Direct Integration and Non-Perturbative Effects in Matrix Models

Understand

We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials.

• In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes.
• This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four.
• There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $\Gamma(2)$.