Journal of Symplectic Geometry

Volume 19 (2021)

Number 4

Lagrangian torus invariants using $ECH = SWF$

Pages: 959 – 992



Chris Gerig (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a $3$‑torus, associated with Lagrangian tori in symplectic $4$‑manifolds and their isotopy classes. They turn out not to be new invariants, instead they repackage the Gromov (and Seiberg–Witten) invariants of various torus surgeries.We then recover a result of Morgan–Mrowka–Szabó on product formulas for the Seiberg–Witten invariants along $3$‑tori.

Received 14 November 2019

Accepted 20 December 2020

Published 8 December 2021