ModMax Oscillators and Root-$T \overline{T}$-Like Flows in Supersymmetric Quantum Mechanics

TL;DR

This paper introduces a deformation technique for supersymmetric quantum mechanics inspired by root-$T ar{T}$ operators, connecting 2D field theories to 1D models and constructing a supersymmetric extension.

Contribution

It constructs a root-$T ar{T}$-like deformation for supersymmetric quantum mechanics and relates it to higher-dimensional theories through dimensional reduction.

Findings

The ModMax oscillator arises as a special case of the deformation.

The deformation operator is related to the dimensional reduction of 2D root-$T ar{T}$.

A supersymmetric extension of the deformation operator is constructed.

Abstract

We construct a deformation of any $(0+1)$-dimensional theory of $N$ bosons with $SO(N)$ symmetry which is driven by a function of conserved quantities that resembles the root-$T \overline{T}$ operator of $2d$ quantum field theories. In the special case of $N=2$ bosons and a harmonic oscillator potential, the solution to the flow equation is the ModMax oscillator of arXiv:2209.06296. We argue that the deforming operator is related, in a particular special regime, to the dimensional reduction of the $2d$ root-$T \overline{T}$ operator on a spatial circle. It follows that the ModMax oscillator is a dimensional reduction of the $4d$ ModMax theory to quantum mechanics, justifying the name. We then show how to construct a manifestly supersymmetric extension of this root-$T \overline{T}$-like operator for any $(0+1)$-dimensional theory with $SO(N)$ symmetry and $\mathcal{N}=2$ supersymmetry by defining a flow equation directly in superspace.