$T\overline{T}$-deformed free energy of the Airy model

Abstract

Sharpening the holographic correspondence of Jackiw-Teitelboim (JT) gravity and its dual matrix model description at a finite radial cutoff λ through the T T deformation is of interest. To proceed, we simplify the problem by considering the Airy model and deform Airy correlators in the same way as in T T -deformed JT gravity. We use those correlators to compute the annealed and quenched free energies for both λ > 0 and λ < 0 from an integral representation of the replica trick. At the leading order in λ and low temperatures, we confirm that the genus-zero quenched free energy monotonically decreases as a function of temperature when perturbation theory is valid. We then study the allgenus quenched free energy at low temperatures, where we discover and discuss subtleties due to non-perturbative effects in the Airy model, as well as the contributions from the non-perturbative branch under the T T deformation.

Note on $T\bar{T}$ deformed matrix models and JT supergravity duals

“…However, the results there don't match the gravity side. Interestingly, recently, another definition of T T deformed matrix model was proposed in [19]. The T T flow equation could be satisfied in this definition.…”

“…For higherdimensional cases, it has been explored in [14]. For lower dimension, i.e., the one-dimensional case or ordinary quantum mechanic system, which is the main interest of present work, the T T deformation was first introduced in [15,16], see also recent developments [17][18][19][20][21][22][23][24].…”

“…For other related recent developments, please refer to [45][46][47][48][49]. In the present work, we are interested in the JT and SJT/ matrix model duality with T T deformation [19,[50][51][52][53][54][55][56][57].…”

$$ T\overline{T} $$ deformations of supersymmetric quantum mechanics

“…Other quantities in the deformed theory -such as the torus partition function [36][37][38], the flat space S-matrix [39,40], torus one-point functions [41], and (to some degree) correlation functions [42][43][44][45][46][47][48][49][50][51][52] -can also be related to quantities in the undeformed theory. One interesting consequence of these results is that the high-energy density of states in a T Tdeformed theory follows Hagedorn rather than Cardy growth, which means that it cannot be a local quantum field theory and instead shares some properties with string theories.…”

“…There have been some interpretations offered for the holographic interpretations of the T T -like deformation of quantum mechanics in these various dual theories, at least in the non-manifestly-supersymmetric context. For instance, connections to cutoff JT gravity and to the Schwarzian have been discussed in [73,74,[102][103][104], related analyses of the dual matrix models have been carried out in [50,105], and a connection to modified boundary conditions in BF gauge theory is discussed in [84].…”