Generalized Integrability and two-dimensional Gravitation

TL;DR

This paper reviews the construction of generalized integrable hierarchies linked to affine Kac-Moody algebras, which can be utilized to develop new models in 2D quantum gravity and $ ext{W}$-algebras.

Contribution

It introduces a unified framework for integrable hierarchies associated with affine Kac-Moody algebras, expanding their applications in 2D gravity and algebraic structures.

Findings

Constructed generalized integrable hierarchies including Drinfel'd and Sokolov's models.

Demonstrated applications to 2D quantum and topological gravity.

Developed new $ ext{W}$-algebras from these hierarchies.

Abstract

We review the construction of generalized integrable hierarchies of partial differential equations, associated to affine Kac-Moody algebras, that include those considered by Drinfel'd and Sokolov. These hierarchies can be used to construct new models of 2D quantum or topological gravity, as well as new $\cal W$-algebras.