Elements of Saito theory via Batalin--Vilkovisky algebras

TL;DR

This paper explores the connection between Saito theory and Batalin--Vilkovisky algebras, providing recursive formulas for primitive forms and R-matrices, enhancing understanding of Frobenius manifolds associated with singularities.

Contribution

It introduces a BV-algebra framework to study primitive forms and Frobenius manifolds, offering recursive computational formulas for key structures.

Findings

Recursive formulas for primitive forms

Explicit computation of R-matrices

Deeper understanding of Frobenius manifold structures

Abstract

Saito theory associates to a quasihomogeneous isolated singularity the structure of a Dubrovin--Frobenius manifold. This structure is not unique, depending on the special choice of a primitive form or, equivalently, a good basis. We study primitive forms and respective Dubrovin--Frobenius manifolds via BV-algebras. In particular, we give recursive formulae for the primitive form of K. Saito and the R-matrix of Givental using BV-algebra computations.