Polynomial Modular Frobenius Manifolds

TL;DR

This paper classifies semi-simple polynomial modular Frobenius manifolds in low dimensions, exploring their symmetries, classifications, and connections to orbifold quantum cohomology.

Contribution

It provides a classification of semi-simple polynomial modular Frobenius manifolds in dimensions three and four, including their construction via folding and relation to orbifold quantum cohomology.

Findings

Complete classification in 3 and 4 dimensions.

Examples obtained from higher dimensions by folding.

Discussion of connections to orbifold quantum cohomology.

Abstract

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class - so-called modular Frobenius manifolds - lie at the fixed points of this symmetry. In this paper a classification of semi-simple modular Frobenius manifolds which are polynomial in all but one of the variables is begun, and completed for three and four dimensional manifolds. The resulting examples may also be obtained from higher dimensional manifolds by a process of folding. The relationship of these results with orbifold quantum cohomology is also discussed.