Functoriality in categorical symplectic geometry (Q6621308)

This paper surveys the expected construction of a symplectic category, called extended Fukaya category, which is an \((A_{\infty},2)\)-category with objects symplectic manifolds and morphisms Lagrangian correspondences between symplectic manifolds, which form \(A_{\infty}\)-categories between symplectic manifolds. \(2\)-morphisms between Lagrangian correspondences form a chain complex whose generators are intersection points between Lagrangian submanifolds and whose differentials are counting quilted pseudo-holomorphic strips, or equivalently witch spheres in the product symplectic manifold. There are higher structures preventing product structures on the given chain complex to be associative, and there is a program in order to show that relations of higher structures, coming from compactification of moduli spaces of witch spheres, can be grouped together to give an \((A_{\infty},2)\)-operad, which is explicitly defined in this paper.\N\NSection 2 of this paper gives a wonderful and comprehensive introduction to Fukaya categories. The authors sketch the construction of (compact) Fukaya categories on compact symplectic manifolds, provide a thoroughly computed example of the Fukaya category of the sphere \(\mathbb{S}^2\) with its standard symplectic structure, and also point out under what assumptions on symplectic manifolds, which part of the construction would work. This section also provides a modern point of view toward the \(A_{\infty}\)-structure on the Fukaya category under the framework of operads, and shows how the classical construction of \(A_{\infty}\)-structures via counting pseudo-holomorphic disks coincides with the modern description of \(A_{\infty}\)-algebras as algebras over the \(A_{\infty}\)-operad.\N\NSection 3 of this paper discusses quilted Floer theory, which is essential for defining \(2\)-morphisms between Lagrangian correspondences. This can be regarded as a generalization of Lagrangian Floer theory, where we count Hamiltonian-perturbed pseudo-holomorphic disks with boundary punctures asymptotic to Hamiltonian chords and boundary edges between punctures lying on corresponding Lagrangian submanifolds. In quilted Floer theory, we consider pseudo-holomorphic quilted strips, which consists of a finite collection of vertical lines in \(\mathbb{C}\), where regions between two vertical lines can be regarded as pseudo-holomorphic strips mapping to one of the symplectic manifolds, where the vertical lines are mapped to the image under projection of the corresponding Lagrangian correspondences. Counts of pseudo-holomorphic quilted strips give operations on morphisms on the given sequence of Lagrangian correspondences, and their degenerations would tell us relations between these operations. The authors also discuss possible degenerations in this section, where apart from standard degenerations similar to pseudo-holomorphic curves, there is an extra degeneration, which they call figure-eight bubbles. They provide an explanation for the appearance of the figure-eight bubbles, that it corresponds to the choice of a bounding cochain for the Lagrangian Floer theory to be well-defined.\N\NSection 4 discusses the underlying operadic structure of the symplectic \(2\)-category, which they call the \((A_{\infty},2)\)-operad relative to the \(A_{\infty}\)-operad. This operad is different from the absolute one as defined in section 2 due to the problem that boundary strata of the moduli space are not always the product of two moduli spaces. Nonetheless, they provide the framework of a relative operad and introduce a program on showing that the \(2\)-category of symplectic manifolds admits such a relative operadic structure.\N\NSection 5 surveys some existing applications of Lagrangian correspondences, including a generalization of Seidel's long exact sequence for Dehn twists, construction of the Floer field theory stepping toward the Atiyah-Floer conjecture, realizing monoidal structures on the Fukaya category mirror to derived tensor products on coherent sheaves, formal group structures on the moduli space of Maurer-Cartan elements, and a Barr-Beck type result for Fukaya categories.