Massive gravity applications for $T\overline{T}$ deformations

TL;DR

This paper builds a unified framework that ties stress-tensor deformations, notably the $T\overline{T}$ program, to massive gravity in arbitrary dimensions. By promoting the deformation to a dynamical flow governed by $\\partial_{\\lambda} S_{\\lambda}=\\mathcal{O}$ and introducing an auxiliary vielbein, the authors derive universal initial conditions, recover the 2D $T\overline{T}$ structure (with trace deformations) and extend it to higher dimensions where the deformation is governed by ghost-free minimal massive gravity. They also demonstrate that 2D deformations can be expressed as infinite sums of hypergeometric functions that resum to algebraic forms, thereby linking stress-tensor flows to algebraic properties of special functions, and they develop a local RG interpretation of the trace-flow equation. Further, the work explores dynamical coordinates, Born-Infeld-like non-linear electrodynamics, and $(\\tr T)^{n}$ deformations, illustrating the broad applicability of the massive-gravity description to diverse theories and highlighting deep connections to holography and higher-spin consistency. Overall, the framework provides new tools for constructing and constraining deformations across dimensions with potential implications for holography, non-linear electrodynamics, and the algebraic theory of special functions.

Abstract

We employ the massive gravity approach to stress-tensor deformations in a variety of scenarios, obtaining novel results and establishing new connections. Starting with perturbation theory, we show that the addition of $\text{tr} T+Λ_{2}$ to $T\overline{T}$ can be recovered and we construct the deformed action of an interacting non-abelian spin-1 along with spin-1/2 seed model, extending previous findings. As a result, a set of algebraic properties for certain hypergeometric functions is derived, allowing us to initiate the algebraic study of special functions directly via stress-tensor deformations and massive gravity. Moreover, we sharpen the connection between the trace-flow equation and the local renormalization group in any dimension. In $d>2$, the usual initial condition for the coupling leads to a potential known as ghost-free, minimal massive gravity. Upon expansion around the reference background, we retrieve Fierz-Pauli at leading order, matching the random geometry and holographic approaches. At the same time, we demonstrate that a change of coordinates interpretation is possible for the corresponding operator, which we verify with a simple example. Finally, we study the family of $(\text{tr} T)^{n}$ deformations advancing earlier work, and illustrate how the massive gravity description of non-linear electrodynamics can be incorporated in our framework.