Hochschild Cohomology of the Fukaya Category via Floer Cohomology with Coefficients

Abstract

Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map ${\mathcal{C}\mathcal{O}}_{\lambda }:{\mathrm{QH}}^{\ast }(X)\to {\mathrm{HH}}^{\ast }(\mathcal{F}(X{)}_{\lambda })$ to a variant of the length-zero closed-open map on $L$ incorporating $\mathbf{k}[H_1(L;\mathbb{Z})]$ coefficients, denoted ${\mathcal{C}\mathcal{O}}_{\mathbf{L}}^0$. The former is categorically important but very difficult to compute, whilst the latter is geometrically natural and amenable to calculation. We further show that, after a suitable completion, injectivity of ${\mathcal{C}\mathcal{O}}_{\mathbf{L}}^0$ implies injectivity of ${\mathcal{C}\mathcal{O}}_{\lambda }$. Via Sheridan's version of Abouzaid's generation criterion, this gives a powerful tool for proving split-generation of the Fukaya category. We illustrate this by showing that the real part of a monotone toric manifold (of minimal Chern number at least 2) split-generates the Fukaya category in characteristic 2. We also give a short new proof (modulo foundational assumptions in the non-monotone case) that the Fukaya category of an arbitrary compact toric manifold is split-generated by toric fibres.