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Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, weconstruct a commutative diagram relating the closed-open string map $CO_\lambda: QH^*(X) \to HH^*(\mathcal{F}(X)_\lambda)$ to a variant of the length-zeroclosed-open map on $L$ incorporating $\mathbf{k}[H_1(L; \mathbb{Z})]$coefficients, denoted $CO^0_\mathbf{L}$. The former is categorically importantbut very difficult to compute, whilst the latter is geometrically natural andamenable to calculation. We further show that, after a suitable completion,injectivity of $CO^0_\mathbf{L}$ implies injectivity of $CO_\lambda$. ViaSheridan's version of Abouzaid's generation criterion, this gives a powerfultool for proving split-generation of the Fukaya category. We illustrate this byshowing that the real part of a monotone toric manifold (of minimal Chernnumber at least 2) split-generates the Fukaya category in characteristic 2. Wealso give a short new proof (modulo foundational assumptions in thenon-monotone case) that the Fukaya category of an arbitrary compact toricmanifold is split-generated by toric fibres.

Hochschild cohomology of the Fukaya category via Floer cohomology with coefficients