Equivariant A-infinity algebras for nonorientable Lagrangians

TL;DR

This paper develops an algebraic framework for studying pseudoholomorphic discs with nonorientable Lagrangians, introducing equivariant $A_ fty$ algebras and minimal models, with applications to open Gromov-Witten invariants.

Contribution

It introduces unital cyclic twisted $A_ fty$ algebras, constructs their equivariant extensions, and develops methods for minimal models, advancing the algebraic tools for nonorientable Lagrangian Floer theory.

Findings

Defined unital cyclic twisted $A_ abla$ algebras.

Constructed equivariant extensions invariant under torus actions.

Established homotopy retraction to equivariant cohomology.

Abstract

We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted $A_\infty$ algebras and prove some basic results about them, including a homological perturbation lemma which allows one to construct minimal models of such algebras. We then construct an equivariant extension of $A_\infty$ algebras which are invariant under a torus action on the underlying complex. Finally, we construct a homotopy retraction of the Cartan-Weil complex to equivariant cohomology, which allows us to construct minimal models for equivariant cyclic twisted $A_\infty$ algebras. In a forthcoming paper we will use these results to define and obtain fixed-point expressions for the open Gromov-Witten theory of $\mathbb{RP}^{2n} \hookrightarrow \mathbb{CP}^{2n}$, as well as its equivariant extension.