On the Quantum Homology of Real Lagrangians in Fano Toric Manifolds

TL;DR

This paper investigates the quantum homology of real Lagrangian submanifolds in Fano toric manifolds, revealing their wide nature and isomorphism with the ambient quantum homology, using Laurent polynomial coefficients over Z/2Z.

Contribution

It demonstrates that real Lagrangians in Fano toric manifolds are wide and their quantum homology is isomorphic to that of the ambient manifold, with coefficients in Laurent polynomials over Z/2Z.

Findings

Real Lagrangians are wide in Fano toric manifolds.

Quantum homology of these Lagrangians matches that of the ambient manifold.

Quantum homology is isomorphic as a module and as a ring to classical homology and ambient quantum homology.

Abstract

We study the Lagrangian quantum homology of real parts of Fano toric manifolds of minimal Chern number at least 2, using coefficients in a ring of Laurent polynomials over Z/2Z. We show that these Lagrangians are wide, in the sense that their quantum homology is isomorphic as a module to their classical homology tensored with this ring. Moreover, we show that the quantum homology is isomorphic as a ring to the quantum homology of the ambient symplectic manifold.