wykresx.eps Acta Polytechnica Vol. 51 No. 1/2011 Five-dimensional N =4 Supersymmetric Mechanics S. Bellucci, S. Krivonos, A. Sutulin Abstract We perform an su(2) Hamiltonian reduction in the bosonic sector of the su(2)-invariant action for two free (4,4,0) supermultiplets. As a result, we get the five dimensional N =4 supersymmetric mechanics describing the motion of an isospin carrying particle interacting with a Yang monopole. Some possible generalizations of the action to the cases of systems with a more general bosonic action constructed with the help of ordinary and twisted N = 4 hypermultiplets are considered. Keywords: supersymmetric mechanics, Hamiltonian reduction, non-Abelian gauge fields. 1 Introduction The supersymmetric mechanics describing the mo- tion of an isospinparticle in backgroundnon-Abelian gauge fields has attracted a lot of attention in the last few years [1, 2, 3, 4, 5, 6, 7, 8, 9], especially due to its close relationwith higher dimensional Hall effects and their extensions [10], as well as with the supersymmetric versions of various Hopf maps (see e.g. [1]). The key point of any possible con- struction is to find a proper realization for semi- dynamical isospin variables, which have to be in- vented for the description of monopole-type inter- actions in Lagrangian mechanics. In supersymmet- ric systems these isospin variables should belong to some supermultiplet, and the main question is what to do with additional fermionic components accom- panying the isospin variables. In [3] fermions of such a kind, together with isospin variables, span an aux- iliary (4,4,0) multiplet with a Wess-Zumino type action possessing an extra U(1) gauge symmetry1. In this framework, an off-shell Lagrangian formu- lation was constructed, with the harmonic super- space approach [11, 12], for a particular class of four- dimensional [3] and three-dimensional [5] N =4 me- chanics, with a self-dual non-Abelian background. The same idea of coupling with an auxiliary semi- dynamical supermultiplet has also been elaborated in [6] within the standard N = 4 superspace frame- work, and then it has been applied for the construc- tion of Lagrangian and Hamiltonian formulations of the N = 4 supersymmetric system describing the motion of the isospin particles in three [7] and four- dimensional [8] conformally flat manifolds, carrying the non-Abelian fields of theWu-Yangmonopole and the BPST instanton, respectively. In both these approaches the additional fermions were completely auxiliary, and they were expressed through the physical ones on themass shell. Another approachbased on the direct use of the SU(2) reduc- tion, firstly considered on the Lagrangian level in the purelybosonic case in [1], hasbeenused in the super- symmetric case in [9]. The key idea of this approach is to perform a direct su(2) Hamiltonian reduction in the bosonic sector of the N = 4 supersymmetric system, with the general SU(2) invariant action for a self-coupled (4,4,0) supermultiplet. No auxiliary superfields are needed within such an approach, and the procedure itself is remarkably simple and auto- matically successful. As concerning the interaction with the non- Abelianbackground, the systemconsidered in [9]was not too illuminating, due to its small number (only one) of physical bosons. In the present Letter, we ex- tend the construction of [9] to the case of the N =4 supersymmetric systemwithfive (and four in the spe- cial case) physical bosonic components. It is not, therefore, strange that the arising non-Abelian back- groundcoincideswith thefieldof aYangmonopole (a BPST instanton field in the four dimensional case). A very important preliminary step, discussed in details in Section 2, is to pass to new bosonic and fermionic variables, which are inert under the SU(2) group, over which we perform the reduction. Thus, the SU(2) group rotates only the three bosonic com- ponents, which enter the action through SU(2) in- variant currents. Just these bosonic fields become the isospin variableswhich the background field cou- ples to. Due to the commutativity of N = 4 super- symmetry with the reduction SU(2) group, it sur- vives upon reduction. In Section 3, we consider some possible generalizations,which include a systemwith more general bosonic action, a four-dimensional sys- tem which still includes eight fermionic components, and the variant of five-dimensional N = 4 mechan- ics constructedwith the help of ordinary and twisted N = 4 hypermultiplets. Finally, in the Conclusion we discuss some unsolved problems and possible ex- tensions of the present construction. 1Note that the first implementation of this idea was proposed in [4] 22 Acta Polytechnica Vol. 51 No. 1/2011 2 (8B,8F) → (5B,8F) reduction and the Yang monopole Asthefirstnontrivial exampleof the SU(2) reduction in N =4 supersymmetric mechanics we consider the reduction from the eight-dimensional bosonic mani- fold to the five dimensional one. To start with, let us choose our basic N = 4 superfields to be the two quartets of real N = 4 superfields Qiα̂A (with i, α̂, A = 1,2) defined in the N = 4 superspace R(1|4) =(t, θia) and subjected to the constraints D(iaQj)α̂A =0, and ( Qiα̂A )† = Qiα̂A, (2.1) where the corresponding covariant derivatives have the form Dia = ∂ ∂θia +iθia∂t, so that{ Dia, Djb } = 2i ij ab∂t. (2.2) These constrained superfields describe the ordinary N = 4 hypermultiplet with four bosonic and four fermionic variables off-shell [12, 13, 14, 15, 16, 17]. The most general action for Qiα̂A superfields is constructed by integrating an arbitrary superfunc- tion F(Qiα̂A ) over thewhole N =4 superspace. Here, we restrict ourselves to the simplest prepotential of the form2 F(Qiα̂A ) = Q iα̂ A Qiα̂A −→ S = ∫ dtd4θ Qiα̂A Qiα̂A. (2.3) The rationale for this selection is, first of all, itsman- ifest invariance under su(2) transformations acting on the “α̂” index of Qiα̂. This is the symmetry over which we are going to perform the su(2) reduction. Secondly, just this form of the prepotential guaran- tees SO(5) symmetry in the bosonic sector after re- duction. In terms of components, the action (2.3) reads S = ∫ dt [ Q̇iα̂A Q̇iα̂A − i 8 Ψ̇aα̂A Ψaα̂A ] (2.4) where the bosonic and fermionic components are de- fined as Qiα̂A = Q iα̂ A | , Ψ aα̂ A = D iaQα̂iA| , (2.5) and, asusually, (. . .)|denotes the θia =0 limit. Thus, from the beginning we have just the sum of two in- dependent non-interacting (4,4,0) supermultiplets. To proceed further, we introduce the following bosonic qiαA and fermionic ψ aα A fields qiαA ≡ Q i Aα̂G αα̂, ψaαA ≡ Ψ a α̂AG αα̂, (2.6) where the bosonic variables Gαα̂, subjected to Gαα̂Gαα̂ =2, are chosen as G11 = e− i 2 φ√ 1+ΛΛ Λ, G21 = − e− i 2 φ√ 1+ΛΛ , G22 = ( G11 )† , G12 = − ( G21 )† . (2.7) The variables Gαα̂ play the role of a bridge relating the twodifferent SU(2) groups realizedon the indices α and α̂, respectively. In terms of the variables given above, the action (2.4) acquires the form S = ∫ dt [ q̇iαA q̇iαA −2q iα A q̇ β iAJαβ + qiαA qiαA 2 J βγ Jβγ − i 8 ψ̇aαA ψaαA + (2.8) i 8 ψaαA ψ β aAJαβ ] , where J αβ = J βα = Gαα̂Ġβα̂. (2.9) As follows from (2.6), the variables qiαA and ψ aα A , which, clearly, contain five independent bosonic and eight fermionic components, are inert under su(2) ro- tations acting on α̂ indices. Under these su(2) rota- tions, realized now only on Gαα̂ variables in a stan- dard way δGαα̂ = γ(α̂β̂)Gα β̂ , (2.10) the fields (φ,Λ,Λ̄) (2.7) transform as [17] δΛ = γ11eiφ(1+ΛΛ), δΛ = γ22e−iφ(1+ΛΛ), (2.11) δφ = −2iγ12 +iγ22e−iφΛ− iγ11eiφΛ. It is easy to check that the forms J αβ (2.9), express- ing in terms of the fields (φ,Λ,Λ̄), J11 = − Λ̇− iΛφ̇ 1+ΛΛ , J22 = − Λ̇+ iΛφ̇ 1+ΛΛ , J12 = −i 1−ΛΛ 1+ΛΛ φ̇ − Λ̇Λ−ΛΛ̇ 1+ΛΛ (2.12) are invariant under (2.11). Hence, the action (2.8) is invariant under the transformations in (2.10). Next, we introduce the standardPoissonbrackets for bosonic fields {π,Λ} =1, { π̄,Λ } =1, {pφ, φ} =1, (2.13) so that the generators of the transformations (2.11), 2We used the following definition of the superspace measure: d4θ ≡ − 1 96 DiaDibD bj Dja. 23 Acta Polytechnica Vol. 51 No. 1/2011 Iφ = pφ, I =e iφ [ (1+ΛΛ)π − iΛpφ ] , Ī = e−iφ [ (1+ΛΛ) π̄ +iΛpφ ] , (2.14) will be the Noether constants of motion for the ac- tion (2.8). To perform the reduction over this SU(2) group we fix the Noether constants as (c.f. [1]) Iφ = m and I = Ī =0, (2.15) which yields pφ = m and π = imΛ 1+ΛΛ , π̄ = − imΛ 1+ΛΛ . (2.16) Performing a Routh transformation over the vari- ables (Λ,Λ, φ), we reduce the action (2.8) to S̃ = S − ∫ dt { π Λ̇+ π̄ Λ̇+ pφφ̇ } (2.17) and substitute the expressions (2.16) in S̃. At the final step, we have to choose the proper parametriza- tion for bosonic components qiαA (2.6), taking into account that they contain only five independent vari- ables. Following [1] we will choose these variables as qiα1 = 1 2 iα √ r + z5, qiα2 = 1√ 2(r + z5) ( x(iα) − 1 √ 2 iαz4 ) , (2.18) where x12 = i√ 2 z3, x 11 = 1√ 2 (z1 + iz2) , x22 = 1√ 2 (z1 − iz2) , r2 = 5∑ i=1 zizi , (2.19) and now the five independent fields are zm. Slightly lengthy but straightforward calculations lead to Sred = ∫ dt [ 1 4r żmżm − i 8 ψ̇aαA ψaαA + i 4r HαβVαβ + 1 128r Hαβ Hαβ − m2 r − m 4r vαv̄β Hαβ − (2.20) 4im r vαv̄β Vαβ + im ( v̇αv̄α − vα ˙̄vα )] . Here Hαβ = ψaαA ψ β aA, v α = Gα1, v̄α = Gα2, vαv̄α =1, (2.21) and to ensure that the reduction constraints (2.16) are satisfiedweaddedLagrangemultiplier terms (the last two terms in (2.20)). Finally, the variables V αβ in the action (2.20) are defined in a rather symmetric way to be V αβ = 1 2 ( qiαA q̇ β iA + q iβ A q̇ α iA ) . (2.22) To clarify the relations of these variables with the potential of the Yang monopole, one has to intro- duce the following isospin currents (which will form an su(2) algebra upon quantization) T I = vα ( σI )β α v̄β , I =1,2,3. (2.23) Now, the (vαv̄β)-dependent terms in theaction (2.20) can be rewritten as − m 4r vαv̄β Hαβ − 4im r vαv̄β Vαβ = m T I ( 1 r(r + z5) ηIμν zμżν + 1 8r HI ) , (2.24) μ, ν =1,2,3,4, where ηIμν = δ I μδν4 − δ I ν δμ4 + A μν4 (2.25) is the self-dual t’Hooft symbol and the fermionic spin currents are introduced HI = Hαβ ( σI )β α . (2.26) Thus we conclude that the action (2.20) describes N = 4 supersymmetric five-dimensional isospin par- ticles moving in the field of the Yang monopole Aμ = − 1 r(r + z5) ηIμν zν T I . (2.27) We stress that the su(2) reduction algebra, realized in (2.11), commutes with all (super)symmetries of the action (2.4). Therefore, all symmetry properties of the theory are preserved in our reduction and the final action (2.20) represents the N = 4 supersym- metric extension of the system presented in [1]. With this, we have completed the classical de- scription of N = 4 five-dimensional supersymmetric mechanics describing the isospin particle interacting with a Yang monopole. Next, we analyze some pos- sible extensions of the present system, together with some possible interesting special cases. In what fol- lows we will concentrate on the bosonic sector only, while the full supersymmetric action could be easily reconstructed, if needed. 3 Generalizations, and cases of special interest Let us consider more general systems with a more complicated structure in the bosonic sector. We will 24 Acta Polytechnica Vol. 51 No. 1/2011 concentrate on the bosonic sector only, while the full supersymmetric action could be easily reconstructed. 3.1 SO(4) invariant systems Our first example is the most general system, which still possesses SO(4) symmetry upon SU(2) reduc- tion. It is specified by the prepotential F (2.3) de- pending on two scalars X and Y F = F(X, Y ), X = Qiα̂1 Q1 iα̂, (3.1) Y = Qiα̂2 Q2 iα̂. Such a system is invariant under SU(2) transforma- tions realized on the “hatted” indices α̂ and thus the SU(2) reduction that we discussed in the Sec- tion 2 goes in the same manner. In addition, the full SU(2) × SU(2) symmetry realized on the superfield Qiα̂2 will survive in the reduction process. So we ex- pected the final system to possess SO(4) symmetry. The bosonic sector of the system with prepoten- tial (3.1) is described by the action S = ∫ dt [( Fx + 1 2 xFxx ) Q̇iα̂1 Q̇1 iα̂ +( Fy + 1 2 yFyy ) Q̇iα̂2 Q̇2 iα̂ + (3.2) 2FxyQ jβ̂ 2 Q1 jα̂Q̇2 iβ̂ Q̇ iα̂ 1 ] . Even with such a simple prepotential, the bosonic action (3.2) after reduction has a rather complicated form. A further, stillmeaningful simplification, could be achieved with the following prepotential F = F(X, Y )= F1(X)+F2(Y ), (3.3) where F1(X) and F2(Y ) are arbitrary functions de- pending on X and Y , respectively. With such a pre- potential the third term in theaction(3.2)disappears and the action acquires a readable form. With our notations (2.18), (2.19) the reduced action reads S = ∫ dt [ HxHy 2((Hx − Hy)z5 +(Hx + Hy)r) żμżμ + (Hx − Hy)2 8r2 ((Hx − Hy)z5 +(Hx + Hy)r) (zμżμ) 2 + Hx − Hy 4r2 (zμżμ) ż5 + 1 8 ( Hx − Hy r2 z5 + Hx + Hy r ) ż25 + im ( v̇αv̄α − vα ˙̄vα ) − (3.4) 2m2 (Hx + Hy)r +(Hx − Hy)z5 − 8imHy (Hx + Hy)r +(Hx − Hy)z5 vαv̄β Vαβ ] , where Hx = F ′ 1(x)+ 1 2 xF ′′1 (x), Hy = F ′ 2(y)+ 1 2 yF ′′2 (y), (3.5) and x = 1 2 (r + z5) , y = 1 2 (r − z5) . (3.6) Let us stress that the unique possibility to have an SO(5) invariant bosonic sector is to choose Hx = Hy = const. This is just the case we considered in Section 2. With arbitrary potentials Hx and Hy we have a more general system with the action (3.4), describing the motion of the N = 4 supersymmet- ric particle in five dimensions and interacting with a Yang monopole and some specific potential. 3.2 Non-linear supermultiplet It has been known for a long time that in some spe- cial cases one could reduce the action for hypermul- tiplets to the action containing one fewer physical bosonic components — to the action of a so-called non-linear supermultiplet [12, 17, 18]. Themain idea of such a reduction is to replace of the time deriva- tive of the “radial” bosonic component of hypermul- tiplet Log(qiaqia) by anauxiliary component B with- out breaking the N =4 supersymmetry [19]: d dt Log(qiaqia) → B. (3.7) Clearly, to perform such a replacement in some ac- tion the “radial”bosonic component has to enter this action only with a time derivative. This condition strictly constraints the variety of the possible hyper- multiplet actions in which this reduction works. For performing the reduction from a hypermulti- plet to the non-linear one, parametrization (2.18) is not very useful. Instead, we choose the following pa- rameterizations for independent components of two hypermultiplets qiα1 and q iα 2 qiα1 = 1 √ 2 iαe 1 2 u, qiα2 = x (iα) − 1 √ 2 iαz4, (3.8) where x12 = i √ 2 z3, x 11 = 1 √ 2 (z1 + iz2) , x22 = 1 √ 2 (z1 − iz2) . (3.9) Thus, the five independent components are u and zμ (μ =1, . . . ,4), and x = q21 = e u, y = q22 = 4∑ μ=1 zμzμ ≡ r24. (3.10) 25 Acta Polytechnica Vol. 51 No. 1/2011 With this parametrization the action (3.4) acquires the form S = ∫ dt [ G1G2e u euG1 + G2r24 żμżμ + G22 euG1 + G2r24 (zμżμ) 2 + 1 4 G1e uu̇2 + im ( v̇αv̄α − vα ˙̄vα ) − m2 euG1 + G2r24 − 4imG2 euG1 + G2r24 vαv̄β Vαβ ] , (3.11) where G1 = G1(u)= F ′ 1(x)+ 1 2 xF ′′1 (x), G2 = G2(r4)= F ′ 2(y)+ 1 2 yF ′′2 (y). (3.12) If we choose G1 = e −u, then the “radial” bosonic component u will enter the action (3.11)only through the kinetic term ∼ u̇2. Thus, performing replacement (3.7) and excluding the auxiliary field B by its equa- tion of motion we will finish with the action S = ∫ dt [ G2 1+ G2r24 ( żμżμ + G2 (zμżμ) 2 ) + im ( v̇αv̄α − vα ˙̄vα ) − m2 1+ G2r24 − (3.13) 4imG2 1+ G2r24 vαv̄β Vαβ ] . The action (3.13) describes the motion of an isospin particle on a four-manifoldwith SO(4) isometry car- rying the non-Abelian field of a BPST instanton and somespecial potential. Ouraction is rather similar to those recently constructed in [3, 8, 2], but it contains twice more physical fermions. 3.3 Ordinary and twisted hypermultiplets One more way to generalize the results presented in the previous Section is to consider simultaneously the ordinary hypermultiplet Qjα̂ obeying (2.1) to- gether with twisted hypermultiplet Vaα̂ — a quartet of N =4 superfields subjected to constraints [17] Di(aVb)α̂ =0, and ( Vaα̂ )† = Vaα̂ . (3.14) Themost general systemwhich is explicitly invariant under SU(2) transformations realized on the “hat- ted” indices is defined, similarly to (3.1), by the su- perspace action depending on two scalars X, Y S = ∫ dtd4θF(X, Y ), X = Qiα̂Qiα̂, Y = Vaα̂Vaα̂. (3.15) The bosonic sector of the action (3.15) is a rather simple S = ∫ dt [( Fx + 1 2 xFxx ) Q̇iα̂Q̇iα̂ −( Fy + 1 2 yFyy ) V̇ aα̂V̇aα̂ ] . (3.16) Thus, we see that the term causes the most compli- cated structure of the action with two hypermulti- plets, which disappear in the case of ordinary and twisted hypermultiplets. Clearly, the bosonic action after SU(2) reduction will have the same form (3.4), but with Hx = Fx + 1 2 xFxx, Hy = − ( Fy + 1 2 yFyy ) . (3.17) Here F = F(x, y) is still a function of two variables x and y. The mostly symmetric situation again cor- responds to the choice Hx = Hy ≡ h(x, y) (3.18) with the action S = ∫ dt [ h 4r żmżm + im ( v̇αv̄α − vα ˙̄vα ) − m2 h r − 4im r vαv̄βVαβ ] . (3.19) Unfortunately, due to definition (3.17), (3.18) the metric h(x, y) cannot be chosen fully arbitrarily. For example, looking for an SO(5) invariant model with h = h(x + y) we could find only two solutions3 h1 = const., h2 =1/(x + y) 3. (3.20) Both solutions describe a cone-like geometry in the bosonic sector, while the most interesting case of the sphere S5 cannot be treated within the present ap- proach. Finally, we would like to draw attention to the fact that with h = const. the bosonic sectors of sys- temswith twohypermultiplets andwith oneordinary andonetwistedhypermultiplets coincide. This is just onemore justification for the claimthat“almost free” systems can be supersymmetrized in various ways. 3The same metric has been considered in [20]. 26 Acta Polytechnica Vol. 51 No. 1/2011 4 Conclusion In this paper, starting with the non-interacting sys- tem of two N = 4 hypermultiplets, we perform a reduction over the SU(2) group which commutes with supersymmetry. The resulting system describes the motion of an isospin carrying particle on a con- formally flat five-dimensional manifold in the non- Abelian field of a Yang monopole and in some scalar potential. Themost important step for this construc- tion passes to new bosonic and fermionic variables, which are inert under the SU(2) group over which weperformthe reduction. Thus, the SU(2) groupro- tates only three bosonic components, which enter the action through SU(2) invariant currents. Just these bosonic fields become the isospin variableswhich the background field couples to. Due to the commuta- tivity of N = 4 supersymmetry with the reduction SU(2) group, it survives upon reduction. We con- sidered some possible generalizations of the action to the cases of systems with a more general bosonic ac- tion, a four-dimensional system which still includes eight fermionic components, and a variant of five- dimensional N = 4 mechanics constructed with the help of ordinary and twisted N =4 hypermultiplets. The main advantage of the proposed approach is its applicability to any systemwhichpossesses SU(2) in- variance. If, in addition, this SU(2) commutes with supersymmetry, then the resulting system will auto- matically be supersymmetric. Possibledirectapplicationsofour construction in- clude a reduction in the cases of systems with non- linear N =4 supermultiplets [21], systemswithmore than two (non-linear) hypermultiplets, in systems with bigger supersymmetry, say for example N =8, etc. However, the most important case, which is still missing within our approach, is the construction of the N =4 supersymmetric particle on the sphere S5 in the field of a Yang monopole. Unfortunately, the use of standard linear hypermultiplets makes the so- lution of this task impossible, because the resulting bosonic manifolds have a different structure (conical geometry) to include S5. Acknowledgement We thank Armen Nersessian and Francesco Top- pan for useful discussions. This work was partially supported by grants RFBF-09-02-01209 and 09-02- 91349, by Volkswagen Foundation grant I/84 496, as well as by ERC Advanced Grant no. 226455, “Su- persymmetry, Quantum Gravity and Gauge Fields” (SUPERFIELDS). 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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 28