Exact triangles, Koszul duality, and coisotopic boundary conditions

TL;DR

This paper introduces a new algebraic framework involving arrowed operads and dioperads to model exact triangles and dualities, applying it to derive cochain-level duality lifts and extend the AKSZ construction for boundary conditions.

Contribution

It develops the theory of arrowed operads and dioperads, linking them to exact triangles and dualities, and applies this to boundary conditions in field theories.

Findings

Provides a cochain-level lift of relative Poincaré duality.

Introduces arrowed operads controlling derived ideals.

Extends the AKSZ construction to boundary settings.

Abstract

We develop a theory of "arrowed" (operads and) dioperads, which are to exact triangles as dioperads are to vector spaces. A central example to this paper is the arrowed operad controlling "derived ideals" for any operad. The Koszul duality theory of arrowed dioperads interacts well with rotation of exact triangles, and in particular with "exact Stars of David," which are pairs of exact triangles drawn on top of each other in an interesting way. Using this framework, we give a cochain-level lift of the "relative Poincar\'e duality" enjoyed by oriented manifolds with boundary; moreover, our cochain-level lift satisfies a natural locality-type condition, and is uniquely determined by this property. We discuss the meaning of the words "relative orientation" and "coisotropic." We extend the AKSZ construction to bulk-boundary settings with Poisson bulk fields and coisotropic boundary conditions.