On $\sqrt{T\overline{T}}$ deformed pathways: CFT to CCFT

TL;DR

This paper explores the effects of a specific marginal deformation on 2D massless scalar theories, revealing symmetry changes and new theoretical insights through Legendre transformations and extreme limit analyses.

Contribution

It introduces a dynamical perspective on the $ oot{Tar{T}}$ deformation, connecting conformal and Carrollian symmetries, and uncovers new electric and magnetic Carroll theories.

Findings

Deformation induces a Legendre transformation between Lagrangians and Hamiltonians.

Conformal symmetry smoothly transitions to Carrollian symmetry at special moduli points.

Extreme limits of the flow parameter reveal electric and magnetic Carroll theories.

Abstract

We discuss the marginal $\sqrt{T\overline{T}}$ deformation of massless scalar field theories in two dimensions from a dynamical perspective. The operator flow equations for such deformations induce a particular Legendre Transformation between flowed Lagrangians and flowed Hamiltonians. The marginal deformation does not change the conformal symmetries of the theory, until some special points in the moduli space are reached, and the relativistic conformal algebra smoothly changes to the Carrollian conformal (equivalently BMS) one. We investigate this change of symmetry from both configuration space and phase space point of view, while keeping the notion of Legendre Transformation unchanged during the flow. By expanding the actions, in the extreme limits of the flow parameter, we recover the usual ``Electric'' Carroll theory and further uncover a novel ``Magnetic'' counterpart. We discuss the intriguing geometric understanding of such dynamical maps for the deformed theories, and also provide a concrete example for the same from a deformed string theory in flat space.