Note on Nonlinear Schr\"odinger Equation, KdV Equation and 2D Topological Yang-Mills-Higgs Theory

TL;DR

This paper explores the deep connections between nonlinear Schr"odinger, KdV equations, and 2D topological Yang-Mills-Higgs theory, revealing dualities at classical and quantum levels through boson/vortex duality and Bethe Ansatz equations.

Contribution

It proposes a novel duality web linking classical and quantum integrable systems with topological gauge theories in the UV regime.

Findings

Classical NLS maps to KdV via boson/vortex duality in the UV limit.

Quantum NLS and KdV correspond to Bethe Ansatz solutions of XXX and XXZ chains.

A duality web connecting NLS, KdV, and 2D topological Yang-Mills-Higgs theory is proposed.

Abstract

In this paper we discuss the relation between the (1+1)D nonlinear Schr\"odinger equation and the KdV equation. By applying the boson/vortex duality, we can map the classical nonlinear Schr\"odinger equation into the classical KdV equation in the small coupling limit, which corresponds to the UV regime of the theory. At quantum level, the two theories satisfy the Bethe Ansatz equations of the spin-$\frac{1}{2}$ XXX chain and the XXZ chain in the continuum limit respectively. Combining these relations with the dualities discussed previously in the literature, we propose a duality web in the UV regime among the nonlinear Schr\"odinger equation, the KdV equation and the 2D $\mathcal{N}=(2,2)^*$ topological Yang-Mills-Higgs theory.