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Commutative $BV_\infty$ algebras, their morphisms and $\frac{\infty}{2}$-variation of Hodge structures

Hao Wen

Abstract

We study morphisms between commutative $BV_\infty$ algebras and show that, under suitable additional assumptions, a quasi-isomorphism of commutative $BV_\infty$ algebras induces an identification of $\frac{\infty}{2}$-variations of Hodge structures with polarizations, and consequently of Frobenius manifolds. An explicit example arising from singularity theory is provided to illustrate the result.

Commutative $BV_\infty$ algebras, their morphisms and $\frac{\infty}{2}$-variation of Hodge structures

Abstract

We study morphisms between commutative algebras and show that, under suitable additional assumptions, a quasi-isomorphism of commutative algebras induces an identification of -variations of Hodge structures with polarizations, and consequently of Frobenius manifolds. An explicit example arising from singularity theory is provided to illustrate the result.
Paper Structure (6 sections, 3 theorems, 70 equations)

This paper contains 6 sections, 3 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Commutative $BV_\infty$ algebra and $BV_\infty$ morphism
  3. Maurer-Cartan elements and twisting procedure
  4. $\frac{\infty}{2}$-variation of Hodge structure
  5. $BV_\infty$ quasi-isomorphism
  6. An example: $A_1$ singularity

Key Result

Proposition 4.6

Let $(A,\Delta)$ be a commutative $BV_\infty$ algebra satisfying the Assumption degeneration, pairing and basis, then there is a Frobenius manifold structure on the formal neighbourhood $\mathcal{M}$ of zero in $H(A,\Delta_0)$.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 4.1
  • Remark 4.2
  • Remark 4.3
  • Example 4.4
  • Example 4.5
  • Proposition 4.6
  • Theorem 5.1
  • proof
  • ...and 4 more