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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., Comment: new in v2: important correction to the D1-D5 NCOS decoupling limit and related statements in section 4.2; an appendix added to discuss this limit pedagogically and more generally; another appendix supplemented with 10d formulae for the conserved charges necessary for this new discussion; references added and other minor modifications; main conclusions unchanged

We derive various properties of symmetric product orbifolds of $T\bar{T}$ and $J\bar{T}$ - deformed CFTs from a field-theoretical perspective. First, we generalise the known formula for the torus partition function of a symmetric orbifold theory in terms of the one of the seed to non-conformal two-dimensional QFTs; specialising this to seed $T\bar{T}$ and $J\bar{T}$ - deformed CFTs reproduces previous results in the literature. Second, we show that the single-trace $T\bar{T}$ and $J\bar{T}$ deformations preserve the Virasoro and Kac-Moody symmetries of the undeformed symmetric product orbifold CFT, including their fractional counterparts, as well as the KdV charges. Finally, we discuss correlation functions in these theories. By extending a previously-proposed basis of operators for $J\bar{T}$ - deformed CFTs to the single-trace case, we explicitly compute the correlation functions of both untwisted and twisted-sector operators and compare them to an appropriate set of holographic correlators. Our derivations are based mainly on Hilbert space techniques and completely avoid the use of conformal invariance, which is not present in these models., Comment: 50 pp, 5 figures

We use the variational principle approach to derive the large NN holographic dictionary for two-dimen-sional T\bar TTT‾-deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the non-dynamical graviton; the boundary conditions for matter fields are undeformed. When the matter fields are turned off and the deformation parameter is negative, the mixed boundary conditions for the metric at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. The holographic stress tensor of the deformed CFT is fixed by the variational principle, and in pure gravity it coincides with the Brown-York stress tensor on the radial bulk slice with a particular cosmological constant counterterm contribution. In presence of matter fields, the connection between the mixed boundary conditions and the radial ``bulk cutoff’’ is lost. Only the former correctly reproduce the energy of the bulk configuration, as expected from the fact that a universal formula for the deformed energy can only depend on the universal asymptotics of the bulk solution, rather than the details of its interior. The asymptotic symmetry group associated with the mixed boundary conditions consists of two commuting copies of a state-dependent Virasoro algebra, with the same central extension as in the original CFT.

We show that T\bar{T}, J\bar{T}TT‾,JT‾ and JT_aJTa - deformed classical CFTs posses an infinite set of symmetries that take the form of a field-dependent generalization of two-dimensional conformal transformations. If, in addition, the seed CFTs possess an affine U(1)U(1) symmetry, we show that it also survives in the deformed theories, again in a field-dependent form. These symmetries can be understood as the infinitely-extended conformal and U(1)U(1) symmetries of the underlying two-dimensional CFT, seen through the prism of the ``dynamical coordinates’’ that characterise each of these deformations. We also compute the Poisson bracket algebra of the associated conserved charges, using the Hamiltonian formalism. In the case of the J\bar{T}JT‾ and JT_{a}JTa deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the T\bar{T}TT‾ deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra.

It has previously been proven that $T\bar T$ - deformed CFTs possess Virasoro $\times$ Virasoro symmetry at the full quantum level, whose generators are obtained by simply transporting the original CFT generators along the $T\bar T$ flow. In this article, we explicitly solve the corresponding flow equation in the classical limit, obtaining an infinite set of conserved charges whose action on phase space is well-defined even when the theory is on a compact space. The field-dependent coordinates that are characteristic of the $T\bar T$ deformation are shown to emerge unambiguously from the flow. Translating the symmetry transformations from the canonical to the covariant formalism, we find that they are different from those that have been previously proposed in the Lagrangian context, but that they agree precisely with those obtained from holography. We also comment on different possible bases for the symmetry generators and on how our explicit classical results could be extended perturbatively to the quantum level., Comment: 26 pages

We show that the three-dimensional asymptotically linear dilaton background that arises in the near-horizon decoupling region of NS5-branes compactified on $T^4$ admits boundary conditions that lead to an infinite set of symmetries. The associated conserved charges, which implement field-dependent coordinate transformations, are found to be identical to the corresponding generators in a symmetric product orbifold of $T\bar T$ - deformed CFTs. Their algebra is a non-linear modification of the $\mathrm{Virasoro \times Virasoro}$ algebra, which precisely coincides with the algebra of the "unrescaled" symmetry generators in $T\bar T$-deformed CFTs. This further strengthens a previously proposed link between the single-trace $T\bar T$ deformation and compactified little string theory., Comment: 31 + 19 pp

$J\bar T$-deformed CFTs provide an interesting example of non-local, yet UV-complete two-dimensional QFTs that are entirely solvable. They have been recently shown to possess an infinite set of symmetries, which are a continuous deformation of the Virasoro-Kac-Moody symmetries of the seed CFT. In this article, we put forth a definition of primary operators in $J\bar T$-deformed CFTs on a cylinder, which are singled out by having CFT-like momentum-space commutation relations with the symmetry generators in the decompatification limit. We show -- based on results we first derive for the case of $J^1 \wedge J^2$-deformed CFTs -- that all correlation functions of such operators in the $J\bar T$-deformed CFT can be computed exactly in terms of the correlation functions of the undeformed CFT and are crossing symmetric in the plane limit. In particular, two and three-point functions are simply given by the corresponding momentum-space correlator in the undeformed CFT, with all dimensions replaced by particular momentum-dependent conformal dimensions. Interestingly, scattering amplitudes off the near-horizon of extremal black holes are known to take a strikingly similar form., Comment: 30 pp

Various holographic set-ups in string theory suggest the existence of non-local, UV complete two-dimensional QFTs that possess Virasoro symmetry, in spite of their non-locality. We argue that $J\bar T$-deformed CFTs are the first concrete realisation of such "non-local CFTs", through a detailed analysis of their classical and quantum symmetry algebra. Classically, the symmetries consist of an infinite set of left-moving conformal and affine $U(1)$ transformations that generate a Witt-Kac-Moody algebra, as well as a set of non-local, field-dependent generalizations of right-moving conformal and affine $U(1)$ transformations, whose algebra depends on the chosen basis. Notably, there exists a basis, denoted as the "flowed" representation, in which the right-moving charge algebra is simply Witt-Kac-Moody. At the quantum level, we provide a concrete prescription for constructing the symmetry generators via a combination of the flow equations they satisfy and the Sugawara construction, and use this to explicitly resolve the ordering ambiguities and the quantum corrections to the generators up to second order in the $J\bar T$ coupling parameter. This construction naturally produces the "flowed" generators, whose algebra is Virasoro-Kac-Moody to all orders in the coupling, with the same central extension as that of the undeformed CFT. We use this input to work out the quantum modifications to the "unflowed" generator algebra. A peculiarity of the Virasoro generators we study is that their zero mode does not equal the Hamiltonian, but is a quadratic function of it; this helps reconcile the Virasoro symmetry with the non-locality of the model. We argue that also $T\bar T$-deformed CFTs posses Virasoro symmetry, and discuss the existence of such a symmetry in more general non-local QFTs., Comment: 34 pp

$J\bar T$-deformed CFTs provide an interesting example of non-local, yet UV-complete two-dimensional QFTs that are entirely solvable. They have been recently shown to possess an infinite set of symmetries, which are a continuous deformation of the Virasoro-Kac-Moody symmetries of the seed CFT. In this article, we put forth a definition of primary operators in $J\bar T$-deformed CFTs on a cylinder, which are singled out by having CFT-like momentum-space commutation relations with the symmetry generators in the decompatification limit. We show -- based on results we first derive for the case of $J^1 \wedge J^2$-deformed CFTs -- that all correlation functions of such operators in the $J\bar T$-deformed CFT can be computed exactly in terms of the correlation functions of the undeformed CFT and are crossing symmetric in the plane limit. In particular, two and three-point functions are simply given by the corresponding momentum-space correlator in the undeformed CFT, with all dimensions replaced by particular momentum-dependent conformal dimensions. Interestingly, scattering amplitudes off the near-horizon of extremal black holes are known to take a strikingly similar form., Comment: 30 pp

Various holographic set-ups in string theory suggest the existence of non-local, UV complete two-dimensional QFTs that possess Virasoro symmetry, in spite of their non-locality. We argue that $J\bar T$-deformed CFTs are the first concrete realisation of such "non-local CFTs", through a detailed analysis of their classical and quantum symmetry algebra. Classically, the symmetries consist of an infinite set of left-moving conformal and affine $U(1)$ transformations that generate a Witt-Kac-Moody algebra, as well as a set of non-local, field-dependent generalizations of right-moving conformal and affine $U(1)$ transformations, whose algebra depends on the chosen basis. Notably, there exists a basis, denoted as the "flowed" representation, in which the right-moving charge algebra is simply Witt-Kac-Moody. At the quantum level, we provide a concrete prescription for constructing the symmetry generators via a combination of the flow equations they satisfy and the Sugawara construction, and use this to explicitly resolve the ordering ambiguities and the quantum corrections to the generators up to second order in the $J\bar T$ coupling parameter. This construction naturally produces the "flowed" generators, whose algebra is Virasoro-Kac-Moody to all orders in the coupling, with the same central extension as that of the undeformed CFT. We use this input to work out the quantum modifications to the "unflowed" generator algebra. A peculiarity of the Virasoro generators we study is that their zero mode does not equal the Hamiltonian, but is a quadratic function of it; this helps reconcile the Virasoro symmetry with the non-locality of the model. We argue that also $T\bar T$-deformed CFTs posses Virasoro symmetry, and discuss the existence of such a symmetry in more general non-local QFTs., Comment: 34 pp

We show that T (T) over bar, J (T) over bar and J T-a - deformed classical CFTs posses an infinite set of symmetries that take the form of a field-dependent generalization of two-dimensional conformal transformations. If, in addition, the seed CFTs possess an affine U (1) symmetry, we show that it also survives in the deformed theories, again in a field-dependent form. These symmetries can be understood as the infinitely-extended conformal and U (1) symmetries of the underlying two-dimensional CFT, seen through the prism of the dynamical coordinates that characterise each of these deformations. We also compute the Poisson bracket algebra of the associated conserved charges, using the Hamiltonian formalism. In the case of the J (T) over bar and J T-a deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the T (T) over bar deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra.

We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a U (1) current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and half a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT differs from the original S-matrix only by an overall phase factor that depends on the charges and momenta of the external particles, thus supporting the conjecture that such QFTs are UV complete, although intrinsically non-local. For the special case of an integrable QFT, we check that this phase factor precisely reproduces the change in the finite-size spectrum via the Thermodynamic Bethe Ansatz equations.

It has been recently shown that classical J¯T - deformed CFTs possess an infinite-dimensional Witt-Ka\v{c}-Moody symmetry, generated by certain field-dependent coordinate and gauge transformations. On a cylinder, however, the equal spacing of the descendants' energies predicted by such a symmetry algebra is inconsistent with the known finite-size spectrum of J¯T - deformed CFTs. Also, the associated quantum symmetry generators do not have a proper action on the Hilbert space. In this article, we resolve this tension by finding a new set of (classical) conserved charges, whose action is consistent with semi-classical quantization, and which are related to the previous symmetry generators by a type of energy-dependent spectral flow. The previous inconsistency between the algebra and the spectrum is resolved because the energy operator does not belong to the spectrally flowed sector.

We use the variational principle approach to derive the large N holographic dictionary for two-dimensional T (T) over bar -deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the non dynamical graviton; the boundary conditions for matter fields are undeformed. When the matter fields are turned off and the deformation parameter is negative, the mixed boundary conditions for the metric at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. The holographic stress tensor of the deformed CFT is fixed by the variational principle, and in pure gravity it coincides with the Brown-York stress tensor on the radial bulk slice with a particular cosmological constant counterterm contribution. In presence of matter fields, the connection between the mixed boundary conditions and the radial bulk cutoff is lost. Only the former correctly reproduce the energy of the bulk configuration, as expected from the fact that a universal formula for the deformed energy can only depend on the universal asymptotics of the bulk solution, rather than the details of its interior. The asymptotic symmetry group associated with the mixed boundary conditions consists of two commuting copies of a state-dependent Virasoro algebra, with the same central extension as in the original CFT.

It has been recently shown that classical $J\bar T$ - deformed CFTs possess an infinite-dimensional Witt-Ka\v{c}-Moody symmetry, generated by certain field-dependent coordinate and gauge transformations. On a cylinder, however, the equal spacing of the descendants' energies predicted by such a symmetry algebra is inconsistent with the known finite-size spectrum of $J\bar T$ - deformed CFTs. Also, the associated quantum symmetry generators do not have a proper action on the Hilbert space. In this article, we resolve this tension by finding a new set of (classical) conserved charges, whose action is consistent with semi-classical quantization, and which are related to the previous symmetry generators by a type of energy-dependent spectral flow. The previous inconsistency between the algebra and the spectrum is resolved because the energy operator does not belong to the spectrally flowed sector., Comment: fixed a minus sign; various minor cosmetic changes; added a brief discussion

We show that $T \bar T, J \bar T$ and $J T_a$ - deformed classical CFTs possess an infinite set of symmetries that take the form of a field-dependent generalization of two-dimensional conformal transformations. If, in addition, the seed CFTs possess an affine $U(1)$ symmetry, we show that it also survives in the deformed theories, again in a field-dependent form. These symmetries can be understood as the infinitely-extended conformal and $U(1)$ symmetries of the underlying two-dimensional CFT, seen through the prism of the "dynamical coordinates" that characterise each of these deformations. We also compute the Poisson bracket algebra of the associated conserved charges, using the Hamiltonian formalism. In the case of the $J \bar T$ and $J T_a$ deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the $T \bar T$ deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra., Comment: 28 + 12 pages, added some corrections and clarifications

We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a $U(1)$ current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and "half" a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT differs from the original S-matrix only by an overall phase factor that depends on the charges and momenta of the external particles, thus supporting the conjecture that such QFTs are UV complete, although intrinsically non-local. For the special case of an integrable QFT, we check that this phase factor precisely reproduces the change in the finite-size spectrum via the Thermodynamic Bethe Ansatz equations., Comment: 21 pages, v2 signs fixed

We use the variational principle approach to derive the large $N$ holographic dictionary for two-dimensional $T\bar T$-deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the non-dynamical graviton; the boundary conditions for matter fields are undeformed. When the matter fields are turned off and the deformation parameter is negative, the mixed boundary conditions for the metric at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. The holographic stress tensor of the deformed CFT is fixed by the variational principle, and in pure gravity it coincides with the Brown-York stress tensor on the radial bulk slice with a particular cosmological constant counterterm contribution. In presence of matter fields, the connection between the mixed boundary conditions and the radial "bulk cutoff" is lost. Only the former correctly reproduce the energy of the bulk configuration, as expected from the fact that a universal formula for the deformed energy can only depend on the universal asymptotics of the bulk solution, rather than the details of its interior. The asymptotic symmetry group associated with the mixed boundary conditions consists of two commuting copies of a state-dependent Virasoro algebra, with the same central extension as in the original CFT., Comment: 23 + 5 pages, 4 figures

The $J\bar T$ deformation, built from the components of the stress tensor and of a $U(1)$ current, is a universal irrelevant deformation of two-dimensional CFTs that preserves the left-moving conformal symmetry, while breaking locality on the right-moving side. Operators in the $J\bar T$-deformed CFT are naturally labeled by the left-moving position and right-moving momentum and transform in representations of the one-dimensional extended conformal group. We derive an all-orders formula for the spectrum of conformal dimensions and charges of the deformed CFT, which we cross-check at leading order using conformal perturbation theory. We also compute the linear corrections to the one-dimensional OPE coefficients and comment on the extent to which the correlation functions in $J\bar T$-deformed CFTs can be obtained from field-dependent coordinate transformations., Comment: 26+7 pp

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator JT, where J is a chiral U(1) current and T is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its JT deformation. We find that the dual spacetime is still AdS(3), but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the U(1) current. We show that when the coefficient of the chiral anomaly for J vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the U(1) Ka-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local.

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator JT, where J is a chiral U(1) current and T is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its JT deformation. We find that the dual spacetime is still AdS(3), but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the U(1) current. We show that when the coefficient of the chiral anomaly for J vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the U(1) Ka-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local., QC 20190219

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator JT, where J is a chiral U(1) current and T is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its JT deformation. We find that the dual spacetime is still AdS(3), but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the U(1) current. We show that when the coefficient of the chiral anomaly for J vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the U(1) Ka-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local.

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator JT, where J is a chiral U(1) current and T is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its JT deformation. We find that the dual spacetime is still AdS(3), but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the U(1) current. We show that when the coefficient of the chiral anomaly for J vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the U(1) Ka-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local.

The J (T) over bar deformation, built from the components of the stress tensor and of a U(1) current, is a universal irrelevant deformation of two-dimensional CFTs that preserves the left-moving conformal symmetry, while breaking locality on the right-moving side. Operators in the J (T) over bar -deformed CFT are naturally labeled by the left-moving position and right-moving momentum and transform in representations of the one-dimensional extended conformal group. We derive an all-orders formula for the spectrum of conformal dimensions and charges of the deformed CFT, which we cross-check at leading order using conformal perturbation theory. We also compute the linear corrections to the one-dimensional OPE coefficients and comment on the extent to which the correlation functions in J (T) over bar -deformed CFTs can be obtained from field-dependent coordinate transformations.

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator JT, where J is a chiral U(1) current and T is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its JT deformation. We find that the dual spacetime is still AdS(3), but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the U(1) current. We show that when the coefficient of the chiral anomaly for J vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the U(1) Ka-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local.

We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a $U(1)$ current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and "half" a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT differs from the original S-matrix only by an overall phase factor that depends on the charges and momenta of the external particles, thus supporting the conjecture that such QFTs are UV complete, although intrinsically non-local. For the special case of an integrable QFT, we check that this phase factor precisely reproduces the change in the finite-size spectrum via the Thermodynamic Bethe Ansatz equations., Comment: 21 pages, v2 signs fixed

We use the variational principle approach to derive the large $N$ holographic dictionary for two-dimensional $T\bar T$-deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the non-dynamical graviton; the boundary conditions for matter fields are undeformed. When the matter fields are turned off and the deformation parameter is negative, the mixed boundary conditions for the metric at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. The holographic stress tensor of the deformed CFT is fixed by the variational principle, and in pure gravity it coincides with the Brown-York stress tensor on the radial bulk slice with a particular cosmological constant counterterm contribution. In presence of matter fields, the connection between the mixed boundary conditions and the radial "bulk cutoff" is lost. Only the former correctly reproduce the energy of the bulk configuration, as expected from the fact that a universal formula for the deformed energy can only depend on the universal asymptotics of the bulk solution, rather than the details of its interior. The asymptotic symmetry group associated with the mixed boundary conditions consists of two commuting copies of a state-dependent Virasoro algebra, with the same central extension as in the original CFT., Comment: 23 + 5 pages, 4 figures

The $J\bar T$ deformation, built from the components of the stress tensor and of a $U(1)$ current, is a universal irrelevant deformation of two-dimensional CFTs that preserves the left-moving conformal symmetry, while breaking locality on the right-moving side. Operators in the $J\bar T$-deformed CFT are naturally labeled by the left-moving position and right-moving momentum and transform in representations of the one-dimensional extended conformal group. We derive an all-orders formula for the spectrum of conformal dimensions and charges of the deformed CFT, which we cross-check at leading order using conformal perturbation theory. We also compute the linear corrections to the one-dimensional OPE coefficients and comment on the extent to which the correlation functions in $J\bar T$-deformed CFTs can be obtained from field-dependent coordinate transformations., Comment: 26+7 pp

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator JT, where J is a chiral U(1) current and T is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its JT deformation. We find that the dual spacetime is still AdS(3), but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the U(1) current. We show that when the coefficient of the chiral anomaly for J vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the U(1) Ka-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local.

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator JT, where J is a chiral U(1) current and T is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its JT deformation. We find that the dual spacetime is still AdS(3), but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the U(1) current. We show that when the coefficient of the chiral anomaly for J vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the U(1) Ka-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local.

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator $J \bar T$, where $J$ is a chiral $U(1)$ current and $\bar T$ is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its $J \bar T$ deformation. We find that the dual spacetime is still AdS$_3$, but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the $U(1)$ current. We show that when the coefficient of the chiral anomaly for $J$ vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the $U(1)$ Ka\v{c}-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local., Comment: v2: added discussion of the chiral anomaly and one appendix, affiliation and grant information updated

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T (T) over bar, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U (1) current, J. The deformation takes the schematic form J (T) over bar and is interesting because it preserves an SL (2, R)xU (1) subgroup of the original global conformal symmetries. For the case of a purely (anti) chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T (T) over bar, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U (1) current, J. The deformation takes the schematic form J (T) over bar and is interesting because it preserves an SL (2, R)xU (1) subgroup of the original global conformal symmetries. For the case of a purely (anti) chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T (T) over bar, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U (1) current, J. The deformation takes the schematic form J (T) over bar and is interesting because it preserves an SL (2, R)xU (1) subgroup of the original global conformal symmetries. For the case of a purely (anti) chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T (T) over bar, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U (1) current, J. The deformation takes the schematic form J (T) over bar and is interesting because it preserves an SL (2, R)xU (1) subgroup of the original global conformal symmetries. For the case of a purely (anti) chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.

Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator $J \bar T$, where $J$ is a chiral $U(1)$ current and $\bar T$ is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its $J \bar T$ deformation. We find that the dual spacetime is still AdS$_3$, but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the $U(1)$ current. We show that when the coefficient of the chiral anomaly for $J$ vanishes, the energy and thermodynamics of black holes obeying these modified boundary conditions precisely reproduce the previously derived field theory spectrum and thermodynamics. Our proposed holographic dictionary can also reproduce the field-theoretical spectrum in presence of the chiral anomaly, upon a certain assumption that we justify. The asymptotic symmetry group associated to these boundary conditions consists of two copies of the Virasoro and one copy of the $U(1)$ Ka\v{c}-Moody algebra, just as before the deformation; the only effect of the latter is to modify the spacetime dependence of the right-moving Virasoro generators, whose action becomes state-dependent and effectively non-local., Comment: v2: added discussion of the chiral anomaly and one appendix, affiliation and grant information updated

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T (T) over bar, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U (1) current, J. The deformation takes the schematic form J (T) over bar and is interesting because it preserves an SL (2, R)xU (1) subgroup of the original global conformal symmetries. For the case of a purely (anti) chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions., QC 20210924

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T (T) over bar, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U (1) current, J. The deformation takes the schematic form J (T) over bar and is interesting because it preserves an SL (2, R)xU (1) subgroup of the original global conformal symmetries. For the case of a purely (anti) chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T (T) over bar, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U (1) current, J. The deformation takes the schematic form J (T) over bar and is interesting because it preserves an SL (2, R)xU (1) subgroup of the original global conformal symmetries. For the case of a purely (anti) chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator $T \bar T$, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that posess a conserved $U(1)$ current, $J$. The deformation takes the schematic form $J \bar T$ and is interesting because it preserves an $SL(2,\mathbb{R}) \times U(1)$ subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions., Comment: v3: institutional and grant information added

We study the null dipole deformation of N=4 super Yang-Mills theory, which is an example of a potentially solvable "dipole CFT": a theory that is non-local along a null direction, has non-relativistic conformal invariance along the remaining ones, and is holographically dual to a Schrodinger space-time. We initiate the field-theoretical study of the spectrum in this model by using integrability inherited from the parent theory. The dipole deformation corresponds to a nondiagonal Drinfeld-Reshetikhin twist in the spin chain picture, which renders the traditional Bethe ansatz inapplicable from the very beginning. We use instead the Baxter equation supplemented with nontrivial asymptotics, which gives the full 1-loop spectrum in the sl(2) sector. We show that anomalous dimensions of long gauge theory operators perfectly match the string theory prediction, providing a quantitative test of Schrodinger holography., Comment: 44 pages

We revisit the holographic construction of (approximately) local bulk operators inside an eternal AdS black hole in terms of operators in the boundary CFTs. If the bulk operator carries charge, the construction must involve a qualitatively new object: a Wilson line that stretches between the two boundaries of the eternal black hole. This operator - more precisely, its zero mode - cannot be expressed in terms of the boundary currents and only exists in entangled states dual to two-sided geometries, which suggests that it is a state-dependent operator. We determine the action of the Wilson line on the relevant subspaces of the total Hilbert space, and show that it behaves as a local operator from the point of view of either CFT. For the case of three bulk dimensions, we give explicit expressions for the charged bulk field and the Wilson line. Furthermore, we show that when acting on the thermofield double state, the Wilson line may be extracted from a limit of certain standard CFT operator expressions. We also comment on the relationship between the Wilson line and previously discussed mirror operators in the eternal black hole.

We study the null dipole deformation of N = 4 super Yang-Mills theory, which is an example of a potentially solvable 'dipole CFT': a theory that is non-local along a null direction, has non-relativistic conformal invariance along the remaining ones, and is holographically dual to a Schrodinger space-time. We initiate the field-theoretical study of the spectrum in this model by using integrability inherited from the parent theory. The dipole deformation corresponds to a nondiagonal Drinfeld-Reshetikhin twist in the spin chain picture, which renders the traditional Bethe ansatz inapplicable from the very beginning. We use instead the Baxter equation supplemented with nontrivial asymptotics, which gives the full 1-loop spectrum in the sl(2) sector. We show that anomalous dimensions of long gauge theory operators perfectly match the string theory prediction, providing a quantitative test of Schrodinger holography., QC 20190212

We revisit the holographic construction of (approximately) local bulk operators inside an eternal AdS black hole in terms of operators in the boundary CFTs. If the bulk operator carries charge, the construction must involve a qualitatively new object: a Wilson line that stretches between the two boundaries of the eternal black hole. This operator more precisely, its zero mode -cannot be expressed in terms of the boundary currents and only exists in entangled states dual to two-sided geometries, which suggests that it is a state-dependent operator. We determine the action of the Wilson line on the relevant subspaces of the total Hilbert space, and show that it behaves as a local operator from the point of view of either CFT. For the case of three bulk dimensions, we give explicit expressions for the charged bulk field and the Wilson line. Furthermore, we show that when acting on the thermofield double state, the Wilson line may be extracted from a limit of certain standard CFT operator expressions. We also comment on the relationship between the Wilson line and previously discussed mirror operators in the eternal black hole., QC 20171009

We revisit the holographic construction of (approximately) local bulk operators inside an eternal AdS black hole in terms of operators in the boundary CFTs. If the bulk operator carries charge, the construction must involve a qualitatively new object: a Wilson line that stretches between the two boundaries of the eternal black hole. This operator - more precisely, its zero mode - cannot be expressed in terms of the boundary currents and only exists in entangled states dual to two-sided geometries, which suggests that it is a state-dependent operator. We determine the action of the Wilson line on the relevant subspaces of the total Hilbert space, and show that it behaves as a local operator from the point of view of either CFT. For the case of three bulk dimensions, we give explicit expressions for the charged bulk field and the Wilson line. Furthermore, we show that when acting on the thermofield double state, the Wilson line may be extracted from a limit of certain standard CFT operator expressions. We also comment on the relationship between the Wilson line and previously discussed mirror operators in the eternal black hole.

We revisit the holographic construction of (approximately) local bulk operators inside an eternal AdS black hole in terms of operators in the boundary CFTs. If the bulk operator carries charge, the construction must involve a qualitatively new object: a Wilson line that stretches between the two boundaries of the eternal black hole. This operator - more precisely, its zero mode - cannot be expressed in terms of the boundary currents and only exists in entangled states dual to two-sided geometries, which suggests that it is a state-dependent operator. We determine the action of the Wilson line on the relevant subspaces of the total Hilbert space, and show that it behaves as a local operator from the point of view of either CFT. For the case of three bulk dimensions, we give explicit expressions for the charged bulk field and the Wilson line. Furthermore, we show that when acting on the thermofield double state, the Wilson line may be extracted from a limit of certain standard CFT operator expressions. We also comment on the relationship between the Wilson line and previously discussed mirror operators in the eternal black hole.

We study the null dipole deformation of N = 4 super Yang-Mills theory, which is an example of a potentially solvable 'dipole CFT': a theory that is non-local along a null direction, has non-relativistic conformal invariance along the remaining ones, and is holographically dual to a Schrodinger space-time. We initiate the field-theoretical study of the spectrum in this model by using integrability inherited from the parent theory. The dipole deformation corresponds to a nondiagonal Drinfeld-Reshetikhin twist in the spin chain picture, which renders the traditional Bethe ansatz inapplicable from the very beginning. We use instead the Baxter equation supplemented with nontrivial asymptotics, which gives the full 1-loop spectrum in the sl(2) sector. We show that anomalous dimensions of long gauge theory operators perfectly match the string theory prediction, providing a quantitative test of Schrodinger holography.

We study the null dipole deformation of N = 4 super Yang-Mills theory, which is an example of a potentially solvable 'dipole CFT': a theory that is non-local along a null direction, has non-relativistic conformal invariance along the remaining ones, and is holographically dual to a Schrodinger space-time. We initiate the field-theoretical study of the spectrum in this model by using integrability inherited from the parent theory. The dipole deformation corresponds to a nondiagonal Drinfeld-Reshetikhin twist in the spin chain picture, which renders the traditional Bethe ansatz inapplicable from the very beginning. We use instead the Baxter equation supplemented with nontrivial asymptotics, which gives the full 1-loop spectrum in the sl(2) sector. We show that anomalous dimensions of long gauge theory operators perfectly match the string theory prediction, providing a quantitative test of Schrodinger holography.

We revisit the holographic construction of (approximately) local bulk operators inside an eternal AdS black hole in terms of operators in the boundary CFTs. If the bulk operator carries charge, the construction must involve a qualitatively new object: a Wilson line that stretches between the two boundaries of the eternal black hole. This operator more precisely, its zero mode -cannot be expressed in terms of the boundary currents and only exists in entangled states dual to two-sided geometries, which suggests that it is a state-dependent operator. We determine the action of the Wilson line on the relevant subspaces of the total Hilbert space, and show that it behaves as a local operator from the point of view of either CFT. For the case of three bulk dimensions, we give explicit expressions for the charged bulk field and the Wilson line. Furthermore, we show that when acting on the thermofield double state, the Wilson line may be extracted from a limit of certain standard CFT operator expressions. We also comment on the relationship between the Wilson line and previously discussed mirror operators in the eternal black hole.

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator $T \bar T$, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that posess a conserved $U(1)$ current, $J$. The deformation takes the schematic form $J \bar T$ and is interesting because it preserves an $SL(2,\mathbb{R}) \times U(1)$ subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions., Comment: v3: institutional and grant information added