Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and $\mathcal{N}=4$ 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 $\mathcal{N}=4$ 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.

arXiv comment1+4 pages; v2: title change, published version

References in corpus (7)

• (Super)conformal many-body quantum mechanics with extended supersymmetry
• N=4 superconformal Calogero models
• N=4 Supersymmetric Multidimensional Quantum Mechanics, Partial SUSY Breaking and Superconformal Quantum Mechanics
• New insight into WDVV equation
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Cited by in corpus (7)

• Special Geometry, Hessian Structures and Applications
• $\mathcal{N}=4$ supersymmetric mechanics on curved spaces
• SU(2$|$1) supersymmetric mechanics on curved spaces
• Superfield approach to higher derivative N=1 superconformal mechanics
• $SU(1,1|N)$ superconformal mechanics with fermionic gauge symmetry
• Geometry and integrability in $\mathcal{N}=8$ supersymmetric mechanics
• Hidden symmetries and nonlinear (super)algebras