We calculate the S-multiplets for two-dimensional Euclidean N = (0, 2) and N = (2, 2) superconformal field theories under the TT¯ deformation at leading order of perturbation theory in the deformation coupling. Then, from these N = (0, 2) deformed multiplets, we calculate two- and three-point correlators. We show the N = (0, 2) chiral ring’s elements do not flow under the TT¯ deformation. Specializing to integrable supersymmetric seed theories, such as N = (2, 2) Landau-Ginzburg models, we use the thermodynamic Bethe ansatz to study the S-matrices and ground state energies. From both an S-matrix perspective and Melzer’s folding prescription, we show that the deformed ground state energy obeys the inviscid Burgers’ equation. Finally, we show that several indices independent of D-term perturbations including the Witten index, Cecotti-Fendley-Intriligator-Vafa index and elliptic genus do not flow under the TT¯ deformation.

Sharpening the correspondence of Jackiw-Teitelboim (JT) gravity and its dual matrix model description at a finite radial cutoff λ through the TT¯ 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 TT¯ -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 all-genus 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 TT¯ deformation.

We develop the holographic dictionary for pure $\mathrm{AdS}_3$ gravity where the Lagrangian of the dual $2d$ conformal field theory has been deformed by an arbitrary function of the energy-momentum tensor. In addition to the $T \overline{T}$ deformation, examples of such functions include a class of marginal stress tensor deformations which are special because they leave the generating functional of connected correlators unchanged up to a redefinition of the source and expectation value. Within this marginal class, we identify the unique deformation that commutes with the $T \overline{T}$ flow, which is the root-$T \overline{T}$ operator, and write down the modified boundary conditions corresponding to this root-$T \overline{T}$ deformation. We also identify the unique marginal stress tensor flow for the cylinder spectrum of the dual CFT which commutes with the inviscid Burgers' flow driven by $T \overline{T}$, and we propose this unique flow as a candidate root-$T \overline{T}$ deformation of the energy levels. We study BTZ black holes in $\mathrm{AdS}_3$ subject to root-$T \overline{T}$ deformed boundary conditions, and find that their masses flow in a way which is identical to that of our candidate root-$T \overline{T}$ energy flow equation, which offers evidence that this flow is the correct one. Finally, we also obtain the root-$T \overline{T}$ deformed boundary conditions for the gauge field in the Chern-Simons formulation of $\mathrm{AdS}_3$ gravity.

We develop the holographic dictionary for pure AdS3 gravity where the Lagrangian of the dual 2D conformal field theory has been deformed by an arbitrary function of the energy-momentum tensor. In addition to the TT¯ deformation, examples of such functions include a class of marginal stress tensor deformations which are special because they leave the generating functional of connected correlators unchanged up to a redefinition of the source and expectation value. Within this marginal class, we identify the unique deformation that commutes with the TT¯ flow, which is the root-TT¯ operator, and write down the modified boundary conditions corresponding to this root-TT¯ deformation. We also identify the unique marginal stress tensor flow for the cylinder spectrum of the dual CFT which commutes with the inviscid Burgers' flow driven by TT¯, and we propose this unique flow as a candidate root-TT¯ deformation of the energy levels. We study BTZ black holes in AdS3 subject to root-TT¯ deformed boundary conditions, and find that their masses flow in a way which is identical to that of our candidate root-TT¯ energy flow equation, which offers evidence that this flow is the correct one. Finally, we also obtain the root-TT¯ deformed boundary conditions for the gauge field in the Chern-Simons formulation of AdS3 gravity.

JT gravity has a first-order formulation as a two-dimensional BF theory, which can be viewed as the dimensional reduction of the Chern-Simons description of 3d3d gravity. We consider {T\overbar{T}}TT¯-type deformations of the (0+1)(0+1)-dimensional dual to this 2d2d BF theory and interpret the deformation as a modification of the BF theory boundary conditions. The fundamental observables in this deformed BF theory, and in its 3d3d Chern-Simons lift, are Wilson lines and loops. In the 3d3d Chern-Simons setting, we study modifications to correlators involving boundary-anchored Wilson lines which are induced by a {T\overbar{T}}TT¯ deformation on the 2d2d boundary; results are presented at both the classical level (using modified boundary conditions) and the quantum-mechanical level (using conformal perturbation theory). Finally, we calculate the analogous deformed Wilson line correlators in 2d2d BF theory below the Hagedorn temperature where the principal series dominates over the discrete series.

We define a manifestly supersymmetric version of the TT¯ deformation appropriate for a class of (0 + 1)-dimensional theories with N = 1 or N = 2 supersymmetry, including one presentation of the super-Schwarzian theory which is dual to JT supergravity. These deformations are written in terms of Noether currents associated with translations in superspace, so we refer to them collectively as f(Q) deformations. We provide evidence that the f(Q)) deformations of N = 1 and N = 2 theories are on-shell equivalent to the dimensionally reduced supercurrent-squared deformations of 2d theories with N = (0, 1) and N = (1, 1) supersymmetry, respectively. In the N = 1 case, we present two forms of the f(Q) deformation which drive the same flow, and clarify their equivalence by studying the analogous equivalent deformations in the non-supersymmetric setting.