Gluing together Modular flows with free fermions

We revisit the calculation of multi-interval modular Hamiltonians for free fermions using a Euclidean path integral approach. We show how the multi-interval modular flow is obtained by gluing together the single interval modular flows. Using this relation, we obtain an exact expression for the multi-interval modular Hamiltonian and entanglement entropy in agreement with existing results. An essential ingredient in our derivation is the introduction of the \emp{modular action}. This determines the non-local field theory describing the free fermion reduced density matrix, and makes manifest it's non-local conformal symmetry and $U(1)$ Kacs-Moody symmetry.