String Theory Journal Club

In our last episode we saw how to derive the loop equations for formal matrix integrals. Now following sections 3.1, 3.2 and 3.3 of B. Eynard's book (Counting Surfaces) we will learn how to solve the loop equations in two main steps. First, we solve for the two leading contributions, the disk and cylinder amplitudes. They are often referred to as unstable geometries because of their Euler characteristic being negative \xi < 0. Here we will introduce the notion of a spectral curve and its fundamental differential of the second kind. We will derive the spectral curves for the Gaussian, cubic and quartic matrix integrals. Second, we compute the remaining contributions. These are often called stable geometries because they are Rieman surfaces of genus g and n marked points with Euler characteristic \xi >=0. This is where the topological recursion appears, as recursion relations depend only on the topology of Riemann surfaces. We will finish with a couple of examples, the Gaussian and Airy matrix integrals. There we will notice that for two very different matrix integrals, they satisfy the same (topological) recursion relations but for different spectral curves. This is what is known as universality of topological recursion. Link: meet.google.com/gbj-tmns-err