Gluing algebraic quantum field theories on manifolds

It has been observed that, given an algebraic quantum field theory (AQFT) on a manifold $M$ and an open cover $\{M_{\alpha }\}$ 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 $\{M_{\alpha }\}$. Instead of gluing local algebras, we introduce a gluing construction for AQFTs subordinate to $\{M_{\alpha }\}$ and we show that for simple examples of AQFTs, constructed out of geometric data, gluing the local AQFTs subordinate to $\{M_{\alpha }\}$ recovers the global AQFT on $M$.