Abstract:
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.