Regular F -manifolds with eventual identities

TL;DR

This paper explores the construction of dual multiplications on regular F-manifolds via eventual identities, providing solutions for their defining equations and establishing dual coordinate systems, with applications to Nijenhuis operators.

Contribution

It introduces methods to solve for eventual identities on non-semi-simple F-manifolds and constructs dual coordinate systems that preserve the dual multiplication structure.

Findings

Solutions for eventual identity equations on regular F-manifolds.

Construction of dual coordinate systems preserving dual multiplication.

Development of families of Nijenhuis operators.

Abstract

Given an F-manifold one may construct a dual multiplication (generalizing the idea of an almost-dual Frobenius manifold introduced by Dubrovin) using a so-called eventual identity, the definition of which ensure that the dual object is also an F-manifold. In this paper we solve the equations for an eventual identity for a regular (so non-semi-simple) F-manifold and construct a dual coordinate system in which dual multiplication is preserved. As an application, families of Nijenhuis operators are constructed.