Two-dimensional Yang-Mills theory is a useful model of an exactly solvable gauge theory with a string theory dual at large N . We calculate entanglement entropy in the 1/N expansion by mapping the theory to a system of N fermions interacting via a repulsive entropic force. The entropy is a sum of two terms: the "Boltzmann entropy", log dim(R) per point of the entangling surface, which counts the number of distinct microstates, and the "Shannon entropy", − p R log p R , which captures fluctuations of the macroscopic state. We find that the entropy scales as N 2 in the large N limit, and that at this order only the Boltzmann entropy contributes. We further show that the Shannon entropy scales linearly with N , and confirm this behaviour with numerical simulations. While the term of order N is surprising from the point of view of the string dual -in which only even powers of N appear in the partition function -we trace it to a breakdown of large N counting caused by the replica trick. This mechanism could lead to corrections to holographic entanglement entropy larger than expected from semiclassical field theory.

We argue that quantum gravity is nonlocal, first by recalling well-known arguments that support this idea and then by focusing on a point not usually emphasized: that making a conventional effective field theory (EFT) for quantum gravity is particularly difficult, and perhaps impossible in principle. This inability to realize an EFT comes down to the fact that gravity itself sets length scales for a problem: when integrating out degrees of freedom above some cutoff, the effective metric one uses will be different, which will itself re-define the cutoff. We also point out that even if the previous problem is fixed, naïvely applying EFT in gravity can lead to problems — we give a particular example in the case of black holes.