Semisimple Frobenius (super)manifolds and quantum cohomology of $P^r$

TL;DR

This paper introduces a superversion of Frobenius manifolds, establishes a correspondence with supersymmetric Schlesinger equations, and computes initial conditions relevant to quantum cohomology of projective spaces.

Contribution

It extends Frobenius manifold theory to supermanifolds and connects it with supersymmetric integrable systems, providing new tools for quantum cohomology analysis.

Findings

Established a correspondence between super Frobenius manifolds and supersymmetric Schlesinger equations.

Calculated initial conditions for solutions related to quantum cohomology of projective spaces.

Extended the framework of Frobenius manifolds to include supergeometry.

Abstract

We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing quantum cohomology of projective spaces.