tt* geometry, Frobenius manifolds, their connections, and the construction for singularities

TL;DR

This paper explores the rich geometric structures of hypersurface singularities, specifically Frobenius manifolds and tt* geometry, and develops methods to construct these structures using oscillating integrals, revealing their deep interrelations.

Contribution

It provides a unified construction of Frobenius manifolds and tt* geometry for any hypersurface singularity, extending previous work and clarifying their connections through oscillating integrals.

Findings

Constructed Frobenius manifolds and tt* geometry for all hypersurface singularities.

Established the relationship between polarized mixed Hodge structures and tt* geometry.

Analyzed flat connections with poles associated with these geometric structures.

Abstract

The base space of a semiuniversal unfolding of a hypersurface singularity carries a rich geometry. By work of K. Saito and M. Saito is can be equipped with the structure of a Frobenius manifold. By work of Cecotti and Vafa it can be equipped with tt* geometry if the singularity is quasihomogeneous. tt* geometry generalizes the notion of variation of Hodge structures. In the second part of this paper (chapters 6-8) Frobenius manifolds and tt* geometry are constructed for any hypersurface singularity, using essentially oscillating integrals; and the intimate relationship between polarized mixed Hodge structures and this tt* geometry is worked out. In the first part (chapters 2-5) tt* geometry and Frobenius manifolds and their relations are studied in general. To both of them flat connections with poles are associated, with distinctive common and different properties. A frame for a simultaneous construction is given.