The symplectic arc algebra is formal

Abstract

We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology, over fields of characteristic zero. The key ingredient is the construction of a degree one Hochschild cohomology class on a Floer A-infinity algebra associated to the (k,k)-nilpotent slice Y, obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification of Y. The partial compactification is obtained as the Hilbert scheme of a partial compactification of a Milnor fibre. A sequel to this paper will prove formality of the symplectic cup and cap bimodules, and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.

58 pages, 15 figures. Final version: minor corrections

References in corpus (6)

• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Symplectic cohomology and duality for the wrapped Fukaya category
• Fukaya categories and deformations
• Hamiltonian handleslides for Heegaard Floer homology
• Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
• Quiver algebras as Fukaya categories

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