Integrable Hierarchy for Multidimensional Toda Equations and Topological-Anti-topological Fusion

TL;DR

This paper develops an integrable hierarchy framework that unifies multidimensional Toda equations with topological-anti-topological fusion, using symmetry flows and loop group automorphisms.

Contribution

It introduces a novel integrable hierarchy incorporating negative symmetry flows and links Toda models with topological fusion through a new algebraic approach.

Findings

Incorporates negative symmetry flows into the Riemann-Hilbert problem.

Defines a sub-hierarchy with only odd symmetry flows.

Connects Toda equations with topological-anti-topological fusion.

Abstract

The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous $A_m$-hierarchy and its $\hat{gl} (m+1, C)$ extension. A loop group automorphism of order two is used to define a sub-hierarchy of $\hat{gl} (m+1, C)$ hierarchy containing only the odd symmetry flows. The positive and negative flows of the $\pm 1$ grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion.