Barcode of a pair of compact exact Lagrangians in a punctured exact two-dimensional symplectic manifold

TL;DR

This paper introduces a modified Floer complex for pairs of exact Lagrangians in a punctured symplectic surface, showing its invariance and relating its barcode to classical invariants, extending Viterbo's conjecture.

Contribution

It develops a new Floer complex that tracks intersections with a distinguished point and proves its barcode invariance, extending spectral norm bounds to punctured cotangent bundles.

Findings

The modified Floer complex is invariant under Hamiltonian isotopy.

The barcode of the modified complex matches classical barcodes.

Extension of Viterbo's conjecture to punctured cotangent bundles.

Abstract

In this article, we modify the classical Floer complex $CF(L_0,L_1)$ of a pair of two compact exact Lagrangian submanifolds $L_0,L_1$ of an exact symplectic 2-manifold $M$ into a $\mathbb{Z}_2[T]$-complex $CF_h(L_0,L_1)$, whose differential keeps track of how many times a pseudo-holomorphic strip passes through a distinguished point $h\in M$. We show that this complex is invariant under Hamiltonian isotopy, and we prove that its barcode, if it exists, is the same as both barcodes $\mathcal{B}(CF(L_0,L_1;M))$ and $\mathcal{B}(CF(L_0,L_1;M\setminus \{h \}))$. This allows us to extend a conjecture of Viterbo, which states that for every Hamiltonian isotopy $\phi^1_H(L)$ in $D^* L$, the spectral norm $\gamma(L,\phi^1_H(L))$ remains bounded independently of $H$, to the case of $D^* L$ with a point removed.