Cyclic $BV_\infty$ algebra and Frobenius manifold

TL;DR

This work develops a framework to produce formal Frobenius manifolds from cyclic, unital commutative $BV_\infty$ algebras under the Hodge-to-de Rham degeneration condition, using a compatible special homotopy retract and the universal solution to the quantum master equation. It extends Barannikov–Kontsevich-type constructions beyond dGBV algebras to broader geometric contexts, including Jacobi and Hermitian manifolds, by establishing cyclic structures and a good basis on cohomology. The main result yields a formal Frobenius manifold structure on the formal neighborhood of $0$ in $H(A)$, with the product and metric encoded via derivatives of the universal solution $\Gamma$ to $\Delta e^{\Gamma/\hbar}=0$. Applications to compact Jacobi and Hermitian manifolds demonstrate the existence of these structures in new geometric settings, broadening the reach of Frobenius-type deformation theory and connecting to broader deformation problems such as those appearing in Landau–Ginzburg models.

Abstract

We describe the construction of Frobenius manifold out of a cyclic (commutative) $BV_\infty$ algebra $(A,Δ)$ under the assumption of a Hodge-to-de Rham degeneration property and the existence of a compatible homotopy retract of $A$ onto its cohomology. We then apply it to Jacobi manifolds and Hermitian manifolds, generalizing known results in literature.

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Corollary 2.6
• Definition 3.1
• Definition 3.2