$T\bar T$ deformation of Calogero-Sutherland model via dimensional reduction

TL;DR

This paper introduces a new $T\bar T$-like deformation for Calogero-Sutherland models and related quantum systems, modifying their spectra while preserving eigenfunctions, through dimensional reduction and coordinate transformations.

Contribution

It derives a novel $T\bar T$-like deformation for non-relativistic quantum mechanics and gauge theories, extending the scope of $T\bar T$ deformations to new models.

Findings

Deformation modifies the spectrum but preserves eigenfunctions.

Derived the classical Lagrangian for the deformed models.

Mapped 2D Yang-Mills to $T\bar T$-perturbed fermions.

Abstract

We perform the dynamical change of coordinates to derive a generalization of the trace relation and apply it to the non-linear Schr\"odinger model. After that, we work out the dimensional reduction of the bilinear $T\bar T$ operator and obtain the new $T\bar T$-like deformation of the quantum mechanics of free non-relativistic fermions and interacting Calogero-Sutherland particles. The deformation modifies the spectrum of the Hamiltonian but does not alter its eigenfunctions. The deformed classical Lagrangian is also obtained. Finally, we study a particular deformation of the two-dimensional Yang-Mills theory that maps the gauge theory onto a system of $T\bar T$-perturbed fermions.