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Register a new userCovariantly functorial wrapped Floer theory on Liouville sectors
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We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaids generation criterion follows.
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In all known explicit computations on Weinstein manifolds, the self-wrapped Floer homology of non-compact exact Lagrangian is always either infinite-dimensional or zero. We show that a global variant of this observed phenomenon holds in broad generality: the wrapped Fukaya category of any positive-dimensional Weinstein (or non-degenerate Liouville) manifold is always either non-proper or zero, as is any quotient thereof. Moreover any non-compact connected exact Lagrangian is always either a (both left and right) non-proper object or zero in such a wrapped Fukaya category, as is any idempotent summand thereof. We also examine criteria under which the argument persists or breaks if one drops exactness, which is consistent with known computations of non-exact wrapped Fukaya categories which are smooth, proper, and non-vanishing (e.g., work of Ritter-Smith).
The Nadler-Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms at infinity, given on the Floer side by Reeb trajectories (also known as wrapping) and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry. More precisely, given a real analytic manifold $M$ and a subanalytic isotropic subset $Lambda$ of its co-sphere bundle $S^*M$, we show that the partially wrapped Fukaya category of $T^*M$ stopped at $Lambda$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $Lambda$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.
We use quilted Floer theory to construct functor-valued invariants of tangles arising from moduli spaces of flat bundles on punctured surfaces. As an application, we show the non-triviality of certain elements in the symplectic mapping class groups of moduli spaces of flat bundles on punctured spheres.
We study threefolds $Y$ fibred by $A_m$-surfaces over a curve $S$ of positive genus. An ideal triangulation of $S$ defines, for each rank $m$, a quiver $Q(Delta_m)$, hence a $CY_3$-category $(C,W)$ for any potential $W$ on $Q(Delta_m)$. We show that for $omega$ in an open subset of the Kahler cone, a subcategory of a sign-twisted Fukaya category of $(Y,omega)$ is quasi-isomorphic to $(C,W_{[omega]})$ for a certain generic potential $W_{[omega]}$. This partially establishes a conjecture of Goncharov concerning `categorifications of cluster varieties of framed $PGL_{m+1}$-local systems on $S$, and gives a symplectic geometric viewpoint on results of Gaiotto, Moore and Neitzke.
We construct partial category-valued field theories in (2+1)-dimensions using Lagrangian Floer theory in moduli spaces of central-curvature unitary connections with fixed determinant of rank r and degree d where r,d are coprime positive integers. These theories associate to a closed, connected, oriented surface the Fukaya category of the moduli space, and to a connected bordism between two surfaces a functor between the Fukaya categories. We obtain the latter by combining Cerf theory with holomorphic quilt invariants.
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