Primitive Forms without Higher Residue Structure and Integrable Hierarchies (I)

TL;DR

This paper introduces primitive forms without higher residue structures and investigates their relation to flat structures and integrable KdV-type hierarchies, extending classical theories to new non-metric cases.

Contribution

It constructs primitive forms without metrics via Birkhoff decomposition and links them to integrable hierarchies, expanding the understanding of primitive forms beyond metric cases.

Findings

Primitive forms without metrics are constructed from Birkhoff decomposition.

Oscillatory integrals of these forms generate KdV-type hierarchies.

Extension of primitive form theory to non-metric cases.

Abstract

We introduce primitive forms with or without higher residue structure and explore their connection with the flat structures with or without a metric and integrable hierarchies of KdV type. Just as the classical case of primitive forms with metric arXiv:1311.1659, the primitive forms without metrics are constructed as the positive part of the Birkhoff decomposition of formal oscillatory integrals with respect to the descendent variable. The oscilating integrals of a primitive form without metric give rise to a hierarchy of commuting PDE of the KdV type as in the case of primitive forms with metric. This shall be studied in (II).