ELSEVIER

CrossMaik

Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 899 (2015) 848-903

www. elsevier. com/locate/nuclphysb

Sasakian quiver gauge theories and instantons on cones

over lens 5-spaces

Olaf Lechtenfelda, Alexander D. Popova, Marcus Sperlinga-*, Richard J. Szabob c d

a Institut für Theoretische Physik and Riemann Center for Geometry and Physics, Leibniz Universität, Hannover Appelstraße 2, 30167 Hannover, Germany b Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK c Maxwell Institute for Mathematical Sciences, Edinburgh, UK d The Higgs Centre for Theoretical Physics, Edinburgh, UK

Received 17 June 2015; accepted 2 September 2015

Available online 7 September 2015

Editor: Herman Verlinde

Abstract

We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, which are projective planes. We describe the Higgs branches of these quiver gauge theories as moduli spaces of spherically symmetric instantons which are SU(3)-equivariant solutions to the Hermitian Yang-Mills equations on the associated Calabi-Yau cones, and further compare them to moduli spaces of translationally-invariant instantons on the cones. We provide an explicit unified construction of these moduli spaces as Kahler quotients and show that they have the same cyclic orbifold singularities as the cones over the lens 5-spaces.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

* Corresponding author.

E-mail addresses: lechtenf@itp.uni-hannover.de (O. Lechtenfeld), alexander.popov@itp.uni-hannover.de (A.D. Popov), marcus.sperling@itp.uni-hannover.de (M. Sperling), R.J.Szabo@hw.ac.uk (R.J. Szabo).

http://dx.doi.org/10.1016/j.nuclphysb.2015.09.001

0550-3213/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction and summary

Sasaki-Einstein 5-manifolds M5 have played a prominent role in developments in string theory. For example, type IIB string theory on AdS5 x M5 is conjecturally dual to the 4-dimensional N = 1 superconformal worldvolume field theory on a stack of D3-branes placed at the apex singularity of the 6-dimensional Calabi-Yau cone C(M5) over M5 [1-6]. They have moreover served as interesting testing grounds for the suggestion that maximally supersymmetric Yang-Mills theory in 5 dimensions contains all degrees of freedom of the 6-dimensional (2, 0) superconformal theory compactified on a circle [7,8]. Metrics on the non-compact spaces C(M5) are also known explicitly [9-11], in contrast to the compact examples of Calabi-Yau string com-pactifications.

In this paper we derive new quiver gauge theories via equivariant dimensional reduction over M5 and describe their vacua in terms of moduli spaces of generalised instantons on the cones C(M5)1; such instantons play a central role in supersymmetric compactifications of heterotic string theory [14]. This extends the constructions of [15] which dealt with the case of 3-dimensional Sasaki-Einstein manifolds, wherein these field theories were dubbed as "Sasakian" quiver gauge theories. The only Sasaki-Einstein 3-manifolds are the ADE orb-ifolds S3/V of the 3-sphere by a discrete subgroup V of SU(2). They have natural extensions as ADE orbifolds M5 = S5/V of the 5-sphere which preserve N = 2 supersymmetry [2,1]. In the following we shall be interested in generalisations of these orbifolds to cases where V is instead a finite subgroup of SU(3). The corresponding affine cones C(S5/V) play a central role in the McKay correspondence for Calabi-Yau 3-folds [16,17]. Moreover, the BPS configurations in the worldvolume gauge theories on D-branes located at points of Calabi-Yau manifolds which are resolutions of the orbifolds C3/V [18,19] are parameterised by moduli spaces of translationally-invariant solutions of Hermitian Yang-Mills equations on C3/V, which coincide with Calabi-Yau spaces that are partial resolutions of these orbifolds [20,21]. Drawing from the situation in the 3-dimensional case [15], it is natural to expect the same sort of similarities between these moduli spaces and those of "spherically symmetric" instantons on cones over any Sasaki-Einstein 5-manifold, where the generalised instanton equations can also be reduced to generalised Nahm equations of the form considered in [22].

On general grounds, any quasi-regular Sasaki-Einstein 5-manifold M5 is a U(1) V-bundle over a 4-dimensional Kahler-Einstein orbifold M4. In this paper we consider the special case where M5 = S5/V with V = Zq+1 c SU(3) a cyclic group. Then M4 = CP2 and we can exploit the constructions from [23] which thoroughly studies SU(3)-equivariant dimensional reduction over the Kahler coset space CP2 = SU(3)/S(U(2) x U(1)). We shall construct the relevant quiver bundles and study the corresponding quiver gauge theories in detail; these quivers are new and we relate them explicitly to those arising from dimensional reduction over the leaf spaces CP2 of the characteristic foliation of S5/Zq+1. In particular, we will compare the moduli spaces of spherically symmetric and translationally-invariant instantons on the cones C(S5/Zq+1) = C3/Zq+1, and show that they contain the same orbifold singularities C3/Zn (where N is the rank of the gauge group) analogously to the cases of [15]. In analogy to the interpretations of [15], our constructions thereby shed light on the interplay between the Higgs branches of the worldvolume quiver gauge theories on D^-branes which probe a set of D(^ + 6)-branes wrapping a (partial) resolution of C(S5/Zq+1), and BPS states of the quiver

1 The analogous instanton moduli spaces were studied by [12] for the 3-dimensional case and by [13] in arbitrary (odd)

dimension.

gauge theories on pairs of + 4)-branes wrapping C(S5/Zq+1) which transversally intersect + 6)-branes at the apex of the cone C(S5/Zq+1). In this scenario, it is the codimensionality of the D-brane bound states which selects both the quiver type and the abelian category in which the quiver representation is realised; in particular, the arrows of the quivers keep track of the massless bifundamental transverse scalars stretching between constituent fractional D-branes at the vertices.

The outline of the remainder of this paper is as follows. In Section 2 we provide a detailed description of the geometry of the orbifold S5/Zq+1 using its realisation as both a coset space and as a Sasaki-Einstein manifold. In Section 3 we give a detailed description of the quiver gauge theory induced via SU(3)-equivariant dimensional reduction over S5/Zq+1, including explicit constructions of the quiver bundles and their connections as well as the form of the action functional. We then describe the Higgs branch vacuum states of quiver gauge theories on the cone C(S5/Zq+l) as SU(3)-equivariant solutions to the Hermitian Yang-Mills equations in Section 4 and as translationally-invariant solutions in Section 5. In Section 6 we compare the two quiver gauge theories in some detail, including a contrasting of their quiver bundles and explicit universal constructions of their instanton moduli spaces as Kahler quotients. Four appendices at the end of the paper contain technical details and results which are employed in the main text.

2. Sasaki-Einstein geometry

In this section we shall introduce the basic geometrical constructions that we shall need throughout this paper.

2.1. Preliminaries

Sasakian manifolds M2n+1 of dimension 2n + 1 are contact manifolds which form a natural bridge between two different Kahler spaces M2n and M2n+2 of dimensions 2n and 2n + 2, respectively. On the one hand, the metric cone over a Sasakian manifold M 2n+1 gives a Kahler space M2n+2 = C(M2n+1). On the other hand, the Reeb vector field on M2n+1 defines a foliation of M2n+1 and the transverse space M2n is also Kahler. For further details, see for example [24].

A Riemannian manifold is called Einstein if its Ricci tensor is a scalar multiple of its metric. A Sasakian manifold which is also Einstein is called a Sasaki-Einstein manifold [24]. Since the cone over an Einstein manifold is also an Einstein space, the metric cone over a Sasaki-Einstein manifold is a Calabi-Yau space and in this case the transverse space M2n is Kahler-Einstein. Because of the R>0 scaling action on the cones we can write the Calabi-Yau metric as

d^C(M2n+i) = dr2 + r2ds2M2n+i , (2.1)

where r e R>0 and the tensor dsM2n+1 defines a metric on the intersection M2n+1 of the cone with the unit sphere in Cn+2.

Given a Riemannian manifold M and a finite group T acting isometrically on M, one can, loosely speaking, define the Riemannian space of T-orbits M/ T, which is called an orbifold or sometimes V-manifold, see for instance [24]. The notion of fibre bundle can be adapted to the category of orbifolds, and we follow [24] in calling them V-bundles. Any quasi-regular Sasaki-Einstein manifold M2n+1 is a principal U(1) V-bundle over its transverse space M2n. In this case the Sasaki-Einstein metric can be expressed as

dsM2n+1 = dsM2n + n ® n , (2.2)

where d^M^ is the (pullback of the) Kahler-Einstein metric of M2n, and n is the contact 1-form which is a connection on the fibration M2n+1 ^ M2n of curvature dn = -2rnM2n with rnM2n the Kahler form of the base M2n.

2.2. Sphere S5

The 5-dimensional sphere S5 has two realisations: Firstly, as the coset space S5 = SU(3)/ SU(2) and, secondly, as a principal U(1)-bundle over the complex projective plane CP2. As such, we have the chain of principal bundles

SU(3) ^ S5 CP2 . (2.3)

Our description of S5 will be based on the principal U(1)-bundle over CP2, and we will construct a flat connection on the principal SU(2)-bundle over S5 by employing this feature.

2.2.1. Connections on CP2

Let us consider a local section U over a patch U0 of CP2 for the principal bundle SU(3) ^ CP2. For this, let G = SU(3) and H = S(U(2) x U(1)) c G, and consider the principal bundle associated to the coset G/H given by

G = SU(3) H=S(U(2)xU(1)); G/H = CP2 . (2.4) By the definition of the complex projective plane

CP2 = C3 / -={[z1 : z2 : z3]e C3 : [z1: z2 : z3]-[Xz1 : Xz2 : Xz3], X e C*}, (2.5) one introduces on the patch U0 = {[z1: z2 : z3] e CP2 : z3 = 0} the coordinates

y2 \ z2/z

Y := ( "2 M . (2.6)

Define a local section on U0 of the principal bundle (2.4) via [23] U : U0 SU(3)

1 ( K Y \ (2.7)

Y U(Y) := ^ (-ft 1) ,

with the definitions

A := y I2--— Y Yt and y := V1 + Yt Y . (2.8)

From these two definitions, one observes the properties

At = A , A2 = y212 - Y Yt, AY = Y and Yt A = ^t. (2.9)

It is immediate from (2.9) that U as defined in (2.7) is SU(3)-valued. One can define a flat connection A0 on the bundle (2.4) via

A0 = UtdU T -fl), (2.10)

with the definitions

B := — (A dA + Y dYt - 112 d(Yt Y )) , (2.11a)

1 - - T 1 -t -

P:=-r A dY and pT := -r dY1A , (2.11b)

a :=-^2 tdl'- dY1 = - . (2.11c)

That U e SU(3) directly implies the vanishing of the curvature 2-form F0 = dA0 + A0 A A0, which is equivalent to the set of relations

dB + B A B = ¡Ha pT and da = -p TA p = pt A p, (2.12a)

dp + B A p = ¡¡A a and dp T + p TA B = a A pT . (2.12b)

As elaborated in [23,25], B can be regarded as a u(2)-valued connection 1-form and a as a u(1)-valued connection. Consequently, one can introduce an su(2)-valued connection B(1) by removing the trace of B. An explicit parametrisation yields

B(1) := B - 2tr(B) 12 ^(Bn BBn) with tr(B) = a , Bn = -Bn . (2.13)

The geometry of CP2 including the properties of the SU(3)-equivariant 1-forms , the instanton connection B(1) and the monopole connection a are described in Appendix A.

2.2.2. Connections on S5

Consider now the principal SU(2)-bundle

G = SU(3) K=SU(2\ G/K = S5 , (2.14)

where K c G. Then the section U from (2.7) can be modified as U: U0 x [0, 2n) SU(3)

ei * 12 0 0 e-

(Y, *) U(Y, *) := U(Y) [ n 2 " ) = U(Y) Z(*), (2.15)

which is a local section of the bundle (2.14) on the patch U0 x [0,2n) with coordinates {y1, y2, *}. Note that Z-1 = Zt = Z and det(Z) = 1, and furthermore Z(*) Z(f) = Z(f) Z(*) = Z(f + *), which implies that Z realises the embedding U(1) ^ SU(3); this also shows that U e SU(3). The modified (flat) connection A on the bundle (2.14) and the corresponding curvature F are given as

.-UtdU- AH(7-1)A^7tH7- ( B + i 12d* p e-3i*

A := UtdU = Ad(Z"1)A0 + ZtdZ ^-¡T e2i* -(a + 2id*J , (2.16a)

F := dA + A a A = Ad(Z-1)F0

dB + Ba B - ¡A pT (dpi + B A p-¡A a) e-3i<p"

(2.16b)

- (dpT + pT A B - a A pT) e3i* -da - pT A p J 0

Again the flatness of A yields the same set of identities (2.12), because F differs from F only by the adjoint action of Z-1.

2.2.3. Contact geometry of S5

By construction, the base space of (2.14) is a 5-sphere. The aim now is to choose a basis of the cotangent bundle T*S5 over the patch U0 x [0, 2n) such that one recovers the Sasaki-Einstein structure on S5. For this, we start with the identifications

Pp := P1 e3ip = e1 + i e2 , Pp := P2e3ip = e3 + i e4 and k e5 := 1 a + idp, (2.17)

where k e C is a constant to be determined and the 1-forms P' originate from the complex cotangent space T*y y^CP2 at a point (Y, Y) e U0 c CP2. Next we define the forms

«1 := e14 + e23 , «2 := e31 + e24 , «3 := e12 + e34 and n := e5 , (2.18)

where generally eav"ak = ea1 A ••• A eak. In the basis (2.17), one obtains

«1 = 2 (Pp A Pp - Pp A Pp) , «2 = (pp A Pi + pp A pp) and

«3 = -!! (Pp A Pp + Pp A Pp) . (2.19)

Note that «3 coincides (up to a normalisation factor) with the Kahler form on CP2, cf. Appendix A. The exterior derivatives of P'p and P'p are given as

dPp = e3ip dP' - 3iPp a dp and dPp = e-3ip dP' + 3i Pp a dp. (2.20)

The distinguished 1-form n is taken to be the contact 1-form dual to the Reeb vector field of the Sasaki-Einstein structure. At this stage, the choice of the quadruple (n, «1, «2, «3) defines an SU(2)-structure on the 5-sphere [26]. For it to be Sasaki-Einstein, one needs the relations

d«i = 3n A «2 , d«2 = —3n A «i and dn = —2«3 (2.21) to hold [27]. Employing (2.12) one arrives at

d«1 = 6iKn A «2 and d«2 = —6iKn A «1 , (2.22a)

dn = K «3 and d«3 = 0 . (2.22b)

Consequently, the coframe {n, Pp, Pp} yields a Sasaki-Einstein structure on S5 if and only if k = —3, and from now on this will be the case.

2.3. Orbifold S5/Zq+1

Our aim is to now construct a principal V-bundle over the orbifold S5/Zq+1 by the following steps: Take the principal SU(2)-bundle n : G = SU(3) ^ SU(3)/SU(2) = S5, which is SU(2)-equivariant. Embed Zq+1 U(1) c SU(3) such that U(1) commutes with SU(2) c SU(3), and define a Zq+1-action y on S5. The action y : Zq+1 x S5 ^ S5 can be lifted to an action y : Zq+1 x G ^ G with an isomorphism on the SU(2) fibres induced by this action. The crucial point is that the fibre isomorphism is trivial as SU(2) commutes with Zq+1 by construction. Hence one can consider the Zq+1-projection of G to the principal SU(2) V-bundle G, which is schematically given as

G-y—^G

(2.23)

S5 /Zq +1

With an abuse of notation, we will denote the V-bundles obtained via Zq+i-projection by the same symbols as the fibre bundles they originate from; only Zq+1-equivariant field configurations survive this orbifold projection.

A section U of the principal V-bundle (2.23) is obtained by a (further) modification of the section (2.7) as

U : Uo x [0,q+i) SU(3)

(Y,q+) — ) := U(Y)

e «+1 12

_2 UPe 2 q+1

= U(Y)Zq+1 (P).

(2.24)

Here * e [0, 2n) is again the local coordinate on the S 1-fibration S5 U-1 CP2; hence e<+ e S1 /Zq+1. Analogously to the q = 0 case of S5 above, one can prove that Zq+1 realises the embedding S1/Zq+11 U(1) c SU(3), and U e SU(3). As before, one computes the connection 1-form A and the curvature F of the flat connection on the V-bundle (2.23). This yields

À := f/t dU = Ad(Z-11)Ao + Zq+1 dZq+1 =

b+12 q+1

T 3 -iP-

?T o3 q+ 1

P e 3 q+T

-HT e3 q+1 - (a + 2 q+D

(2.25a)

F := dÀ + U A À = Ad(Z-+11)Fo

dB + B A B - P A PT

— (dpT + PT A B - a A P T e3

(dpp + B A P - P A a) e 3 «'+1 -da - PT A P

= 0. (2.25b)

Again the flatness of the connection /1 yields the very same relations (2.12).

2.3.1. Local coordinates

Our description of the orbifold S5/Zq+1 follows [15]. The key idea is the embedding S5 = SU(3)/SU(2) R6 = C3 via the relation

r2 = Stv x» xv = |Z1|2 + |Z2|2 + |z3|

(2.26)

where x'x (fi = 1, ..., 6) are coordinates of R6 and za (a = 1, 2, 3) are coordinates of C3; here r e R>0 gives the radius of the embedded 5-sphere. In general, on the coordinates za the Zq+1-action is realised linearly by a representation h 1 (hp such that

haPzP and za

h a ZP = (h-1)aP~zP

(2.27)

where h is the generator of the cyclic group Zq+1. In this paper the action of the finite group Zq+1 is chosen to be realised by the embedding of Zq+1 in the fundamental 3-dimensional complex representation C1,0 of SU(3) given by

(hap) =

'Zq+1 0 0

Zq+1 0 I e SU(3) with Zlq+1 := eq+

(2.28)

Since CP2 is naturally defined via a quotient of C3, see (2.5), one can deduce the Z^-action on the local coordinates (y1, y2) of the patch U0 to be

- a Z-1 Za

ya = Z3+1 ya and ya -q+i^ = Zq+1 ya for a = 1,2 . (2.29)

Zq+1Z -q+1 Z

Next consider the action of Zq+1 on the S1 coordinate f. By (2.28) one naturally has

i f zq+1 i(_f__, 2n±\ i _£_ ,

ei q+1 ei(q+1 + q+1 ) = eiq+1 -lq+1 for I e{0, 1,...,q}, (2.30)

i+ —) r t i f

i.e. the transformed coordinate e (q+1 q+1' lies in the Zq+1-orbit [e q+1J of e q+1.

2.3.2. Lens spaces

The spaces S5/Zq+1 are known as lens spaces, see for instance [24]. For this, one usually embeds S5 into C3 and chooses the action of p e{0, 1, ..., q} as

Zq+1 x C3 C3

1 a o 1 a o 2nip 2n i p _ 2n i p .

(p, (z1 ,z2,z3)) P ■ (z1 ,z2,z3) := (e~ z1, e~r1 z2, e~r2 z3) (2.31)

where the integers r1 and r2 are chosen to be coprime to q +1. The coprime condition is necessary for the Zq+1-action to be free away from the origin of C3. The quotient space S5/Zq+1 with the action (2.31) is called the lens space L(q + 1, r1, r2) or L2(q + 1, r1, r2). It is a 5-dimensional orbifold with fundamental group Zq+1.

We choose the Zq+1-action to be given by (2.28), i.e. r1 = 1 and r2 = -2. Then r1 is always coprime to q + 1, but r2 is coprime to q + 1 only if q is even. Thus for q + 1 e 2Z>0 + 1 the only singular point in C3/Zq+1 is the origin, and its isotropy group is Zq+1. However, for q + 1 e 2Z>0 there is a singularity at the origin and also along the circle {z1 = z2 = 0 , |z31 = 1} c S5 of singularities with isotropy group {0, q+1} = Z2 c Zq+1. Hence for the chosen action (2.28) we are forced to take q e 2Z>0 in all considerations.

2.3.3. Differential forms

Similarly to the previous case, one can construct locally a basis of differential forms. However, one has to work with a uniformising system of local charts on the orbifold S5/Zq+1 instead of local charts for the manifold S5. Choosing the identifications

pq+1 := p1 e^ = e1 + ie2 , p2q+x := p2 e^ = e3 + ie4 and n := e5 = ia - q+T

(2.32)

and by means of the relations imposed by the flatness of (2.25a), one can study the geometry of S5/Zq+1. Defining the three 2-forms

:= 2 (pq+1 a p2+1- pzq+1 a pq+1), «2 :=-2 (pq+1 a p2q+1+pq+1 a ^+1)

and «3 :=-2 (pq+1 a pq+1 + Pq2+1 a ^ (2.33)

and employing (2.12) implied by the flatness of A, one obtains the correct Sasaki-Einstein relations (2.21).

2.3.4. Zq+1-action on 1-forms

Consider the Zq+1-action on the forms j'q +1, j^+p and n. Firstly, recall the definitions (2.32) and (A.1), from which one sees that

jq+1 --4 jq+1 and pq+1 --4 r3-+3i vq+1. (2.34)

This follows directly from the transformation (2.29). Moreover, it agrees with the monodromy of jq+1 and j3q+1 along the S1 fibres, i.e.

jq +1 = j e3 q+ —+4 jq +1 ?q3+1 . (2.35)

Secondly, for the 1-form n from (2.32) we know that a is a U(1) connection. As any U(1) connection is automatically U(1)-invariant, due to the embedding Zq+1 4 U(1) one also has Zq+1-invariance of a.2 We conclude that

n-> n . (2.36)

From the definition (2.26) of the radial coordinate, one observes that r is invariant under Zq+1. The same is true for the corresponding 1-form, so that

dr ---4 dr . (2.37)

Following [15], let T be a Zq+1-invariant 1-form on the metric cone C(S5/Zq+1) which is locally expressed as

T = + Tr dr = Wi jq +1 + Wi j3q +1 + W5 e5 + Wr dr (2.38)

with i = 1, 2 and n = 1, ..., 5, where W1 = 1 (T1 - i T2), W2 = 2 (T3 - i T4), W5 = T5 and Wr = Tr. This induces a representation n of Zq+1 in ^^C(S5^ as

Wi n(h)(Wi) = fq"+31 Wi , Wi n(h)(Wi) = Zq+1 Wi , (2.39a)

W5 -4 n(h)(W5) = W5 , Wr -4 n(h)(Wr) = Wr . (2.39b)

3. Quiver gauge theory

In this section we define quiver bundles over a d-dimensional manifold Md via equivari-ant dimensional reduction over Md x S5/Zq+1, and derive the generic form of a G-equivariant connection. For this, we recall some aspects from the representation theory of G = SU(3), and exemplify the relation between quiver representations and homogeneous bundles over S5/Zq+1. Then we extend our constructions to G-equivariant bundles over Md x S5/Zq+1, which will furnish a quiver representation in the category of vector bundles instead of vector spaces. We shall also derive the dimensional reduction of the pure Yang-Mills action on Md x S5 to obtain a Yang-Mills-Higgs theory on Md from our twisted reduction procedure (for the special case q = 0).

2 Alternatively, one can work out the transformation behaviour of a directly from the explicit expression (2.11c).

3.1. Preliminaries

3.1.1. Cartan-Weyl basis of su(3)

Our considerations are based on certain irreducible representations of the Lie group G = SU(3), which are decomposed into irreducible representations of the subgroup H = SU(2) x U(1) c SU(3). For this, we recall the root decomposition of the Lie algebra su(3). There is a pair of simple roots a1 and a2, and the non-null roots are given by ± a1, ± a2, and ± (a1 + a2). The Lie algebra su(3) is 8-dimensional and has a 2-dimensional Cartan subalgebra spanned by Ha1 and Ha2. We distinguish one su(2) subalgebra, which is spanned by Ha1 and E± a1 with the commutation relations

[Ha1 ,E± aj = ±2E± a1 and [Ea1 ,E-aJ = Ha1 . (3.1a)

The element Ha2 generates a u(1) subalgebra that commutes with this su(2) subalgebra, i.e.

Ha2 ,Ha1 ] = 0 and Ha2 ,E± a1 ] = 0 . (3.1b)

In the Cartan-Weyl basis, the remaining generators E±a2 and E± (a1+a2) satisfy non-vanishing commutation relations with the su(2) generators given by

[Ha1 ,E± a2] = TE± a2 and [E± a1, E± a2] = ± E± (a1+a2) ,

[Ha1 ,E± (a1+a2^ = ± E± (a1+a2) and [E± a1 ,E=p (a1+a2^ = T EF a2

with the u(1) generator given by

[Ha2 ,E± a2] = ±3E± a2 and [Ha2 ,E± (a1+a2^ = ±3E± (a1+a2) , and amongst each other given by

[Ea2 ,E-a^ = 1 (Ha2 Ha1) and [Ea1 +a2 , E-a

2] — 2 (H®1 +

± Œ2 ,EF (&I+&2)

] = ±E

F ai •

(3.1c) (3.id)

(3.1e)

(3.1f) (3.1g)

3.1.2. Skew-Hermitian basis of sl(3, C)

Equivalently, we introduce the complex basis given by

— Ea1+a2 E-a1-a2 = Ea2 E-a2 >

— -i H

— 2 Ha2 ,

— Ea1 E-a1 7

^2 ■— i (Ea1+a2 + E-a

I4 ■— —i (Ea2 + E-a2) , I7 ■— —i (Eœ1 + E—a1) ,

¡8 ■— i Ha1 ,

which reflects the splitting su(3) — su(2) © m in which

¡i e su(2) for i — 6,7, 8 and IM e m for /x — 1 ,•••, 5 •

This representation of generators is skew-Hermitian, i.e. — -1/ for / — 1,

Ii — —¡J for i — 6, 7, 8, in contrast to the Cartan-Weyl basis. The chosen Cartan subalgebra is spanned by I5 and I8, and [I5,Ii] — 0. From the commutation relations (3.1) one can infer the non-vanishing structure constants of these generators as

(3.2a) (3.2b) (3.2c) (3.2d) (3.2e)

(3.3) , 5 and

f678 = -2 plus cyclic ,

f631 = f642 = r 4 r 2 r 1 -c 4 1 /71 = /73 = /82 = /83 = 1 plus cyclic ,

f125 = f345 = 2,

f251 = -f152 /■3 /-4 3 = /45 = —/35 = 2 .

The Killing form Kab := IacdIdbc (with A, B,... = 1,. agonal but not proportional to the identity, and is given by

(3.4a) (3.4b) (3.4c) (3.4d)

8) associated to this basis is di-

Kab = 12 Sab for a,b = 1, 2, 3, 4, K55 = 9 and Kij = 12 Sij for i, j = 6, 7, 8 . Introducing the 't Hooft tensors nab for a, b = 1, 2, 3, 4 and a = 1, 2, 3 one has

fab5 = 2 nib and fa5b = -| nib .

3.1.3. Biedenharn basis

The irreducible SU(3)-representations Ck,t are labelled by a pair of non-negative integers (k, l) and have (complex) dimension

po := dim(Ck'kJ) = 1 (k +1 + 2)(k + 1)(l + 1) . (3.7)

We decompose Ck^ with respect to the subgroup H = SU(2) x U(1) c G, just as in [23]. A particularly convenient choice of basis for the vector space Ck,t is the Biedenharn basis [28-30], which is defined to be the eigenvector basis given by

m j = q

m) = n(n + 2)

where L2 := 2 (Ea1 E-a1 + E-a1 Ea1) + H2X is the isospin operator of su(2). Define the representation space (n, m) as the eigenspace with definite isospin n e Z>0 and magnetic monopole charge mr for m e Z. Then the SU(3)-representation ^^ decomposes into irreducible SU(2) x U(1)-representations (n, m) as

Ck,1 = 0 (n,m)

(n,m)eQo(k,l)

where Q0(k, l) parameterises the set of all occurring representations (n, m). In Appendix B.1 we summarise the matrix elements of all generators in the Biedenharn basis.

3.1.4. Representations of Zq+i

As the cyclic group Zq+1 is abelian, each of its irreducible representations is 1-dimensional. There are exactly q + 1 inequivalent irreducible unitary representations pl given by

•q +1

-4 S1 C C*

2^ i (p+l) 1—> e q+1

for l = 0, 1,..

(3.10)

3.2. Homogeneous bundles and quiver representations

Consider the groups G = SU(3), H = SU(2) x U(1), K = SU(2), K = SU(2) x Zq+1 c H and a finite-dimensional K-representation R which descends from a G-representation. Associate to the principal bundle (2.14) the K-equivariant vector bundle Vr := G xK R. Due to the embedding Zq+1 U(1) c SU(3) and the origin of R from a G-module, it follows that R is also a Zq+1-module. Consequently, as in Section 2.3, the Zq+1 -action y : Zq+1 x S5 ^ S5 can be lifted to a Zq+1-action y : Zq+1 x Vr ^ Vr wherein the linear Zq+1-action on the fibres is trivial. Thus one can define the corresponding K-equivariant vector V-bundle vr by suitable Zq+1-projection as3

Vr-^ yR (3.11)

S 5 /Zq+1

and again we denote the vector V-bundle yR by the same symbol Vr whenever the context is clear.

It is known [31] that the category of such holomorphic homogeneous vector bundles Vr is equivalent to the category of finite-dimensional representations of certain quivers with relations. We use this equivalence to associate quivers to homogeneous bundles related to an irreducible SU(3)-representation R = Ck'1, which is evidently a finite-dimensional (and usually reducible) representation of SU(2) x Zq+1 SU(2) x U(1).

3.2.1. Flat connections

Inspired by the structure of the flat connection (2.25a) on the V-bundle (2.23), one observes that it can be written as4

Ao = [#11 Ha1 + B12 Ea1 - (B12 Ea^] - 2 qH^ + £«1+^2 + P\+\ E»2

- pq+1 E-a1-a2 - p2+1 E-a2 , (3.12a)

or equivalently

Ao = r + (3.12b)

with the coframe (eM}M=1,...,5 defined in (2.32) and the definition

r := r Ii with r6 = 2 (B12 - B12) , r7 = 1 (B12 + B12) , r8 = -i B11. (3.13)

Note that r is an su(2)-valued connection 1-form. The flatness of Ao, i.e. F0 = dA + Ao a Ao = o, is encoded in the relation

Fo = Fr + In de^ + r a e» + 2 [I,„IV] e^v = 0 , (3.14a)

Fo|su(2) = 0 : Fr = -1 f^he^, (3.14b)

Fo|m = 0 : de^ = -P f./ A ev - 1 fp? epa , (3.14c)

See also the treatment in [15].

4 Note that (2.25a) implicitly uses the fundamental representation C1'0 of SU(3).

where Fr = dr + r a r. The equivalent information can be cast in a set of relations starting from (3.12a) and using the Biedenharn basis; see Appendix B.2 for details.

3.2.2. Zq+1 -equivariance

Consider the principal V-bundle (2.23), where the Zq+1-action is defined on S5 as in Section 2.3. The connection (3.12) is SU(3)-equivariant by construction, but one can also check its Zq+1-equivariance explicitly. For this, one needs to specify an action of Zq+1 on the fibre Ck^, which decomposes as an SU(2)-module via (3.9). Demanding that the Zq+1-action commutes with the SU(2)-action on C'k,t forces it to act as a multiple of the identity on each irreducible SU(2)-representation by Schur' s lemma. Hence we choose a representation y : Zq+1 —> U(p0) of Zq+1 on Ck^1- such that Zq+1 acts on (n, m) as

Y(h)\{n,m) = C+1 !n+1 e U(1). (3.15)

Consider the two parts of the connection (3.12): The connection r and the endomorphism-valued 1-form I/A, eIn terms of matrix elements, r is completely determined by the 1-forms B(n,m) e (su(2), End(n, m )) which are instanton connections on the K-equivariant vector V-bundle

^ (n,m) ^ -

K(n,m) -^ G/K = S5/Zq+1 with V(n,m) := G xk (n, m), (3.16)

simply because they are K-equivariant by construction and Zq+1 — U(1) c SU(3) commutes with this particular SU(2) subgroup (see also Appendix A). More explicitly, taking (3.15) one observes that Zq+1 acts trivially on the endomorphism part,

Y(h)B(n,m)Y(h)~1 = B(n,m), (3.17)

as well as on the 1-form parts H because they are horizontal in the V-bundle (2.23). For Zq+1-equivariance of the second term I^e^, from (3.12a) and the representation n defined in (2.39) one demands the conditions

Y(h)Ew Y(h)-1 = n(h)-1 (Ew) = Zq+1 E„ for w = a2,a1 + a2 , (3.18a)

Y(h)E-w Y(h)-1 = n(h)-1 (E-w) = £"+3 E-w for w = a2a + a2 , (3.18b)

Y(h)Ha2 Y(h)-1 = n(h)-1 (Ha2) = Ha2 . (3.18c)

One can check that these conditions are satisfied by our choice of representation (3.15), due to the explicit components of the generators (B.2). We conclude that, due to our ansatz for the connection (3.12) on the principal V-bundle (2.23) and the embedding Zq+1 — U(1) c SU(3), the 1-form A0 is indeed Zq+1-equivariant.

3.2.3. Quiver representations

Recall from [23] that one can interpret the decomposition (3.9) and the structure of the connection (3.12) as a quiver associated to ^^ as follows: The appearing H-representations (n,m) form a set Q0(k, l) of vertices, whereas the actions of the generators Ea2 and Ea1+a2 intertwine the H-modules. These H-morphisms, together with Ha2, constitute a set Q1(k, l) of arrows (n, m) — (n', m') between the vertices. The quiver Qk,l is then given by the pair Qk,i = Q0(k, l), Q1(k, l); the underlying graph of this quiver is obtained from the weight diagram of the representation by collapsing all horizontal edges to vertices, cf. [23]. See Appendix C for an explicit treatment of the examples C1,0, C2,0 and C

3.3. Quiver bundles and connections

In the following we will consider representations of quivers not in the category of vector spaces, but rather in the category of vector bundles. We shall construct a G-equivariant gauge theory on the product space

Md xK G := Md x G/K = Md x S5/Z3+1 (3.19)

where G and all of its subgroups act trivially on a d-dimensional Riemannian manifold Md. The equivariant dimensional reduction compensates isometries on G/K with gauge transformations, thus leading to quiver gauge theories on the manifold Md.

Roughly speaking, the reduction is achieved by extending the homogeneous V-bundles (3.11) by K-equivariant bundles E ^ Md, which furnish a representation of the corresponding quiver in the category of complex vector bundles over Md. Such a representation is called a quiver bundle and it originates from the one-to-one correspondence between G-equivariant Hermitian vector V-bundles over Md x G/K and K-equivariant Hermitian vector bundles over Md, where K acts trivially on the base space Md [31].

3.3.1. Equivariant bundles

For each irreducible H-representation (n, m) in the decomposition of C_k'1, construct the (trivial) vector bundle

, (n,m) ,

(n,m) Md := Md xK (n,m) =^ Md (3.20)

of rank n + 1, which is g-equivariant due to the trivial K-action on Md and the linear action on the fibres. For each module (n, m) introduce also a Hermitian vector bundle

Cp(n,m) ,

EP(n,m) -► Md with rk(Emm)) = P(n,m) (3.21)

with structure group U(p(n,m)) and a u(p(n,m))-valued connection A(n,m), and with trivial K-action. Denote the identity endomorphism on the fibres of EP(n m) by n(n,m). With these data one constructs a KK-equivariant bundle

EkJ = 0 Epnnm, ® (nm)Md Md (3.22)

(n,m)eQo(k,l)

whose rank P is given by

P = P(n,m) dim (n,m) = ^ P(n,m)(n +1). (3.23)

(n,m)&Qo(k,l) (n,m)&Qo(k,l)

Following [23], the bundle Ek,]i is the K-equivariant vector bundle of rank p associated to the representation |k of K, and (3.22) is its isotopical decomposition. This construction breaks the structure group U(p) of Ek,]i via the Higgs effect to the subgroup

GkJ := n U(p(n,m))n+' (3.24)

(n,m)eQo(k,l)

which commutes with the SU(2)-action on the fibres of (3.22).

On the other hand, one can introduce K-equivariant V-bundles over S5/Zq+1 by . On V(nm) one has the su(2)-valued 1-instanton connection B(nm) in the (n + 1)-dimensional irreducible representation. The aim is to establish a G-equivariant V-bundle Ek,]i over Md x S5/ Zq+i as an extension of the g-equivariant bundle Ek,]l. By the results of [31] such a V-bundle Ek,l exists and according to [23] it is realised as

EkJ := G xg Ekl = 0 Ep^m) H Vnm-^ Md x S5/Zq+1 , (3.25)

(n,m)eQo(k,l)

yk,l = 0 cp(n,m) 0 (n,m) (3.26)

(n,m)eQo(k,l)

is the typical fibre of (3.25).

3.3.2. Generic G-equivariant connection

The task now is to determine the generic form of a G-equivariant connection on (3.25). Since the space of connections on Eis an affine space modelled over ^1(End(Ek,l))G, one has to study the G-representations on this vector space. Recall from [23] that the decomposition of ^1(End(Ek,l))G with respect to G yields a "diagonal" subspace which accommodates the connections A(n,m) on (3.21) twisted by G-equivariant connections on (3.16), and an "off-diagonal" subspace which gives rise to bundle morphisms.

In other words, K-equivariance alone introduces only the connections A(n,m) on each bundle (3.21) as well as the SU(2)-connections B(nm) on the V-bundles (3.16). On the other hand, G-equivariance additionally requires one to introduce a set of bundle morphisms

tfn,m) e H°m(EP(n,m) ,EP(n±1,m+3)) (3.27a)

and their adjoint maps

(^±))(n,m] e Hom(EP(n±1,m+3),EP(n,m)) , (3.27b)

for all (n, m) e Qo(k, l); one further introduces the bundle endomorphisms

f(n,m) e EM(Ep{nm)) (3.27c)

at each vertex (n, m) e Qo(k, l) with m = o. The morphisms m) and f(n,m) are collectively called Higgs fields, and they realise the G-action in the same way that the generators IM (or more precisely the 1-forms m) and 1Jm n ^(n,m)) do in the case of the flat connection (3.12). The "new" Higgs fields f(nm) implementing the vertical connection components on the (orbifold of the) Hopf bundle S5 ^ CP2 must be Hermitian, i.e. ty(n,m) = m), by construction in order for the connection to be u(p)-valued.

3.3.3. Ansatz for connection

The ansatz for a G-equivariant connection on the equivariant V-bundle (3.25) is given by

A = A + ? + X^e^ (3.28)

wherein the u(p(n,m))-valued connections A(nm) and the su(2)-valued connection r are extended as

A := 0 A{n,m) ® n{n,m) = A® 1 and f := 0 n{n,m) ® p//n,m) = Pf = 1 ® r ,

(n,m) (n,m)

(3.29)

together with f = 0(n m) n(n,m) ® i(n,m). The matrices XM are required to satisfy the equivariance condition [22,32]

[îi,X^] = /i/Xv for i = 6, 7, 8 and д = 1.....5 .

(3.30)

As explained in [32], the equivariance condition ensures that XM are frame-independently defined endomorphisms that are the components of an endomorphism-valued 1-form, which is here given as the difference A - (A + Г).

The general solution to (3.30) expresses Xin terms of Higgs fields and generators as

2 (X1 + i X2) = 0 ф±т) ® \

±,(n,m)

2 (Xi - i X2) = - 0 (Ф±)}п,т) ® E±

±,(n,m)

2 (X3 + i X4) = 0 Ф±пт) ® E±2(n,m)

±,(n,m)

1 (X3 - iX4) = - 0 ^¡nm) ® E±

±,(n,m)

± (n,m) ai — a2

± (n,m) a2

(3.31a)

X5 = — 2 0 f(n,m) ® Han'

(3.31b) (3.31c)

Altogether the G-equivariant connection takes the form

A = 0 ^A(n,m) ® ^-(nm) + n(n,m) ® B(n,m) - f(n,m) ® ^V^inm) (n,m)eQo (k,l)

+ Km) ® e~(n,m) + Ф(n,m) ® e~{n,m) - ^+)(n,m) ® e("+,m) - (<£-)(n,m) ® e(-,m)) •

(3.32)

3.3.4. Zq+1-equivariance

One needs to extend the Zq(1-representation у of (3.15) to act on the fibres (3.26) of the equivariant V-bundle (3.25). Since by construction K = SU(2) x Zq+i acts trivially on the fibres of the bundles (3.21), one ends up with the representation у : Zq+1 4 U(p) given by

Y(h)= 0 1p(n,m) ® Y(h)\(jbm1 = 0 1p(n,m) ® zm+i 1n+i • (3.33)

(n,m)eQo(k,l) (n,m)eQo(k,l)

To prove Zq(1-equivariance of (3.28) one again needs to show two things. Firstly, the connections A ® 1 and 1 ® Г have to be Zq(1-equivariant. This can be seen as follows: For A ® 1 the representation y of (3.33) acts trivially on each bundle Ep(n m), and thus

Y(h) (A ® 1) Y(h) — 1 = A ® 1.

(3.34)

Furthermore, 1 ® r is Zq+i-equivariant because r is by (3.17), and hence the connection A ® 1 + 1 ® r satisfies the equivariance conditions.

Secondly, the endomorphism-valued 1-form X^e^ = A - A - V needs to be Zq+1-equivariant as well. Due to its structure, one needs to consider a combination of the adjoint action of y from (3.33) and the Zq+1-action on forms from (2.39). As y acts trivially on each bundle EP(nm), the Zq+1-equivariance conditions

y(h)X^y(h)-1 = n(h)-1(X/) for / = 1,...,5 (3.35)

hold also for the quiver connection A just as they hold for the flat connection Ao by (3.18).

Thus the chosen representations (2.39) and (3.33) render the quiver connection (3.28) equiv-ariant with respect to the action of Zq+1. On each irreducible representation (n, m) the generator h of Zq+1 is represented by t^+x 1n+1 which depends on the U(1) monopole charge but not on the SU(2) isospin. This comes about as follows: The bundle morphisms associated to j'q+1 map between bundles EP(nm) ® (n,m) Md that differ in m by -3 (from source to target vertex), but differ in n by either +1 or -1. Thus the representation y should only be sensitive to m and not to n. We shall elucidate this point further in Section 6.1.

3.3.5. Curvature

The curvature F = dA + A a A of the connection (3.28) is given by

F = Fa ® 1 + 1 ® Fv + (dX/x + [ A, X/]) a e» + X/ dex + [f, X/] A e»

+ 2 [X/,XV] e/v, (3.36a)

where FA = dA + A A A. Employing the relations (3.14) then yields

F = Fa ® 1 + (dX/ + [f, X/]) A ef + V ([f ,XM] - ft/ Xv) A ex

+ 2 ([X/,Xv] - f/vp Xp - f/J f ) e/v . (3.36b)

Since the matrices Xx satisfy the equivariance relation (3.3o), the final form of the curvature reads

F = Fa ® 1 + (DX)/ A ex + 2 ([X/,Xv] - / Xp - f/J f) e/v , (3.36c)

where we defined the bifundamental covariant derivatives as

(DX)/ := dX/ + [ f,X/] . (3.36d)

Inserting the explicit form (3.31) for the scalar fields Xleads to the curvature components in the Biedenharn basis; the detailed expressions are summarised in Appendix B.3.

3.3.6. Quiver bundles

Let us now exemplify and clarify how the equivariant bundle Ek,]l ^ Md from (3.22) realises a quiver bundle from our constructions above. Recall that the quiver consists of the pair Qk,]l = (Qo(k, l), Q1(k, l), with vertices (n, m) e Qo(k, l) and arrows (n, m) ^ (n', m') e Q1(k, l) between certain pairs of vertices( which are here det)ermined by the decomposition (3.9). We consider a representation Qk,l = (<go(k, l), Q1(k, l) ) of this quiver in the category of complex vector bundles. The set of vertices is

Qo(k,l) = {Ep^ Md, (n,m) e Qo(k,l)}, (3.37)

i.e. the set of Hermitian vector bundles each equipped with a unitary connection A(n,m). The set of arrows is

e Hom(Ep(nm), Ep(n±hm+3)), (n, m) e Qo(k, l) U [f(n , m) e End (Ep(n , m}), (n, m) e Q°(k , l) , m = °J , (3.38)

which is precisely the set of bundle morphisms, i.e. the Higgs fields. These quivers differ from those considered in [23] by the appearance of vertex loops corresponding to the endomorphisms f(n,m). See Appendix C for details of the quiver bundles based on the representations CC2,0 and Cu.

These constructions yield representations of quivers without any relations. We will see later on that relations can arise by minimising the scalar potential of the quiver gauge theory (see Section 3.4) or by imposing a generalised instanton equation on the connection A (see Section 4).

3.4. Dimensional reduction of the Yang-Mills action

Consider the reduction of the pure Yang-Mills action from Md x S5 to Md. On S5 we take as basis of coframes {fij, fij}j=1,2 and e5 = n, and as metric

d4 = R2 fi ® fil + fil ® fil + ® + ® fil) + r2 n ® n. (3.39)

The Yang-Mills action is given by

S =—^ f tr F A *F, (3.40)

Md xS5

with coupling constant g and * the Hodge duality operator corresponding to the metric on Md x S5 given by

ds2 = ds2Md + d^ . (3.41)

We denote the Hodge operator corresponding to the metric ds2Md on Md by *Md. The reduction

of (3.40) proceeds by inserting the curvature (3.36c) and performing the integrals over S5, which

can be evaluated by using (3.39) and the identities of Appendix D.2. One finally obtains for the

reduced action

2^3 r R4 if, )

tr (Fa a*MdFA) ® 1

+ 2R2 f ¿tr (DX)a A *Md(DX)a + ^ f tr (DX)5 A *Md(DX)5

Md a=1 Md

+ 8R4 J M J2 H[Xa,Xb] - fab5 X5 - fab Î)2 Md a,b=1 1 f 4 + 8R2T2 J *MdJ2 tr([Xa,X5] - fa5bXb)•

Md a=1

Here the explicit structure constants (3.4), i.e. fabc = fa55 = fa5! = 0, have been used. One may detail this action further by inserting the G-equivariant solution (3.31) for the scalar

fields XM in the Biedenharn basis, which allows one to perform the trace over the SU(2) x U(1)-representations (n, m). The explicit but lengthy formulas are given in Appendix D.3.

3.4.1. Higgs branch

On the Higgs branch of the quiver gauge theory where all connections A(n,m) are trivial and the Higgs fields are constant, the vacuum is solely determined by the vanishing locus of the scalar potential. The vanishing of the potential gives rise to holomorphic F-term constraints as well as non-holomorphic D-term constraints which read as

[Xa,Xb] = fab5 X5 + fab'f and [Xa,X5] = fa5b Xb , (3.43)

for a, b = 1, 2, 3, 4. The equivariance condition (3.3o) implies that XM lie in a representation of the su(2) Lie algebra. Hence the BPS configurations of the gauge theory XM, together with f, furnish a representation of the Lie algebra su(3) in the representation space of the quiver in u(p). These constraints respectively give rise to a set of relations and a set of stability conditions for the corresponding quiver representation. The details can be read off from the explicit expressions in Appendix D.3.

4. Spherically symmetric instantons

In this section we specialise to the case where the Riemannian manifold Md = M1 is 1-dimensional. We investigate the Hermitian Yang-Mills equations on the product M1 x S5/ Zq+1 for the generic form of G-equivariant connections derived in Section 3.3.

4.1. Preliminaries

Consider the product manifold M1 x S5/Zq+1 with M1 = R such that M1 x S5 /Zq+1 = C(S5/Zq+1) is the metric cone over the Sasaki-Einstein space S5/Zq+1, which is an orbifold of the Calabi-Yau manifold C(S5). The Calabi-Yau space C(S5) is conformally equivalent to the cylinder R x S5 with the metric

d^C(S5) = dr2 + r2di2 = r2 (dr2 + di^) = e2r (dr2 + S^e^ ® e^ (4.1)

where r = log r. The Kahler 2-form is given by

mC(S5) = e2r («3 + n A dr) . (4.2)

4.1.1. Connections

As R is contractible, each bundle EP(n m) ^ R is necessarily trivial and hence one can gauge away the (global) connection 1-forms A(n,m) = A(n,m)(r) dr; explicitly, there is a gauge transformation g : R ^ Qk,t such that

A(n,m) = Ad(g-1)A(n,m) + g-1dg = o with g = exp (-J A(n,m)(r) dr) . (4.3)

The ansatz for the connection on the equivariant V-bundle then reads

A = 1 ® T + X/e^, (4.4)

where the Higgs fields m) and f(n,m) depend only on the cone coordinate r (compare also with [32, Section 4.1]). The curvature of this connection can be read off from (3.36c) and is

evaluated to

F = X dT A e^ + 1 ( [X^,XV] - fM/ Xp - V . (4.5)

4.2. Generalised instanton equations

The ansatz (4.4) restricts the space of all connections on the SU(3)-equivariant vector V-bundle over C(S5/Zq+1) to SU(3)-equivariant and Zq+1-equivariant connections.

4.2.1. Quiver relations

On this subspace of connections one can further restrict to holomorphic connections, i.e. connections which allow for a holomorphic structure.5 For this, one requires the holomorphicity condition F0'2 = 0 = F2'0 which for the connection (4.4) is equivalent to

F14 + F23 = 0 , F1t + F25 = 0 , F3t + F45 = 0 , (4.6a)

F13 - F24 = 0 , F15 - F2t = 0 , F35 - F4t = 0 . (4.6b)

Substituting the explicit components of the curvature (4.5), one finds relations for the endomor-phisms XM given by

[X1,X4] + [X2,X3] = 0 = [X1,X3] - [X2,X4] and

b t 2 dXb \

[Xa,X5] = fa5b[Xb + 3 X ) (4.7)

for a = 1' 2' 3,4.

4.2.2. Stability conditions

By well-known theorems from algebraic geometry [33-35], a holomorphic vector bundle admits solutions to the Hermitian Yang-Mills equations if and only if it is stable. This condition can be translated into a condition on the remaining (1, 1)-part of the curvature F: One demands that F is a primitive (1, 1)-form, i.e. mC(S5) _i F = 0, or in components

F12 + F34 + F5t = 0 . (4.8)

Using the explicit components (4.5) one can deduce the matrix differential equation for XM given by

[X1,X2] + [X3,X4] = 4X5 + ^ . (4.9)

One can also regard the stability condition in terms of a moment map \x from the space of holomorphic connections to the dual of the Lie algebra of the gauge group [36]. The dual fi* then acts on a connection A via f*(A) = &>C(S5) 1F, which is well-defined as the curvature F is a Lie algebra-valued 2-form. Then the stability conditions correspond to the level set of zeroes f *—1(0); we shall return to this interpretation in Section 6.2.2.

5 For a Hermitian connection A on a complex vector bundle, the requirement for it to induce a holomorphic structure is equivalent to the (0, 1)-part A0'1 of A being integrable, i.e. the corresponding curvature F is of type (1, 1).

4.3. Examples

We shall now apply these considerations to the three simplest examples: The quivers based on the representations C1'0, C20 and C11. For each example we explicitly provide the representation of the generators and the form of the matrices XM, followed by the quiver relations and the stability conditions.

4.3.1. C1'0-quiver

The generators in the fundamental representation C1'0, which splits as in (C.1), are given as / 0 j(0'-2) \ / ,(1,1) 0 \ '• = (-(ji°h, a 0 ) and J5 =(50 jf-2^ (410a)

for a = 1, 2, 3, 4, with components

j(0'-2) = (0) = i '2°,-2) and '3(0,-2) = (0) = i 'f-), (4.10b)

'5(0,-2) = i 12 and '5(1,1) = -2 • (4.10c)

The endomorphisms XM read as

02 $®'a0 2)\ v _(f1 ®'5(1,1) 0

Xa =( , 02(0 -2),t W 1 and X5 = ( ^ «'5

® (/i0'-2^f 0 ) ^ 0 f ® /f-2V

(4.11)

where the Higgs fields from Appendix C give a representation of the quiver

iQ?):---H(Q)i (4.12)

The Zq+1 -representation (3.33) reads

/ 1p(11) ® 12 Zq+1 0 \

' 0 ^ 1p(0, -2)® Zq+0' (4J3)

where h is the generator of the cyclic group Zq+1.

4.3.1.1. Quiver relations The first two equations from (4.7) are trivially satisfied without any further constraints. The second set of equations all have the same non-trivial off-diagonal component (and its adjoint) which yields

2 = -3 - + 2-f0 + fi(4.14) dr

Thus for the C1 0-quiver there are no purely algebraic quiver relations.

4.3.1.2. Stability conditions From (4.9) we read off the two non-trivial diagonal components which yield

1 / , At A 1 dYr . 4.

- — = —o + 0' 0 and - — = -Y + 00' . 4 dr 4 dr

(4.15)

By taking t0 and ti to be identity endomorphisms, we recover the Higgs branch BPS equations from equivariant dimensional reduction over CP2: In this limit (4.14) implies that the scalar field 0 is independent of t, while (4.15) correctly reproduces the D-term constraints of the quiver gauge theory for constant matrices [23,25].

4.3.2. C2'0-quiver

The generators in the 6-dimensional representation C2'0, which splits as in (C.3), are given by

- ('¿w0

r(1,-1)

(0,-4h t

r(0,-4)

/i(2'2) 0

and I5 =

(1,-1) 5

(0,-4)

for a = 1, 2, 3,4, with components

(1,-1)

(1,-1)

'V2 0N 0 1 0 0,

'0 0 1 0 ,0 V2y

(1,-1)

(0,-4) 1

(1,-1)

(0,-4) .

(0,-4)

(0,-4)

I5(2,2) = -i 13

(1,-1) _ i

= 212 and I5(

(0,-4) _

The endomorphisms Xß read

-0t ® I 0

(1,-1)Nt

01 ® I

(1,-1)

00 ® I

(4.16a)

(4.16b)

(4.16c)

(4.16d)

(0,-4)

-0t ® (i(0,-4)) ' 0 y*

(Yi ® l52,2)

Ï1 ® i5 0

(1,-1) 0 Y ® i50,-4)

(4.17)

with the Higgs field content from Appendix C that furnishes a representation of the quiver

to tl t2

The representation (3.33) in this case reads

( ^(2,2) ® 13 Z2+1

Y : hi

!p(1,-1) ® 12 Zq+1

(4.18)

(4.19)

,-4) ® Zq-H/

4.3.2.1. Quiver relations Again the first two equations of (4.7) turn out to be trivial, while the second set of equations have two non-vanishing off-diagonal components (plus their conjugates) which yield

= -3 0o - fi 00 + 40o fo and 2^ = -3 01 + 01 f + 2 f2 01 (4.20) dr dr

and the C2,0-quiver has no purely algebraic quiver relations either.

4.3.2.2. Stability conditions From (4.9) one obtains three non-trivial diagonal components that yield

1 df0 ,

= -f0 + 0(

1 dfi = -f1 - 2 00 0t + 3 0t 01 , 4 dr 0 1

1 df22 I A At

T — = f 2 + 01 01 • 4 dr 1

(4.21a) (4.21b) (4.21c)

Taking f0, f 1 and f2 again to be identity morphisms, from (4.20) we obtain constant matrices 00 and 01 which by (4.21c) obey the expected D-term constraints from equivariant dimensional reduction over CP2 [23,25].

4.3.3. C1,1 -quiver

The decomposition of the adjoint representation C1'1, which splits as given in (C.5), yields

02 j(0,0) la j(2,0) la 0

-(la(0,0))t 0 0 j - (1,-3) la

-(la(2,0))t 0 03 j + (1,-3) la

0 -(la" (1-3))t (l+ (1-3))t 02

(4.22a)

(0,0) 5

0 0 5 \ 0 0 0 for a = 1 , 2, 3, 4, with components

j(1'-3)

(4.22b)

l(0'0) =

■(1,-3) _

j(0,0) _ l3 =

and l1(2,0) =

(0 -/§) = i l2-(1-3) and I

0 -1 /1 0 \

(2,0), 2,

+ (1,-3) 1=

= i l+(1,-3)

V0 0 /

and l3(2,0) =

'1 0 0N ,0 0

; j(2,0)

(1,-3)

(4 0)=ij

- (1,-3)

+(1,-3)

/ 0 0\

2 0 0 V

+(1,-3)

(4.22c)

, (4.22d)

(4.22e)

, (4.22f)

/(1.3) = _|i l2 , /(0,0) = 0 , /(2,0) = 03 and I(1'-3) = 3LÎ12 .

(4.22g)

The matrices XM are given by

_(00_)t ® (li2,0))t V 0

00 » Ia 0 0

00 » /a 0 03

+ » I5(1,3) 0 0 0 0 0

_(0_)t ® (Ia_ (1._3^t _(0+)t ® (Ia+ (1._3^t 0 \

01 » I_ (1,- 3)

0+< » I+ (1,- 3)

0 0 0 03

(4.23a)

(4.23b)

0 0 f I5(1, 3)/

This example involves the collection of Higgs fields from Appendix C which furnish a representation of the quiver

,+ (1. +3))

(4.24)

01 (I, _3):

In this case the Zq+1-representation (3.33) has the form

Y : h i

^(1,3) » ^2 zq+1 0 0 1

P(0,0)

1P(2,0) » 13

1P(1,_3) » 12 Zq+1 }

(4.25)

4.3.3.1. Quiver relations For this 8-dimensional example, one finds that the first two equations of (4.7) have the same single non-trivial off-diagonal component (plus its adjoint) which yields

0+ = 00 0+ .

(4.26)

This equation is precisely the anticipated algebraic relation for the C11-quiver expressing equality of paths between the vertices (1, ± 3), cf. [23]. The second set of equations have four non-trivial off-diagonal components (plus their conjugates) which yield

2 d0± + + + 2 d0± . . --00- = _ 00± + f + 0± and - -0L = _0± + 0± f 3 dr 0 0 3 dr 11

(4.27)

4.3.3.2. Stability conditions From (4.9) one computes four non-vanishing diagonal components that yield

(ffi = ^r - (4.28a)

1 ddr = — ++1K (0o+)t+- (4.28b)

1ddT = —- + 1 ((^-)t + (0+)t• (4.28c)

We thus obtain two non-holomorphic purely algebraic conditions, which coincide with D-term constraints of the quiver gauge theory for the C1,1 -quiver, and two further differential equations which for identity endomorphisms f ± reproduce the remaining stability equations for constant matrices and in equivariant dimensional reduction over CP2 [23,25].

5. Translationally-invariant instantons

In this section we study translationally-invariant instantons on a trivial vector V-bundle over the orbifold C3/Z3+i. In contrast to the G-equivariant Hermitian Yang-Mills instantons of Section 4, the generic form of a translationally-invariant connection is determined by Zq+1-equivariance alone and is associated with a different quiver.

5.1. Preliminaries

Consider the cone C(S5)/Zq+1 = C3/Zq+i, with the Zq+i-action given by (2.28), and the (trivial) vector V-bundle

E -► C3/Zq+1 (5.1)

of rank p which is obtained by suitable Zq+1-projection from the trivial vector bundle C3 x Vk,]i 4 C3. The fibres of (5.1) can be regarded as representation spaces

VkJ = 0 Cp(nm) ® (n,m) = 0 (Cp(n.m) ® cn+1) ® Vm. (5.2)

(n,m)eQo(k,l) (n,m)eQo(k,l)

Here Vm is the [m]-th irreducible representation p[m] of Zq+1 (cf. (3.10)), with [m] e {0, 1, ..., q} the congruence class of m e Z modulo q + 1, and the vector space Cp(n,m) ® Cn+1 serves as the multiplicity space of this representation. The structure group of the bundle Ek,]i is

Gkl := Y\ U(p{n,m)(n +1)) , (5.3)

(n,m)eQo(k,l)

because the fibres are isomorphic to (5.2) and hence it carries a natural complex structure J; this complex structure is simply multiplication with i on each factor Vm. Consequently, the structure group is reduced to the stabiliser of J.

On the base the canonical Kahler form of C3 is given by

MC3 = 2 Sap dza a dz^ . (5.4)

This Kahler form is compatible with the standard metric ds^3 = 2 8ap (dza ® dZ^ + dZa ® dz^) and the complex structure J(dza) = i dza, J(dza) = —i dza.

5.1.1. Connections

Consider a connection 1-form

A = Wa dza + Wa dza (5.5)

on Ek,t, and impose translational invariance along the space C3. For the coordinate basis {dza, dZa} of T*z z)C3 at any point (z, Z) e C3, this translates into the condition

dWa = 0 = dWa for a = 1,2, 3 . (5.6)

Thus the curvature F = dA + A a A simplifies to

F = A a A = 1 [Wa,Wp] dza a dz^ + [Wa, Wp] dza a dz^ + ± [ Wa,Wp] dza a dz^ .

5.1.2. Zq+1-action

As before one demands Zq+1-invariance due to the projection from the trivial vector bundle C3 x Vk,r ^ C3 to the trivial V-bundle Ek,r ^ C3/Zq+1. Again one needs to choose a representation of Zq+1 on the fibres (5.2). For reasons that will become clear later on (see Section 6.1), this time one chooses

Y(h)= 0 1pin,m) » zq+11n+1 e Center(©k,z) . (5.8)

(n,m)eQ0(k,l)

One immediately sees that all elements of ©k^ commute with the action of Zq+1 given by (5.8), i.e. Y(Zq+1) c Center (G^1). The action of Zq+1 on the coordinates za defined in (2.28) induces a representation n of Zq+1 in Œ1 (C3), which on the generator h of Zq+1 is given by

Z;h Wi , i = 1,2 _ i Zq+1 Wi, i = 1,2

Zq\, W3 " I Z"+1 W3

n(h)(Wa) = \ 12 and n(h)(Wa) = _2 _ . (5.9)

zq+1 "3 I ^q+1

The requirement of Zq+1-equivariance of the connection A reduces to conditions similar to (3.35), i.e. the equivariance conditions read as

y(h)WaY(h) -1 = n(h)-1(Wa) and y(h)Way(h)-1 = n(h) -1 (Wa) (5.10)

for a = 1, 2, 3, but this time with different Zq+1-actions y and n.

5.1.3. Quiver representations

For a decomposition of the endomorphisms

Wa = 0 (Wa>(nm)M'm')' (n,m),(n/,m;)

(Wa){n,m)M'M) e Hom(C^m> ® (n,m), Cp<nW) ® (n',m')) (5.11)

as before, the equivariance conditions imply that the allowed non-vanishing components are given by

®{n,m) := W^nmUn'rn') = 0 for n'- n = 1 (mod q + 1), (5.12a)

n,m) := ^^m)»') = 0 for n - n = -2 (mod q + 1), (5.12b)

for i = 1, 2, together with the analogous conjugate decomposition for Wa; in each instance m' is implicitly determined by n and m via the requirement (n', m') e Q0(k, l). The structure of these endomorphisms thus determines a representation of another quiver Qk,l with the same vertex set Q0(k, l) as before for the quiver Qk,]i but with new arrow set consisting of allowed components

(n, m) 4 (n', m').

5.2. Generalised instanton equations

Similarly to Section 4.2, the Hermitian Yang-Mills equations on the complex 3-space can be regarded in terms of holomorphicity and stability conditions.

5.2.1. Quiver relations

The condition that the connection A defines an integrable holomorphic structure on the bundle (5.1) is, as before, equivalent to the vanishing of the (2, 0)- and (0, 2)-parts of the curvature F, i.e. F0,2 = 0 = F2,0, which in the present case is equivalent to

[Wa,Wp ] = 0 and [ Wa,Wp ] = 0 . (5.13)

The general solutions (5.12) to the equivariance conditions allow for a decomposition of the generalised instanton equations (5.13) into components given by

(W1)(n,m),(n+1,m') (W2)(n-1,m"),(n,m) = (W2)(n,m),(n+1,m') (W1)(n-1,m"),(n,m) , (5.14a)

(Wi)(n,m),(n+1,m') (W3)(n+2,m"),(n,m) = 0 = (W3)(n,m),(n-2,m') (Wi)(n-1,m"),(n,m) ,

(5.14b)

for (n, m) e Q0(k, l) and i = 1, 2, together with their conjugate equations. Note that in (5.14a) both combinations are morphisms between the same representation spaces and hence the commutation relation [Wi, W2] = 0 requires only that their difference vanish. On the other hand, in (5.14b) the two terms are morphisms between different spaces and so the relation [W¡-, W3] = 0 implies that they each vanish individually; in particular, in the generic case the solution has W3 = 0.

5.2.2. Stability conditions

For invariant connections there is a peculiarity involved in formulating stability of a holomorphic vector bundle, see for example [20]. On a 2n-dimensional Kahler manifold with Kahler form m, the stability condition is usually formulated through the identity

F A mn-1 = (m _ F)mn (5.15)

by demanding that m _F e Center(g), where g is the Lie algebra of the structure group. For generic connections the center of g is trivial and the usual stability condition m _ F = 0 follows. However, for invariant connections the structure group is smaller and the center can be non-trivial. This implies that there are several moduli spaces of translationally-invariant (and Zq+1-equivariant) instantons depending on a choice of element in Center(g).

Analogously to Section 4.2, the stability condition is associated to the moment map on the space of translationally-invariant and Zq+1-equivariant connections as we elaborate on in Section 6.2.3. In this case one can use any gauge-invariant element

S := 0 !p(n,m) ® i t(n,m) 1n+1 e Center(g^1)

(n,m)eQ0(k,l)

(5.16)

from the center of the Lie algebra

:= 0 u(pM(n +1}) , (5.17)

(n,m)eQo(k,l)

where ^(n,m) e R are called Fayet-Iliopoulosparameters. Thus the remaining instanton equations mC3 _i F =—i S read

[W1,W 1] + [W2,W2] + [W3,W3] = -i S. (5.18)

Again by substituting the general solutions (5.12a) and (5.12b) to the equivariance conditions we can decompose the generalised instanton equation (5.18) explicitly into component equations

J2 ((Wi)(nm) ,(n+1,m/} (Wi}(n+1 - (Wi}(n,m) ,(n — 1,m') (Wi}(n-1,m'},(n,m})

+ (W3}(n,m)

,(n—2,m'} (W 3)( n—2,m'},(n,m} — (W 3}(n,m} ,(n+2,m'} (W3}(n+2 ,m' ),(n,m) = 1p(n,m} ® 1n+1 ^(n,m) (5.19)

for (n, m} e Q0(k, l}.

5.3. Examples

We shall now elucidate this general construction for the three examples C1,0, C2,0 and C1,1. In each case we highlight the non-vanishing components of the matrices Wa and the representation (5.8).

5.3.1. C 1,0-quiver

The decomposition of the fundamental representation C1,0 into irreducible SU(2)-representa-tions is given by (C.1). The

non-vanishing components can be read off to be (Wi)(0,— 2),(1,1) and their adjoints ( Wi}(1,1},(0,—2). Thus there are two complex Higgs fields

$i := (Wi)№,—2),(U) for i = 1,2 , (5.20)

which determine a representation of the 2-Kronecker quiver

(0, ———2}C —d, 1) (5.21)

By (5.8) the representation of the generator h is given by

--CP.....V' ^ «p„,% 0 . (5-22)

5.3.1.1. Quiver relations The mutual commutativity of the matrices Wa is trivial in this case, and thus there are no quiver relations among the arrows of (5.21).

5.3.1.2. Stability conditions Choosing Fayet-Iliopoulos parameters f0, e R, the requirement of a stable quiver bundle yields non-holomorphic matrix equations given by

$1 + $2 $2 = 1Pd,1) ® & and $1 + $I $2 = 1p(0,—2} ® 12 £1. (5.23)

5.3.2. C2'°-quiver

The representation C2,0 is decomposed according to (C.3). The non-vanishing components can be determined as before to be (Wi)(0,_4),(i,_i), (^¡)(i,-i),(2,2) and (W3)(2,2),(0,_4), together with their adjoints ( Wf)(1,_1),(0,_4), ( Wi)(2,2),(1,_1) and ( W3)(0,_4),(2,2). Thus there are five complex Higgs fields

$i := W)(0,_4),(1 ,_1) , $¿+2 := (Wi)(1,_1),(2,2) and V := (W3)(2,2),(0,_4) for i = 1, 2, which can be encoded in a representation of the quiver

(5.24a)

(5.25)

As before the representation (5.8) for this example is

/ ^(2,2) ® !3 Z2+1 0 0

Y: h I 0 1pa _d ® I2 Zq+1 0 J . (5.26)

V 0 0 1p(0,_4) ® 1,

5.3.2.1. Quiver relations The holomorphicity condition yields

$iV = 0 , V$i+2 = 0 and $3 $2 = $4 $1 (5.27)

for i = 1, 2, plus the conjugate equations. The first two sets of quiver relations of (5.27) each describe the vanishing of a path of the quiver (5.25); an obvious trivial solution of these equations is V = 0. The last relation expresses equality of two paths with source vertex (0, _4) and target vertex (2, 2).

5.3.2.2. Stability conditions Choosing Fayet-Iliopoulos parameters f0, , f2 e R, the stability conditions yield

$1 $1 + $1 $2 _ V V1 = 1p(0,_4) ® & , (5.28a)

$1 $1 + $2 $2 _ $3 $3 _ $1 $4 = 1pa,_1) ® 12 £1 , (5.28b)

$3 $1 + $4 $1 _ V1 V = 1p(2,2) ® I312 . (5.28c)

5.3.3. C1,1-quiver

The decomposition of the adjoint representation C1,1 is given by (C.5). The non-vanishing components are (W)0,0),(1,3), (W)0,0),(1,_3), (W)(1,3),(2,0), (Wi)jx_3),(2,0) and (W3)(2,0),(0,0), together with their adjoint maps ( Wi)(1,3),(0,0), ( Wi)(1,_3),(0,0), ( Wi)(2,0),(1,3), ( Wi)(2,0),(1,_3) and ( W 3)(0,0),(2,0). Thus there are nine complex Higgs fields

$±:= (Wi)(0,0),(1,±3) , $±+2 := (Wi)(1,±3,2,0) and V := (W3)(2,0),(0,0) , (5.29) for i = 1 , 2, which can be assembled into a representation of the quiver

a; 0);

In this example the generator h of Z3+i has the representation

/ 1p(i,3) ® 12 Zq+1 0 0

Y ■ h \—> p(0,0)

'h^^ 0 0

V,0) ® 13 Zq+1 0

(5.30)

1P(1,-3) ® 12 Zq+1)

(5.31)

5.3.3.1. Quiver relations In this case the holomorphicity condition yields the relations

$± V = 0 , V$±+2 = 0 and $+ $+ + $_ = $+ $+ + (5.32)

for i = 1, 2. Again the first two sets of relations of (5.32) each describe the vanishing of a path in the associated quiver (5.30) (with the obvious trivial solution V = 0), while the last relation equates two sums of paths.

5.3.3.2. Stability conditions Introducing Fayet-Iliopoulos parameters f2, f3 e R, from the stability conditions one obtains

($+)t $+ + ($+)t $+ + ($-)t + ($-)t _ v Vt = 1^(0,0) ® & , (5.33a)

($±)t + ($±)t _ ($±)t _ )t = 1p(1,±3) ® 12 , (5.33b)

$+ ($+)t + $+ ($+)t + $3_ ($3_)t + ($_)t _ Vt V = 1p(2,0) ® 13 12 • (5.33c)

6. Quiver gauge theories on cones: comparison

In Sections 4 and 5 we defined Higgs branch moduli spaces of vacua of two distinct quiver gauge theories on the Calabi-Yau cone over the orbifold S5 /Zq+1. In this section we shall explore their constructions in more detail, and describe their similarities and differences.

6.1. Quiver bundles

6.1.1. SU(3)-equivariance

Consider the quiver bundle £k,t over R x S5/Zq+1 (as a special case of (3.25)). By construction the space of all connections is restricted to those which are both SU(3)-equivariant and Zq+1-equivariant. For holomorphic quiver bundles, one additionally imposes the holomorphicity condition on the allowed connections. The general solution to these constraints (up to gauge equivalence) is given by the ansatz (4.4), where the matrices XM satisfy the equivariance conditions (3.30) and (3.35) as well as the quiver relations (4.7). The induced quiver bundles have the following structure:

• A single morphism (arrow) m) between two Hermitian bundles (vertices) Ep(n m) and

P(n' ,m')

if n _ n =± 1 and m _ m =± 3.

• An endomorphism (vertex loop) f(nm) at each Hermitian bundle (vertex) EP(n m) with nontrivial monopole charge i|.

The reason why there is precisely one arrow between any two adjacent vertices is SU(3}-equi-variance, which forces the horizontal component matrices Xa for a = 1, 2, 3,4 to have exactly the same Higgs fields $fn m}, i.e. SU(3)-equivariance intertwines the horizontal components. The vertical component X5 can be chosen independently as it originates from the Hopf fibration S5 4 CP2. No further constraints arise from Z^-equivariance as we embed Zq+1 4 U(1) c SU(3). These quivers are a simple extension of the quivers obtained by [23,25] from dimensional reduction over CP2, because the additional vertical components only contribute loops on vertices with m = 0. This structure is reminescent of that of the quivers of [15] which arise from reduction over 3-dimensional Sasaki-Einstein manifolds.

The Hermitian Yang-Mills equations can be considered as the intersection of the holomor-phicity condition (4.7) and the stability condition (4.9). In this way their form can be recognised as Nahm-type equations of the sort considered in [22]. We will come back to this point in Section 6.2.2.

6.1.2. C3-invariance

Consider the V-bundle l over C3/Zq+1 from (5.1). Recall that C(S5) = C3. In contrast to the former case, we now impose invariance under the translation group C3 acting on the base as well as Zq+1 -equivariance. We demand that these invariant connections also induce a holomor-phic structure as previously. The general solution to these constraints is given by the ansatz (5.5) where the matrices Wa are constant along the base by (5.6), they commute with each other, and they solve the Zq+1-equivariance conditions (5.12). The induced quiver representations have the following characteristic structure:

• Two morphisms (arrows) m) (i = 1, 2) between each pair of Zq+1-representations (vertices) Cp(n,m) ^ (n, m) and Cp(n,m>) ® (n',m') if n - n' = ± 1 in Zq+1.

• One homomorphism (arrow) ^(„,m) between each pair of Zq+1-representations (vertices)

Cp(n,m) ® (n,m) and CP(n',m') ® (n',m') if n - n' = ± 2 in Zq+1.

The reason why there are exactly two arrows between adjacent vertices is that the chosen representation (5.8) does not intertwine W1, W2 and acts in the same way on both of them. Thus both endomorphisms have the same allowed non-vanishing components independently of one another, which gives rise to two independent sets of Higgs fields. The next novelty, compared to the former case, is the additional arrow associated to W3; its existence is again due to the chosen Zq+1-action. Translational invariance plus Zq+1-equivariance are weaker constraints than SU(3)-equivariance plus Zq+1-equivariance, and consequently the allowed number of Higgs fields is larger. On the other hand, holomorphicity seems to impose the constraint W3 = 0 for generic non-trivial endomorphisms W1 and W2 as discussed in Section 5. Hence there are two arrows between adjacent vertices, i.e. with n - n' = ± 1, but no vertex loops as in the former case.

It follows that the generalised instanton equations (5.13) and (5.18) give rise to non-linear matrix equations similar to those considered in [20] for moduli spaces of Hermitian Yang-Mills-type generalised instantons and in [15] for instantons on cones over 3-dimensional Sasaki-Einstein orbifolds. We will analyse these equations further in Section 6.2.3.

6.1.3. Fibrewise Zq+1 -actions

We shall now explain the origin of the difference between the choices of Zq+1-representations (3.33) and (5.8). Consider the generic linear Zq+1-action on C3: Letting h denote the generator of the cyclic group Zq+1, and choosing (0a) = (01,02, 03) e Z3 and (za) = (z1, z2, z3) e C3, one has

h ■ (za) = (hapzj) with (hj

This defines an embedding of Zq+1 into SU(3) if and only if 01 + 02 + 03 = 0 mod q + 1.

However, we also have to account for the representation y of Zq+1 in the fibres of the bundles (3.25) and (5.1). These bundles are explicitly constructed from SU(3)-representations Ck^ which decompose under SU(2) x U(1) into a sum of irreducible representations (n, m) from (3.9). If (n,m) and (n',m') both appear in the decomposition (3.9), then there exists (r, s) e Z^0 such that n _ n' = ± r and m _ m' = ± 3s.

6.1.3.1. SU(3)-equivariance The 1-forms j'q+1 transform under the generic Zq+1-action (6.1) as

j'q +1 Zq0+_03 j'q +1 for i = 1, 2 , (6.2)

while n and dr are invariant. Thus the equivariance condition for the connection (3.28) becomes

Y(h) (X2i_1 _ iX2i) Y(h)_1 = Zq"+01+03 (X2i_1 _ iX2i) for i = 1,2 , (6.3a)

Y(h) (X2i_1 + iX2i) Y(h)_1 = Zq0+_03 (X2i_1 + iX2i) for i = 1, 2 , (6.3b)

Y(h)X5 Y(h)_1 = X5 • (6.3c)

In this case the aim is to embed Zq+1 in such a way that the entire quiver decomposition (3.25) is automatically Zq+1-equivariant; hence the non-vanishing components of the matrices Xa and X5 are already prescribed by SU(3)-equivariance. For generic 0a it seems quite difficult to realise this embedding, because if one assumes a diagonal Zq+1-action on the fibre of the form

Y(h)= 0 1pnm) ® Z^ 1n+1 with Y(n,m)e Z, (6.4)

(n,m)eQ0(k,l)

then these equivariance conditions translate into

Y(n± 1,m + 3) _ Y(n, m) = 0' _ 03 mod q + 1 for i = 1, 2 (6.5)

on the non-vanishing components of Xa, a = 1, 2, 3, 4.

In this paper we specialise to the weights (0a) = (1, 1, _ 2) and obtain (2.34) for the Zq+1-action on SU(3)-equivariant 1-forms. From this action we naturally obtain factors Zq+31 for the induced representation n(h). This justifies the choice of y in (3.33), as m changes by integer multiples of 3 while n in (6.5) does not have such uniform behaviour.

Zq+1 0

Z03 Zq+1

6.1.3.2. C3-invariance The modified equivariance condition under (6.1) is readily read off to be

Y(h}WaY(h}—l = 1 Wa for a = 1, 2, 3 . (6.6)

In contrast to the SU(3)-equivariant case above, no particular form of the matrices Wa is fixed yet, i.e. here the choice of realisation of the Zq+1-action on the fibres determines the field content. By the same argument as above, a representation of Zq+1 on the fibres of the form (6.4) allows the component (Wa}(n,m),(n',m') to be non-trivial if and only if

Y(n',m'} — Y(n,m} = 0a mod q + 1 for a = 1, 2, 3 . (6.7)

For the weights (0a} = (1, 1, —2} we then pick up factors of Zq+11 or Zq+21, which excludes the choice (3.33). However, the modification to (5.8) is allowed as n changes in integer increments.

6.1.4. McKay quiver

In [15,37] the correspondence between the Hermitian Yang-Mills moduli space for transla-tionally-invariant and Zq+1-equivariant connections and the representation moduli of the McKay quiver is employed. The McKay quiver associated to the orbifold singularity C3/Zq+1 and the weights (0a} = (1, 1, —2} is constructed in exactly the same way as the C1,1 -quivers from Section 5, except that it is based on the regular representation of Zq+1 rather than the representations Ck^ considered here. It is a cyclic quiver with q + 1 vertices labelled by the irreducible representations of Zq+1, whose underlying graph is the affine extended Dynkin diagram of type Aq, and whose arrow set coincides with those of the C^1-quivers. See [21,38,39] for explicit constructions of instanton moduli on C3/Zq+1 in this context.

6.2. Moduli spaces

We shall now formalise the treatment of the instanton moduli spaces. We will first present an account of the general construction following [36,40], and then discuss the individual scenarios.

6.2.1. Kahler quotient construction

Let M be a Kahler manifold of complex dimension n and G a compact Lie group with Lie algebra g. Assume that G acts in the cotangent bundle T*M preserving the complex structure J and the metric g; hence G also preserves the Kahler form m. Let P = P(M, G} be a principal G-bundle over M, A a connection 1-form and F = Fa = dA + A a A its curvature.

Let Ad(P} := P xg G be the group adjoint bundle (where G acts on itself via the adjoint action, i.e. by the inner automorphism h 4 ghg—1), and let ad(P} := P xg g be the algebra adjoint bundle (where G acts on g via the adjoint action, i.e. by X 4 Ad(g}X = gX g—1). Let E := P xg F be the complex vector bundle associated to a G-representation F.

Denote the space of all connections A on P by A = A(P} and note that all associated bundles E inherit their space of connections A(E} from P. On A(P} there is a natural action of the gauge group G, i.e. the group of automorphisms of P which are trivial on the base M. One can identify the gauge group with the space of global sections

0 = (M, Ad(P}} (6.8)

of the group adjoint bundle, and the action is realised via the gauge transformations

A —4 g ■ A = Ad(g}A + g—1dg for g e tt0(M, Ad(P}}. (6.9)

The Lie algebra of the gauge group can then be identified with the space of sections

"0 = Q0(M, ad(P)) (6.10)

of the algebra adjoint bundle, and the infinitesimal gauge transformations are given by

A 5X A = dAx := dx + [A,x] for x e tt0(M, ad(P)) . (6.11)

Since A(P) is an affine space, its tangent space T^A at any point A e A can be canonically identified with Ul(M, ad(P)). If the structure group is a matrix Lie group, i.e. there is an embedding G ^ U(N) for some N e Z>0, then 0 is a matrix Lie algebra and the trace defines an Ad(G)-invariant inner product on 0. The induced invariant inner product on Ul(M, ad(P)) is

<X1,X2>:=y tr (X1 A*X2) for X1,X2 e ft1(M, ad(P)), (6.12a)

which gives rise to a gauge-invariant metric on A(P) via the pointwise definition

g|A(X1,X2) :=<X1,X2>|a for X1X e TAA . (6.12b)

The space A(P) moreover carries a gauge-invariant symplectic structure defined by

w|a(X1,X2) = j tr (X1 A X2) A rnn_1 for X1,X2 e TAA. (6.13)

Note that the 2-form w is completely independent of the base point A e A. Let D denote the exterior derivative acting on forms on A. Then by computing

Dw|a(X0,X1,X2) = X0(«|a(X1,X2)) _ X1 («|a(X0,X2)) + X2(w|A(Xo,Xl))

_ W|A([X0,X1],X2) + W|A([X0,X2],X1) _ w|a([X1,X2],X0),

(6.14)

one observes that Dw = 0 as X^wAXj, Xk)) = 0 due to base point independence and

>]A([Xi,Xj ],Xk) = y tr ([Xt,Xj ] A Xk ) A of-1 = 0 (6.15)

as tr ([Xi,Xj] a Xk) e Q3(M) which renders the integrand into a form of degree larger than the top degree. It follows that « is a symplectic form, which promotes A to _an infinite-dimensional Riemannian symplectic manifold (A, g, «) equipped with a compatible (/-action.

6.2.1.1. Holomorphic structure Consider now the restriction to connections on E ^ M which are generalised instanton connections. Recall that one part of the Hermitian Yang-Mills equations can be interpreted as holomorphicity conditions, and the corresponding subspace is

Au = e A(E) : J^2 =-(FA0)t = 0} c A(E). (6.16)

This definition employs the underlying complex structure on M. As before, this condition is equivalent to the existence of a holomorphic structure on E, i.e. a Cauchy-Riemann operator dE := d + A0,1 that satisfies the Leibniz rule as well as dE o dE = 0. Thus a G -bundle with only holomorphic connections induces a GC-bundle where GC = G ® C. One can show that A1,1 is an infinite-dimensional Kahler manifold, i.e. the metric g is Hermitian and the symplectic form « is Kahler. These tensor fields descend from A to A1,1 simply by restriction.

6.2.1.2. Moment map The space A1,1 inherits a G-action from A and since it has a G-invariant symplectic form, i.e. the Kahler form «, one can introduce a moment map

j: A1,1 —4-g* = U2n(M, ad(P}}

A —4 FA a mn—1 . (6.17)

For this to be a moment map of the G- -action one needs to verify the defining properties, generalising the arguments presented in [36]. For this, note that j is obviously G-equivariant. Next let 0 e (M, ad(P}} be an element of the gauge algebra, 0^ the corresponding vector field on A1,1 and f e Ul(M, ad(P}} a tangent vector at the base point A. Then the condition to verify is

(0, Dj\A}(f} = 0&\A(f}, (6.18)

wherein i denotes contraction and the dual pairing (■, ■} of — with —* is defined via integration over M of the invariant inner product on g. Firstly, in the definition of j only Fa is base point dependent, and a standard computation gives F^+tf = Fa + t ¡Af + 2t2 f a f so that DF\a = (¡d FA+tf)\t=0 = ¡Af. Thus the left-hand side of (6.18) is (0, Dj\A}(f} = fM tr((dAf} a 0) a mn—1. Secondly, the vector field 0^ can be read off from (6.11) to be 0A = ¡a0 e ^(M, ad(P}}. Hence the right-hand side is 0v\A(f} = fm tr((da0} A f) A an 1. But from j'M d (tr (f A 0} A af—1) = 0 and dm = 0 one has j'M tr((¡a f} A 0) A mn—1 = — fm tr (f A (da0}) A mn—1, and therefore the relation (6.18) holds, i.e. f is a moment map of the G-action on A1,1.

We will use the dual moment map defined by

fj*: A1,1 —4 — = Q0(M, ad(P}}

A —4 m _i Fa , (6.19)

which is equivalent to the definition (6.17) due to the identification—=—* given by (5.15) (gener-ically by a choice of metric). Thus we will no longer explicitly distinguish between the moment map j and its dual j*.

For regular elements S e the centraliser of S in G is the maximal torus and j—1(E} c A1,1 defines a submanifold which carries a G-action. The quotient of the level sets6

A1,1 //S -:= j—1 (S} / — (6.20)

is well-defined, and moreover it defines a Kahler space as the Kahler form and the complex structure descend from A1,1 by gauge-invariance. The level set of zeroes is precisely the Hermitian Yang-Mills moduli space.

6.2.1.3. Complex group action As the G-action in T*M preserves the Kahler structure, one can extend it to a GC-action in T *M. The same is true for the extension to the complexification of the --action on A1,1, i.e. the holomorphicity conditions FJ^ = 0 are invariant under the action of the complex gauge group

GC = -® C. (6.21)

6 One must in fact take S e Center(—} for a well-defined quotient.

For A e A1,1 the orbit of the "^-action is given by

= {A'e A1,1 : A'= g • A ,g e GC } . (6.22)

A point A e A1,1 is called stable if "c n ll_1(S) = 0. Denote by A^S) c A1,1 the set of all stable points (for a given regular element S). Then the Kahler quotient can be identified with the GIT quotient (see for instance [41])

A1,1 ¡Is G = A^1 (S) / . (6.23)

In the following we discuss applications of this Kahler quotient construction to SU(3)-equiv-ariant and Z3+1-equivariant instantons on the Calabi-Yau cone M = C(S5/Zq+1), as well as to the C3-invariant and Zq+1-equivariant case. These vacuum moduli spaces are special cases of those constructed above, as we do not consider generic connections but rather equivariant connections. For instance, equivariance reduces the gauge groups.

6.2.2. SU (3)-equivariance

Consider the space of SU(3)-equivariant connections A(£k,t) on the bundle (3.25) (for d = 1), which is an affine space modelled on Ul{C(S5/Zq+1), Endu^) (Vk,z)). The structure group Gk,t of the bundle (3.25) is given by (3.24). An element X e ft1(C(S5/Zq+1), EndU(1)(Vk,1)) can be expressed as

X = X^e^ + XT dT = Xj0j + Xj 0j , (6.24)

once one has chosen the coframe (eM, dT} of the conformally equivalent cylinder R x S5/Zq+1 with r = eT. One can equivalently use the complex basis 0j = e2j_1 + i e2j for j = 1, 2, 3, where e6 := dT; then (Xj)^ = _Xj. Thus once one has fixed a choice of coframe on the Calabi-Yau cone C(S5/Zq+1), the tangent space to A(£k,t) at a point A is described by all 6-tuples ({XM}, XT) or equivalently ({Xj}, { Xj}). Here XM and XT depend only on the cone coordinate t by SU(3)-equivariance.7

6.2.2.1. Instanton equations One can eliminate the linear terms in (4.7) and (4.9) via the redefinitions

Xa = e_ 2 T Xa for a = 1,2, 3,4 and X5 = e_4T X5 , Xt = e_4T X6 . (6.25) Using 't Hooft tensors the matrix equations read

[Xa, Xb] = 0 and r,2ab [Xa, Xs] = 0 , (6.26a)

-T = -n3ab [Xb, X5] _ [Xa, X6] , (6.26b)

= _Hs) ([X1, X2] + [X3, X4^ _ [X5, X6] , (6.26c)

where s := 4 e_4T e R>0 and k(s) = ()4. The equations (6.26) are automatically satisfied in the temporal gauge XT = 0 by taking constant scalar fields XM for \x = 1, ..., 5 satisfying the Higgs branch BPS equations (3.43) of the quiver gauge theory.

7 Recall that the equivariance condition (3.30) makes the endomorphisms X^ base point independent on S5/Zq+1 ; hence it is consistent to have solely t-dependent matrices X^ in any coframe.

Changing to a complex basis as before and defining

Yj = 1 (Xj-i - i Xij) and Yj = 2 (X2j-i + iX2j) for j = 1,2, 3 , (6.27) the resulting holomorphicity conditions are

[Y1, Y2] = 0 and [ Y1, Y2] = 0 , (6.28a)

dY1 = —2i [Yi, Y3] and dY- = 2i[Yi, Y3] for i = 1,2 , (6.28b) while the stability condition yields

+ dY1 = 2i[Y3, Y3] + 2U(s) ([Yi, Y1] + [Y2, Y2]) • (6.28c)

dY3 + dY3 ds ds

Analogously to the generic situation, we define the subspace

A1,1 (EkJ) = j ({Yj}, {Yj}) e A(EkJ) : (6.28a) and (6.28b) hold} • (6.29)

6.2.2.2. Real gauge group On the space A1,1 (£k,v) there is an action of the gauge group

dkJ := ^°(R>o, GkJ) , (6.30)

with Gk,t 4 U(p), given by8

Yi —4 Ad(g)Yi for i = 1,2 and Y3 —4 Ad(g)Y3 + , (6.31a)

for g e Gk,'t. One readily checks that the full set of equations (6.28) is invariant under these "real" gauge transformations. Moreover, one can always find a gauge transformation g e G?1,1 such that g ■ X6 = 0 or equivalently g ■ Y3 = g ■ Y3.

6.2.2.3. Complex gauge group The space A1,1(Ek,t) also admits an action of the complex gauge group

(Gk,l)C := ^0(R>0,(Gk,')C) , (6.32)

with (Gk,t)C 4 GL(p, C). However, only the equations (6.28a) and (6.28b) are invariant under the "complex" gauge transformations given by

Yi Ad(g)Yi and Yi —4 Ad(g*-1)Yi for i = 1, 2 , (6.33a)

Y3 —4 Ad(g)Y3 + (ddS) g—1 and Y3 —4 Ad(g*—1)Y3 + 2,

S S (6.33b)

where g e (?k,l)c and g*—1 = (g—1)f.

8 We assume that the paths g(s) : (0, to) 4 Qk'1 are sufficiently smooth.

6.2.2.4. Kahler structure Following the construction of Section 6.2.1, the next step is to define a Kahler structure on A1,1(£k,t). The tangent space tam^'1) at point A is C(S5/Zq+1), Endu(1)(VkJ)), so a tangent vector x = xjOj + Xj9j over Au(£k>1) is defined by linearisation of the holomorphicity equations (6.28a) and (6.28b) for paths xj(s) : (0, œ) ^ EndU(1)(Vk,l)C. The gauge transformations are given by xj ^ Ad(g)xj for j = 1, 2, 3.

A metric on A1,1(Ek,l) can be defined from (6.12) as

g\A(x' y) := 2/ds J2 tr(x î yj + xj y Î ) '

2J -vj yj + Xj y )) , (6.34)

0+ j=1

where the integral over S5/Zq+1 drops out here as the tangent vectors at equivariant connections are independent of the coordinates of S5/Zq+1. A symplectic form on A1,1 (£k,v) can likewise be defined from (6.13) as

^\a(x' y) := 2 J ds J2 tt(xî/ yj- xj yP • 0+ j=1

yj - Xj y )) • (635)

Both g and w are gauge-invariant by construction. Moreover, we immediately see that for the choice of complex structure9 J( Xj) — i xj the symplectic form w and the metric g are compatible, i.e. g(-, J ■) = w(-, ■).

6.2.2.5. Moment map On the Kahler manifold A1,1 (Ek'1) we define a moment map by Z : AU(£k'l) Endu(1) (VkJ)

({Yj}, {Yj}) + d7 - 2i[Y3, Y3] - 2U(s) ([Y1, Y1] + [Y2, Y2]) , (6.36)

which readily gives us the Kahler quotient for the instanton moduli space

^SU(3) = f-1(0) / Gk,1 • (6.37)

6.2.2.6. Stable points We can alternatively describe the moduli space M^3 via the stable points

AsV(£k,r) := {({Yj}, {Yj}) € A1,1^1) : (Çkl^,y.„ n f-1(0) = 0} , (6.38)

and by taking the GIT quotient as before to get

MS?3 = As\,1(fk,l)/(^,l)C • (6.39)

We show below that it is sufficient to solve the holomorphicity equations (subject to certain boundary conditions), as the solution to the stability equation then follows automatically by a complex gauge transformation. More precisely, for every point in A1,1(Ek,v) there exists a unique point in its complex gauge orbit which satisfies the stability equation, i.e. every point in A1,1(£k,v) is stable.

9 We essentially use the complex structure J of C3.

6.2.2.7. Solutions of the holomorphicity equations Following [42] one can regard the holomor-phicity equations as being locally trivial. For this, we use a complex gauge transformation (6.33) to eliminate Y3 via

3 = Ad(g)Y3 + = 0 • (6.40)

.— i

2 V ds

From the holomorphicity equations (6.28b) and (6.28a) one obtains in this gauge

— = 0 and Yi = Ti with [ T1, T2] = 0 , (6.41)

ds L J

where T; are constant for i = 1, 2. Consequently the general local solution of the holomorphicity equations (6.28a) and (6.28b) is

Yi = Ad(g—1)Ti with [T1, T2] = 0 and Y3 = — g—1dg, (6.42)

with g e (G^^)c. A solution to the commutator constraint chooses T; for i = 1, 2 as elements of the Cartan subalgebra of the complex Lie algebra Endu(1)(Vk,l)C of the structure group (3.24).

6.2.2.8. Solutions of the stability equation We also need to solve the stability equation (6.28c), for which we follow again [42]. Recall that the complete set of instanton equations (6.28) is ^^-invariant, and for each g e (GGk,l)C define

h = h(g) = ggf : (0, to) —4 (GkJ)c / GkJ 4 GL(p, C) /U(p)• (6.43)

Fix a 6-tuple {Yj, Yj}j=123 and define the gauge transformed 6-tuple {Yj, Yj}j=123. We will study the critical points of the functional

Le [g] = ^ f ds tr(+ Y3|2 + 2X(s)J2 | Yi for 0 <e< 1 • (6.44)

e i = 1

As the instanton equations are invariant under U(p)-valued gauge transformations, we can restrict g to take values in the quotient GL(p, C) /U(p) which may be identified with the set of positive Hermitian p x p matrices [42]. Hence it is sufficient to consider variations with Sg = 5gt around g = 1p (and with Sg = 0). Then the gauge transformations (6.33) imply that

SY3 = [Sg, Y3] + and SYi = [Sg, Yi ] for i = 1,2 • (6.45)

The variation then leads to

SLe[g] = —i f ds tr ([¿({Yj}, {Yj}) Sg) , (6.46)

i.e. the critical points of (6.44) form the zero-level set of the moment map.

Now we use the solution (6.42) as an initial evaluation of Le. Then we obtain the functional of h given by

Le [h] = 2 i ds T 4 tr (h—1 dh )2 + V(h^ , (6.47)

where the potential V(h) = 2X(s) J^2=1 tr(h 1 T; h T;is positive. This implies that for any boundary values h± e (Gk,l)C/Gk,]l there exists a continuous path10

(Gk,1 )C / Gk,1 with h(e) = h— and h(1) = h+ , (6.48)

which is smooth on (e, 1) and minimises the functional Le. Hence for any choice of complex gauge transformation g such that gg} = he, the triple g ■ ({T;}i=1,2, 0) = ({Ad(g)T;}i=1,2, 2 (dF) g—1) satisfies the stability equation [({Yj}, { Yj}) = 0 on (e, 1) for any 0 < e < 1.

The uniqueness of the solution he and its extension to the limit e 4 0 follows from [42]

ii 1 similarly to the proof of [43, Lemma 3.17]. The gauge transformation gTO = (hTO)2 is obtained

from hTO = lime40 he. However, the corresponding complex gauge transformation g = g(he) is

not unique. Similarly to [42,43], given a solution { Yj}j=1,2,3 of the holomorphicity equations

one can define two solutions { Yj'} j=1,2,3 = {g1 ■ Yj}j=1,2,3 and { Yj"}j=1,2,3 = {g2 ■ Yj}j=1,2,3

of the stability equation for any g1, g2 e (Gk,l)C. By uniqueness one has g1 g} = g2 g2j; thus

there exists g e Gk,]l such that g1(s) = g2(s)g(s). This ambiguity in the choice of g = g(he)

can be removed as follows: The complete set of instanton equations is invariant under ty^ and

a Gk,]l gauge transformation is sufficient to eliminate X6. Hence one can demand that the gauge

transformation { Yj'}j=1,2,3 = {g ■ Yj}j=1,2,3 of a solution { Yj}j=1,2,3 satisfies Y3' = Y3'. This

fixes g = g(he) uniquely.

6.2.2.9. Boundary conditions A trivial solution of (6.26) is given by

X6(s) = 0 and X[(s) = T[ with [T[,TV] = 0 for ¿,v = 1,^,5, (6.49)

where T[ are constant elements in the Cartan subalgebra u(1)p of EndU(1)(Vk,v). From the rescaling (6.25) we then see that the original scalar fields XM are singular at the origin r = 0 (corresponding to t 4 —to or s 4 to). Following [43,44], in the generic case we choose boundary conditions for Xn such that12 x^(t) 4 0 as t 4 +to for [ = 1, • ••, 5. Arguing as in [43], this implies the existence of the limit of X^(s) for s 4 0 and hence the solutions extend over the half-closed interval R>0. Since (6.26) is a system of first order ordinary differential equations, it suffices to impose one additional boundary condition for the matrices X^(s') on [0, to) which we take to be

lim X[(s) = Ad(g0)T[, (6.50)

10 See for instance the note under [42, Corollary 2.13]: Since GL(p, C)/U(p) satisfies all necessary conditions for the existence of a unique stationary path between any two points, the quotient (Gk'l)C/= n(nm) GL(p(n,m), C)/ U(P(n,m)) x GL(n + 1, C)/U(n + 1) inherits these properties.

11 We omit a description of the required differential inequality as well as a treatment of potential pole contributions from l(s); see [13, Section 3] for a general discussion of these issues.

12 From now on we will no longer deal with the scalar field X6 as it can always be gauged away.

for suitable g0 e Gk,]l ensuring compatibility with the SU(3)-equivariant structure from (3.31) (cf. Section 4 for explicit examples). Then the value of X^(s') at s = 0 is completely determined by the solution.

From (6.28b) it follows that the paths Y¡(s) for i = 1, 2 each lie respectively in the same adjoint orbits Oi of the complex Lie algebra EndU(1)(Vk,l)C for all s e [0, x). Let Ti = 2 (T2i-1 + iT2i) for i = 1, 2, and denote by O^. the adjoint orbit of Ti in EndU(1)(Vk,l)C. Then the boundary conditions (6.50) imply that the closures O^. contain Oi for i = 1, 2. If the quintuple (TM|M=i,...,5 is regular in the Cartan subalgebra of Endu^V^1), i.e. the joint centraliser of T^ in Gk,]i is the maximal torus U(1)p, then O^. = O^. are regular orbits and hence O^. = Oi [43]. By our previous results, there exists a unique complex gauge transformation g, which is bounded and framed, such that {g ■ Yj}j=1,2,3 satisfies (6.28c) and g ■ Y3 is skew-Hermitian. Employing (6.28a), it follows that in this case there is a map

^SU(3) OT1 x OT2

({Yj(r)}j = 1,2,3 , {Yj(r)}j = 1,2,3}) (Y 1(0), Y2(0)) (6.51)

from the moduli space of solutions satisfying the boundary conditions (6.50) together with the equivariance condition imposed by our construction. Arguing as in [43], by our construction of local solutions to the complex equations, and the existence of a unique solution to the real equation within the complex gauge orbit of these elements, this map is a bijection which moreover preserves the holomorphic symplectic structure. This space is naturally a complex symplec-tic manifold of (complex) dimension 2 dim(Gk,l)C -J22=1 dim(Er.) with the product of the standard Kirillov-Kostant-Souriau symplectic forms on the orbits, where Z^. C (Gk,l)C is the subgroup that commutes with Ti for i = 1, 2. By our general constructions it is a Kahler manifold. In the cases that SU(3)-equivariance forces Ti = 0 for some i e {1, 2}, the corresponding orbit closure Ot. should be replaced by the nilpotent cone N of dimension dim(Gk,l)C - p which consists of all nilpotent elements of EndU(1) (Vk,l)C. The variety N has singularities corresponding to non-regular nilpotent orbits, and in particular it contains the locus of Kleinian singularities C2/Zp in complex codimension 2; see [15] for further details. Thus in this case the moduli space is singular and by SU(3)-equivariance we expect that it contains the singular subvariety C3/Zp.

6.2.3. C3-invariance

Now we turn our attention to the space of translationally-invariant connections A(Ek,h) on the bundle (5.1). The structure group Gk,t of (5.1) (which in this case coincides with the gauge group) is given by (5.3) and its Lie algebra by (5.17). A generic element of the tangent space TAA(Ek,1) at a point A e A(Ek,l) is given by

W = Wa dza + Wa dza e Q1(C3/Zq+1, 0k,r) , (6.52)

with constant Wa, Wa for a = 1, 2, 3. As before, let us define a metric g on A(£k,lX Gauge transformations of tangent vectors w = wa dza + wa dza are given by wa ^ Ad(g)wa for a = 1,2, 3.

We deduce the metric to be

1 3 ( )

g\A(w, v) := 2 D trW va + wa vD , (6.53)

and a symplectic form via i 3

V\A(w, v) := 2 J2 tr(wfa Va — wa vD • (6.54)

These definitions follow directly from the translationally-invariant limit of (6.12) and (6.13) (and agree with those of [20]). Evidently the metric and symplectic structure are gauge-invariant.

Define the subspace of invariant connections that satisfy the holomorphicity conditions (5.13)

A1,1^) = {({Wa}, {Wa}) e A^) : [Wa,Wp ] = 0 for a, j = 1,2, 3} , (6.55) which is a finite-dimensional Kahler space by the general considerations of Section 6.2.1.

6.2.3.1. Moment map The corresponding moment map can be introduced as before via [ : A1,1(ek,l) —4 fl^

({Wa }, { Wa })—4 iJ2 [ Wa, ^^^ , (6.56)

but in this case it is possible to choose various gauge-invariant levels S from (5.16) and consequently define different moduli spaces

M%(S) = [^(S) / ©^ • (6.57)

6.2.3.2. Real gauge group The complete set of instanton equations (5.13) and (5.18) is invariant under the action of the gauge group (5.3) with the usual transformations

Wa —4 Ad(g)Wa for a = 1, 2, 3 (6.58)

for g e ©k,l4 U(p).

6.2.3.3. Complex gauge group Recalling that the holomorphicity conditions allow for the introduction of a (©k,l)C-bundle, we find that the corresponding equations are invariant under (©k,l)C gauge transformations. Again the stability equation is not invariant under the action of the complex gauge group.

6.2.3.4. Stable points The set of stable points is defined as before to be

A1; 1(Ek l; S) := |({Wa} , {Wa}) e a1, ^ l) : (©k l^}, {¥a„ n M—1(S) = 0} ,

(6.59)

and by taking the GIT quotient one obtains the S -dependent moduli spaces13

mZ(E) = A^1; s) / (©k,l)C • (6.60)

13 This description is analogous to the quiver GIT quotients used by [21,37] to describe instanton moduli on C3/Z?+1 as representation moduli of the McKay quiver.

The moment map (6.56) transforms under g e (Gk,l)C as

fl({Wa}, { Wa})=

iAd(g)^][h-1 Wah,Wa], (6.61)

where we introduced h = h(g) = gt g e (Gk^)C/Gk,t. Similarly to before, h can be identified with a positive Hermitian p x p matrix. Moreover, Ad(g' )S = S for any g' e Gk'1. By the embedding ©k^ ^ U(p) and the polar decomposition of an element g e (Gk,l)C into g = h' exp(i X) for Hermitian h' e ©^l and skew-adjoint X e gk,]l, we have

Ad(g)S = Ad(h')(Ad(exp(iX))S) = Ad(h') (S + i [X, S]) = Ad (h')S = S, (6.62)

where we used the Baker-Campbell-Hausdorff formula and the fact that S is central in gk,]i. It follows that Center(gk,1) c Center((gk,1)^. Hence a point ({Wa}, { Wa^ e A1,1(Ek,1) is stable if and only if there exists a positive Hermitian matrix h (modulo unitary transformations) satisfying the equation

J2[h-1 Wah,Wa] = -i S. (6.63)

By our general constructions the moduli spaces MjCl(S) are Kahler spaces, which however are generically not smooth manifolds but have a complicated scheme structure with branches of varying dimension that should be analysed within the context of a perfect obstruction theory; such an analysis is beyond the scope of the present paper. Generally, the canonical map M^S) ^ MC3(0) is a partial resolution of singularities for generic S. For example, in the case p(n, m) = 1 for all (n, m) e Q0(k, l) (so that Vk,l = C_k,l and p = p0), for generic levels S = 0 the moduli spaces M'Cl(S) are schemes akin to the Zp-Hilbert scheme of p = dim( Ck^ ) points on C3 for the Zp-action given by (2.28) (with q = p - 1), which are partial resolutions of the singular spaces M'Cl(0) that correspond to configurations of p points of C3 given as unions of Zp-orbits (cf. [21,20] for the case of the McKay quiver)14; these are precisely the same types of singularities encountered in the moduli spaces M^3 ^ above.

Acknowledgements

This work was partially supported by the Grant LE 838/13 from the Deutsche Forschungsgemeinschaft (DFG, Germany), by the Consolidated Grant ST/L000334/1 from the UK Science and Technology Facilities Council (STFC), and by the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST).

14 In the special case q = 1 the Z2-Hilbert scheme is the product M1 x C where M1 is the total space of the canonical line bundle O^p 1 (-2) ^ CP1 (Eguchi-Hanson space), whereas for q = 2 the Z3-Hilbert scheme is the total space of the canonical line bundle O^p2 (-3) ^ CP2 (local del Pezzo surface of degree 0).

Appendix A. Bundles on CP2

A.1. Geometry of CP2

A.1.1. SU(3)-equivariant 1-forms

Consider the row vector jT = (j1, j2). The relations (2.11) and (2.12) dictate the explicit form of the 1-forms j' and their exterior derivatives as

; 1 ; 1

j' = - dy; - -

j Y y Y2 (Y + 1)

y1 y^ yj dyj , j=1

jS' = - dy' - -

j Y y Y2 (Y + 1)

y' y^ yj dyj j=1

(A.1a)

dj1 = - j1 A (Bu + § a) + j2 A B12 , djS1 = -(Bn + 2 a) A jS1 - B12 A j32 ,

= -j1 A B12 + j2 A (Bu - § a),

(A.1b)

= B12 A jS1 + (B11 - 2 a) A j2 . (A.1c)

One can regard j' as a basis for the (1, 0)-forms and ¡3' as a basis for the (0, 1)-forms of the complex cotangent bundle over the patch U0 of CP2 with respect to an almost complex structure J. The canonical 1-forms d/ and dy' could equally well be used for a holomorphic decomposition with respect to J, but the forms ¡' and ¡3' are SU(3)-equivariant.

A.1.2. Hermitian Yang-Mills equations

The canonical Kahler 2-form on the patch U0 is given by

MCP2 = -i R2 jT A jS = i R2 (p1 A jS1 + j2 A «2

where R is the radius of the linearly embedded projective line CP1 c CP2. The 1-form B(\) is then an instanton connection by the following argument: Locally, one can define a (2, 0)-form Œ proportional to j1 A j2. The Hermitian Yang-Mills equations for a curvature 2-form F are

Q A F = 0 and

ojCp 2

which translate to F = F1,1 being a (1, 1)-form for which tr(F1,1) = 0; here the contraction j between two forms n and n' is defined as n j n' := * (n A . The curvature FB = dB + B A B = ¡3 A jT is a (1, 1)-form which is u(2)-valued, i.e. tr(FB) = 2a = 0. However Fa = da = ¡t A ¡3 is also a (1, 1)-form. Thus the curvature of the connection B(1) = B — 2 a 12 given by FB(1) = FB — 1 Fa 12 is a (1, 1)-form and by construction traceless; hence B(1) is an su(2)-valued connection satisfying the Hermitian Yang-Mills equations, i.e. it is an instanton connection.

A.2. Hopf fibration and associated bundles

Consider the principal U(1)-bundle S5 = SU(3)/SU(2) 4 CP2. One can associate to it a complex vector bundle whose fibres carry any representation of the structure group U(1), i.e. a complex vector space V together with a group homomorphism p : U(1) 4 GL(V). Then the associated vector bundle E is given as E := S5 xp V 4 CP2. In particular, one can choose V = m to be the one-dimensional irreducible representation of highest weight m e Z. Following [25], one then generates associated complex line bundles Lm := (L®m)2.

A.2.1. Chern classes and monopole charges

Using the conventions of [25] for CP2, there is a normalised volume form

Pvoi :=— P1 A p1 A p2 A p2 with J Pvol = 1 , (A.4)

and the canonical Kahler 2-form (A.2) with

MCp2 A MCp2 = -2n R j pvol . (A.5)

Consider the connection a from (2.11c) on the line bundle L associated to the Hopf bundle S5 ^ CP2 and the fundamental representation. Since its curvature is Fa = Rj mCP2, the total Chern character of the monopole bundle L is

ch(L) = exp (Fa) = exp(f) (A.6)

where f := -^nR MCP2. Then one immediately reads off the first Chern class

cl(L) = f with j f Af = -1 . (A.7)

Since [f] = [ci(L)] generates H2(CP2, Z) = Z [23], this identifies the first Chern number of L as -1. Thus L = L1 exists globally, and the dual bundle L-1 has first Chern class c1(L-1) = -c1(L) and hence first Chern number +1. For all other bundles Lm one takes the connection to be m a, which changes the first Chern class accordingly to

c1 (Lm) = m f> (A.8)

and the first Chern number to -m. Hence only for even values of m do the line bundles Lm exist globally in the sense of conventional bundles. On the other hand, for odd values of m the line bundles Lm (and also the instanton bundles In for odd values of the isospin n [25]) are examples of twisted bundles. The obstruction to the global existence of these bundles is the failure of the cocycle condition for transition functions on triple overlaps of patches, which is given by a non-trivial integral 3-cocycle representing the Dixmier-Douady class of an abelian gerbe; see for example [45] for more details. As argued in [25], the Chern number m of the line bundle L-m should be taken as the monopole charge rather than the Ha2 -eigenvalue m in the Biedenharn basis.

Appendix B. Representations

B.1. Biedenharn basis

Let us summarise the relevant details we need concerning the Biedenharn basis [28-30], which is defined as the basis of eigenvectors according to (3.8); we follow [23,25] for the presentation and notation.

B.l.l. Generators

The remaining generators of su(3) act on this eigenvector basis as

E± a1

|= ^(n t q)(n ± q + 2) q ± 2 m,

qmj = 2V (n T

nm) = 7 WW)^ti(nm)

n — 1 m + 3) + 7sn+T) A—,l(n,m)

n—1 q—1

a1 +a2

m = 7W++2 A+,l(n,m) n + 1 m + 3) ^ A/2(n—iq1TA—,l(n,m)

(B.1a)

m + 3|, (B.1b)

n — 1 q + 1

(B.1c)

with Ea2 = Et2 = E—a2 and Ea1+a2 = ET+a2 = E—(a1+a2). It is convenient to express the generators as

E+ (n,m) _ ^ j n+q+2 A+ , m) Ea1+a2 = V 2(n+1) Ak,l(n,m)

Ea1+a2 m = V2{n+1) Ak,l(n, m)

E+2in,m) = E/liA+l(„,m)

Ea2(n,m) = E fnkAli(nm)

n + 1 q + 1

n — 1 q + 1

q — 1

n — 1

q — 1

m + 3 m q

m + 3 m q

m + 3 m q

m + 3 m q

(B.2a) (B.2b) (B.2c) (B.2d)

where Qn := {—n, —n + 2, • ••, n — 2, n} and

+ 1 k+2l n m ( k—l n m ) 2k+l n m

Akl(n,m) = vn+IV + 5 + m + V + n + m + ^ — 2 — ej

(B.3a)

A—nm)— n+m+0 (^+n — m) +n — m+0, ^

with A—1(0, m) := 0 [25]. The identity operator ^(n,m) of the representation (n, m) is given by

B.1.2. Fields

The 1-instanton connection (2.13) is represented in the Biedenharn basis by B(1) = B11 Ha, + B12 Ea1 — (B 12 Ea1

(Bnq nmjln™ +1B12V(n — q)(n + q + 2) q + 2m)lnm

— 1 B 12Tf(n+q)(n—q+Y)

= ^^ B(n,m) , (n,m)eQ0(k,l)

q — 2 q

where B(nm) e (su(2), End( (n, m))). One further introduces matrix-valued 1-forms given by

fiq+1 = @q+1 Ea1+a2 + /^+1 Ea2 = 0 (P(«,m) + P(n,m))

(n,m)eQo(kJ)

with the morphism-valued 1-forms

Knm) e S5/Zq+1, Hom( (n,m), (n ± 1,m + 3))) , and the corresponding adjoint maps

Pt«,m) e S5/Zq+1, Hom( (n ± 1,m + 3), (n,m))) . They have the explicit form

ß±nm) = L Wn ± q + 1 ± 1 ¿W

\J2(n + 1)

n ± 1 \/n

m + 3 m q + 1 \q

+ у/n T q + 1 ± 1 ßq2+1

n ± 1 A In ) , m + 3 )( m .

q -1 /\q /

(B.6a)

(B.6b)

(B.6c)

B.1.3. Skew-Hermitian basis

Similarly to [46], for a given representation C_k'1 of the generators Ii and defined in (3.2) the decomposition into the Biedenharn basis yields

j _ /T\ j(n,m) _ ST\ ( E± (n,m) _ E± (n,m)\

4 — j1 — tt7 \Eai+a2 E-ai-a2) ,

(n,m) ± , (n,m)

i2—0 i(n,m)—-i 0 (E±+nam)+E±an^m2),

±, (n,m)

I3 — 0 I(n,m) — 0 (e,

± (n,m) _ e± (n

± , (n,m)

I4 — 0 I(tm) — -i © (E±2(n'm) + E±(n,m)),

±, (n,m)

I5 — 0 I^1^) — -2 0 Я(

(n,m) 2 -a2

(B.8a) (B.8b) (B.8c) (B.8d) (B.8e)

The commutation relations [Ii,Ia] = fiab Ib and [/¡,/5] = 0 induced by (3.4) respectively imply relations among the components given by

j(n',m') j(n,m) _ j(n,m) j(n,m) f b j(n,m)

Ii Ia — Ia Ii + Jia Ib

j(n,m) j(n,m) _ j(n,m) j(n,m)

Ii I5 — I5 Ii ,

(B.9a) (B.9b)

where i e {6, 7, 8}, a e {1, 2, 3, 4}, Ii — 0(nm) I(n,m) and (n', m' ) — (n ± 1, m + 3).

B.2. Flat connections

One can compute the matrix elements of A from (3.12) with respect to the Biedenharn basis. By choosing an SU(3)-representation C^^, which induces an SU(2)-representation by restriction, one induces a connection A on the vector V-bundle

VCk,l —4 G/K with VCk,l := G xk CkJ (B.10)

associated to the principal V-bundle (2.23). Then the connection A can be decomposed into morphism-valued 1-forms

A = 0 (B(n,m) - limn^(n,m) + 3~+n,m) + 3(—n,m) - 3+n,m) - 3(—n,m)) (B.11) (n,m)eQ0(k,l)

with respect to this basis. The computation of the vanishing curvature F0 = 0 yields relations between the different matrix elements given by

dB(n,m) + B(n,m) A B(n,m) - If dn ^-(n,m)

= 3+ —1,m—3) A 3+-1,m-3) + 3(n+1,m—3) A 3-+1,m—3)

+ 3+nm) A 3+nm) + 3-n,m) A 3-n,m) , (B.12a)

0 = d3±,m) + B(n+1,m+3) A 3%m) + ¡3(±,m) A B(n,m) - 3 n n(n±1,m+3) A 3%m),

(B.12b)

0 = 3~+i,m) A 3(n+1,m—3) + 3(n+2,m) A 3++1,m-3) , (B.12c)

0 = 3tn,m) A 3(—n,m) + 3 —+1,m+3) A 3+-1,m+3) , (B.12d)

0 = 3±n,m) A 3(«T1,m—3) , (B.12e)

plus their conjugate equations.

B.3. Quiver connections

One can also compute the matrix elements of the curvature (3.36c) in the Biedenharn basis. For this, the curvature F = dA + A a A is arranged into components

(F )(n,m),(n' ,m') e ^2(skJ , End(Ep (nm),Ep M) ® End( (n,m), (n',m') )) , (B.13)

which can be simplified by using the relations (B.12). We denote the curvature of the connection A(n,m) on the bundle (3.21) by

F(n,m) := dA(n,m) + A(n,m) A A(n,m) (B.14a)

and the bifundamental covariant derivatives of the Higgs fields as

D$fn,m) := d^tn,m) + A(n±1,m+3) 4±m) - tfn,m) A(n,m) , (B.14b)

Df(n,m) := df(n,m) + A(n,m) f(n,m) - t(n,m) A(n,m) • (B.14c)

Then the non-zero curvature components read as

(F)(n,m),(n,m) = F(n,m) ® ^(n,m) D0(n,m) A ~m~ n ^(n,m)

- 1P(n,m) ) ® Lf dn^(n,m)

- (f(n,

+ p(n,m

+ ("1p(n,m

+ ("1p(n,m

+ p(n,m

- 0+-1,m-3) (0+)(n-1,m-3^ ® P(n-1,m-3) A P+-1,m-3)

0(n+1,m-3) (0 )(n+1,m-3)) ® ^(n+1,m-3) A P(n+1,m-3) - ($+)\n,m) K,m)) ® £+,m) A fi~(n,m)

(0-)(n,m) 0(-,m)) ® P-n,m) A ' (B-15a)

(F)(n,m),(n±1,m+3) = D0±,m) A ¿*±,m) - ((m + 3)0(n±1,m+3) 0(n,m)

- m0(±,m) 0(n,m) - 30±,m)) ® 22 * H(n±1,m+3) A P±m) ' (B-15b) (F )(n+1 ,m-3),(n+1,m+3)

= (^(+,m) 0 (n+1,m—3) - 0-n+2,m) 0++1,m-3)) ® ^(n,m) A ^(n+1,m-3) ' (B.l5c)

(F ) (n-1,m+3),(n+1,m+3)

= -$+n,m) (0~)}n,m) - (0-)tn+1,m+3) 0+-1,m+3)) ® P+n,m) A P -m) ' (B.15d)

which are accompanied by the anti-Hermiticity conditions

(F)(n! ,m!),(n,m) = - (F ) ))T • (B.15e)

By setting f(nm) = 1P(nm) for all (n, m) e Qo(k, l), these curvature matrix elements correctly reproduce those computed in [25] for equivariant dimensional reduction over CP2.

Appendix C. Quiver bundle examples

C.1. C1,0-quiver

Consider the fundamental 3-dimensional representation C1,0 of G = SU(3). Its decomposition into irreducible SU(2) -representations is given by

C 1,0|SU(2) = (0, -2) ® (1,1), (C.1)

wherein (0, -2) is the SU(2)-singlet and (1,1) is the SU(2)-doublet. Using the general quiver construction of Section 3.3, the G-action dictates the existence of bundle morphisms

0 := 0+,-2) e Hom( EP(0,-2) ,EP(1,1)^ , 0( := №+) (0,-2) e HoM EP(1,1),EP(0,-2)),

(C.2a)

f0 := f(0,-2) e End(Ep^)) , 01 := 0(1,1) e End(Ep(hr)) • (C.2b)

C.2. C2,0-quiver

The 6-dimensional representation C2,0 of SU(3) splits under restriction to SU(2) as

c2,0|su(2) = (22 ® (1,-1) ® (0, -4) • (C.3)

The SU(3)-action intertwines the irreducible SU(2)-modules and the corresponding bundles. The actions of Ea1+a2 and Ea2 respectively yield Higgs fields

00 := 0+0,-4) e Hom(EP(0,-4),EP(1,-1^ , 01 := 0+,-1) e Hom(EP(1,-1),EP(2,2)) • (C.4a)

Due to the non-zero restrictions of Ha2 to its eigenspaces (0, -4), (1, -1) and (2,2), one further has three bundle endomorphisms

00 := 0(0,-4) e End(Ep(0,-4)), 01 := 0(1,-1) e En^Epa,-1^,

02 := 0(2,2) e End(EP(2,2) • (C.4b)

C.3. C1,1-quiver

The 8-dimensional adjoint representation of SU(3) splits under restriction to SU(2) as C 1,1|SU(2) = (h3) ® (00 ® (2,0) ® (1, -3)^ (C.5)

The action of SU(3) implies the existence of the following bundle morphisms: The actions of Ea1 +a2 and Ea2 translate into the Higgs fields

0+ := 0+1,-3) e Hom(EP(l,-з),EP(2o), 0— := 0--3) e Hom(EP(1,-3),EP(0,0)),

(C.6a)

0+ := 0+0,0) e HoMEP(0,0),EP(1,33)^ , 00- := 0-,0) e HoM EP(2,0),EP(1,3)), (C.6b)

whereas the action of Ha2 generates

0± := 0(1,±3) e End(Ep^^) • (C.6c)

Note that Ha2 neither introduces endomorphisms on (0,0) and (2,0) nor does it intertwine these SU(2)-multiplets. This follows from the fact that these representations are subspaces of the kernel of Ha2, and that Ha2 commutes with the entire Lie algebra su(2).

Appendix D. Equivariant dimensional reduction details

D.1. 1-form products on CP2

The metric on Md x CP2 is given as

ds2 = ds2Md + dsCP2 , (D.1)

ds2Md = G^' dx>"'® dxv' (D.2)

with (x^ ) local real coordinates on the manifold Md and ¡x', v',... = 1, • ••, d. The metric on CP2 is written as

gCP2 := dsCP2 = R2 (31 ® ~3X + 331 ® 31 + 32 ® 32 + 32 ® 32) • (D.3)

This metric is compatible with the Kahler form (A.2), and by defining the complex structure J via mcp2(■, ■) = gCP2 (■, J ■) on the cotangent bundle of CP2 one obtains J3' = i 3' and J3l =

-i 3' for i = 1, 2. The corresponding Hodge operator is denoted *CP2, with the non-vanishing 1-form products

*CP21 = R4 31 a 31 A 32 A 32 = 2(n R2)2 3vol, (D.4a)

31 A *CP2 31 = 32 A *CP2 32 = 31 A *CP2 31 = 32 A *CP2 32 = 2n2 R2 3vol , (D.4b) *CP2 31A 31 = 32 A 32, *CP2 32 A 32 = 31A 31, (D.4c)

*CP2 31A 32 = 31A 32, *CP2 32 A 31 = 32 A 31 • (D.4d)

For later use we shall also need to compute various products involving matrix-valued 1-forms. Firstly, we have15

3(±,m) A *cP23(±,m)

tr --75-= 2n2 R2 (n + 1 ± 1)3vol, (D.5a)

A(j(n,m)2

tr 3(n,m) A 3(n,m) A *cp2 (3(n,m) A 3(n,m)) = 2^2 (n + 1 ± 1 )3vol, (D.5b) A±l(n,m)4

t 3(:,m) A 3(«,m) A *cP2 (3(tm) A 3(lm))f 0 2 (n + 1 ( 1)2 a _ c ,

tr --—.- = 2n -—-Pvol , (D.5c)

A^^nmr n +1

t 3(0,m) A 3(n+1,m-3) A *cP2 (3+,m) A 3(n+1,m-3))f , 2 n + 1

= 2n —-— 3vol , (D.5d)

A+l(n,m)2 A- (n + 1,m- 3)2 3

t 3(0,m) A 3(n,m) A *cP2(3+,m) A 3(n,m))f , 2 n(n + 2)

tr-T+-"-,9 . _ ,--2-= 2n -—J— Pvol • (D.5e)

Aj°l(n,m)2 A-l(n,m)2 n + 1

The trace formulas (D.5) will have to be supplemented by

t 3+ , m) A 3+,m) A *cP2(3(n , m) A 3(n, m))f , 2 2n(n + 2)

tr-T+-"-,9 . _ ,--2-= 2n —TTT 3v°l , (D.6a)

A+l(n,m)2 Akl(n,m)2 3(n + 1)

t 3+-1,m-3) A 3+-1,m-3) A *cP2 (3(n+1,m-3) A 3(n+1,m-3))f , 2 2(n + 1)

tr -;----- = 2n - Pv°l ,

A+^n - 1,m- 3)2 Ak(l(n + 1,m- 3)2 3

(D.6b)

tr 3(tm) A 3(tm) A *cp2 (3(:t1,m-3) A 3(±т1,m-3))f = -2n2 n(n + 2) 3 (D 6c)

A(:J(n,m)2 A^n t 1,m — 3)2 n + 1 T 1 PУ°l , .

tr ^ A 3tn,m) A *cP^3t±1,m-3) A ^m-3)'

A(:ll(n,m)2 Aj^n ( 1,m - 3)2

(n + 1^3vol (D.6d)

2 / n(n + 2)

v 3(n +1: 1)

15 The expressions (D.5) correct the trace formulas from [25, eq. (B.7)].

and one additionally needs the traces

^tn,m) A P(n,m) _

Af,,(n,m)2

i (P(« m) A P(n m])

2R2 (n + 1 ± ^«cp2 = *cp2tr AtMm)2

(D.7a)

tr P(»T1,m-3) A P(nT1,m-3) i ( ,, , tr lP(«T1,m-3)

tr -(-—— = 2R2 (n + 1)mcp2 = *c,2 tr

± a ft± ^

-3) A P((

J(n^1,m-3)J

A((n T 1,m - 3)2

A((n t 1, m- 3)2

(D.7b)

D.2. 1-form products on S5

Let us write the metric (3.39) in the forms

d4 = gij P ® Pj + Pip ® Pip) + g55 n ® n = 2R2 Sab ea ® eb + r2 e5 ® e5 , (D.8)

for i, j = 1, 2 and a, b = 1, 2, 3, 4, where r is the radius of the S1-fibre of the Hopf bundle S5 ^ CP2; the corresponding Hodge operator is denoted *S5. Define the normalised volume form nvol on S5 as

*S51 = -(2k)1 r R4 nvol with Pvol A n = -4n nvol = —22

e12345 and

nvol = 1 •

In the computation of the reduced action (3.42) we use the identities

ej A *s5 ev = gjv e12345 =

4n3 r R2 nvol

(2n)3 R4 r

nvol ,

j = v = a ,

j = v = 5 j = v ,

ejv A *S5 epa =

^ggJP gva e12345 , j = p, v = a , -VggJagVPe12345 , j = a, v = p , 0 , otherwise ,

(D.10a)

(D.10b)

eab A *S5 eab = 2n3 rnvol and ea5 A *S5 ea5 = nvol.

(D.10c)

We can reduce the action of the Hodge operator in 5 dimensions to the action of *CP2 from Appendix D.1 to get

*s5pP = r (*cp2pp) A n, *s5pi = r (*c,2pi) A n, (D.11a)

*S5 (Pp A Pj) = r (*c,2Pp A Pj) A n , *S5 (Pp A j = r (*c,2Pp A j A n ,

*S5 (n A pp) = 1 *c,2 PP , *S5 (n A pp) = 1 *c,2 PP

*S5 n = ^ Pv0l

n A *S5 n = - ^j1 R4 nvol

(D.11b) (D.11c)

(D.11d)

We can additionally compute

dn = -2« = i (pi A Pi + Pi A Pl) = -R* , (D.12a)

*55dn = -R2 *S5 «CP2 = -R2 (*CP2«CP2) A n = R2 «CP2 A n , (D.12b)

dn A *S5dn = -2(2n)) r nvol , (D.12c)

wherein we used *CP2mcp2 = -<wCP2 and (A.5). Note that due to the structure of the extension from CP2 to S5, the matrices accompanying contributions from n or dn are always proportional to the identity operators N.(nm); thus their inclusion does not alter the trace formulas of Appendix D.1.

D.3. Yang-Mills action

The reduction of (3.40) proceeds by writing

tr F A *F =- V tr(F A *Ff), w (D.13)

Z_j V / (n,m),(n,m) v y

(n,m)eQo(k,l)

We insert the explicit non-vanishing components (B.15), rescale the horizontal Higgs fields

Km) l^^tm) (D.14)

as in [25] (but not the vertical Higgs fields 0(n,m)), and evaluate the traces over the representation spaces (n, m) for each weight (n, m) e Q0(k, l) using the matrix products from Appendix D.1 and the relations of Appendix D.2. Finally, one then integrates over S5 using the unit volume form nvol introduced in Appendix D.2. The dimensionally reduced Yang-Mills action on Md then reads as16

2n3 r R4

£=""_'" I ddx

2 I ddx Vo J2 tt((n + D {F(n,m)Y

Md (n,m)eQo(k,l)

n + 2 , , ) + ± n + 1 , , ,,/1

-¿T Dm'Km))' DM Km) + -J2T D '

n + 2 / + )t m' + n + 1 + / M' + )t

+ R2 $(n,m)) D $(n,m) + r2 DM'$(n-1,m-3) \D $(n-1,m-3))

n / - )t m' - n +1 - / m - )t

+ R2 \DM'&(n,m)) D $(n,m) + r2 DM'0(«+1,m-3) \D $(n+1,m-3))

+ nR2 (a+I(" ^1 t

(A+,l(n>m)2 1P(n,m) - (0+)(n,m)0+,m)) (A-l(n'm)2 1 P(n,m) - ODÎnm^-nm))

n +1 / + 2 + + t A2

+ \Ak,l (n - 1,m - 3) 1 P(n,m) - $(n-1,m-3) (r)(n-1,m-3))

(n + 1)2 / 2 _ t

(n + 2)R4 {Ak,l(n + 1'm- 3) 1P(n,m) - 0(„+1,m-3)(0 V

16 By setting = 1p(n m) f°r (n, m) e Qo(k, l) and r = in(D.15)we obtain the quiver gauge theory action

for equivariant dimensional reduction over the complex projective plane CP2; this reduction eliminates the last nine lines of (D.15) and the resulting expression corrects [25, eq. (3.5)].

2(n + 3) +

+ 3R4 ^(n^m) ф(n+1,m—3) A+(n, m) A—(n + l,m - 3) A+(n + l,ñ — 3)A—l(n + 2,m) V(n+2m) ф(n+1,m—3) 2n(n + 2)

ф(n+-,ñ) Ф+

(n + 1)R4

t^(n,m) (Ф )(n,m)

A~+l(n, ñ) A- l(n, ñ) t +

k,l k,l SI - W A\ +

A+(n - l,m + 3)A-l(n + l,m + 3) (Ф )(n+l,m+3)^n—l,m+3)

k,l(n - )Ak,l 4n(n + 2) /(+ 2 . t + \

+ 3(n + 1)R4 {iAk,l(n,ñ) 1 P(n,m) - (r)(n^(nm)) X (-A-l(n,ñ)2 1P(n,m) - (4>—\n,m)^n,m)))

2(n + 2) 2 . t + \

--R^ {iAk,l(n, ñ) 1 P(n,m) - (r )(n,m) {P(n,m))

X (A+l(n - l,ñ - 3)2 1p(n,m) - Ф+ —l,m—3) (Ф+)tn-l,m-3

+ ^ ( n - n - 0 ( A+l (n, ñ)2 1P(n,m) - (ф+i,m) Ф^))

- n - 111'

( 2 t X iAk,l(n + l,ñ- 3) 1P(n,m) - ф(n+1,m-3)(ф V+l,«

2 z n + 2 W(_ 2 _ t - \

+ --n - VI\Ak,l(n, ñ) 1 P(n,m) - (ф )(n,m)ф(nm))

X (A+l(n - l,ñ - 3)2 1p(n,m) - Ф+ñ-l,m-3) (Ф+)(n-l,m

2n /( - 2 -t - \

- Rïi(Ak,l(n,ñ) 1 P(n,m) - (ф )(n,m) (^(n,m))

( 2 t X (Ak,l(n + l,ñ- 3) 1 P(n,m) - ф(n+1,m-3)(Ф Wm

4(n + 1) 2 + . t ч

+ 3r4 {{Ak,l(n - l,ñ - 3) 1P(n,m) - ф(n-1,m-3) (r)(n-l,m-3))

( 2 t X (Ak,l(n + l,ñ - 3) 1P(n,m) - Ф(n+1,m—3) (Ф )(n+lñ

(n + l)ñ2 ( , ч t 2(n + l)ñ2 z \2

+ -4^2-Dß' t(n,m) D f(n,m)) + -R- [Ф(n,m) — 1p{nm )

ñ(n + 2) /( + 2 . t + ч( ч\ --{{Ak,l(n,ñ) 1P(n,ñ) — (ф^nm^M) №(n,m) — 1P(n,m)))

R/4 ч( mmn((A—,l(n, ñ)2 1 P(n,m) — W-)\n,m)^n,m)) ty(n,m) — 1P(n,m)))

ñ(n + 1 )

X ((A+l(n — l,ñ — 3)2 1P(n,m) — ф+ —l,m—3) (Ф+)(п-l,m—3)) (f(n,m) — 1P(n,m)))

ñ(n + 1 )

X ((A—l(n + l,ñ — 3)2 1P(n,m) — Ф——+l,m—3) (ф — )tn+l,m—3)) №(n,m) — 1P(n,m)))

4R2 r2

4R2 r2

4R2 r2

"0(n,m) <P+,-1,m-3) - (m - 3)0+-1,m-3) 0(n-1,m-3) - 30+n-1,m-3)

4R2 r 2

m0(n,m)0(n+1,m-3) - (m - 3)0(n+1,m-3) 0(n+1,m-3) - 30(„+1,m-3)

(n,m) - "lV(n,m) V(n,m) - 3V(n,m)

(" + 3)0(n+1,m+3) 0+™) - "0+,m) 0(n,m) - 30+

(" + 3)0(n-1,m+3) 0(n,m) - m$(n,m) XHn,m) - 30

(D.15)

Note that while the trace in (3.42) is taken over the full fibre space Vk,l of the equivariant vector bundle (3.22), in (D.15) the trace over the SU(2) x U(1)-representations (n, m) has already been evaluated.

References

[1] S. Kachru, E. Silverstein, 4D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855-4858, arXiv:hep-th/9802183.

[2] A.E. Lawrence, N.A. Nekrasov, C. Vafa, On conformal field theories in 4-dimensions, Nucl. Phys. B 533 (1998) 199-209, arXiv:hep-th/9803015.

[3] A. Kehagias, New type IIB vacua and their F-theory interpretation, Phys. Lett. B 435 (1998) 337-342, arXiv:hep-th/9805131.

[4] I.R. Klebanov, E. Witten, Superconformal field theory on 3-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199-218, arXiv:hep-th/9807080.

[5] D.R. Morrison, M.R. Plesser, Nonspherical horizons 1, Adv. Theor. Math. Phys. 3 (1999) 1-81, arXiv:hep-th/ 9810201.

[6] D. Martelli, J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Commun. Math. Phys. 262 (2006) 51-89, arXiv:hep-th/0411238.

[7] J. Kallen, M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, J. High Energy Phys. 1205 (2012) 125, arXiv:1202.1956.

[8] J. Qiu, M. Zabzine, 5D super Yang-Mills on Yp'q Sasaki-Einstein manifolds, Commun. Math. Phys. 333 (2015) 861-904, arXiv:1307.3149.

[9] J.P. Gauntlett, D. Martelli, J.F. Sparks, D. Waldram, Sasaki-Einstein metrics on S2 x S3 , Adv. Theor. Math. Phys. 8 (2004) 711-734, arXiv:hep-th/0403002.

[10] J.P. Gauntlett, D. Martelli, J.F. Sparks, D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys. 8 (2006) 987-1000, arXiv:hep-th/0403038.

[11] M. Cvetic, H. Lu, D.N. Page, C. Pope, New Einstein-Sasaki spaces in 5 and higher dimensions, Phys. Rev. Lett. 95 (2005) 071101, arXiv:hep-th/0504225.

[12] T.A. Ivanova, O. Lechtenfeld, A.D. Popov, R.J. Szabo, Orbifold instantons, moment maps and Yang-Mills theory with sources, Phys. Rev. D 88 (2013) 105026, arXiv:1310.3028.

[13] M. Sperling, Instantons on Calabi-Yau cones, arXiv:1505.01755.

[14] P. Candelas, G.T. Horowitz, A. Strominger, E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46-74.

[15] O. Lechtenfeld, A.D. Popov, R.J. Szabo, Sasakian quiver gauge theories and instantons on Calabi-Yau cones, arXiv:1412.4409.

[16] Y. Ito, M. Reid, The McKay correspondence for finite subgroups of SL(3, C), in: M. Andreatta, T. Peternell (Eds.), Higher Dimensional Complex Varieties, de Gruyter, 1996, pp. 220-240, arXiv:alg-geom/9411010.

[17] Y. Ito, H. Nakajima, McKay correspondence and Hilbert schemes in dimension 3, Topology 39 (2000) 1155-1191, arXiv:math/9803120.

[18] M.R. Douglas, B.R. Greene, D.R. Morrison, Orbifold resolution by D-branes, Nucl. Phys. B 506 (1997) 84-106, arXiv:hep-th/9704151.

[19] M.R. Douglas, B. Fiol, C. Romelsberger, The spectrum of BPS branes on a noncompact Calabi-Yau, J. High Energy Phys. 0509 (2005) 057, arXiv:hep-th/0003263.

[20] A.V. Sardo-Infirri, Partial resolutions of orbifold singularities via moduli spaces of HYM-type bundles, arXiv:alg-geom/9610004.

[21] M. Cirafici, A. Sinkovics, R.J. Szabo, Instantons, quivers and noncommutative Donaldson-Thomas theory, Nucl. Phys. B 853 (2011) 508-605, arXiv:1012.2725.

[22] T.A. Ivanova, A.D. Popov, Instantons on special holonomy manifolds, Phys. Rev. D 85 (2012) 105012, arXiv:1203.2657.

[23] O. Lechtenfeld, A.D. Popov, R.J. Szabo, SU(3)-equivariant quiver gauge theories and nonabelian vortices, J. High Energy Phys. 0808 (2008) 093, arXiv:0806.2791.

[24] C.P. Boyer, K. Galicki, Sasakian Geometry, Oxford University Press, Oxford, 2008.

[25] B.P. Dolan, R.J. Szabo, Dimensional reduction and vacuum structure of quiver gauge theory, J. High Energy Phys. 0908 (2009) 038, arXiv:0905.4899.

[26] D. Conti, S. Salamon, Generalized Killing spinors in dimension 5, Trans. Am. Math. Soc. 359 (2007) 5319-5343, arXiv:math/0508375.

[27] M. Fernandez, S. Ivanov, V. Munoz, L. Ugarte, Nearly hypo structures and compact nearly Kâhler 6-manifolds with conical singularities, J. Lond. Math. Soc. 78 (2008) 580-604, arXiv:math/0602160.

[28] L.C. Biedenharn, Invariant operators of the Casimir type, Phys. Lett. 3 (1962) 69-70.

[29] G.E. Baird, L.C. Biedenharn, On the representations of semisimple Lie groups. 2, J. Math. Phys. 4 (1963) 1449-1466.

[30] N. Mukunda, L.K. Pandit, Tensor methods and a unified representation theory of SU(3), J. Math. Phys. 6 (1965) 746-765.

[31] L. Alvarez-Consul, O. Garcia-Prada, Dimensional reduction and quiver bundles, J. Reine Angew. Math. 556 (2003) 1-46, arXiv:math/0112160.

[32] S. Bunk, O. Lechtenfeld, A.D. Popov, M. Sperling, Instantons on conical half-flat 6-manifolds, J. High Energy Phys. 1501 (2015) 030, arXiv:1409.0030.

[33] S.K. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1-26.

[34] K. Uhlenbeck, S.-T. Yau, On the existence of Hermitian Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) S257-S293.

[35] S.K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231-247.

[36] M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond. A 308 (1982) 523-615.

[37] A.V. Sardo-Infirri, Resolutions of orbifold singularities and flows on the McKay quiver, arXiv:alg-geom/9610005.

[38] M. Cirafici, A. Sinkovics, R.J. Szabo, Instanton counting and wall-crossing for orbifold quivers, Ann. Henri Poincaré 14 (2013) 1001-1041, arXiv:1108.3922.

[39] M. Cirafici, R.J. Szabo, Curve counting, instantons and McKay correspondences, J. Geom. Phys. 72 (2013) 54-109, arXiv:1209.1486.

[40] A. Deser, O. Lechtenfeld, A.D. Popov, Sigma-model limit of Yang-Mills instantons in higher dimensions, Nucl. Phys. B 894 (2015) 361-373, arXiv:1412.4258.

[41] R.P. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, Surv. Differ. Geom. 10 (2006) 221-273, arXiv:math/0512411.

[42] S.K. Donaldson, Nahm's equations and the classification of monopoles, Commun. Math. Phys. 96 (1984) 387-407.

[43] P.B. Kronheimer, A Hyper-Kâhlerian structure on coadjoint orbits of a semisimple complex group, J. Lond. Math. Soc. 42 (1990) 193-208.

[44] N.J. Hitchin, Hyper-Kâhler manifolds, Astérisque 206 (1992) 137-166.

[45] M.K. Murray, M.A. Singer, Gerbes, Clifford modules and the index theorem, Ann. Glob. Anal. Geom. 26 (2004) 355-367, arXiv:math/0302096.

[46] A.D. Popov, R.J. Szabo, Double quiver gauge theory and nearly Kâhler flux compactifications, J. High Energy Phys. 1202 (2012) 033, arXiv:1009.3208.