The Fukaya $A_\infty$ algebra of a non-orientable Lagrangian

TL;DR

This paper constructs a family of $A_ abla$ structures on differential forms of a non-orientable Lagrangian, extending Fukaya's algebra to non-orientable cases using orientor calculus.

Contribution

It introduces a novel $A_ abla$ algebra framework for non-orientable Lagrangians, incorporating local systems and relative cohomology.

Findings

Constructed cyclic unital curved $A_ abla$ structures on differential forms

Extended Fukaya algebra to non-orientable Lagrangians

Utilized orientor calculus to handle non-orientability issues

Abstract

Let $L\subset X$ be a not necessarily orientable relatively $Pin$ Lagrangian submanifold in a symplectic manifold $X$. We construct a family of cyclic unital curved $A_\infty$ structures on differential forms on $L$ with values in the local system of graded non-commutative rings given by the tensor algebra of the orientation local system of $L$. The family of $A_\infty$ structures is parameterized by the cohomology of $X$ relative to $L$ and satisfies properties analogous to the axioms of Gromov-Witten theory. On account of the non-orientability of $L,$ the evaluation maps of moduli spaces of $J$-holomorphic disks with boundary in $L$ may not be relatively orientable. To deal with this problem, we use recent results on orientor calculus.