The cyclic Deligne conjecture and Calabi-Yau structures

Speaker: Chris Brav (SIMIS)

Time: 2025-12-20 09:00-10:30
Location: 1110, SIMIS
Zoom Meeting ID: 886 8880 9649 Passcode: SIMIS

Abstract:

The Deligne conjecture, many times a theorem, states that for a dg category C, the dg endomorphisms
End(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 that the 2-algebra
structure and the circle action together give a framed 2-algebra structure on Hochschild cochains. We establish the cyclic Deligne conjecture,
as well as a variation that works for relative Calabi-Yau structures on dg functors D –> C,
more generally for functors between stable infinity categories. We discuss examples coming from
oriented manifolds with boundary, Fano varieties with anticanonical divisor, and doubled quivers with preprojective relation.