Deformations of Lagrangian \(NQ\)-submanifolds (Q6634745)

The authors study Lagrangian submanifolds (and Lagrangian \(Q\)-submanifolds) in non-negatively graded symplectic manifolds (and symplectic \(NQ\)-manifolds). Like in the classical non-graded case, a graded symplectic manifold is a graded manifold \(\mathcal M\) together with a symplectic form, i.e., a \(2\)-form \(\omega\) which is additionally closed under the de Rham differential and non-degenerate, meaning that the flat map from vector fields to \(1\)-forms is an isomorphism of graded modules. A symplectic \(NQ\)-manifold is a graded symplectic manifold \((\mathcal M, \omega)\) with coordinates concentrated in non-negative degrees, additionally equipped with a homological vector field \(Q\) preserving \(\omega\): The Lie derivative of \(\omega\) along \(Q\) vanishes. Like in the classical non-graded case, a Lagrangian submanifold in a graded symplectic manifold \((\mathcal M, \omega)\) is a coisotropic submanifold \(S \subseteq \mathcal M\) (i.e., the vanishing ideal of \(S\) is closed under the Poisson bracket of \(\omega\)) of (total) dimension half equal to half the (total) dimension of \(\mathcal M\). A Lagrangian \(NQ\)-submanifold in a symplectic \(NQ\)-manifold \((\mathcal M, \omega, Q)\) is a Lagrangian submanifold \(S\) of \((\mathcal M, \omega)\) additionally preserved by the homological vector field, i.e., \(Q\) preserves the vanishing ideal of \(S\), yet in other words, \(Q\) is tangent to \(S\).\N\NThe main results of the paper are the following (clearly listed in the introduction):\N\begin{itemize}\N\item[1)] versions of the Darboux lemma and the Weinstein Lagrangian tubular neighborhood theorem for non-negatively graded symplectic manifolds;\N\item[2)] the construction of an \(L_\infty\)-algebra \(\mathfrak L\) from a tubular neighborhood of a Lagrangian \(NQ\)-submanifold, together with a proof that \(\mathfrak L\) is independent of the tubular neighborhood up to \(L_\infty\)-isomorphisms;\N\item[3)] a proof that the \(L_\infty\)-algebra \(\mathfrak L\) controls formal deformations of the Lagrangian \(NQ\)-submanifold inside the symplectic \(NQ\)-manifold.\N\item[4)] a geometric characterization of gauge equivalence of Maurer-Cartan elements in \(\mathfrak L\).\N\end{itemize}\NPoints 2) and 3) greatly generalize older results by \textit{A. S. Cattaneo} and \textit{G. Felder} [Adv. Math. 208, No. 2, 521--548 (2007; Zbl 1106.53060)] and \textit{F. Schätz} and \textit{M. Zambon} [Lett. Math. Phys. 103, No. 7, 777--791 (2013; Zbl 1282.53073)] on coisotropic submanifolds of a Poisson manifold and their deformations.