PHYSICAL REVIEW D 88, 105026 (2013) Orbifold instantons, moment maps, and Yang-Mills theory with sources
Tatiana A. Ivanova,1,* Olaf Lechtenfeld,2,† Alexander D. Popov,2,‡ and Richard J. Szabo3,§ 1Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia 2Institut fu¨r Theoretische Physik and Riemann Center for Geometry and Physics, Leibniz Universita¨t Hannover, Appelstraße 2, 30167 Hannover, Germany 3Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, United Kingdom; Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom; and The Tait Institute, Edinburgh, United Kingdom (Received 20 October 2013; published 26 November 2013) We revisit the problem of constructing instantons on ADE orbifolds R4= and point out some subtle relations with the complex structure on the orbifold. We consider generalized instanton equations on R4= which are BPS equations for the Yang-Mills equations with an external current. The relation between level sets of the moment maps in the hyper-Ka¨hler quotient construction of the instanton moduli space and sources in the Yang-Mills equations is discussed. We describe two types of spherically symmetric -equivariant connections on complex V bundles over R4= which are tailored to the way in which the orbifold group acts on the fibers. Some explicit Abelian and non-Abelian SU(2)-invariant solutions to the instanton equations on the orbifold are worked out.
DOI: 10.1103/PhysRevD.88.105026 PACS numbers: 11.15. q, 11.27.+d
I. INTRODUCTION AND SUMMARY ½W2;W3 þ½W1;W4 ¼ 1;
Instantons in Yang-Mills theory [1] and gravity [2,3] ½W3;W1 þ½W2;W4 ¼ 2; (1.1) play an important role in modern field theory [4–6]. They are nonperturbative configurations which solve first-order ½W1;W2 þ½W3;W4 ¼ 3; (anti-)self-duality equations for the gauge field and the where is a finite subgroup of the Lie group SU(2) acting Riemann curvature tensor, respectively. The construction on the fundamental representation C2 ffi R4; W , with ¼ of gauge instantons can be described systematically in the 1, 2, 3, 4, are matrices taking values in, e.g.,1 the Lie algebra framework of twistor theory [7,8] and by the ADHM uðNÞ;and , with a ¼ 1, 2, 3, are matrices in the center h construction [9]. There are also many methods for con- a of a subalgebra g of uðNÞ.For ¼ 0,Eqs.(1.1)givethe structing gravitational instantons including twistor theory a anti-self-dual Yang-Mills equations on the orbifold C2= [8] and the hyper-Ka¨hler quotient construction [10] based reduced by translations. Their solutions satisfy the full on the hyper-Ka¨hler moment map introduced in Ref. [11]. Yang-Mills equations. In the general case, the equations In this paper, we revisit the problem of constructing (1.1) are interpreted as hyper-Ka¨hler moment map quotient instantons on the ADE orbifolds R4= . The corresponding equations, and Hitchin shows [13] that one can similarly instanton moduli spaces are of special interest in type-II interpret the Bogomolny monopole equations and vortex string theory, where they can be realized as Higgs branches equations. Kronheimer shows that the moduli space of of certain quiver gauge theories which appear as world- solutions to Eqs. (1.1) in the Coulomb branch is a hyper- volume field theories on Dp branes in a Dp-Dðp þ 4Þ Ka¨hler ALE space M , which is the minimal resolution system with the Dðp þ 4Þ branes located at the fixed point of the orbifold [12]. The ADHM equations can be identi- M ! M0 (1.2) fied with the vacuum equations of the supersymmetric C2 gauge theory, and the structure of the vacuum moduli space of the orbifold M0 ¼ = . Here are parameters in the provides an important example of resolution of spacetime matrices a of Eqs. (1.1). Similar results were obtained in singularities by stringy effects in the form of D-brane Refs. [14,15] for SU(2)-invariant Yang-Mills instantons on probes. We point out in particular some salient relations R4 (see also Ref. [16]). Moreover, it was shown by between the construction of instantons and complex Kronheimer and Nakajima [17] that there exists a bundle R4 E E E structures on = . !M with Chern classes c1ð Þ¼0 and c2ð Þ¼ð# 1Þ= Kronheimer [10] considers -equivariant solutions of # such that the moduli space of framed instantons on the matrix equations: E satisfying the anti-self-dual Yang-Mills equations coin- cides with the base manifold M itself. In the limit ¼ 0, one obtains C2= as the moduli space of minimal *[email protected] †[email protected] 1In type-II string theory in the presence of orientifold Oðp þ 4Þ ‡[email protected] planes, one should use instead the Lie algebras of orthogonal or §[email protected] symplectic Lie groups.
1550-7998=2013=88(10)=105026(10) 105026-1 Ó 2013 American Physical Society IVANOVA et al. PHYSICAL REVIEW D 88, 105026 (2013) fractional instantons on the V bundle E over the orbifold though a detailed description of these moduli spaces is C2 M0 ¼ = . beyond the scope of the present work. In this paper, we consider gauge instanton equations 2 4 4 with matrices a on the orbifold C = ffi R = and II. INSTANTON EQUATIONS ON R show that the choices of a 0 correspond to sources in R4 the Yang-Mills equations. For gauge potentials on R4= A. Euclidean space Z with ¼ kþ1, we analyze solutions of equivariance Consider the two-forms conditions in two different SUð2Þ-invariant bases adapted 1 to the spherical symmetry. Recall that one can write a !a :¼ a dy ^ dy ; (2.1) 2 realization of the Lie algebra soð4Þffisuð2ÞÈsuð2Þ in 4 R4 a a terms of vector fields on R = as where y are coordinates on and ! :¼ are components of the ’t Hooft tensors given by the formulas a a @ a a @ E ¼ y and E~ ¼ y ; (1.3) a a a a a @y @y bc ¼ "bc and b4 ¼ 4b ¼ b: (2.2) 1 where a and a are components of the self-dual and Here "23 ¼ 1, ; ; ...¼ 1,2,3,4,anda; b; ...¼ 1,2,3. a anti-self-dual ’t Hooft tensors [18], and y are local coor- The forms ! are symplectic and self-dual, dinates on R4= . The commutation relations between these d!a ¼ 0 and à !a ¼ !a; (2.3) vector fields are where à is the Hodge duality operator for the flat metric a b abc c ½E ;E ¼2" E ; g ¼ dy dy (2.4) ½E~a; E~b ¼2"abcE~c; and (1.4) on R4. a ~b ½E ; E ¼0: Using the metric (2.4), we introduce three complex structures Ja ¼ !a g 1 on R4 with components Introducing complex coordinates z1 ¼ y1 þ iy2 and z2 ¼ 3 4 R4 C2 a a y þ iy on = ffi = , one finds that the vector fields ðJ Þ ¼ ! ; (2.5) E~a preserve this complex structure, but the vector fields Ea R4 a C2 R4 do not; i.e., the group SU(2) acting on C2 is generated by so that ð ;J Þffi Ja . The space is hyper-Ka¨hler; i.e., ðE~aÞ. Furthermore, the actions of the corresponding Lie it is Ka¨hler with respect to each of the complex structures 3 R4 derivatives are given by in Eq. (2.5). We choose one of them, J :¼ J, to identify and C2 ffiðR4;JÞ. With respect to J, the complex two-form L a L a a c E~b e ¼ 0 and E~b e~ ¼ 2"bce~ ; (1.5) 1 2 !C ¼ ! þ i! (2.6) where ea ¼ ea dy and e~a ¼ e~a dy are one-forms dual to is closed and holomorphic; i.e., !C is a (2, 0)-form. the vector fields Ea and E~a, respectively. We show that a a 2 both bases of the one-forms ðe ; drÞ and ðe~ ; drÞ, with r ¼ B. Instanton equations y y , can be used for describing spherically symmetric Let E be a rank-N complex vector bundle over R4 ffi C2. instanton configurations, but due to Eq. (1.5), the basis We endow this bundle with a connection A ¼ A dy ðea; drÞ is more suitable for connections on V bundles E of curvature F ¼ dA þ A ^ A ¼ 1 F dy ^ dy with trivial action of the finite group SUð2Þ, while the 2 ð a Þ taking values in the Lie algebra uðNÞ. Let us constrain basis e~ ; dr is more suitable for connections on V bundles F E with nontrivial action on the fibers of E. Explicit the curvature by the equations ð Þ a examples of Abelian and non-Abelian SU 2 -invariant ÃF þ F ¼ 2! a; (2.7) R4 Z instanton solutions on = kþ1 are worked out below. The structure of the remainder of this paper is as follows: where the functions a belong to uðNÞ. Solutions to this In Sec. II, we consider generalized instanton equations on equation of finite topological charge are called (general- R4 which reduce to Eq. (1.1) and show that they corre- ized) instantons. If a belong to the center uð1Þ of uðNÞ spond to BPS-type equations for Yang-Mills theory with and d a ¼ 0, then solutions to the equations (2.7) satisfy R4 sources. In Sec. III, we extend these equations to the ADE the Yang-Mills equations on .If a do not belong to this 2 quotient singularities R4= , focusing on the special case center, then Eqs. (2.7) are BPS-type equations for Z ¼ kþ1. In Sec. IV, we study the moduli spaces of Yang-Mills theory with sources which vanish only if a 4 translationally invariant instantons on R = via the are constant and a 2 uð1ÞuðNÞ for a ¼ 1,2,3. hyper-Ka¨hler quotient construction. In Sec. V, we consider Indeed, from Eqs. (2.7), we get the construction of spherically symmetric instanton solu- 4 tions on R = and make some preliminary comments 2Later on, we will consider an important example of such concerning the structure of the instanton moduli spaces, noncentral elements a.
105026-2 ORBIFOLD INSTANTONS, MOMENT MAPS, AND YANG- ... PHYSICAL REVIEW D 88, 105026 (2013) d à F þ A ^ÃF ÃF ^ A (2.15) becomes constant and the term in Eq. (2.14)is topological. Constant matrices of the form ¼ i 1 a A a a N ¼ 2! ^ðd a þ½ ; a Þ; (2.8) correspond to D3 branes in a nonzero B field in string which after taking the Hodge dual can be rewritten as theory and can be described in terms of a noncommutative deformation of Yang-Mills theory on the space R4 a @ F þ½A ; F ¼4! ð@ a þ½A ; a Þ: (see e.g., Refs. [19,20]). (2.9) III. INSTANTON EQUATIONS ON R4= The current A. Orbifold R4= j :¼ 4!a D with D :¼ @ þ½A ; a a a a The complex structure J ¼ J3, introduced in Eq. (2.5), (2.10) defines the complex coordinates satisfies the covariant continuity equation z1 ¼ y1 þ iy2 and z2 ¼ y3 þ iy4 (3.1) R4 C2 D j ¼ 0; (2.11) on ffi , where y are real coordinates. The Lie group SU(2) naturally acts on the vector space C2 with the as required for minimal coupling of an external current in coordinates in Eq. (3.1). We are interested in the Kleinian the Yang-Mills equations. orbifolds C2= , where is a finite subgroup of SU(2). They have an ADE classification in which is associated C. Variational equations with the extended Dynkin diagram of a simply laced To formulate the generalized instanton equations (2.7)as simple Lie algebra. For the Ak-type simple singularities, ¼ Z absolute minima of Euler-Lagrange equations derived corresponding to the cyclic group kþ1 of order þ from an action principle, we note that the presence of k 1, explicit descriptions of instantons will be readily the current [Eq. (2.10)] in the Yang-Mills equations (2.9) available. However, most of our results can be generalized requires the addition of the term to the other ADE groups corresponding to non-Abelian orbifolds C2= . 1 The action of ¼ Z on C2 is given by trj A (2.12) kþ1 2 ðz1;z2Þ ° ð z1; 1z2Þ; (3.2) in the standard Yang-Mills Lagrangian where 1 LYM ¼ trF F : (2.13) 2 i 8 ¼ exp with kþ1 ¼ 1 (3.3) k þ 1 Up to a total derivative, the term in Eq. (2.12) is equivalent to the term is a primitive (k þ 1)-th root of unity. This action has a single isolated fixed point at the origin ðz1;z2Þ¼ð0; 0Þ. 1 C2 !a trF : (2.14) The orbifold = is defined as the set of equivalence 2 a classes on C2 with respect to the equivalence relation After adding the term in Eq. (2.14), together with the ð z1; 1z2Þðz1;z2Þ; (3.4) nondynamical term and it has a singularity at the origin. The metric on C2= is 3 tr a a (2.15) g ¼ dz1 d z1 þ dz2 d z2; (3.5) and the topological density where the coordinates z 1, z 2 are complex-conjugated to 1 z1, z2. " trF F ; (2.16) 2 C2= we obtain the Lagrangian B. V bundles on A V bundle on C2= is a -equivariant bundle over C2; 1 þ a þ a C2 L ¼ trðF ! aÞðF ! aÞ; (2.17) i.e., a vector bundle on with a action on the fibers 4 which is compatible with the action of on C2. The Z where orbifold group ¼ kþ1 has k þ 1 one-dimensional irreducible representations such that the generator of 1 F þ ¼ ðÃF þ F Þ (2.18) Z acts on the ‘th module as multiplication by ‘ 2 kþ1 for ‘ ¼ 0; 1; ...;k. Let us denote by E‘ complex V bundles 2 is the self-dual part of the curvature two-form F . In the over C = of rank N‘ on which acts in the ‘th irreducible following, we will consider constant matrices a for which representation as
105026-3 IVANOVA et al. PHYSICAL REVIEW D 88, 105026 (2013)
‘ N‘ v‘ ° v‘ for v‘ 2 C (3.6) 1 1 W 1 ¼ ðW iW Þ and W 2 ¼ ðW iW Þ; (3.14) z 2 1 2 z 2 3 4 N on a generic fiber C ‘ of E‘. Then any complex V bundle E over C2= of rank N can be decomposed into isotopical the action of is given by components as a Whitney sum ° 1 ° Wz1 Wz1 and Wz2 Wz2 : (3.15) Mk A E ¼ E‘; (3.7) The action of on the components of any unitary ‘¼0 connection A ¼ A dy on a Hermitian V bundle and its structure group is of the form [Eq. (3.7)] is given by a combination of the spacetime action [Eq. (3.15)] and the adjoint action generated by Yk Xk matrices from Eq. (3.9)as UðN‘Þ with N‘ ¼ N: (3.8) A ° 1 A 1 A ° A 1 ‘¼0 ‘¼0 z1 z1 and z2 z2 : From Eq. (3.6), it follows that the action of the point group (3.16) on the V bundle [Eq. (3.7)] is given by the unitary matrices The corresponding -equivariance conditions require that Mk the connection defines a covariant representation of the v ° ðvÞ with ¼ ‘1 (3.9) N‘ orbifold group, in the sense that ‘¼0 A 1 A A 1 1A k CN z1 ¼ z1 and z2 ¼ z2 : Lon vectors v ¼ðv‘Þ‘¼0 in the generic fiber ¼ k CN‘ E ‘¼0 of . (3.17) Simplifying the situation discussed in the previous It is easy to see that the solutions to the constraint equations section, we choose matrices a in the form (3.17) are given by block off-diagonal matrices Mk 0 1 ¼ ‘1 ÁÁÁ c a i a N‘ ; (3.10) 00 0 kþ1 ‘¼ B C 0 B . C B c 0 . . 00C where ‘ 2 R are constants. The matrices in Eq. (3.10) B 1 C a B . . C belong to the center of the Lie algebra of the gauge group A 1 ¼ B c . . C and z B 0 2 . . 0 C [Eq. (3.8)]. The diagonal U(1) subgroup of scalars in B C B . . . C Eq. (3.8) acts trivially on ðE; AÞ, so we can factor the @ . . . . . 00A gauge group in Eq. (3.8) by this Uð1Þ subgroup to get the 0 ÁÁÁ 0 c 0 quotient group 0 k 1 0 1 0 ÁÁÁ 0 Yk B C B . . C G :¼ UðN‘Þ =Uð1Þ: (3.11) B . . C B 00 2 . . C ‘¼0 B C A 2 ¼ B . . . . C; (3.18) z B ...... 0 C Then, the Lie algebra g of G is the traceless part of the B C ‘ B C Lie algebra of Eq. (3.8), and one should impose on a in @ 00ÁÁÁ 0 k A Eq. (3.10) the tracelessness condition kþ1 0 ÁÁÁ 00 Xk ‘ y y A 1 ¼ A A 2 ¼ A aN‘ ¼ 0; (3.12) together with z z1 and z z2 . Here the ‘¼0 c E E E E bundle morphisms ‘þ1: ‘ ! ‘þ1 and ‘þ1: ‘þ1 ! ‘ which defines the center h of g. are bifundamental scalar fields given fiberwise by matrices
c CN‘ CN‘þ1 ‘þ1 2 Homð ; Þ and C. -equivariant connections CN‘þ1 CN‘ (3.19) ‘þ1 2 Homð ; Þ Consider a one-form for ‘ ¼ 0; 1; ...;k (with indices read modulo k þ 1). 1 2 1 2 W ¼ W dy ¼ Wz1 dz þ Wz2 dz þ Wz 1 d z þ Wz 2 d z Substitution of Eq. (3.18) into Eq. (2.7) then yields 2 (3.13) the generalized instanton equations on the orbifold C = . The transformations in Eq. (3.15) are defined for the holo- on R4 ffi C2 which is invariant under the action of nomic basis dy of one-forms on R4= and can differ for SUð2ÞSOð4Þ defined by Eq. (3.2). Then, on the other bases of one-forms, leading to modifications of the components formulas in Eqs. (3.16), (3.17), and (3.18).
105026-4 ORBIFOLD INSTANTONS, MOMENT MAPS, AND YANG- ... PHYSICAL REVIEW D 88, 105026 (2013) IV. TRANSLATIONALLY INVARIANT we can consider the G-invariant level set INSTANTONS 1ð Þ (4.8) A. Matrix equations which defines a submanifold of the manifold M, where Consider translationally invariant connections A on the ¼ð ; ; Þ2R3 hà and h is the center of g. V bundle in Eq. (3.7) over C2= satisfying the equations 1 2 3 Then one can define the hyper-Ka¨hler quotient as (see (2.7) with given in Eq. (3.10); i.e., we assume that A a e.g., Refs. [10,11,13]) are independent of the coordinates y , which reduces A 1 Eq. (2.7) to the matrix equations (1.1) with W :¼ . M ¼ ð Þ===G; (4.9) :¼ A 1 :¼ A 2 A Denoting B1 z and B2 z for given by ‘ Eq. (3.18) with constant matrices c and for where ¼ð aÞ are parameters defining ¼ð aÞ2 ‘þ1 ‘þ1 R3 à ‘ ¼ 0; 1; ...;k, we obtain the equations h . The hyper-Ka¨hler metric on M descends to a hyper-Ka¨hler metric on the quotient M . When the group 1 a action is free, the reduced space M is a hyper-Ka¨hler ½W ;W ¼ a; (4.1) 2 manifold of dimension dim M ¼ dim M 4 dim G. which can be rewritten as In the case of the matrix model in Eq. (4.1), the manifold M is the flat hyper-Ka¨hler manifold i ½B ;B ¼ ð i Þ¼: C; (4.2) R4 1 2 4 1 2 M ¼ uðNÞ; (4.10) the group G is given in Eq. (3.11) and the three moment y y i 4 ½B ;B þ½B ;B ¼ ¼: R: (4.3) maps are 1 1 2 2 2 3 a 1 a Solutions to these equations satisfy the reduced Yang-Mills ðWÞ¼ ½W ;W 2uðNÞ: (4.11) 2 equations (2.9) with the external source Solutions of Eqs. (4.2) and (4.3) form a submanifold a j ¼ 4 ½W ; a ; (4.4) 1ð Þ of the manifold in Eq. (4.10), and by factoring with the gauge group in Eq. (3.11) [which for generic where W is given by Eq. (3.18) and a by Eq. (3.10). ‘ parameters ¼ð aÞ acts freely on the solutions], we obtain the moduli space [Eq. (4.9)]. This moduli space B. Hyper-Ka¨hler quotients was studied by Kronheimer [10], who showed that for Z The reduced equations (4.1) [and also the instanton ¼ kþ1 and the Coulomb branch N0 ¼ N1 ¼ÁÁÁ¼ equations (2.7)] can be interpreted as hyper-Ka¨hler mo- Nk ¼ 1, it is a smooth, four-dimensional, asymptotically a ment map equations. For this, recall that if ðM; g; ! Þ is a locally Euclidean (ALE) hyper-Ka¨hler manifold M with a ‘ hyper-Ka¨hler manifold with an action of a Lie group G metric defined by the parameters ¼ð aÞ. The ALE 3 which preserves the metric g and the three Ka¨hler forms condition means that at asymptotic infinity of M , the !a, then one can define three moment maps metric approximates the Euclidean metric on the orbifold C2= . Kronheimer also shows that M is diffeomorphic to a à : M ! g (4.5) the minimal smooth resolution of the Kleinian singularity C2 taking values in the dual gà of the Lie algebra g of G such M0 ¼ = , regarded as the affine algebraic variety kþ1 2 2 C3 that, for each 2 g with a triholomorphic Killing vector x þ y þ z ¼ 0 in . For the Hilbert-Chow map field L generated by the G action on M, the functions in : M ! M0; (4.12) Eq. (4.5) satisfy the equations the exceptional divisor of the blowup is the set a a hd ; i¼L 5 ! ; (4.6) [k à 1 where h ; i is the dual pairing between elements of g ð0Þ¼ ‘; (4.13) and g, and 5 denotes the contraction of vector fields and ‘¼0 differential forms. Denoting by ¼ð 1; 2; 3Þ the 1 5 where ‘ ffi CP and k ¼ # 1. The parameters vector-valued moment map determine the periods of the three symplectic forms !a as : M ! R3 gÃ; (4.7) 4We identify uðNÞà and uðNÞ. 5 Z Recall that we consider ¼ kþ1 for definiteness here, but 3They are Ka¨hler with respect to the three complex structures many of these considerations generalize to the other Kleinian a a 1 3 J ¼ ! g . With respect to the complex structure J , the groups SUð2Þ. In the general case, N‘ are the dimensions 3 1 2 two-form !R ¼ ! is Ka¨hler, and !C ¼ ! þ i! is of the irreducible representations of the finite group in holomorphic. Kronheimer’s construction.
105026-5 IVANOVA et al. PHYSICAL REVIEW D 88, 105026 (2013) Z a ‘ V. SPHERICALLY SYMMETRIC INSTANTONS ! ¼ a: (4.14) 3 ‘ A. Cone CðS = Þ 4 In the general case N‘ 0, one can also define a map The Euclidean space R can be regarded as a cone over 3 M ! M0 which is a resolution of singularities [21]. the three-sphere S , R4 nf0g¼CðS3Þ; (5.1) C. Hermitian Yang-Mills connections
The matrix C in Eq. (4.2) parametrizes deformations of with the metric E the complex structure on the V bundle , and it can be g ¼ dy dy ¼ dr2 þ r2 ea eb; (5.2) reabsorbed through a nonanalytic change of coordinates on ab 2 a the space in Eq. (4.10)[10,22]. Therefore, we may take where r ¼ y y and ðe Þ give a basis of left C ¼ 0 without loss of generality; in this case, the ALE SU(2)-invariant one-forms on S3. One can define ea by space M is biholomorphic to the minimal resolution. the formula In fact, the moduli spaces M and M 0 are diffeomorphic 1 0 a a for distinct and such that R 0 for both sets of e :¼ y dy ; (5.3) r2 parameters. For C ¼ 0,wehave ¼ ¼ 0, and 1 2 a the equations (2.7) become the Hermitian Yang-Mills where the ’t Hooft tensors are defined in Eq. (2.2). The equations [23,24] one-forms ea are dual to the vector fields Ea from Eq. (1.3). By using the identities F F 3 Ã þ ¼ ! 3: (4.15) a b c a a a a "bc ¼ þ ; A E A connection on satisfying Eq. (4.15) is said to be a (5.4) Hermitian Yang-Mills connection. It defines a holomorphic E structure on , since from Eq. (4.15) it follows that the a b ¼ þ " ; (5.5) curvature F is of type (1, 1) with respect to the complex ab structure J, i.e., one can easily verify the Maurer-Cartan equations F 2;0 F 0;2 a a b c ¼ 0 ¼ ; (4.16) de þ "bce ^ e ¼ 0 (5.6) and the third equation from Eq. (4.15), and 3 F 1 1 ! ¼ 3; (4.17) !a ¼ a dy ^ dy ¼ a e^ ^ e^ ; (5.7) 2 2 1 E means that, for 3 ¼ i N, the V bundle is (semi)stable where [23,24]. In the special case 3 ¼ 0, we get the standard anti-self-dual Yang-Mills equations e^a :¼ rea and e^4 :¼ dr: (5.8) ÃF ¼ F : (4.18) The relation (5.2) between the metric in Cartesian and spherical coordinates can be readily checked as well. All formulas (5.2), (5.3), (5.4), (5.5), (5.6), (5.7), and D. Translationally equivariant instantons (5.8) are also valid for the orbifold C2= after imposing the equivalence relation in Eq. (3.4), and the orbifold is a Instead of constant matrices A which reduce Eq. (2.7) cone over the lens space S3= ,6 to the matrix equations (4.1), one can also consider the gauge potential ðC2 nf0gÞ= ¼ CðS3= Þ; (5.9) 1 A ¼ !a y dy ; (4.19) with the metric (5.2). The one-forms [Eq. (5.3)] in the 2 a complex coordinates [Eq. (3.1)] have the form where the commuting matrices a are given in Eq. (3.10). i e1 þ ie2 ¼ ðz1dz2 z2dz1Þ and The connection in Eq. (4.19) is translationally invariant r2 up to a gauge transformation and can be extended to i (5.10) the orbifold T4= , where T4 is a four-dimensional torus. e3 þ ie4 ¼ ðz 1 dz1 þ z 2 dz2Þ; r2 The curvature of A is plus their complex-conjugated expressions. Hence, 1 F ¼ dA ¼ !a dy ^ dy ; (4.20) Eq. (5.10) defines two complex one-forms which are 2 a a providing in essence the three symplectic structures ! 6The orbifolds S3= for arbitrary ADE point groups exhaust from Eq. (2.1). the possible Sasaki-Einstein manifolds in three dimensions.
105026-6 ORBIFOLD INSTANTONS, MOMENT MAPS, AND YANG- ... PHYSICAL REVIEW D 88, 105026 (2013) J ¼ J3 (1, 0)-forms with respect to the complex structure A ¼ X e with X ¼ rX^ ; (5.17) defined in Eq. (2.5). The symplectic two-forms [Eq. (5.7)] and the complex structures [Eq. (2.5)] have the same @ dXa components in the holonomic ðdy ; Þ and nonholonomic F ¼ þ½X ;X and @y 4a d a ðe^a; E^ Þ bases, where E^ 5 e^b ¼ b. From Eqs. (1.5), (3.2), (5.18) a a a F and (5.10), it follows that ab ¼ 2"abcXc þ½Xa;Xb ; dr and Eq. (2.7) reduces to a form of the generalized Nahm ea and e4 :¼ ¼ d with ¼ log r (5.11) r equations given by are invariant under the action of the finite group dX 1 a ¼ ½X ;X 2X þ " ½X ;X : (5.19) SUð2Þ. d 4 a a 2 abc b c a Introducing B. Nahm equations E C2 1 Consider the complex V bundle over = described Y :¼ e2 X and s ¼ e 2 ¼ ; (5.20) in Sec. III. Let r2
A ¼ X^ e^ we obtain the equations 1 1 dY 1 1 ¼ ðX^ iX^ Þðe^1 þ ie^2Þþ ðX^ iX^ Þ 2 a ¼½Y ;Y " ½Y ;Y þ : (5.21) 2 1 2 2 3 4 ds 4 a 2 abc b c s2 a Âðe^3 þ ie^4ÞþH:c: (5.12) For a ¼ 0, these equations coincide with the Nahm be a connection on E written in the basis (5.8). The corre- equations [25]. Choosing a ¼ 0 and defining sponding -equivariance conditions are 1 1 :¼ ðY þ iY Þ and :¼ ðY þ iY Þ; (5.22) ^ ^ 1 ^ ^ 3 4 1 2 ðX1 iX2Þ ¼ X1 iX2 and 2 2 (5.13) ^ ^ 1 ^ ^ ðX3 iX4Þ ¼ X3 iX4: we obtain the equations Solutions to these equations are given by d ð þ yÞþ½ ; y þ½ ; y ¼0; (5.23) 1 ds ðX^ iX^ Þ¼diagð ; ; ...; Þ and 2 1 2 0 1 k d 1 þ½ ; ¼0 (5.24) ðX^ iX^ Þ¼diagð’ ;’ ; ...;’ Þ; ds 2 3 4 0 1 k (5.14) 1 considered by Kronheimer [14,15] (see also Ref. [16]) in ðX^ þ iX^ Þ¼ diagð y; y; ...; yÞ and 2 1 2 0 1 k the description of SU(2)-invariant instantons. The equa- 1 tions (5.21) have three obvious solutions, which we now ðX^ þ iX^ Þ¼ diagð’y;’y; ...;’yÞ; 2 3 4 0 1 k consider in turn. where ‘ and ’‘ are N‘  N‘ complex matrices. Thus, the C. Abelian instantons with a 0 -equivariance conditions in the basis (5.8) force the block-diagonal form [Eq. (5.14)] of the connection com- For the first solution, we choose ^ A ponents X ; i.e., the connection is reducible, or else 1 r2 N ¼ 0 for ‘ 0,if acts trivially on E. Y ¼ ¼ and Y ¼ 0: (5.25) ‘ a 2s a 2 a 4 The instanton equations (2.7) are conformally invariant, and it is more convenient to consider them on the cylinder Then we get the solution R 3  S = (5.15) 1 2 a 1 a A ¼ r e a and F ¼ ae^ ^ e^ (5.26) with the metric 2 2 of the equations (2.7), which coincide with Eqs. (4.19) and dr2 1 2 a b a b (4.20). This configuration can also be regarded as a trans- gcyl ¼ d þ abe e ¼ 2 þ abe e ¼ 2 g: r r lationally equivariant solution of the self-dual Yang-Mills (5.16) equations a In the basis ðe Þ¼ðe ; d Þ, the SUð2Þ-invariant (spheri- ÃF ¼ F ; (5.27) cally symmetric) connection A and its curvature F have components depending only on r ¼ e and are given by i.e., as an anti-instanton on R4= or T4= .
105026-7 IVANOVA et al. PHYSICAL REVIEW D 88, 105026 (2013) D. Abelian instantons with poles gauge group (3.11); i.e., there are k þ 1 instanton solutions with gauge group SUð2ÞUðN‘Þ if N‘ 2 for all ‘ ¼ For the second solution, considered in Ref. [15], we set a dY 0; 1; ...;k. From the explicit form of e in Eq. (5.3), it ¼ 0 and a ¼ 0. Then the constant matrices Y satisfy a ds follows that each of these solutions is the standard ’t the reduced anti-self-dual Yang-Mills equations Hooft instanton generalized from R4 to R4= .Forframed 1 instantons,7 there are four moduli: the scale parameter ½Y ;Y þ " ½Y ;Y ¼0 (5.28) a 4 2 abc b c and three global SU(2) rotational parameters (see e.g., Ref. [22]). considered in Ref. [10] and discussed in Sec. IV. Solutions to Eq. (5.28) are necessarily given by commuting matrices [10], and one can choose them in the forms F. Moduli spaces of SU(2)-invariant instantons In the special case where is the trivial group, we obtain 2 ^ Ya ¼ 2 a and Y4 ¼ 0; (5.29) SU(2)-invariant solutions of the anti-self-dual Yang-Mills equations (4.18)onR4 nf0g¼CðS3Þ. The moduli spaces where ^ have the form (3.10) and is a scale parameter. a of these framed instantons (subject to appropriate bound- For the corresponding gauge potential and its field ary conditions) are four-dimensional hyper-Ka¨hler ALE strength, we obtain C2 0 spaces M resolving M0 ¼ = as in Eq. (4.12), where 2 2 0 is a finite subgroup of the group SU(2) related to A ¼ X ea ¼ ^ e^a and a r3 a boundary conditions for the solutions [14–16]. This is the (5.30) 2 2 moduli space of the spherically symmetric instanton F a ^ E 0 ¼ 4 ae^ ^ e^ ; which has the minimal topological charge c2ð Þ¼ð# r 1Þ=# 0. In our reducible case, we obtain a product of a where e^ are given in Eq. (5.8) and are the anti-self- hyper-Ka¨hler moduli spaces dual ’t Hooft tensors defined by  ÂÁÁÁ M 0 M 1 M k : (5.35) a ¼ "a and a ¼ a ¼ a: (5.31) bc bc b4 4b b ¼ ¼ Note that M ‘ is a point if N‘ 1.ForN‘ 1, one can Thus, we obtain singular Abelian solutions with delta- also use the singular Abelian solution from Eq. (5.30), function sources in the Maxwell equations, as discussed A 2 2 by Ref. [15]. The gauge potential from Eq. (5.30) can be F ‘ ¼ a ‘e^ ^ e^ ; (5.36) regarded as an asymptotic approximation of a smooth r4 a solution. Note also that ‘ with a 2 R. 2 2 We have seen that for constant matrices Ya, Y , the a a 2 ! :¼ e^ ^ e^ (5.32) moduli space is the orbifold M ¼ C = .For r4 0 s-dependent solutions Ya, Y , similarly to Refs. [10,16], can be viewed as three additional anti-self-dual symplectic one can choose boundary conditions such that each block forms on the cone ðR4 nf0gÞ= ¼ CðS3= Þ, complemen- 1 tends to a constant multiple of the identity N‘ in the limits tary to those given in Eq. (2.1). !Æ1, while as ! 0 the solutions define a represen- tation of SU(2). For N0 ¼ N1 ¼ÁÁÁ¼Nk ¼ 1,itis E. ’t Hooft instantons on C2= natural to expect that the corresponding moduli space of solutions is a resolution of the orbifold C2= . For the third solution, we choose Y4 ¼ Y ¼ 0 ¼ a to get G. BPST instantons on C2= 2 2 2r2 Y ¼ I ¼ I with 2 R and Instead of the one-forms [Eq. (5.3)], one can introduce a a s þ 2 a r2 þ 2 a basis of right SU(2)-invariant one-forms on S3= given by c ½Ia;Ib ¼"abIc: (5.33) 1 e~a :¼ a y dy : (5.37) Then, for the anti-self-dual connection and curvature we r2 obtain They are dual to the vector fields E~a given in Eq. (1.3), and 2 2 they satisfy the relations A ¼ eaI and r2 þ 2 a (5.34) de~a þ "a e~b ^ e~c ¼ 0; (5.38) 2 2 bc F ¼ a I e^ ^ e^ ; ðr2 þ 2Þ2 a 7Framed instantons are instanton solutions modulo SU(2)- 2 where we used the relation s ¼ r . Here, Ia are the invariant gauge transformations which approach the identity at generators of the group SU(2) embedded in the broken asymptotic infinity.
105026-8 ORBIFOLD INSTANTONS, MOMENT MAPS, AND YANG- ... PHYSICAL REVIEW D 88, 105026 (2013)
2 2 a b 2 2k 2 2q 3 2q 1 g ¼ dy dy ¼ dr þ r abe~ e~ ; (5.39) diagð1; ;...; Þ¼diagð1; ;...; ; ; ;...; Þ: which are similar to those for ea and can be proven by (5.45) using identities for a analogous to Eqs. (5.4) and (5.5). Then, by using the matrix The complex combinations ¼ ð1 21 2q1 1 diag N0 ; N1 ; ...; Nq ; Nqþ1 ; 1 2 i 1 2 2 1 e~ þ ie~ ¼ ðz d z z dz Þ and 31 2q 11 r2 N ; ...; N Þ (5.46) (5.40) qþ2 2q i e~3 þ ie~4 ¼ ðz 1 dz1 þ z2d z2 Þ in Eq. (5.43), we obtain the solution r2 0 1 0 1 0 ÁÁÁ 0 are neither (1, 0)- nor (0, 1)-forms with respect to the B C complex structure J ¼ J3. One can show that the forms B . . . C B 00 2 . . C in Eq. (5.40) are (1, 0)-forms with respect to the complex B C X~ þ iX~ ¼ B . . . . C and structure 1 2 B ...... 0 C B C ~ ~3 3 B C J ¼ J :¼ð Þ; (5.41) @ 00ÁÁÁ 0 k A
kþ1 0 ÁÁÁ 00 which is used in the consideration of self-duality equations 0 1 4 (and anti-instantons) on R = . The one-forms in 0 0 ÁÁÁ 0 Eqs. (5.10) and (5.40) are related by the coordinate change B C B . . C 2 ° 2 B 0 . 0 C z z or, equivalently, by the change of orientation X~ þ iX~ ¼ B 1 C; (5.47) x4 ° x4 of R4= . For a fixed orientation, this inequiva- 3 4 B . . . C @B . . . AC lence becomes more apparent in the case of the CP2, K3, . . . 0 a and ALE hyper-Ka¨hler manifolds. Note that exactly e~ 0 ÁÁÁ 0 k (but not ea) form a basis of one-forms on the Sasaki- CN‘þ1 CN‘ CN‘ 3 R4 where ‘þ1 2 Homð ; Þ and ‘ 2 Endð Þ. Einstein space S = = , since the complex structure a C 1 3 In the basis ðe~ Þ¼ðe~ ; d Þ, the SU(2)-invariant connec- on P ,! S = is matched with Eqs. (5.39), (5.40), and A F (5.41) but not with Eqs. (2.5)or(5.10). In any case, e~a tion and the curvature are given by R4 are suitable one-forms on = which can be used in the A ¼ X~ e~ ¼ X~^ e~^ with ansatz for instanton solutions. 1 Let X~^ ¼ X~ and r (5.48) A ¼ X~ e~ (5.42) e~^ ¼ re~ ; be an SU(2)-invariant connection on the V bundle E over R4 a 1 1 = given in Eq. (3.7). Here e~ are given in Eq. (5.37), F ~ ~ ~ a 4 ¼ 2 "abc½Xb; Xc Xa dy ^ dy e~ :¼ d ¼ dr=r and X~ depend only on r ¼ e .Theex- r 4 plicit form [Eq. (5.40)] of e~ and the action [Eq. (3.2)] dX~ 1 a ~ ~ ~ ~ ~ 4 a imply -equivariance conditions for the components X~ þ þ½X4; Xa 2Xa þ "abc½Xb; Xc e~ ^ e~ ; d 2 given by (5.49) ~ ~ 1 2 ~ ~ ðX1 þ iX2Þ ¼ ðX1 þ iX2Þ and (5.43) where we used the identity ~ ~ 1 ~ ~ ðX3 þ iX4Þ ¼ X3 þ iX4: a 1 a ~ e~ ^ e~ ¼ dy ^ dy : (5.50) For k 2, the nonzero blocks of X solving Eq. (5.43) are r2 given by the matrix elements From Eq. (5.49), it follows that F is anti-self-dual, ‘;‘þ2 CN‘þ2 CN‘ F F ~ ðX~1 þ iX~2Þ 2 Homð ; Þ and à ¼ ,ifXa satisfy the Nahm equations (5.44) ðX~ þ iX~ Þ‘;‘ 2 EndðCN‘ Þ dX~ 1 3 4 a ¼ 2X~ " ½X~ ; X~ ½X~ ; X~ : (5.51) d a 2 abc b c 4 a for ‘ ¼ 0; 1; ...;k, together with corresponding nonzero ~ ~ ~ ~ y ~ ~ ~ ¼ blocks of X1 iX2 ¼ ðX1 þ iX2Þ and X3 iX4 ¼ We obtain a solution by choosing X4 0 and taking ð ~ þ ~ Þy X3 iX4 . 2r2 ~ c In the following, we consider only the case of even Xa ¼ 2 2 Ia with ½Ia;Ib ¼"abIc; (5.52) rank k ¼ 2q, since the odd case k ¼ 2q þ 1 can be re- r þ duced to a ‘‘doubling’’ of the even case. Using the property where the Ia are SU(2) generators in the irreducible repre- 2qþ1 CN ¼ 1, one has sentationonthespace with N ¼ N0 þ N1 þÁÁÁþNk
105026-9 IVANOVA et al. PHYSICAL REVIEW D 88, 105026 (2013) that fits with the -equivariant form [Eq. (5.47)]. For in- Eq. (5.53) to a ’t Hooft-type solution in a singular gauge, stance, one can work in the Coulomb branch with N‘ ¼ 1 but this transformed solution will not be compatible with 4 for all ‘ ¼ 0; 1; ...;kso that Ia embed the group SU(2) into -equivariance; i.e., it cannot be projected from R to SUðk þ 1Þ. We thus obtain the configuration R4= . On the other hand, ’t Hooft-type solutions are well E R4 2 defined on V bundles over the orbifold = if the group A a E E E ¼ 2 2 Iay dy and acts trivially on the fibers of ; i.e., if ¼ 0, N ¼ N0, r þ 1 (5.53) and ¼ N . The explicit form of such solutions for 2 2 0 F ¼ a I dy ^ dy ; N ¼ N0 ¼ 2 can be found, e.g., in Refs. [22,26,27]. ðr2 þ 2Þ2 a R4 which is exactly the BPST instanton extended from to ACKNOWLEDGMENTS R4= . We again have four moduli: the scale parameter and the three parameters of global SU(2) rotations. The work of T. A. I. and O. L. was partially supported by The ’t Hooft instanton [Eq. (5.34)] is gauge equivalent to the Heisenberg-Landau program. The work of O. L. and the BPST instanton [Eq. (5.53)] on the Euclidean space R4. A. D. P. was supported in part by the Deutsche However, this is not so on the orbifold R4= . For instance, Forschungsgemeinschaft under Grant No. LE 838/13. taking N‘ ¼ 1 for ‘ ¼ 0; 1; ...;k, one can obtain only The work of R. J. S. was partially supported by Abelian solutions in the ’t Hooft ansatz [Eq. (5.17)] Consolidated Grant No. ST/J000310/1 from the U.K. while one has irreducible non-Abelian BPST instantons Science and Technology Facilities Council, and by Grant [Eq. (5.53)]. Of course, one can transform the solution in No. RPG-404 from the Leverhulme Trust.
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