Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and N=4 mechanics

Abstract

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.

Superfield approach to higher derivative $$ \mathcal{N} $$ = 1 superconformal mechanics

“…The first term reproduces the inappropriate potential function (18) revealed above, while the requirement for the second term in (19) to be Grassmann-odd is satisfied provided the coupling constant γ is Grassmann-even. Let us consider the superfield action…”

Hidden symmetry and (super)conformal mechanics in a monopole background

“…If so, it would be an interesting result from the point of view of three-dimensional (or more generally higher-dimensional) supersymmetric quantum mechanics, because contrary to the one-dimensional case [30,31,32], there is no canonical way to obtain such systems. Although there are particular and elegant constructions, see for example [7,8,27,33,34,35,36], it is in general a non-trivial task to produce such theories. In the present context, a possible approach would be to look for a general (3 + 0) dimensional Dirac type operator as a supercharge (square root) of a Klein-Gordon type "super-Hamiltonian".…”

Geometry and integrability in N=8 supersymmetric mechanics

“…Another way to increase the number of supersymmetries (above N = 2 supersymmetry) is the doubling of fermionic degrees of freedom, supplied with introducing of additional geometric objects. Say, to construct the N = 4 supersymmetric extension of free-particle system on generic configuration space we have to double the number of fermionic degrees of freedom from 2N to 4N and introduce the third-rank symmetric tensor F ijk (x)dx i dx j dx k which satisfies the curved WDVV equations [1] F kmj;i = F kmi;j , F jkp g pq F imq − F ikp g pq F jmq = R ijkm , (1.2) where R ijkl are the components of Riemann tensor of (M 0 , g ij dx i dx j ), and the subscript ; denotes covariant derivative with Levi-Civita connection. Similarly, to construct the N = 8 supersymmetric extension of free-particle system on Kähler manifold we have to increase the number of the (real) fermionic variables from 4N to 8N and introduce the third-rank holomorphic symmetric tensor f abc (z)dz a dz b dz c which satisfies the equations [2] These manifolds are known as the special Kähler manifolds of the rigid type [3] and they have been extensively studied since their introduction within the context of Seiberg-Witten duality [4].…”

“…In this paper we show that this similarity holds for the supersymmetric mechanics with the potential term as well. Namely, after reviewing the main properties of N = 4 supersymmetric mechanics connected with the solution of modified WDVV equations [1,5,6](Section 2), we construct on the special Kähler manifold of the rigid type, the N = 8 supersymmetric mechanics with potential term (Section 3). We find that for the doubling of the supersymmetries the prepotentials W (x), U (z) in the bosonic Hamiltonians (1.1) should satisfy the following equations…”