wykresx.eps Acta Polytechnica Vol. 51 No. 1/2011 N =4, d =1 Supersymmetric Hyper-Kähler Sigma Models and Non-Abelian Monopole Background S. Bellucci, S. Krivonos, A. Sutulin Abstract Weconstruct aLagrangian formulation of N =4supersymmetricmechanicswithhyper-Kähler sigmamodels in abosonic sector in a non-Abelian background gauge field. The resulting action includes a wide class of N = 4 supersymmetric mechanics describing the motion of an isospin-carrying particle over spaces with non-trivial geometry. In two examples that we discuss in details, the background fields are identified with the field of BPST instantons in flat and Taub-NUT spaces. Keywords: supersymmetric mechanics, Hyper-Kähler spaces, non-Abelian gauge fields. 1 Introduction N = 4 supersymmetric mechanics provides a nice framework for the study ofmany interesting features of higherdimensional theories. At the same time, the existence of a variety of off-shell N = 4 irreducible linear supermultiplets in d = 1 [1, 2, 3, 4, 5] makes the situation in one dimension evenmore interesting, and this is what prompted us to investigate such su- persymmetric models themselves, without reference to higher dimensional counterparts. Being a super- symmetric invariant theory, N =4mechanics admits a natural formulation in terms of superfields living in a standard and/or in a harmonic superspace [6], adapted to one dimension [7]. In any case, the pre- ferred approach for discussing supersymmetric me- chanics is theLagrangianone. Beingquite useful, the Lagrangian approach has one subtle point, when we try todescribe the system inanarbitrarygaugeback- ground. While the inclusion of the Abelian gauge background can be done straightforwardly [7], the non-Abelian background reqiures new ingredients — isospin variables — which have to be included in the description [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. These isospin variables become purely internal de- grees of freedomafter quantization and form an aux- iliary N = 4 supermultiplet, together with the aux- iliary fermions. Thereare variousapproaches for introducing such auxiliary superfields and couplings with them, but until nowall constructedmodels havebeen restricted to have conformally flat sigma models in the bosonic sector. This restriction has an evident source — it has been known for a long time that all linear N =4 supermultiplets can be obtained through a dualiza- tion procedure from the N = 4 “root” supermulti- plet — the N =4 hypermultiplet [18, 19, 20, 21, 22, 23], while the bosonic part of the general hypermulti- plet action is conformal to the flat one. The onlyway to escape this flatness situation is tousenonlinear su- permultiplets [24, 25, 26], instead of linear ones. The main aim of this paper is to construct the Lagrangian formulation of N = 4 supersymmetric mechanics on conformal to hyper-Kähler spaces in non-Abelianbackgroundgaugefields. Toachieve this goal we combine two ideas • We introduce the coupling ofmatter supermulti- plets with an auxiliary fermionic supermultiplet Ψα̂ containing on-shell four physical fermions and four auxiliary bosons playing the role of isospin variables. The very specific coupling results in a component action which contains only time derivatives of the fermionic compo- nents present in Ψα̂. Then, we dualize these fermions into auxiliary ones, ending up with the proper action for matter fields and isospin vari- ables. This procedure was developed in [11]. • As the next step, starting from the action for the N = 4 tensor supermultiplet [27, 28] cou- pled with the superfield Ψα̂, following [24], we dualize the auxiliary component A into a fourth physical boson, finishing with the action having a geometry conformal to the hyper-Kähler one in the bosonic sector. The resulting action contains a wide class of N = 4 supersymmetric mechanics describing the motion of an isospin-carrying particle over spaces with non- trivial geometryand in the presence of a non-Abelian backgroundgaugefield. In two examples thatwedis- cuss in details, these backgrounds correspond to the field of the BPST instanton in the flat and Taub- NUT spaces. In order to make our presentation self- sufficient, we include in Section 2 a sketchy descrip- tion of our construction applied to the linear tensor and hypermultiplet. We also discuss the relation be- tween these supermultiplets in the context of our ap- proach (Section 3), which immediately leads to the generalized procedure presented in Section 4. 13 Acta Polytechnica Vol. 51 No. 1/2011 2 Isospin particles in conformally flat spaces One way to incorporate the isospin-like variables in the Lagrangian of supersymmetric mechanics is to couple the basic superfields with auxiliary fermionic superfields Ψα̂,Ψ̄α̂, which contain these isospin vari- ables [11]. Such a coupling, being written in a stan- dard N = 4 superspace, has to be rather special, in order to provide a kinetic term of the first order in time derivatives for the isospin variables and to de- scribe the auxiliary fermionic components present in Ψα̂,Ψ̄α̂. Following [11], we introduce the coupling of auxiliary Ψ superfields with some arbitrary, for the time being, N =4 supermultiplet X as Sc =− 1 32 ∫ dtd4θ(X + g)Ψα̂Ψ̄α̂, g=const. (2.1) The Ψ supermultiplet is subjected to the irreducible conditions [5] DiΨ1 =0, DiΨ2 + DiΨ1 =0, DiΨ 2 =0, (2.2) and thus it contains four fermionic and four bosonic components ψα̂ =Ψα̂|, ui = −DiΨ̄2|, ūi = DiΨ1|, (2.3) where the symbol | denotes the θ = θ̄ = 0 limit and N =4 covariant derivatives obey standard relations{ Di, Dj } =2iδij ∂t. (2.4) It has been demonstrated in [11] that if the N = 4 superfield X is subjected to the constraints [29, 5] DiDiX =0, DiD iX =0, [ Di, Di ] X =0, (2.5) then the component action which follows from (2.1) can be written as Sc = ∫ dt [ −(x + g) ( ρ1ρ̄2 − ρ2ρ̄1 ) − i 4 (x + g) ( u̇iūi − ui ˙̄ui ) + 1 4 Aij u iūj + (2.6) 1 2 ηi ( ūiρ̄2 + uiρ2 ) + 1 2 η̄i ( uiρ 1 + ūiρ̄ 1 )] , where the new fermionic components ρα̂, ρ̄α̂ are de- fined as ρα̂ = ψ̇α̂, ρ̄α̂ = ψ̇α̂. (2.7) The components of the superfield X entering the ac- tion (2.6) have been introduced as x = X|, Aij = A(ij) = 1 2 [ Di, Dj ] X|, ηi = −iDiX|, η̄i = −iDiX|. (2.8) What makes the action (2.6) interesting is that, de- spite the non-local definition of the spinors ρα̂, ρ̄α̂ (2.7), the action is invariant under the following N =4 supersymmetry transformations: δρ1 = − ̄i ˙̄ui, δρ2 = i ˙̄ui, δui = −2i iρ̄1 +2i ̄iρ̄2, δūi = −2i iρ1 +2i ̄iρ2, δx = −i iηi − i ̄iη̄i, δηi = − ̄iẋ − i ̄j Aij , δη̄i = − iẋ + i jAji , δAij = − (iη̇j) + ̄(i ˙̄ηj). (2.9) In the action (2.6) the fermionic fields ρα̂, ρ̄α̂ are aux- iliary ones, and thus they can be eliminated by their equations of motion ρ1 = 1 2(x + g) ηiū i, ρ2 = − 1 2(x + g) η̄iūi. (2.10) Finally, the actiondescribing the interactionofΨand X supermultiplets acquires a very simple form Sc = 1 4 ∫ dt [ −i(x + g) ( u̇iūi − ui ˙̄ui ) + Aij u iūj + 1 x + g ηiη̄j ( uiūj + ujūi )] . (2.11) Thus, in the fermionic superfieldsΨ only the bosonic components ui, ūi, entering the action with a kinetic term linear in time-derivatives, survive. After quan- tization, these variables become purely internal de- grees of freedom. In order to be meaningful, action (2.1) has to be extended by the action for the supermultiplet X it- self. If the superfield X obeying (2.5) is considered as an independent superfield, then the most general action reads S = Sx + Sc = − 1 32 ∫ dtd4θF(X)+ Sc, (2.12) where F(X) is an arbitrary function of X. In this case the components Aij (2.8) are auxiliary ones, and they have to be eliminated by their equations of mo- tion. The resulting action describes N = 4 super- symmetric mechanics with one physical boson x and four physical fermions ηi, η̄j interacting with isospin variables ui, ūi. This is the system that has been considered in [8, 10, 11]. It is clear that treating the scalar bosonic super- field X as an independent one is too restrictive, be- cause the constraints (2.5) leave in this supermulti- plet only one physical bosonic component x, which is not enough to describe the isospin particle. In the present approach, a way to overcome this limi- tation was proposed in [14, 17]. The key point is to treat superfield X as a composite one, constructed from N = 4 supermultiplets with a larger number of physical bosons. The two reasonable superfields 14 Acta Polytechnica Vol. 51 No. 1/2011 fromwhich it is possible to construct superfield X are N = 4 tensor supermultiplet Vij [27, 28] and a one- dimensional hypermultiplet Qiα [18, 19, 20, 21, 7]. Tensor supermultiplet The N =4 tensor supermultiplet is described by the triplet of bosonic N = 4 superfields Vij = Vij sub- jected to the constraints D(iVjk) = D(iVjk) =0, ( Vij )† = Vij , (2.13) which leave in Vij the following independent compo- nents: va = − i 2 (σa)i jVij|, λ i = 1 3 DjVij|, (2.14) λ̄i = 1 3 DjVji |, A = i 6 DiDjVij|. Thus, its off-shell component field content is (3,4,1), i.e. three physical va and one auxiliary A bosons and four fermions λi, λ̄i [27, 28]. Under N =4 supersym- metry these components transform as follows: δva = i i(σa)ji λ̄j − iλ i(σa)ji ̄j , δA = ̄iλ̇ i − i ˙̄λi, δλi = i iA + j(σa)ij v̇ a, (2.15) δλ̄i = −i ̄iA +(σa)ji ̄j v̇ a. Now one may check that the composite superfield X = 1 |V| ≡ 1 √ VaVa , (2.16) where Va = − i 2 (σa)i jVij, obeys (2.5) in virtue of (2.13). Clearly, now all components of the X super- field, i.e. the physical boson x, fermions ηi, η̄i and auxiliary fields Aij (2.8) are expressed through the components of the Vij supermultiplet (2.14) as x = 1 |v| , ηi = va |v|3 (λσa)i, η̄i = va |v|3 (σaλ̄)i, Aij = −3 vavb |v|5 (λσa)i(σbλ̄)j − va(σa)ij |v|3 A + (2.17) 1 |v|3 abcvav̇b(σc)ij + 1 |v|3 ( δij λ kλ̄k − λj λ̄i ) . In what follows, we will also need the expression for Aij components (2.17) in terms of η i, η̄i fermions (2.8), which reads Aij = − va(σa)ij |v|3 A + 1 |v|3 abcvav̇b(σc)ij − (2.18) |v| ( ηiη̄j + ηj η̄ i ) − 1 |v| va(ησaη̄) vb(σb)ij . Finally, one should note that, while dealing with the tensor supermultiplet Vij, onemay generalize the Sx action (2.12) to have the full action in the form S = Sv + Sc = − 1 32 ∫ dtd4θF(V)+ Sc, (2.19) whereF(V) is nowanarbitrary functionofVij. After eliminating the auxiliary component A in the compo- nent form of (2.19)wewill obtain the action describ- ing the N = 4 supersymmetric three-dimensional isospin particle moving in the magnetic field of a Wu-Yang monopole and in some specific scalar po- tential [14]1. Hypermultiplet Similarly to the tensor supermultiplet one may con- struct the superfield X starting from the N =4 hy- permultiplet. The N = 4 , d = 1 hypermultiplet is described in N =4 superspace by the quartet of real N =4 superfields Qiα subjected to the constraints D(iQj)α = 0, D(iQj)α =0,( Qiα )† = Qiα. (2.20) This supermultiplet describes four physical bosonic and four physical fermionic variables qiα = Qiα|, ηi = −iDi ( 2 QjαQjα ) |, η̄i = −iDi ( 2 QjαQjα ) |, (2.21) and it does not contain any auxiliary components [7, 18, 19, 20, 21]. One may easily check that if we define the com- posite superfield X as X = 2 QiαQiα , (2.22) then it will obey (2.5) in virtue of (2.20) [5]. For the hypermultiplet Qiα we defined the fermionic compo- nents to coincide with those present in the X super- field (2.8),while the former auxiliary components Aij are now expressed via the components of Qiα as Aij = − 4i (qkβ qkβ)2 ( q̇αi qjα + q̇ α j qiα ) − (qkβ qkβ) 2 (ηiη̄j + ηj η̄i) . (2.23) As in the case of the tensor supermultiplet, one may write the full action with the hypermultiplet self- interacting part Sq added as S = Sq + Sc = − 1 32 ∫ dtd4θF(Q)+ Sc, (2.24) 1An alternative description of the same system has recently been constructed in [16] 15 Acta Polytechnica Vol. 51 No. 1/2011 where now F(Q) is an arbitrary function of Qiα. The action (2.24) describes the motion of an isospin par- ticle on a conformally flat four-manifold carrying the non-Abelian field of a BPST instanton [17]. This systemhas recently been obtained in different frame- works in [13, 15]. To close this Section one should mention that, while dealing with the tensor supermultiplet Vij and the hypermultiplet Qiα, the structure of the action Sc (2.1) can be generalized to be [17] Sc = − 1 32 ∫ dtd4θ Y Ψα̂Ψ̄α̂, (2.25) with Y obeying �3Y =0 in case of the tensor supermultiplet �4Y =0 in case of the hypermultiplet. (2.26) Here �n denotes the Laplace operator in a flat Eu- clidean n-dimensional space. Clearly, our choice Y = X + g with X defined in (2.16), (2.22) corre- sponds to spherically-symmetric solutions of (2.26). 3 From the hypermultiplet to the tensor supermultiplet and back Oneof themost attractive features of our approach is the unified structure of the action Sc (2.1) which has the same form for any type of supermultiplets that we are using to construct a composite superfield X. This iswhat opens theway to relate the different sys- tems via duality transformations. Indeed, it has been known for a long time [1, 2, 3, 4, 22, 23] that in one dimension one may switch between supermultiplets with a different number of physical bosons, by ex- pressing the auxiliary components through the time derivative of physical bosons, and vice versa. Here we will use this mechanism to obtain the action of the tensor multiplet (2.19) from the hypermultiplet (2.24) action and then, alternatively, action (2.24) (with some restrictions) from (2.19). Inwhat follows, to make some expressions more transparent, we will use, sometimes, the following stereographic coordi- nates for the bosonic components of hypermultiplet (2.21) and tensor supermultiplet (2.14): q11 = e 1 2(u−iφ)√ 1+ΛΛ Λ, q21 = − e 1 2(u−iφ)√ 1+ΛΛ , q22 = ( q11 )† , q21 = − ( q12 )† , (3.1) V 11 = 2i eu 1+ΛΛ Λ, V 22 = −2i eu 1+ΛΛ Λ, V 12 = −ieu ( 1−ΛΛ 1+ΛΛ ) . (3.2) One may easily check that these definitions are com- patible with (2.16) and (2.22). From hypermultiplet to tensor supermultiplet Themain ingredient for getting the tensor supermul- tiplet action from the hypermultiplet one is provided by the expression for “auxiliary” components Aij in termsof the components of superfields Vij andQiα in (2.18) and (2.23), respectively. Identifying the right hand sides of (2.18) and (2.23), one may find the ex- pression of the auxiliary component A present in the superfield Vij in terms of components of Qiα: A = i ( q̇i1q2i + q̇ i2q1i ) + 1 4 (qkαqkα) 2 qi1qj2 (ηiη̄j + ηj η̄i) . (3.3) Another way, and probably the easiest one, to check the validity of (3.3) is to use the following superfield representation for the tensor supermultiplet [7]: Vij = i ( Qi1Qj2 +Qj1Qi2 ) . (3.4) This “composite” superfield Vij automatically obeys (2.13) as a consequence of (2.20). Being partially rewritten in terms of components (3.1), expression (3.3) reads φ̇ = e−uA − i Λ̇Λ−ΛΛ̇ 1+ΛΛ − (3.5) 1 4 e−u(qkαqkα) 2 qi1qj2 (ηiη̄j + ηj η̄i) . Thus, we see that, in order to get the action for the tensor supermultiplet, onehas to replace, in the com- ponentaction for thehypermultiplet, the timederiva- tive of field φ by the combination on the r.h.s. of (3.5), which includes the new auxiliary field A. An additional restriction comes from the Sq part of the action (2.24), which now has to depend only on the “composite” superfield Vij (3.4). If it is so, then in the full action (2.24) component φ will enter only through φ̇, and the discussed replacement will be valid. From the tensor supermultiplet to the hypermultiplet It is clear that the backward procedure also exists. Indeed, from (2.17) and (2.23) one may get the fol- lowing expression for A: A = 1 f [ φ̇ + ∂ ∂va f(λσaλ̄)− Bav̇a ] , (3.6) where f = 1 |v| , and , B1 = − v2(v3 + |v|) (v21 + v 2 2)|v| , B2 = v1(v3 + |v|) (v21 + v 2 2)|v| , B3 =0. (3.7) 16 Acta Polytechnica Vol. 51 No. 1/2011 It is easy to check that in the coordinates (2.14), (3.2) we have Bav̇a = −i Λ̇Λ−ΛΛ̇ 1+ΛΛ and |v| = eu (3.8) in full agreement with (3.5). Thus, to get the hy- permultiplet action (2.24) from that for the tensor supermultiplet (2.19), one has to dualize the auxil- iary component A into a new physical boson φ using (3.6). Of course, we do not expect to get the most general action for thehypermultiplet interactingwith the isospin-containing supermultiplet Ψ, because the Sv part in (2.19) depends only on the Vij supermul- tiplet. But we will surely get a particular class of hypermultiplet actions with one isometry, with the Killing vector ∂φ. 4 Hyper-Kähler sigma model with isospin variables The considerationwe carried out in the previous Sec- tion has one subtle point. Indeed, if we rewrite (3.7) as φ̇ = Bav̇a − f,a(λσaλ̄)+ f A, f,a ≡ ∂ ∂va f, (4.1) then the r.h.s. of (4.1) has to transform as a full time derivative under supersymmetry transforma- tions (2.15). One may check that it is so, if f and Ba are chosen as in (3.7). However, this choice is not unique. It has been proved in [24] that the r.h.s. of (4.1) transforms as a full time derivative, if the functions f and Ba satisfy the equations �3f ≡ f,aa =0, f,a = abcBc,b. (4.2) Thus, one may construct a more general action for four-dimensional N = 4 supersymmetric mechanics using the component action for the tensor supermul- tiplet and substituting there thenewdualizedversion of the auxiliary component A (4.1). Integrating over theta’s in (2.19) and eliminating the auxiliary fermions ρα̂ (2.10), (2.17), we will get the following component action for the tensor super- multiplet: S = 1 8 ∫ dt [ F ( v̇av̇a + A 2 ) +i ( ξ̇iξ̄i − ξi ˙̄ξi ) + i abc F,a F v̇bΣc − i F,a F ΣaA − 1 6 �3F F2 ΣaΣa − 2i ( ẇiw̄i − wiw̄i ) + 4 1+3g|v| F(1+ g|v|)2|v|4 (vaIa)(vbΣb)− 4 g F(1+ g|v|)2|v| (IaΣa)− 4i (1+ g|v|)|v|2 (vaIa) A + 4i (1+ g|v|)|v|2 abcvav̇bIc ] , (4.3) where F = �3 F(V)|, Ia = i 2 (wσaw̄) , (4.4) Σa = −i ( ξσaξ̄ ) , and the re-scaled fermions and isospin variables are chosen to be ξi = √ F λi, wi = √ g + 1 |v| ui. (4.5) Substituting (4.1) into (4.3), we obtain the resulting action S = 1 8 ∫ dt [ F ( v̇av̇a + 1 f2 ( φ̇ − Bav̇a )2) + i ( ξ̇iξ̄i − ξi ˙̄ξi ) −2i ( ẇiw̄i − wiw̄i ) − i [ 1 f δab ( φ̇ − Bcv̇c ) + abcv̇c ] ·( F,a F Σb + 4 (1+ g|v|)|v|2 vaIb ) + 4 F 1+3g|v| (1+ g|v|)2|v|4 (vaIa)(vbΣb)− 1 F 4g (1+ g|v|)2|v| (IaΣa)+ (4.6) 1 3F2 ( F,af,a f − F f,af,a f2 − 1 2 �3F ) ΣbΣb ] . Action (4.6) is our main result. It describes the mo- tion of a N = 4 supersymmetric four-dimensional isospin carrying particle in a non-Abelian field of some monopole. The metric of this four-dimensional space isdefined in termsof two functions: thebosonic part of our pre-potential F (4.4) and the harmonic function f (4.2). The supersymmetric version of the coupling with the monopole (second line in action (4.6)) is defined by the same harmonic function f and the coupling constant g. In the more general case (2.25), we will have two harmonic functions — f and Y , besides the pre-potential F . Among all possible systems with action (4.6) there is a very interesting sub-class which corre- sponds to hyper-Kähler sigma models in the bosonic sector. This case is distinguished by the condition F = f. (4.7) Clearly, in this case the bosonic kinetic term of ac- tion (4.6)acquires the familiar formof the onedimen- sional version of the general Hawking-Gibbons solu- tion for four-dimensional hyper-Kähler metrics with 17 Acta Polytechnica Vol. 51 No. 1/2011 one triholomorphic isometry [30]: Skin = 1 8 ∫ dt [ f v̇av̇a + 1 f ( φ̇ − Bav̇a )2] , �3f = 0, rot �B = �∇f. (4.8) It is worth noting that the bosonic part of N =4 su- persymmetric four dimensional sigma models in one dimension does not necessarily have to be a hyper- Kähler one. This fact is reflected in the arbitrariness of the pre-potential F in action (4.6). Only under the choice F = f is the bosonic kinetic term reduced to theGibbons-Hawking form (4.8). Let us note that for hyper-Kähler cases the four-fermionic term in ac- tion (4.6) disappears. This fact has been previously established in [24]. Now we can see that the addi- tional interaction with the background non-Abelian gauge field does not destroy these nice properties. Among all possible bosonic metrics one may eas- ily find the following interesting ones. Conformally flat spaces There are two choices for the function f which corre- spond to the conformally flat metrics in the bosonic sector. The first choice is realized by f = 1 |v| . (4.9) This is just the casewe have considered in Section 2. The gauge field in this case is the field of BPST in- stanton [17]. Next, an almost trivial solution, also correspond- ing to theflatmetrics in thebosonic sector, is selected by the condition f =const., Ba =0. (4.10) Note that the relation with the tensor supermulti- plet, in this case, is achieved through the following “composite” construction of Vij [31] Vij = Q(iα). (4.11) Onemay check that the constraints on Vij (2.13) fol- low directly from (4.11) and (2.20). Let us recall that in both these cases we have not specified the pre-potential F yet. Therefore, the full metrics in the bosonic sector is defined up to this function. Taub-NUT space One should stress that the previous two cases are unique, because only for these choices of f can the resulting action (4.6) be obtained directly from the hypermultiplet action (2.24). With other solutions for f wecome to the theorywith thenonlinear N =4 hypermultiplet [24, 25]. Among the possible solu- tions for f which belong to this new situation, the simplest one corresponds to one center Taub-NUT metrics with f = p1 + p2 |v| , p1, p2 =const. (4.12) In order to achieve the maximally symmetric case, we will choose these constants as p1 = g, p2 =1 → f = g + 1 |v| . (4.13) With such a definition, f coincides with the function Y = g + 1 |v| (2.25) entering in our basic action Sc in (2.1), (2.16). To get the Taub-NUT metrics, one has also to fix the pre-potential F to be equal to f. The resulting action which describes the N = 4 su- persymmetric isospin carrying particle moving in a Taub-NUT space reads ST aub−N UT = 1 8 ∫ dt [( g + 1 |v| ) v̇av̇a + 1( g + 1|v| ) (φ̇ − Bav̇a)2 +i(ξ̇iξ̄i − ξi ˙̄ξi) − 2i ( ẇiw̄i − wiw̄i ) + i (1+ g|v|)|v|2 ·⎡ ⎣ va( g + 1|v| ) (φ̇ − Bcv̇c) − abcvbv̇c ⎤ ⎦(Σa −4Ia)+ 4(1+3g|v|) (1+ g|v|)3|v|3 (vaIa)(vbΣb)− 4g (1+ g|v|)3 (IaΣa) ] . (4.14) Thebosonic term in the second line of this action can be rewritten as AaIa = i 2 [ 1 f ∂ logf ∂va ( φ̇ − Bav̇a ) − abc ∂ logf ∂vb v̇c ] Ia, (4.15) where f is defined in (4.13). In this form the vec- tor potential Aa coincides with the potential of a Yang-Mills SU(2) instanton in the Taub-NUT space [32, 33], if we may view Ia, as defined in (4.4), as proper isospin matrices. The remaining terms in the second and third lines of (4.14) provide a N = 4 supersymmetric extension of the instanton. Finally, to close this Section, let us note that more general non-Abelian backgrounds can be ob- tained from the multi-centered solutions of the equa- tion for the harmonic function Y (2.26), which de- fined the coupling of the tensor supermultiplet with 18 Acta Polytechnica Vol. 51 No. 1/2011 auxiliary fermionic ones. Thus, thevarietymodelswe constructed are defined through three functions: pre- potential F (2.19)which is an arbitrary function, 3D harmonic function Y (2.25), (2.26) defining the cou- pling with isospin variables and, through the again 3D harmonic function f (4.1), (4.2), which appeared during the dualization of the auxiliary component of the tensor supermultiplet. It is clear that we can always redefine F to be F = F̃ f. Thus, all our mod- els are conformal to hyper-Kähler sigmamodels with N = 4 supersymmetry describing the motion of a particle in the background non-Abelian field of the corresponding instantons. 5 Conclusion In this paperwehaveconstructed theLagrangian for- mulation of N = 4 supersymmetric mechanics with hyper-Kähler sigma models in the bosonic sector in the non-Abelian background gauge field. The result- ing action includes thewide class of N =4supersym- metricmechanicsdescribing themotionof an isospin- carrying particle over spaces with non-trivial geom- etry. In two examples that we discussed in detail, the background fields are identified with the field of BPST instantons in the flat and Taub-NUT spaces. The approach used in the paper has utilized two ideas: (i) the coupling ofmatter supermultipletswith an auxiliary fermionic supermultiplet Ψα̂ contain- ing on-shell four physical fermions and four auxil- iary bosons playing the role of isospin variables, and (ii) the dualization of the auxiliary component A of the tensor supermultiplet into a fourth physical bo- son. The final action that we constructed contains three arbitrary functions: the pre-potential F, a 3D harmonic function Y which defines the couplingwith isospin variables and, again 3D harmonic, a function f which appeared during the dualization of the aux- iliary component of the tensor supermultiplet. The usefulness of the proposed approach is demonstrated by the explicit example of the simplest system with non-trivial geometry — the N = 4 supersymmetric action for one-center Taub-NUT metrics. We identi- fied the background gauge field in this case, which appears automatically in our framework, with the field of the BPST instanton in the Taub-NUT space. Thus, one may hope that the other actions will pos- sess the same structure. Of course, the presented results are just prelim- inary in the quest for full understanding of N = 4 supersymmetric hyper-Kähler sigma models in non- Abelian backgrounds. Interesting questions that re- main unanswered include, in particular: • The full analysis of the general coupling with an arbitrary harmonic function Y has yet to be carried out. • The structure of the background gauge field has to be further clarified: is this really the field of some monopole (instanton) for some hyper- Kähler metrics? • The Hamiltonian construction is really needed. Let us note that the Supercharges have to be very specific, because the four-fermions coupling is absent in the case of HK metrics! • It is quite interesting to check the existence of the conservedRunge-Lenz vector in the fully su- persymmetric version. • Explicit examples of other hyper-Kählermetrics (say, multi-centered Eguchi-Hanson and Taub- NUT ones) would be very useful. • Questions of quantization and analysis of the spectra, at least in the cases of well known, sim- plest hyper-Kählermetrics, are doubtless urgent tasks. Finally, let us stress that our construction is re- stricted to the case of hyper-Kählermetrics with one translational (triholomorphic) isometry. It will be very nice to find a similar construction applicable to the case of geometries with rotational isometry. We hope thismaybe donewithin the approachdiscussed in [34]. Acknowledgement WethankAndreyShcherbakov for useful discussions. 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Lett.B645 (2007) 299, arXiv:hep-th/0611248. Stefano Bellucci INFN – Frascati National Laboratories Via E. Fermi 40, 00044 Frascati, Italy Sergey Krivonos Bogoliubov Laboratory of Theoretical Physics JINR, 141980 Dubna, Russia Anton Sutulin E-mail: sutulin@theor.jinr.ru Bogoliubov Laboratory of Theoretical Physics JINR, 141980 Dubna, Russia 21