We present N ¼ 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 WittenDijkgraaf-Verlinde-Verlinde equations and related prepotentials defining N ¼ 4 superconformal mechanics in flat space can be lifted to soðnÞ-invariant Riemannian manifolds. For the Hamiltonian this lift generates an additional potential term which, on spheres and (two-sheeted) hyperboloids, becomes a Higgsoscillator potential. In particular, the sum of n copies of one-dimensional conformal mechanics results in a specific superintegrable deformation of the Higgs oscillator.

Abstract:We derive the component on-shell action of the space-filling D3-brane, i.e. N = 1 supersymmetric Born-Infeld action, within the nonlinear realization approach. The covariant Bianchi identity defining the N = 1, d = 4 vector supermultiplet has been constructed by introducing a new bosonic Goldstone superfield associated with the generator of the U(1) group, which transforms to each other the spinor generators of unbroken and spontaneously broken N = 1, d = 4 supersymmetries. The first component of this Goldstone superfield is the auxiliary field of the vector supermultiplet and, therefore, the Bianchi identity can be properly defined. The component action of the D3-brane has a very simple form, being written in terms of derivatives covariant with respect to spontaneously broken supersymmetry -it just mimics its bosonic counterpart.

Based on the connection between partial breaking of global supersymmetry, coset approach, which realized the given pattern of supersymmetry breaking, and the Nambu-Goto actions for the extended objects, we have constructed on-shell component action for N = 1, D = 5 supermembrane and its dual cousins. We demonstrate that the proper choice of the components and the use of the covariant (with respect to broken supersymmetry) derivatives drastically simplify the action: it can be represented as a sum of four terms each having an explicit geometric meaning.

We couple N = 4 chiral supermultiplet with an auxiliary N = 4 fermionic supermutiplet containing on-shell four physical fermions and four auxiliary bosons. The latter ones play the role of isospin variables. We choose the very specific coupling which results in a component action containing only time derivatives of fermionic components presented in the auxiliary supermultiplet, which therefore may be dualized into auxiliary ones. The resulting component action describes the interaction of the chiral supermultiplet with a magnetic field constant on the pseudo-sphere SU (1, 1)/U (1). Then we specify the prepotential of our theory to get, in the bosonic sector, the action for the particle moving over the pseudo-sphere -Lobachevsky space. We provided also the Hamiltonian formulation of this system and show that the full symmetry group of our system is SU (1, 1) × U (1). The currents forming the su(1, 1) algebra are modified, as compared to the bosonic case, by the fermionic and isospin terms, while the additional u(1) current contains only isospin variables.One of the most important features of our construction is the presence in the Hamiltonian and supercharges of all currents of the isospin group SU (2). Despite the fact that two of the su(2) currents {T, T } enter the Hamiltonian only through the Casimir operator of the SU (2) group, they cannot be dropped out, even after fixing the total isospin of the system, because these currents themselves enter into the supercharges.We also present the Hamiltonian and supercharges describing the motion of a particle over the sphere S 2 in the background of constant magnetic field. In this case the additional isospin currents form the su(1, 1) algebra.

We propose a generalization of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation from R n to an arbitrary Riemannian manifold. Its form is obtained by extending the relation of the WDVV equation with N = 4 supersymmetric n-dimensional mechanics from flat to curved space. The resulting 'curved WDVV equation' is written in terms of a third-rank Codazzi tensor. For every flat-space WDVV solution subject to a simple constraint we provide a curved-space solution on any isotropic space, in terms of the rotationally invariant conformal factor of the metric.