Complex geometry in the entanglement entropy of fermionic chains

The geometry of Riemann surfaces plays a relevant role in the study of entanglement entropy in the ground state of a free fermionic chain. Recently, a new symmetry for the entropy in non critical theories has been discovered. It is based on the Mobius transformations in a compact Riemann surface associated to the Hamiltonian of the system. Here, we argue how to extend it to critical theories supporting our conjectures with numerical tests. We also highlight the intriguing parallelism that exists with conformal symmetry in real space.