January 01, 1970
We study the moduli space of non-Hermitian Yang–Mills connections over a compact Kähler manifold. Using normalized harmonic metrics, we construct a natural Hermitian metric on the smooth locus and show that, near the Hermitian locus, the smooth locus carries an almost hypercomplex structure which is compatible with the associated Riemannian metric.
Non-Hermitian Yang–Mills (NHYM) connections were introduced by Kaledin and Verbitsky [[1]]. Let \(M\) be a compact Kähler manifold and \(E\to M\) be a complex vector bundle. A connection \(\nabla\) on \(E\) is called non-Hermitian Yang–Mills if its curvature \(\Theta\) satisfies \[\left\{ \begin{align} \Lambda \Theta &= \lambda \,\operatorname{Id}_E,\\ \Theta &\in \Omega^{1,1}(\operatorname{End}E), \end{align} \right.\] for some constant \(\lambda\). This definition is standard; see [[2], [3]]. However, usually \(\nabla\) is assumed to be compatible with some Hermitian metric on \(E\). This is why we use the term “non-Hermitian Yang–Mills” to denote Yang–Mills connections which are not necessarily Hermitian.
Kaledin and Verbitsky denote by \(\mathcal{M}^s\) the moduli space of \((0,1)\)-stable NHYM connections on \(E\), modulo the complex gauge group \(G=\textrm{Maps}(M, \textrm{Aut} E)\). Here \((0,1)\)-stability means that the holomorphic structure induced by the \((0,1)\)-part of the connection is stable. This holomorphic bundle admits a Hermitian-Yang-Mills (HYM) connection. It is useful because the induced geometric structures near the HYM locus are better behaved, which will be recalled below in our discussion.
The motivation for Kaledin and Verbitsky’s study of NHYM connections comes from the following observation.
When \(c_1(E)=c_2(E)=0\), HYM connections are flat [[4]], and the unitary flat connections are in one-to-one correspondence with those holomorphic structures on \(E\) which make it a polystable holomorphic bundle [[3], [4]].
For an arbitrary bundle \(E\), the analogous correspondence is the Uhlenbeck–Yau theorem which states that polystable holomorphic structures on \(E\) are exactly those which admit HYM connections. Thus it is natural to weaken the flatness condition and consider instead all HYM connections.
The moduli space of flat, but not necessarily unitary, bundles is a beautiful subject, well studied in the literature [[5], [6]]. This space has dimension twice that of the moduli space of unitary flat bundles and carries a natural holomorphic symplectic form. Also, the generic part of the moduli space of non-unitary flat bundles is equipped with a holomorphic Lagrangian fibration over the space of unitary flat connections.
This analogy motivates the passage from HYM connections to NHYM connections in arbitrary Chern classes. Namely, one keeps the HYM curvature equations but drops the Hermitian compatibility condition, just as one passes from unitary flat connections to flat connections which are not necessarily unitary. The resulting question is whether the corresponding NHYM moduli space admits analogues of the structures known in the flat case. Kaledin and Verbitsky showed that near the Hermitian locus, \(\mathcal{M}^s\) has dimension twice that of the moduli space of HYM connections and is naturally equipped with a holomorphic symplectic form. As in the case of flat bundles, the generic part of \(\mathcal{M}^s\) admits a holomorphic Lagrangian fibration over the space of HYM connections [[1]].
When the base manifold is hyperkähler, it provides a further layer of structure. In this case, the quaternionic structure induces an \(SU(2)\)-action on differential forms. Kaledin and Verbitsky single out autodual connections, namely connections whose curvature is \(SU(2)\)-invariant, and showed that autodual connections are NHYM [[1]]. Under the assumption that \(c_1(E)\) and \(c_2(E)\) are \(SU(2)\)-invariant, every NHYM connection sufficiently close to the Hermitian locus is autodual [[1]]. This led them to ask whether NHYM connections over hyperkähler manifolds are necessarily autodual, and more importantly to formulate their hyperkähler conjecture for the NHYM moduli space.
We now turn to the conjecture studied in this paper. Kaledin and Verbitsky constructed a natural closed holomorphic two-form \(\Omega\) on the smooth part of \(\mathcal{M}^s\). Since a hyperkähler manifold, after choosing one of its complex structures, carries a canonical holomorphic symplectic form [[7]], it is natural to ask whether the form \(\Omega\) arises from a hyperkähler metric.
Conjecture 1 ([[1], Conjecture 8.1]). There exists a hyperkähler metric on \(\mathcal{M}^s\) such that \(\Omega\) is the associated holomorphic symplectic form.
When \(\dim_{\mathbb{C}}M=1\), the type condition is automatic and the NHYM equation reduces to projective flatness. In the degree-zero case, this is the usual flatness equation, which is already well understood [[8]]. Therefore, in this paper we focus on the case \(\dim_{\mathbb{C}}M>1\). Note that the construction of the form \(\Omega\) is completely parallel to the construction of a holomorphic symplectic form on the Hitchin–Simpson moduli space \(\mathcal{M}^s_{\mathrm{DR}}\) of flat connections on \(E\) [[5], [6]]. The analogue of Conjecture 1.1 for \(\mathcal{M}^s_{\mathrm{DR}}\) is known.
Kaledin and Verbitsky state that there is also a formal reason for this expectation. After choosing a Hermitian metric on the underlying bundle, the space \(\mathcal{A}\) of all connections carries a natural infinite-dimensional hyperkähler structure, and the complex gauge group acts compatibly with this structure. The corresponding complex moment map is given by \(\mu_\mathbb{C}(\nabla)=\Lambda\nabla^2,\) so that it is exactly one of the NHYM equations. They conjecture that \(\mu_{\mathbb{C}}^{-1}(0)/G\) admits a hyperkähler structure, and the embedding \(\mathcal{M}^s\hookrightarrow \mu_\mathbb{C}^{-1}(0)/G\) gives a hyperkähler structure on \(\mathcal{M}^s\) by restriction.
The hyperkähler-reduction viewpoint also suggests an analogue of the Uhlenbeck–Yau theorem for NHYM bundles. This leads to the following conjecture.
Conjecture 2 ([[1], Conjecture 8.7]). Let \(( E,\nabla)\) be a bundle with NHYM connection \(\nabla\). Then there exists a harmonic metric \(h\) on \(E\) if and only if \(E\) is a direct sum of \(\nabla\)-stable bundles. Also, if \(E\) itself is \(\nabla\)-stable, then \(h\) is unique up to a constant factor.
Pan, Shen, and Zhang later proved this conjecture in [[9]]. Their result is the NHYM analogue of harmonic metrics on flat bundles [[10], [11]]. Their theorem provides the real moment map \(\mu_\mathbb{R}(\nabla)=\Lambda \Xi\) required for hyperkähler reduction. However, this picture should be regarded as a heuristic rather than as a completed conclusion. In the NHYM setting, the relevant complex gauge action does not preserve any fixed background metric, and the formal ambient hyperkähler structure does not immediately descend to \(\mathcal{M}^s\). We defer a detailed analysis of this approach to the final section.
Rather than giving a direct hyperkähler interpretation of the conjecture, we separate the two ingredients which would enter such a structure. Namely, we study the existence of a Hermitian metric on the smooth locus and, independently, the existence of an almost hypercomplex structure near the Hermitian locus. Our approach is modeled on Itoh’s construction [[12]]. The relevant geometric objects are first constructed on local slices of the moduli space, then we check their compatibility with the transition maps on overlaps, so that the local constructions glue to globally defined objects.
The main results are the following.
Theorem 1. Let \(M\) be a compact Kähler manifold and let \(E\to M\) be a complex vector bundle. Let \(\widehat{\mathcal{M}}^s\) denote the smooth locus of the moduli space of \((0,1)\)-stable NHYM connections. Then \(\widehat{\mathcal{M}}^s\) carries a natural Hermitian metric.
Theorem 2. There exists a neighborhood of the Hermitian locus in \(\widehat{\mathcal{M}}^s\) such that the neighborhood carries an almost hypercomplex structure \((\mathcal{I}_\alpha,\mathcal{J}_\alpha,\mathcal{K}_\alpha)\) compatible with the above Hermitian metric.
These results suggest that the expected hyperkähler structure, if it exists, cannot be obtained directly from the harmonic metrics together with the natural quaternionic almost complex structures.
The main difficulty in the NHYM case is that the relevant geometric structures cannot be obtained by directly descending fixed ambient structures. To obtain well-defined objects on the moduli space, one has to let the Hermitian metric and the almost complex structures vary with the base point and introduce correction terms ensuring compatibility with the transition maps between slices. These corrections are precisely what distinguishes the NHYM case from the classical settings. They make the constructions well defined, but at the same time produce the closedness obstruction of the fundamental form and integrability obstructions of \(\mathcal{J}_\alpha,\mathcal{K}_\alpha\).
This paper is organized as follows.
Section 2 reviews the description of the NHYM moduli space. In particular, we give the elliptic deformation complex, the slice expression, and the unobstructed smooth locus \(\widehat{\mathcal{M}}^s\), which will be the main setting for the constructions in the subsequent sections.
Section 3 recalls the existence and uniqueness of harmonic metrics for stable NHYM bundles. This result is one of the main inputs in our construction of a Hermitian metric on \(\widehat{\mathcal{M}}^s\). We also establish several elementary compatibility properties of the Hermitian inner product under rescaling of Hermitian metrics and under gauge transformations. These properties will be used later to prove that the locally defined metrics are compatible with the transition maps between slices.
Section 4 is devoted to the construction of the Hermitian metric on \(\widehat{\mathcal{M}}^s\). We assign to each point of a local slice a normalized harmonic metric and use it to define an \(L^2\)-inner product of tangent vectors. After proving the smooth dependence of this construction and its gauge-equivariance, we show that the resulting local metrics are compatible with the transition maps between slices. This yields a well-defined Hermitian metric on the smooth locus.
Section 5 follows a similar strategy for almost complex structures. We start from the standard quaternionic structures on the ambient space of connections, as in the usual hyperkähler picture. However, this ambient triple does not in general preserve the tangent spaces of \(\widehat{\mathcal{M}}^s\). It does so only along the Hermitian locus. We are therefore led to work in a neighborhood of the Hermitian locus. To extend the construction away from this locus, we introduce certain correction terms so that the resulting operators define an almost hypercomplex structure on the smooth locus under consideration.
In Section 6, we recall the hyperkähler reduction construction and discuss whether the standard hyperkähler reduction method can be applied to \(\mathcal{M}^s\) . Indeed, \(\mathcal{M}^s\) is only a subset of a formal hyperkähler space, and its tangent spaces are not, in general, preserved by the ambient complex structures. Therefore the hyperkähler structure of the ambient space cannot be inherited by \(\mathcal{M}^s\) through the usual reduction argument.
We begin by recalling some basic definitions and properties of the moduli space of NHYM connections [[1]].
Suppose \(E\) is a complex vector bundle over a compact Kähler manifold \(M\). Let \(\mathcal{A}\) denote the space of all connections on \(E\). \(\mathcal{A}_0\) is the subspace of NHYM connections in \(\mathcal{A}\), i.e., its curvature \(\Theta\) is of Hodge type (1,1) and satisfies \(\Lambda \Theta=\lambda \textrm{Id}\). The Lie group \(G=\textrm{Maps}(M, \textrm{Aut} E)\) acts on \(\mathcal{A}\) by \(u\cdot\nabla=u^{-1}\nabla u.\) Equivalently, if \(\nabla'=\nabla+\eta\) with \(\eta\in\Omega^1(\mathop{\mathrm{End}}E)\), then \(u\cdot\nabla' = u^{-1}\nabla u+u^{-1}\eta u.\) This action preserves the subset \(\mathcal{A}_0\subset\mathcal{A}\). The symbol \(u\) is used either for an element of the gauge group or for the induced gauge action, the intended meaning should be clear from the context. To simplify exposition, we will always consider only NHYM connections with the constant \(\lambda=0\).
Let \(\mathcal{A}^s \subset \mathcal{A}_0\) be the open subset of (0,1)-stable NHYM connections.
Definition 1 ([[1]]). Let \(\mathcal{M}^s=\mathcal{A}^s/G\) be the moduli space of NHYM connections on \(E\) endowed with the quotient topology.
Proposition 3 ([[1], Proposition 2.7]). The deformation complex \(\mathcal{C}_\nabla\) associated with the NHYM connection \(\nabla\), \[0 \to \Omega^0(\mathop{\mathrm{End}}E) \xrightarrow{\,d_1\,} \Omega^1(\mathop{\mathrm{End}}E)\xrightarrow{\,d_2\,} \Omega^{2,0}(\mathop{\mathrm{End}}E) \oplus \Omega^{0,2}(\mathop{\mathrm{End}}E) \oplus \Omega^0(\mathop{\mathrm{End}}E)\] is elliptic, where \(d_{1}\xi=\nabla\xi, \;d_{2}\beta = \bigl( (\nabla\beta)^{2,0}, (\nabla\beta)^{0,2}, \Lambda\nabla\beta \bigr)\). We write \(\nabla=\nabla'+\nabla^{''}\) for the decomposition of \(\nabla\) into its \((1,0)\)- and \((0,1)\)-components.
For later computations, we fix the following notational convention. Let \(\nabla\) be a fixed NHYM connection. Unless otherwise stated, the operators \(d_1\) and \(d_2\) always denote the operators in the deformation complex at \(\nabla\). For \(\alpha\in\Omega^1(\mathop{\mathrm{End}}E)\), we write \(d_{1,\alpha}\) and \(d_{2,\alpha}\) for the corresponding operators at \(\nabla+\alpha\). Throughout the paper, the subscript \(\alpha\) will indicate that the corresponding object is taken with respect to the base connection \(\nabla+\alpha\).
Lemma 1. For \(\xi\in\Omega^0(\mathop{\mathrm{End}}E), \beta\in\Omega^1(\mathop{\mathrm{End}}E)\), \[d_{1,\alpha}\xi = \nabla\xi+[\alpha,\xi],\] \[d_{2,\alpha}\beta = \left( \bigl(\nabla\beta+[\alpha,\beta]\bigr)^{2,0}, \bigl(\nabla\beta+[\alpha,\beta]\bigr)^{0,2}, \Lambda\bigl(\nabla\beta+[\alpha,\beta]\bigr) \right).\] Here \([\cdot,\cdot]\) denotes the graded commutator of \(\mathop{\mathrm{End}}E\)-valued forms.
Suppose \(h\) is a Hermitian structure on \(E\). The spaces \(\Omega^l(\mathop{\mathrm{End}}E)\) are endowed with the Sobolev \(W^{k,2}\)-norms, denoted by \(|\cdot |_k\), where \(k\) is chosen sufficiently large relative to \(\dim M\).
The following proposition is obtained by the same Kuranishi-type argument as in [[13]].
Proposition 4 ([[13], (7.3.9)]). The slice of \(\nabla\) in \(\Omega^1(\mathop{\mathrm{End}}E)\) is \[S^h_{\nabla,\varepsilon}=\{\alpha \in \Omega^1(\mathop{\mathrm{End}}E) : |\alpha|_k <\varepsilon, \nabla^{*,h} \alpha=0, (N_\alpha)^{2,0}=(N_\alpha)^{0,2}=\Lambda (N_\alpha)=0\},\] where \(N_\alpha=\nabla \alpha+\alpha \wedge \alpha\).
Note that every \((0,1)\)-stable \(\nabla\) is simple, and obviously, if we choose another Hermitian structure \(h'\), the following proposition still holds.
Proposition 5 ([[13], Theorem 7.3.17],[[12], Proposition 2.3]). The natural map \(p: S^h_{\nabla,\varepsilon} \to \mathcal{M}^s, \;\alpha \to [\nabla +\alpha]\) gives a homeomorphism of a neighborhood of \(0\) in \(S^h_{\nabla,\varepsilon}\) onto a neighborhood of \([\nabla]\) in \(\mathcal{M}^s\).
After replacing \(\varepsilon\) by a smaller positive number, we may assume that the above homeomorphism is defined on the whole slice \(S^h_{\nabla,\varepsilon}\).
Proposition 6 ([[1], Corollary 2.9]). \(\mathcal{M}^s\) is a complex-analytic space.
Let \(\mathop{\mathrm{End}}_0E\subset \mathop{\mathrm{End}}E\) denote the trace-free endomorphism bundle. \(H^i(\widetilde{\mathcal{C}}_\nabla)\) is the \(i\)-th cohomology group of the complex for the trace-free deformation complex \[0 \longrightarrow A^0(\mathop{\mathrm{End}}_0E) \xrightarrow{d_{1}} A^1(\mathop{\mathrm{End}}_0E) \xrightarrow{d_{2}} A^{2,0}(\mathop{\mathrm{End}}_0E)\oplus A^{0,2}(\mathop{\mathrm{End}}_0E)\oplus A^0(\mathop{\mathrm{End}}_0E) .\]
We denote the smooth locus by \(\widehat{\mathcal{M}}^s = \left\{ [\nabla]\in\mathcal{M}^s \;\middle|\; H^2(\widetilde{\mathcal{C}}_\nabla)=0 \right\} .\) \(\mathcal{M}^s\) is nonsingular at every point of \(\widehat{\mathcal{M}}^s\), and \(T_{[\nabla]}\mathcal{M}^s \cong H^1(\mathcal{C}_\nabla).\) This is the standard smoothness criterion for such elliptic deformation problems, compare Kobayashi’s corresponding statements [[13]].
In the following sections, we shall mainly work with the smooth locus \(\widehat{\mathcal{M}}^s\subset \mathcal{M}^s\).
In this section, we recall the existence of harmonic metrics for semisimple NHYM bundles and prove several properties needed later. This prepares the ground for the construction of a Hermitian metric on \(\mathcal{M}^s\) in the next section.
Let \(E\) be a complex vector bundle over a compact Kähler manifold \((M,\omega)\) and \(\nabla\) is a connection on \(E\). Given a Hermitian metric \(h\) on \(E\), there is a unique decomposition \(\nabla = \nabla_h + \psi^{\nabla}_h\), where \(\nabla_h\) is an \(h\)-unitary connection and \(\psi_h^{\nabla} \in \Omega^1(\mathop{\mathrm{End}}(E))\) is self-adjoint with respect to \(h\).
For later reference, we first introduce the notation and conventions used throughout the sequel. Readers already familiar with them may skip it.
Definition 2. Let \(h\) be a Hermitian metric on \(E\). We use the following conventions. Throughout, Hermitian pairings are taken to be linear in the first argument.
For \(s,t \in \Gamma(E)\), the fiberwise Hermitian pairing on \(E\) is denoted by \(h(s,t)\).
For \(A \in \Gamma(\mathop{\mathrm{End}}E)\), its fiberwise \(h\)-adjoint is denoted by \(A^{*,h}\) and is defined by \(h(As,t)=h(s,A^{*,h}t).\) The same notation is used for \(\mathop{\mathrm{End}}E\)-valued forms: \((\eta\otimes A)^{*,h} = \bar\eta\otimes A^{*,h}.\)
The fiberwise Hermitian pairing on \(\mathop{\mathrm{End}}E\) is denoted by \(\langle A,B\rangle_{\mathop{\mathrm{End}},h}=\operatorname{tr}(B^{*,h}A), A,B \in \Gamma(\mathop{\mathrm{End}}E)\).
For \(F\)-valued forms \(\beta_1,\beta_2\) of the same degree where \(F=E\) or \(F=\mathop{\mathrm{End}}E\), we write \(\langle \beta_1,\beta_2\rangle_{F,h}\) for the pointwise pairings induced by the fixed Kähler metric on \((M,\omega)\) and by the corresponding fiber metrics. The associated \(L^2\)-pairing is \((\beta_1,\beta_2)_{F,h} = \int_M \langle \beta_1,\beta_2\rangle_{F,h}\,\frac{\omega^n}{n!}.\)
For a connection \(\nabla\) on \(E\), define a connection \(\nabla^{\dagger_h}\) by \(d\bigl(h(s,t)\bigr) = h(\nabla s,t)+h(s,\nabla^{\dagger_h}t), \;s,t\in\Gamma(E).\)
For \(F=E\) or \(F=\mathop{\mathrm{End}}E\), we denote by \(\nabla^{*,h}\) the \(L^2\)-formal adjoint of the induced covariant derivative \(\nabla:\Omega^k(F)\to\Omega^{k+1}(F)\), characterized by \((\nabla\beta_1,\beta_2)_{F,h} = (\beta_1,\nabla^{*,h}\beta_2)_{F,h},\;\beta_1\in\Omega^k(F),\; \beta_2\in\Omega^{k+1}(F)\).
The formal adjoint of \(d_{1,\nabla}:\Omega^0(\mathop{\mathrm{End}}E)\longrightarrow \Omega^1(\mathop{\mathrm{End}}E)\) with respect to \((\;,\;)_{\mathop{\mathrm{End}},h}\) is denoted by \(d_{1,\nabla}^{*,h}:\Omega^1(\mathop{\mathrm{End}}E)\longrightarrow \Omega^0(\mathop{\mathrm{End}}E),\) and is characterized by \(\bigl(d_{1,\nabla}\beta_1,\beta_2\bigr)_{\mathop{\mathrm{End}},h} = \bigl(\beta_1,d_{1,\nabla}^{*,h}\beta_2\bigr)_{\mathop{\mathrm{End}},h}, \; \beta_1\in\Omega^0(\mathop{\mathrm{End}}E),\; \beta_2\in\Omega^1(\mathop{\mathrm{End}}E)\).
For \(u\in G\), the pullback Hermitian metric \(u^*h\) is defined by \((u^*h)(s,t)=h(u s,u t),\;s,t\in\Gamma(E)\).
When applied to endomorphisms or endomorphism-valued forms, the symbol \(^{*,h}\) denotes the fiberwise Hermitian adjoint. When applied to a differential operator, it denotes the formal \(L^2\)-adjoint with respect to the indicated metric. When no confusion is possible, the subscripts \(E,\mathop{\mathrm{End}}\) and \(h\) are suppressed.
With these conventions, we have \(\nabla_h=\frac{1}{2}(\nabla+\nabla^{\dagger_h})\), \(\psi^{\nabla}_h=\frac{1}{2}(\nabla-\nabla^{\dagger_h})\).
Definition 3 ([[9], (2.13)]). Let \((E,\nabla)\) be an NHYM bundle over a compact Kähler manifold \((M,\omega)\), and let \(h\) be a Hermitian metric on \(E\). We say that \(h\) is a harmonic metric if it satisfies either, and hence both, of the following equivalent definitions:
\(h\) satisfies \(\nabla_h^{*,h}\psi^{\nabla}_h=0.\)
Let \(\Xi_h\) be the pseudocurvature of \((E,\nabla,h)\), then \(\Lambda \Xi_h=0.\)
Theorem 3 ([[9], Theorem 1.2]). Let \((X,\omega)\) be a compact Kähler manifold, and let \((E,\nabla)\) be a NHYM bundle over \(X\). Then \((E,\nabla)\) admits a harmonic metric if and only if it is semisimple.
Remark 7 ([[9], Remark 2.4],[[1], Remark 8.4]). The definition of \(\nabla\)-simplicity used here agrees with Kaledin–Verbitsky’s notion of \(\nabla\)-stability. Consequently, \(\nabla\)-semisimplicity means that the NHYM bundle is a direct sum of \(\nabla\)-stable NHYM bundles, as in Conjecture 8.7 of [[1]]. Moreover, every \((0,1)\)-stable NHYM connection is \(\nabla\)-stable.
We restrict to (0,1)-stable NHYM connections \(\nabla\). In this case, the corresponding harmonic metric is unique up to a positive scalar multiple.
For each \(\nabla \in \widehat{\mathcal{M}}^s\), let \([h_\nabla]\) be the positive-scale class of the harmonic metric associated with \(\nabla\). Since \(c>0\) is a constant, multiplying \(h\) by \(c\) does not change the fiberwise adjoint of an endomorphism. Indeed, if \(A\in \mathop{\mathrm{End}}(E_x)\), then for any \(s,t\in E_x\), \[(ch)(As,t) = c\,h(As,t) = c\,h(s,A^{*{h}}t) = (ch)(s,A^{*,{h}}t).\] Hence \(A^{*,{ch}} = A^{*,{h}}\). The pointwise Hermitian inner product on \(\mathop{\mathrm{End}}E\) is \(\langle A,B\rangle_{ch} = \operatorname{tr}(B^{*,{ch}}A) = \operatorname{tr}(B^{*,h}A) = \langle A,B\rangle_h.\) Therefore the pointwise Hermitian pairing on \(\mathop{\mathrm{End}}(E)\), and hence the induced pairing on \(\Omega^1(\mathop{\mathrm{End}}E)\), is unchanged.
The following result will be used later in the paper.
Lemma 2. Suppose \(h\) is a Hermitian metric on \(E\), \(c>0\) is a constant, and \(u \in G\), then for every \(\xi \in \Omega^0(\mathop{\mathrm{End}}E)\), \(\eta,\eta_1,\eta_2 \in \Omega^1(\mathop{\mathrm{End}}E)\) and \(A \in \Gamma(\mathop{\mathrm{End}}E)\),
\((\eta_1,\eta_2)_{c h} = (\eta_1,\eta_2)_h,\; d_{1,\nabla}^{*,ch} = d_{1,\nabla}^{*,h}.\)
\((u^{-1} A u)^{*,u^*h}=u^{-1} A^{*,h} u\)
\((u\cdot \eta_1,\, u\cdot \eta_2)_{u^*h} = (\eta_1,\eta_2)_h, \; d_{1,u\cdot \nabla}^{*,u^*h} (u \cdot \eta) = u\cdot ( d_{1,\nabla}^{*,h}\eta).\)
As a consequence, the gauge action preserves the space of harmonic forms: \(u\cdot \mathcal{H}_\nabla^1=\mathcal{H}_{u\cdot \nabla}^1 .\)
Proof. (i) is clear since \((\xi,d_{1,\nabla}^{*,ch}\eta)_{ch}=(d_{1,\nabla}\xi,\eta)_{ch} = (d_{1,\nabla}\xi,\eta)_h = (\xi,d_{1,\nabla}^{*,h}\eta)_h,\) and (ii) follows by \[\begin{align} \bigl((u^{-1}Au)s,t\bigr)_{u^*h} &= \bigl(u(u^{-1}Au)s,ut\bigr)_h \\ &= (Aus,ut)_h \\ &= (us,A^{*,h}ut)_h \\ &= \bigl(us,u(u^{-1}A^{*,h}u)t\bigr)_h \\ &= \bigl(s,(u^{-1}A^{*,h}u)t\bigr)_{u^*h}. \end{align}\] We now prove the first assertion in (iii). \[\begin{align} \langle u\cdot A,u\cdot B\rangle_{u^*h} &= \operatorname{tr}\left((u^{-1}Bu)^{*,{u^*h}}(u^{-1}Au)\right)\\ &= \operatorname{tr}\left(u^{-1}B^{*,h}u\,u^{-1}Au\right)\\ &= \operatorname{tr}\left(u^{-1}B^{*,h}Au\right)\\ &= \operatorname{tr}(B^{*,h}A)\\ &= \langle A,B\rangle_h. \end{align}\] This immediately implies that \((u\cdot \eta_1,u\cdot \eta_2)_{u^*h} = (\eta_1,\eta_2)_h.\)
It remains to prove the second assertion in (iii). By the definition of the formal adjoint, \[\begin{align} (u\cdot\xi,\;d_{1,u\cdot \nabla}^{*,u^*h}(u\cdot\eta))_{u^*h} &= (d_{1,u\cdot \nabla}(u\cdot\xi),\;u\cdot\eta)_{u^*h}\\ &= (u\cdot d_{1,\nabla}\xi,\;u\cdot\eta)_{u^*h}\\ &= (d_{1,\nabla}\xi,\eta)_h\\ &= (\xi,d_{1,\nabla}^{*,h}\eta)_h\\ &= (u\cdot\xi,\;u\cdot d_{1,\nabla}^{*,h}\eta)_{u^*h}. \end{align}\] ◻
In this section, we construct a Hermitian metric on \(\widehat{\mathcal{M}}^s\). The construction is carried out locally on gauge slices and consists of the following steps:
Define a Hermitian pairing on each local slice by using the harmonic metric associated with the corresponding \((0,1)\)-stable NHYM connection.
Show that the local Hermitian pairing is well-defined and varies smoothly on each slice.
Prove that these local Hermitian pairings are compatible with the transition maps between slices.
Consequently, the local pairings descend to a Hermitian metric on \(\widehat{\mathcal{M}}^s\).
The construction is analogous to Itoh’s construction in [[12]], except that in the present setting the Hermitian metric is chosen pointwise on the moduli space rather than fixed once and for all, since the complex gauge action does not preserve a fixed Hermitian metric, a single background metric cannot be used in the construction. Instead, we choose the Hermitian metric pointwise, in such a way that it transforms equivariantly under gauge transformations up to a positive scalar. Since such a scalar rescaling does not change the induced pairing on \(\mathop{\mathrm{End}}E\)-valued forms, the resulting slice metric is invariant under gauge-induced transition maps, see Lemma 3(i).
We begin by recalling the slice of the NHYM moduli space. The following definitions are independent of the choice of representative element in the harmonic metric class \([h_\nabla]\). \[S^{h_\nabla}_{\nabla,\varepsilon}=\{\alpha \in \Omega^1 : |\alpha|_k <\varepsilon, \nabla^{*,h_\nabla} \alpha=0, (N_\alpha)^{2,0}=(N_\alpha)^{0,2}=\Lambda (N_\alpha)=0\},\] \[T_\alpha S^{h_\nabla}_{\nabla,\varepsilon}=\{\beta \in \Omega^1 : \nabla^{*,h_\nabla} \beta=0, d_{2,\alpha}(\beta)=0\}.\]
In particular, \[T_\nabla S^{h_\nabla}_{\nabla,\varepsilon}=\{\beta \in \Omega^1 : \nabla^{*,h_\nabla} \beta=0, d_{2,\nabla}(\beta)=0\}=\ker d^*_{1,\nabla}\cap \ker d_{2,\nabla}\cong \mathcal{H}^1_{\nabla}.\]
Lemma 3. Fix a reference Hermitian metric \(h\) on \(E\). For every \((0,1)\)-stable NHYM connection \(\nabla\), let \([h_\nabla]\) denote the class of harmonic metrics associated with \(\nabla\). Then there exists a unique representative \(h_\nabla^{\mathrm{n}}\in [h_\nabla]\) satisfying the normalization condition \(\int_M \log\det h^{-1}h_\nabla^{\mathrm{n}}\frac{\omega^n}{n!} =0.\) Thus the assignment \[\nabla\longmapsto h_\nabla^{\mathrm{n}}\] is an assignment to actual Hermitian metrics on \(E\). Moreover, it satisfies the following properties:
\(h_{u\cdot \nabla}^{\mathrm{n}}= c(u,\nabla) u^*h_\nabla^{\mathrm{n}}\) for a uniquely determined positive constant \(c(u,\nabla)>0\).
After possibly shrinking \(\varepsilon_0\), the assignment \(\nabla\longmapsto h_\nabla^{\mathrm{n}}\) is smooth on \(S_{\nabla_0,\varepsilon_0}\).
Proof. We first prove (i). In this step only, \(k\) denotes a Hermitian metric. Let \(k=h_\nabla^{\mathrm{n}},\;k'=u^* k\), then by definition, \[d (k'(s,t))=d k(u s,u t)=k(\nabla_k(u s),u t)+k(u s,\nabla_k(u t)).\]
Since \(u ((u \cdot \nabla_k)s)=u ((u^{-1}\circ \nabla_k \circ u)s)=\nabla_k (u s)\), we have \[d (k'(s,t))=k(u ((u \cdot \nabla_k)s),u t)+k(u s,u ((u \cdot \nabla_k)t))=k'((u \cdot \nabla_k)s,t)+k'(s,(u \cdot \nabla_k)t).\]
Similarly, \(u\cdot\psi_k\) is \(k'\)-self-adjoint, since \[k'((u\cdot\psi_k)s,t) = k(\psi_k(us),u t) = k(us,\psi_k(ut)) = k'(s,(u\cdot\psi_k)t).\] If we write \(u \cdot \nabla=(u \cdot \nabla)_{k'}+(u \cdot \psi)_{k'}\), then \((u \cdot \nabla)_{k'}=u \cdot \nabla_{k}\) and \((u \cdot \psi)=u \cdot \psi_{k}\).
Consequently, \[(u\cdot \nabla_k)^{*,k'}(u\cdot \psi_k) = u\cdot (\nabla_k^{*,k}\psi_k).\]
Thus, if \(k\) is harmonic for \(\nabla\), then \(u^*k\) is harmonic for \(u\cdot \nabla\). The constant \(c(u,\nabla)\) is uniquely determined by the normalized condition.
It remains to prove (ii). Consider the slice \(S^{h_{\nabla_0}^{\mathrm{n}}}_{\nabla_0,\varepsilon_0}\) and let \(h_0=h_{\nabla_0}^{\mathrm{n}}\). The elements of the slice are of form \(\nabla_\alpha=\nabla_0+\alpha\in S^{h_{0}}_{\nabla_0,\varepsilon_0}.\)
Every Hermitian metric sufficiently close to \(h_0\) can be written uniquely as \(h_f=h_0e^f,\) where \(f=f^{*,{h_0}}\in \mathop{\mathrm{\textrm{Herm}}}(E,h_0)=\{f\in \mathop{\mathrm{End}}(E)|f^{*,{h_0}}=f\}.\) Since harmonic metrics are unique only up to positive constant multiples, we impose the normalization \(\int_M \operatorname{tr}(f)\,\frac{\omega^n}{n!}=0.\) After Sobolev completion, set \[B_k= \left\{ f\in W^{k,2}(\mathop{\mathrm{\textrm{Herm}}}(E,h_0)) \;\middle|\; \int_M \operatorname{tr}(f)\,\frac{\omega^n}{n!}=0 \right\}.\] Let \(F(\nabla_\alpha,f)=e^{f/2}(\nabla_{\alpha,h_f})^{*,h_f}\psi^{\nabla_{\alpha}}_{h_f}e^{-f/2}\) denote the harmonic metric equation for \(\nabla_\alpha\) with respect to the metric \(h_f=h_0e^f\) with \(F(\nabla_0,0)=0\). If an endomorphism \(A\) is self-adjoint with respect to \(h_f\), then \(e^{f/2} A e^{-f/2}\) is self-adjoint with respect to \(h_0\). For \(k\) sufficiently large, \[F: S^{h_0}_{\nabla_0,\varepsilon_0}\times B_k \longrightarrow B_{k-2}\] is a smooth map.
By [[14]], we obtain \(LF := D_f F \big|_{(\nabla_0,0)}= -\frac{1}{2} (\nabla_0^{\dagger_{h_0}})^{*,h_0}\nabla_0^{\dagger_{h_0}}: B_k \longrightarrow B_{k-2}\).
It is elliptic, self-adjoint, and Fredholm of index zero. Its kernel consists of \(\nabla_0^{\dagger_h}\)-parallel \(h_0\)-self-adjoint endomorphisms satisfying the above trace normalization.
Since \(\nabla_0\) is simple, \(\mathop{\mathrm{End}}_{\nabla_0}(E)=\mathbb{C}\mathop{\mathrm{\textrm{Id}}}.\) The equations \(s=s^{*,h_0}\) and \(\nabla_0^{\dagger_{h_0}}s=0\) imply \(\nabla_0 s=0\), and thus \(s\in \mathbb{C} \mathop{\mathrm{\textrm{Id}}}\). Thus the \(h_0\)-self-adjoint part of the kernel is \(\mathbb{R}\mathop{\mathrm{\textrm{Id}}}\), and the normalization removes this one-dimensional kernel. Hence \(\ker LF=0.\) Since \(LF\) is Fredholm of index zero, it follows that \(LF:B_k\longrightarrow B_{k-2}\) is an isomorphism.
The implicit function theorem therefore gives, after shrinking the slice if necessary, a unique map from \(S^{h_0}_{\nabla_0,\varepsilon_0}\) to \(B_k\), \(\alpha\longmapsto f(\alpha)\) such that \(F(\nabla_\alpha,f(\alpha))=0\), and \(f(0)=0.\) By elliptic regularity, \(f(\alpha)\) is in fact smooth.
Therefore the normalized harmonic metric \(h_{\alpha}=h_0e^{f(\alpha)}\) varies smoothly on the local slice. It is easy to prove \(h_{\alpha}\) satisfies the normalization condition since \(\log\det\bigl(h^{-1}h_\alpha\bigr) = \log\det\bigl(h^{-1}h_0\bigr)+\log\det\bigl(h_0^{-1}h_\alpha\bigr) = \operatorname{tr}\bigl(f(\alpha)\bigr)\). ◻
Throughout the rest of the paper, the metric entering the definition of the slice is always the normalized representative \(h_\nabla^{\mathrm n}\). Thus, whenever \(h_\nabla\) appears in the slice notation, it is understood to mean \(h_\nabla^{\mathrm n}\). When no confusion is likely, we suppress this metric from the notation and write \(S_{\nabla,\varepsilon}\) for \(S_{\nabla,\varepsilon}^{h_{\nabla}^{\mathrm{n}}}.\)
Lemma 4. Fix \(S^{h_\nabla}_{\nabla,\varepsilon}\), let \(S^{h_{\nabla'}}_{\nabla',\varepsilon'}\) be another slice centered at \(\nabla'=\nabla +\alpha_0 \in S^{h_\nabla}_{\nabla,\varepsilon}\), then there is a smooth mapping \[u: U_{\nabla \nabla'}:=S^{h_\nabla}_{\nabla,\varepsilon} \cap p^{-1}(p(S^{h_{\nabla'}}_{\nabla',\varepsilon'})) \to G, \;\alpha \mapsto u_\alpha\] such that \(u_\alpha \cdot(\nabla+\alpha) \in S^{h_{\nabla'}}_{\nabla',\varepsilon'}\) for \(\alpha \in U_{\nabla \nabla'}\) near \(\alpha_0\), and \(u_{\alpha_0}\cdot (\nabla+\alpha_0)=\nabla+\alpha_0\).
Proof. The proof is similar to the proof of [[13], Theorem 7.3.17]. Define \[\begin{align}\Psi:W^{k,2}(\Omega^1(\mathop{\mathrm{End}}E)) \times U_{k+1} &\longrightarrow U_{k-1}\\ (\alpha,\xi)&\mapsto d_{1,\nabla'}^{*,h_{\nabla'}}(\exp(\xi)\dot{(}\nabla+\alpha)-\nabla'),\end{align}\] where we set \(U_k = \left\{ \xi\in W^{k,2}(\mathop{\mathrm{End}}(E)) \;\middle|\; \int_M \operatorname{tr}(\xi)\,\frac{\omega^n}{n!}=0 \right\},\) then \(\left. D_\xi \Psi \right|_{(\alpha_0,0)}=d^{*,h_{\nabla'}}_{\nabla'}d_{\nabla'}: U_{k+1} \to U_{k-1}\) is elliptic and self-adjoint. It is an isomorphism. Therefore the Banach implicit function theorem gives, for \(\alpha\) near \(\alpha_0\), a unique smooth map \(\xi=\xi(\alpha)\in U_{k+1}\), with \(\xi(\alpha_0)=0,\) such that \(\Psi(\alpha,\xi(\alpha))=0.\) Set \(u_\alpha=\exp(\xi(\alpha)),\) then \(d_{1,\nabla'}^{*,h_{\nabla'}} \left( u_\alpha\cdot(\nabla+\alpha)-\nabla' \right)=0,\) so \(u_\alpha\cdot(\nabla+\alpha)\in S_{\nabla',\varepsilon'}^{h_{\nabla'}}.\) The smoothness assertion follows from elliptic regularity. ◻
The preceding lemma gives an explicit formula for the transition map between the slices: \(\tau_{\nabla\nabla'} : U_{\nabla\nabla'} \longrightarrow U_{\nabla'\nabla}\) by \(\tau_{\nabla\nabla'}(\nabla+\alpha) = u_\alpha\cdot(\nabla+\alpha).\)
The tangent space at \(\nabla+\alpha\) is \(T_{\nabla+\alpha}S_{\nabla,\varepsilon}=\ker d^*_{1,\nabla} \cap \ker d_{2,\alpha}\) , we define an inner product by \[\langle \beta_1,\beta_2\rangle^\mathrm{h}_{\nabla+\alpha} =(\beta_1^\mathrm{h},\beta_2^\mathrm{h})_{\mathop{\mathrm{End}}E,h_{\alpha}} = \int_M \langle \beta_1^\mathrm{h},\beta_2^\mathrm{h}\rangle_{\mathop{\mathrm{End}}E,h_{\alpha}}\,\frac{\omega^n}{n!}.\] Here \(\beta_1^\mathrm{h}, \beta_2^\mathrm{h} \in \ker d_{2,\alpha} \cap \ker d_{1,\alpha}^*\) are the harmonic parts with respect to the splitting \(\Omega^1(\mathop{\mathrm{End}}(E)) = \operatorname{Im} d_{1,\alpha} \oplus \ker d_{1,\alpha}^*\)[[12]]. The definition is consistent with [[12]]. The Hermitian pairing \(\langle \cdot,\cdot \rangle^{\mathrm h}_{\nabla+\alpha}\) is also denoted by \((\cdot,\cdot)_{h_\alpha^{\mathrm h}}.\) The corresponding Riemannian metric \(g_\alpha^{\mathrm h}\) is defined by taking the real part.
It remains only to prove that this construction is independent of the choice of slice. The proof of the following proposition follows from [[12]].
Proposition 8. Let \(\beta \in T_{\nabla+\alpha} S_{\nabla,\varepsilon},\) and let \(\alpha(t)\) be a smooth curve in \(S_{\nabla,\varepsilon}\) with \(\alpha(0)=\alpha\) and \(\dot{\alpha}(0)=\beta\). Then the differential of \(\tau\) at \(\nabla+\alpha\) in the direction \(\beta\) is \((\tau_{\nabla \nabla'})_{*,\nabla+\alpha} : T_{\nabla+\alpha}S_{\nabla,\varepsilon} \longrightarrow T_{\tau(\nabla+\alpha)}S_{\nabla',\varepsilon'}\) given by \[\tau_{*,\nabla+\alpha}(\beta)=d_{1,{\nabla'+\alpha'}}\psi+u_\alpha(\beta)\] for some \(\psi \in \Omega^0(\mathop{\mathrm{End}}E)\).
Thus \[\begin{align} \bigl\langle \tau_*\beta_1,\tau_*\beta_2\bigr\rangle^\mathrm{h}_{\nabla'+\alpha'} &= \int_M \bigl( (\tau_*\beta_1)^\mathrm{h}, (\tau_*\beta_2)^\mathrm{h} \bigr)_{\nabla'+\alpha'} \,\frac{\omega^n}{n!} \\[2mm] &= \int_M \bigl( u_\alpha(\beta_1^\mathrm{h}), u_\alpha(\beta_2^\mathrm{h}) \bigr)_{u_\alpha\cdot (\nabla+\alpha)} \,\frac{\omega^n}{n!} \\[2mm] &= \int_M \bigl( \beta_1^\mathrm{h}, \beta_2^\mathrm{h} \bigr)_{\nabla+\alpha} \,\frac{\omega^n}{n!} \\[2mm] &= \bigl\langle \beta_1,\beta_2\bigr\rangle^\mathrm{h}_{\nabla+\alpha}. \end{align}\]
Thus it is invariant under changes of slice. Here \(\langle\cdot,\cdot\rangle_{\nabla+\alpha}\) indicates that the pairing is defined using \(h^{\mathrm{n}}_{\alpha}\). Consequently, the local pairings glue to a well-defined Hermitian metric on \(\widehat{\mathcal{M}}^s\) with respect to the complex structure induced by \(I\).
Remark 9. The metric constructed above should be regarded as a Hermitian metric which is not expected, in general, to be Kähler. There is no a priori reason for its associated fundamental form to be closed. Indeed, this point can be seen by comparing the construction with the proof of [[12], Proposition 4.2]. In that proof, the \(L^2\)-inner product is defined using a single fixed background metric throughout the local model. This fixedness is a key reason why the corresponding fundamental form is closed. By contrast, in the present construction the harmonic metric entering the definition varies with the base point of the slice. Consequently, when one differentiates the associated fundamental form, the first variation of this harmonic metric produces additional terms. There is no apparent reason for these extra terms to cancel in general.
In this section, we construct a quaternionic-type triple of operators \(\mathcal{I}_{\alpha},\mathcal{J}_{\alpha},\mathcal{K}_{\alpha}\) on the NHYM moduli space. More precisely, \(\mathcal{I}_{\alpha}\) defines a genuine complex structure, whereas \(\mathcal{J}_{\alpha}\) and \(\mathcal{K}_{\alpha}\) are only almost complex structures and are not asserted to be integrable. Our emphasis will not be on the complex structure \(\mathcal{I}_{\alpha}\) alone. We will treat the triple \(\mathcal{I}_{\alpha},\mathcal{J}_{\alpha},\mathcal{K}_{\alpha}\) simultaneously.
We study almost complex structures on the moduli space in the same way as the above section: namely, we first define the \(\mathcal{I}_{\alpha},\mathcal{J}_{\alpha},\mathcal{K}_{\alpha}\) on each local slice and then check that they are compatible.
Our strategy will be to construct the almost complex structure \(\mathcal{I}_{\alpha},\mathcal{J}_{\alpha},\mathcal{K}_{\alpha}:T_\alpha S_{\nabla,\varepsilon} \longrightarrow T_\alpha S_{\nabla,\varepsilon}\), as the composition \[\mathcal{L}_{\alpha}:T_\alpha S_{\nabla,\varepsilon} \xrightarrow{\;Q_\alpha\;} \mathcal{H}_\alpha^1 \xrightarrow{\;\tilde{L}_{\alpha}^{H}\;} \mathcal{H}_\alpha^1 \xrightarrow{\;Q_\alpha^{-1}\;} T_\alpha S_{\nabla,\varepsilon},\;\mathcal{L}_{\alpha} \in \{\mathcal{I}_{\alpha},\mathcal{J}_{\alpha},\mathcal{K}_{\alpha}\}\]
Following Hitchin’s hyperkähler construction, we introduce three formal candidate complex-structure operators. For every Hermitian metric \(h_\alpha\), define real-linear maps \(I_{\alpha},J_{\alpha},K_{\alpha}:\Omega^1(\mathop{\mathrm{End}}E) \to \Omega^1(\mathop{\mathrm{End}}E)\) by \[I_{\alpha}(a,b)=(ia,ib),\; J_{\alpha}(a,b)=\bigl(i b^{*,h_\alpha},-i a^{*,h_\alpha}\bigr),\; K_{\alpha}(a,b)=\bigl(-b^{*,h_\alpha},a^{*,h_\alpha}\bigr),\] where we use the decomposition \(\Omega^1({\mathop{\mathrm{End}}E})=\Omega^{0,1}(\mathop{\mathrm{End}}E)\oplus \Omega^{1,0}(\mathop{\mathrm{End}}E)\)
It follows by a straightforward argument that \(I_\alpha^2=J_\alpha^2=K_\alpha^2=-\mathop{\mathrm{\textrm{Id}}}\) and \(I_\alpha J_\alpha=K_\alpha\) and they satisfy the quaternionic identities. These operators are equivariant with respect to the gauge transformation \(u_\alpha\). We shall also need the following proposition.
Proposition 10. Let \(g_\alpha=\operatorname{Re}h_\alpha\) be the corresponding Riemannian metric, then \(g_\alpha(L_\alpha s,L_\alpha t)=g_\alpha(s,t),\; L_\alpha\in\{I_\alpha,J_\alpha,K_\alpha\}.\)
Proof. Let \(s=(a,b),t=(c,d)\in \Omega^1(\mathop{\mathrm{End}}E).\) It is immediate that \(I_\alpha\) preserves \(g_\alpha\). Since \(K_\alpha=I_\alpha J_\alpha\), it remains to prove the assertion for \(J_\alpha\).
We use the identity \(\bigl(\eta^{*,{h_\alpha}},\theta^{*,{h_\alpha}}\bigr)_{h_\alpha} = \overline{(\eta,\theta)_{h_\alpha}}\), then \[\begin{align} g_\alpha(J_\alpha s,J_\alpha t) &= \operatorname{Re}\left[ \bigl( (i b^{*,h_\alpha},-i a^{*,h_\alpha}), (i d^{*,h_\alpha},-i c^{*,h_\alpha}) \bigr)_{h_\alpha}\right]\\ &= \operatorname{Re} \left[ (b^{*,h_\alpha},d^{*,h_\alpha})_{h_\alpha} + (a^{*,h_\alpha},c^{*,h_\alpha})_{h_\alpha} \right]\\ &= \operatorname{Re} \left[ \overline{(b,d)_{h_\alpha}} + \overline{(a,c)_{h_\alpha}} \right]\\ &= g_\alpha(s,t). \end{align}\] ◻
We begin with the almost complex structure on \(\mathcal{H}^1_\alpha\). Take \(I^H_{\alpha}=I_{\alpha}\big|_{\mathcal{H}^1_\alpha},J^H_{\alpha}=J_{\alpha}\big|_{\mathcal{H}^1_\alpha},K^H_{\alpha}=K_{\alpha}\big|_{\mathcal{H}^1_\alpha}\). In general, they do not satisfy the quaternionic relations. Nevertheless, since \(I\) preserves harmonic forms, they still satisfy some identities. For instance, \(I_\alpha J_\alpha^H=K_\alpha^H, J^H_\alpha I^H_\alpha=-I_\alpha J^H_\alpha\) and similarly for the other cyclic relations. In particular, when \(\alpha=0\), the three operators are defined on \(T_\nabla S_{\nabla,\varepsilon}^{h_\nabla}\cong \mathcal{H}^1_\nabla\).
Proposition 11. \(I^H_{\nabla}\) preserves the tangent space, i.e. \(I^H_{\nabla}\bigl(T_\nabla S_{\nabla,\varepsilon}\bigr) \subset T_\nabla S_{\nabla,\varepsilon}.\) If, in addition, \(\nabla\) is unitary with respect to \(h_\nabla\), then \(J^H_\nabla\) and \(K^H_\nabla\) also preserve the tangent space, i.e. \(J^H_\nabla\bigl(T_\nabla S_{\nabla,\varepsilon}\bigr) \subset T_\nabla S_{\nabla,\varepsilon}, K^H_\nabla\bigl(T_\nabla S_{\nabla,\varepsilon}\bigr) \subset T_\nabla S_{\nabla,\varepsilon}. \;\)
Proof. For notational convenience, we suppress the subscript \(h_\nabla\) from the fiberwise adjoint. Let \(\beta=(a,b)\in T_\nabla S_{\nabla,\varepsilon}= \ker d_{1,\nabla}^{*}\cap \ker d_{2,\nabla}\), then \[\nabla' b=0,\; \nabla''a=0,\; \Lambda(\nabla'a+\nabla''b)=0, \; d_{1,\nabla}^{*}(a,b)=0.\]
Since both \(d_{1,\nabla}^{*,h_\nabla}\) and \(d_{2,\nabla}\) are complex-linear, we have \(I^H_\nabla\beta\in T_\nabla S_{\nabla,\varepsilon}.\)
Since \(\nabla\) is unitary with respect to \(h_\nabla\), then \((\nabla'a)^*=\nabla''(a^*), (\nabla''b)^*=\nabla'(b^*), (\nabla''a)^*=\nabla'(a^*), (\nabla'b)^*=\nabla''(b^*).\) and the Kähler identities give \((\nabla'')^*a=-i\Lambda\nabla'a, (\nabla')^*b=i\Lambda\nabla''b.\) Therefore \[0=d_{1,\nabla}^{*,h_\nabla}(a,b)=(\nabla'')^{*,h_\nabla} a +(\nabla')^{*,h_\nabla}b=i\Lambda\nabla''b-i\Lambda\nabla'a.\]
Set \(J^H_\nabla\beta=(a_J,b_J)=(i b^*,-i a^*).\) Since \(\nabla''a=0\) and \(\nabla'b=0\), we compute, \[\nabla' b_J = -i\nabla'(a^*) = -i(\nabla''a)^* = 0,\; \nabla'' a_J = i\nabla''(b^*) = i(\nabla'b)^* = 0.\] \[\begin{align} \Lambda(\nabla'a_J+\nabla''b_J) &= \Lambda\bigl(i\nabla'(b^*)-i\nabla''(a^*)\bigr)\\ &= i\Lambda\bigl((\nabla''b)^*-(\nabla'a)^*\bigr)\\ &= i\bigl(\Lambda(\nabla''b-\nabla'a)\bigr)^*=0. \end{align}\] \[\begin{align} d_{1,\nabla}^{*}(J^H_\nabla\beta) &= (\nabla'')^*(i b^*)+(\nabla')^*(-i a^*)\\ &= \Lambda\nabla'(b^*)+\Lambda\nabla''(a^*)\\ &= \Lambda\bigl((\nabla''b)^*+(\nabla'a)^*\bigr)\\ &= \bigl(\Lambda(\nabla''b+\nabla'a)\bigr)^*=0. \end{align}\]
Hence \(J^H_\nabla\beta\in T_\nabla S_{\nabla,\varepsilon}.\)
Finally, since \(K^H_\nabla=I^H_\nabla J^H_\nabla\), and both \(I^H_\nabla\) and \(J^H_\nabla\) preserve \(T_\nabla S_{\nabla,\varepsilon}\) under the Hermitian assumption, the same is true for \(K^H_\nabla\). This proves the proposition. ◻
In view of the preceding proposition, we shall henceforth restrict our attention to slices centered at the Hermitian locus, namely slices \(S_{\nabla,\varepsilon}\) with \(\nabla\) Hermitian.
Thus, we are led to study almost complex structures on the open subset \[\mathcal{U} = \bigcup p\bigl(S_{\nabla,\varepsilon_\nabla}\bigr)\subset \mathcal{M}^s,\] where the union is taken over all \((0,1)\)-stable HYM connections satisfying \(H^2(\widetilde{\mathcal{C}}_\nabla)=0\).
The transition maps on \(\mathcal{U}\) considered here satisfy the following properties.
Lemma 5. Let \(\nabla\) and \(\nabla'\) be HYM connections, and consider the Hermitian-centered slices \(S_{\nabla,\varepsilon}^{h_\nabla}\) and \(S_{\nabla',\varepsilon'}^{h_{\nabla'}}.\) On the non-empty overlap \(U_{\nabla\nabla'} = S_{\nabla,\varepsilon}^{h_\nabla} \cap p^{-1}\bigl(p(S_{\nabla',\varepsilon'}^{h_{\nabla'}})\bigr),\) there exists \(V \subset U_{\nabla\nabla'}\) and a smooth map \(u:V \to G,\; \alpha\mapsto u_\alpha,\) such that \(u_\alpha\cdot(\nabla+\alpha)\in S_{\nabla',\varepsilon'}^{h_{\nabla'}}\). Thus the transition map is \(\tau_{\nabla\nabla'}(\nabla+\alpha) = u_\alpha\cdot(\nabla+\alpha).\)
Thus far, we have constructed three almost complex structures on the tangent space at the base point \(\nabla\). These structures, however, do not admit a natural extension to the tangent spaces at points of the form \(\nabla+\alpha\). The idea is to use the harmonic space as an intermediate model and define the corrected candidate almost complex structure by conjugation through \(Q_\alpha=\mathbb{H}_\alpha|_{T_\alpha S_{\nabla,\varepsilon}}\), where \(\mathbb{H}_\alpha:\Omega^1(\mathop{\mathrm{End}}E)\longrightarrow \mathcal{H}_{\alpha}^1\) denotes the harmonic projection, namely: \[\label{cs} \mathcal{L}_{\alpha}:T_\alpha S_{\nabla,\varepsilon} \xrightarrow{\;Q_\alpha\;} \mathcal{H}_\alpha^1 \xrightarrow{\;\tilde{L}_{\alpha}^{H}\;} \mathcal{H}_\alpha^1 \xrightarrow{\;Q_\alpha^{-1}\;} T_\alpha S_{\nabla,\varepsilon},\; \mathcal{L}_\alpha\in\{\mathcal{I}_\alpha,\mathcal{J}_\alpha,\mathcal{K}_\alpha\}.\tag{1}\]
The use of the harmonic projection and of \(Q_\alpha^{-1}\) generally destroys integrability. Thus the construction does not imply the integrability of \(\mathcal{J}_\alpha\) and \(\mathcal{K}_\alpha\). We now explain how each of the maps in this construction is obtained. The next two propositions describe the operators \(Q_\alpha\) and \(\tilde{L}_{\alpha}^{H}\), respectively.
Proposition 12. \(Q_\alpha=\mathbb{H}_\alpha|_{T_\alpha S_{\nabla,\varepsilon}}:T_\alpha S_{\nabla,\varepsilon} \to \mathcal{H}_{\alpha}^1\) is invertible for small \(\alpha\).
Proof. \(Q_\alpha\) is well-defined since every \(\beta\in T_\alpha S_{\nabla,\varepsilon} \subset \ker d_{2,\alpha}\) admits a decomposition \(\beta=\beta^{\mathrm{h}}+d_{1,\alpha}\xi,\; \beta^{\mathrm{h}}\in \mathcal{H}_{\alpha}^{1}, \xi\in \Omega^{0}(\mathop{\mathrm{End}}E).\)
Given \(\gamma \in \mathcal{H}^1_\alpha\), we need to find \(\beta \in T_\alpha S_{\nabla,\varepsilon}=\ker d^*_{1,\nabla} \cap \ker d_{2,\alpha}\) such that \(Q_\alpha(\beta)=\gamma\). Clearly, \(\beta\) has form \(\beta=\gamma+d_{1,\alpha}\xi\) and \(d_{2,\alpha}\beta=d_{2,\alpha}\gamma+d_{2,\alpha}d_{1,\alpha}\xi=0\).
It remains only to impose the fixed slice condition \(\nabla^{*,h_{\nabla}}\beta=0.\) Substituting \(\beta\) into this condition, we obtain \[\nabla^{*,h_{\nabla}}\bigl(\gamma+d_{1,\alpha}\xi\bigr)=0.\] Define \(B_\alpha=d_{1,\nabla}^{*,h_{\nabla}}d_{1,\alpha}:\Omega^0(\mathop{\mathrm{End}}E)\to \Omega^0(\mathop{\mathrm{End}}E)\), it remains to solve the equation \[B_\alpha\xi=-\nabla^{*,h_{\nabla}}\gamma.\] Claim: \(\widetilde{B}_\alpha=B_\alpha|_{\Omega^{0}_{\perp}}:\Omega^{0}_{\perp}(\mathop{\mathrm{End}}E)\to \Omega^{0}_{\perp}(\mathop{\mathrm{End}}E)\) is an isomorphism for small \(\alpha\) where \(\Omega^{0}_{\perp}(\mathop{\mathrm{End}}E)=(\ker d_{1,\nabla})^{\perp}=(\mathbb{C} \mathop{\mathrm{\textrm{Id}}})^{\perp}.\)
Since \(\widetilde{B}_0=d_{1,\nabla}^{*,h_{\nabla}}d_{1,\nabla}:\Omega^{0}_{\perp} \to \Omega^{0}_{\perp}\) is an isomorphism, and for small \(\alpha\), \(\widetilde{B}_\alpha\) is also an isomorphism. This is the standard fact that the set of invertible bounded operators between Banach spaces is open in the operator norm topology. A point worth noting is that \(\Omega^0_{\perp}\) does not depend on the choice of the Hermitian metric since \(\nabla\) here is simple.
Note that \(\nabla^{*,h_{\nabla}}\gamma \in \Omega^{0}_{\perp}(\mathop{\mathrm{End}}E)\) since \((\nabla^{*,h_{\nabla}}\gamma,\eta)=(\gamma,\nabla \eta)=0 \textrm{ for } \eta \in \ker d_{1,\nabla}.\) Thus \(\xi=-\widetilde{B}^{-1}_\alpha \nabla^{*,h_{\nabla}}\gamma\) is well-defined.
To see the injectivity of \(Q_\alpha\), suppose \(Q_\alpha(\beta)=0\). There exists \(\gamma \in \mathcal{H}^1_\alpha, \xi_{\perp} \in \Omega^0_{\perp}\) and \(\xi_0 \in \ker \nabla =\mathbb{C} \mathop{\mathrm{\textrm{Id}}}\) such that \(\beta=\gamma+d_{1,\alpha}\xi_{\perp}+d_{1,\alpha}\xi_0\). Then \(0=Q_\alpha(\beta)=\gamma\) and clearly \(d_{1,\alpha}\xi_0=0\), \(0= d^{*,h_{\nabla}}_{1,\nabla}\beta=d^{*,h_{\nabla}}_{1,\nabla}d_{1,\alpha}\xi_{\perp}=\widetilde{B}_\alpha \xi_{\perp} \Rightarrow \xi_{\perp}=0\Rightarrow \beta=0\). ◻
Proposition 13. Set \(S_{\alpha}=\mathbb{H}_\alpha J^H_\alpha:\mathcal{H}_\alpha^1 \to \mathcal{H}_\alpha^1, R_{\alpha}=\bigl(-S_{\alpha}^2\bigr)^{1/2},\) then \(\tilde{J}^H_{\alpha}=S_{\alpha} R_{\alpha}^{-1}:\mathcal{H}_\alpha^1\to \mathcal{H}_\alpha^1\) is a well-defined almost complex structure on the underlying real vector space of \(\mathcal{H}^1_\alpha\).
Proof. The proposition follows from the following steps.
\(S_\alpha=\mathbb{H}_\alpha J^H_\alpha:\mathcal{H}^1_\alpha \to\mathcal{H}^1_\alpha\) is well-defined.
\(S_\alpha\) is skew-adjoint, i.e. \(S_{\alpha}^{*,g_\alpha^{\mathrm{h}}}=-S_\alpha\).
Indeed, for \(\beta_1,\beta_2 \in \mathcal{H}^1_\alpha\), \(\langle S_\alpha \beta_1,\beta_2 \rangle_{g_\alpha}^\mathrm{h} =\langle\mathbb{H}_\alpha J^H_\alpha \beta_1,\beta_2 \rangle_{g_\alpha}^\mathrm{h} =\langle J^H_\alpha \beta_1,\beta_2 \rangle_{g_\alpha}^\mathrm{h} =\langle \beta_1,-J^H_\alpha \beta_2 \rangle_{g_\alpha}^\mathrm{h} =\langle \beta_1,- \mathbb{H}_\alpha J^H_\alpha \beta_2 \rangle_{g_\alpha}^\mathrm{h} =-\langle \beta_1,S_\alpha \beta_2 \rangle_{g_\alpha}^\mathrm{h} .\)
\(-S^2_\alpha\) is positive definite self-adjoint.
Obviously, \(S^{*,g_\alpha}_\alpha S=-S^2_\alpha\). \(\langle -S^2_\alpha \beta,\beta\rangle_{g_\alpha}^\mathrm{h}=||S_\alpha \beta||^2 \geq 0\). If \(-S^2_\alpha \beta=0\), then \(S_\alpha \beta=0\). The following claim implies \(\beta=0\).
Claim: \(S_\alpha\) is still an isomorphism for \(\alpha\) sufficiently small.
Let \(\Phi_\alpha = \mathbb{H}_\alpha\big|_{\mathcal{H}_\nabla^1}:\mathcal{H}_\nabla^1 \longrightarrow \mathcal{H}_\alpha^1\), we define \(\widetilde{S}_\alpha:\mathcal{H}_\nabla^1 \to \mathcal{H}_\nabla^1\) by the composition \(\widetilde{S}_\alpha = \Phi_\alpha^{-1}\circ S_\alpha\circ \Phi_\alpha,\) that is, \(\mathcal{H}_\nabla^1 \xrightarrow{\;\Phi_\alpha\;} \mathcal{H}_\alpha^1 \xrightarrow{\;S_\alpha\;} \mathcal{H}_\alpha^1 \xrightarrow{\;\Phi_\alpha^{-1}\;} \mathcal{H}_\nabla^1\).
Here we show \(\Phi_\alpha\) is an isomorphism for \(\alpha\) sufficiently small. If \(\Phi_\alpha(v)=0\), then \(v=\mathbb{H}_\nabla\big|_{\mathcal{H}_\nabla^1}(v)-\Phi_\alpha(v)=(\mathbb{H}_\nabla\big|_{\mathcal{H}_\nabla^1}-\Phi_\alpha)v\). Comparing the norms of both sides gives \(v=0\). Moreover, Hodge theory for elliptic complexes implies both \(\mathcal{H}^1_\nabla\) and \(\mathcal{H}^1_\alpha\) are finite-dimensional vector spaces of the same dimension. Hence \(\Phi_\alpha\) is an isomorphism. Since \(\widetilde{S}_0=S_0=J^H_\nabla:\mathcal{H}^1_\nabla \to \mathcal{H}^1_\nabla\) is an isomorphism. For small \(\alpha\), \(\widetilde{S}_\alpha\) is still an isomorphism. Therefore \(S_\alpha=\Phi_\alpha\circ \widetilde{S}_\alpha\circ \Phi_\alpha^{-1}\) is also an isomorphism.
Since \(\mathcal{H}^1_\alpha\) is finite-dimensional and \(-S^2_\alpha\) is positive definite self-adjoint. We choose the unique positive square root \(R_\alpha\) of \(-S^2_\alpha\) and \(R_\alpha\) commutes with \(S_\alpha\) since \(R_\alpha\) commutes with any bounded linear operator that commutes with \(-S^2_\alpha\) by [[15], Theorem 6.6.4]. \[(\widetilde{J}^H_\alpha)^2=S_{\alpha} R_{\alpha}^{-1}S_{\alpha} R_{\alpha}^{-1}=S_{\alpha}S_{\alpha} R_{\alpha}^{-1} R_{\alpha}^{-1}=S^2_\alpha (-S^2_\alpha)^{-1}=-\mathop{\mathrm{\textrm{Id}}}.\]
Moreover, we have \((\widetilde{J}^{H}_{\alpha})^{*,g_\alpha^{\mathrm{h}}} = (S_{\alpha}R_{\alpha}^{-1})^{*,g_\alpha^{\mathrm{h}}} = R_{\alpha}^{-1}S_{\alpha}^{*,g_\alpha^{\mathrm{h}}} = -R_{\alpha}^{-1}S_{\alpha} = -\widetilde{J}^{H}_{\alpha}.\) Thus the operator \(\widetilde{J}^{H}_{\alpha}\) is \(g_\alpha^{\mathrm{h}}\)-orthogonal. ◻
The above proposition gives \(\mathcal{I}_\alpha^2=\mathcal{J}_\alpha^2=\mathcal{K}_\alpha^2=-\mathop{\mathrm{\textrm{Id}}}\).
Proposition 14. The three operators \(\mathcal{I}_\alpha,\mathcal{J}_\alpha,\mathcal{K}_\alpha\) defined in 1 are almost complex structures satisfying \(\mathcal{I}_\alpha\mathcal{J}_\alpha=\mathcal{K}_\alpha\), and thus constitute an almost hypercomplex structure on \(T_\alpha S_{\nabla,\varepsilon}\).
Proof. We first verify that \(\mathcal{I}_\alpha,\mathcal{J}_\alpha,\mathcal{K}_\alpha\) satisfy the quaternionic relations and then check that these structures are compatible with the transition maps between the slices. To distinguish these operators, we attach the subscripts \(I,J,K\) to \(S_\alpha\) and \(R_\alpha\), according to the corresponding almost complex structures.
First, since \(I\) is a complex structure already, it preserves the harmonic forms, and it follows immediately that \(\mathbb{H}_\alpha I_\alpha=I_\alpha \mathbb{H}_\alpha=I_\alpha^H \mathbb{H}_\alpha,S_{I,\alpha}=\mathbb{H}_\alpha I^H_\alpha=I^H_\alpha, S^2_{I,\alpha}=-\mathop{\mathrm{\textrm{Id}}}_{\mathcal{H}^1_\alpha},R_{I,\alpha}=\mathop{\mathrm{\textrm{Id}}}_{\mathcal{H}^1_\alpha},\widetilde{I}^H_{\alpha}=I^H_\alpha\).
\(S_{K,\alpha}=I^H_\alpha S_{J,\alpha}\).
\(S_{K,\alpha} = \mathbb{H}_{\alpha}K_{\alpha}^{H} = \mathbb{H}_{\alpha}I_{\alpha}J_{\alpha}^{H} = I_{\alpha}\mathbb{H}_{\alpha}J_{\alpha}^{H} = I_{\alpha}^{H}S_{J,\alpha} .\)
\(S_{J,\alpha}I_\alpha^H = -I_\alpha^H S_{J,\alpha}\).
\(S_{J,\alpha}I_\alpha^H = \mathbb{H}_\alpha J_\alpha^H I_\alpha^H = -\mathbb{H}_\alpha I_\alpha J_\alpha^H = -I_\alpha^H\mathbb{H}_\alpha J_\alpha^H = -I_\alpha^H S_{J,\alpha} .\)
\(-S_{K,\alpha}^2 = -S_{J,\alpha}^2\).
\(S_{K,\alpha}^2= \bigl(I_\alpha^H S_{J,\alpha}\bigr) \bigl(I_\alpha^H S_{J,\alpha}\bigr) = I_\alpha^H S_{J,\alpha}I_\alpha^H S_{J,\alpha} = I_\alpha^H\bigl(-I_\alpha^H S_{J,\alpha}\bigr)S_{J,\alpha} = -\bigl(I_\alpha^H\bigr)^2S_{J,\alpha}^2 = S_{J,\alpha}^2.\)
\(R_{K,\alpha} = R_{J,\alpha}\) by the uniqueness of the positive square root.
Then we have \(\widetilde{K}_\alpha^H = S_{K,\alpha}R_{K,\alpha}^{-1} = I_\alpha^H S_{J,\alpha}R_{J,\alpha}^{-1} = I_\alpha^H \widetilde{J}_\alpha^H=\widetilde{I}_\alpha^H \widetilde{J}_\alpha^H,\) and \(\widetilde{J}^{H}_{\alpha} \widetilde{I}^{H}_{\alpha} = S_{J,\alpha} R^{-1}_{J,\alpha} \widetilde{I}^{H}_{\alpha} = S_{J,\alpha} \widetilde{I}^{H}_{\alpha} R^{-1}_{J,\alpha} = - \widetilde{I}^{H}_{\alpha} S_{J,\alpha} R^{-1}_{J,\alpha} = - \widetilde{I}^{H}_{\alpha} \widetilde{J}^{H}_{\alpha}.\)
Thus \(\mathcal{I}_\alpha\mathcal{J}_\alpha=Q_\alpha^{-1} \widetilde{I}^H_\alpha \widetilde{J}^H_\alpha Q_\alpha=Q_\alpha^{-1} \widetilde{K}^H_\alpha Q_\alpha=\mathcal{K}_\alpha\) and \(\mathcal{I}_\alpha\mathcal{J}_\alpha=-\mathcal{J}_\alpha\mathcal{I}_\alpha\).
It is enough to prove the compatibility for \(\mathcal{J}_\alpha\), since the arguments for the other almost complex structures are analogous. Thus it suffices to establish the commutativity of the following diagram: \[\begin{tikzcd}[column sep=large,row sep=large] T_{\alpha}S_{\nabla,\varepsilon} \arrow[r, "\mathcal{J}^{\nabla}_{\alpha}"] \arrow[d, "(\tau_{\nabla\nabla'})_{*,\alpha}"'] & T_{\alpha}S_{\nabla,\varepsilon} \arrow[d, "(\tau_{\nabla\nabla'})_{*,\alpha}"] \\ T_{\alpha'}S_{\nabla',\varepsilon'} \arrow[r, "\mathcal{J}^{\nabla'}_{\alpha'}"] & T_{\alpha'}S_{\nabla',\varepsilon'} \end{tikzcd}\] where \(\mathcal{J}_{\alpha}=Q_{\alpha}^{-1}\widetilde{L}_{\alpha}^{H} Q_{\alpha}\) and the subscript indicates the slice on which the operator is defined. Since \(\mathcal{H}_\alpha^1\) is a complex subspace, we have \(\bigl(\mathcal{H}_\alpha^1\bigr)^{\perp,g_\alpha}=\bigl(\mathcal{H}_\alpha^1\bigr)^{\perp,h_\alpha}.\)
The following identities establish the commutativity of the diagram. Here and below, \(u_\alpha\) acts on tangent vectors by the gauge action. Thus, for any operator \(L\), we use the shorthand \((u_\alpha\circ L)(\beta)=u_\alpha\cdot(L\beta)\), and \((L\circ u_\alpha)(\beta)=L(u_\alpha\cdot\beta).\)
\(\mathbb{H}_{\alpha'}\circ u_\alpha = u_\alpha \circ \mathbb{H}_{\alpha}\). It implies \(Q_{\alpha'}\circ \tau_*=u_\alpha \circ Q_\alpha\).
We need to prove \(u_\alpha (\mathcal{H}^1_\alpha) \subset \mathcal{H}^1_{\alpha'}\) and \(u_\alpha ((\mathcal{H}^1_\alpha)^{\perp,g_\alpha}) \subset (\mathcal{H}^1_{\alpha'})^{\perp,g_{\alpha'}}\). It is clear since \(g_{\alpha'}(u_\alpha \beta_1,u_\alpha \beta_2)=g_\alpha(\beta_1,\beta_2)\) and \(u_\alpha (\mathcal{H}^1_{\alpha})=\mathcal{H}^1_{\alpha'}\).
\(J^H_{\alpha'}\circ u_\alpha=u_\alpha \circ J^H_{\alpha}\).
\(J_{\alpha'}^{H}u_{\alpha}(a,b)= J_{\alpha'}^{H} \bigl( u_{\alpha}^{-1}a u_{\alpha}, u_{\alpha}^{-1}b u_{\alpha} \bigr)= \left( i\bigl( u_{\alpha}^{-1}b u_{\alpha}\bigr)^{*,h_{\alpha'}}, -i\bigl( u_{\alpha}^{-1}a u_{\alpha}\bigr)^{*,h_{\alpha'}} \right)= \left( i u_{\alpha}^{-1}b^{*,h_{\alpha}} u_{\alpha}, -i u_{\alpha}^{-1}a^{*,h_{\alpha}} u_{\alpha} \right)= u_{\alpha} \bigl( ib^{*,h_{\alpha}}, -ia^{*,h_{\alpha}} \bigr)= u_{\alpha}J_{\alpha}^{H}(a,b).\)
\(S_{\alpha'} \circ u_\alpha=u_\alpha \circ S_{\alpha}\).
\(S_{\alpha'}u_\alpha= \mathbb{H}_{\alpha'}\, J^H_{\alpha'}\, u_\alpha = \mathbb{H}_{\alpha'}\,u_\alpha\,J^H_{\alpha} = u_\alpha\,\mathbb{H}_{\alpha}\,J^H_{\alpha} = u_\alpha\,S_{\alpha}.\) \(S_{\alpha'}=u_\alpha\,S_{\alpha}u_\alpha^{-1}\). \(-S^2_{\alpha'}=u_\alpha\,(-S^2_{\alpha})u_\alpha^{-1}\). Since \(u_\alpha\) is an isometry, and the positive square root is unique, then \(R_{\alpha'}=u_\alpha\,R_{\alpha}u_\alpha^{-1}\).
\(\widetilde{J}^{H}_{\alpha'}\circ u_\alpha=u_\alpha \circ \widetilde{J}^{H}_\alpha\).
\(\widetilde{J}^{H}_{\alpha'}\,u_\alpha = S_{\alpha'}\,R_{\alpha'}^{-1}\,u_\alpha = u_\alpha\,S_{\alpha}\,u_\alpha^{-1} \cdot u_\alpha\,R_{\alpha}^{-1}\,u_\alpha^{-1} \cdot u_\alpha = u_\alpha\,S_{\alpha}\,R_{\alpha}^{-1} = u_\alpha\,\widetilde{J}^{H}_\alpha.\)
Using the above identities, we compute, \[\begin{align} Q_{\alpha'} \Bigl( (\tau_{\nabla\nabla'})_{*,\alpha}\, \mathcal{J}_\alpha^\nabla \beta \Bigr) &= Q_{\alpha'} \Bigl( u_\alpha \cdot (\mathcal{J}_\alpha^\nabla\beta) + d_{1,\alpha'}\psi_{J_\alpha^\nabla\beta} \Bigr) \\ &= (\mathbb{H}_{\alpha'}\circ u_\alpha) (\mathcal{J}_\alpha^\nabla\beta) \\ &= (u_\alpha \circ \mathbb{H}_\alpha)(\mathcal{J}_\alpha^\nabla\beta) \\ &= (u_\alpha \circ Q_\alpha) ( Q_\alpha^{-1}\widetilde{J}_\alpha^H Q_\alpha\beta ) \\ &= (u_\alpha \circ \widetilde{J}_\alpha^H Q_\alpha)\beta \\ &= (\widetilde{J}_{\alpha'}^H \circ u_\alpha\circ Q_\alpha)\beta \\ &= \widetilde{J}_{\alpha'}^H Q_{\alpha'} \Bigl( (\tau_{\nabla\nabla'})_{*,\alpha}\beta \Bigr)\\ &= Q_{\alpha'} Q_{\alpha'}^{-1}\widetilde{J}_{\alpha'}^H Q_{\alpha'}\Bigl( (\tau_{\nabla\nabla'})_{*,\alpha}\beta \Bigr)\\ &= Q_{\alpha'} \mathcal{J}_{\alpha'}^{\nabla'}\Bigl( (\tau_{\nabla\nabla'})_{*,\alpha}\beta \Bigr). \end{align}\]
Since \(Q_{\alpha'}\) is an isomorphism for small \(\alpha'\), the diagram is commutative. We now verify that the almost complex structure constructed above is smooth. It is enough to show that the assignment \(\alpha \longmapsto \mathcal{J}_\alpha\) is smooth. By construction, this reduces to the smoothness of the following two ingredients.
First, the map \(\alpha \longmapsto \mathbb{H}_\alpha\) is smooth. This follows from [[16]], applied to the family of formally self-adjoint strongly elliptic operators \(L_s=\Delta_\alpha^1\). In that theorem, \(F_s\) denotes the harmonic projector, which corresponds in our notation to \(\mathbb{H}_\alpha\).
Second, the map \(\alpha \longmapsto R_\alpha\) is smooth. Indeed, \(R_\alpha\) is defined as the principal square root of the positive self-adjoint operator \(-S_\alpha^2.\) After choosing a local orthonormal frame, \(-S_\alpha^2\) is represented by a smooth family of real symmetric positive definite matrices. Therefore the smoothness of \(R_\alpha\) follows from [[17]], applied to \(Q=-S_\alpha^2\). ◻
Remark 15. The underlying Riemannian metric of the Hermitian metric is compatible with the almost hypercomplex structure since \[\begin{align} \langle \mathcal{L}_\alpha\beta_1, \mathcal{L}_\alpha\beta_2 \rangle_{g_\alpha}^{\mathrm{h}} &= \langle Q_\alpha Q_\alpha^{-1} \widetilde{L}^H_\alpha Q_\alpha \beta_1, Q_\alpha Q_\alpha^{-1} \widetilde{L}^H_\alpha Q_\alpha \beta_2 \rangle^{\mathrm{h}}_{g_\alpha}\\ &= \langle \widetilde{L}^H_\alpha Q_\alpha \beta_1, \widetilde{L}^H_\alpha Q_\alpha \beta_2 \rangle^{\mathrm{h}}_{g_\alpha}\\ &= \langle Q_\alpha \beta_1, Q_\alpha \beta_2 \rangle^{\mathrm{h}}_{g_\alpha}\\ &= \langle \beta_1, \mathcal{\beta}_2 \rangle_{g_\alpha}^{\mathrm{h}}. \end{align}\]
In this section, we discuss whether the hyperkähler quotient method proposed in [[1]] can be applied to the NHYM moduli space.
We begin by recalling the hyperkähler quotient construction of [[18]], which provides a mechanism for producing new hyperkähler manifolds from hyperkähler manifolds equipped with tri-Hamiltonian group actions.
Theorem 4. Let \((M,g,I,J,K)\) be a hyperkähler manifold and let \(G\) be a compact Lie group acting on \(M\) by triholomorphic isometries. Suppose that the action admits an equivariant hyperkähler moment map \(\mu=(\mu_1,\mu_2,\mu_3):M\longrightarrow \mathfrak g^*\otimes\mathbb{R}^3 .\) If \(0\) is a regular value of \(\mu\) and \(G\) acts freely on \(\mu^{-1}(0)\), then \(\mu^{-1}(0)/G\) is a smooth hyperkähler manifold. Its metric and three Kähler forms are induced by restricting the corresponding tensors on \(M\) to the horizontal distribution orthogonal to the \(G\)-orbits.
This construction has several standard applications. Kronheimer’s construction realizes hyperkähler ALE spaces [[19]]. In the same spirit, Nakajima realizes quiver varieties as hyperkähler quotients, relating them to ALE instantons and Kac–Moody representation theory [[20]].
Hitchin’s construction is the basic infinite-dimensional gauge-theoretic analogue of the hyperkähler quotient theorem. The ambient space is the configuration space of pairs consisting of a connection and a Higgs field, modelled on infinite-dimensional spaces of sections, and the unitary gauge group acts on it by triholomorphic isometries. The self-duality equations are the corresponding hyperkähler moment-map equations. Unlike the finite-dimensional case, however, the quotient requires analytic input: one works with Sobolev completions and uses the slice theorem, ellipticity, and regularity to obtain the smooth hyperkähler moduli space [[21]]. This gauge-theoretic infinite-dimensional viewpoint also underlies Konno’s construction of parabolic Higgs bundle moduli spaces and Fujiki’s study of flat-bundle moduli in higher dimensional Kähler geometry [[8], [22]].
For the purposes of the following discussion, we ignore the issue of whether the objects involved satisfy the assumptions of the hyperkähler quotient theorem and proceed at a purely formal level.
The NHYM moduli space considered here belongs to the infinite dimensional setting. We therefore introduce related frameworks. The space of all connections \(\mathcal{A}\) is an affine space modelled on the complex vector space \(\Omega^1(\mathop{\mathrm{End}}E)\). Choose a Hermitian metric \(h\) on \(E\). The decomposition \(\Omega^1(\mathop{\mathrm{End}}E) = \Omega^{0,1}(\mathop{\mathrm{End}}E)\oplus \Omega^{1,0}(\mathop{\mathrm{End}}E)\) allows one to define a quaternionic structure on \(\Omega^1(M,\mathop{\mathrm{End}}E)\). Together with the natural trace metric and a suitable Sobolev completion, this structure makes \(\mathcal{A}\) an infinite-dimensional hyperkähler manifold.
Kaledin and Verbitsky suggest that the map \(\mu_\mathbb{C}(\nabla)= \Lambda \Theta\) on \(\mathcal{A}\) should be regarded as a complex moment map. This leads formally to the expectation that the quotient \(\mu_\mathbb{C}^{-1}(0)/ G\) should carry a hyperkähler structure by the principle of hyperkähler reduction. However, this argument has to be interpreted with care. A genuine hyperkähler quotient involves not only the complex moment map, but also the real moment map, and the quotient is taken by a unitary gauge group rather than by the full complex gauge group. Thus the formal quotient \(\mu_\mathbb{C}^{-1}(0)/ G\) is not automatically a hyperkähler quotient in the standard sense. For this reason, the hyperkähler quotient picture must be reformulated more carefully.
Define the quaternionic structure on \(\Omega^1(\mathop{\mathrm{End}}E)\) by \[I(a,b)=(ia,ib),\; J(a,b)=\bigl(i b^{*,h},-i a^{*,h}\bigr),\; K(a,b)=\bigl(-b^{*,h},a^{*,h}\bigr).\]
It is useful to emphasize a basic difference between the NHYM and HYM settings. In the NHYM case, the \((1,0)\) and \((0,1)\)-parts are independent variables. This additional freedom makes it possible, at least formally, to define a quaternionic triple \(I,J,K\) on the ambient space of connections. By contrast, in the HYM setting one works with unitary connections with respect to a fixed Hermitian metric. Hence the \((1,0)\)-part is determined by the \((0,1)\)-part through the Hermitian adjoint. The same flat hyperkähler ambient model is not available in this formulation.
The complex moment map is \(\mu_\mathbb{C}=\Lambda \nabla^2\) and the real moment map is \(\mu_\mathbb{R}=\Lambda \Xi_{\nabla,h}\) [[1], pp. 41–42]. We use \(U(E,h)\) to denote the unitary gauge group with respect to the fixed Hermitian metric \(h\), because \(\mu_{\mathbb{R}}^{-1}(0)\) is \(U(E,h)\)-invariant. Hence the formal hyperkähler quotient is \[\mathcal{M}^{hk}=(\mu^{-1}_\mathbb{C}(0)\cap \mu^{-1}_\mathbb{R}(0))/U(E,h)\]
An important consequence of Theorem 3 is that it provides, by choosing harmonic metrics, a set-theoretic injection \[\iota:\mathcal{M}^s \hookrightarrow \mathcal{M}^{hk} .\]
For \(\nabla \in [\nabla]_{G} \in \mathcal{M}^s\), we choose the harmonic metric \(h_\nabla\) such that \(\Lambda \Xi_{\nabla,h_{\nabla}}=0\), and there exists \(q_\nabla \in \Gamma(\mathop{\mathrm{\textrm{Herm}}}^+_{h}(E))\) with \(h_\nabla=(q_\nabla)^* h\), i.e., \(h_\nabla(s,t)=h(q_\nabla s,q_\nabla t)\). Since \((\nabla,h_\nabla)\) is a harmonic pair, the transformed pair \((q_\nabla^{-1}\nabla ,(q_\nabla^{-1})^* h_\nabla)\) is also harmonic. Moreover, \((q_\nabla^{-1})^* h_\nabla=h\) and \(\mu_\mathbb{R}(q_\nabla^{-1}\cdot \nabla)=\Lambda \Xi_{q_\nabla^{-1}\nabla,h}=0\). Thus \([q_\nabla^{-1}\cdot \nabla]_{U(E,h)} \in \mathcal{M}^{hk}\).
We then define \(\iota([\nabla]_{G})=[q_\nabla^{-1} \cdot \nabla ]_{U(E,h)}\). It is well-defined. First, if one chooses another \(\widehat{h}_\nabla=c h_\nabla\) with \(\Lambda \Xi_{\nabla,\widehat{h}_\nabla}=0\), then \(\widehat{h}_\nabla=\widehat{q}^*_\nabla h=(\sqrt{c}q_\nabla)^* h\). Hence \(\widehat{q}^{-1}_\nabla \cdot \nabla=(\sqrt{c}q_\nabla)^{-1}\cdot \nabla=q_\nabla^{-1}\cdot \nabla\). Therefore \([q_\nabla^{-1} \cdot \nabla ]_{U(E,h)}=[\widehat{q}_\nabla^{-1} \cdot \nabla ]_{U(E,h)}\). Second, if \(\nabla'=u \cdot \nabla \in [\nabla]_{G}\) for some \(u \in G\), then \((\nabla',u^* h_\nabla)\) is harmonic. By the uniqueness, \(h_{\nabla'}=c u^* h_\nabla=c u^* q^*_\nabla h =(\sqrt{c}q_\nabla u)^* h\). Thus \(q_{\nabla'}\) and \(\sqrt{c}q_\nabla u\) induce the same Hermitian metric from \(h\). There exists \(v \in U(E,h)\) such that \(q_{\nabla'}=v \sqrt{c}q_\nabla u\). Therefore \(q_{\nabla'}^{-1}\cdot \nabla'=(q^{-1}_{\nabla}v^{-1})\cdot \nabla=v^{-1}\cdot (q^{-1}_\nabla \cdot \nabla)\). It implies \([q_\nabla^{-1} \cdot \nabla ]_{U(E,h)}=[q_{\nabla'}^{-1} \cdot \nabla' ]_{U(E,h)}\). Here we use the convention that \(a\cdot(b\cdot\nabla)=(ba)\cdot\nabla\).
To see the injectivity, if \(\iota([\nabla_1]_G)=\iota([\nabla_2]_G)\), then there exists \(v\in U(E,h)\) such that \(q_{\nabla_2}^{-1}\cdot\nabla_2 = v\cdot(q_{\nabla_1}^{-1}\cdot\nabla_1)= (q_{\nabla_1}^{-1}v)\cdot\nabla_1.\) Therefore, \(\nabla_2=(q_{\nabla_1}^{-1}v q_{\nabla_2})\cdot\nabla_1\) where \(q_{\nabla_1}^{-1}v q_{\nabla_2} \in G\). It follows that \([\nabla_1]_G=[\nabla_2]_G\).
The actual NHYM condition is stronger. Besides the moment-map equation, it also requires the curvature-type conditions \(\Theta^{2,0}=0, \Theta^{0,2}=0\). Thus this map is not expected to be surjective in general. \(\mathcal{M}^s\) should be regarded, at most, as a proper subspace of \(\mathcal{M}^{hk}\) via its image.
The question is therefore whether \(\mathcal{M}^s\) can inherit a hyperkähler structure from the ambient space. The following two examples illustrate both possibilities. The point is whether the subspace is invariant under the ambient hypercomplex structure. In the NHYM case, the additional curvature-type conditions are not preserved by the full hypercomplex structure, and hence \(\mathcal{M}^s\) cannot inherit the ambient hyperkähler structure in this direct way.
In Fujiki’s work on flat bundles over compact Kähler manifolds, the desired finite-dimensional moduli space is obtained as a distinguished subspace of a larger hyperkähler quotient. The hyperkähler structure restricts to this subspace precisely because its tangent spaces are preserved by the three ambient complex structures [[8]].
By contrast, in Dey’s study of the dimensionally reduced Seiberg–Witten equations with a Higgs field, a subset of the equations gives a hyperkähler moduli space, but the moduli space defined by the full system does not inherit this hyperkähler structure. Nevertheless, it still carries a natural symplectic structure, together with an almost complex structure [[23]]. This comparison is relevant for the NHYM problem: even if the natural ambient construction does not yield a hyperkähler metric on the NHYM moduli space, the existence of a natural symplectic structure remains compatible with this general pattern.
Besides the hyperkähler quotient method, another important source of hyperkähler metrics is the twistor construction. Hitchin, Karlhede, Lindström, and Roček explain how suitable twistor data can be used to reconstruct hyperkähler manifolds [[18]]. Feix proved that the cotangent bundle of any real-analytic Kähler manifold carries a canonical hyperkähler metric in a neighborhood of the zero section [[24]]. More recently, Mayrand generalized this local construction: in particular, he showed that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type admits a hyperkähler structure in a neighborhood of its identity section [[25]]. This suggests a different possible approach to the NHYM problem: if the NHYM moduli space near the Hermitian locus could be identified with an appropriate twistor model, then the above construction might provide another route to a hyperkähler metric. We leave this possible approach for future work.