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Elements of Saito theory via Batalin--Vilkovisky algebras

Alexey Basalaev

TL;DR

The paper develops a BV-algebra framework to study Saito theory and the Dubrovin--Frobenius structure associated with quasihomogeneous singularities. By working with the BV algebra of polyvector fields and its extension to unfoldings, it constructs a topological trivialization and a BV-based route to the Gauss--Manin connection, higher residue pairing, and good bases, leading to a formal Dubrovin connection and potential ${\mathcal{F}}^{\omega}$. It provides recursive formulas for the primitive form and the R-matrix of the resulting Dubrovin--Frobenius manifold, showing that BV-generated correlators reproduce the Saito-theory potential, thus unifying approaches to primitive forms, good bases, and Frobenius structures. The methods yield computational tools for explicit calculation of primitive forms and $R$-matrices from BV-algebra data, with implications for mirror symmetry and integrable systems through the Givental framework and WDVV equations.

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.

Elements of Saito theory via Batalin--Vilkovisky algebras

TL;DR

The paper develops a BV-algebra framework to study Saito theory and the Dubrovin--Frobenius structure associated with quasihomogeneous singularities. By working with the BV algebra of polyvector fields and its extension to unfoldings, it constructs a topological trivialization and a BV-based route to the Gauss--Manin connection, higher residue pairing, and good bases, leading to a formal Dubrovin connection and potential . It provides recursive formulas for the primitive form and the R-matrix of the resulting Dubrovin--Frobenius manifold, showing that BV-generated correlators reproduce the Saito-theory potential, thus unifying approaches to primitive forms, good bases, and Frobenius structures. The methods yield computational tools for explicit calculation of primitive forms and -matrices from BV-algebra data, with implications for mirror symmetry and integrable systems through the Givental framework and WDVV equations.

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.
Paper Structure (41 sections, 10 theorems, 82 equations)

This paper contains 41 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Dubrovin--Frobenius manifold of a singularity
  3. Saito--Frobenius manifold
  4. Primitive form and good basis
  5. BV--algebra for a hypersurface singularity
  6. BV operator trivialization
  7. In this paper
  8. Acknowledgements
  9. Brieskorn lattice of a singularity
  10. Jacobian algebra
  11. Weight and Hessian
  12. Brieskorn lattice
  13. BV algebras and homotopic resolution
  14. BV--algebra of the polyvector fields
  15. BV--algebra of a hypersurface singularity
  16. ...and 26 more sections

Key Result

Theorem 4.1

If $R$, assumed as an operator on $H^\ast(V,d)$, satisfies the condition $R(z)\eta^{-1}R^T(-z) = \eta^{-1}$, then the generating functions ${\mathcal{F}}^{\Phi}$ and ${\mathcal{F}}^{R\Phi}$ are connected by the Givental action of $R$ (see Section section: givental's action).

Theorems & Definitions (18)

  • Theorem 4.1: Theorem 6.3 in KMS
  • Proposition 6.1
  • proof
  • Remark 6.2
  • Proposition 6.3
  • proof
  • Theorem 7.1: Theorem 4.15 of LLS and Theorem 3.7 of LLSS
  • Theorem 7.2
  • proof
  • Proposition 8.1: Proposition of Section 1.3 G04
  • ...and 8 more