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

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.

arXiv comment58 pages, includes summary of relevant background from arXiv:2211.05117