Five-dimensional N=4 supersymmetric mechanics

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 \Nf 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 the ordinary and twisted \Nf hypermultiplets are considered.

1 Introduction 2 (8 B , 8 F ) → (5 B , 8 F ) reduction and Yang monopole As the first nontrivial example of the SU (2) reduction in N = 4 supersymmetric mechanics we consider the reduction from the eight-dimensional bosonic manifold 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 Q iα A (with i,α, A = 1, 2) defined in the N = 4 superspace R (1|4) = (t, θ ia ) and subjected to the constraints where the corresponding covariant derivatives have the form so that D ia , D jb = 2iǫ ij ǫ 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 Q iα A superfields is constructed by integrating an arbitrary superfunction F (Q iα A ) over the whole N = 4 superspace. Here, we restrict ourselves to the simplest prepotential of the 3) The rationale for this selection is, first of all, its manifest invariance under su(2) transformations acting on the "α" index of Q iα . This is the symmetry over which we are going to perform the su(2) reduction. Secondly, just this form of the prepotential guarantees SO(5) symmetry in the bosonic sector after reduction. In terms of components the action (2.3) reads where the bosonic and fermionic components are defined as and, as usually, (. . .)| denotes the θ ia = 0 limit. Thus, from beginning we have just the sum of two independent non-interacting (4, 4, 0) supermultiplets.
To proceed further we introduce the following bosonic q iα A and fermionic ψ aα A fields where the bosonic variables G αα , subjected to G αα G αα = 2, are chosen as The variables G αα play the role of bridge relating two different SU (2) groups realized on the indices α andα, respectively. In terms of the given above variables the action (2.4) acquires the form As follows from (2.6), the variables q iα A and ψ aα A , which, clearly, contain five independent bosonic and eight fermionic components, are inert under su(2) rotations acting onα indices. Under these su(2) rotations, realized now only on G αα variables in a standard way the fields (φ, Λ,Λ) (2.7) transform as [17] It is easy to check that the forms J αβ (2.9), expressing in terms of the fields (φ, Λ,Λ), will be the Noether constants of motion for the action (2.8). To perform the reduction over this SU (2) group we fix the Noether constants as (c.f. [1]) Performing a Routh transformation over the variables (Λ, Λ, φ), we reduce the action (2.8) to and substitute the expressions (2.16) inS. At the final step, we have to choose the proper parametrization for bosonic components q iα A (2.6), taking into account that they contain only five independent variables. Following [1] we will choose these variables as and now the five independent fields are z m . A slightly lengthy but straightforward calculations lead to Here 21) and to ensure that the reduction constraints (2.16) are satisfied we added Lagrange multiplier 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 To clarify the relations of these variables with the potential of Yang monopole, one has to introduce the following isospin currents (which will form su(2) algebra upon quantization) Now, the (v αvβ )-dependent terms in the action (2.20) can be rewritten as is the self-dual t'Hooft symbol and the fermionic spin currents is introduced Thus we conclude, that the action (2.20) describes N = 4 supersymmetric five-dimensional isospin particles moving in the field of Yang monopole 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 supersymmetric extension of the system presented in [1]. With this, we completed the classical description of N = 4 five-dimensional supersymmetric mechanics describing the isospin particle interacting with a Yang monopole. Next, we analyze some possible extensions of the present system, together with some possible interesting special cases. In what follows we will concentrate on the bosonic sector only, while the full supersymmetric action could be easily reconstructed, if needed.

Generalizations and the cases of a special interest
Let us consider the more general systems with more complicated structure in the bosonic sector. We will concentrate on the bosonic sector only, while the full supersymmetric action could be easily reconstructed.

SO(4) invariant systems
Our first example is the most general system, which still possesses SO(4) symmetry upon SU (2) reduction. It is specified by the prepotential F (2.3) depending on two scalars X and Y Such a system is invariant under SU (2) transformations realized on the "hatted" indicesα and thus the SU (2) reduction we discussed in the Section 2 goes in the same manner. In addition the full SU (2)×SU (2) symmetry realized on the superfield Q iα 2 will survive in the reduction process. So we expected the final system will possess SO(4) symmetry.
The bosonic sector of the system with prepotential (3.1) is described by the action Even with a such simple prepotential the bosonic action (3.2) after reduction has a rather complicated form. Next, still meaningful simplification, could be achieved with the following prepotential where F 1 (X) and F 2 (Y ) are arbitrary functions depending on X and Y , respectively. With a such prepotential the third term in the action (3.2) disappeared and the action acquires readable form. With our notations (2.18), (2.19) the reduced action reads where and x = 1 2 (r + z 5 ) , y = 1 2 (r − z 5 ) . Let us stress, that the unique possibility to have SO(5) invariant bosonic sector is to choose H x = H y = const. This is just the case we considered in the Section 2. With arbitrary potentials H x and H y we have a more general system with the action (3.4), describing the motion of the N = 4 supersymmetric particle in five dimensions and interacting with Yang monopole and some specific potential.

Non-linear supermultiplet
It is known for a long time that in some special cases one could reduce the action for hypermultiplets to the action containing one less physical bosonic components -to the action of so-called non-linear supermultiplet [12,17,18]. The main idea of such reduction is replacement of the time derivative of the "radial" bosonic component of hypermultiplet Log(q ia q ia ) by an auxiliary component B without breaking of N = 4 supersymmetry [19]: Clearly, to perform such replacement in some action the "radial" bosonic component has to enter this action only with time derivative. This condition is strictly constraints the variety of the possible hypermultiplet actions in which this reduction works.
To perform the reduction from hypermultiplet to the non-linear one, the parametrization (2.18) is not very useful. Instead, we choose the following parameterizations for independent components of two hypermultiplets q iα 1 and q iα Thus, the five independent components are u and z µ , µ = 1, ..., 4, and With this parametrization the action (3.4) acquires the form where If we choose G 1 = e −u , than the "radial" bosonic component u will enter the action (3.11) only through kinetic term ∼u 2 . Thus, performing replacement (3.7) and excluding the auxiliary field B by its equation of motion we will finish with the action (3.13) The action (3.13) describes the motion of an isospin particle on four-manifold with SO(4) isometry carrying the non-Abelian field of a BPST instanton and some special potential. Our action is rather similar to those one recently constructed in [3,8,2], but it contains twice more physical fermions.

Ordinary and twisted hypermultiplets
One more possibility to generalize the results we presented in the previous Section is to consider simultaneously ordinary hypermultiplet Q jα obeying to (2.1) together with twisted hypermultiplet V aα -a quartet of N = 4 superfields subjected to constraints [17] D i(a V b)α = 0 , and V aα † = V aα . (3.14) The most general system which is explicitly invariant under SU (2) transformations realized on the "hatted" indices is defined, similarly to (3.1), by the superspace action depending on two scalars X, Y The bosonic sector of the action (3.15) is a rather simple Thus, we see that the term causes most complicated structure of the action with two hypermultiplets, disappeared 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 Here F = F (x, y) is still function of two variables x and y. The mostly symmetric situation again corresponds to the choice with the action Both solutions describe a cone-like geometry in the bosonic sector, while the most interesting case of the sphere S 5 can not be treated within the present approach. Finally, we would like to point the attention to the fact that with h = const the bosonic sectors of the systems with two hypermultiplets and with one ordinary and one twisted hypermultiplets are coincide. This is just one more justification that "almost free" systems could be supersymmetrized in the different ways.

Conclusion
In the present paper, starting with the non-interacting system 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 conformally flat five-dimensional manifold in the non-Abelian field of a Yang monopole and in some scalar potential. The most important step for this construction is passing 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 three bosonic components, which enter the action through SU (2) invariant currents. Just these bosonic fields become the isospin variables, which the background field couples to. Due to the commutativity of N = 4 supersymmetry with the reduction SU (2) group, it survives upon reduction. Some possible generalizations of the action to the cases of systems with a more general bosonic action, a four-dimensional system which still includes eight fermionic components, and a variant of five-dimensional N = 4 mechanics constructed with the help of the ordinary and twisted N = 4 hypermultiplets were considered. The main preference of the proposed approach is its applicability to any system which possesses SU (2) invariance. If, in addition, this SU (2) commutes with supersymmetry, then the resulting system will be automatically supersymmetric.
Among possible direct applications of our construction there are the reduction in the cases of systems with non-linear N = 4 supermultiplets [21], systems with more than two (non-linear)hypermultiplets, in the 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 S 5 in the field of a Yang monopole. Unfortunately, the use of standard linear hypermultiplets makes the solution of this task impossible because the resulting bosonic manifolds have a different structure (the conical geometry) to include S 5 .