Gluing algebraic quantum field theories on manifolds
Angelos Anastopoulos, Marco Benini
TL;DR
This work tackles descent of AQFTs on manifolds by demonstrating that gluing local algebras over open covers does not generally yield the global theory. It develops a gluing construction at the level of AQFTs using colored ∗-operads to encode both algebraic structure and causality, and shows that the construction is compatible with the extension-restriction framework of AQFTs over open subsets. The key result is that, for simple geometric AQFTs built from natural vector bundles, gluing the local AQFTs along an open cover recovers the global AQFT, while gluing the underlying local algebras fails in general. This highlights the necessity of preserving the full AQFT structure during descent and provides a robust operadic method for assembling global theories from local data, with connections to Fredenhagen's universal algebra via operadic left Kan extensions. The framework offers a precise, computable approach to global reconstruction in AQFT and clarifies when local-to-global principles hold in the presence of causality constraints.
Abstract
It has been observed that, given an algebraic quantum field theory (AQFT) on a manifold $M$ and an open cover $\{M_α\}$ of $M$, it is typically not possible to recover the global algebra of observables on $M$ by simply gluing the underlying local algebras subordinate to $\{M_α\}$. Instead of gluing local algebras, we introduce a gluing construction for AQFTs subordinate to $\{M_α\}$ and we show that for simple examples of AQFTs, constructed out of geometric data, gluing the local AQFTs subordinate to $\{M_α\}$ recovers the global AQFT on $M$.