Open WDVV equations and $\bigvee$-systems

TL;DR

This paper extends the concept of $igvee$-systems to open WDVV equations, providing algebraic and geometric conditions for rational solutions and exploring their connections to superpotentials and duality.

Contribution

It generalizes $igvee$-systems to open WDVV equations and develops conditions for rational solutions in the rank-one case.

Findings

Established algebraic/geometric conditions for solutions

Provided examples linking to superpotentials

Explored relation to Dubrovin almost-duality

Abstract

The idea of a $\bigvee$-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the WDVV equations. Here, this idea is generalized to open WDVV equations, which are an additional set of PDEs originating from open Gromow-Witten Theory. We develop -- for rank-one extensions -- algebraic/geometric conditions on the covectors that supplement the $\bigvee$-system to give rational solutions to the open WDVV equations. Examples, and the relation to superpotentials and to Dubrovin almost-duality, are given.