Hochschild Cohomology of the Fukaya Category via Floer Cohomology with Coefficients

Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map $\mathcal{CO}_\lambda \colon \operatorname{QH}^*(X) \to \operatorname{HH}^*(\mathcal{F} (X)_\lambda)$ to a variant of the length-zero closed-open map on $L$ incorporating $\mathbf{k}[\operatorname{H}_1(L; \mathbb{Z})]$ coefficients, denoted $\mathcal{CO}^0_\mathbf{L}$. 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 $\mathcal{CO}^0_\mathbf{L}$ implies injectivity of $\mathcal{CO}_\lambda$. 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.