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Expansions in matrix models, topological recursion and applications

Le : 11/10/2016 14h00
Par : Gaëtan Borot (MPI)
Lieu : I 001
Lien web :
Résumé : The topological recursion, initially developed by Eynard and Orantin in the context of matrix models, has found since then many applications, in enumerative geometry, integrable systems, topological field theories and topological strings, ... After a short presentation of the framework of the topological recursion, this talk will review the results obtained in the last 10 years on formal matrix integrals and asymptotics of convergent matrix integrals. They both satisfy the same Schwinger-Dyson equations -- coming from invariance under diffeomorphism of the integration map -- but their nature differ. In the formal setting, there is by construction an formal series expansion in 1/N (N is the size of the matrix), while in the convergent setting, one can show 1/N or oscillatory all-order asymptotic expansion. Although the techniques to study them differ (analytic combinatorics/probability and functional analysis), the final result is the same, i.e. the coefficients of expansion (formal or asymptotic) are governed by a universal topological recursion. This will be illustrated by several applications among TQFT, topological gravity, Chern-Simons theory on Seifert spaces. I will also indicate the limits of this approach in relation with two examples coming from quantum integrable systems and matrix models with discrete eigenvalues.