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=À and point out some subtle relations with the complex structure on the orbifold. We consider generalized instanton equations on R4=À 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 À-equivariant connections on complex V bundles over R4=À 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.Àq, 11.27.+d I. INTRODUCTION AND SUMMARY ½W2;W3þ½W1;W4¼Ä1; Instantons in Yang-Mills theory [1] and gravity [2,3] ½W3;W1þ½W2;W4¼Ä2; (1.1) play an important role in modern field theory [4–6]. They are nonperturbative configurations which solve first-order ½W1;W2þ½W3;W4¼Ä3; (anti-)self-duality equations for the gauge field and the where À 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;andÄ , 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.ForÄ ¼ 0,Eqs.(1.1)givethe structing gravitational instantons including twistor theory a anti-self-dual Yang-Mills equations on the orbifold C2=À [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=À. 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 =À. Here are parameters in the provides an important example of resolution of spacetime matrices Äa 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 =À. !M with Chern classes c1ð Þ¼0 and c2ð Þ¼ð#À1Þ= Kronheimer [10] considers À-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=À 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 =À. beyond the scope of the present work. In this paper, we consider gauge instanton equations 2 4 4 with matrices Äa on the orbifold C =À ffi R =À and II. INSTANTON EQUATIONS ON R Þ show that the choices of Äa 0 correspond to sources in R4 the Yang-Mills equations. For gauge potentials on R4=À A. Euclidean space Z with À ¼ kþ1, we analyze solutions of À 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 =À as where y are coordinates on and ! :¼ are components of the ’t Hooft tensors given by the formulas a a @ a a @ E ¼y and E~ ¼ y ; (1.3) a a a a a @y @y bc ¼ "bc and b4 ¼4b 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=À. 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~b¼2"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 gÀ1 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 =À ffi =À, 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 yy, 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 À 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 À 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! Äa; (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 Äa 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 Äa belong to the center uð1Þ of uðNÞ spond to BPS-type equations for Yang-Mills theory with and da 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 .IfÄa do not belong to this 2 quotient singularities R4=À, focusing on the special case center, then Eqs.