Ascending the attractor flow in the D1-D5 system

We study maximally supersymmetric irrelevant deformations of the D1-D5 CFT that correspond to following the attractor flow in reverse in the dual half-BPS black string solutions of type IIB supergravity on K3. When a single, quadratic condition is imposed on the parameters of the 22 such irrelevant deformations, the asymptotics of the solution degenerate to a linear dilaton like spacetime. We identify each such degeneration limit with a known decoupling limit of string theory, which yields little string theory or deformations thereof (the so-called open brane LST, or ODp theories), compactified to two dimensions. This suggests that a 21-parameter family of the above deformations leads to UV-complete theories, which are string theories decoupled from gravity that are continuously connected to each other. All these theories have been argued to display Hagedorn behaviour; we show that including the F1 strings leads to an additional Cardy term. The resulting entropy formula closely resembles that of single-trace $T\bar T$-deformed CFTs, whose generalisations could provide possibly tractable effective two-dimensional descriptions of the above web of theories. We also consider the asymptotically flat black strings. At fixed temperature, the partition function is dominated by thermodynamically stable, small black string solutions, similar to the ones in the decoupled backgrounds. We show that certain asymptotic symmetries of these black strings bear a striking resemblance with the state-dependent symmetries of single-trace $T\bar T$, and break down precisely when the background solution reaches the large black string threshold. This suggests that small, asymptotically flat black strings may also admit a $T\bar T$ - like effective description.