Symplectic quasi-states on the quadric surface and Lagrangian submanifolds

TL;DR

This paper demonstrates the existence of distinct symplectic quasi-states on the monotone complex quadric surface, supported on disjoint Lagrangian submanifolds, using spectral sequences and symplectic field theory techniques.

Contribution

It provides a proof that the unities of the split quantum homology fields induce different symplectic quasi-states supported on separate Lagrangian submanifolds.

Findings

Quantum homology of the quadric surface splits into two fields.

Distinct symplectic quasi-states are supported on disjoint Lagrangian submanifolds.

Spectral sequence approach with symplectic field theory computes differentials.

Abstract

The quantum homology of the monotone complex quadric surface splits into the sum of two fields. We outline a proof of the following statement: The unities of these fields give rise to distinct symplectic quasi-states defined by asymptotic spectral invariants. In fact, these quasi-states turn out to be "supported" on disjoint Lagrangian submanifolds. Our method involves a spectral sequence which starts at homology of the loop space of the 2-sphere and whose higher differentials are computed via symplectic field theory, in particular with the help of the Bourgeois-Oancea exact sequence.