Equivariant A-infinity algebras for nonorientable Lagrangians

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.

Revised subsection 5.2 to simplify the argument and correct a typo in the proof of Lemma 51 (of v1). This does not affect the validity of the statement of Lemma 51 (or any other parts of the paper)

References in corpus (3)

• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Point-like bounding chains in open Gromov-Witten theory

Cited by in corpus (1)

• Point-like bounding chains in open Gromov-Witten theory