[Paper Review] Symplectic cohomology and duality for the wrapped Fukaya category

Sheel Ganatra|arXiv (Cornell University)|Apr 27, 2013

Geometric and Algebraic TopologyMathematics24 references91 citations

TL;DR

This paper establishes a non-compact version of Kontsevich's conjecture by proving that, under a non-degeneracy condition, the natural geometric maps from Hochschild homology to symplectic cohomology and from symplectic cohomology to Hochschild cohomology of the wrapped Fukaya category are isomorphisms, compatible with ring and module structures. The key result is a duality between Hochschild homology and cohomology for the wrapped Fukaya category, realized via a new geometric Poincaré duality isomorphism and generalized Fourier-Mukai theory using wrapped holomorphic quilts.

ABSTRACT

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) Fourier-Mukai theory for W via a wrapped version of holomorphic quilts, (2) new geometric operations, coming from discs with two negative punctures and arbitrary many positive punctures, (3) a generalization of the Cardy condition, and (4) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal.

Motivation & Objective

• To establish a non-compact analogue of Kontsevich's conjecture relating quantum cohomology to Hochschild cohomology of the Fukaya category.
• To prove that symplectic cohomology is isomorphic to both Hochschild homology and cohomology of the wrapped Fukaya category under a non-degeneracy condition.
• To construct a geometric Poincaré duality isomorphism between Hochschild homology and cohomology for the wrapped Fukaya category with arbitrary coefficient bimodules.
• To relate the non-degeneracy condition to smoothness of the wrapped Fukaya category via A-infinity categorical techniques.
• To generalize the Cardy condition and introduce new geometric operations from holomorphic discs with two negative punctures and multiple positive punctures.

Proposed method

• Introduce a wrapped version of holomorphic quilts to develop Fourier-Mukai theory for the wrapped Fukaya category.
• Define new geometric operations using holomorphic discs with two negative punctures and arbitrary positive punctures to model higher operations in the category.
• Generalize the Cardy condition to incorporate these new operations and establish compatibility with the duality structure.
• Use homotopy units and A-infinity shuffle products to relate the non-degeneracy condition to a resolution of the diagonal in the wrapped Fukaya category.
• Construct a direct geometric Poincaré duality isomorphism between Hochschild homology and cohomology, bypassing symplectic cohomology.
• Perform detailed orientation calculations using moduli spaces of holomorphic discs with multiple asymptotic inputs, tracking sign contributions via orientation lines and lambda-classes.

Experimental results

Research questions

• RQ1 Does a non-compact version of Kontsevich's conjecture hold, with symplectic cohomology isomorphic to Hochschild cohomology of the wrapped Fukaya category?
• RQ2 Can a geometric Poincaré duality isomorphism be constructed directly between Hochschild homology and cohomology of the wrapped Fukaya category?
• RQ3 How does the non-degeneracy condition—implying existence of enough Lagrangians—relate to the smoothness of the wrapped Fukaya category?
• RQ4 What is the role of new geometric operations from discs with two negative punctures in realizing duality structures?
• RQ5 How do orientation signs in moduli spaces of holomorphic discs contribute to the consistency of the duality isomorphism?

Key findings

• The natural maps from Hochschild homology to symplectic cohomology and from symplectic cohomology to Hochschild cohomology are isomorphisms when the Liouville manifold is non-degenerate.
• A direct geometric Poincaré duality isomorphism exists between Hochschild homology and cohomology of the wrapped Fukaya category with arbitrary coefficient bimodules.
• The wrapped Fukaya category is smooth if the underlying Liouville manifold is non-degenerate, as established via resolution of the diagonal using A-infinity shuffle products.
• The non-degeneracy condition ensures that the collection of Lagrangians generates symplectic cohomology, in particular hitting the identity class.
• The orientation sign calculations in the moduli spaces of holomorphic discs are consistent and lead to a well-defined duality structure, with all sign contributions combining to a canonical sign twist.
• The generalized Cardy condition and new operations from discs with two negative punctures are essential for realizing the duality and ensuring compatibility with ring and module structures.