${\cal N}{=}\,4$ supersymmetric mechanics on curved spaces

We present ${\cal N}{=}\,4$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds constructed within the Hamiltonian approach. The structure functions entering the supercharges and the Hamiltonian obey modified covariant constancy equations as well as modified Witten-Dijkgraaf-Verlinde-Verlinde equations specified by the presence of the manifold's curvature tensor. Solutions of original Witten-Dijkgraaf-Verlinde-Verlinde equations and related prepotentials defining ${\cal N}{=}\,4$ superconformal mechanics in flat space can be lifted to $so(n)$-invariant Riemannian manifolds. For the Hamiltonian this lift generates an additional potential term which, on spheres and (two-sheeted) hyperboloids, becomes a Higgs-oscillator potential. In particular, the sum of $n$ copies of one-dimensional conformal mechanics results in a specific superintegrable deformation of the Higgs oscillator.


I. INTRODUCTION
The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations were introduced a few decades ago in the context of two-dimensional topological field theories [1]. In their initial version they read Clearly, these equations are not covariant with respect to general coordinate transformations, so their covariantization requires introducing additional geometric structures. Finding solutions to the WDVV equations is a nontrivial task, which was considered in numerous papers in various contexts (see, e.g. [2][3][4][5][6][7][8] and references in them). The WDVV equations appear also when constructing N ¼ 4 supersymmetric/superconformal extensions of mechanics on n-dimensional Euclidean space (see [9][10][11][12][13][14] and references therein). Indeed, as it was firstly demonstrated in [9], if we insist that the N ¼ 4 supercharges Q a andQ b with a, b ¼ 1, 2, in the form then the totally symmetric structure functions F ð0Þ ijk ðxÞ entering the supercharges (1.2) have to satisfy (1.1), while the prepotential W ð0Þ is found by solving We should add that, when evaluating the brackets between the supercharges in (1.3), the basic variables were taken to obey the standard Dirac brackets fx i ; p j g ¼ δ i j and fψ ai ;ψ j b g ¼ i 2 δ a b δ ij : ð1:5Þ Exploiting this relation, one can postulate a general coordinate invariant version of the WDVV equations simply by constructing N ¼ 4 supersymmetric mechanics on arbitrary Riemannian spaces. This was achieved in our recent paper [15]. To construct such mechanics, one has to generalize the ansatz for the supercharges (1.2) and the Poisson brackets (1.5) to be covariant under general coordinate transformations, and then to check the conditions on the structure functions implied by the N ¼ 4 super-Poincaré algebra relations (1.3) these supercharges should obey. While [15] accomplished this for the special case of W i ¼ 0-no potential term-its result for the structure functions F ijk remains true in general: These are the curved WDVV equations on spaces with a metric g ij . It was also shown there that solutions of the flat WDVV equations (1.1) can be extended to solutions of the curved WDVV equations (1.6) on isotropic spaces.
The main goal of the present paper is to construct N ¼ 4 supersymmetric mechanics with a nonzero potential on arbitrary Riemannian spaces. As a first step, in Sec. II we introduce generalized Poisson brackets which are covariant with respect to general coordinate transformations. Then we write down the most general ansatz for the supercharges (linear and cubic in the fermionic variables, for the case with and without additional spin variables) and analyze the conditions on the structure functions. We obtain equations on the prepotentials which generalize (1.4). In Sec. III we specialize to isotropic spaces and extend the general solution found in [15] to include the prepotential W i . In Sec. IV we present some explicit solutions for the most interesting spaces-spheres and pseudospheres. Finally, we conclude with a few comments and remarks.

II. SYMPLECTIC STRUCTURE, SUPERCHARGES AND HAMILTONIAN
We are going to construct the supercharges Q a , for n-dimensional systems in which each bosonic degree of freedom x i ði ¼ 1; …; nÞ is accompanied by four fermionic ones ψ ai ,ψ j b ¼ ðψ bj Þ † . The extended phase space, parametrized by the bosonic coordinates x i and momenta p i and the fermionic coordinates ψ ai ,ψ j b , can be equipped by the symplectic structure Ω ¼ dp i ∧ dx i þ idðψ ia g ij Dψ j a −ψ ia g ij Dψ j a Þ ¼ dp i ∧ dx i þ iR ijkl ψ iaψ j a dx k ∧ dx l þ 2ig ij Dψ ia ∧ Dψ j a ; ð2:2Þ where Dψ ia ≡ dψ ia þ Γ i jk ψ ja dx k , and Γ i jk and R i jkl are the components of the Levi-Civita connection and curvature of the metric g ij ðxÞ defined in a standard way as and This symplectic structure is manifestly invariant with respect to the transformations The Poisson brackets between the basic variables can be immediately extracted from the symplectic structure (2.2): To construct the supercharges Q a ,Q b we have two possibilities.
(i) One may construct the standard supercharges in terms of the variables x i , p j , ψ ai ,ψ j b only, mainly following to the line of the paper [11]. (ii) One may extend the set of the basic variables by the additional bosonic spin variables fu a ;ū a ja ¼ 1; 2g parametrizing an internal two-sphere and obeying the brackets These new variables will appear in the supercharges only through the suð2Þ currents [14,16] J ab ¼ i 2 ðu aūb þ u būa Þ ⇒ fJ ab ; J cd g ¼ −ϵ ac J bd − ϵ bd J ac : ð2:7Þ Let us consider these possibilities separately.

A. Standard supercharges
The most general ansatz for the standard supercharges reads: Here, W i , F ijk and G ijk are arbitrary, for the time being, real functions depending on n coordinates x i . In addition, we assume that the functions F ijk and G ijk are symmetric and antisymmetric over the first two indices, respectively: The conditions that these supercharges span an N ¼ 4 super Poincaré algebra (2.1) result in the following equations on the functions involved: where, as usual, Once the equations (2.10)-(2.14) are satisfied, the Hamiltonian H acquires the form

B. Supercharges with spin variables
Following [14,16], the spin variables fu a ;ū a g may be utilized to slightly modify the prepotential term in the supercharges to be These supercharges form an N ¼ 4 super Poincaré algebra if the functions F ijk and G ijk obey the same constraints (2.10)-(2.12), while the constraints (2.13), (2.14) which include the prepotential are changed to When the constraints (2.10)-(2.12) and (2.18), (2.19) are satisfied, the Hamiltonian reads  [17], while (2.12) is the curved WDVV equation proposed in [15], for which the Eqs. (2.14), (2.19) are the curved analogs of the flat equations on the prepotentials discussed in [11,14].
Summarizing, one may conclude that to construct N ¼ 4 supersymmetric n-dimensional mechanics with a given bosonic metric g ij one has (i) To solve the curved WDVV equations (2.11), (2.12) for the fully symmetric function F ijk , (ii) To find admissible prepotentials as solutions of the Eqs. (2.13), (2.14) and/or (2.18), (2.19). In what follows, we will use this procedure to construct N ¼ 4 supersymmetric n-dimensional mechanics.

A. Solution of curved WDVV equation
In [15] a large class of solutions to the Eqs. (2.11), (2.12) has been constructed on isotropic spaces. Such spaces admit an soðnÞ-invariant metric with components The key point of the analysis performed in [15] is the following ansatz on the structure function F ijk : ijk is an arbitrary solution of the flat WDVV equation, i.e., ð3:3Þ One may check that the linear equation (2.11) is satisfied if We note that these equations already imply the condition (3.4).
The equations (3.5) may be easily solved as It should be noted that the solution for a in (3.6) becomes 0=0 indeterminate if ð3:9Þ Another exceptional case corresponds to c ¼ AE1 and the metric function For this case, the function aðrÞ is not restricted while bðrÞ has the form b ¼ − c μ 2 r 4 : ð3:11Þ

B. Searching for the prepotentials
Having at hands the solution for the curved WDVV equations (2.11), (2.12), one may try to solve the equations for the prepotentials (2.13), (2.14) or (2.18), (2.19), respectively. It should be noted that even in the flat case, where a variety of the solutions to the WDVV equations is known [3][4][5][6][7], such a task is far from being completely solved. Nevertheless, many particular solutions are known for the standard supercharges [11][12][13] as well as for the case with spin variables [14,16]. Leaving the full analysis of the admissible prepotentials for the future, let us demonstrate that each prepotential found for the flat WDVV equations can be embedded into isotropic spaces with a metric (3.1).

Lifting a "flat" prepotential W ð0Þ
In this case we have to solve the equation (2.13) which for the metric (3.1) and for the F ijk given in (3.2) acquires the form ijm δ mn ∂ n W ¼ 0: ð3:12Þ Let us also suppose that we know the solution W ð0Þ of the flat equation ijm δ mn ∂ n W ð0Þ ¼ 0: ð3:13Þ All such solutions found in [11][12][13] obey the additional condition i.e., the parameter c in (3.6) fixed to be equal to one.
If we now choose the following ansatz for the prepotential W, W ¼WðrÞ þ W ð0Þ ; ð3:15Þ then the "flat" prepotential W ð0Þ will appear in the Eq. (3.12) only through the constant α, except for the second term To kill this term we have to choose b ¼ − f 0 rf 3 . This choice corresponds to the following solution in (3.7) Finally, it is a matter of straightforward calculations to check that the prepotential solves the Eq. (3.12). Correspondingly, the bosonic potential in the Hamiltonian (2.16) reads ð3:18Þ

Lifting a "flat" prepotential with spin variables
For the supercharges with spin variables the solutions for the Eq. (2.19) may be found by using the relation (2.21). However, the additional constraint for the flat prepotential (3.14) we used above is quite unconventional. Indeed, in [14] the additional condition on the solution of the flat equation ð3:20Þ Therefore, we need to reconsider the solution of (2.19) using an ansatz U ¼ŨðrÞ þ U ð0Þ : ð3:21Þ We will use the same ansatz for the F ijk (3.2) with the same conditions on the functions aðrÞ, bðrÞ (3.5) and the constraint (3.20). Now, it is rather easy to check that the prepotential U (3.21) obeys the Eqs. (2.19) if Therefore, the resulting potential in the Hamiltonian (2.20) reads with b given in (3.6).

Lifting a vanishing "flat" prepotential
This case corresponds to the absence of the "flat" prepotential, i.e., to the case with W ð0Þ ¼ 0. To simplify the analysis we suppose that the prepotential W depends on r only, while the "flat" WDVV solution F ð0Þ ijk still obeys the constraint (3.14), i.e., With these assumptions the Eq. (2.13) reads To kill the first term in (3.25) we have to choose b ¼ −2fþrf 0 r 2 f 3 . This choice corresponds to the following solution in (3.7), With such functions a, b the equation which defines the prepotential acquires the form The bosonic potential in the Hamiltonian (2.16) now reads It is interesting that, after passing to the prepotential U ¼ − logðWÞ (2.21), the potential term in the Hamiltonian with spin variables (2.20) acquires the form

A. Supersymmetric black holes
In this section we present some interesting prepotentials on (pseudo)spheres which admit N ¼ 4 supersymmetry. As we can see from the previous section, any "flat" system obeying the constraints (3.14), i.e., ð4:1Þ has its image on isotropic spaces with the potential (3.18) ð4:2Þ In the case of a (pseudo)sphere with the metric the potential V is simplified to be Thus we see that, besides getting multiplied with the standard factor ð1 þ ϵr 2 Þ 2 , the "flat" potential is shifted by the potential of a Higgs oscillator [18], ð4:5Þ This means that even a mutually noninteracting system, being placed on the (pseudo)sphere, becomes interacting via the Higgs potential. A prominent example comes from the sum of several N ¼ 4 supersymmetric mechanics on flat space with conformal prepotentials ð4:6Þ With such almost "free" prepotentials we obtain the following potential for the system on the (pseudo)sphere, ð4:7Þ A system with such a potential was obtained as a reduced angular (compact) part of conformal mechanics describing the motion of a particle in a near-horizon Myers-Perry black-hole background with coinciding rotational parameters in [19], and its superintegrability was proven there as well.

B. (Pseudo)sphere image of a free system
This case is analogous to the one previously considered, but, unfortunately, we cannot just substitute W ð0Þ ¼ 0 into (4.4) because this is in contradiction with the constraint x i ∂ i W ð0Þ ¼ α. Thus, we have to use the general consideration in Sec. III B 3.
For the (pseudo)sphere with metric f ¼ 1 þ ϵr 2 the potential in (3.29) reads ð4:8Þ At the same time, the potential for the supercharges with spin variables can be easily obtained by passing to the prepotential U (2.21), and by evaluating the potential for the case with spin variables in the Hamiltonian (2.20) we get Choosing now Thus, the Higgs oscillator in the Hamiltonian with spin variables on the (pseudo)sphere is the image of a completely free system.

C. (Pseudo)sphere image of the isotropic harmonic oscillator
Due to the exceptional role the standard harmonic oscillator plays among integrable systems, it is interesting to find its image in the case of N ¼ 4 supersymmetric mechanics on the (pseudo)sphere. Unfortunately, in this case our consideration of the previous sections does not help too much, because if we choose W ð0Þ ¼ mr 2 with m ¼ const; ð4:13Þ then the condition x i ∂ i W ð0Þ ¼ α we used previously will not be valid anymore. However, plugging the expression (4.13) into (3.13) we will get Thus, in the general solution (3.2), (3.6) for the curved WDVV equation we have to substitute c ¼ −1. Thus, everything is simplified, and we have two solutions for the (pseudo)sphere with f ¼ 1 þ ϵr 2 :

ð4:16Þ
Now, the basic equation which defines the admissible prepotentials (3.12) for the prepotentialsW ¼WðrÞ acquires the form ðW 0 þ 2mrÞ: ð4:17Þ Now we see that, to kill the δ ij term, we have to choose the first solution (4.15). Therefore, the equation for the prepotentialW reads ð4:19Þ where M 1 and M 2 are integration constants. Thus, the (pseudo)sphere image of the oscillator potential W ð0Þ (4.13) reads One may see that this potential coincides, modulo redefinition of the coupling constants, with the image of the "free" system (4.8), despite the fact that the prepotential (4.19) is different from the one in (3.28). This difference plays an essential role after passing to the Hamiltonian with spin variables, in which the corresponding potential acquires the form ð4:21Þ Special cases of this potential are known. For example,

V. CONCLUSIONS
We constructed n-dimensional N ¼ 4 supersymmetric mechanics on arbitrary spaces with a metric g ij . Besides reproducing the curved WDVV equations already found in [15] for the first prepotential, we obtained the curved version of the equations defining the second prepotential for the cases of supercharges with and without spin variables. For any solution of the curved WDVV equations on soðnÞ-invariant conformally flat spaces we constructed admissible prepotentials on these spaces. A nice feature of our construction is the possibility to lift any "flat" prepotentials to isotropic spaces. Finally, we provided some interesting potentials for N ¼ 4 mechanics on the (pseudo)sphere.
A still unsolved task is a superspace description of our mechanics. To generalize the superspace approach developed in [20,21] clearly one will have to find new superspace irreducibility constraints for the (1,4,3) supermultiplets which are covariant under general coordinate transformations in the target space. A related task is to understand how the real target-space Kähler metric which necessarily appears in the superspace approach [20,21] is related with our component Hamiltonian description. These tasks will be considered elsewhere.