Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and ${\cal N}{=}\,4$ mechanics

TL;DR

This paper extends the WDVV equation to curved Riemannian manifolds, linking it with ${ m N}=4$ supersymmetric mechanics, and provides solutions on isotropic spaces using conformal factors.

Contribution

It introduces a curved version of the WDVV equation related to ${ m N}=4$ mechanics and offers explicit solutions on isotropic manifolds.

Findings

Curved WDVV equation expressed via Codazzi tensor.

Solutions constructed on isotropic spaces using conformal factors.

Extension from flat to curved space in supersymmetric mechanics.

Abstract

We propose a generalization of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation from ${\mathbb R}^n$ to an arbitrary Riemannian manifold. Its form is obtained by extending the relation of the WDVV equation with ${\cal N}{=}\,4$ 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.