Research

Drinfeld-Sokolov hierarchies

An integrable system is a particularly simple kind of a Poisson manifold. The most famous example used in geometry is the Hitchin system, whose quantum version is the main player in the geometric Langlands program. While the classical Hitchin system is finite-dimensional, there are several infinite-dimensional integrable systems as well. One example is the so-called Drinfeld-Sokolov hierarchy, which is defined for any complex curve and a semisimple group. When the curve has genus 0 and the group is SL₂, the corresponding integrable system is the well-known Korteweg-de Vries hierarchy.

Although the Drinfeld-Sokolov system is integrable and the global structure of the space of solutions is known, it is hard to pinpoint any particular solution. One class of solutions is given by the algebro-geometric solutions of Krichever. The data for these solutions is a Higgs bundle on the base curve with a special ramification structure at infinity. However, the famous solution of Kontsevich-Witten corresponding to the partition function of 2d quantum gravity is not of this sort and obeys a collection of equations known as the Virasoro constraints, string equation being the first. I describe geometrically that these "string" solutions are obtained from the data of a flat principal bundle, such that the connection has a special structure at infinity. In particular, it allows one to obtain Virasoro constraints on the tau-function of arbitrary Drinfeld-Sokolov systems.

A preliminary version is available at arXiv:1302.3540. Any comments are welcome.

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