Instantons on Calabi-Yau cones

The Hermitian Yang-Mills equations on certain vector bundles over Calabi-Yau cones can be reduced to a set of matrix equations; in fact, these are Nahm-type equations. The latter can be analysed further by generalising arguments of Donaldson and Kronheimer used in the study of the original Nahm equations. Starting from certain equivariant connections, we show that the full set of instanton equations reduce, with a unique gauge transformation, to the holomorphicity condition alone.


Introduction
Instantons have proven to be interesting both for mathematicians and physicists. Starting from the seminal work [1] by Donaldson, anti-self-dual Yang-Mills connections provided a new topological invariant for four-manifolds. However, the moduli spaces of higher-dimensional instantons are still not fully understood.
From a physics perspective, instantons describe non-perturbative Yang-Mills configurations in various settings [2][3][4]. Focusing, for example, on heterotic string theory and compactifications thereof, the notion of instantons appears naturally in the so-called BPS-equations. In the simplest case, it is necessary to specify a 6-dimensional Calabi-Yau manifold as well as a Hermitian Yang-Mills (HYM) instanton on a gauge bundle over that manifold [5]. However, due to the appearance of phenomenologically problematic moduli, it is physically desirable to relax the strict Calabi-Yau condition to one of more general SU(3)-manifolds (SU(3)-structures with intrinsic torsion), for which the instanton notion needs to be adjusted. Examples of those are nearly Kähler and half-flat manifolds. For further details on so-called flux compactifications see for instance [6,7].
have been reduced to a set of matrix equations. 1 The aim of this paper is to discuss the resulting matrix equations on Calabi-Yau cones over a generic Sasaki-Einstein manifold M 2n+1 , which carries an SU(n)-structure. In particular, the HINP matrix equations conceptually comprise three types of equations: (i) the so-called equivariance condition, (ii) the holomorphicity condition, and (iii) a stability-like condition. Starting form solutions to (i), i.e. decomposing the matrices into irreducible representations of su(n), we show that it suffices to solve (ii) for certain boundary conditions, because (iii) then follows by a unique gauge transformation. The arguments presented are a generalisation of [22][23][24].
Instantons on Calabi-Yau cones and their resolutions have also been studied in [25,26] and, for the particular orbifolds C n /Z n , in [27]. However, their setting and ansatz are different: on the one hand, [25,26] considered instantons on the tangent bundle of a (2n+2)-dimensional Calabi-Yau cone whose structure was largely determined by the 2n-dimensional Einstein-Kähler manifold underlying the Sasaki-Einstein manifold in between. The ansatz for the connection was adapted to the isometry of the Calabi-Yau cone, and the "seed" was the spin connection in the Einstein-Kähler space, which is an instanton. On the other hand, certain gauge backgrounds for heterotic compactifications were constructed in [27] by extending a flat connection on CP n−1 to U(1) and U(n−1)-valued instanton connections on the orbifolds. In contrast, the approach of [19], which is further discussed here, can conceptually take any instanton on the Sasaki-Einstein manifold as a starting point, and the bundle does not need to be the tangent bundle anymore. This paper is organised as follows: the relevant details on Sasaki-Einstein manifolds and the Calabi-Yau cone over it are briefly summarised in Section 2. In addition, the notion of Hermitian Yang-Mills instantons is recalled. The main body is Section 3 where we firstly recapitulate the ansatz for the connection that reduces the HYM equations to matrix equations. The subsequent paragraphs consider the geometry, symmetries and solutions to these equations. Section 4 concludes.

Sasaki-Einstein manifolds
Sasakian geometry can be understood as odd-dimensional analogue of Kähler geometry; in particular, an odd-dimensional manifold M 2n+1 with a Sasakian structure is naturally sandwiched between two different types of Kähler geometry in the neighbouring dimensions 2n and 2n+2.
A Sasaki-Einstein manifold is Sasakian and Einstein simultaneously; thus, the defining property is that the metric cone is Calabi-Yau, i.e. the holonomy group on the cone is reduced to SU(n+1).
Moreover, there exists a canonical connection Γ P on T M 2n+1 which is metric compatible, is an instanton with respect to the SU(n)-structure, and has non-vanishing torsion. 2 The torsion components are given by [18] T 2n+1

Calabi-Yau metric cone
First of all, recall the basic properties of a Calabi-Yau manifold M 2n+2 : as a Calabi-Yau space is Kähler, one has the Kähler form, which is an exact (1, 1)-form on M 2n+2 . In addition, the Calabi-Yau condition enforces the canonical bundle to be trivial, i.e.
Let us compare the choice (2.6) with the "canonical choice" θ j can = e 2j−1 +ie 2j and the canonical complex structure J can θ j can = iθ j can . The conventions used here correspond to J = −J can such that the (1, 0) and (0, 1)-forms are interchanged, which implies that ω(·, ·) = g(J can ·, ·) = − g(·, J can ·) = g(·, J·) is consistent with the above. The reasons for this choice are that we desire a resemblance to the treatment of [22][23][24], while at the same time we treat dt as the (2n+2)th basis 1-form instead of the 0th.

Hermitian Yang-Mills instantons
For the later analysis, the geometric properties of the space of connections and the HYM instanton moduli space over a Kähler manifold are recalled. This brief account is inspired from [29,30].
Space of connections Let M 2n be a (closed) Kähler manifold of dim C (M ) = n and G a compact matrix Lie group. Let P (M 2n , G) be a G-principal bundle over M 2n , A a connection 1-form and F A = dA + A ∧ A the curvature.
Let Int(P ) := P × G G be the group bundle (where G acts via the internal automorphism h → ghg −1 ), let Ad(P ) := P × G g be the Lie algebra bundle (where G acts on g via the adjoint action), and E := P × G F be an associated vector bundle (where the vector space F , the typical fibre, carries a G-representation).
Denote the space of all connections on P by A(P ) and note that all associated bundles E inherit their space of connections A(E) from P . On A(P ) there is a natural action of the gauge group G, i.e. the set of automorphisms on P which are trivial on the base. With one has an identification with the space of global sections of the group bundle. The action is realised via The Lie algebra of the gauge group is then given as g = Γ(M 2n , Ad(P )) , (2.10) and the infinitesimal gauge transformations are given by for χ ∈ Γ(M 2n , Ad(P )) . (2.11) Since A(P ) is an affine space, the tangent space T A A for any A ∈ A(P ) is canonically identified with Ω 1 (M 2n , Ad(P )). Further, assuming G ֒→ U(N ) for some N ∈ N, implies that the trace is an Ad-invariant inner product. Hence, a metric on A(P ) is defined via with ⋆ the Hodge-dual on M 2n . Moreover, the space A(P ) allows for a symplectic structure Since ω is completely base-point independent (on A), ω is in fact a symplectic form. In addition, one can check that X ∧ ω n−1 (n−1)! = ⋆J(X) holds for any X ∈ T A A, where J, the (canonical) complex structure of M 2n , acts only the 1-form part of X. This allows to reformulate the symplectic structure as Moreover, it implies that ω is non-degenerate as ω |A (X 1 , X 2 ) = g |A (X 1 , J(X 2 )) holds for any X 1 , X 2 and any A. Consequently, (A, g, ω) is an infinite-dimensional Riemannian, symplectic manifold, which is equipped with compatible G-action.
Holomorphic structure Next, consider the restriction to connections on E ≃F − − → M which satisfy the so-called holomorphicity condition 14) It is well-known that this condition is equivalent to the existence of a holomorphic structure on E, i.e. a Cauchy-Riemann operator∂ E :=∂ +A 0,1 that satisfies the Leibniz-rule as well as∂ E •∂ E = 0. Thus, having a G-bundle with a holomorphic connection induces a holomorphic G C -bundle. If M 2n is also Calabi-Yau, then the condition (2.14) is equivalent to Ω ∧ F A = 0, where Ω is a holomorphic (n, 0)-form.
Define the subspace of holomorphic connections as This definition employs the underlying complex structure on M 2n . Moreover, one can show that A 1,1 is an infinite-dimensional Kähler space, i.e. g is a Hermitian metric and the symplectic form ω is Kähler. We note that these objects descend from A to A 1,1 simply by restriction. The compatible complex structure J (with ω(·, ·) = g(J ·, ·)) can be read off from (2.12) and (2.13) to be i.e. it is base point independent.
Moment map The space A 1,1 inherits the G-action from A and since it has a symplectic form, i.e. the Kähler form, one can introduce a moment map We see that µ is G-equivariant by construction. Nonetheless, for this to be a moment map of the G-action, one needs to verify the defining property where φ ∈ Γ(M 2n , Ad(P )) an element of the gauge Lie algebra, φ ♮ be the corresponding vector field on A 1,1 and ψ ∈ Ω 1 (M 2n , Ad(P )) a tangent vector at the base point A. Moreover, the duality pairing (·, ·) of g and its dual is defined via the integral over M 2n and the invariant product on g. Generalising the arguments from [29], one can prove that µ is indeed a moment map for the G-action on A 1,1 . Firstly, in the definition of µ only F A is base point dependent, and a standard computation gives (n−1)! . Secondly, the vector field φ ♮ can be read off from (2.11) to be φ ♮ |A = d A φ ∈ Ω 1 (M, Ad(P )). Hence the right-hand side (n−1)! = 0 by Stokes' theorem 3 and dω = 0 one has M tr (d A ψ) ∧ φ ∧ ω n−1 (n−1)! = − M tr ψ ∧ (d A φ) ∧ ω n−1 (n−1)! , and therefore the relation (2.18) holds, i.e. µ is a moment map of the G-action on A 1,1 .
However, one can equally well use the dual map defined by which is equivalent to µ of (2.17) due to Thus, we will no longer explicitly distinguish between µ and µ * .
For Ξ ∈ Centre( g), we know µ −1 (Ξ) ⊂ A 1,1 defines a sub-manifold which allows for a G-action. The quotient is well-defined and, moreover, is a Kähler manifold, as the Kähler form and the complex structure descend from A 1,1 .
We recognise the zero-level set as the Hermitian Yang-Mills moduli space. In other words, the HYM equations consist of the holomorphicity conditions (2.14) together with the so-called stability condition By well-known theorems [31][32][33], a holomorphic vector bundle admits a solution to the HYM equations if and only if these bundles are (poly-)stable in the algebraic geometry sense.
Complex group action As the G-action on A 1,1 preserves the Kähler structure, one can extend to an G C -action on A 1,1 . In other words, the holomorphicity conditions F 0,2 A = 0 are invariant under the action of the complex gauge group the set of all stable points (for a given Ξ). Then, a well-known result (see for example [34]) is Remark A peculiarity arises for holomorphic bundles E over a compact Kähler manifold M 2n with non-empty boundary [35]. Due to the prescription of boundary conditions, the stability condition is automatically satisfied for a unitary connection whose curvature is of type (1, 1). Hence, all points in A 1,1 are stable in this case.
In the following we will consider the HYM equations (2.14) and (2.22) on the non-compact Calabi-Yau cones. For these, the holomorphicity conditions still imply the existence of a holomorphic structure; while the notion of stability is not applicable anymore. Nonetheless, we will continue referring to ω F A = 0 as stability-like condition.

Equivariant instantons
The main focus of this paper lies on the description of the instantons on certain vector bundles E. However, instead of generic connections the set-up will be restricted to connections that arise from an instanton on the Sasaki-Einstein space M 2n+1 by an extension X ∈ Ω 1 (Cyl(M 2n+1 ); End(E)). This extension has to satisfy a certain invariance condition.
The arguments presented in what follows are a generalisation of [22][23][24]: i.e. we generalise from spherically symmetric instantons on vector bundles over C(S 3 ) ∼ = R 4 \{0} with an SU(2)-structure to SU(n+1)-equivariant instantons on vector bundles over C(M 2n+1 ) with an SU(n+1)-structure, where M 2n+1 is an arbitrary Sasaki-Einstein manifold. Analogous to Donaldson and Kronheimer, it will be necessary to consider boundary conditions for the components of the connection 1-form, i.e. for the Yang-Mills fields.

Ansatz
Let us recall the ansatz presented in [19] and explicitly discussed in [21]. Start from any Sasaki-Einstein manifold M 2n+1 , i.e. the manifold carries an SU(n)-structure together with a canonical connection Γ P on the tangent bundle. The metric cone is Calabi-Yau with holonomy SU(n+1), i.e. an integrable SU(n+1)-structure. By conformal equivalence one can consider Cyl(M 2n+1 ).
Consider a complex vector bundle E → Cyl(M 2n+1 ) of rank p which has structure group SU(n+1); in particular, that is a Hermitian vector bundle where F † = −F and tr(F) = 0 hold for the curvature F of a compatible connection. (In the compact case, one would have a vanishing first Chern class.) For example, the (holomorphic) tangent bundle of the Calabi-Yau cone is such a bundle, but one does not have to restrict to this case.
We recall that the connection 1-forms are su(n+1)-valued 1-forms on Cyl(M 2n+1 ) for any connection A on E. The ansatz for a connection is where Γ P is the lifted su(n)-valued connection on E obtained from Γ P , i.e. one essentially has to change the representation on the fibres. Moreover, on a patch U ⊂ Cyl(M 2n+1 ) with the co-frame {eμ} we employ the local description where Xμ |x ∈ End(C p ) for x ∈ U . Usually X 2n+2 is eliminated by a suitable gauge transformation, but there is no harm in not doing so.
The ansatz (3.1) is a generic connection in the sense that the Xμ are base-point dependent, skew-Hermitian, traceless matrices with nontrivial transformation behaviour under change of trivialisation. Hence, any connection A on E can be reached starting from Γ P .
Next, we simplify the ansatz by demanding Xμ = Xμ(t); i.e. not all connections A on E can be reached anymore. Moreover, this demand is only valid in any trivialisation if the following conditions hold (see [21] for further details) The (f ν αµ ) can be interpreted as the matrix elements (ρ * (I α )) ν µ of a (suitably chosen) representation ρ of SU(n) on the typical fibre of T M 2n+1 . The representation theoretic content of (3.4) is that the matrix-valued functions Xμ have to transform in a representation of su(n).
Computing the curvature F A for the ansatz (3.1) together with the equivariance condition (3.4) then yields with F Γ P is the curvature of Γ P , and a, b = 1, . . . , 2n. The HYM instanton equations (2.14) and (2.22) reduce for the ansatz to a set of matrix equations for the Xμ, which are given in [19] (note that X 2n+2 = 0 for this case). Moreover, F Γ P already satisfies the HYM equations, as the Γ P is the lift of an SU(n)-instanton and the corresponding SU(n)-principal bundle is a subbundle in the SU(n + 1)-principal bundle associated to E.
Matrix equations: real basis For completeness, the resulting matrix HINP-equations in the real basis {eμ} are the holomorphicity conditions for j, k = 1, . . . , n and the stability-like condition Matrix equations: complex basis For the intents and purposes here, it is more convenient to switch to the complex basis {θ j ,θ j } defined in (2.6) and introduce 8a) and the adjoint equations thereof. The stability-like condition reads The equivariance conditions for the complex matrices are for j, = 1, . . . , n. For these calculations we have used the choice of structure constants f ν αµ = 0 if µ or ν = 2n + 1 and f b αa ∝ ω ab , see for instance [19][20][21].
Change of trivialisation The remaining nontrivial effects of a change of trivialisation of the bundle E over Cyl(M 2n+1 ) are given by the set of functions {g(t)| g : R → SU(p)} that act as . 4 Due to their adjoint transformation behaviour, the X µ are sometimes called Higgs fields, for example in quiver gauge theories. The inhomogeneous transformation of X 2n+2 is crucial to be able to "gauge away" this connection component. Furthermore, these gauge transformations (and their complexification) will be used to study the solutions of the matrix equations.
Yang-Mills with torsion The instanton equations (on the cone and the cylinder) are equivalently given by where ω is the corresponding (1, 1)-form (dω = 0 on the cone, but dω = 0 on the cylinder). An immediate consequence is that the instanton equation for the integrable SU(n+1)-structure implies the Yang-Mills equations, while this is not true for the SU(n + 1)-structure with torsion. In detail cone: These torsionful Yang-Mills equations (3.12b), which arise in the context of non-integrable Gstructures (with intrinsic torsion), have been studied in the literature before [13][14][15][36][37][38]. In particular, the torsion term does not automatically vanish on instantons because dω contains (2, 1) and (1, 2)-forms. This is, for instance, in contrast to the nearly Kähler case discussed in [39], in which nearly Kähler instantons were found to satisfy the ordinary Yang-Mills equations.
It is known that the appropriate functional for the torsionful Yang-Mills equations comprises the ordinary Yang-Mills functional plus an additional Chern-Simons term which is a gauge-invariant functional. The properties of S YM+T are the following: firstly and most importantly, instanton connections satisfying (3.11) have S YM+T (A) = 0, i.e. the action is finite. Secondly, the stationary points of (3.13) are the vanishing locus of the torsionful Yang-Mills equations (up to boundary terms). For this, we use F A+zΨ = F A + zd A Ψ + 1 2 z 2 Ψ ∧ Ψ for any Ψ ∈ T A A(E) and compute the variation The boundary term would vanish for closed manifolds. In our case, if one assumes M 2n+1 to be closed, the vanishing of the boundary term requires certain assumptions on the fall-off rate of F A for t → ±∞. Moreover, it is interesting to observe that the boundary term in (3.14) vanishes for instanton configurations.

Rewriting the instanton equations
Real equations Returning to the instanton equations for the X-matrices (3.6), the linear terms can be eliminated via a change of coordinates: (3.15c) Note that the exponent 2 − n+1 n 2 vanishes for n = 1 and is strictly positive for any n > 1. The matrix HINP equations (3.6) read now as follows: (3.16c) Complex equations Completely analogous, the change of coordinates for the complex equations is performed via Y j =: e − n+1 n t Y j for j = 1, . . . , n and Y n+1 =: e −2nt Z . We will refer to this set of matrices simply by (Y, Z). In summary, the instanton equations are now comprised by the "complex equations" for j, k = 1, . . . , n , (3.18a) and the "real equation" These equations are reminiscent to the considerations of the instantons on R 4 \{0} of [22][23][24], and, in fact, they reduce to the same system for n = 1, but in general one a Calabi-Yau 2-fold C 2 /Γ. To see this, we recall [28] that all 3-dimensional Sasaki-Einstein spaces are given by S 3 /Γ, where Γ is a finite subgroup of SU(2) (and commutes with U(1) ⊂ SU(2)) which acts freely and isometrically from the left on S 3 ∼ = SU (2).
Moreover, the rescaling has another salient feature: the matrices {Xμ} or ({Y j }, Z) (as well as their derivatives) are bounded (see for instance [23]); in contrast, the original connection components will develop a pole at the origin r = 0. This will become apparent once the boundary conditions are specified. For further details, see Appendix A.1.
In addition, we observe that the exponents on the rescaling (3.15) reflect the torsion components (2.3). The choice of a flat "starting point" Γ = 0 would lead to Nahm-type equations straight away, but solutions to the resulting matrix equations would not interpolate between any (nontrivial) lifted instanton from M 2n+1 and instantons on the Calabi-Yau space C(M 2n+1 ), cf. [14,19].

Real gauge group The full set of instanton equations (3.18) is invariant under the action of the gauge group
wherein the action is defined via Note that only the real equation (3.18b) requires g −1 = g † for it to be gauge invariant. Moreover, one can always find a gauge transformation g ∈ G such that Z g = (Z g ) † (Hermitian) or, equivalently, X g 2n+2 = 0. In summary, these properties follow from (3.10) as the X-matrices are extensions to a connection. However, the gauge group (3.19) still contains a nontrivial centre {g(s)|g(s) = φ(s)1 p×p with φ : R − → U(1)}, such that (3.10) corresponds to the quotient of G by its centre.
Complex gauge group Moreover, the complex equations (3.18a) allow for an action of the complexified gauge group given by The extension to G C -invariance for the holomorphicity conditions exemplifies the generic situation discussed in Section 2.3.
Equivariance condition Actually, one needs to be a bit more careful in considering these equations and their symmetries. Recall that we restrict ourselves to the matrices Xμ which satisfy the equivariance conditions (3.4). However, if the equivariance conditions are not invariant under the gauge transformations (3.20), then a solution obtained by gauge transformation might not be equivariant anymore.
The real gauge transformations can be interpreted as change of basis on the fibres E x ∼ = C p or, more appropriately, change of trivialisation. Since the I α are representations of the generators I α on these fibres, the same transformation acts on them as well. In order to preserve the Lie algebra (3.3), all generators have to transform as for g ∈ G and A = 1, . . . (n + 1) 2 − 1 . (3.23) The same transformation behaviour is adopted when passing to the complexified gauge group. This renders [ I α , X µ ] = f ν αµ X ν into a gauge invariant condition for both G and G C -transformations; the G C -invariance follows as (3.9) does not intertwine {Y k } and {Yk} for any k = 1, . . . , n + 1. Unfortunately, [ I α , X 2n+2 ] = 0 is not gauge invariant, due the inhomogeneous transformation behaviour (3.10) of X 2n+2 . However, the way out is that we will only impose this last condition at the very end, i.e. once we have chosen a gauge transformation g such that X g 2n+2 = 0, the last equivariance condition follows trivially.

Boundary conditions We observe that a trivial solution of (3.16) is
where the (constant) T µ are elements in the Cartan subalgebra of su(p); i.e. the (real) (p−1)dimensional space spanned by the diagonal, traceless matrices with purely imaginary values. From the rescaling (3.15) of the Xμ, it is apparent that these matrices become singular as r → 0 (t → −∞ or s → −∞). Following [23,40], it is appropriate to choose the boundary conditions for X µ to be 5 s → 0 : X µ (s) → 0 for µ = 1, . . . , 2n + 1 and (3.25a) s → −∞ : ∃g 0 ∈ U(p) such that X µ (s) → Ad(g 0 )T µ for µ = 1, . . . , 2n + 1 . One can show [23] that this implies the existence of the limit of X µ for s → 0. Hence, the solutions extend to the interval (−∞, 0], see also Appendix A.1. Thus, we are led to consider (3.16) for matrices X µ (s) over (−∞, 0] with one remaining boundary condition: ∃g 0 ∈ U(p) such that ∀µ = 1, . . . , 2n + 1 : lim Moreover, since one has first order differential equations it suffices to impose this one boundary condition, here at s = −∞. Thus, the values of Y k at s = 0 are completely determined by the solution. Following [23], we observe that (3.18a) implies that Y k (s) lies entirely in a single adjoint orbit O(k) of the complex group G C , for each k = 1, . . . , n. Next, assuming that T k = 1 2 (T 2k − iT 2k−1 ) for k = 1, . . . , n is a regular tuple in the Cartan subalgebra of g C in the sense of [23](that is the joint stabiliser of the T µ in SU(p) is the maximal torus), one obtains that Note that the case n = 1 is reminiscent to the instantons with poles considered in [23].

Geometric structure
Space of connections under consideration Consider the space of su(n+1)-valued connections A(E) in which any element can be parameterised as in (3.1). Due to the ansatz of Section 3.1, we restrict ourselves to the subspace A equi (E) ⊂ A(E) of connections which satisfy (3.4). Specialising the considerations of Section 2.3, we will now establish certain (formal) geometric structures.
Kähler structure The first step is to establish a Kähler structure on A equi (E). Since A equi (E) descends from the space of all connection A(E), one can simply obtain the geometric structures by restriction. A tangent vector at a point A ∈ A equi (E) is defined by the linearisation of (3.8) for paths y j : R → su(p). Their gauge transformations are y j → y g j := Ad(g)y j for j = 1, . . . , n + 1 .

(3.29)
Taking the generic expressions for the metric (2.12) and the symplectic structure (2.13), we can specialise to the case at hand by transition to the cylinder and neglecting the volume integral of M 2n+1 . Thus, for a metric on A equi we obtain Similarly, the symplectic form reads as Moreover, a complex structure J on A(E) equiv has been given in (2.16). Keeping in mind that (2.6) implies J = −J can , we obtain As before, the symplectic form ω and the metric g are compatible, i.e g(J ·, ·) = ω(·, ·). We note that both structures are gauge-invariant by construction.

Moment map
The subspace of holomorphic connections A 1,1 equi (E) ⊂ A equi (E) is defined by the condition (3.18a). This condition only restricts the allowed endmorphism-valued 1-forms, because Γ P is already a (1, 1)-type connection, since it is an HYM-instanton. Again, the metric g and Kähler form ω descend to A 1,1 equi (E) from the corresponding objects on A equi (E). Moreover, on the Kähler space A 1,1 equi (E), one defines a moment map µ : where G 0 is the corresponding framed gauge group. That is It is an important realisation that on the non-compact Calabi-Yau cone (and the conformally equivalent cylinder) one has to compensate the appearing boundary terms in Stokes' theorem by the transition to the framed gauge transformations. The details of the proof that ( The part of instanton moduli space that is connected with the lift Γ P (in the sense of our ansatz (3.1)) is then readily obtained by the Kähler quotient Stable points Alternatively, one can describe this part of the moduli space via the stable points where the tuple (Y, Z) is obtained from X via complex linear combinations and rescaling as before.
The moduli space arises then by taking the G C 0 -quotient We argue in the next couple of paragraphs that it suffices to solve the complex equations (3.18a), because the solution to the real equation (3.18b) follows from a framed complex gauge transformation. More precisely: for every point in A 1,1 equi (E) there exists a unique point in the complex gauge orbit such that the real equation is satisfied. In other words, every point in A 1,1 equi (E) is stable.

Solutions to matrix equations
Solutions to complex equation In the spirit of [22], one can also understand the complex equations as being locally trivial. That is, take (3.22) and demand the gauge transformed Z to be zero From the holomorphicity equations (3.18a) one obtains for j, k = 1, . . . , n and g 0 is a constant gauge transformation 6 . Consequently, the general local solution of the complex equations (3.18a) is for any g ∈ G C . A solution to the commutator constraint is choosing T k for k = 1, . . . , n as elements of the Cartan subalgebra of the Lie algebra gl(p, C), which are all diagonal (complex) p×p matrices.
Solution to the real equation In any case, one can in principle solve the complex equations; now, the real equation (3.18b) needs to be solved as well. Following the ideas of [22], the considerations are split in two steps: (i) a variational description and (ii) a differential inequality. We provide the details of (i) in this paragraph, while we postpone the details of (ii) to the Appendix A.4. Let us recall that the complete set of instanton equations is gauge-invariant under G. Thus, define for each g ∈ G C the map The quotient GL(p, C)/U(p) can be identified with the set of positive, self-adjoint p × p matrices. Then, fix a tuple (Y, Z) and define the functional L ǫ [g] for g The variation then leads to Following [22], the potential V (h) = 2λ n (s) n k=1 tr hT k h −1 T † k is positive 7 , implying that for any boundary values h − , h + ∈ GL(p, C)/U(p) there exists a continuous path 8 which is smooth in I ǫ = − 1 ǫ , −ǫ and minimising the functional. Hence, for any choice of gauge transformation g such that g † g = h one has that ({T k } k=1,...,n , 0) g = {Ad(g)T k } k=1,...,n , − 1 2 ( d ds g)g −1 satisfies the real equation in I ǫ for any 0 < ǫ < 1. From now on, we restrict the attention to h + = h − = 1, i.e. h is framed.
The uniqueness of the solution h on each interval I ǫ and the existence of the limit h ∞ for ǫ → 0 follows from the aforementioned differential inequality similar to [22] and the discussion of [23,Lemma 3.17]. The details are presented in Appendix A.4. The relevant (framed) gauge transformation is then simply given by g = √ h ∞ . 9 However, we need to emphasise two crucial points. Firstly, the construction of a solution for the limit ǫ → 0 relies manifestly on the use of the boundary conditions (3.26), and the fact that these give rise to a (constant) solution of both the complex equations and the real equation. Secondly, the corresponding complex gauge transformation g = g(h ∞ ) is only determined up to unitary gauge transformations, i.e. it is not unique. This ambiguity in the choice of g can be removed, when we recall that a G gauge transformation suffices to eliminate X 2n+2 . Hence, one can demand that the gauge-transformed system (Y g , Z g ) of a solution (Y, Z) satisfies Z g = (Z g ) † . This fixes g = g(h) uniquely, see also Appendix A.4.4 for further details.
Result In summary, it is sufficient to search for solutions (Y ′ , Z ′ ) of the complex equations (3.18a) on the interval (−∞, 0] such that the boundary conditions (3.26) are satisfied. Then one has the existence of a unique complex gauge transformation g such that (ii) Z is Hermitian (i.e. X 2n+2 = 0) and (iii) g is bounded and framed.
In other words, it suffices to solve the complex equations subject to some boundary conditions and the real equation will be satisfied automatically.
Moreover, the above indicates that any point in A 1,1 equi is stable, which we recall to be exactly the condition that every complex gauge orbit intersects µ −1 (0). We believe that this circumstance holds because we restricted ourselves to the space of equivariant connections. The benefit is then, that one, in principle, only has to show the solvability of the holomorphicity conditions in order to solve the instanton (matrix) equations. Nevertheless, one still has to find an ansatz that satisfies the equivariance conditions (3.4).

Further directions
Before concluding we can further exploit the results collected so far as well as illustrate another viewpoint of the HINP matrix equations.
Relation to coadjoint orbits Let us denote by M n (E) the moduli space of solutions to the complex and real equations satisfying the boundary conditions (3.26) together with the equivariance condition. From the considerations above, we can establish the following map 10 where O T k denotes the adjoint orbit of T k in gl(p, C). Analogous to [23], this map is a bijection due to the construction of the local solution (3.40) and the uniqueness of the corresponding solution of the real and complex equations. Moreover, one knows that the orbit of an element T k of the Cartan subalgebra is of the form GL(p, C)/Stab(T k ). The product of coadjoint orbits in (3.47) is a complex symplectic manifold of complex dimension n dim(GL(p, C)) − n j=1 dim(Stab(T j )). Each orbit is equipped with the so-called Kirillov-Kostant-Souriau symplectic form and the product thereof gives the symplectic structure on the total space. In addition, the bijection above preserves the holomorphic symplectic structure.
Relation to quiver representations The HINP matrix equations can be seen to define quiver representations, depending on the chosen SU(n+1)-representation on the typical fibre C p . Then, by the employed ansatz, we decompose this representation with respect to SU(n) into where C nw carries a n w -dimensional irreducible SU(n)-representation. More explicitly, w should be understood as pair of labels: let φ label the irreducible SU(n)-representations and recall that the centraliser of SU(n) inside SU(n+1) is a U(1). Then each representation space C nw carries also a U(1)-representation characterised by a "charge" q. Therefore, the decomposition is labelled by pairs w = (φ, q).
As a consequence, the equivariance condition (3.4) dictates the decomposition of the X µmatrices into homomorphisms The quiver representation then arises as follows: the set Q 0 of vertices is the set {C nw |w ∈ J} of vector spaces and the set Q 1 of arrows is given by the non-vanishing homomorphisms {(X µ ) w,w ′ |w, w ′ ∈ J , µ = 1, . . . , 2n + 1}.
The instanton equations (or HINP equations) then lead to relations on the quiver representation. Examples for the arising quiver diagrams as well as their relations for the case n = 1 and M 3 = S 3 can be found in [41] and for n = 2 and M 5 = S 5 in [42]. To study the representations of a quiver one would rather use the constructions of [41,42], instead of the ansatz employed here. Because once the bundle E and the action of SU(n+1) on the fibres is chosen, there is no freedom to change the quiver representation anymore.

Conclusions
It is known that the instanton moduli space over a Kähler manifold is a Kähler space. Therefore, also the moduli space of certain invariant connections should inherit this property. The overall situation remains unknown.
In the ansatz presented here, we restricted ourselves to a subset of all possible connections by, firstly, imposing an equivariance condition and simplifying to t-dependence only and by, secondly, fixing an instanton Γ P as a starting point. Hence, by this construction one can only reach a particular part of the full instanton moduli space by the solutions of the HINP matrix equations.
The arguments presented in this paper show that the reduced HINP matrix equations can be treated similarly to the Nahm-equations of SU(2) monopoles. As a consequence, one gains local solvability of the holomorphicity conditions together with the fact that any solution can be uniquely gauge-transformed into a solution of the stability-like condition. Moreover, the structure of the (framed) moduli space shares, at least locally, all features of a Kähler space due to the Kähler quotient construction or the GIT quotient.
It is of interest to extend the ansatz presented here from cones to their smooth resolutions as in [19,26], and to consider quiver gauge theories which can be associated to Calabi-Yau cones along the lines of [41,42].

A Details
In this appendix, we provide the proofs of the statements made in Sections 3.2-3.4. Although the steps are similar to those performed in [22][23][24], we believe that these are necessary because the HINP equations are generalisations of the Nahm equations.

A.1 Boundedness of rescaled matrices
Recall the boundary conditions (3.26) for the original matrices t → +∞ : Evaluating the asymptotic behaviour for t → +∞ of (3.6), one finds the leading behaviour of (the real and imaginary part) of each matrix element to be because the commutator terms vanish faster than linear order. These results imply the following: (i) The rescaled matrices X µ of (3.15) are bounded for s → 0.
(iii) The derivatives d dt e n+1 n t X a and d dt e 2nt X 2n+1 are integrable, which follows by the use of the equations (3.6).
In conclusion, the Xμ as well as their derivatives are bounded.

A.2 Well-defined moment map
We need to prove (2.18) for µ defined in (3.33); recall that µ(A) := F A ∧ ω n−1 (n−1)! and we identified µ * with µ. Moreover, it is crucial to use the closed Kähler 2-form from the cone, i.e. ω = e 2t ω on the cylinder. We will work with the original connection components Y k defined in (3.7).
For the left-hand-side we proceed as follows: Let φ ∈ g 0 and Ψ = Ψ k θ k − Ψ † kθ k be a tangent vector at A. The duality pairing of Lie-and dual Lie-algebra is realised by the integration over the cylinder and the subsequent invariant product on u(p).
Hence, for the dual moment map one can neglect the volume integral over M 2n+1 and the dual pairing is defined via the first integral over t.
The compute the right-hand-side of (2.18) we need to take a step back and derive the symplectic form on A from (2.13) as follows Again, we can drop the volume of the Sasaki-Einstein space. Next, we need the infinitesimal gauge transformation generated by an (framed) Lie-algebra element φ. From (3.20) we obtain which then brings us to A close inspection of the boundary term reveals that vanishes provided lim t→±∞ φ(t) = 0 i.e. the map defined in (3.33) is a moment map for the action of the framed gauge group G 0 = {g(t)|g : R → U(p), s.t. lim t→±∞ g(t) = 1}.

A.3 Notation
We need to introduce some notation, which is relevant for the proofs later.
∂,∂-operators Following [22], we define the following ∂,∂-operators on C p -valued functions f on R − which is well-defined. The claim is that the inequalities hold in a weak sense.
Now, we consider the diagonal elements where we used Similarly, one derives Then, one proceeds To get the estimate for Φ(h) = 2t1 take a = 1 and use 1 − e 2(t 1 −t b ) − 1 − e −2(t 1 −t b ) < 0 as t1 > t b for all b > 1. Then Similarly, the estimate for Φ(h −1 ) is obtained by taking a = p and 1 − e 2(tp−t b ) − 1 − e −2(tp−t b ) > 0 for all b < p. Then Thus, the claim (A.15) holds.
Then h 1 = h 2 in I ǫ .

A.4.3 Boundedness
Next, we need to show the boundedness of µ(Y, Z). The only critical term is λ n (s), which diverges for s → 0. However, it is straight forward to derive the pole structure of the gauge transformed operator Υ to be (A. 25) But recall that we will consider framed gauge transformation, i.e. h = 1 at the boundaries, and Y(s = 0) are elements of a Cartan subalgebra. Hence, the potential pole vanishes for any gauge transformation once the correct boundary conditions (3.26) are imposed. Thus, µ(Y, Z) is bounded.
A.4.4 Limit ǫ → 0 Finally, we need to show that the limit ǫ → 0 exists, for which we follow [23,24]. Let (Y, Z) be any solution of the complex equation, then for each ǫ > 0 there exists a unique complex gauge transformation g ǫ such that (Y gǫ , Z gǫ ) satisfies the real equation in I ǫ . Associate h ǫ = g † ǫ g ǫ . We start by constructing a solution ( Y, Z) of the complex equations with the properties where (T j , T n+1 ) correspond to the complex linear combinations of the T µ of the boundary condition (3.26), i.e. they lie in a Cartan subalgebra of su(n + 1). The τ j are arbitrary points in the complex orbits O(T j ), because we know that the boundary values at s → 0 are in gauge orbits of the T j .
The existence of such a solution follows from the local triviality of the complex equations. Note that this solution is constant in (− 1 ǫ , −1) and µ( Y, Z) = 0 for − 1 ǫ < s < −1. The claim then is: Starting from ( Y, Z) as above, for each ǫ > 0 there exists a unique gauge transformation g ǫ such that (i) ( Y gǫ , Z gǫ ) satisfies the real equation everywhere in I ǫ , (ii) ( Y gǫ , Z gǫ ) has the correct boundary conditions (3.26), (iii) g = 1 at the boundaries and Z gǫ is Hermitian, Thus, by the uniform bound, one has the existence a C ∞ limit h ∞ := lim ǫ→0 h ǫ such that g ∞ := √ h ∞ has all desired properties on the negative half-line.
Proof: The existence and the uniqueness of such a gǫ follows from the above. Using the differential inequalities (A.15) and the boundedness of µ we derive at independent of ǫ. This uniform bound leads to the existence of the limit ǫ → 0 of hǫ.