Fukaya categories of Lagrangian cobordisms and duality

We introduce a new type of duality structure for $A_\infty$-categories called a relative weak Calabi-Yau pairing which generalizes Kontsevich and Soibelman's notion of a weak (proper) Calabi-Yau structure. We prove the existence of a relative weak Calabi-Yau pairing on Biran and Cornea's Fukaya category of Lagrangian cobordisms $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$. Here $M$ is a symplectic manifold which is closed or tame at infinity. This duality structure on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ extends the relative Poincar\'e duality satisfied by Floer complexes for pairs of Lagrangian cobordisms. Moreover, we show that the relative weak Calabi-Yau pairing on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ satisfies a compatibility condition with respect to the usual weak Calabi-Yau structure on the monotone Fukaya category of $M$. The construction of the relative weak Calabi-Yau pairing on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ is based on counts of curves in $\mathbb{C}\times M$ satisfying an inhomogeneous nonlinear Cauchy-Riemann equation. In order to prove the existence of this duality structure and to verify its properties, we extend the methods of Biran and Cornea to establish regularity and compactness results for the relevant moduli spaces. We also consider the implications of the existence of the relative weak Calabi-Yau pairing on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ for the cone decomposition in the derived Fukaya category of $M$ associated to a Lagrangian cobordism, and we present an example involving Lagrangian surgery.