$A_\infty$ relations in orientor calculus on moduli of stable disk maps

TL;DR

This paper introduces orientors and orientor calculus to handle orientation issues in moduli spaces of holomorphic disks, enabling the construction of Fukaya $A_infty$ algebras for non-orientable Lagrangians.

Contribution

It develops the concept of orientors and their calculus to systematically address orientation problems in moduli spaces of holomorphic disks.

Findings

Defined the notion of an orientor for non-orientable Lagrangians.

Established relations between orientors via orientor calculus.

Set the stage for constructing Fukaya $A_infty$ algebras using these tools.

Abstract

Let $L\subset X$ be a not necessarily orientable relatively $Pin$ Lagrangian submanifold in a symplectic manifold $X$. Evaluation maps of moduli spaces of $J$-holomorphic disks with boundary in $L$ may not be relatively orientable. To deal with this problem, we introduce the notion of an orientor. Interactions between the natural operations on orientors are governed by orientor calculus. Orientor calculus gives rise to a family of orientors on moduli spaces of $J$-holomorphic stable disk maps with boundary in $L$ that satisfy natural relations. In a sequel, we use these orientors and relations to construct the Fukaya $A_\infty$ algebra of $L.$