Sasakian quiver gauge theory on the Aloff-Wallach space $X_{1,1}$

We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form $M^d \times X_{1,1}$ with d-dimensional Riemannian manifold $M^d$ and the Aloff-Wallach space $X_{1,1}$= SU(3)/U(1) endowed with its Sasaki-Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff-Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly and then compare the results with the quiver gauge theory on $Q_3$ =SU(3)/(U(1) x U(1)), the leaf space underlying the Sasaki-Einstein manifold $X_{1,1}$. Moreover, we study instanton solutions on the metric cone $C(X_{1,1})$ by evaluating the Hermitian Yang-Mills equation. We briefly discuss some features of the moduli space thereof, following the main ideas of a treatment of Hermitian Yang-Mills instantons on cones over generic Sasaki-Einstein manifolds in the literature.


Introduction
The emergence of extra dimensions in string theory and the typical ansatz for compactifications make a detailed understanding of higher-dimensional gauge theories desirable. Inspired by the seminal investigation of four-dimensional manifolds by self-dual connections [1], generalized self-duality equations and instantons in higher dimensions have been studied [2,3,4,5]. Their significance in physics is evident in heterotic string theory where an instanton equation is part of the BPS equations [5,6].
Often the manifolds modelling the internal degrees of freedom are chosen as coset spaces G/H, and dimensional reduction of the gauge theory on M d × G/H to a theory on M d is known as coset space dimensional reduction [7]. On those spaces one can demand G-equivariance of the vector bundles the gauge connection takes values in, and this equivariant dimensional reduction yields systematic restrictions which can be depicted as quiver diagrams, i.e. directed graphs. A detailed mathematical treatment for Kähler manifolds can be found in [8,9] and short physical reviews are given e.g. in [10,11].
These quiver gauge theories have been studied for the Kähler cosets CP 1 [9,12,13,14], CP 1 × CP 1 [15], and SU(3)/H [14,16]. The odd-dimensional counterparts of Kähler spaces are Sasaki manifolds [17], and among them Sasaki-Einstein manifolds [18] are of particular interest for compactifications in string theory because, by definition, their metric cones are Calabi-Yau [19,20]. In the literature, Sasakian quiver gauge theory has been studied on the orbifold S 3 /Γ [22], on orbifolds S 5 /Z q+1 of the five-sphere [23] and on the space T 1,1 [24], the base space of the conifold. The five-dimensional Sasaki-Einstein coset spaces as well as the new examples [25,26] are of interest for versions of the AdS/CFT correspondence. In dimension seven, one can encounter the following typical examples: the seven-sphere S 7 , the Aloff-Wallach space X 1,1 [27], and also a new class of spaces constructed in [26]. They could play a role for compactifications of 11-dimensional supergravity. In this article we will consider the Sasakian quiver gauge theory on the Aloff-Wallach space X 1,1 . The mathematical properties of the generic Aloff-Wallach spaces X k,l [27] -basically their G 2 and Spin(7) structure and, for the special case of X 1,1 , being Sasaki-Einstein and even 3-Sasakian -are well known [28,29]. Moreover, instanton solutions on these spaces have been constructed in [30,31]. Due to the special geometry, more precisely the existence of Killing spinors, they have been intensively studied in M-theory or supergravity [32].
This article is organized as follows: Section 2 reviews the geometry of the space X 1,1 , providing local coordinates, the structure equations, the Sasaki-Einstein properties as well as a comment on the closely related Kähler space Q 3 := SU(3)/U(1) × U(1). The subsequent section begins with a short review of equivariant vector bundles over homogeneous spaces and the arising quiver diagrams. Then we study the equivariant gauge theory on X 1,1 , placing the focus on the evaluation of the equivariance condition, already known from [30,31], in terms of quiver diagrams. We discuss the general construction for the quiver diagrams associated to X 1,1 and clarify it by considering three examples with a small number of vertices. The resulting Yang-Mills functional of the equivariant gauge theory is provided, and the reduction to the quiver gauge theory on Q 3 is discussed in the last part of Section 3. Subsequently, we study instanton solutions of the quiver gauge theory by evaluating the Hermitian Yang-Mills equations on the metric cone C (X 1,1 ). We briefly sketch the techniques used by Donaldson [33] and Kronheimer [34] for the discussion of the Nahm equations and the application of those methods to Hermitian Yang-Mills instantons on generic Calabi-Yau cones [35]. We discuss the modifications that appear in our setup, due to using a different instanton connection in the ansatz for the gauge connection, in comparison with the general results of [35]. The appendix provides some technical details.
2 Geometry of the Aloff-Wallach space X 1,1 In this section we review the geometric properties of the Aloff-Wallach space X 1,1 and its metric cone C (X 1,1 ) which are necessary for the discussion in this article. Among the huge number of articles on the geometry of Aloff-Wallach spaces X k,l [27], we follow the exposition given in the article [30], in which G 2 and Spin(7)-instantons on the spaces have been considered. In particular, we employ their choice of SU(3) generators, structure constants and the ansatz for the gauge connections. Since we are aiming only at the Sasaki-Einstein structure of X 1,1 , we will not consider general spaces X k,l . For details on theses structures we refer to [30] and the references therein.

Local coordinates and structure equations
The Aloff-Wallach spaces [27], denoted as X k,l , for coprime integers k and l, are defined as quotients where the embedding of elements h ∈ U (1) k,l into SU(3) is given by It is known that the homogeneous space X 1,1 is not only Sasaki-Einstein but moreover admits a 3-Sasakian structure 1 . Due to [36] a homogeneous 3-Sasakian manifold different from a sphere is a SO(3) ∼ = SU(2)/Z 2 bundle over a quaternionic Kähler manifold; in the case of X 1,1 the underlying space is CP 2 . Using this result, we can construct local coordinates 2 by starting from a local section of the fibration SU(3) → CP 2 , as it can be found e.g. in [16,21]. Given a local patch and a local section of the bundle SU(3) → CP 2 is given by Furthermore, an arbitrary element g of SU(2) can be written as where z andz are stereographic coordinates on CP 1 . Putting both expressions (2.4) and (2.6) together, one gets a local section of the bundle SU(3) −→ X 1,1 as (y 1 , y 2 , z, ϕ) −→Ṽ : Hence, the manifold can be locally described by the coordinates {y 1 ,ȳ 1 , y 2 ,ȳ 2 , z,z, ϕ}, and the Maurer-Cartan form provides SU(3) left-invariant 1-forms Θ α and e i , defined by Here we have defined the forms such that the generators of SU(3) (see Appendix A.1) coincide with those from [30]. Due to the flatness of the connection, dA 0 + A 0 ∧ A 0 = 0, one obtains the structure equations together with the complex conjugated equations for Θᾱ, α = 1, 2, 3. By construction, the group U(1) k,l in the definition (2.1) is generated by I 8 in (A.1), and the remaining group U(1) inside X 1,1 is associated to I 7 and the local coordinate ϕ.

Sasaki-Einstein structure
Following [30], the Einstein metric is chosen to be 10) and the Sasaki structure is defined by declaring the forms Θ α to be holomorphic,JΘ α = i Θ α . Herẽ J denotes the complex structure of the leaf space orthogonal to the contact direction e 7 . Then the fundamental form ω associated to it satisfies the Sasaki condition 11) which implies that ω is the Kähler form of the leaf space. The metric cone C (X 1,1 ) has by definition the metric where one has defined a fourth holomorphic form (2.13) Equation (2.12) establishes the correspondence between the metric cone and the conformally equivalent cylinder 3 . The definition of Θ 4 yields an integrable complex structure J on the metric cone whose fundamental form Ω (X, Y ) := g (JX, Y ) is then given by (2.14) Due to the Sasaki condition de 7 = 2ω this form is closed and the cone C (X 1,1 ), thus, carries a Kähler structure. For the cone to be Calabi-Yau, the holonomy U(4) of the Kähler manifold must be reduced further to SU(4), which is ensured by the closure of the 4-form [30] Consequently, the geometric structure is that of a Calabi-Yau 4-fold, which implies the Sasaki-Einstein structure of X 1,1 . As a Sasakian manifold, X 1,1 is a U(1)-bundle over an underlying Kähler manifold, namely the leaf space of the foliation along the Reeb vector field, with fundamental form ω. The Kähler manifold underlying X 1,1 is denoted as Q 3 or F 3 [16, 21] From the (local) section in (2.7) one has locally Q 3 ∼ = CP 2 × CP 1 , and the space is described by the coordinates {y 1 ,ȳ 1 , y 2 ,ȳ 2 , z}.
3 Quiver gauge theory on X 1,1 Quiver diagrams are a powerful tool in representation theory, and this motivates their appearance in gauge theories, where the field content can be described by these directed graphs. In this section we will demonstrate the basic features of quiver gauge theories by considering them on the spaces X 1,1 and Q 3 . We start the survey with a brief review of how quiver diagrams arise in the context of gauge theories 4 on reductive homogeneous spaces G/H .

Preliminaries of quiver gauge theory
The condition generating the quiver diagrams, which we will usually refer to as equivariance condition, can be understood from two point of views: On the one hand, one could consider equivariant vector bundles in a rigorous algebraic fashion as it is done in [8,9], purely based on the representation theory of the Lie algebras involved. On the other hand, the equivariance condition occurs quite naturally in the context of instanton studies, e.g. [38,39,40,41,42], as invariance condition on gauge connections on reductive homogeneous spaces G/H.

Equivariant vector bundles
We sketch the basics of equivariant vector bundles and their relation to quiver gauge theories, following roughly [8,10]. For the application of this approach we refer also to the examples in [16,23]. Let G/H be a Riemannian coset space modelling the internal degrees of freedom, M d a d-dimensional Riemannian manifold, and let π : E → M d × G/H be a Hermitian vector bundle 5 of rank k, i.e. a vector bundle with structure group U(k). Suppose that the Lie group G acts trivially on M d and in the usual way on the coset space. Then the bundle is called G-equivariant if the action of G on the base space and on the total space, respectively, commutes with the projection map π and induces isomorphisms among the fibers E x ≃ C k . By restriction and induction of bundles, E = G × H E, G-equivariant bundles E → M d × G/H are in one-to-one correspondence with H-equivariant bundles E → M d [8].
Since the action of the closed subgroup H on the base space is trivial, the equivariance of the bundle implies that the fibers must carry representations of H. We assume that these H-representations stem from the restriction of an irreducible 6 G-representation D which decomposes under restriction to H as follows where the ρ i 's are irreducible H-representations. This yields an isotopical decomposition 7 of the vector bundle E as a Whitney sum in the very same way and induces a breaking of the generic structure group U (k) of the bundle to The action of the entire group G on the decomposition (3.2) connects different representations ρ i , i.e. it leads to homomorphisms from Hom C k i , C k j . In this way, the fibres of the G-equivariant bundle are representations of a quiver 8 (Q 0 , Q 1 ), where Q 0 denotes the set of vertices and Q 1 the set of arrows. Each vertex v i ∈ Q 0 carries a vector space isomorphic to C k i with an H-representation, and the arrows are represented by linear maps among these spaces. The entire G-equivariant bundle thus carries a representation of the quiver, and this contruction is called a quiver bundle. Since the allowed arrows of the quiver diagram arise from the commutation relations of the generators with the elements of the subalgebra h, this approach is entirely based on the representation theory of h and g, and it can be realized using (parts of) the weight diagram of the Lie algebra g.

Invariant gauge connections
The equivariance condition leading to the quiver diagrams also occurs naturally when studying instanton solutions of invariant gauge connections on reductive homogeneous spaces, e.g. in [30,41,42]. Let G/H be a reductive homogeneous space with the Ad(H)-invariant splitting where the generators satisfy the space m can be identified with the tangent space of G/H. Let e µ be the 1-forms dual to the generators I µ , which obey the structure equation where Γ µ ν are the connection 1-forms describing a (metric) connection Γ on the homogeneous space, and T µ is its torsion. Due to a known result from differential geometry [45] and following the approach used for example in [41], we can express a G-invariant connection A on the homogeneous space as where the skew-hermitian matrices X a , the Higgs fields, describe the endomorphism part. The connection 9 Γ := I j ⊗e j takes values entirely in the vertical component h and is obtained by declaring the torsion to be T (X, is then given by For the connection to be G-invariant, terms containing the mixed 2-forms e j ∧ e a must not occur, so that one obtains -assuming that the last term in (3.8) does not yield incompatible contributions 10 -the equivariance condition [41,45] [ Thus the equivariance forces the endomorphisms X a to act (with respect to the adjoint action) on the fibres of the bundle as the generators I a in (3.5) do.
Construction procedure Based on the outline above, we can construct an equivariant gauge connection and the corresponding quiver bundle for X 1,1 = SU(3)/U(1) 1,1 in the following way. Let be a decomposition of the representations on the fibres in (m + 1) terms, which yields the breaking of the structure group (3.3) and the isotopical decomposition as in (3.2). Since the irreducible representations ρ i of the abelian subgroup H = U(1) 1,1 are 1-dimensional, the group H acts as on the vectors (3.10). The constants ζ i can be obtained from an irreducible representation of the U(1) 1,1 -generator on an (m + 1)-dimensional vector space. This fact and the way how the quiver diagrams arise motivate to consider the gauge connection as a block matrix of size (m + 1) 2 , whose structure is determined by the (m + 1)-dimensional G-representation in which the entries are (implicity) replaced by endomorphisms. By construction and due to the equivariance condition (3.9), the quiver diagram is then based on (parts of) the underlying weight diagram of the chosen G-representation. If the subgroup H is a maximal torus, the quiver coincides with the weight diagram because all Cartan generators occur as operators I j in (3.9). For smaller subgroups there might be degeneracies as double arrows in the diagram, while larger groups require a collapsing of vertices in the weight diagram along the action of the ladder operators of h as it is done, for instance, in [11,16]. We will clarify this procedure for the abelian subgroup H = U(1) 1,1 in the following.

Equivariance condition and quiver diagrams of X 1,1
The aforementioned approach is now applied to the space X 1,1 . Following the outline above and according to (3.7), we write an SU (3) where A is a connection on M d . Moreover, we have defined complex endomorphisms In terms of the structure constants (A.3) the field strength of the connection A is given by [30] (3.14) Following some notation in the literature, e.g. in [16], we call φ (α) := Yᾱ for α = 1, 2, 3, and X 7 (3.15) the Higgs fields and set 11 The equivariance condition (3.9), equivalent to the vanishing of the terms in the last line of (3.14), then reads Consequently, the endomorphisms φ (1) and φ (2) † will have the same block form and the form of φ (3) coincides with that of X 7 , but their entries are still arbitrary and not related to each other. The commutation relations (3.17) provide the action of the Higgs fields on the quantum numbers (ν 7 , ν 8 ) associated to the two Cartan generatorsÎ 7 andÎ 8 of SU(3) Since the quantum number ν 7 does not enter the equivariance condition, it is reasonable 12 to label the vertices in the quiver diagram only by the number ν 8 , so that one obtains effectively a modified version of the holomorphic chain [9]: a diagram consisting of double arrows between adjacent vertices and double loops at each vertex, where the black two headed arrows denote the contributions by φ (1) and φ (2) † , while the endomorphisms φ (3) and X 7 are represented by the blue two headed loops 13 . Here, the integer p denotes the highest weight (with respect to ν 8 ) of the representation D. The endomorphism part of the invariant connection associated to this modified holomorphic chain of length m + 1 is then given by where we have defined the abbreviations for j = 0, 1, . . . , m, and the indices label the tail of the arrow. The remaining contribution to the invariant connection (3.12) is given by the diagonal parts

Examples
We consider three explicit examples of SU(3) representations and the quiver diagrams associated to them.
We skip the explicit index structure of the invariant gauge connection which can be read from the quiver diagram, Figure 2, and provide only the result for the modified holomorphic chain It is interesting to compare this block matrix of size 3 × 3 with that of the adjoint representation in the last example, which -on the level of the modified holomorphic chain -only differs in the occurring quantum numbers and, thus, the connection Γ.
Adjoint representation 8 The U(1) 1,1 -generator in the adjoint representation is given byÎ 8 = diag (3, 3, 0, 0, 0, 0, −3, −3) and the weight diagram is a hexagon with two degenerated points at the origin 15 . The Higgs fields must thus have the shape  31) and the quiver diagram Figure 3 contains a large number of arrows. The identification leading to the modified holomorphic chain yields as connection Figure 3: Quiver diagram for the adjoint representation 8 of SU(3). Note that due to the degeneracy of (0, 0) each arrow involving the origin must be counted twice (depicted as arrows consisting of two lines), i.e. there are, for instance, four arrows between (0, 0) and (1, −3) etc.
As mentioned before, this modified holomorphic chain of length 3 is different from that of the six-dimensional representation, (3.30), only due to the quantum numbers that appear.
The huge number of arrows in the last two examples have shown that it is advantageous to use only the relevant quantum number ν 8 rather than the entire weight diagram of G, but for comparisons with Q 3 the latter description is also useful. The occurrence of degeneracies in the entire weight diagram of SU(3) due to the weaker equivariance condition is similar [15,24] to the case of the five-dimenional Sasaki-Einstein manifold T 1,1 := (SU(2) × SU(2)) /U(1) in comparison with its underlying manifold CP 1 × CP 1 .

Dimensional reduction of the Yang-Mills action
In the previous section we have completely characterized the form of a G-invariant gauge connection by applying the rules (3.18) in the weight diagram and in terms of the results (3.20) and (3.22). Given such a gauge connection A on M d ×X 1,1 with field strength F, we now determine its standard Yang-Mills action yielding the usual Yang-Mills Lagrangian 16 where we denoteĝ := det g X 1,1 det g M d . Using the Sasaki-Einstein metric (2.10), g X 1,1 αβ = 1 2 δ αβ and g X 1,1 77 = 1, and the field strength components from (3.14), one obtains as Lagrangian Here, we have defined the covariant derivatives D µ φ (α) := dφ (α) + A, φ (α) µ for α = 1, 2, 3 and D µ X 7 := (dX 7 + [A, X 7 ]) µ , the field strength F µν := (dA + A ∧ A) µν and we write |X| 2 := XX † . Since the fields φ (α) and X 7 are assumed to be independent from internal coordinates of X 1,1 (due to equivariance), the additional dimensions can be integrated out easily, which yields only a prefactor vol (X 1,1 ) for the dimensional reduction of the Lagrangian. In this way, one obtains from a pure Yang-Mills theory on M d × X 1,1 a Yang-Mills-Higgs action on M d , where the endomorphisms φ (a) and X 7 constitute a non-trivial potential provided by the internal geometry of X 1,1 .

Reduction to quiver gauge theory on Q 3
The equivariance condition and the examples of the quiver diagrams in the previous section have shown that the quiver gauge theory on X 1,1 depends on only one of the two quantum numbers of SU (3). This yields effectively a modified holomorphic chain as quiver diagram or, considered in the original weight diagram of SU(3), a diagram with multiple arrows and degeneracies. As mentioned in the discussion of the Sasaki-Einstein structure on X 1,1 in Section 2.2, the space is a U(1)-bundle over the (Kähler) space Q 3 , so that it is natural to consider the reduction from the gauge theory on X 1,1 to that on Q 3 by removing the contact direction as a degree of freedom. Since we then divide by a Cartan subalgebra, the quiver diagram is simply the weight diagram of SU(3) without the degeneracies which have been caused by the weaker conditions on X 1,1 . This reduction can be performed by setting the terms containing e 7 ∧ Θ α or e 7 ∧ Θᾱ in the field strength (3.14) to zero. This provides the additional equivariance conditions (2) , and For the reduction to Q 3 , the field X 7 must be proportional to I 7 and setting X 7 = I 7 fixes the action of the Higgs fields to be This, indeed, requires the quiver diagrams in Figure 4 to coincide with the weight diagrams of the chosen representations and yields the results 17 from [16,21]. The endomorphism part of the gauge connection, e.g. for the fundamental representation, reads The Lagrangian of the gauge theory on M d × Q 3 is then given by that on M d × X 1,1 without the terms containing commutators with X 7 , 40) because the vanishing of them is subject to the further equivariance conditions (3.37).
4 Instantons on the metric cone C (X 1,1 ) The implementation of the equivariance condition (3.17) has determined the general form of the gauge connection, expressed in the associated quiver diagram, and the action functional, but has not restricted the entries of the endomorphisms. Further conditions and relations among the endomorphisms can be imposed by studying vacua of the gauge theory, i.e. by minimizing the action functional (3.33). To this end, we will evaluate the Hermitian Yang-Mills equations -a certain form of generalized self-duality equations -on the metric cone C(X 1,1 ), as it has been done in similar setups, e.g. [23,24], and describe their moduli space, following [33,34,35].

Generalized self-duality equation
A very useful tool for obtaining minima of a Yang-Mills functional in gauge theory is to evaluate a first-order equation implying the second-order Yang-Mills equations [2,3,4]. Given a connection A on an n-dimensional manifold whose curvature F satisfies the generalized self-duality equation for a 4-form Q, one obtains by taking the differential [5] which is the usual Yang-Mills equation with torsion term (d * Q) ∧ F. Explicit formulae for the choice of the form Q, in dependence of the geometry of the manifold, such that the torsion term vanishes even if the form Q is not co-closed have been given in [5]. Their construction is based on the existence of (real) Killing spinors, and thus also applies to Sasaki-Einstein structures. A connection A whose curvature satisfies (4.1) for the form Q given by [5] is called a (generalized) instanton. For a Sasaki-Einstein manifold the form Q reads [5] The corresponding instanton equation (4.1) on X 1,1 is solved by the connection Γ = I 8 ⊗ e 8 , which we used for expressing the G-invariant connection in (3.12); see Appendix A.2. The form Q Z occurring in the instanton equation on the cylinder, which is conformally equivalent to the metric cone, over a Sasaki-Einstein manifold reads [5] 18 and one thus obtains where Ω is the Kähler form of the Calabi-Yau cone and the cylinder, respectively. Since the Calabi-Yau manifold is of complex dimension 4 and as we have chosen the standard form of the Kähler form, the 4-form Q Z is self-dual, such that d * Q Z = dQ Z = 0, and the Yang-Mills equation without torsion follows from the instanton equation (4.1). We evaluate the instanton equation (4.1) with the form Q Z by imposing the (equivalent) Hermitian Yang-Mills equations (HYM) [42,47,48] F (2,0) = 0 = F (0,2) and Ω F := * (Ω ∧ * F) = 0, (4.7) where F (2,0) refers to the (2, 0)-part with respect to the complex structure J . The first equation is a holomorphicity condition and the second one can (sometimes) be considered as a stability condition on vector bundles; they are also known as Donaldson-Uhlenbeck-Yau equations.

Hermitian
Yang-Mills instantons on C (X 1,1 ) We consider the same ansatz (3.7) [30], now including also the additional form e τ := dτ := dr r on the cylinder, where we set 19 Due to the equivariance, the endomorphisms are "spherically symmetric", X a = X a (r). After the implementation of the same equivariance conditions as before, the non-vanishing components of the field strength read Constant endomorphisms: For the special case of constant matrices X a , the situation corresponds to that of the underlying Sasaki-Einstein manifold X 1,1 with the parameter τ (or r, respectively) just as a label of the foliation along the preferred direction of the cone. Gauging the field X τ to zero, one recovers then from (4.13) and (4.14) exactly the additional equivariance conditions (3.38), which appeared in the discussion of the gauge theory on Q 3 . Thus the equivariant gauge theory on Q 3 can be considered as a special instanton solution 20 of the more general setup on C (X 1,1 ).

Moduli space of SU(3)-equivariant instantons
For a desription of the moduli space of the equations (4.14) and (4.13), (4.12) under the equivariance conditions (4.10), it is advantageous to re-write them in a form similar to the Nahm equations. Then one can employ the techniques used by Donaldson [33] and Kronheimer [34] for the discussion thereof. We will briefly sketch the application of these methods to our system of flow equations, following [35], where the framed moduli space of solutions to the Hermitian Yang-Mills equations on metric cones over generic Sasaki-Einstein manifolds is discussed in this way. Note that the treatment [35] uses the canonical connection of [5] as starting point Γ for the gauge connection and that our connection Γ = I 8 ⊗ e 8 in (4.8) differs from it (see Appendix A.2). This is why some modifications, in comparison with [35], will appear in our discussion 21 .
Changing the argument in the flow equations to τ = ln(r) and setting 22  The equation (4.19) shall be referred to as the real equation and the equations (4.17) and (4.18) as complex equations. The discussion of the moduli space is based on the invariance of the complex 20 The vanishing of the contributions stemming from the form e 7 is obvious from the Yang-Mills action (3.33) and the instanton condition (4.1). Due to * 7Q ∝ e 7 those terms do not contribute to the action for instanton solutions, and this is equivalent to the further equivariance conditions (3.37). 21 Of course, using the canonical connection of [5] yields the results of [35] also for X1,1. However, for the discussion of the quiver diagrams in Section 3 the connection Γ = I8 ⊗e 8 was more suitable because it is valued in the subalgebra h and, thus, adapted to the setup of a homogeneous space. The canonical connection, in contrast, is adapted to the Sasaki-Einstein structure of X1,1; see Appendix A.2. 22 For the canonical connection (A.14) of a seven-dimensional Sasaki-Einstein manifold, the matrices scale as [35] Yᾱ = e − 4 3 τ Wα for α = 1, 2, 3 and Y4 = e −6τ Z. (4.15) equations under the complexified gauge transformation [33] W α −→ W g α := gW α g −1 , for α = 1, 2, 3 and Z −→ Z g := gZg −1 − 1 2 dg ds g −1 (4.21) with g ∈ C ((−∞, 0], GL(C, k)). A local solution of (4.17) can be attained by applying the gauge so that -due to the complex equations (4.17) -the gauge transformed matrices W g α must be constant, (4.23) To obtain solutions, one has to choose these constant matrices such that they satisfy (4.18). One special choice, for instance, could be to set T 3 = 0 and take for T 1 and T 2 elements of a Cartan subalgebra. Note that not only the scaling in (4.16) is different from that in [35], but also the conditions (4.18): There all three matrices have to commute with each other and, thus, also T 3 can be chosen as arbitrary element of a Cartan subalgebra. Adapting Donaldson's arguments [33,35], the real equation (4.19) can be -locally on an interval I ⊂ (−∞, 0] -considered as the equation of motion (i.e. δL ∝ µ) of the Lagrangian Since the potential term in this Lagrangian is non-negative, the existence of a solution to (4.19) as equation of motion follows from a variational problem [33]. One still has to ensure some technical aspects: the uniqueness of the solutions, the existence of the gauge transformation and the Lagrangian on the entire interval (−∞, 0], as well as the boundedness of µ. In the reference [35] these properties are proven, given that for framed instantons, i.e. those with h = 1 at the boundary of the interval (−∞, 0], the following condition is satisfied for constant matrices obeying the conditions (4.18). For their constraints, i.e. mutually commuting matrices T α , it is shown that the moduli space can be expressed as diagonal orbit in a product of coadjoint orbits [35]. In our case, however, due to the different constraints (4.12), the situation might be more involved. But we can at least conclude that (4.23) provides local solutions of the Nahm-type equations (4.17)-(4.19).
Moreover, it was shown in the references (see again [35]) that the real equation (

Summary and conclusions
In this article we studied the SU(3)-equivariant dimensional reduction of gauge theories over the Sasaki-Einstein manifold X 1,1 . We interpreted the condition of equivariance, which had already occurred in articles [30,31] on Spin(7)-instantons on cones over Aloff-Wallach spaces X k,l , in terms of quiver diagrams, and we discussed the general construction of the quiver bundles. This yielded a new class of Sasakian quiver gauge theories. The associated quiver diagram of this gauge theory is a "doubled modified holomorphic chain", consisting of two arrows between adjacent vertices and two loops at each vertex, and three explicit examples thereof were considered in the article. For the comparison with the gauge theory on the underlying Kähler manifold Q 3 we studied the quivers also in the entire weight diagram of G = SU (3), which implied degeneracies of the arrows. This behavior is similar to the case [15,24] of the five-dimensional Sasaki-Einstein manifold T 1,1 over CP 1 × CP 1 . The reduction to the gauge theory on Q 3 led to the correct, expected result for the quiver diagram [16]: the weight diagram of SU (3).
For the investigation of the vacua described by this gauge theory we imposed the Hermitian Yang-Mills equations on the metric cone C (X 1,1 ). The resulting flow equations have been re-written in a form similar to Nahm's equations, which allowed a discussion based on Kronheimer's [34] and Donaldson's [33] work and its generalized application to equivariant HYM instantons on Calabi-Yau cones [35]. Since we formulated the quiver gauge theory by using an instanton connection different from that of [5] in the gauge connection, some modifications appeared. While the real equation can be still interpreted as a moment map for framed instanton solutions, as in [35], and, thus, leads to a description of the moduli space as a Kähler quotient, the description based on coadjoint orbits is more involved: The HYM equations impose a non-trivial commutation relation on the gauge transformed matrices, in contrast to [35], where they have to commute with each other. Thus, the behavior is more complicated and further effort would be needed to study the consequences thereof in detail.
The non-vanishing structure constants are [30]  By the Maurer-Cartan equations, That the Lagrangian can be defined for the entire range s ∈ (−∞, 0] and other technical issues can be found in [35]. The only quantitive difference is the concrete form of the factors λ α (s) but this does not affect the general line of reasoning.