# The curved WDVV equations and superfields

## TL;DR

This paper reformulates ${
m N}=4$ supersymmetric mechanics on curved spaces using superfield methods, establishing a link between the geometry, modified WDVV equations, and supersymmetry conditions.

## Contribution

It introduces a superfield approach to ${
m N}=4$ mechanics on curved spaces, connecting the metric, Codazzi tensor, and modified WDVV equations for supersymmetry.

## Key findings

- Superfield Lagrangian constructed from metric and Codazzi tensor.
- Irreducibility conditions are consistent iff modified WDVV equations hold.
- Provides a new geometric framework for ${
m N}=4$ supersymmetric mechanics.

## Abstract

We reproduce the ${\cal N}=4$ supersymmetric mechanics on curved spaces, constructed earlier within the Hamiltonian formalism, using the superfield methods. We show that for any such mechanics, given by the metric and the third order Codazzi tensor, it is possible to construct a suitable modification of irreducibility conditions of linear ${\cal N}=4$ multiplets and obtain the superfield Lagrangian by solving a simple differential equation. Also, we prove that the constructed irreducibility conditions are consistent if and only if the metric and Codazzi tensor satisfy the modification of the WDVV equations, which are the conditions of existence of ${\cal N}=4$ supersymmetry.