Topological expansion of the chain of matrices

We solve the loop equations to all orders in $1/N^2$, for the Chain of Matrices matrix model (with possibly an external field coupled to the last matrix of the chain). We show that the topological expansion of the free energy, is, like for the 1 and 2-matrix model, given by the symplectic invariants of the associated spectral curve. As a consequence, we find the double scaling limit explicitly, and we discuss modular properties, large $N$ asymptotics. We also briefly discuss the limit of an infinite chain of matrices (matrix quantum mechanics).