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16120015

About a decade ago Eynard and Orantin proposed a powerful computation algorithm

known as topological recursion. Starting with a ``spectral curve" and some ``initial data"

(roughly, meromorphic differentials of order one and two) the topological recursion produces by induction

a collection of symmetric meromorphic differentials on the spectral curves parametrized by

pairs of non-negative integers (g,n) (g should be thought of as a genus and n as the number of punctures).

Despite of many applications of the topological recursion (matrix integrals, WKB expansions, TFTs, etc.etc.)

the nature of the recursive relations was not understood.

Recently, in a joint work with Maxim Kontsevich we found a simple underlying structure of the recursive relations of Eynard and Orantin. We call it ``Airy structure". In this talk I am going to define this notion and explain how the recursive relations of Eynard and Orantin follow from the quantization of

a quadratic Lagrangian subvariety in a symplectic vector space.

©2012 Perimeter Institute for Theoretical Physics