Orbifold instantons, moment maps and Yang-Mills theory with sources
Tatiana A. Ivanova, Olaf Lechtenfeld, Alexander D. Popov, Richard J. Szabo
aa r X i v : . [ h e p - t h ] O c t EMPG–13–15ITP–UH–18/13
Orbifold instantons, moment mapsand Yang-Mills theory with sources
Tatiana A. Ivanova , Olaf Lechtenfeld , Alexander D. Popov and Richard J. Szabo Bogoliubov Laboratory of Theoretical Physics, JINR141980 Dubna, Moscow Region, Russia
Email: [email protected] Institut f¨ur Theoretische Physik and
Riemann Center for Geometry and PhysicsLeibniz Universit¨at HannoverAppelstraße 2, 30167 Hannover, Germany
Email: [email protected], [email protected] Department of Mathematics, Heriot-Watt UniversityColin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K. and
Maxwell Institute for Mathematical Sciences, Edinburgh, U.K. and
The Tait Institute, Edinburgh, U.K.
Email:
Abstract
We revisit the problem of constructing instantons on ADE orbifolds R / Γ and point out somesubtle relations with the complex structure on the orbifold. We consider generalized instantonequations on R / Γ which are BPS equations for the Yang-Mills equations with an externalcurrent. The relation between level sets of the moment maps in the hyper-K¨ahler quotientconstruction of the instanton moduli space and sources in the Yang-Mills equations is discussed.We describe two types of spherically-symmetric Γ-equivariant connections on complex V-bundlesover R / Γ which are tailored to the way in which the orbifold group acts on the fibres. Someexplicit abelian and nonabelian SU(2)-invariant solutions to the instanton equations on theorbifold are worked out.
Introduction and summary
Instantons in Yang-Mills theory [1] and gravity [2, 3] play an important role in modern field the-ory [4, 5, 6]. They are nonperturbative configurations which solve first order (anti-)self-dualityequations for the gauge field and the Riemann curvature tensor, respectively. The constructionof gauge instantons can be described systematically in the framework of twistor theory [7, 8] andby the ADHM construction [9]. There are also many methods for constructing gravitational in-stantons including twistor theory [8] and the hyper-K¨ahler quotient construction [10] based on thehyper-K¨ahler moment map introduced in [11].In this paper we revisit the problem of constructing instantons on the ADE orbifolds R / Γ. Thecorresponding instanton moduli spaces are of special interest in type II string theory, where theycan be realized as Higgs branches of certain quiver gauge theories which appear as worldvolumefield theories on D p -branes in a D p -D( p +4) system with the D( p +4)-branes located at the fixedpoint of the orbifold [12]. The ADHM equations can be identified with the vacuum equationsof the supersymmetric gauge theory, and the structure of the vacuum moduli space provides animportant example of resolution of spacetime singularities by stringy effects in the form of D-braneprobes. We point out in particular some salient relations between the construction of instantonsand complex structures on R / Γ.Kronheimer [10] considers Γ-equivariant solutions of the matrix equations[ W , W ] + [ W , W ] = Ξ , [ W , W ] + [ W , W ] = Ξ , [ W , W ] + [ W , W ] = Ξ , (1.1)where Γ is a finite subgroup of the Lie group SU(2) acting on the fundamental representation C ∼ = R , W µ with µ = 1 , , , the Lie algebra u ( N ), and Ξ a with a = 1 , , h of a subalgebra g of u ( N ). For Ξ a = 0 the equations(1.1) are the anti-self-dual Yang-Mills equations on the orbifold C / Γ reduced by translations.Their solutions satisfy the full Yang-Mills equations. In the general case, the equations (1.1) areinterpreted as hyper-K¨ahler moment map quotient equations, and Hitchin shows [13] that one cansimilarly interpret the Bogomolny monopole equations and vortex equations. Kronheimer showsthat the moduli space of solutions to (1.1) in the Coulomb branch is a hyper-K¨ahler ALE space M ξ , which is the minimal resolution M ξ −→ M (1.2)of the orbifold M = C / Γ. Here ξ are parameters in the matrices Ξ a of (1.1). Similar resultswere obtained in [14, 15] for SU(2)-invariant Yang-Mills instantons on R (see also [16]). Moreover,it was shown by Kronheimer and Nakajima [17] that there exists a bundle E → M ξ with Chernclasses c ( E ) = 0 and c ( E ) = ( − / E satisfying the anti-self-dual Yang-Mills equations coincides with the base manifold M ξ itself. In thelimit ξ = 0 one obtains C / Γ as the moduli space of minimal fractional instantons on the V-bundle E over the orbifold M = C / Γ.In this paper we consider gauge instanton equations with matrices Ξ a on the orbifold C / Γ ∼ = R / Γ and show that the choices of Ξ a = 0 correspond to sources in the Yang-Mills equations. Forgauge potentials on R / Γ with Γ = Z k +1 we analyse solutions of Γ-equivariance conditions in twodifferent SU(2)-invariant bases adapted to the spherical symmetry. Recall that one can write a In type II string theory in the presence of orientifold O( p +4) planes one should use instead the Lie algebras oforthogonal or symplectic Lie groups. so (4) ∼ = su (2) ⊕ su (2) in terms of vector fields on R / Γ as E a = − η aµν y µ ∂∂y ν and ˜ E a = − ¯ η aµν y µ ∂∂y ν , (1.3)where η aµν and ¯ η aµν are components of the self-dual and anti-self-dual ’t Hooft tensors [18] and y µ are local coordinates on R / Γ. The commutation relations between these vector fields are[ E a , E b ] = 2 ε abc E c , [ ˜ E a , ˜ E b ] = 2 ε abc ˜ E c and [ E a , ˜ E b ] = 0 . (1.4)Introducing complex coordinates z = y + i y and z = y + i y on R / Γ ∼ = C / Γ, one findsthat the vector fields ˜ E a preserve this complex structure but the vector fields E a do not, i.e. thegroup SU(2) acting on C is generated by ( ˜ E a ). Furthermore, the actions of the corresponding Liederivatives are given by L ˜ E b e a = 0 and L ˜ E b ˜ e a = 2 ε abc ˜ e c , (1.5)where e a = e aµ d y µ and ˜ e a = ˜ e aµ d y µ are one-forms dual to the vector fields E a and ˜ E a , respectively.We show that both bases of one-forms ( e a , d r ) and (˜ e a , d r ) with r = δ µν y µ y ν can be used fordescribing spherically-symmetric instanton configurations, but due to (1.5) the basis ( e a , d r ) ismore suitable for connections on V-bundles E with trivial action of the finite group Γ ⊂ SU(2),while the basis (˜ e a , d r ) is more suitable for connections on V-bundles E with non-trivial Γ-actionon the fibres of E . Explicit examples of abelian and nonabelian SU(2)-invariant instanton solutionson R / Z k +1 are worked out below.The structure of the remainder of this paper is as follows. In Section 2 we consider generalizedinstanton equations on R which reduce to (1.1) and show that they correspond to BPS-typeequations for Yang-Mills theory with sources. In Section 3 we extend these equations to the ADEquotient singularities R / Γ, focusing on the special case Γ = Z k +1 . In Section 4 we study the modulispaces of translationally-invariant instantons on R / Γ via the hyper-K¨ahler quotient construction.In Section 5 we consider the construction of spherically-symmetric instanton solutions on R / Γ andmake some preliminary comments concerning the structure of the instanton moduli spaces, thougha detailed description of these moduli spaces is beyond the scope of the present work. R Euclidean space R . Consider the two-forms ω a := η aµν d y µ ∧ d y ν , (2.1)where y µ are coordinates on R and ω aµν := η aµν are components of the ’t Hooft tensors given bythe formulas η abc = ε abc and η ab = − η a b = δ ab . (2.2)Here ε = 1, µ, ν, . . . = 1 , , , a, b, . . . = 1 , ,
3. The forms ω a are symplectic and self-dual,d ω a = 0 and ∗ ω a = ω a , (2.3)where ∗ is the Hodge duality operator for the flat metric g = δ µν d y µ ⊗ d y ν (2.4)on R .Using the metric (2.4) we introduce three complex structures J a = ω a ◦ g − on R with com-ponents ( J a ) µν = ω aνλ δ λµ , (2.5)2o that ( R , J a ) ∼ = C J a . The space R is hyper-K¨ahler, i.e. it is K¨ahler with respect to each of thecomplex structures (2.5). We choose one of them, J =: J , to identify R and C ∼ = ( R , J ). Withrespect to J the complex two-form ω C = ω + i ω (2.6)is closed and holomorphic, i.e. ω C is a (2 , Instanton equations.
Let E be a rank N complex vector bundle over R ∼ = C . We endowthis bundle with a connection A = A µ d y µ of curvature F = d A + A ∧ A = F µν d y µ ∧ d y ν takingvalues in the Lie algebra u ( N ). Let us constrain the curvature F by the equations ∗F + F = 2 ω a Ξ a , (2.7)where the functions Ξ a belong to u ( N ). Solutions to this equation of finite topological charge arecalled (generalized) instantons. If Ξ a belong to the center u (1) of u ( N ) and dΞ a = 0, then solutionsto the equations (2.7) satisfy the Yang-Mills equations on R . If Ξ a do not belong to this center, then (2.7) are BPS-type equations for Yang-Mills theory with sources which vanish only if Ξ a areconstant and Ξ a ∈ u (1) ⊂ u ( N ) for a = 1 , ,
3. Indeed, from (2.7) we getd ∗ F + A ∧ ∗F − ∗F ∧ A = 2 ω a ∧ (cid:0) dΞ a + [ A , Ξ a ] (cid:1) , (2.8)which after taking the Hodge dual can be rewritten as ∂ µ F µν + [ A µ , F µν ] = 4 ω aµν (cid:0) ∂ µ Ξ a + [ A µ , Ξ a ] (cid:1) . (2.9)The current j µ := 4 ω aνµ D ν Ξ a with D µ Ξ a := ∂ µ Ξ a + [ A µ , Ξ a ] (2.10)satisfies the covariant continuity equation D µ j µ = 0 , (2.11)as required for minimal coupling of an external current in the Yang-Mills equations. Variational equations.
To formulate the generalized instanton equations (2.7) as absoluteminima of Euler-Lagrange equations derived from an action priniciple, we note that the presenceof the current (2.10) in the Yang-Mills equations (2.9) requires the addition of the term tr j µ A µ (2.12)in the standard Yang-Mills lagrangian L YM = − tr F µν F µν . (2.13)Up to a total derivative the term (2.12) is equivalent to the term ω aµν tr F µν Ξ a . (2.14)After adding the term (2.14), together with the non-dynamical term − a Ξ a (2.15)and the topological density − ε µνλσ tr F µν F λσ , (2.16) Later on we will consider an important example of such non-central elements Ξ a .
3e obtain the lagrangian L = − tr (cid:0) F + µν − ω aµν Ξ a (cid:1) (cid:0) F + µν − ω aµν Ξ a (cid:1) , (2.17)where F + = ( ∗F + F ) (2.18)is the self-dual part of the curvature two-form F . In the following we will consider constant matricesΞ a for which (2.15) becomes constant and the term (2.14) is topological. Constant matrices of theform Ξ a = i ξ a N correspond to D3-branes in a non-zero B -field in string theory and can bedescribed in terms of a noncommutative deformation of Yang-Mills theory on the space R (seee.g. [19, 20]). R / Γ Orbifold R / Γ . The complex structure J = J , introduced in (2.5), defines the complexcoordinates z = y + i y and z = y + i y (3.1)on R ∼ = C , where y µ are real coordinates. The Lie group SU(2) naturally acts on the vectorspace C with the coordinates (3.1). We are interested in the Kleinian orbifolds C / Γ where Γis a finite subgroup of SU(2). They have an ADE classification in which Γ is associated with theextended Dynkin diagram of a simply-laced simple Lie algebra. For the A k -type simple singularities,corresponding to the cyclic group Γ = Z k +1 of order k +1, explicit descriptions of instantons willbe readily available. However, most of our results can be generalized to the other ADE groups Γcorresponding to nonabelian orbifolds C / Γ.The action of Γ = Z k +1 on C is given by( z , z ) (cid:0) ζ z , ζ − z (cid:1) , (3.2)where ζ = exp (cid:0) π i k +1 (cid:1) with ζ k +1 = 1 (3.3)is a primitive ( k +1)-th root of unity. This action has a single isolated fixed point at the origin( z , z ) = (0 , C / Γ is defined as the set of equivalence classes on C with respectto the equivalence relation (cid:0) ζ z , ζ − z (cid:1) ≡ ( z , z ) , (3.4)and it has a singularity at the origin. The metric on C / Γ is g = d z ⊗ d¯ z ¯1 + d z ⊗ d¯ z ¯2 , (3.5)where the coordinates ¯ z ¯1 , ¯ z ¯2 are complex conjugated to z , z . V-bundles on C / Γ . A V-bundle on C / Γ is a Γ-equivariant bundle over C , i.e. a vector bundleon C with a Γ-action on the fibres which is compatible with the action of Γ on C . The orbifoldgroup Γ = Z k +1 has k +1 one-dimensional irreducible representations such that the generator of Z k +1 acts on the ℓ -th Γ-module as multiplication by ζ ℓ for ℓ = 0 , , . . . , k . Let us denote by E ℓ complex V-bundles over C / Γ of rank N ℓ on which Γ acts in the ℓ -th irreducible representation as v ℓ ζ ℓ v ℓ for v ℓ ∈ C N ℓ (3.6)4n a generic fibre C N ℓ of E ℓ . Then any complex V-bundle E over C / Γ of rank N can be decomposedinto isotopical components as a Whitney sum E = k M ℓ =0 E ℓ , (3.7)and its structure group is of the form k Y ℓ =0 U( N ℓ ) with k X ℓ =0 N ℓ = N . (3.8)From (3.6) it follows that the action of the point group on the V-bundle (3.7) is given by the unitarymatrices v γ Γ ( v ) with γ Γ = k M ℓ =0 ζ ℓ N ℓ (3.9)on vectors v = ( v ℓ ) kℓ =0 in the generic fibre C N = L kℓ =0 C N ℓ of E .Simplifying the situation discussed in the previous section, we choose matrices Ξ a in the formΞ a = k M ℓ =0 i ξ ℓa N ℓ , (3.10)where ξ ℓa ∈ R are constants. The matrices (3.10) belong to the center of the Lie algebra of thegauge group (3.8). The diagonal U(1) subgroup of scalars in (3.8) acts trivially on ( E , A ), so wecan factor the gauge group (3.8) by this U(1) subgroup to get the quotient group G := (cid:16) k Y ℓ =0 U( N ℓ ) (cid:17) (cid:14) U(1) . (3.11)Then the Lie algebra g of G is the traceless part of the Lie algebra of (3.8), and one should imposeon ξ ℓa in (3.10) the tracelessness condition k X ℓ =0 ξ ℓa N ℓ = 0 , (3.12)which defines the center h of g .Γ -equivariant connections. Consider a one-form W = W µ d y µ = W z d z + W z d z + W ¯ z ¯1 d¯ z ¯1 + W ¯ z ¯2 d¯ z ¯2 (3.13)on R ∼ = C which is invariant under the action of Γ ⊂ SU(2) ⊂ SO(4) defined by (3.2). Then onthe components W z = ( W − i W ) and W z = ( W − i W ) (3.14)the action of Γ is given by W z ζ − W z and W z ζ W z . (3.15)The action of Γ on the components A µ of any unitary connection A = A µ d y µ on a hermitianV-bundle (3.7) is given by a combination of the spacetime action (3.15) and the adjoint actiongenerated by (3.9) as A z ζ − γ Γ A z γ − Γ and A z ζ γ Γ A z γ − Γ . (3.16)5he corresponding Γ-equivariance conditions require that the connection defines a covariant repre-sentation of the orbifold group, in the sense that γ Γ A z γ − Γ = ζ A z and γ Γ A z γ − Γ = ζ − A z . (3.17)It is easy to see that the solutions to the constraint equations (3.17) are given by block off-diagonalmatrices A z = · · · ψ k +1 ψ ψ . . . ... 0... . . . . . . 0 00 · · · ψ k and A z = φ · · ·
00 0 φ . . . ...... ... . . . . . . 00 0 · · · φ k φ k +1 · · · (3.18)together with A ¯ z ¯1 = −A † z and A ¯ z ¯2 = −A † z . Here the bundle morphisms ψ ℓ +1 : E ℓ → E ℓ +1 and φ ℓ +1 : E ℓ +1 → E ℓ are bifundamental scalar fields given fibrewise by matrices ψ ℓ +1 ∈ Hom (cid:0) C N ℓ , C N ℓ +1 (cid:1) and φ ℓ +1 ∈ Hom (cid:0) C N ℓ +1 , C N ℓ (cid:1) (3.19)for ℓ = 0 , , . . . , k (with indices read modulo k +1). Substitution of (3.18) in (2.7) then yields thegeneralized instanton equations on the orbifold C / Γ. The transformations (3.15) are defined forthe holonomic basis d y µ of one-forms on R / Γ and can differ for other bases of one-forms, leadingto modifications of the formulas (3.16)–(3.18).
Matrix equations.
Consider translationally-invariant connections A on the V-bundle (3.7)over C / Γ satisfying the equations (2.7) with Ξ a given in (3.10), i.e. we assume that A µ areindependent of the coordinates y µ , which reduces (2.7) to the matrix equations (1.1) with W µ := A µ .Denoting B := A z and B := A z for A given by (3.18) with constant matrices ψ ℓ +1 and φ ℓ +1 for ℓ = 0 , , . . . , k , we obtain the equations η aµν [ W µ , W ν ] = Ξ a , (4.1)which can be rewritten as [ B , B ] = − i4 (Ξ − i Ξ ) =: Ξ C , (4.2) (cid:2) B , B † (cid:3) + (cid:2) B , B † (cid:3) = − i2 Ξ =: Ξ R . (4.3)Solutions to these equations satisfy the reduced Yang-Mills equations (2.9) with the external source j µ = − η aµν [ W ν , Ξ a ] , (4.4)where W µ is given by (3.18) and Ξ a by (3.10). Hyper-K¨ahler quotients.
The reduced equations (4.1) (and also the instanton equations (2.7))can be interpreted as hyper-K¨ahler moment map equations. For this, recall that if (
M, g, ω a ) is ahyper-K¨ahler manifold with an action of a Lie group G which preserves the metric g and the threeK¨ahler forms ω a , then one can define three moment maps µ a : M −→ g ∗ (4.5) They are K¨ahler with respect to the three complex structures J a = ω a ◦ g − . With respect to the complexstructure J , the two-form ω R = ω is K¨ahler and ω C = ω + i ω is holomorphic. g ∗ of the Lie algebra g of G such that, for each ξ ∈ g with triholomorphicKilling vector field L ξ generated by the G -action on M , the functions (4.5) satisfy the equations h d µ a , ξ i = L ξ y ω a , (4.6)where h− , −i is the dual pairing between elements of g ∗ and g , and y denotes contraction of vectorfields and differential forms. Denoting by µ = ( µ , µ , µ ) the vector-valued moment map µ : M −→ R ⊗ g ∗ , (4.7)we can consider the G -invariant level set µ − (Ξ) (4.8)which defines a submanifold of the manifold M , where Ξ = (Ξ , Ξ , Ξ ) ∈ R ⊗ h ∗ and h is thecenter of g . Then one can define the hyper-K¨ahler quotient as (see e.g. [10, 11, 13]) M ξ = µ − (Ξ) (cid:14)(cid:14)(cid:14) G , (4.9)where ξ = ( ξ ℓa ) are parameters defining Ξ = (Ξ a ) ∈ R ⊗ h ∗ . The hyper-K¨ahler metric on M descends to a hyper-K¨ahler metric on the quotient M ξ . When the group action is free, the reducedspace M ξ is a hyper-K¨ahler manifold of dimension dim M ξ = dim M − G .In the case of the matrix model (4.1), the manifold M is the flat hyper-K¨ahler manifold M = R ⊗ u ( N ) , (4.10)the group G is given in (3.11) and the three moment maps are µ a ( W ) = η aµν [ W µ , W ν ] ∈ u ( N ) . (4.11)Solutions of the equations (4.2)–(4.3) form a submanifold µ − (Ξ) of the manifold (4.10), and byfactoring with the gauge group (3.11) (which for generic parameters ξ = ( ξ ℓa ) acts freely on thesolutions) we obtain the moduli space (4.9). This moduli space was studied by Kronheimer [10],who showed that for Γ = Z k +1 and the Coulomb branch N = N = · · · = N k = 1 it is a smoothfour-dimensional asymptotically locally euclidean (ALE) hyper-K¨ahler manifold M ξ with metricdefined by the parameters ξ = ( ξ ℓa ). The ALE condition means that at asymptotic infinity of M ξ the metric approximates the euclidean metric on the orbifold C / Γ. Kronheimer also shows that M ξ is diffeomorphic to the minimal smooth resolution of the Kleinian singularity M = C / Γ,regarded as the affine algebraic variety x k +1 + y + z = 0 in C . For the Hilbert-Chow map π : M ξ −→ M (4.12)the exceptional divisor of the blow-up is the set π − (0) = k [ ℓ =0 Σ ℓ , (4.13)where Σ ℓ ∼ = C P and k = − The parameters ξ determine the periods of the three symplecticforms ω a as Z Σ ℓ ω a = ξ ℓa . (4.14) We identify u ( N ) ∗ and u ( N ). Recall that we consider Γ = Z k +1 for definiteness here, but many of these considerations generalize to the otherKleinian groups Γ ⊂ SU(2). In the general case, N ℓ are the dimensions of the irreducible representations of the finitegroup Γ in Kronheimer’s construction.
7n the general case N ℓ ≥
0, one can also define a map M ξ → M which is a resolution of singulari-ties [21]. Hermitian Yang-Mills connections.
The matrix Ξ C in (4.2) parametrizes deformations of thecomplex structure on the V-bundle E and it can be reabsorbed through a non-analytic change ofcoordinates on the space (4.10) [10, 22]. Therefore we may take Ξ C = 0 without loss of generality;in this case the ALE space M ξ is biholomorphic to the minimal resolution. In fact, the modulispaces M ξ and M ξ ′ are diffeomorphic for distinct ξ and ξ ′ such that Ξ R = 0 for both sets ofparameters. For Ξ C = 0 we have Ξ = Ξ = 0 and the equations (2.7) become the hermitianYang-Mills equations [23, 24] ∗F + F = ω Ξ . (4.15)A connection A on E satisfying (4.15) is said to be a hermitian Yang-Mills connection. It defines aholomorphic structure on E since from (4.15) it follows that the curvature F is of type (1 ,
1) withrespect to the complex structure J , i.e. F , = 0 = F , , (4.16)and the third equation from (4.15), ω µν F µν = Ξ , (4.17)means that for Ξ = i ξ N the V-bundle E is (semi-)stable [23, 24]. In the special case Ξ = 0 weget the standard anti-self-dual Yang-Mills equations ∗F = −F . (4.18) Translationally-equivariant instantons.
Instead of constant matrices A µ which reduce (2.7)to the matrix equations (4.1), one can also consider the gauge potential A = ω aµν Ξ a y µ d y ν , (4.19)where the commuting matrices Ξ a are given in (3.10). The connection (4.19) is translationally-invariant up to a gauge transformation and can be extended to the orbifold T / Γ, where T is afour-dimensional torus. The curvature of A is F = d A = ω aµν Ξ a d y µ ∧ d y ν , (4.20)providing in essence the three symplectic structures ω a from (2.1). Cone C ( S / Γ) . The euclidean space R can be regarded as a cone over the three-sphere S , R \ { } = C ( S ) (5.1)with the metric g = δ µν d y µ ⊗ d y ν = d r + r δ ab e a ⊗ e b , (5.2)where r = δ µν y µ y ν and ( e a ) give a basis of left SU(2)-invariant one-forms on S . One can define e a by the formula e a := − r η aµν y µ d y ν , (5.3)8here the ’t Hooft tensors η aµν are defined in (2.2). The one-forms e a are dual to the vector fields E a from (1.3). By using the identities ε abc η bµν η cλσ = δ µλ η aνσ − δ µσ η aνλ − δ νλ η aµσ + δ νσ η aµλ , (5.4) δ ab η aµν η bλσ = δ µλ δ νσ − δ µσ δ νλ + ε µνλσ , (5.5)one can easily verify the Maurer-Cartan equationsd e a + ε abc e b ∧ e c = 0 (5.6)and ω a = η aµν d y µ ∧ d y ν = η aµν ˆ e µ ∧ ˆ e ν , (5.7)where ˆ e a := r e a and ˆ e := d r . (5.8)The relation (5.2) between the metric in cartesian and spherical coordinates can be readily checkedas well.All formulas (5.2)–(5.8) are also valid for the orbifold C / Γ after imposing the equivalencerelation (3.4), and the orbifold is a cone over the lens space S / Γ, (cid:0) C \ { } (cid:1) (cid:14) Γ = C (cid:0) S / Γ (cid:1) , (5.9)with the metric (5.2). The one-forms (5.3) in the complex coordinates (3.1) have the form e + i e = i r (cid:0) z d z − z d z (cid:1) and e + i e = i r (cid:0) ¯ z ¯1 d z + ¯ z ¯2 d z (cid:1) , (5.10)plus their complex conjugated expressions. Hence (5.10) defines two complex one-forms which are(1 , J = J defined in (2.5). The symplectic two-forms (5.7) and the complex structures (2.5) have the same components in the holonomic (d y µ , ∂∂y µ )and non-holonomic (ˆ e a , ˆ E a ) bases, where ˆ E a y ˆ e b = δ ba . From (1.5), (3.2) and (5.10) it follows that e a and e := d rr = d τ with τ = log r (5.11)are invariant under the action of the finite group Γ ⊂ SU(2).
Nahm equations.
Consider the complex V-bundle E over C / Γ described in Section 3. Let A = ˆ X µ ˆ e µ = (cid:0) ˆ X − i ˆ X (cid:1) (cid:0) ˆ e + i ˆ e (cid:1) + (cid:0) ˆ X − i ˆ X (cid:1) (cid:0) ˆ e + i ˆ e (cid:1) + h.c. (5.12)be a connection on E written in the basis (5.8). The corresponding Γ-equivariance conditions are γ Γ (cid:0) ˆ X − i ˆ X (cid:1) γ − Γ = ˆ X − i ˆ X and γ Γ (cid:0) ˆ X − i ˆ X (cid:1) γ − Γ = ˆ X − i ˆ X . (5.13)Solutions to these equations are given by (cid:0) ˆ X − i ˆ X (cid:1) = diag( χ , χ , . . . , χ k ) and (cid:0) ˆ X − i ˆ X (cid:1) = diag( ϕ , ϕ , . . . , ϕ k ) , (cid:0) ˆ X + i ˆ X (cid:1) = − diag (cid:0) χ † , χ † , . . . , χ † k (cid:1) and (cid:0) ˆ X + i ˆ X (cid:1) = − diag (cid:0) ϕ † , ϕ † , . . . , ϕ † k (cid:1) , (5.14) The orbifolds S / Γ for arbitrary ADE point groups Γ exhaust the possible Sasaki-Einstein manifolds in threedimensions. χ ℓ and ϕ ℓ are N ℓ × N ℓ complex matrices. Thus the Γ-equivariance conditions in the basis(5.8) forces the block-diagonal form (5.14) of the connection components ˆ X µ , i.e. the connection A is reducible or else N ℓ = 0 for ℓ = 0 if Γ acts trivially on E .The instanton equations (2.7) are conformally invariant and it is more convenient to considerthem on the cylinder R × S / Γ (5.15)with the metric g cyl = d τ + δ ab e a ⊗ e b = d r r + δ ab e a ⊗ e b = 1 r g . (5.16)In the basis ( e µ ) = ( e a , d τ ) the SU(2)-invariant (spherically-symmetric) connection A and itscurvature F have components depending only on r = e τ and are given by A = X µ e µ with X µ = r ˆ X µ , (5.17) F a = d X a d τ + [ X τ , X a ] and F ab = − ε abc X c + [ X a , X b ] , (5.18)and (2.7) reduce to a form of the generalized Nahm equations given byd X a d τ = − [ X , X a ] − X a + 12 ε abc [ X b , X c ] − Ξ a . (5.19)Introducing Y µ := e τ X µ and s = e − τ = r , (5.20)we obtain the equations 2 d Y a d s = [ Y , Y a ] − ε abc [ Y b , Y c ] + 1 s Ξ a . (5.21)For Ξ a = 0 these equations coincide with the Nahm equations [25]. Choosing Ξ a = 0 and defining α := ( Y + i Y ) and β := ( Y + i Y ) , (5.22)we obtain the equations dd s (cid:0) α + α † (cid:1) + (cid:2) α, α † (cid:3) + (cid:2) β, β † (cid:3) = 0 , (5.23)d β d s + [ α, β ] = 0 (5.24)considered by Kronheimer [14, 15] (see also [16]) in the description of SU(2)-invariant instantons.The equations (5.21) have three obvious solutions which we now consider in turn. Abelian instantons with Ξ a = 0 . For the first solution, we choose Y a = − s Ξ a = − r Ξ a and Y = 0 . (5.25)Then we get the solution A = − r e a Ξ a and F = η aµν Ξ a ˆ e µ ∧ ˆ e ν (5.26)of the equations (2.7), which coincide with (4.19) and (4.20). This configuration can also beregarded as a translationally-equivariant solution of the self-dual Yang-Mills equations ∗F = F , (5.27)10.e. as an anti-instanton on R / Γ or T / Γ. Abelian instantons with poles.
For the second solution, considered in [15], we put Ξ a = 0and d Y a d s = 0. Then the constant matrices Y µ satisfy the reduced anti-self-dual Yang-Mills equations[ Y a , Y ] + ε abc [ Y b , Y c ] = 0 (5.28)considered in [10] and discussed in Section 4. Solutions to (5.28) are necessarily given by commutingmatrices [10], and one can choose them in the form Y a = 2Λ ˆΞ a and Y = 0 , (5.29)where ˆΞ a have the form (3.10) and Λ is a scale parameter. For the corresponding gauge potentialand its field strength, we obtain A = X a e a = 2Λ r ˆΞ a ˆ e a and F = − r ¯ η aµν ˆΞ a ˆ e µ ∧ ˆ e ν , (5.30)where ˆ e µ are given in (5.8) and ¯ η aµν are the anti-self-dual ’t Hooft tensors defined by¯ η abc = ε abc and ¯ η ab = − ¯ η a b = − δ ab . (5.31)Thus we obtain singular abelian solutions with delta-function sources in the Maxwell equations, asdiscussed by [15]. The gauge potential A from (5.30) can be regarded as an asymptotic approxi-mation of a smooth solution. Note also that¯ ω a := − r ¯ η aµν ˆ e µ ∧ ˆ e ν (5.32)can be viewed as three additional anti-self-dual symplectic forms on the cone ( R \ { } ) / Γ = C ( S / Γ), complimentary to those given in (2.1). ’t Hooft instantons on C / Γ . For the third solution we choose Y = Y τ = 0 = Ξ a to get Y a = 2 s + Λ − I a = 2Λ r r + Λ I a with Λ ∈ R and [ I a , I b ] = ε cab I c . (5.33)Then for the anti-self-dual connection and curvature we obtain A = 2Λ r + Λ e a I a and F = − (cid:0) r + Λ (cid:1) ¯ η aµν I a ˆ e µ ∧ ˆ e ν , (5.34)where we used the relation s = r − . Here I a are the generators of the group SU(2) embedded in thebroken gauge group (3.11), i.e. there are k +1 instanton solutions with gauge group SU(2) ⊂ U( N ℓ )if N ℓ ≥ ℓ = 0 , , . . . , k . From the explicit form of e a in (5.3) it follows that each of thesesolutions is the standard ’t Hooft instanton generalized from R to R / Γ. For framed instantons there are four moduli: the scale parameter Λ and three global SU(2) rotational parameters (seee.g. [22]). Moduli spaces of SU(2)-invariant instantons.
In the special case where Γ is the trivial group,we obtain SU(2)-invariant solutions of the anti-self-dual Yang-Mills equations (4.18) on R \ { } = C ( S ). The moduli spaces of these framed instantons (subject to appropriate boundary conditions) Framed instantons are instanton solutions modulo SU(2)-invariant gauge transformations which approach theidentity at asymptotic infinity. M ξ resolving M = C / Γ ′ as in (4.12), where Γ ′ is a finite subgroup of the group SU(2) related to boundary conditions for the solutions [14]–[16].This is the moduli space of the spherically-symmetric instanton which has the minimal topologicalcharge c ( E ) = ( ′ − / ′ . In our reducible case we obtain a product of hyper-K¨ahler modulispaces M ξ × M ξ × · · · × M ξ k . (5.35)Note that M ξ ℓ is a point if N ℓ = 1. For N ℓ = 1 one can also use the singular abelian solution from(5.30), F ℓ = − r ¯ η aµν ξ ℓa ˆ e µ ∧ ˆ e ν , (5.36)with ξ ℓa ∈ R .We have seen that for constant matrices Y a , Y τ the moduli space is the orbifold M = C / Γ. For s -dependent solutions Y a , Y τ , similarly to [10, 16] one can choose boundary conditions such thateach block tends to a constant multiple of the identity N ℓ in the limits τ → ± ∞ , while as τ → N = N = · · · = N k = 1 it is natural to expectthat the corresponding moduli space of solutions is a resolution of the orbifold C / Γ. BPST instantons on C / Γ . Instead of the one-forms (5.3), one can introduce a basis of rightSU(2)-invariant one-forms on S / Γ given by˜ e a := − r ¯ η aµν y µ d y ν . (5.37)They are dual to the vector fields ˜ E a given in (1.3), and they satisfy the relationsd˜ e a + ε abc ˜ e b ∧ ˜ e c = 0 , (5.38) g = δ µν d y µ ⊗ d y ν = d r + r δ ab ˜ e a ⊗ ˜ e b (5.39)which are similar to those for e a and can be proven by using identities for ¯ η aµν analogous to (5.4)–(5.5).The complex combinations˜ e + i ˜ e = i r (cid:0) z d¯ z ¯2 − ¯ z ¯2 d z (cid:1) and ˜ e + i ˜ e = i r (cid:0) ¯ z ¯1 d z + z d¯ z ¯2 (cid:1) (5.40)are neither (1 , , J = J . One can showthat the forms (5.40) are (1 , J = ˜ J := (cid:0) ¯ η µλ δ λν (cid:1) (5.41)which is used in the consideration of self-duality equations (and anti-instantons) on R / Γ. Theone-forms (5.10) and (5.40) are related by the coordinate change z ¯ z ¯2 or, equivalently, by thechange of orientation x
7→ − x of R / Γ. For a fixed orientation, this inequivalence becomes moreapparent in the case of the C P , K e a (butnot e a ) form a basis of one-forms on the Sasaki-Einstein space S / Γ ⊂ R / Γ, since the complexstructure on C P ֒ → S / Γ is matched with (5.39)–(5.41) but not with (2.5) or (5.10). In any case,˜ e a are suitable one-forms on R / Γ which can be used in the ansatz for instanton solutions.Let A = ˜ X µ ˜ e µ (5.42)12e an SU(2)-invariant connection on the V-bundle E over R / Γ given in (3.7). Here ˜ e a are givenin (5.37), ˜ e := d τ = d r/r and ˜ X µ depend only on r = e τ . The explicit form (5.40) of ˜ e µ and theΓ-action (3.2) imply Γ-equivariance conditions for the components ˜ X µ given by γ Γ (cid:0) ˜ X + i ˜ X (cid:1) γ − Γ = ζ − (cid:0) ˜ X + i ˜ X (cid:1) and γ Γ (cid:0) ˜ X + i ˜ X (cid:1) γ − Γ = ˜ X + i ˜ X . (5.43)For k ≥ X µ solving (5.43) are given by the matrix elements (cid:0) ˜ X + i ˜ X (cid:1) ℓ,ℓ +2 ∈ Hom (cid:0) C N ℓ +2 , C N ℓ (cid:1) and (cid:0) ˜ X + i ˜ X (cid:1) ℓ,ℓ ∈ End (cid:0) C N ℓ (cid:1) (5.44)for ℓ = 0 , , . . . , k , together with corresponding non-zero blocks of ˜ X − i ˜ X = − ( ˜ X + i ˜ X ) † and˜ X − i ˜ X = − ( ˜ X + i ˜ X ) † .In the following we consider only the case of even rank k = 2 q , since the odd case k = 2 q +1can be reduced to a “doubling” of the even case. Using the property ξ q +1 = 1, one hasdiag (cid:0) , ζ , . . . , ζ k (cid:1) = diag (cid:0) , ζ , . . . , ζ q , ζ, ζ , . . . , ζ q − (cid:1) . (5.45)Then by using the matrix γ Γ = diag (cid:0) N , ζ N , . . . , ζ q N q , ζ N q +1 , ζ N q +2 , . . . , ζ q − N q (cid:1) (5.46)in (5.43), we obtain the solution˜ X + i ˜ X = φ · · ·
00 0 φ . . . ...... ... . . . . . . 00 0 · · · φ k φ k +1 · · · and ˜ X + i ˜ X = ρ · · · ρ . . . 0... . . . . . . 00 · · · ρ k (5.47)where φ ℓ +1 ∈ Hom( C N ℓ +1 , C N ℓ ) and ρ ℓ ∈ End( C N ℓ ).In the basis (˜ e µ ) = (˜ e a , d τ ) the SU(2)-invariant connection A and the curvature F are given by A = ˜ X µ ˜ e µ = b ˜ X µ b ˜ e µ with b ˜ X µ = r ˜ X µ and b ˜ e µ = r ˜ e µ , (5.48) F = 1 r (cid:16) ε abc [ ˜ X b , ˜ X c ] − ˜ X a (cid:17) ¯ η aµν d y µ ∧ d y ν + (cid:16) d ˜ X a d τ + [ ˜ X , ˜ X a ] − X a + 12 ε abc [ ˜ X b , ˜ X c ] (cid:17) ˜ e ∧ ˜ e a (5.49)where we used the identity η aµν ˜ e µ ∧ ˜ e ν = 1 r ¯ η aµν d y µ ∧ d y ν . (5.50)From (5.49) it follows that F is anti-self-dual, ∗F = −F , if ˜ X a satisfy the Nahm equationsd ˜ X a d τ = 2 ˜ X a − ε abc [ ˜ X b , ˜ X c ] − [ ˜ X , ˜ X a ] . (5.51)We obtain a solution by choosing ˜ X = 0 and taking˜ X a = 2 r r + Λ I a with [ I a , I b ] = ε cab I c , (5.52)where the I a are SU(2) generators in the irreducible representation on the space C N with N = N + N + · · · + N k that fits with the Γ-equivariant form (5.47). For instance, one can work in13he Coulomb branch with N ℓ = 1 for all ℓ = 0 , , . . . , k so that I a embed the group SU(2) intoSU( k +1). We thus obtain the configuration A = − r + Λ ¯ η aµν I a y µ d y ν and F = − (cid:0) r + Λ (cid:1) ¯ η aµν I a d y µ ∧ d y ν , (5.53)which is exactly the BPST instanton extended from R to R / Γ. We again have four moduli: thescale parameter Λ and the three parameters of global SU(2) rotations.The ’t Hooft instanton (5.34) is gauge equivalent to the BPST instanton (5.53) on the eu-clidean space R . However, this is not so on the orbifold R / Γ. For instance, taking N ℓ = 1 for ℓ = 0 , , . . . , k , one can obtain only abelian solutions in the ’t Hooft ansatz (5.17) while one hasirreducible nonabelian BPST instantons (5.53). Of course, one can transform the solution (5.53) toa ’t Hooft-type solution in a singular gauge, but this transformed solution will not be compatiblewith Γ-equivariance, i.e. it cannot be projected from R to R / Γ. On the other hand, ’t Hooft-typesolutions are well-defined on V-bundles E over the orbifold R / Γ if the group Γ acts trivially onthe fibres of E , i.e. if E = E , N = N and γ Γ = N . The explicit form of such solutions for N = N = 2 can be found e.g. in [22, 26, 27]. Acknowledgements
The work of TAI and OL was partially supported by the Heisenberg-Landau program. The workof OL and ADP was supported in part by the Deutsche Forschungsgemeinschaft under grant LE838/13. The work of RJS was partially supported by the Consolidated Grant ST/J000310/1 fromthe UK Science and Technology Facilities Council, and by Grant RPG-404 from the LeverhulmeTrust.
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