Invariants for Lagrangian tori (Q1882838)

Let \(X\) be a symplectic \(4\)-manifold and \(L\) an embedded null-homologous Lagrangian torus inside \(X\). There are two obvious trivilizations of a tubular neighbourhood of \(L\), the Lagrangian trivialization and the null-homologous trivialization. The difference of both defines an invariant \(\lambda(T)\). This is used to show that some symplectic \(4\)-manifolds have infinitely many pairwise symplectically inequivalent null-homologous Lagrangian tori. The symplectic \(4\)-manifolds used to show this behaviour are obtained by connected sum along a torus of a symplectic \(4\)-manifold with \(S^1\times M_K\), where \(M_K\) is the \(0\)-surgery of \(S^3\) along a fibered knot \(K\), see \textit{R. Fintushel} and \textit{R. Stern} [J. Differ. Geom. 52, No.2, 203-222 (1999; Zbl 0981.53085)]. In some examples, \(\lambda(T)\) is detected by the Seiberg-Witten invariants of the given \(4\)-manifold and those obtained by surgery along \(T\), using the results of \textit{J. Morgan, T. Mrowka} and \textit{Z. Szabó} [Math. Res. Lett. 4, No.6, 915-929 (1997; Zbl 0892.57021)]. In this case, \(\lambda(T)\) is a diffeomorphism invariant. In addition, the invariant \(\lambda(T)\) is used to show that many symplectic \(4\)-manifolds have non-trivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved in [\textit{S. Vidussi}, Lagrangian surfaces in a fixed homology class: existence of knotted Lagrangian tori, J. Differ. Geom. (to appear)].