  The Deligne conjecture (many times a theorem) endows Hochschild cochains of a
linear category with the structure of an $E_2$-algebra, that is, of an algebra
over the little 2-disks operad. In this paper, we prove the cyclic Deligne
conjecture, stating that for a linear category equipped with a Calabi-Yau
structure (a kind of non-commutative orientation), the Hochschild cochains is
endowed with the finer structure of a framed $E_2$-algebra, that is, of a
circle-equivariant algebra over the little 2-disks operad. Our approach applies
simultaneously to both smooth and proper linear categories, as well as to
linear functors equipped with a relative Calabi-Yau structure, and works for a
very general notion of linear category, including any dualizable presentable
$\infty$-category. As a particular application, given a compact oriented
manifold with boundary $\partial M \subset M$, our construction gives
chain-level genus zero string topology operations on the relative loop homology
$H_{*}(LM,L\partial M)$.
