The cyclic Deligne conjecture for relative Calabi-Yau structures

The Deligne conjecture, many times a theorem, states for a dg category $C$, the dg endomorphisms $\mathrm{End}(\mathrm{Id}_C)$ of the identity functor – that is, the Hochschild cochains – carries a natural structure of $2$-algebra. When $C$ is endowed with a Calabi-Yau structure, then Hochschild cochains and Hochschild chains are identified up to a shift, and we may transport the circle action from Hochschild chains onto Hochschild cochains. The cyclic Deligne conjecture states the $2$-algebra structure and the circle action together give a framed $2$-algebra structure on Hochschild cochains. We establish a generalization of the cyclic Deligne conjecture that works for relative Calabi-Yau structures on dg functors $D \to C$. We discuss examples coming from oriented manifolds with boundary, Fano varieties with anticanonical divisor, and doubled quivers with preprojective relation. This is joint work with Nick Rozenblyum.