Analyticity of the total ancestor potential in singularity theory

TL;DR

This paper proves Givental's conjecture that the total ancestor potential extends analytically to non-semisimple points in singularity theory, using the Eynard--Orantin recursion, thus advancing the understanding of Frobenius manifolds.

Contribution

It confirms Givental's conjecture on the analytic extension of the total ancestor potential in singularity theory, employing the Eynard--Orantin recursion method.

Findings

Proves Givental's conjecture for non-semisimple points

Uses Eynard--Orantin recursion to establish analyticity

Enhances understanding of Frobenius manifold structures

Abstract

K. Saito's theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated singularity. On the other hand, Givental introduced the notion of a total ancestor potential for every semi-simple point of a Frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non-semisimple points as well. In this paper we prove Givental's conjecture by using the Eynard--Orantin recursion.