[Paper Review] Lagrangian Cobordism and Fukaya Categories

Paul Biran, Octav Cornea|arXiv (Cornell University)|Apr 22, 2013

Geometric and Algebraic Topology24 references5 citations

TL;DR

This paper establishes a functorial correspondence between geometric Lagrangian cobordisms in symplectic manifolds and iterated triangular decompositions in the derived Fukaya category. Using a monoidal category of cobordisms and a triangulated resolution category, it constructs a monoidal functor from cobordisms to the derived Fukaya category, proving that any Lagrangian cobordism induces a sequence of exact triangles, thereby embedding the positive end into the triangulated subcategory generated by the negative ends.

ABSTRACT

Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M in a way that takes into account the triangulated structure of the latter.

Motivation & Objective

• To establish a functorial, algebraic correspondence between geometric Lagrangian cobordisms and triangulated structures in the derived Fukaya category.
• To resolve the non-geometric nature of the derived Fukaya category by showing that its triangulated structure arises naturally from cobordism geometry.
• To construct a monoidal category Cob(M) of Lagrangian cobordisms and a target category T SDFuk(M) encoding iterated exact triangles.
• To prove that the derived Fukaya category captures the topological and algebraic structure of cobordisms via a canonical functor eF.

Proposed method

• Define a category Cob(M) whose objects are finite ordered families of Lagrangians and morphisms are isotopy classes of admissible Lagrangian cobordisms with multiple ends.
• Construct the category T SDFuk(M) as a triangulated resolution category derived from the derived Fukaya category, parametrizing resolutions via iterated exact triangles.
• Use the Yoneda embedding and mapping cone construction in A∞-categories to define exact triangles algebraically in the derived Fukaya category.
• Establish a monoidal functor eF: Cob(M) → T SDFuk(M) that sends each Lagrangian to itself and each cobordism to a sequence of exact triangles.
• Leverage the homological unitality and triangulated closure of A∞-categories to ensure the functor respects the algebraic structure of the derived category.
• Apply Gromov-Witten theory and J-holomorphic curve counting to ensure the geometric cobordism data lifts consistently to chain-level structures.

Experimental results

Research questions

• RQ1 How can Lagrangian cobordisms be algebraically encoded in the derived Fukaya category?
• RQ2 What is the precise algebraic structure induced by a cobordism in the derived Fukaya category?
• RQ3 Does the triangulated structure of the derived Fukaya category arise functorially from geometric cobordisms?
• RQ4 Can the composition of cobordisms correspond to refinement of triangular decompositions in the derived category?
• RQ5 Is there a canonical monoidal functor from the geometric cobordism category to the derived category's resolution category?

Key findings

• The main result, Theorem B, establishes a monoidal functor eF: Cob(M) → T SDFuk(M) that maps each Lagrangian to itself and each cobordism to a sequence of exact triangles.
• Theorem A follows as a corollary: any Lagrangian cobordism with k negative ends L1,…,Lk and one positive end L induces a sequence of k−1 exact triangles in DFuk(M), showing L is in the triangulated subcategory generated by L1,…,Lk.
• The construction shows that concatenation of cobordisms corresponds to refinement of triangular decompositions, providing a geometric realization of algebraic resolutions.
• The derived Fukaya category DFuk(M) captures the triangulated structure induced by cobordisms, making the abstract triangulated closure geometrically meaningful.
• The Yoneda embedding and mapping cone construction in A∞-categories provide the algebraic machinery to realize exact triangles from cobordism data.
• The framework is independent of auxiliary choices (e.g., almost complex structures), ensuring the functor eF is well-defined and invariant under isotopy.