The Zero Curvature Formulation of TB, sTB Hierarchy and Topological Algebras

TL;DR

This paper presents a zero curvature approach to deriving the TB and sTB hierarchies, revealing their connection to topological algebras and 2D topological field theories.

Contribution

It introduces a zero curvature formulation for the TB and sTB hierarchies using specific gauge groups, linking integrable systems with topological algebras.

Findings

Derived TB hierarchy from zero curvature condition on SL(2,R)×U(1)

Extended to supersymmetric sTB hierarchy using OSp(2|2)

Identified topological algebras as second Hamiltonian structures

Abstract

A particular dispersive generalization of long water wave equation in $1+1$ dimensions, which is important in the study of matrix models without scaling limit, known as two--Boson (TB) equation, as well as the associated hierarchy has been derived from the zero curvature condition on the gauge group $SL(2,R)\otimes U(1)$. The supersymmetric extension of the two--Boson (sTB) hierarchy has similarly been derived from the zero curvature condition associated with the gauge supergroup $OSp(2|2)$. Topological algebras arise naturally as the second Hamiltonian structure of these classical integrable systems, indicating a close relationship of these models with 2d topological field theories.