Lagrangian torus invariants using $ECH = SWF$

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.

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Received 14 November 2019

Accepted 20 December 2020

Published 8 December 2021