L-manifolds

TL;DR

This paper explores the extension of Frobenius manifold theory to cyclic Lie-infinity algebras, highlighting similarities with the classical Com-infinity algebra framework in algebraic geometry.

Contribution

It initiates the development of a parallel theory for Frobenius manifolds based on cyclic Lie-infinity algebras, expanding the algebraic structures involved.

Findings

Identifies parallels between Com-infinity and cyclic Lie-infinity algebra descriptions.

Proposes a foundational framework for Frobenius manifolds using cyclic Lie-infinity algebras.

Abstract

The notion of a Frobenius manifold appears in relation to various topics in algebraic and analytic geometry, such and quantum cohomology, deformation of meromorphic connections, unfolding of singularities and others. In the local setting the structure of a Frobenius manifold admits two other equivalent descriptions, either as an algebra over a cyclic operad Com-infinity, or alternatively as a (genus zero) cohomological field theory. In this paper we make the first steps towards outlining the parallel theory, when one starts with the cyclic Lie-infinity algebras instead of Com-infinity, and highlight the striking similarities between the two pictures.