I am going to argue that the non-vanishing gravitational anomaly in 2D CFT obstructs the existence of the well-defined notion of entanglement. As a corollary, we will also see that the non-vanishing gravitational anomaly implies the non-existence of the lattice regulator generalising the Nielsen-Ninomiya theorem. I will also generalise this to higher dimensions and cases with global symmetries.
I will also take time to discuss the modified Villain construction of compact free boson with on-site U(1)_m \times U(1)_w symmetries (which has a mixed anomaly), proposed by Gorantla, Lam, Seiberg and Shao. I will argue that this construction seemingly contradicts the global symmetry version of our theorem, but it doesn't.
Finally, I will conclude the talk with possible future directions, in particular the implication it might have for the island conjecture. The talk is based on my recent paper with Simeon Hellerman and Domenico Orlando [2101.03320].