[Paper Review] Deformations of Lagrangian $NQ$-submanifolds

Miquel Cueca, Jonas Schnitzer|arXiv (Cornell University)|Sep 11, 2023

Homotopy and Cohomology in Algebraic TopologyMathematics3 citations

TL;DR

This paper establishes a deformation theory for Lagrangian $Q$-submanifolds in $ atsym$-graded symplectic $Q$-manifolds by proving graded versions of the Darboux and Weinstein tubular neighborhood theorems. It constructs an $L_{inity}$-algebra that controls deformations, showing invariance under tubular neighborhood choice and providing a geometric characterization of gauge equivalence.

ABSTRACT

In this paper we prove graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem in order to study the deformation theory of Lagrangian $NQ$-submanifolds of degree $n$ symplectic $NQ$-manifolds. Using Weinstein's Lagrangian tubular neighbourhood Theorem, we attach to every Lagrangian $NQ$-submanifold an $L_\infty$-algebra, which controls its deformation theory. The main examples are coisotropic submanifolds of Poisson manifolds and (higher) Dirac structures with support in (higher) Courant algebroids.

Motivation & Objective

• To develop a rigorous deformation theory for Lagrangian $Q$-submanifolds in $ atsym$-graded symplectic $Q$-manifolds.
• To generalize classical theorems—Darboux and Weinstein’s tubular neighborhood—to the graded setting.
• To construct an $L_{inity}$-algebra that governs the formal deformation problem of such submanifolds.
• To show that the $L_{inity}$-algebra is independent of the choice of tubular neighborhood up to isomorphism.
• To provide a geometric interpretation of the gauge equivalence relation in the deformation moduli space.

Proposed method

• Prove a graded Darboux theorem to locally model the symplectic structure of a $ atsym$-graded $Q$-manifold near a Lagrangian $Q$-submanifold.
• Establish a graded version of Weinstein’s tubular neighborhood theorem, showing that a neighborhood of a Lagrangian $Q$-submanifold is $Q$-symplectomorphic to a model neighborhood in the shifted cotangent bundle.
• Construct an $L_{inity}$-algebra from the normal bundle data of the tubular neighborhood using derived bracket constructions.
• Use Voronov’s derived bracket formalism to define the $L_{inity}$-structure on the space of sections of the normal bundle, with curvature determined by the $Q$-action.
• Show that different tubular neighborhoods yield isomorphic $L_{inity}$-algebras via $L_{inity}$-isomorphisms induced by gauge-like flows.
• Characterize the gauge action on Maurer-Cartan elements geometrically as a flow generated by Hamiltonian vector fields associated to degree-zero sections.

Experimental results

Research questions

• RQ1 How can the Weinstein tubular neighborhood theorem be generalized to the setting of $ atsym$-graded symplectic $Q$-manifolds?
• RQ2 What $L_{inity}$-algebra structure governs the deformation problem of Lagrangian $Q$-submanifolds in this graded context?
• RQ3 How does the $L_{inity}$-algebra depend on the choice of tubular neighborhood?
• RQ4 What is the geometric meaning of gauge equivalence in the moduli space of deformations?
• RQ5 How do classical geometric structures—such as coisotropic submanifolds and Dirac structures—emerge as special cases of Lagrangian $Q$-submanifolds?

Key findings

• A graded version of the Weinstein tubular neighborhood theorem is proven, showing that every Lagrangian $Q$-submanifold admits a $Q$-symplectomorphic neighborhood in the shifted cotangent bundle $T^*[1] u o u$.
• An $L_{inity}$-algebra is constructed from the normal bundle of a Lagrangian $Q$-submanifold using derived brackets, which controls the formal deformation problem.
• Two different tubular neighborhoods yield isomorphic $L_{inity}$-algebras, proving the invariance of the deformation theory under choice of model.
• The gauge action on Maurer-Cartan elements is geometrically characterized as a flow generated by Hamiltonian vector fields associated to degree-zero sections of the normal bundle.
• The construction recovers known deformation theories: coisotropic submanifolds of Poisson manifolds and Dirac structures in Courant algebroids as special cases of Lagrangian $Q$-submanifolds.
• The $L_{inity}$-algebra structure arises from a curved $L_{inity}$-algebra via the derived bracket construction on a $V$-algebra, with curvature determined by the $Q$-action.