On the cohomology ring of narrow Lagrangian 3-manifolds, quantum Reidemeister torsion, and the Landau-Ginzburg superpotential

TL;DR

This paper investigates the algebraic and geometric properties of monotone Lagrangian 3-manifolds, revealing a dichotomy in their cohomology rings and linking quantum invariants to Gromov-Witten invariants.

Contribution

It establishes a dichotomy in the cohomology ring structure of certain Lagrangian 3-manifolds and connects quantum Reidemeister torsion to open Gromov-Witten invariants.

Findings

Cohomology ring exhibits a parity-dependent dichotomy.

Landau-Ginzburg superpotential is either non-vanishing or constant.

Quantum Reidemeister torsion is invariant and expressible via Gromov-Witten invariants.

Abstract

Let $L$ be a closed, orientable, monotone Lagrangian 3-manifold of a symplectic manifold $(M, \omega)$, for which there exists a local system such that the corresponding Lagrangian quantum homology vanishes. We show that its cohomology ring satisfies a certain dichotomy, which depends only on the parity of the first Betti number of $L$. Essentially, the triple cup product on the first cohomology group is shown to be either of maximal rank or identically zero. This in turn influences the Landau-Ginzburg superpotential of $L$: either one of its partial derivatives do not vanish on the corresponding local system, or it is globally constant. We use this to prove that quantum Reidemeister torsion is invariant and can be expressed in terms of open Gromov-Witten invariants of $L$.