Manifolds with multiplication on the tangent sheaf

TL;DR

This paper surveys the theory of F-manifolds, which are manifolds with a tangent sheaf multiplication satisfying integrability, highlighting their role in quantum K-theory, moduli spaces, and singularity theory.

Contribution

It provides a comprehensive overview of the development and current state of F-manifold theory, emphasizing their applications and structural properties.

Findings

F-manifolds generalize Frobenius manifolds with weaker conditions.

They are useful in quantum K-theory and moduli space theory.

The survey summarizes key developments and open problems.

Abstract

This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf $\Cal{T}_M$, satisfying an integrability condition. $F$--manifolds and compatible flat structures on them furnish a useful weakening of Dubrovin's Frobenius structure which naturally arises in the quantum $K$--theory, theory of extended moduli spaces, and unfolding spaces of singularities.