Mini Workshop at IPMU
A New Recursion from Random Matrices and Topological String Theory
This mini workshop is devoted to the search for
a mathematical foundation of a newly discovered
topological recursion by Eynard and Orantin, which
was further generalized by Bouchard, Klemm, Marino
and Pasquetti, in the context of topological string
theory. The recursion is originated in
Random Matrix Theory as an effective tool to
calculate free energies and npoint correlation
functions of the resolvent of Hermitian matrix
models. Here the technique involved is a simple complex
analysis on a hyperelliptic curve that appears as
the resolvent set of Hermitian random matrices.
When applied as an axiom to an arbitrary plane analytic curve, the topological recursion calculates an infinite sequence of differentials and symplectic invariants of the curve. A natural question arises: What are these quantities calculating?
Miraculously, a particular choice of the curve, the spectral curve of the theory, reconstructs the WittenKontsevich theory. Moreover, by deforming this spectral curve, the recursion reproduces the generalizations of the WittenKontsevich theory due to MulaseSafnuk and LiuXu. Amazingly, a particular specialization of the deformation also gives the Mirzakhani recursion formula for the WeilPetersson volume of the moduli spaces of bordered hyperbolic surfaces.
BKMP further show that when the curve is defined by the Lambert Wfunction, the recursion provides an effective formula to compute linear Hodge integrals and Hurwitz numbers. This formula has been unknown to the mathematics community.
Even more miraculous is the claim of BKMP that the EynardOrantin recursion computes the GromovWitten invariants of toric CalabiYau 3folds through mirror symmetry. An attempt of understanding the BCOV anomaly equation is also proposed by DijkgraafVafa.
In this mini workshop we provide an ample time to learn the origin of the topological recursion from its discoverer, Bertrand Eynard. He will also talk about the new derivation of WittenKontsevich theory and its various generalizations, as well as his most recent results.
Mulase will talk about an integrable system approach to the topological recursion using a deformation theory of taufunctions (joint work with Safnuk, in preparation).
When applied as an axiom to an arbitrary plane analytic curve, the topological recursion calculates an infinite sequence of differentials and symplectic invariants of the curve. A natural question arises: What are these quantities calculating?
Miraculously, a particular choice of the curve, the spectral curve of the theory, reconstructs the WittenKontsevich theory. Moreover, by deforming this spectral curve, the recursion reproduces the generalizations of the WittenKontsevich theory due to MulaseSafnuk and LiuXu. Amazingly, a particular specialization of the deformation also gives the Mirzakhani recursion formula for the WeilPetersson volume of the moduli spaces of bordered hyperbolic surfaces.
BKMP further show that when the curve is defined by the Lambert Wfunction, the recursion provides an effective formula to compute linear Hodge integrals and Hurwitz numbers. This formula has been unknown to the mathematics community.
Even more miraculous is the claim of BKMP that the EynardOrantin recursion computes the GromovWitten invariants of toric CalabiYau 3folds through mirror symmetry. An attempt of understanding the BCOV anomaly equation is also proposed by DijkgraafVafa.
In this mini workshop we provide an ample time to learn the origin of the topological recursion from its discoverer, Bertrand Eynard. He will also talk about the new derivation of WittenKontsevich theory and its various generalizations, as well as his most recent results.
Mulase will talk about an integrable system approach to the topological recursion using a deformation theory of taufunctions (joint work with Safnuk, in preparation).
Date:  December 11  13, 2008  
Place:  Seminar Room at IPMU Prefab. B, Kashiwa Campus of the University of Tokyo  
Organizer:  Akihiro Tsuchiya (IPMU) and Motohico Mulase (Davis)  
Speaker: 


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