Weak Frobenius manifolds

TL;DR

This paper introduces the concept of weak Frobenius manifolds, establishing a universal relation between Lie brackets and multiplication, and explores their connection to Virasoro algebra representations without relying on a metric.

Contribution

It defines weak Frobenius manifolds independent of metrics and links Euler field powers to Virasoro algebra on general Frobenius supermanifolds.

Findings

Universal relation between Lie bracket and multiplication on Frobenius manifolds

Introduction of weak Frobenius manifolds without metric dependence

Generation of Virasoro algebra by Euler field powers

Abstract

We establish a new universal relation between the Lie bracket and $\circ$-multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of ``weak Frobenius manifold'' which does not involve metric as part of structure. As another application, we show that the powers of an Euler field generate (a half of) the Virasoro algebra on an arbitrary, not necessarily semisimple, Frobenius supermanifold.