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

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 $\text{tr} T+\Lambda_{2}$ 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.