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Nicolas Orantin, Instituto Superior Técnico

Generalized loop equations and their formal solution on deformed
spectral curves

Many formal matrix models have been proved to be solvable by the so-called topological recursion formalism. However, the universality of this solution was still mysterious since it had to be derived case by case. In this talk, I will explain how to solve many formal matrix models by deriving some generalized loop equations for sets of normal random matrices whose eigenvalues have an arbitrary interaction. I will show how the deformation of the interaction among eigenvalues maps to a deformation of the spectral curve associated. This talk will be mainly based on the example of loop models on random surfaces to explicit this construction. If time allows, I will also explain why this procedure can be seen as a matrix model counterpart of Givental group action in the context of Gromov-Witten theory.

Based on a joint work with G. Borot and B. Eynard.