Sasakian quiver gauge theory on the Aloff-Wallach space X 1,1
aa r X i v : . [ h e p - t h ] M a y ITP–UH–12/16
Sasakian quiver gauge theory on the Aloff-Wallach space X , Jakob C. Geipel
Institut f¨ur Theoretische PhysikLeibniz Universit¨at HannoverAppelstraße 2, 30167 Hannover, Germany
Email: [email protected]
Abstract
We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form M d × X , with d -dimensional Riemannian manifold M d and the Aloff-Wallach space X , =SU(3) / U(1) endowed with its Sasaki-Einstein structure. The condition of SU(3)-equivarianceof vector bundles, which has already occurred in the studies of Spin(7)-instantons on conesover Aloff-Wallach spaces, is interpreted in terms of quiver diagrams, and we construct thecorresponding quiver bundles, using (parts of) the weight diagram of SU(3). We considerthree examples thereof explicitly and then compare the results with the quiver gauge theoryon Q = SU(3) / (U(1) × U(1)), the leaf space underlying the Sasaki-Einstein manifold X , .Moreover, we study instanton solutions on the metric cone C ( X , ) by evaluating the HermitianYang-Mills equation. We briefly discuss some features of the moduli space thereof, followingthe main ideas of a treatment of Hermitian Yang-Mills instantons on cones over generic Sasaki-Einstein manifolds in the literature. ontents X , X , X , . . . . . . . . . . . . . . . . . . . 73.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Dimensional reduction of the Yang-Mills action . . . . . . . . . . . . . . . . . . . . . 123.4 Reduction to quiver gauge theory on Q . . . . . . . . . . . . . . . . . . . . . . . . . 14 C ( X , ) C ( X , ) . . . . . . . . . . . . . . . . . . . . . . 164.3 Moduli space of SU(3)-equivariant instantons . . . . . . . . . . . . . . . . . . . . . . 17 A.1 SU(3) generators and structure constants . . . . . . . . . . . . . . . . . . . . . . . . 19A.2 Connections and instanton equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20A.3 Details of the moduli space description . . . . . . . . . . . . . . . . . . . . . . . . . . 21
The emergence of extra dimensions in string theory and the typical ansatz for compactificationsmake a detailed understanding of higher-dimensional gauge theories desirable. Inspired by the sem-inal investigation of four-dimensional manifolds by self-dual connections [1], generalized self-dualityequations and instantons in higher dimensions have been studied [2, 3, 4, 5]. Their significancein physics is evident in heterotic string theory where an instanton equation is part of the BPSequations [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 cosetspace dimensional reduction [7]. On those spaces one can demand G -equivariance of the vectorbundles the gauge connection takes values in, and this equivariant dimensional reduction yieldssystematic restrictions which can be depicted as quiver diagrams , i.e. directed graphs. A detailedmathematical treatment for K¨ahler manifolds can be found in [8, 9] and short physical reviews aregiven e.g. in [10, 11].These quiver gauge theories have been studied for the K¨ahler cosets C P [9, 12, 13, 14], C P × C P [15], and SU(3) /H [14, 16]. The odd-dimensional counterparts of K¨ahler spaces are Sasaki manifolds[17], and among them Sasaki-Einstein manifolds [18] are of particular interest for compactificationsin string theory because, by definition, their metric cones are Calabi-Yau [19, 20]. In the literature,1 asakian quiver gauge theory has been studied on the orbifold S / Γ [22], on orbifolds S / Z q +1 ofthe five-sphere [23] and on the space T , [24], the base space of the conifold. The five-dimensionalSasaki-Einstein coset spaces as well as the new examples [25, 26] are of interest for versions of theAdS / CFT correspondence. In dimension seven, one can encounter the following typical examples:the seven-sphere S , the Aloff-Wallach space X , [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 wewill consider the Sasakian quiver gauge theory on the Aloff-Wallach space X , . The mathematicalproperties of the generic Aloff-Wallach spaces X k,l [27] – basically their G and Spin(7) structureand, for the special case of X , , 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 tothe special geometry, more precisely the existence of Killing spinors, they have been intensivelystudied in M-theory or supergravity [32].This article is organized as follows: Section 2 reviews the geometry of the space X , , providing localcoordinates, the structure equations, the Sasaki-Einstein properties as well as a comment on theclosely related K¨ahler space Q := SU(3) / U(1) × U(1). The subsequent section begins with a shortreview of equivariant vector bundles over homogeneous spaces and the arising quiver diagrams.Then we study the equivariant gauge theory on X , , placing the focus on the evaluation of theequivariance condition, already known from [30, 31], in terms of quiver diagrams. We discuss thegeneral construction for the quiver diagrams associated to X , and clarify it by considering threeexamples with a small number of vertices. The resulting Yang-Mills functional of the equivariantgauge theory is provided, and the reduction to the quiver gauge theory on Q is discussed in thelast part of Section 3. Subsequently, we study instanton solutions of the quiver gauge theory byevaluating the Hermitian Yang-Mills equations on the metric cone C ( X , ). We briefly sketch thetechniques used by Donaldson [33] and Kronheimer [34] for the discussion of the Nahm equationsand the application of those methods to Hermitian Yang-Mills instantons on generic Calabi-Yaucones [35]. We discuss the modifications that appear in our setup, due to using a different instantonconnection in the ansatz for the gauge connection, in comparison with the general results of [35].The appendix provides some technical details. X , In this section we review the geometric properties of the Aloff-Wallach space X , and its metriccone C ( X , ) which are necessary for the discussion in this article. Among the huge number ofarticles on the geometry of Aloff-Wallach spaces X k,l [27], we follow the exposition given in thearticle [30], in which G 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 gaugeconnections. Since we are aiming only at the Sasaki-Einstein structure of X , , we will not considergeneral spaces X k,l . For details on theses structures we refer to [30] and the references therein. The Aloff-Wallach spaces [27], denoted as X k,l , for coprime integers k and l , are defined as quotients X k,l = G/H := SU (3) / U (1) k,l (2.1)where the embedding of elements h ∈ U (1) k,l into SU(3) is given by h = diag (cid:16) e i ( k + l ) ϕ , e − i kϕ , e − i lϕ (cid:17) . (2.2)2t is known that the homogeneous space X , is not only Sasaki-Einstein but moreover admits a3-Sasakian structure . Due to [36] a homogeneous 3-Sasakian manifold different from a sphere is aSO(3) ∼ = SU(2) / Z bundle over a quaternionic K¨ahler manifold; in the case of X , the underlyingspace is C P . Using this result, we can construct local coordinates by starting from a localsection of the fibration SU(3) → C P , as it can be found e.g. in [16, 21]. Given a local patch U := (cid:8) [ w : w : w ] ∈ C P | w = 0 (cid:9) of C P , one can introduce coordinates Y := (cid:18) y y (cid:19) ∼ (cid:18) , w w , w w (cid:19) T , (2.3)and a local section of the bundle SU(3) → C P is given by C P ∋ Y V := 1 γ (cid:18) Y † − ¯ Y Λ (cid:19) ∈ SU(3) (2.4)with γ := p Y † ¯ Y ,
Λ ¯ Y = ¯ Y , ¯ Y † Λ = ¯ Y † , Λ := γ − γ + 1 ¯ Y ¯ Y † , Λ = γ − ¯ Y ¯ Y † . (2.5)Furthermore, an arbitrary element g of SU(2) can be written as g = 1(1 + z ¯ z ) / (cid:18) − ¯ zz (cid:19) (cid:18) e i ϕ
00 e − i ϕ (cid:19) , (2.6)where z and ¯ z are stereographic coordinates on C P . Putting both expressions (2.4) and (2.6)together, one gets a local section of the bundle SU(3) −→ X , as ( y , y , z, ϕ ) ˜ V := V · g = 1 γ (cid:18) Y † − ¯ Y Λ (cid:19) z ¯ z ) / − ¯ z z i ϕ
00 0 e − i ϕ . (2.7) Hence, the manifold can be locally described by the coordinates { y , ¯ y , y , ¯ y , z, ¯ z, ϕ } , and theMaurer-Cartan form provides SU(3) left-invariant 1-forms Θ α and e i , defined by A := ˜ V − d ˜ V =: √ e √ −√ ¯1 −√ ¯2 − i √ e − i e − Θ ¯3 √ Θ − i √ e + i e . (2.8)Here we have defined the forms such that the generators of SU(3) (see Appendix A.1) coincidewith those from [30]. Due to the flatness of the connection, d A + A ∧ A = 0, one obtains thestructure equations dΘ = − i e ∧ Θ + √ e ∧ Θ − Θ ¯23 , dΘ = − i e ∧ Θ − √ e ∧ Θ + Θ ¯13 , dΘ = − e ∧ Θ − , (2.9)d e = − i (cid:16) Θ + Θ + Θ (cid:17) , d e = √ (cid:16) Θ − Θ (cid:17) , together with the complex conjugated equations for Θ ¯ α , α = 1 , ,
3. By construction, the groupU(1) k,l in the definition (2.1) is generated by I in (A.1), and the remaining group U(1) inside X , is associated to I and the local coordinate ϕ . This means that the (Riemannian) holonomy of the metric cone C ( X , ) can be reduced from SU(4) to Sp(2). Since we will work entirely on Lie algebra level, a local description is sufficient for our purposes. .2 Sasaki-Einstein structure Following [30], the Einstein metric is chosen to bed s X , = g µν e µ ⊗ e ν = Θ ⊗ Θ ¯1 + Θ ⊗ Θ ¯2 + Θ ⊗ Θ ¯3 + e ⊗ e , (2.10)and the Sasaki structure is defined by declaring the forms Θ α to be holomorphic, ˜ J Θ α = i Θ α . Here˜ J denotes the complex structure of the leaf space orthogonal to the contact direction e . Then thefundamental form ω associated to it satisfies the Sasaki condition2 ω = d η := d e = − i (cid:16) Θ + Θ + Θ (cid:17) , (2.11)which implies that ω is the K¨ahler form of the leaf space. The metric cone C ( X , ) has by definitionthe metricd s C ( X , ) = r d s X , + d r ⊗ d r = r (cid:18) d s X , + d rr ⊗ d rr (cid:19) = r X α =1 Θ α ⊗ Θ ¯ α , (2.12)where one has defined a fourth holomorphic formΘ := d rr − i e . (2.13)Equation (2.12) establishes the correspondence between the metric cone and the conformally equiv-alent cylinder . The definition of Θ yields an integrable complex structure J on the metric conewhose fundamental form Ω ( X, Y ) := g ( J X, Y ) is then given byΩ = − i2 r X α =1 Θ α ∧ Θ ¯ α = r ω + r d r ∧ e . (2.14)Due to the Sasaki condition d e = 2 ω this form is closed and the cone C ( X , ), thus, carries aK¨ahler structure. For the cone to be Calabi-Yau, the holonomy U(4) of the K¨ahler manifold mustbe reduced further to SU(4), which is ensured by the closure of the 4-form [30]Ω , := r Θ ∧ Θ ∧ Θ ∧ Θ . (2.15)Consequently, the geometric structure is that of a Calabi-Yau 4-fold, which implies the Sasaki-Einstein structure of X , . As a Sasakian manifold, X , is a U(1)-bundle over an underlying K¨ahlermanifold, namely the leaf space of the foliation along the Reeb vector field, with fundamental form ω . The K¨ahler manifold underlying X , is denoted as Q or F [16, 21] X , Q := SU(3)U(1) × U(1)
U(1) (2.16)From the (local) section in (2.7) one has locally Q ∼ = C P × C P , and the space is described bythe coordinates { y , ¯ y , y , ¯ y , z } . Considering the metric cone is tantamount to studying the conformally equivalent cylinder for the discussion inthis article. One can obtain an orthonormal basis by rescaling the forms ˜ e µ := r e µ . We will mainly use the cylinderfor the description here. Quiver gauge theory on X , Quiver diagrams are a powerful tool in representation theory, and this motivates their appearancein gauge theories, where the field content can be described by these directed graphs. In this sectionwe will demonstrate the basic features of quiver gauge theories by considering them on the spaces X , and Q . We start the survey with a brief review of how quiver diagrams arise in the contextof gauge theories on reductive homogeneous spaces G/H . The condition generating the quiver diagrams, which we will usually refer to as equivariance condi-tion , can be understood from two point of views: On the one hand, one could consider equivariantvector bundles in a rigorous algebraic fashion as it is done in [8, 9], purely based on the represen-tation theory of the Lie algebras involved. On the other hand, the equivariance condition occursquite naturally in the context of instanton studies, e.g. [38, 39, 40, 41, 42], as invariance conditionon gauge connections on reductive homogeneous spaces
G/H . Equivariant vector bundles
We sketch the basics of equivariant vector bundles and their re-lation to quiver gauge theories, following roughly [8, 10]. For the application of this approach werefer also to the examples in [16, 23]. Let
G/H be a Riemannian coset space modelling the internaldegrees of freedom, M d a d -dimensional Riemannian manifold, and let π : E → M d × G/H be aHermitian vector bundle of rank k , i.e. a vector bundle with structure group U( k ). Suppose thatthe Lie group G acts trivially on M d and in the usual way on the coset space. Then the bundleis 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 . Byrestriction and induction of bundles, E = G × H E , G -equivariant bundles E → M d × G/H are inone-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 bundleimplies that the fibers must carry representations of H . We assume that these H -representationsstem from the restriction of an irreducible G -representation D which decomposes under restrictionto H as follows D | H = m M i =0 ρ i (3.1)where the ρ i ’s are irreducible H -representations. This yields an isotopical decomposition of thevector bundle E as a Whitney sum in the very same way E = M i E i with ( E i ) x ∼ = C k i carrying ρ i (3.2) Note that for us the term quiver gauge theory always refers to the structures arising from the bundle equivariance.Thus our definition is not directly related to other forms of quiver gauge theories in the literature, e.g. [37], whichare based on brane physics. One should keep in mind that the fundamental objects of a gauge theory are principal bundles ( P, p, X ; K ) withtotal space P , base space X , projection map p , and gauge group K although we will work completely in terms ofvector bundles in this article. They can be thought of as associated to the relevant principal bundle P . This assumption is not mandatory for the approach, but simplifies the situation due to the classification ofirreducible representations of semisimple Lie algebras. In general, one can split the summands further into E i = ˜ E i ⊗ V i , where V i is an irreducible H -representation andthe subgroup H acts trivially on ˜ E i [15]. Since we consider an abelian subgroup H , the irreducible representationsare 1-dimensional, so that H acts as multiple of the identity on the entire space E i . For an example of a non-abeliansubgroup H , consider for instance [23]. k ) of the bundle toU ( k ) −→ m Y i =0 U ( k i ) with m X i =0 k i = k. (3.3)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 (cid:0) C k i , C k j (cid:1) . In this way, the fibres of the G -equivariantbundle are representations of a quiver ( Q , Q ), where Q denotes the set of vertices and Q the setof arrows. Each vertex v i ∈ Q 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 bundlethus carries a representation of the quiver, and this contruction is called a quiver bundle . Since theallowed arrows of the quiver diagram arise from the commutation relations of the generators withthe 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 alsooccurs naturally when studying instanton solutions of invariant gauge connections on reductivehomogeneous spaces, e.g. in [30, 41, 42]. Let
G/H be a reductive homogeneous space with theAd( H )-invariant splittingspan h I µ i := g = h ⊕ m =: span h I j i ⊕ span h I a i , (3.4)where the generators satisfy[ I j , I k ] = C ljk I l , [ I j , I a ] = C bja I b , and [ I a , I b ] = C cab I c + C jab I j ; (3.5)the space m can be identified with the tangent space of G/H . Let e µ be the 1-forms dual to thegenerators I µ , which obey the structure equationd e µ = − C µρσ e ρσ = − Γ µν ∧ e ν + T µ , (3.6)where Γ µν 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 theapproach used for example in [41], we can express a G -invariant connection A on the homogeneousspace as A = I j ⊗ e j + X a ⊗ e a , (3.7)where the skew-hermitian matrices X a , the Higgs fields , describe the endomorphism part. Theconnection Γ := I j ⊗ e j takes values entirely in the vertical component h and is obtained by declaringthe torsion to be T ( X, Y ) := − [ X, Y ] m for X, Y ∈ T e ( G/H ). The curvature F = d A + A ∧ A of(3.7) is then given by F = F Γ + (cid:16) [ I j , X a ] − C bja X b (cid:17) e j ∧ e a + 12 ([ X a , X b ] − C cab X c ) e ab + d X a ∧ e a . (3.8)For the connection to be G -invariant, terms containing the mixed 2-forms e j ∧ e a must not occur, sothat one obtains – assuming that the last term in (3.8) does not yield incompatible contributions – the equivariance condition [41, 45] [ I j , X a ] = C bja X b . (3.9) For details on representations of quiver diagrams, see for example [43, 44] In principle one could also use different connections Γ as starting point in the ansatz (3.7). See the comments inSection 4. This holds true e.g. for constant matrices or those with X a = X a ( r ), as we will consider on the metriccone C ( G/H ) in Section 4. X a to act (with respect to the adjoint action) onthe 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 gaugeconnection and the corresponding quiver bundle for X , = SU(3) / U(1) , in the following way. Let C k = (cid:16) C k , C k , . . . , C k m (cid:17) T (3.10)be a decomposition of the representations on the fibres in ( m + 1) terms, which yields the breakingof the structure group (3.3) and the isotopical decomposition as in (3.2). Since the irreduciblerepresentations ρ i of the abelian subgroup H = U(1) , are 1-dimensional, the group H acts as( ζ k , ζ k , . . . , ζ m k m ) (3.11)on the vectors (3.10). The constants ζ i can be obtained from an irreducible representation ofthe U(1) , -generator on an ( m + 1)-dimensional vector space. This fact and the way how thequiver diagrams arise motivate to consider the gauge connection as a block matrix of size ( m + 1) ,whose structure is determined by the ( m + 1)-dimensional G -representation in which the entriesare (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 weightdiagram because all Cartan generators occur as operators I j in (3.9). For smaller subgroups theremight be degeneracies as double arrows in the diagram, while larger groups require a collapsingof vertices in the weight diagram along the action of the ladder operators of h as it is done, forinstance, in [11, 16]. We will clarify this procedure for the abelian subgroup H = U(1) , in thefollowing. X , The aforementioned approach is now applied to the space X , . Following the outline above andaccording to (3.7), we write an SU(3)-invariant connection A on M d × X , as [30] A = A + I ⊗ e + X a =1 X a ⊗ e a =: A + I ⊗ e + X α =1 (cid:0) Y α ⊗ Θ α + Y ¯ α ⊗ Θ ¯ α (cid:1) + X ⊗ e , (3.12)where A is a connection on M d . Moreover, we have defined complex endomorphisms Y := 12 ( X + i X ) , Y := 12 ( X + i X ) , Y := 12 ( X + i X ) (3.13)with Y ¯ α := − Y † α . In terms of the structure constants (A.3) the field strength of the connection A is given by [30] F = d A + A ∧ A + (d Y α + [ A, Y α ]) ∧ Θ α + (d Y ¯ α + [ A, Y ¯ α ]) ∧ Θ ¯ α + (d X + [ A, X ]) ∧ e + 12 (cid:16) [ Y α , Y β ] − C γαβ Y γ (cid:17) Θ αβ + (cid:16)(cid:2) Y α , Y ¯ β (cid:3) − C γα ¯ β Y γ − C ¯ γα ¯ β Y ¯ γ + i C α ¯ β X + i C α ¯ β I (cid:17) Θ α ¯ β + 12 (cid:16)(cid:2) Y ¯ α , Y ¯ β (cid:3) − C ¯ γ ¯ α ¯ β Y ¯ γ (cid:17) Θ ¯ α ¯ β + (cid:16) [ X , Y α ] − i C β α Y β (cid:17) e ∧ Θ α + (cid:16) [ X , Y ¯ α ] − i C ¯ β α Y ¯ β (cid:17) e ∧ Θ ¯ α + (cid:16) [ I , Y α ] − i C β α Y β (cid:17) e ∧ Θ α + (cid:16) [ I , Y ¯ α ] − i C ¯ β α Y ¯ β (cid:17) e ∧ Θ ¯ α + [ I , X ] e . (3.14)7ollowing some notation in the literature, e.g. in [16], we call φ ( α ) := Y ¯ α for α = 1 , , , and X (3.15)the Higgs fields and set ˆ I := −√ I = diag (2 , − , −
1) and ˆ I := − i I = diag (0 , − , . (3.16)The equivariance condition (3.9), equivalent to the vanishing of the terms in the last line of (3.14),then reads h ˆ I , φ (1) i = 3 φ (1) , h ˆ I , φ (2) i = − φ (2) , and h ˆ I , φ (3) i = 0 = h ˆ I , X i . (3.17)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 , 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 , ν ) associated to the two Cartan generators ˆ I and ˆ I of SU(3) φ (1) : ( ν , ν ) ( ∗ , ν + 3) ,φ (2) : ( ν , ν ) ( ∗ , ν − , (3.18) φ (3) : ( ν , ν ) ( ∗ , ν ) ,X : ( ν , ν ) ( ∗ , ν ) . Since the quantum number ν does not enter the equivariance condition, it is reasonable to labelthe vertices in the quiver diagram only by the number ν , so that one obtains effectively a modifiedversion of the holomorphic chain [9]: a diagram consisting of double arrows between adjacentvertices and double loops at each vertex, • ( p − ) • ( p − + ) . . . • ( p − ) • ( p ) (3.19)where the black two headed arrows denote the contributions by φ (1) and φ (2) † , while the endomor-phisms φ (3) and X are represented by the blue two headed loops . Here, the integer p denotesthe highest weight (with respect to ν ) of the representation D . The endomorphism part of theinvariant connection associated to this modified holomorphic chain of length m + 1 is then givenby X a e a = Ψ p Φ p − . . . − Φ † p − Ψ p − Φ p − . . . ...0 − Φ † p − . . . . . . 0... ... . . . Ψ p − Φ p − m . . . − Φ † p − m Ψ p − m , (3.20) As mentioned above, we implicity interpret the numbers in the Cartan generators as numbers times identityoperators. This corresponds to the isotopical decomposition (3.2): The representation D is restricted under the subgroupU(1) , rather than under a maximal torus U(1) × U(1). Using one arrow with two heads as symbol for two arrows improves the readibility of the more complicateddiagrams like Figure 3 significantly. p − j := φ (1) p − j ⊗ Θ ¯1 − φ (2) † p − j ⊗ Θ and Ψ p − j := φ (3) p − j ⊗ Θ ¯3 + ( X ) p − j ⊗ e (3.21)for j = 0 , , . . . , m , and the indices label the tail of the arrow. The remaining contribution to theinvariant connection (3.12) is given by the diagonal parts A + Γ = diag (cid:18) k l ⊗ l − p √ e + A p − l (cid:19) l =0 ,...,m , (3.22)where A p − l is a component – according to the isotopical decomposition (3.2) of the fibres – ofa connection on the bundle E → M d . Equations (3.20) and (3.22) describe the general solution.For comparisons with gauge theories of similar geometric structures like Q , it is advantageousto consider not only the decomposition under the subgroup H , i.e. labelling the vertices onlyby ν as we did, but to study the equivariance conditions in the entire weight diagram of G .Since the weight diagrams of the relevant Lie algebras are well-known, one can quickly constructthe invariant connection by implementing the rules (3.18) and can then project to the relevantquantum numbers. In the following, we will consider the triangular/hexagonal weight diagram ofSU(3), spanned by the root system ( − , − , − − , I ¯1 I ¯2 I ¯3 (3.23) We consider three explicit examples of SU(3) representations and the quiver diagrams associatedto them.
Fundamental representation
Applying the prescription (3.18) to the single triangle of theweight diagram of the defining representation provides the quiver diagram in Figure 1. Of course,this diagram could be also obtained by direct evalution of the commutator of ˆ I = diag(2 , − , − × • ) h ˆ I , ( • ) i = • • •− • • •− • • • ! . (3.24)Then the equivariance condition requires the Higgs fields to be of the form φ (1) = ∗ ∗ , φ (2) = ∗ ∗ , φ (3) = ∗ ∗ ∗ ∗ ∗ , X = ∗ ∗ ∗ ∗ ∗ , (3.25) For representation theory of su (3) see e.g. [46]. , ) ( , − )( − , − ) ( ) ( − )projectionprojectionFigure 1: Quiver diagram of X , for the fundamental representation of SU(3): The left diagramstems from the implementation of the equivariance condition in the weight diagram of SU(3) andthe right one is the holomorphic chain with the loop modification and the double arrows, obtainedfrom the projection by forgetting about the second quantum number ν , i.e. identifying pointsalong horizontal lines.which again yields the quiver diagram Figure 1. This translates into the invariant gauge connection A = √ i e ⊗ + Ψ , , Φ − , − , Φ , − , − Φ †− , − , − √ i e ⊗ + Ψ − , − − , − +Ψ , − − , − − Φ † , − , − Ψ † , − − , − − √ i e ⊗ + Ψ , − , − , (3.26) where we have defined Φ i,j ; k,l := ( φ (1) ) i,j ; k,l ⊗ Θ ¯1 − ( φ (2) † ) i,j ; k,l ⊗ Θ , Ψ i,j ; k,l := ( φ (3) ) i,j ; k,l ⊗ Θ ¯3 + ( X ) i,j ; k,l ⊗ e ; (3.27)the U(1) × U(1)-charges ( i, j ) denote the tail of the arrow, and ( k, l ) its head. Going back to theeffective quiver diagram, i.e. the modified holomorphic chain, yields A = √ i e ⊗ + Ψ ˜Φ − − ˜Φ †− − √ i e ⊗ + ˜Ψ − − ! , (3.28)which agrees with the general result (3.20). The anti-fundamental representation ¯ , of course, leadsto an analogous diagram and connection. Representation 6
The six-dimensional representation of SU(3) causes the more complicatedquiver diagram depicted in Figure 2, which could be also obtained by the direct evaluation of thecommutation relation with the U(1) , -generator ˆ I = diag (4 , , , − , − , − φ (1) and φ (2) † = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , φ (3) and X = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . (3.29) , ) ( , )( − , ) ( , − )( , − )( − , − ) ( ) ( ) ( − ) Figure 2: Quiver diagram for the representation with the same notation as before.We skip the explicit index structure of the invariant gauge connection which can be read from thequiver diagram, Figure 2, and provide only the result for the modified holomorphic chain A = √ i e ⊗ + Ψ ˜Φ − ˜Φ † √ i e ⊗ ˜ + ˜Ψ ˜Φ − − ˜Φ †− − √ i e ⊗ ˜ + ˜Ψ − − . (3.30)It is interesting to compare this block matrix of size 3 × Adjoint representation 8
The U(1) , -generator in the adjoint representation is given by ˆ I =diag (3 , , , , , , − , −
3) and the weight diagram is a hexagon with two degenerated points atthe origin . The Higgs fields must thus have the shape φ (1) and φ (2) † = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , φ (3) and X = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , (3.31) and the quiver diagram Figure 3 contains a large number of arrows. The identification leading tothe modified holomorphic chain yields as connection A = √ e ⊗ + Ψ ˜Φ − ˜Φ † ˜Ψ ˜Φ − − ˜Φ †− −√ e ⊗ ˜ + ˜Ψ − − . (3.32) The representation of the other Cartan generator reads ad( − i I ) = diag ( − , , − , , , , − , , , )( − , ) ( , )( , ) ( − . ) ( , − )( − , − ) ( ) ( ) ( − ) Figure 3: Quiver diagram for the adjoint representation of SU(3). Note that due to the degeneracyof (0 ,
0) each arrow involving the origin must be counted twice (depicted as arrows consisting oftwo 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 thesix-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 onlythe relevant quantum number ν rather than the entire weight diagram of G , but for comparisonswith Q the latter description is also useful. The occurrence of degeneracies in the entire weightdiagram of SU(3) due to the weaker equivariance condition is similar [15, 24] to the case of thefive-dimenional Sasaki-Einstein manifold T , := (SU(2) × SU(2)) / U(1) in comparison with itsunderlying manifold C P × C P . In the previous section we have completely characterized the form of a G -invariant gauge connectionby 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 , with field strength F , we now determine its standardYang-Mills action S YM = − Z M d × X , tr F ∧ ∗F , (3.33)yielding the usual Yang-Mills Lagrangian L YM = − p ˆ g tr F ˆ µ ˆ ν F ˆ µ ˆ ν , (3.34)where we denote ˆ g := det g X , det g M d . Using the Sasaki-Einstein metric (2.10), (cid:0) g X , (cid:1) α ¯ β = 12 δ αβ and (cid:0) g X , (cid:1) = 1 , (3.35) We use the set of indices { ˆ µ } = { µ, α, ¯ α, } with µ referring to M d . ) ( , ) ( , − )( − , − ) ( , )( − , ) ( , )( , ) ( − , ) ( , − )( − , − ) c ) b ) ( , ) ( , )( − , ) ( , − )( , − )( − , − )Figure 4: Quiver diagrams of Q for a) fundamental representation , b) representation , and c)adjoint representation (with the degenerated origin) of SU(3). The arrows denote the Higgs fields φ (1) (black), φ (2) (red), and φ (3) (blue), according to the condition (3.38).and the field strength components from (3.14), one obtains as Lagrangian L YM = p ˆ g tr k ( F µν ( F µν ) † + 2 X α =1 (cid:12)(cid:12)(cid:12) D µ φ ( α ) (cid:12)(cid:12)(cid:12) + 12 | D µ X | + 2 (cid:12)(cid:12)(cid:12)h φ (1) , φ (1) † i − i X + √ I (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)h φ (2) , φ (2) † i − i X − √ I (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)h φ (3) , φ (3) † i − i X (cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (2) i − φ (3) (cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (3) i(cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (2) , φ (3) i(cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (2) † i(cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (3) † i + φ (2) † (cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (2) , φ (3) † i − φ (1) † (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)h φ (1) , X i − i φ (1) (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)h φ (2) , X i − i φ (2) (cid:12)(cid:12)(cid:12) +2 (cid:12)(cid:12)(cid:12)h φ (3) , X i − φ (3) (cid:12)(cid:12)(cid:12) ) . (3.36) Here, we have defined the covariant derivatives D µ φ ( α ) := (cid:0) d φ ( α ) + (cid:2) A, φ ( α ) (cid:3)(cid:1) µ for α = 1 , , D µ X := (d X + [ A, X ]) µ , the field strength F µν := (d A + A ∧ A ) µν and we write | X | := XX † .Since the fields φ ( α ) and X are assumed to be independent from internal coordinates of X , (due toequivariance), the additional dimensions can be integrated out easily, which yields only a prefactorvol ( X , ) for the dimensional reduction of the Lagrangian. In this way, one obtains from a pureYang-Mills theory on M d × X , a Yang-Mills-Higgs action on M d , where the endomorphisms φ ( a ) and X constitute a non-trivial potential provided by the internal geometry of X , .13 .4 Reduction to quiver gauge theory on Q The equivariance condition and the examples of the quiver diagrams in the previous section haveshown that the quiver gauge theory on X , depends on only one of the two quantum numbers ofSU(3). This yields effectively a modified holomorphic chain as quiver diagram or, considered in theoriginal weight diagram of SU(3), a diagram with multiple arrows and degeneracies. As mentionedin the discussion of the Sasaki-Einstein structure on X , in Section 2.2, the space is a U(1)-bundleover the (K¨ahler) space Q , so that it is natural to consider the reduction from the gauge theoryon X , to that on Q by removing the contact direction as a degree of freedom. Since we thendivide by a Cartan subalgebra, the quiver diagram is simply the weight diagram of SU(3) withoutthe degeneracies which have been caused by the weaker conditions on X , . This reduction can beperformed by setting the terms containing e ∧ Θ α or e ∧ Θ ¯ α in the field strength (3.14) to zero.This provides the additional equivariance conditions h X , φ (1) i = − i φ (1) , h X , φ (2) i = − i φ (2) , and h X , φ (3) i = − φ (3) . (3.37)For the reduction to Q , the field X must be proportional to I and setting X = I fixes theaction of the Higgs fields to be φ (1) : ( ν , ν ) ( ν − , ν + 3) φ (2) : ( ν , ν ) ( ν − , ν −
3) (3.38) φ (3) : ( ν , ν ) ( ν − , ν ) . This, indeed, requires the quiver diagrams in Figure 4 to coincide with the weight diagrams of thechosen representations and yields the results from [16, 21]. The endomorphism part of the gaugeconnection, e.g. for the fundamental representation, reads A = ⊗ √ i e − Φ (2) † , − , − Φ (1)1 , − , Φ (2)0 , − , − ⊗ (cid:16) − √ i e − i e (cid:17) Φ (3)1 , − − , − − Φ (1) † , − , − Φ (3) † , − − , − ⊗ (cid:16) − √ i e + i e (cid:17) (3.39)with Φ ( α ) := φ ( α ) ⊗ Θ ¯ α . Since the quiver diagram is the weight diagram of SU(3), the Higgs fieldshave the block shape of the generators (A.1) and the central idea of quiver gauge theory becomesevident: One modifies the bundle (2.8) by inserting compatible endomorphisms φ ( α ) as entries inthe block matrices describing the gauge connection.The Lagrangian of the gauge theory on M d × Q is then given by that on M d × X , without theterms containing commutators with X , L Q = p ˆ g tr k ( F µν ( F µν ) † + 2 X α =1 (cid:12)(cid:12)(cid:12) D µ φ ( α ) (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)h φ (1) , φ (1) † i − i I + √ I (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)h φ (2) , φ (2) † i − i I − √ I (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)h φ (3) , φ (3) † i − i I (cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (2) i − φ (3) (cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (3) i(cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (2) , φ (3) i(cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (2) † i(cid:12)(cid:12)(cid:12) + 4 (cid:12)(cid:12)(cid:12)h φ (1) , φ (3) † i + φ (2) † (cid:12)(cid:12)(cid:12) +4 (cid:12)(cid:12)(cid:12)h φ (2) , φ (3) † i − φ (1) † (cid:12)(cid:12)(cid:12) ) , (3.40) because the vanishing of them is subject to the further equivariance conditions (3.37). Note that the orientation of the Higgs fields depends on the chosen convention of the holomorphic structure; wedenote as Higgs fields φ α the endomorphisms accompanying the anti-holomorphic forms Θ ¯ α . Instantons on the metric cone C ( X , ) The implementation of the equivariance condition (3.17) has determined the general form of thegauge connection, expressed in the associated quiver diagram, and the action functional, but hasnot restricted the entries of the endomorphisms. Further conditions and relations among theendomorphisms can be imposed by studying vacua of the gauge theory, i.e. by minimizing theaction functional (3.33). To this end, we will evaluate the
Hermitian Yang-Mills equations – acertain form of generalized self-duality equations – on the metric cone C ( X , ), as it has been donein similar setups, e.g. [23, 24], and describe their moduli space, following [33, 34, 35]. A very useful tool for obtaining minima of a Yang-Mills functional in gauge theory is to evaluate afirst-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 ∗ F = − ∗ Q ∧ F (4.1)for a 4-form Q , one obtains by taking the differential [5] ∇ A ∧ ∗F + (d ∗ Q ) ∧ F = 0 with ∇ A ∧ ∗F := d ∗ F + A ∧ ∗F + ( − n − ∗ F ∧ A , (4.2)which is the usual Yang-Mills equation with torsion term (d ∗ Q ) ∧ F . Explicit formulae for thechoice of the form Q , in dependence of the geometry of the manifold, such that the torsion termvanishes even if the form Q is not co-closed have been given in [5]. Their construction is basedon the existence of (real) Killing spinors, and thus also applies to Sasaki-Einstein structures. Aconnection 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] Q = 12 ω ∧ ω, (4.3)such that we have Q = − (cid:16) Θ + Θ + Θ (cid:17) = e + e + e . (4.4)The corresponding instanton equation (4.1) on X , is solved by the connection Γ = I ⊗ e , whichwe 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 metriccone, over a Sasaki-Einstein manifold reads [5] Q Z = d τ ∧ P + Q with P = η ∧ ω (4.5)and one thus obtains Q Z = 12 Ω ∧ Ω , (4.6)where Ω is the K¨ahler 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 K¨ahlerform, the 4-form Q Z is self-dual, such that d ∗ Q Z = d Q Z = 0, and the Yang-Mills equation without They provide the form for a whole familiy of compatible metrics and we consider one special value here. Q Z by imposing the (equivalent) Hermitian Yang-Mills equations (HYM) [42, 47, 48] F (2 , = 0 = F (0 , and Ω F := ∗ (Ω ∧ ∗F ) = 0 , (4.7)where F (2 , refers to the (2 , J . The first equation is aholomorphicity condition and the second one can (sometimes) be considered as a stability conditionon vector bundles; they are also known as Donaldson-Uhlenbeck-Yau equations . C ( X , )We consider the same ansatz (3.7) [30], now including also the additional form e τ := d τ := d rr onthe cylinder, A = I e + Y α Θ α + Y ¯ α Θ ¯ α + X e + X τ e τ = I e + Y α Θ α + Y ¯ α Θ ¯ α + Y Θ + Y ¯4 Θ ¯4 , (4.8)where we set Y := 12 ( X τ + i X ) . (4.9)Due to the equivariance, the endomorphisms are “spherically symmetric”, X a = X a ( r ). After theimplementation of the same equivariance conditions as before,[ I , Y ¯1 ] = −√ Y ¯1 , [ I , Y ¯2 ] = √ Y ¯2 , and [ I , Y ¯3 ] = 0 = [ I , Y ¯4 ] , (4.10)the non-vanishing components of the field strength read F αβ = [ Y α , Y β ] − C γαβ Y γ , F ¯ α ¯ β = (cid:2) Y ¯ α , Y ¯ β (cid:3) − C ¯ γ ¯ α ¯ β Y ¯ γ , F α ¯ β = (cid:2) Y α , Y ¯ β (cid:3) − C γα ¯ β Y γ − C ¯ γα ¯ β Y ¯ γ + C α ¯ β Y − C α ¯ β Y ¯4 + i C α ¯ β I , F α = [ Y α , Y ] − r ˙ Y α − C β α Y β , F α ¯4 = [ Y α , Y ¯4 ] − r ˙ Y α + 12 C β α Y β , (4.11) F ¯ α = [ Y ¯ α , Y ] − r ˙ Y ¯ α − C ¯ β α Y ¯ β , F ¯ α ¯4 = [ Y ¯ α , Y ¯4 ] − r ˙ Y ¯ α + 12 C ¯ β α Y ¯ β , F = [ Y , Y ¯4 ] − r (cid:16) ˙ Y − ˙ Y ¯4 (cid:17) . Evaluating the condition F ¯ α ¯ β = 0 leads to[ Y ¯1 , Y ¯2 ] = 2 Y ¯3 , [ Y ¯1 , Y ¯3 ] = 0 = [ Y ¯2 , Y ¯3 ] (4.12)(together with their complex conjugates from F αβ = 0). Thus, this part of the holomorphicitycondition imposes algebraic relations on the quiver. In contrast, from F ¯ α ¯4 = 0 we obtain thefollowing flow equations r ˙ Y ¯1 = − Y ¯1 + 2 [ Y ¯1 , Y ¯4 ] , r ˙ Y ¯2 = − Y ¯2 + 2 [ Y ¯2 , Y ¯4 ] , r ˙ Y ¯3 = − Y ¯3 + 2 [ Y ¯3 , Y ¯4 ] . (4.13)The remaining equation Ω F = 0 requires r (cid:16) ˙ Y − ˙ Y ¯4 (cid:17) = 2 [ Y , Y ¯1 ] + 2 [ Y , Y ¯2 ] + 2 [ Y , Y ¯3 ] + 2 [ Y , Y ¯4 ] − Y − Y ¯4 ) . (4.14) The field X τ associated to the radial direction could be gauged to zero [30]. onstant endomorphisms: For the special case of constant matrices X a , the situation cor-responds to that of the underlying Sasaki-Einstein manifold X , with the parameter τ (or r ,respectively) just as a label of the foliation along the preferred direction of the cone. Gauging thefield X τ to zero, one recovers then from (4.13) and (4.14) exactly the additional equivariance con-ditions (3.38), which appeared in the discussion of the gauge theory on Q . Thus the equivariantgauge theory on Q can be considered as a special instanton solution of the more general setupon C ( X , ). SU(3) -equivariant instantons
For a desription of the moduli space of the equations (4.14) and (4.13), (4.12) under the equivarianceconditions (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 discussionthereof. 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 equationson metric cones over generic Sasaki-Einstein manifolds is discussed in this way. Note that thetreatment [35] uses the canonical connection of [5] as starting point Γ for the gauge connectionand that our connection Γ = I ⊗ e in (4.8) differs from it (see Appendix A.2). This is why somemodifications, in comparison with [35], will appear in our discussion .Changing the argument in the flow equations to τ = ln( r ) and setting Y ¯ α =: e − τ W α , for α = 1 , , Y ¯3 =: e − τ W , and Y ¯4 =: e − τ Z (4.16)eliminates the linear terms in (4.13) and (4.14). Defining s := − e − τ = − r − ∈ ( −∞ ,
0] yieldsNahm-type equationsd W d s = 2 [ W , Z ] , d W d s = 2 [ W , Z ] , d W d s = 2 [ W , Z ] , (4.17)[ W , W ] = 2 W and [ W , W ] = 0 = [ W , W ] , (4.18)(from F (2 , = 0) and µ ( W α , Z ) := dd s (cid:16) Z + Z † (cid:17) + 2 X α =1 λ α ( s ) h W α , W † α i + 2 h Z, Z † i = 0 (4.19)(from Ω F = 0), with the non-negative functions λ ( s ) = λ ( s ) := ( − s ) − and λ ( s ) := ( − s ) − . (4.20)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 The vanishing of the contributions stemming from the form e is obvious from the Yang-Mills action (3.33) andthe instanton condition (4.1). Due to ∗ Q ∝ e those terms do not contribute to the action for instanton solutions,and this is equivalent to the further equivariance conditions (3.37). Of course, using the canonical connection of [5] yields the results of [35] also for X , . However, for the discussionof the quiver diagrams in Section 3 the connection Γ = I ⊗ e 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 theSasaki-Einstein structure of X , ; see Appendix A.2. For the canonical connection (A.14) of a seven-dimensional Sasaki-Einstein manifold, the matrices scale as [35] Y ¯ α = e − τ W α for α = 1 , , Y ¯4 = e − τ Z. (4.15) W α W gα := gW α g − , for α = 1 , , Z Z g := gZg − − (cid:18) d g d s (cid:19) g − (4.21)with g ∈ C (( −∞ , , GL( C , k )). A local solution of (4.17) can be attained by applying the gauge Z g = 0 ⇒ Z = 12 g − d g d s , (4.22)so that – due to the complex equations (4.17) – the gauge transformed matrices W gα must beconstant, W α = g − T α g. (4.23)To obtain solutions, one has to choose these constant matrices such that they satisfy (4.18). Onespecial choice, for instance, could be to set T = 0 and take for T and T elements of a Cartansubalgebra. Note that not only the scaling in (4.16) is different from that in [35], but also theconditions (4.18): There all three matrices have to commute with each other and, thus, also T canbe 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 equationof motion (i.e. δ L ∝ µ ) of the Lagrangian L = 12 Z I d s n | Z + Z † | + 2 λ ( s ) | W | + 2 λ ( s ) | W | + 2 λ ( s ) | W | o . (4.24)Employing (4.22) and (4.23), one can re-write this Lagrangian as [33, 35] L ( h ) = 12 Z I d s (
14 tr (cid:18) h − d h d s (cid:19) + 2 X α =1 λ α tr (cid:16) hT α h − T † α (cid:17)) with h := g † g. (4.25)Since the potential term in this Lagrangian is non-negative, the existence of a solution to (4.19) asequation of motion follows from a variational problem [33]. One still has to ensure some technicalaspects: the uniqueness of the solutions, the existence of the gauge transformation and the La-grangian on the entire interval ( −∞ , µ . In the reference [35] theseproperties are proven, given that for framed instantons, i.e. those with h = 1 at the boundary ofthe interval ( −∞ , ∃ g ∈ U( k ) : lim s →−∞ W α = Ad( g ) T α (4.26)is satisfied for constant matrices obeying the conditions (4.18). For their constraints, i.e. mutuallycommuting matrices T α , it is shown that the moduli space can be expressed as diagonal orbit in aproduct of coadjoint orbits [35]. In our case, however, due to the different constraints (4.12), thesituation might be more involved. But we can at least conclude that (4.23) provides local solutionsof the Nahm-type equations (4.17)-(4.19).Moreover, it was shown in the references (see again [35]) that the real equation (4.19) can be consid-ered as a moment map µ : A , → Lie ( G ) from the space A , of framed solutions to the complexequations into the Lie algebra of the framed gauge group G . This result still holds here, despitethe difference in the connections that are used. Hence the moduli space of equivariant HermitianYang-Mills instantons on metric cones over Sasaki-Einstein manifolds admits the description asK¨ahler quotient [35] M = µ − (0) / G . (4.27)18 Summary and conclusions
In this article we studied the SU(3)-equivariant dimensional reduction of gauge theories over theSasaki-Einstein manifold X , . We interpreted the condition of equivariance, which had alreadyoccurred in articles [30, 31] on Spin(7)-instantons on cones over Aloff-Wallach spaces X k,l , in termsof quiver diagrams, and we discussed the general construction of the quiver bundles. This yieldeda new class of Sasakian quiver gauge theories. The associated quiver diagram of this gauge theoryis a “doubled modified holomorphic chain” , consisting of two arrows between adjacent vertices andtwo loops at each vertex, and three explicit examples thereof were considered in the article. For thecomparison with the gauge theory on the underlying K¨ahler manifold Q we studied the quiversalso in the entire weight diagram of G = SU(3), which implied degeneracies of the arrows. Thisbehavior is similar to the case [15, 24] of the five-dimensional Sasaki-Einstein manifold T , over C P × C P . The reduction to the gauge theory on Q led to the correct, expected result for thequiver 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 , ). The resulting flow equations have been re-written ina form similar to Nahm’s equations, which allowed a discussion based on Kronheimer’s [34] andDonaldson’s [33] work and its generalized application to equivariant HYM instantons on Calabi-Yaucones [35]. Since we formulated the quiver gauge theory by using an instanton connection differentfrom that of [5] in the gauge connection, some modifications appeared. While the real equation canbe still interpreted as a moment map for framed instanton solutions, as in [35], and, thus, leads toa description of the moduli space as a K¨ahler quotient, the description based on coadjoint orbitsis more involved: The HYM equations impose a non-trivial commutation relation on the gaugetransformed matrices, in contrast to [35], where they have to commute with each other. Thus, thebehavior is more complicated and further effort would be needed to study the consequences thereofin detail. Acknowledgements
The author thanks Olaf Lechtenfeld, Alexander Popov, and Marcus Sperling for fruitful discussionsand comments. This work was done within the project supported by the Deutsche Forschungs-gemeinschaft (DFG, Germany) under the grant LE 838/13 and was supported by the ResearchTraining Group RTG 1463 “Analysis, Geometry and String Theory” (DFG).
A Appendix
A.1
SU(3) generators and structure constants
The generators defined by the choice of the 1-forms in (2.8) read I − := √ , I − := √ , I − := , I := i − , (A.1) I +¯1 := √ −
10 0 00 0 0 , I +¯2 := √ − , I +¯3 := −
10 0 0 , I := i √ − − , and we define the structure constants via the commutation relations (cid:2) − i I j , I − α (cid:3) = C βjα I − β , (cid:2) − i I j , I +¯ α (cid:3) = C ¯ βj ¯ α I +¯ β , h I − α , I − β i = C γαβ I − γ , h I +¯ α , I +¯ β i = C ¯ γ ¯ α ¯ β I +¯ γ , h I − α , I +¯ β i = − i C jα ¯ β I j + C γα ¯ β I − γ + C ¯ γα ¯ β I +¯ γ . (A.2)19he non-vanishing structure constants are [30] C = − C = − − C ¯12¯3 = C ¯21¯3 , C = 2 = C ¯3¯1¯2 ,C = C = 1 = − C ¯17¯1 = − C ¯27¯2 , C = 2 = − C ¯37¯3 , (A.3) C = − C = −√ − C ¯18¯1 = C ¯28¯2 , C = 0 = C ¯38¯3 ,C = C = C = − , C = − C = √ . By the Maurer-Cartan equations,dΘ α = − i C αjβ − C αβγ Θ βγ − C αβ ¯ γ Θ β ¯ γ , d e j = i C jβ ¯ γ Θ β ¯ γ , (A.4)they yield again the structure equations (2.9). In terms of real formsΘ =: e − i e , Θ =: e − i e , and Θ =: e − i e , (A.5)the structure equations readd e = √ e − e − e − e , d e = −√ e + e − e + e , d e = −√ e − e + e + e , d e = √ e + e + e − e , d e = − e − e − e , d e = 2 e − e − e , d e = 2 e + 2 + 2 , d e = − √ e + 2 √ e . (A.6) A.2 Connections and instanton equation
On the homogeneous space X , = G/H = SU(3) / U(1) , we consider the connection with torsion T ( X, Y ) := − [ X, Y ] m (A.7)for vector fields X , Y on G/H , where [ · , · ] m denotes the projection of the commutator to thecomplement m ; this yields the following torsion components T µρσ = − C µρσ for µ, ρ, σ = 1 , . . . , . (A.8)Using the structure equations and the Maurer-Cartan equationd e µ = − C µρσ e ρσ = − C µ ρ e ∧ e ρ + 12 T µρσ e ρσ (A.9)=: − Γ µρ ∧ e ρ + T µ , one obtains the conection 1-formsΓ µρ = C µ ρ e ⇒ Γ = I ⊗ e , (A.10)which is the U(1)-connection used in the ansatz for the gauge connection in (4.8). Its curvature F Γ = dΓ + Γ ∧ Γ = 2 √ I (cid:0) e − e (cid:1) (A.11)satifies the instanton equation ∗ F Γ = − (cid:0) e + e + e (cid:1) ∧ e ∧ F Γ = − ∗ Q ∧ F Γ . (A.12)20or the 4-form Q = e + e + e from (4.4). Because of(d ∗ Q ) ∧ F Γ ∝ (cid:0) e + e + e (cid:1) ∧ (cid:0) e − e (cid:1) = 0 (A.13)the torsion term in (4.2) vanishes, so that the usual torsion-free Yang-Mills equation is obtained.This is the intention of using special geometric structures. Note, however, that our U(1)-connectiondoes not coincide with what is defined as canonical connection of a Sasaki-Einstein manifold in [5].Its torsion for a seven-dimensional Sasaki-Einstein manifold is defined via T a = 23 P aµν e µν for a = 1 , . . . , T = P µν e µν with P := η ∧ ω = e ∧ ω. (A.14)Since this definition does not require a homogeneous space, but only exploits the Sasaki-Einsteinstructure, it allows for general discussions of gauge theories on those spaces, as used for examplein [35, 38]. On X , this canonical connection is expressed by the connection matrixd Θ Θ Θ e = i e + √ e − Θ ¯2 i e − √ e Θ ¯1 − Θ − i e
00 0 0 0 ∧ Θ Θ Θ e + ~T . (A.15)Thus, the canonical connection is adapted to the SU(3) structure of X , .On the metric cone (with the rescaled forms ˜ e µ := re µ ) or on the conformally equivalent cylinder,respectively, the connection Γ = I ⊗ e is still an instanton for the form Q Z = 12 Ω ∧ Ω = r (cid:0) e + e + e τ + e + e τ + e τ (cid:1) (A.16)= ˜ e + ˜ e + ˜ e τ + ˜ e + ˜ e τ + ˜ e τ = ∗ Q Z because we have ∗ (cid:0) ˜ e − ˜ e (cid:1) = − (cid:0) ˜ e − ˜ e (cid:1) ∧ ˜ e τ = − Q Z ∧ (cid:0) ˜ e − ˜ e (cid:1) . (A.17) A.3 Details of the moduli space description
This section provides some technical aspects of the description in Section 4.3. For details, the readershould consult the references, in particular [35]. To show that the real equation follows (over somerange) as equation of motion of the Lagrangian (4.24), one considers [33] the variation of thematrices W a with respect to g close to the identity. Writing g = 1 + δg , where δg is self-adjoint,one obtains from the gauge transformation (4.21) δW α = (1 + δg ) W α (1 + δg ) − − W α = [ δg, W α ] for α = 1 , , δZ = (1 + δg ) Z (1 + δg ) − − Z −
12 dd s (1 + δg ) (1 + δg ) − = [ δg, Z ] −
12 dd s δg. (A.19)Using the result (A.18), one derives the following variation δ Z d s | W α | := δ Z d s tr W α W † α = 2Re Z d s tr δ ( W α ) W † α = 2Re Z d s tr [ δg, W α ] W † α = 2Re Z d s tr δg h W α , W † α i for α = 1 , , δ Z d s | Z + Z † | = 2Re Z d s tr (cid:18)h δg, Z − Z † i − dd s δg (cid:19) (cid:16) Z + Z † (cid:17) = 2Re Z d s tr δg (cid:18) dd s (cid:16) Z + Z † (cid:17) + 2 h Z, Z † i(cid:19) . (A.21)Putting the results from (A.20) and (A.21) together with the prefactors λ α ( s ), (4.20) yields theLagrangian (4.24) and shows that the real equation is the equation of motion thereof. That theLagrangian can be defined for the entire range s ∈ ( −∞ ,
0] and other technical issues can be foundin [35]. The only quantitive difference is the concrete form of the factors λ α ( s ) but this does notaffect the general line of reasoning. References [1] M.F. Atiyah, N.J. Hitchin, and I.M. Singer, “Self-duality in four-dimensional Riemannian geometry”, Proc.R. Soc. Lond. A. (1978) 425.[2] E. Corrigan, C. Devchand, D.B. Fairlie, and J. Nuyts, “First-order equations for gauge fields in spaces ofdimension greater than four”, Nucl. Phys. B (1983) 452.[3] R.S. Ward, “Completely solvable gauge-field equations in dimensions greater than four”, Nucl. Phys. B (1984) 381.[4] C.M. Hull, ”Higher dimensional Yang-Mills theories and topological terms”, Adv. Theor. Math. Phys. (1998)619, [arXiv:hep-th/9710165v2].[5] D. Harland and C. N¨olle, “Instantons and Killing spinors”, JHEP (2012) 082, [arXiv:1109.3552].[6] U. Gran, G. Papadopoulos, and D. Roest, “Supersymmetric heterotic string backgrounds”, Phys. Lett. B (2007) 119, [arXiv:0706.4407].[7] D. Kapetanakis and G. Zoupanos, “Coset space dimensional reduction of gauge theories”, Phys. Rept. (1992) 4.[8] L. Alvarez-C´onsul and O. Garc´ıa-Prada, “Dimensional reduction and quiver bundles”, J. Reine Angew. Math. (2003) 1, [arXiv:math-dg/0112160].[9] L. Alvarez-C´onsul and O. Garc´ıa-Prada, “Dimensional reduction, SL(2 , C )-equivariant bundles and stableholomorphic chains”, Int. J. Math. (2001) 159, [arXiv:math-dg/0112159].[10] B.P. Dolan and R.J. Szabo, “Equivariant dimensional reduction and quiver gauge theories”, Gen. Rel. Grav. (2010) 2453, [arXiv:hep-th/1001.2429].[11] O. Lechtenfeld, A.D. Popov, and R.J. Szabo, “Quiver gauge theory and noncommutative vortices”, Prog.Theor. Phys. Suppl. (2007) 258, [arXiv:0706.0979][12] A.D. Popov and R.J. Szabo, “Quiver gauge theory of nonabelian vortices and noncommutative instantons inhigher dimensions”, J. Math. Phys. (2006) 012306, [arXiv:hep-th/0504025].[13] I. Biswas, “Holomorphic Hermitian vector bundles over the Riemann sphere”, Bull. Sci. Math. (2008)246.[14] B.P. Dolan and R.J. Szabo, “Dimensional reduction, monopoles and dynamical symmetry breaking”, JHEP (2009) 059, [arXiv:0901.2491].[15] O. Lechtenfeld, A.D. Popov, and R.J. Szabo, “Rank two quiver gauge theory, graded connections and non-commutative vortices,” JHEP (2006) 054, [arXiv:hep-th/0603232].[16] O. Lechtenfeld, A.D. Popov, and R.J. Szabo, “SU(3)-equivariant quiver gauge theories and nonabelian vor-tices”, JHEP (2008) 093, [arXiv:0806.2791v2].[17] C. Boyer and K. Galicki, Sasakian Geometry (Oxford University Press, 2008).
18] J. Sparks, “Sasaki-Einstein manifolds”, Surv. Diff. Geom. (2011) 265, [arXiv:1004.2461][19] D. Joyce, “Lectures on Calabi-Yau and special Lagrangian geometry”, Preprint arXiv: math/0108088.M. Gross, D. Huybrechts, and D. Joyce, Calabi-Yau manifolds and related geometries , (Springer, 2003).[20] B.R. Greene, “String theory on Calabi-Yau manifolds”, Preprint arXiv: hep-th/9702155.[21] A.D. Popov and R.J. Szabo, “Double quiver gauge theory and nearly Kahler flux compactifications”, JHEP (2012) 033, [arXiv:1009.3208v2].[22] O. Lechtenfeld, A.D. Popov, and R.J. Szabo, “Sasakian quiver gauge theories and instantons on Calabi-Yaucones”, Preprint arXiv:1412.4409.[23] O. Lechtenfeld, A.D. Popov, M. Sperling, and R.J. Szabo, “Sasakian quiver gauge theories and instantons oncones over lens 5-spaces”, Nucl. Phys. B (2015) 848, [arXiv:1506.02786].[24] J.C. Geipel, O. Lechtenfeld, A.D. Popov, and R.J. Szabo, “Sasakian quiver gauge theories and instantons onthe conifold”, Nucl. Phys. B (2016), doi:10.1016/j.nuclphysb.2016.04.016, [arXiv:1601.05719].[25] J.P. Gauntlett, D. Martelli, J. Sparks, and D. Waldram, “Sasaki-Einstein metrics on S × S ”,Adv. Theor. Math. Phys. (2004) 711, [arXiv:hep-th/0403002].[26] D. Martelli and J. Sparks, “Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFTduals”, Commun. Math. Phys. (2006) 51, [arXiv:hep-th/0411238].[27] S. Aloff and N.R. Wallach, “An infinite family of distinct 7-manifolds admitting positively curved Riemannianstructures”, Bull. Amer. Math. Soc. (1975) 93.[28] F.M. Cabrera, M.D. Monar, and A.F. Swann, “Classification of G -Structures”, J. London Math. Soc. (2) (1996) 407.[29] T. Friedrich, I. Kath, A Moroianu, and U. Semmelmann, “On nearly parallel G -structures”, J. Geom. Phys. (1997) 259.[30] A.S. Haupt, T.A. Ivanova, O. Lechtenfeld, and A.D. Popov, “Chern-Simons flows on Aloff-Wallach spaces andSpin(7) instantons”, Phys. Rev. D (2011) 105028, [arXiv:1104.5231v1].[31] A.S. Haupt, ”Yang-Mills solutions and Spin(7)-instantons on cylinders over coset spaces with G -structure“,JHEP (2016) 038, [arXiv:1512.07254v1].[32] L. Castellani, L.J. Romans, and N.P. Warner, “A classification of compactifying solutions for d = 11 super-gravity”, Nucl. Phys. B (1984) 429.[33] S.K. Donaldson, “Nahm’s equations and the classification of monopoles”, Commun. Math. Phys. (1984)387.[34] P.B. Kronheimer, “A hyper-K¨ahlerian structure on coadjoint orbits of a semisimple complex group”, J. LondonMath. Soc. (1990) 193.[35] M. Sperling, “Instantons on Calabi-Yau cones”, Nucl. Phys. B (2015) 354, [arXiv:1505.01755].[36] C.P. Boyer, K. Galicki, and B.M. Mann, “The geometry and topology of 3-Sasakian manifolds”, J. reine angew.Math. (1994) 183.[37] B. Florea, S. Kachru, J. McGreevy, and N. Saulina, “Stringy instantons and quiver gauge theories”, JHEP (2007), 024, [arXiv:hep-th/0610003v3].[38] T.A. Ivanova and A.D. Popov, “Instantons on special holonomy manifolds”, Phys. Rev. D (2012) 105012,[arXiv:1203.2657].[39] S. Bunk, O. Lechtenfeld, A.D. Popov, and M. Sperling, “Instantons on conical half-flat 6-manifolds”, JHEP (2015) 030, [arXiv:1409.0030].[40] S. Bunk, T.A. Ivanova, O. Lechtenfeld, A.D. Popov, and M. Sperling, “Instantons on sine-cones over Sasakianmanifolds”, Phys. Rev. D (2014) 065028, [arXiv:1407.2948].
41] I. Bauer, T.A. Ivanova, O. Lechtenfeld, and F. Lubbe, “Yang-Mills instantons and dyons on homogeneous G -manifolds”, JHEP (2010) 044, [arXiv:1006.2388].[42] A.D. Popov, “Hermitian Yang-Mills equations and pseudo-holomorphic bundles on nearly K¨ahler and nearlyCalabi-Yau twistor 6-manifolds”, Nucl. Phys. B (2010) 594, [arXiv:0907.0106].[43] H. Derksen and J. Weyman, “Quiver representations”, Notices of the AMS (2) (2005) 200.[44] R. Schiffler, Quiver representations , (Springer, 2014).[45] S. Kobayashi and K. Nomizu,
Foundations of Differential Geometry , Volume 1 (Interscience Publishers, 1963).[46] W. Fulton and J. Harris,
Representation Theory , (Springer, 1991).[47] S.K Donaldson, “Anti-self dual Yang-Mills connections over complex algebraic surfaces and stable vectorbundles”, Proc. Lond. Math. Soc. (1985) 1.[48] K. Uhlenbeck and S.-T. Yau, “On the existence of Hermitian Yang-Mills connections in stable vector bundles”,Commun. Pure Appl. Math. (1986) 257.(1986) 257.