$ to $\beta$ and using (4.32), we obtain

[TABLE]

so, if $\delta[D\alpha,\alpha]=0,$ then $\mathscr{F}^{-1}$ maps $\Psi^{-1}(0)$ into $S$. To see that $\delta[D\alpha,\alpha]=0,$ put $x=\delta[D\alpha,\alpha]$ and note that

[TABLE]

i.e., $x=\delta[x,\alpha]$. Now, for each $m>3$, there exists $K>0$ such that

[TABLE]

so we can take $c$ small enough that $\|\alpha\|_{m}<\frac{1}{K}$, hence $x=0$, as claimed.

Conversely, let $\alpha\in S$ and set $\beta=\mathscr{F}(\alpha)$, as above. Since $D^{\ast}\beta=D^{\ast}\alpha=0$,

[TABLE]

The left-hand side in the above equality is $D$-exact, whereas the right-hand side is harmonic, hence $D\beta=-{\bf H}[\alpha,\alpha]=\Psi(\beta)=0.$ Therefore $\beta$ is harmonic and lies in $\Psi^{-1}(0).$ ∎

In the context of Lemma 4.23, as shown in [Donaldson1990, §4.2.3], the tangent model is characterised as follows

Proposition 4.24**.**

Let $[A]\in\mathcal{M}_{m}$ be a SDCI gauge modulus, and denote its isotropy by

[TABLE]

where $Z(G)$ is the centre of $G$. Let $c>0$ be small enough so that the inverse $\mathscr{F}^{-1}$ is defined on $U_{\varepsilon}$ (4.30). Then $\Psi^{-1}(0)/\Gamma_{A}$ is a neighborhood of $[A]$ in $\mathcal{M}_{m}$. Furthermore, for $m>3$, the natural map $\mathcal{M}_{m+1}\to\mathcal{M}_{m}$ is a homeomorphism, so we may suppress the subscript $m$ in $\mathcal{M}_{m}$.

Remark 4.25*.*

The Maurer-Cartan condition $D\alpha+\frac{1}{2}[\alpha,\alpha]=0$ [cf. Lemma 4.19] is equivalent to

[TABLE]

Hence, by the slice Lemma 3.3, a chart about $[A]\in\mathcal{M}^{\ast}$ is given by $\mathscr{F}(S_{A,\varepsilon}^{+})$, where $S_{A,\varepsilon}^{+}:=S_{A,\varepsilon}\cap\mathcal{A}^{+}$.

Corollary 4.26**.**

If $[A]\in\mathcal{M}^{\ast}$ is an irreducible SDCI such that the obstruction map (4.31) vanishes, then $\mathcal{M}^{\ast}$ is a smooth manifold near $[A]$.

4.3.2 Cohomological vanishing of obstruction

We now obtain sufficient cohomological conditions for the vanishing of the obstruction map (4.31), and thus the local smoothness of $\mathcal{M}^{\ast}$ (1.8). The transverse Laplacian $\Delta_{T}$ from (4.17) satisfies, by ellipticity,

[TABLE]

Denote by ${\bf H}_{T}\colon{\rm L}^{k}\to\ker(\Delta_{T})$ the projection onto $\Delta_{T}$-harmonic sections, by $G_{T}$ the Green operator of $\Delta_{T}$, and set $\delta_{T}:=D_{T}^{\ast}G_{T}$. By the same argument applied to (4.29) in Lemma 4.22, the map

[TABLE]

is an isomorphism near $0\in{\rm L}^{1}_{m}$, for $m>3$. Taking $c>0$ small enough, we set

[TABLE]

such that $\mathscr{F}^{-1}_{T}$ is defined on $U_{\varepsilon}$.

Proposition 4.27**.**

At an irreducible SDCI such that $H_{B}^{2}=0$, the obstruction $\Psi$ in (4.31) vanishes.

Proof.

By (1.11) and Corollary 4.18, infinitesimal contact instanton deformations are parametrised by $H^{1}=H^{1}_{B}$. Fix $c>0$ small enough that $\mathscr{F}^{-1}_{T}$ is defined on $U_{\varepsilon}$, as in (4.33). Since $D_{T}=0$ on ${\rm L}^{2}_{m-1},$ $\Delta_{T}$ simplifies to $\Delta_{T}=D_{T}D_{T}^{\ast}-D_{V}^{2}$, and, using the fact that $D_{V}$ commutes with the Green operator, we obtain

[TABLE]

Furthermore, the assumption and Lemma 4.11 together imply that $0=H_{B}^{2}\cong\mathcal{H}^{2}_{T}$, so ${\bf H}_{T}$ vanishes on ${\rm L}^{2}_{m-1}$, and the above equation simplifies to

[TABLE]

For $\gamma\in U_{\varepsilon}$, let $\alpha:=\mathscr{F}_{T}^{-1}(\gamma)$, so $\gamma=\alpha+\frac{1}{2}\delta_{T}[\alpha,\alpha]$. Since $\gamma$ is $\Delta_{T}$-harmonic, it follows from (4.18) that

[TABLE]

Now, since $D_{V}$ commutes with $D_{T}$ (4.11) and also with $D_{T}^{\ast}$ and $G_{T}$, a similar argument to Lemma 4.23) shows that $\delta_{T}[D_{V}\alpha,\alpha]=0$, and so

[TABLE]

Thus $D_{T}\alpha=D\alpha$, and we conclude from (4.34) that

[TABLE]

i.e., $\alpha$ is a Maurer-Cartan element [cf. Lemma 4.19]. Furthermore, since $D_{T}^{2}=-\omega\wedge D_{V}$ [cf. (4.12)] and $D_{V}^{\ast}=-D_{V}$ [cf. (4.13)], we have $(D_{T}^{\ast})^{2}=D_{V}\Lambda(\cdot)$, where $\Lambda$ is the formal adjoint of $L_{\omega}$ (4.5). This implies

[TABLE]

In summary, $\alpha=\mathscr{F}^{-1}_{T}(\gamma)\in S\cap\mathcal{A}^{+}=\{\alpha\in{\rm L}^{1}_{m}|D^{\ast}\alpha=0,\;D\alpha+\frac{1}{2}[\alpha,\alpha]=0\}$, and it follows that $\mathscr{F}_{T}^{-1}$ defines a map from $U_{\varepsilon}$ into $S$. In Lemma 4.23 it was shown that $\mathscr{F}$ maps a neighborhood of $0\in S$ to a neighborhood of $\Psi^{-1}(0)$, hence, for $c>0$ small enough,

[TABLE]

since the linearisation of $\mathscr{F}\circ\mathscr{F}^{-1}$ at [math] is the identity. Therefore $\Psi=0$ on $U_{\varepsilon}$, for sufficiently small $c$ . ∎

If $A$ is a contact instanton, the transverse index of $A$ is defined as the index of the basic complex in (1.12), namely,

[TABLE]

When $A$ is irreducible we have $H^{0}_{B}=\{0\}$. If moreover $H^{2}_{B}=\{0\}$, it follows from Proposition 4.27 and Corollary 4.26 that $\mathcal{M}^{\ast}$ is locally a smooth manifold of dimension computed by the transverse index (4.35):

[TABLE]

5 Geometric structures on the contact instanton moduli space $\mathcal{M}^{\ast}$

Arguably the most attractive prospect in studying the moduli space of contact instantons is the future construction of topological invariants for Sasakian $7$-manifolds, so we are particularly interested in geometric structures on $\mathcal{M}^{\ast}$. By means of comparison, we know from [Baraglia2016, § 4.3] that, under suitable assumptions, the moduli space of ASD contact instantons on a Sasakian $5$-manifold $(M^{5},\eta,\xi,g,\Phi)$ is Kähler, and moreover hyper-Kähler in the transverse Calabi-Yau case. In the same spirit, we will show that $\mathcal{M}^{\ast}$ carries a natural Kähler strucutre.

5.1 Riemannian metric on $\mathcal{M}^{\ast}$

We define a Riemannian metric on the (smooth stratum of the) moduli space of SDCI $\mathcal{M}^{\ast}$ following a standard approach, e.g. [itoh1988geometry, Baraglia2016]. From the slice Lemma 3.3, we know that neighborhoods of a smooth point $[A]\in\mathcal{M}^{\ast}\subset\mathcal{B}^{\ast}$ have the form $\pi(S_{A,\varepsilon})\subset\mathcal{B}^{\ast}$, where $\pi\colon\mathcal{A}\to\mathcal{B}$ is the quotient map and

[TABLE]

In view of Proposition 4.5, ${\rm L}^{1}\cong\Omega^{1}({\mathfrak{g}}_{E})$ can be endowed with the inner product (2.9):

[TABLE]

Since $(\cdot,\cdot)$ is $\mathcal{G}$-invariant, it induces naturally an inner product on the quotient space $\mathcal{B}$, described explicitly on each neighbourhood $S_{A,\varepsilon}$ as follows. By elementary Hodge theory, each tangent space has a unique orthogonal decomposition:

[TABLE]

For a basic $1$-form $\beta\in\Omega^{1}_{B}({\mathfrak{g}})$, let us denote, accordingly, $\beta=\beta^{v}+\beta^{h},$ with $\beta^{v}\in\mathop{\mathrm{im}}(D_{A})$ and $\beta^{h}\in\operatorname{ker}(D^{\ast}_{A})$ (Figure 1). We may now define a positive semi-definite inner product on $\Omega^{1}({\mathfrak{g}}_{E})$ which, a priori, depends on the choice of slice neighborhood:

[TABLE]

Fix a slice $S_{A,\varepsilon}$. On the tangent space $T_{A+\alpha}S_{A,\varepsilon}$ at $A+\alpha\in S_{A,\varepsilon}$, an inner product is defined by the restriction of (5.1):

[TABLE]

Although the choice of the slice $S_{A,\varepsilon}$ is not unique, we nonetheless have the following:

Proposition 5.1**.**

[itoh1988geometry, Proposition 3.1]* The restriction to a slice neighborhood of the inner product (5.1) defines a positive definite inner product on $\mathcal{B}^{\ast}$, which is independent of the choice of slice neighborhoods.*

Sketch of Proof.

For clarity, we outline the main ideas of the proof, referring the reader to the original reference for the full argument. Given another slice $S_{A^{\prime},\varepsilon^{\prime}}$ such that $\emptyset\neq\pi(S_{A,\varepsilon})\cap\pi(S_{A^{\prime},\varepsilon^{\prime}})\subset\mathcal{B}$, then there is a transition smooth map

[TABLE]

satisfying $g_{\alpha}(A+\alpha)=A^{\prime}+\alpha^{\prime}\in S_{A^{\prime},\varepsilon^{\prime}}.$ This induces a diffeomorphism between slice neighborhoods

[TABLE]

whose derivative is

[TABLE]

where $\gamma:=\left.\frac{d}{dt}\right|_{t=0}\gamma_{t}\in\Omega^{0}({\mathfrak{g}}_{E})$ for some curve $\gamma_{t}\in\Omega^{0}({\mathfrak{g}}_{E})$ with $\gamma_{0}=0$. Hence $\Psi_{\ast}(\beta)^{h}=g_{\alpha}(\beta)$. Since $(\cdot,\cdot)$ in (5.1) is gauge-invariant, the induced inner product (5.2) does not depend on the choice of the slice neighborhood. ∎

Proceeding as in Lemma 4.22, the linearisation of the Kuranishi map is given by:

[TABLE]

where $\delta:=D^{\ast}_{A}G$ was defined in (4.28). Hence, the tangent space to a slice neighborhood at $A+\alpha$ is characterised as

[TABLE]

where $S_{A,\varepsilon}^{+}:=S_{A,\varepsilon}\cap\mathcal{A}^{+}$ [cf. Notation 4.20] denotes the elements in ${\rm L}^{1}$ satisfying the Maurer-Cartan condition. Taking $\alpha=0$ in (5.4), we see that $T_{A}(S_{A,\varepsilon}^{+})=\mathcal{H}^{1}$.

Proposition 5.2**.**

The differential $\mathscr{F}_{\ast}$ of the Kuranishi Map [cf. Definition 4.21] restricted to $T_{A}(S_{A,\varepsilon}^{+})$ is given by

[TABLE]

where we identify $A+\alpha$ with $\alpha$ and $U_{\varepsilon}$ is the neighbourhood on which the inverse $\mathscr{F}^{-1}$ is defined (4.30) and ${\bf H}_{A}\beta$ is the harmonic part of $\beta$ (4.27).

Proof.

Given $\beta\in T_{A}(S_{A,\varepsilon}^{+})$, from (5.4) we have $D_{A}\beta=-[\alpha,\beta]$. On the other hand, $D^{\ast}_{A}$ commutes with the green operator $G_{A}$, so the restriction of (5.3) to $T_{A}(S_{A,\varepsilon}^{+})$ takes the following form:

[TABLE]

Following the argument of [itoh1988geometry, Proposition 3.3], we split $\beta\in T_{\alpha}(S_{A,\varepsilon}^{+})$ like in (5.2) to see that $\beta^{h}\in\mathcal{H}_{A+\alpha}^{1}$, then the inner product $(\cdot,\cdot)$ on $T_{\alpha}(S_{A,\varepsilon}^{+})$ is written as

[TABLE]

Thus, in view of (5.5), we can compute the inner product (5.2) of elements $\alpha$ and $\beta$ in the deformation space $H^{1}({\rm L})$ by taking their harmonic representatives in $\mathcal{H}^{1}_{T}$.

5.2 Complex structure on $\mathcal{M}^{\ast}$

5.2.1 A natural almost-complex structure

As a straightforward application of the transverse Kähler identities [Lemma 2.2], we can show the following result, which will lead to a well-defined complex structure on $\mathcal{M}^{\ast}$ [cf. Definition 5.4 below]:

Lemma 5.3**.**

Let $H^{1}_{B}({\mathfrak{g}}_{E})$ be the first cohomology group of the basic complex (1.12), and denote by

[TABLE]

the Laplacian associated to $\bar{\partial}_{A}$ acting on basic forms, and by $\mathcal{H}^{p,q}_{\bar{\partial}_{A}}:=\operatorname{ker}\Delta_{\bar{\partial}_{A}}\subset\Omega^{p,q}_{B}(M,{\mathfrak{g}}_{E})$ its space of basic $\bar{\partial}_{A}$-harmonic sections. Then there exists an isomorphism:

[TABLE]

Proof.

We decompose a basic $1$-form as follows:

[TABLE]

Recall from Lemma 4.11 that there is an isomorphism $H^{1}_{B}({\mathfrak{g}}_{E})\cong\mathcal{H}^{1,0}_{T}$, and that, acting on basic forms, $D_{T}=D|_{\Omega^{k}_{B}({\mathfrak{g}}_{E})}=D_{B}$. Therefore, a basic $1$-form $\alpha$ is $D_{T}$-harmonic if, and only if, $D^{\ast}\alpha=0$ and $D\alpha=0$, firstly from

[TABLE]

we obtain

[TABLE]

and

[TABLE]

Secondly, applying the transverse Kähler identities [cf. Lemma 2.2] to $\bar{\partial}_{A}^{\ast}\gamma^{0,1}$ and $\partial_{A}^{\ast}\overline{\beta^{0,1}}$, respectively we obtain:

[TABLE]

hence, using the above equalities, the condition $0=D^{\ast}\alpha$ becomes

[TABLE]

by comparing the last equality with (5.7) we concluded that $\Lambda\partial_{A}\gamma^{0,1}=0$ and $\Lambda\bar{\partial}_{A}\overline{\beta^{0,1}}=0$, and therefore $\bar{\partial}_{A}^{\ast}\gamma^{0,1}=0$ and $\bar{\partial}_{A}^{\ast}\beta^{0,1}=0$. In summary, from (5.6) and the last assertion

[TABLE]

i.e., $\alpha=\gamma^{0,1}+\overline{\beta^{0,1}}\in\mathcal{H}^{0,1}_{\bar{\partial}_{A}}\oplus\overline{\mathcal{H}^{0,1}_{\bar{\partial}_{A}}}.$ ∎

We seek to define a complex structure on the tangent space $T_{[A]}\mathcal{M}^{\ast}$. We recall its identification $T_{[A]}\mathcal{M}^{\ast}\cong H^{1}({\rm L})$ with the first cohomology group of the complex (4.3) [See Theorem 1.2 item $(ii)$] which, in turn, is isomorphic to the first basic cohomology group $H_{B}^{1}$ [see Corollary 4.18], and, finally, that $H_{B}^{1}\cong\mathcal{H}^{1,0}_{T}$ [see Lemma 4.11]:

[TABLE]

Definition 5.4**.**

We define an almost complex structure $\mathcal{J}$ on $T_{[A]}\mathcal{M}^{\ast}\cong H_{B}^{1}$ to be that one induced by the complex structure $J:=\Phi|_{\rm H}$, where ${\rm H}$ is the horizontal bundle induced by the Sasakian structure of $M$ [cf. (A.1)].

To see that the almost complex structure $\mathcal{J}$ in Definition 5.4 is well defined, it suffices to check that the space of $D_{T}$-harmonic $1$–forms is closed under the action of $J$, i.e., $J(\mathcal{H}^{1,0}_{T})\subset\mathcal{H}^{1,0}_{T}$, in view of the identification $H^{1}_{B}\cong\mathcal{H}^{1,0}_{T}$. Indeed, from the isomorphism $H^{1}_{B}({\mathfrak{g}}_{E})\cong\mathcal{H}^{0,1}_{\bar{\partial}_{A}}\oplus\overline{\mathcal{H}^{0,1}_{\bar{\partial}_{A}}}$ [cf. Lemma 5.3], a basic $1$–form $\alpha\in\Omega^{1}({\mathfrak{g}}_{E})$ is $D_{T}$–harmonic if, and only if, $\alpha=\gamma+\bar{\beta}$, for some $\bar{\partial}_{A}$–harmonic forms $\alpha,\beta\in\Omega^{0,1}_{B}({\mathfrak{g}}_{E})$, therefore $J\alpha\in\mathcal{H}^{0,1}_{\bar{\partial}_{A}}\oplus\overline{\mathcal{H}^{0,1}_{\bar{\partial}_{A}}}$.

5.2.2 Integrability of $\mathcal{J}$

For each $A\in\mathcal{A}$, we defined operators $D,D_{T}$ and $D_{V}$, cf. (4.10). Analogously, to a deformation $A+\alpha\in S_{A,\varepsilon}^{+}$ there correspond operators $D_{A+\alpha},D_{T,A+\alpha}$ and $D_{V,A+\alpha}$, which we abbreviate respectively by

[TABLE]

These differentials give rise to cohomology groups, which we denote for clarity by

[TABLE]

Furthermore, for $\alpha\in U_{\varepsilon}$ as in (4.30), we denote by $\alpha^{+}:=\mathscr{F}^{-1}(\alpha)\in S_{A,\varepsilon}^{+}$ its image under the inverse Kuranishi map. Since $\mathcal{H}^{1}_{T}$ embeds into the affine space ${\rm L}^{1}\cong\Omega^{1}({\mathfrak{g}})$ [Proposition 4.5], each tangent vector $\lambda$ at $A$ induces a vector field $\overline{\lambda}$ canonically extended over $U_{\varepsilon}$. Since $\mathscr{F}$ is a diffeomorphism from $S_{A,\varepsilon}^{+}$ onto $U_{\varepsilon}$, $\overline{\lambda}$ provides a locally defined vector field over $S_{A,\varepsilon}^{+}$ (see Figure 2). We denote the push-forward of $\overline{\lambda}$ by

[TABLE]

The Riemannian metric in (5.5) is defined in terms of the projection onto the harmonic part, so we need formulas for the harmonic representatives of tangent vectors to the slice $S_{A,\varepsilon}^{+}$. The next results are an adaptation of [Baraglia2016, Proposition 4.7] to Sasakian $7$-manifolds, but we include full proofs for the sake of completeness.

Lemma 5.5**.**

Let $\lambda\in\mathcal{H}^{1}_{T}$ be a tangent vector at $\alpha\in U_{\varepsilon}\subset H^{1}_{B}({\mathfrak{g}}_{E})$, representing a vector field $\overline{\lambda}$ canonically extended to $U_{\varepsilon}$. Then, the harmonic representative of $\lambda^{+}:=\mathscr{F}^{-1}_{\ast\alpha}\left(\overline{\lambda}\right)$ is given by

[TABLE]

for some $f(\alpha,\lambda)\in\Omega^{0}_{B}({\mathfrak{g}}_{E})$ which is a solution of

[TABLE]

Proof.

Let $\alpha\in U_{\varepsilon}\subset H^{1}_{B}({\mathfrak{g}}_{E})$. Since $H^{1}_{B}({\mathfrak{g}}_{E})\cong\mathcal{H}^{1,0}_{T}$ [cf. Lemma 4.11] we can assume $\alpha$ to be $D_{T}$–harmonic. By Definition 4.21 of the Kuranishi map, $\alpha^{+}=\mathscr{F}^{-1}(\alpha)\in W\subset S_{A,\varepsilon}$ satisfies:

[TABLE]

Since the derivative of the Kuranishi map at $\alpha=0$ is the identity [cf. Lemma 4.22], we can write

[TABLE]

where $o(|\alpha|^{2})$ has vanishing $2$-jet at $\alpha=0$. Take $\lambda\in\mathcal{H}^{1}_{T}$ a tangent vector at $A$, by the affine structure of the tangent space, that vector represent also a canonically extended vector field $\overline{\lambda}$ on $U_{\varepsilon}$. Let $c(t)$ be a integral curve of $\lambda$, then $c(0)=\alpha$ and $\left.\frac{d}{dt}\right|_{t=0}c(t)=\lambda$. So using (5.10) we can compute

[TABLE]

where $o(|\alpha|,\lambda)$ has vanishing $1$–jet at $\alpha=0$, for fixed $\lambda$. Note that, since $\mathscr{F}^{-1}$ is bijective from $S_{A,\varepsilon}$ onto $U_{\varepsilon}$, we get a vector field $\lambda^{+}:=(\mathscr{F}^{-1}_{\ast})_{\alpha}(\overline{\lambda})$ along $S_{A,\varepsilon}$. More precisely, $\lambda^{+}$ corresponds to a local vector field on $\mathcal{M}^{\ast}$ under the natural projection $\pi_{|_{S_{A,\varepsilon}}}\colon S_{A,\varepsilon}\to\pi(S_{A,\varepsilon})\subset\mathcal{B}$ [cf. Lemma 3.3], so $\lambda^{+}$ represents a class in $H^{1}_{B,\alpha^{+}}$. Note that $\lambda^{+}$ is not necessarily harmonic, so we denote by $\gamma:=\gamma(\alpha,\lambda)$ the harmonic representative in $[\lambda^{+}]\in H^{1}_{B,\alpha^{+}}$. Hence there exists $f(x,\lambda)\in\Omega^{0}_{B}({\mathfrak{g}}_{E})$ such that

[TABLE]

Note that $D^{\ast}_{T,\alpha^{+}}\gamma=0$, because $\gamma$ is $D_{T,\alpha^{+}}$–harmonic, so, applying $D^{\ast}_{T,\alpha^{+}}$ to the above equality, we see that $f$ indeed satisfies (5.9). ∎

Lemma 5.6**.**

Under the assumptions of Lemma 5.5, the following assertions hold:

1. (i)

The harmonic representative of $\lambda^{+}$ can be expressed, to first order, as:

[TABLE] 2. (ii)

Let $f\in\Omega^{0}_{B}({\mathfrak{g}}_{E})$ be a solution of (5.9), then $f$ is the unique solution of the elliptic equation

[TABLE]

Proof.

Item $(i)$ follows by direct computation, using (5.11):

[TABLE]

Now for $(ii)$, recall from Definition 4.10 that $\Delta_{T,\alpha^{+}}=D^{\ast}_{T,\alpha^{+}}\circ D_{T,\alpha^{+}}+D_{T,\alpha^{+}}\circ D^{\ast}_{T,\alpha^{+}}-D_{V,\alpha^{+}}^{2}$. Since $f$ is basic with respect to $A$, and $\alpha^{+}$ is itself basic with respect to $A$, then $A$ and $A+\alpha^{+}$ define the same spaces of basic forms, so $f$ is basic with respect to $\alpha^{+}$, i.e., $D_{V,\alpha^{+}}f=0$. As a result, (5.9) can be rewritten as the elliptic equation (5.13). Denote by $f(\alpha,\lambda)\in\Omega^{0}_{B}({\mathfrak{g}}_{E})$ a solution of (5.13), and observe that $f$ is unique for fixed $\alpha$ and $\lambda$, since any two such solutions differ by an element of $\mathcal{H}^{0}_{B,\alpha^{+}}$, which is zero because $A+\alpha^{+}$ is irreducible. By standard elliptic theory, $f(\alpha,\lambda)$ depends smoothly on $\alpha$. For $\alpha=0$, the derivative of $\mathscr{F}^{-1}$ is the identity, so $\lambda^{+}$ is harmonic and by uniqueness $f(0,\lambda)=0$. Clearly, $f$ is linear in $\lambda$. ∎

We’re now in position to show the integrability of $\mathcal{J}$ [cf. Definition 5.4], using Lemma 5.5 and the formula (5.12) for the harmonic representative of $\lambda^{+}$:

Proposition 5.7**.**

The almost complex structure $\mathcal{J}$ [cf. Definition 5.4] on $\mathcal{M}^{\ast}$ is integrable.

Proof.

We show that the assignment $\alpha\mapsto\alpha^{+}=\mathscr{F}^{-1}(\alpha)$ under the Kuranishi map [cf. Definition 4.21] is a system of normal coordinates for the metric (5.1) about $[A]$ [cf. [moroianu2007lectures, Theorem 11.6]]. Let $\lambda_{1},\lambda_{2}\in\mathcal{H}^{1}_{T}$ be tangent vectors and denote by $\bar{\lambda}_{i},i=1,2$ the canonically extended vector fields on $U_{\varepsilon}$. We determine the $1$–jet of the inner product (5.1) at $\alpha=0$, in terms of the expression for harmonic representatives in (5.12):

[TABLE]

In the above we have used the fact that each $\lambda_{i},i=1,2$ is $D_{T}$–harmonic, thus (4.18) implies $D_{T}\lambda_{i}=D_{T}\lambda_{i}^{\ast}=0$. Finally, for fixed $\lambda_{i},i=1,2$, the $1$–jet $o(|\alpha|)$ vanishes at $\alpha=0$, hence $\mathscr{F}^{-1}$ defines normal coordinates at $[A]$. ∎

5.3 Kähler structure on $\mathcal{M}^{\ast}$

Definition 5.8**.**

Let $\mathcal{J}$ be the complex structure on $\mathcal{M}^{\ast}$ from Definition 5.4; using the inner product (5.1), we define a bilinear form on $T\mathcal{M}^{\ast}$ by:

[TABLE]

From Lemma 5.9, we obtain immediately:

Lemma 5.9**.**

Let $\alpha$ and $\beta$ be the harmonic representatives of its classes in $H^{1}_{B}({\mathfrak{g}}_{E})$, then $\Omega$ is the skew-symmetric $2$-form associated to $\mathcal{J}$ via the inner product (2.9):

[TABLE]

Now, we show that $\Omega_{[A]}$ is closed using the

Proposition 5.10**.**

The skew-symmetric form $\Omega$ associated to $\mathcal{J}$ [cf. Definitions 5.4 and 5.8] is closed.

Proof.

In normal coordinates at $[A]\in\mathcal{M}^{\ast}$, given by the inverse of the Kuranishi map [cf. Proposition 5.7], we use (5.12) and (5.14):

[TABLE]

In the last equality we used the fact that $J\lambda_{i}$, $i=1,2$ is $D_{T}$–harmonic. At the point $[A]$ we have $\alpha=0$, so the $1$-jet $o(|\alpha|)$ is constant and $d\Omega=0$. ∎

It follows from Propositions 5.7 and 5.10 that $(\mathcal{J},\Omega)$ is a Kähler structure on $\mathcal{M}^{\ast}$, which concludes the proof of Theorem 1.3.

Remark 5.11*.*

If $M$ is a $5$-dimensional contact Calabi-Yau manifold, there exists $\{J_{i}\}_{i=1}^{3}$ a hypercomplex structure with associated Kähler forms $\{\omega_{i}\}_{i=1}^{3}$ on the distribution ${\rm H}$ (A.1), so Proposition 5.7 can be applied to each $J_{i},i=1,2,3$ to provide a hypercomplex structures $\{\mathcal{J}_{i}\}_{i=1}^{3}$ on $\mathcal{M}^{\ast}$. Analogously, Proposition 5.10 can be applied to each $\mathcal{J}_{i},i=1,2,3$ to obtain a Sp$(m)$–structure on $\mathcal{M}^{\ast}$ ($\dim(\mathcal{M}^{\ast})=4m$). As a result the moduli space $\mathcal{M}^{\ast}$ would be hyper-Kähler in this case [cf. [Baraglia2016]]. For $7$-dimensional Sasakian manifold $M$, even in the contact Calabi-Yau case, we can not hope the existence of a transverse hyper-Kähler structure on the distribution ${\rm H}$ (A.1), just like in the $5$–dimensional case, because ${\rm H}$ (A.1) should have congruent zero rank module $4$.

Afterword: upcoming developments

$3$–Sasakian instantons

Despite the difficulty to endow $\mathcal{M}^{\ast}$ with a hyper-Kähler structure in the general $7$–dimensional Sasakian case [cf. Remark 5.11], we present an approach for an interesting special case of Sasaki-Einstein manifolds. A Riemannian $(4n+3)$-manifold $(M,g)$ is said to be $3$–Sasakian if $n\geq 1$ and the natural cone $(C(M),\bar{g})$ is hyper-Kähler. Equivalently, Sasakian structures $\{(\xi_{j},\eta_{j},\Phi_{j})\}_{j=1}^{3}$ on $M$ which satisfy ‘quaternionic relations’ [Boyer2008, Proposition 1.2.2]:

[TABLE]

Conjecture 5.12**.**

On a $3$–Sasakian $7$-manifold, the ‘transverse quaternionic structure’ (5.15) induces a hyper-Kähler structure on $\mathcal{M}^{\ast}$.

Since a $3$–Sasakian manifold admits actually a $\mathbb{S}^{2}$-family of Sasakian structures, one should expect to obtain a family $\mathcal{M}^{\ast}(\tau),\tau\in\mathbb{S}^{2}$ of moduli spaces of irreducible contact instantons, with respect to the $3$-forms $\eta(\tau)\wedge d\eta(\tau)$, cf. (2.4). It would then be interesting to assess the relations between the moduli spaces in such a family; there is no a priori reason to expect them to be all isomorphic.

An orientation on the moduli space

Defining an orientation on a moduli space $\mathcal{M}^{\ast}$ of connections on a principal bundle $P\to M$ cut out by some elliptic condition is an essential step in defining enumerative invariants. The recent article [joyce2020orientations] provides a package of general techniques to obtain a canonical orientation for a large class of such gauge-theoretic problems. This choice occurs as a global section of the principal $\mathbb{Z}_{2}$–bundle $\pi^{\ast}\colon i^{\ast}(\mathcal{O}^{E_{\bullet}}_{P})\to\mathcal{M}^{\ast}$, pulled back from the orientation bundle $\pi\colon\mathcal{O}^{E_{\bullet}}_{P}\to\mathcal{B}_{P}$ under the inclusion $i\colon\mathcal{M}^{\ast}\to\mathcal{B}_{P}$. In view of that important progress, it should then be relatively straightforward to apply their techniques and prove [joyce2020orientations, Problem 1.3] for our moduli space of contact instantons.

Sufficient conditions for the obstruction vanishing $H^{2}_{B}=0$

In view of Proposition 4.27, one might apply Bochner-type methods to arrange the vanishing of $H^{2}_{B}$, see e.g. [itoh1983moduli]. Then it is not very hard to establish a result along the following lines, which we state here as an announcement, while postponing the proof to a more detailed upcoming work:

Conjecture 5.13**.**

Let ${\bf E}\to M$ be a Sasakian $G$-bundle [Definition A.5] over a compact Sasakian $7$-manifold with positive transverse scalar curvature. Then, at each irreducible selfdual contact instanton $A$ on ${\bf E}$, the second basic cohomology group $H_{B}^{2}$ vanishes.

In particular, since the transverse scalar curvature on a Sasaki-Einstein manifold is always positive [cf [boyer19983, Proposition 1.1.9]]:

Corollary 5.14**.**

The moduli space of irreducible SDCI over a Sasaki-Einstein manifold is smooth.

Computations of transverse index in particular examples

Determining the dimension of a moduli space of irreducible and unobstructed contact instantons amounts to computing the transverse index (4.35). In [Baraglia2016, § 5], this is performed by replacing the foliated complex [Baraglia2016, (3.3)]

[TABLE]

with a complex involving a suitable lifted ${\bf T}$-action on ${\mathfrak{g}}_{E}$ [Baraglia2016, Propositon 2.8]. Hence the dimension of $\mathcal{M}^{\ast}$ is computed from the index of the symbol complex

[TABLE]

associated to the transverse elliptic operator [Baraglia2016, (5.1)]

[TABLE]

The transverse index can then be computed in several interesting cases (see [Baraglia2016, §§ 5.2 & 5.3]). We expect the same approach to be applicable to the index of the symbol complex

[TABLE]

associated to the transverse elliptic operator $\mathbbm{D}_{A}:=d_{7}\oplus d^{\ast}_{A}$ defined in (3.10):

[TABLE]

Appendix A Sasakian vector bundles

We gather here some general results and definitions on ‘holomorphic’ vector bundles over Sasakian manifolds. This concept obviously does not make strict sense as in classical complex geometry, but it admits a straightforward adaptation in terms of the transverse complex structure. All results and definitions in this Appendix stem from the original insights in [Biswas2010]. We adopt the usual notation for a $(2n+1)$-dimensional Sasakian manifold $(M,\eta,\xi,\Phi,g)$. Standard references for Sasakian geometry are [Boyer2008, blair2010contact].

A.1 Differential forms on Sasakian manifolds

The contact structure induces an orthogonal decomposition of the tangent bundle,

[TABLE]

where $\ker(\eta)=:H\subset TM$ is the distribution of rank $2n$ transverse to the Reeb field $\xi$, and the restriction $J=\Phi|_{H}$ defines an almost complex structure on $H$. We write indistinctly $\xi\lrcorner\alpha$ or $i_{\xi}(\alpha)$ to denote the interior product by $\xi$. We denote the complexification of the tangent bundle by $TM_{\mathbb{C}}:=TM\otimes_{R}\mathbb{C}$ and also, respectively, $N_{\xi}^{\mathbb{C}}$ and $H_{\mathbb{C}}$.

Definition A.1**.**

A differential form $\alpha\in\Omega^{k}(M)$ is called transverse if $i_{\xi}\alpha=0.$  If moreover $i_{\xi}d\alpha=0$, then $\alpha$ is said to be basic (i.e., $S^{1}$-invariant).

Let $\Omega^{k}_{H}(M)=\Gamma(M,\Lambda^{k}H^{\ast})$ denote the space of transverse $k$-forms. Let $\alpha\in\Omega^{p}_{H}(M)$ a locally defined, transverse complex form and $x\in M$. Since $i_{\xi}\alpha_{x}=0$, the evaluation of $\alpha_{x}$ on $\Lambda^{p}((TM_{\mathbb{C}})_{x})$ is determined by the values of $\alpha_{x}$ on the subbundle $\Lambda^{p}((H_{\mathbb{C}})_{x})\subset\Lambda^{p}((TM_{\mathbb{C}})_{x})$. We denote the complexification of the transverse complex structure $\Phi$ by

[TABLE]

Since $(\Phi|_{H_{x}})^{2}=-\mathbbm{1}$, the eigenvalues of $\Phi_{x}^{\mathbb{C}}$ are $\pm{\bf i}:=\pm\sqrt{-1}$, and the complexified horizontal distribution splits accordingly:

[TABLE]

For $p,q\geq 0$ we set

[TABLE]

therefore

[TABLE]

A section $\alpha$ is said to be of type $(a,d-a)$. If the evaluation of $\alpha$ on $H_{x}^{i,d-i}$ is zero for all $i\neq a$. The decomposition (A.4) gives a decomposition into a direct sum of vector bundles

[TABLE]

where $\Omega^{pq}_{H_{\mathbb{C}}}(M)$ denotes $\Gamma(M,\Lambda^{p}(H_{\mathbb{C}})^{\ast}\otimes\Lambda^{q}(H_{\mathbb{C}})^{\ast})$. Taking into account the dimension of $N_{\xi}$ and $H$, it follows that

[TABLE]

hence, combining with the decomposition (A.4),

[TABLE]

Proposition A.2**.**

The transverse $(1,1)$-form $\omega:=d\eta|_{H}$ is the fundamental symplectic form on $H$.

A.2 Partial connections

Consider an integrable subbundle $S\subset TM_{\mathbb{C}}$, i.e. the sections of $S$ are closed under the Lie bracket. Of course we have in mind the particular case in which $S=N_{\xi}^{\mathbb{C}}\subset TM_{\mathbb{C}}$, but we state the first few definitions in general terms.

Definition A.3**.**

Consider a complex vector bundle $E\to M$, a partial connection on $E$ in the direction of $S$ is a smooth operator $D\colon\Gamma(E)\to\Gamma(\Lambda^{1}S^{\ast})\otimes\Gamma(E)$, satisfying the ‘Leibniz rule’

[TABLE]

where the projection $q_{S}\colon(TM_{\mathbb{C}})^{\ast}\to S^{\ast}$ is the dual of the inclusion $S\hookrightarrow TM_{\mathbb{C}}$.

Since the distribution $S$ is integrable, smooth sections of $\operatorname{ker}(q_{S})$ are closed under the exterior derivation [Biswas2010, Section 3.2], this induces an exterior derivative on the smooth sections of $S^{\ast}$:

[TABLE]

Consider a partial connection $D$ on $S$ and the operator $D_{1}\colon\Gamma(\Lambda^{1}S^{\ast})\otimes\Gamma(E)\to\Gamma(\Lambda^{2}S^{\ast})\otimes\Gamma(E)$ defined by

[TABLE]

Their composition

[TABLE]

defines a torsion $D_{1}\circ D:=K(D)\in\Gamma\left(\Lambda^{2}S^{\ast}\otimes{\mathrm{End}}(E)\right)$. This section is called the curvature of $D$, we will say that $D$ is a flat connection if $K(D)\equiv 0$. We denote the extended anti-holomorphic $(n+1)$–dimensional foliation

[TABLE]

where $H^{0,1}$ is defined in (A.3), it is shown in [Biswas2010, Lemma 3.2] that the distribution in (A.6) is integrable.

Definition A.4**.**

A (complex) Sasakian vector bundle on a Sasakian manifold $(M,g,\xi)$ is a pair $(E,D_{0})$. Where $D_{0}$ is a partial connection in the direction of subbundle $N_{\xi}\subset TM$ and $E\to M$ is a complex vector bundle.

Notice that $N_{\xi}^{\mathbb{C}}\subset\widetilde{H}^{0,1},$ therefore, we can consider partial connections along $\widetilde{H}^{0,1}$, which define by restriction a partial connection along $N_{\xi}^{\mathbb{C}}$. Furthermore, $N_{\xi}$ is a $1-$dimensional foliation on M, so any partial connection along $N_{\xi}$ is flat. When the context is clear, we denote by $\bar{\partial}$ a flat partial connection $D$ along $\widetilde{H}^{0,1}$ such that $\bar{\partial}|_{N_{\xi}}=D_{0}$, and we abbreviate the notation by $\textbf{E}:=(E,D_{0})$.

Definition A.5**.**

A holomorphic Sasakian vector bundle on a Sasakian manifold $(M,g,\xi)$ is a pair $\mathcal{E}:=({\bf E},\bar{\partial})$, where ${\bf E}=(E,D_{0})$ is a Sasakian vector bundle and $\bar{\partial}$ is a flat connection on $E$ along $\widetilde{H}^{0,1}$ (A.6).

A.3 Hermitian and holomorphic structures

We define a Hermitian structure on E as a smooth Hermitian structure $h$ on $E$ which is compatible with $D_{0}$:

[TABLE]

A unitary connection on $(\textbf{E},h)$ is a connection $A$ on $E$ such that $d_{A}$ preserves $h$ in the usual sense.

A connection $A\in\mathcal{A}(E)$ induces a partial connection along $\widetilde{H}^{0,1}$ (A.6) given by $D_{\widetilde{H}^{0,1}}:=d_{A}|_{\widetilde{H}^{0,1}}$. If it coincides with $\bar{\partial}$, then $A$ is called a integrable connection on $\mathcal{E}$. Let $\mathcal{A}(\mathcal{E})\subset\mathcal{A}(E)$ denote the subset of integrable connections on E. The class of connections mutually compatible with the holomorphic structure and the Hermitian metric is the natural analogue of the concept of Chern connection.

Proposition A.6**.**

Let $(\mathcal{E},\bar{\partial})$ be a holomorphic Sasakian bundle with Hermitian structure, then there exists a unique unitary and integrable Chern connection $A_{h}$ on $\mathcal{E}$ and $F_{A_{h}}\in\Omega^{1,1}$, moreover the expression

[TABLE]

defines closed Chern forms $c_{j}(\mathcal{E},h)\in\Omega^{j,j}(M).$