Recursion structures in topological string theory and enumerative geometry

June 8 to June 12, 2009

at the

American Institute of Mathematics, San Jose, California

organized by

Vincent Bouchard, Motohico Mulase, and Brad Safnuk

Original Announcement

This workshop will be devoted to the study of a newly discovered recursion structure in topological string theory and its miraculous applications to Gromov-Witten theory and enumerative geometry.

The recursion formula was originally discovered in Random Matrix Theory as a tool to compute the genus expansion of the free energy and the n-point correlation functions of resolvents. The recursion formula is entirely geometric, relying only on simple complex analysis of the complex resolvent set of the matrices known as the spectral curve.

One can however take an opposite point of view: Start with an arbitrary plane curve C, and apply the recursion to it, as an axiom. The topological recursion then produces an infinite tower of meromorphic differentials of the curve C and symplectic invariants of the geometric data. The generating function of these quantities is always a tau-function of a KP-type integrable system. An obvious mathematical question arises: What is the recursion computing in this context? It turns out to have deep and unexpected interconnections with various areas of mathematics and physics, notably topological string theory and enumerative geometry.

When applied to their mirror curves, it has been conjectured that the recursion should govern open and closed Gromov-Witten theory on toric Calabi-Yau threefolds. When applied to the analytic curve which defines the Lambert W-function, the recursion computes generating functions of Hurwitz numbers. These exciting developments rely on physical insights, and cry for a mathematical explanation.

It is the aim of this workshop to lay down the mathematical foundations behind the appearance of the recursion in Gromov-Witten theory and topological string theory, which seem to involve new exciting interactions between geometry and integrable systems.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers
by  Bertrand Eynard, Motohico Mulase, and Brad Safnuk