Strong homotopy inner product of an A-infinity algebra

TL;DR

This paper introduces a strong homotopy concept of cyclic symmetric inner products for A-infinity algebras, linking it to symplectic structures and invariance properties of associated potentials, advancing the understanding of algebraic and geometric structures.

Contribution

It defines a new strong homotopy notion of cyclic inner products and proves its equivalence to symplectic structures on non-commutative supermanifolds, extending previous work.

Findings

Characterization theorem for strong homotopy inner products

Equivalence with non-constant symplectic structures

Invariance of open Gromov-Witten potentials under homomorphisms

Abstract

We introduce a strong homotopy notion of a cyclic symmetric inner product of an A-infinity algebra and prove a characterization theorem in the formalism of the infinity inner products by Tradler. We also show that it is equivalent to the notion of a non-constant symplectic structure on the corresponding formal non-commutative supermanifold. We show that (open Gromov-Witten type) potential for a cyclic filtered A-infinity algebra is invariant under the cyclic filtered A-infinity homomorphism up to reparametrization, cyclization and a constant addition, generalizing the work of Kajiura.