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

TL;DR

The paper studies the marginal $\sqrt{T\overline{T}}$ deformation of two-dimensional massless scalar field theories, revealing a dynamical Legendre Transformation that links flowed Lagrangians and flowed Hamiltonians across the entire flow. It shows that conformal symmetry is maintained along the flow until the parameter $\alpha$ reaches special values where the symmetry algebra contracts to Carrollian (BMS$_3$) structure, producing distinct Electric and Magnetic Carroll limits in both configuration and phase spaces. The authors develop a geometric interpretation of the dynamical maps, demonstrate the LT’s persistence at the Carroll point, and illustrate the framework with a concrete deformed string worldsheet example that inherits Carrollian residual symmetry. These results provide a unified, dual description of the CFT to CCFT flow and expose new Carrollian scalar theories with nonlinear derivative structure, enriching the landscape of exactly solvable deformations and their symmetry content.

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.

Figures (10)

• Figure 1: The Lagrangian ($\mathcal{L}$) and Hamiltonian ($\mathcal{H}$) flows stay related by a valid Legendre Transform(LT) across all values of the deformation parameter $\alpha$. As $\alpha$ reaches extreme values, these flows bifurcate into electric and magnetic Carroll theories, both in configuration space and in phase space, with very specific maps between them.
• Figure 2: The canonical momentum $\pi_i$ derived from the deformed Lagrangian, valid for all values of $\alpha$, serves as conjugate momentum field $\pi_i$ in the deformed Hamiltonian theory. The LT makes sure of the sanctity of this momentum definition throughout the flow.
• Figure 3: An illustration of the 'boosted' nature of the transformations in $\mathcal{H}-K$ space and $\mathcal{L}-\mathcal{K}$ space. Reaching the corresponding 'lightcones' signify Carroll symmetries setting in.
• Figure 4: Hyperbolas in the $C_1$-$C_2$ plane for different $\alpha$. Note that the branch moving right signifies $\alpha \to \infty$ and the one moving left signifies $\alpha \to -\infty$.
• Figure 5: Hyperbolas in the scaled$\hat{C}_1$-$\hat{C}_2$ plane for different $\alpha$.