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Abstract |
For some decades, deep connections have been forming among enumerative geometry,
complex geometry, intersection theory and integrability. The topological recursion
is a universal procedure that helps building these connections that was introduced
around 2007 by Chekhov, Eynard and Orantin. It associates to some initial data
called spectral curve, consisting of a Riemann surface and some extra data, a
doubly indexed family of differentials on the curve, which often encode some
enumerative geometric information, such as volumes of moduli spaces, matrix model
correlation functions and intersection numbers. After an introduction to
topological recursion and its relation to different topics, I will focus on the
integrability part. The quantum curve conjecture claims that one can associate to
a spectral curve a differential equation, whose solution can be reconstructed by
the topological recursion applied to the original spectral curve. I will present
this problem in some simple cases and comment on some of the technicalities that
arise when proving the conjecture for algebraic spectral curves of arbitrary rank,
like having to consider non-perturbative corrections. The last part will be based
on joint work with B. Eynard, N. Orantin and O. Marchal. |
