TsT Transformations Research Articles

Topological gauging and double current deformations

We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved U(1) currents Ja. We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the S-matrix of the system, similarly as what happens in the case of a textrm{T}overline{textrm{T}} deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents Ja are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the textrm{T}overline{textrm{T}} deformation to the central extension of the two-dimensional Poincaré algebra.

Marginally deformed Schrödinger/dipole CFT correspondence

We construct and thoroughly study a new integrable example of the AdS/CFT correspondence with Schrödinger symmetry. On the gravity side, the supergravity solution depends on two parameters and is obtained by marginally deforming the internal space of the Schrödinger background through a series of TsT transformations. On the field theory side, we identify the dual field theory which also depends on two parameters.We find a point-like string solution and derive its dispersion relation. By using the Landau-Lifshitz coherent state Lagrangian, which originates from field theory, we reproduce the leading, in the deformation parameters, terms of the string theory prediction. This constitutes a non-trivial test of the correspondence. Then, we calculate the Wilson loop, describing the quark/anti-quark potential at strong coupling. It exhibits confining behaviour when the separation length is much less than the Schrödinger parameter. When the separation length is much greater than the Schrödinger parameter the behaviour is that of a conformal theory. Subsequently, we take the Penrose limit along a certain null geodesic of the constructed background and calculate the bosonic spectrum. Based on that spectrum, we make an educated guess for the exact, in the ’t Hooft coupling, dispersion relation of the magnon excitations in the original doubly deformed background. This provides us with an exact prediction for the dimensions of the dual field theory operators. This applies to operators of large length, for which finite size corrections are suppressed.

Do Drinfeld twists of $AdS_5 \times S^5$ survive light-cone quantization?

We study how a wide class of Abelian Yang-Baxter deformations of the AdS_5 \times S^5AdS5×S5 string behave at the quantum level. These deformations are equivalent to TsT transformations and conjectured to be dual to beta, dipole, and noncommutative deformations of SYM. Classically they correspond to Drinfeld twists of the original theory. To verify this expectation at the quantum level we compute and match (1) the bosonic two-body tree-level worldsheet scattering matrix of these deformations in the uniform light-cone gauge, and (2) the Bethe equations of the equivalent model with twisted boundary conditions. We find that for a generalization of gamma deformations of the BMN string the we are able to express the S matrix either through a Drinfeld twist or a shift of momenta. For deformations of the GKP string around the null-cusp solution we encounter calculational obstacles that prevent us from calculating the scattering matrix.

TsT, black holes, and Toverline{T} + Joverline{T} + Toverline{J}

We generate a class of string backgrounds by a sequence of TsT transformations of the NS1-NS5 system that we argue are holographically dual to states in a single-trace Toverline{T} + Joverline{T} + Toverline{J} -deformed CFT2. The new string backgrounds include general rotating black hole solutions with two U(1) charges as well as a smooth solution without a horizon that is interpreted as the ground state. As evidence for the correspondence we (i) derive the long string spectrum and relate it to the spectrum of the dual field theory; (ii) show that the black hole thermodynamics can be reproduced from single-trace Toverline{T} + Joverline{T} + Toverline{J} -deformed CFTs; and (iii) show that the energy of the ground state matches the energy of the vacuum in the dual theory. We also study geometric properties of these new spacetimes and find that for some choices of the parameters the three-dimensional Ricci scalar in the Einstein frame can become positive in a region outside the horizon before reaching closed timelike curves and singularities.

Penrose limit for holographic duals of deformations

We explore bosonic string sigma models on warped BTZ × S 3 both in the plane wave as well as beyond plane wave limit. Using the light cone gauge, we obtain the corresponding Hamiltonian and therefore the spectrum associated with the pp wave strings. Our analysis reveals a constant shift in the spectrum arising as a result of TsT transformations along the isometries of the target space manifold. Imposing the energy positivity constraint on the CFT2 spectrum, we estimate an upper bound on the shift. We also estimate corrections as we go beyond the plane wave limit. Finally, we perform calculations using conformal gauge which reveals identical spectrum for the pp wave strings and thereby shows the equivalence between the two approaches.

Marginal deformations from type IIA supergravity

We study marginal deformations of a class of three-dimensional N=2 SCFTs that admit a holographic dual description in massive type IIA supergravity. We compute the dimension of the conformal manifold in these SCFTs and identify a special submanifold with enhanced flavor symmetry. We show how to apply the TsT transformation of Lunin-Maldacena to construct explicit supersymmetric AdS_44 solutions of massive type IIA supergravity with non-trivial internal fluxes that are the holographic dual of these marginal deformations. Finally, we also briefly comment on a class of RG flows between some of these SCFTs that also admit a holographic realization.

String backgrounds of the Yang-Baxter deformed AdS4 \xd7 \u2102\u21193 superstring

We build string backgrounds for Yang-Baxter deformations of the AdS4 × ℂℙ3 superstring generated by r-matrices satisfying the classical Yang-Baxter equation. We obtain the metric and the NSNS two-form of the gravity dual corresponding to noncommutative and dipole deformations of ABJM theory, as well as a deformed background with Schrödinger symmetry. The first two backgrounds may also be found by TsT transformations while for the last background we get a new family of non-relativistic ABJM theories with Schrödinger symmetry.

Non-relativistic duality and Toverline{T} deformations

We make some observations connecting non-relativistic limits of string theory with Toverline{T} deformations and TsT transformations.

The first \u03b1\u2032-correction to homogeneous Yang-Baxter deformations using O(d, d)

We use the O(d, d)-covariant formulation of supergravity familiar from Double Field Theory to find the first α′-correction to (unimodular) homogeneous Yang-Baxter (YB) deformations of the bosonic string. A special case of this result gives the α′-correction to TsT transformations. In a suitable scheme the correction comes entirely from an induced anomalous double Lorentz transformation, which is needed to make the two vielbeins obtained upon the YB deformation equal. This should hold more generally, in particular for abelian and non-abelian T-duality, as we discuss.

TsT, mathrm{T}overline{mathrm{T}} and black strings

We study the relationship between TsT transformations, marginal deformations of string theory on AdS3 backgrounds, and irrelevant deformations of 2d CFTs. We show that TsT transformations of NS-NS backgrounds correspond to instantaneous deformations of the worldsheet action by the antisymmetric product of two Noether currents, holographically mirroring the definition of the Toverline{T},Joverline{T},Toverline{J} , and Joverline{J} deformations of 2d CFTs. Applying a TsT transformation to string theory on BTZ ×S3 × M4 we obtain a general class of rotating black string solutions, including the Horne-Horowitz and the Giveon-Itzhaki-Kutasov ones as special cases, which we show are holographically dual to thermal states in single-trace Toverline{T} -deformed CFTs. We also find a smooth solution interpolating between global AdS3 in the IR and a linear dilaton background in the UV that is interpreted as the NS-NS ground state in the dual Toverline{T} -deformed CFT. This background suggests the existence of an upper bound on the deformation parameter above which the solution becomes complex. We find that the worldsheet spectrum, the thermodynamics of the black strings (in particular their Bekenstein-Hawking entropy), and the critical value of the deformation parameter match the corresponding quantities obtained from single-trace Toverline{T} deformations.

TT¯ deformations as TsT transformations

The relationship between $T\bar{T}$ deformations and the uniform light-cone gauge, first noted in arXiv:1804.01998, provides a powerful generating technique for deformed models. We recall this construction, distinguishing between changes of the gauge frame, which do not affect the theory, and genuine deformations. We investigate the geometric interpretation of the latter and argue that they affect the global features of the geometry before gauge fixing. Exploiting a formal relation between uniform light-cone gauge and static gauge in a T-dual frame, we interpret such a change as a TsT transformation involving the two light-cone coordinates. In the static-gauge picture, the $T\bar{T}$ CDD factor then has a natural interpretation as a Drinfel'd-Reshetikhin twist of the worldsheet S matrix. To illustrate these ideas, we find the geometries yielding a $T\bar{T}$ deformation of the worldsheet S matrix of pp-wave and Lin-Lunin-Maldacena backgrounds.

O(d,d) transformations preserve classical integrability

In this note, we study the action of O(d,d) transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the O(d,d)-transformed model and find that they are in general non-local because they depend on the winding modes. We conclude that every O(d,d;R) deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as JJ¯ marginal deformations and TsT transformations of the three-sphere with H-flux.

Unimodular jordanian deformations of integrable superstrings

We find new homogeneous rr matrices containing supercharges, and use them to find new backgrounds of Yang-Baxter deformed superstrings. We obtain these as limits of unimodular inhomogeneous rr matrices and associated deformations of AdS_2\times2×S^2\times2×T^66 and AdS_5\times5×S^55. Our rr matrices are jordanian, but also unimodular, and lead to solutions of the regular supergravity equations of motion. In general our deformations are equivalent to particular non-abelian T duality transformations. Curiously, one of our backgrounds is also equivalent to one produced by TsT transformations and an S duality transformation.

SUSY and the bi-vector

In this note we give an explicit formula for the preserved Killing spinors in deformed string theory backgrounds corresponding to integrable Yang–Baxter deformations realized via (sequences of) TsT transformations. The Killing spinors can be expressed only in terms of the bi-vector Θ which encodes the deformation. This formula is applicable to deformed backgrounds related to r-matrices of various ranks, including those that do not satisfy the unimodularity condition and give rise to backgrounds in generalized supergravity. We conjecture that our formula also remains valid for integrable deformations which are not realized via TsT transformations and motivate this conjecture by explicit examples.

Yang-Baxter deformations of the AdS4\u2009\xd7\u2009\u2102\u21193 superstring sigma model

The gravity dual of β-deformed ABJM theory can be obtained by a TsT transformation of AdS4 × ℂℙ3. We present a supercoset construction of ℂℙ3 to obtain this gravity dual theory as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to get an Abelian r-matrix satisfying the classical Yang-Baxter equation. Our results provide another illustration of the relation between Abelian r-matrices and TsT transformations.

Killing spinors from classical r-matrices

The Yang–Baxter deformation (yb deformation) is a systematic way of performing integrable deformations of 2D symmetric non-linear sigma models. The deformations can be labeled by classical r-matrices satisfying the classical yb equation. This yb deformation is also applicable to type IIB superstring theory defined on AdS. In this case, a simple class of yb deformations is associated with TsT transformations of string backgrounds. The data of this transformation is encoded in an antisymmetric bi-vector Θ (which is often called β field or non-commutativity parameter). In this article, we give a simple recipe for obtaining the explicit expression for the Killing spinors of the TsT transformed background starting from Θ. We moreover discuss the M-theory equivalent of the TsT transformation, allowing us to also give Killing spinors of 11D backgrounds. We discuss examples of TsT transformed backgrounds starting from flat space, and . We find that in this way we can relate the Ω-deformation to yb deformations.

On Yang–Baxter models, twist operators, and boundary conditions

We discuss homogeneous Yang–Baxter deformations of integrable sigma models in terms of twist operators. We show that the twist operators behave as the classical analogue of a Drinfeld twist, for all abelian and almost abelian deformations. We also use twist operators to rederive the well-known interpretation of TsT transformations—equivalent to abelian deformations—in terms of twisted boundary conditions. We discuss complications in extending this boundary condition picture to non-abelian deformations.

Yang-Baxter deformations beyond coset spaces (a slick way to do TsT)

Yang-Baxter string sigma-models provide a systematic way to deform coset geometries, such as AdSp × Sp, while retaining the σ-model integrability. It has been shown that the Yang-Baxter deformation in target space is simply an open-closed string map that can be defined for any geometry, not just coset spaces. Given a geometry with an isometry group and a bivector that is assumed to be a linear combination of antisymmetric products of Killing vectors, we show the equations of motion of (generalized) supergravity reduce to the Classical Yang-Baxter Equation associated with the isometry group, proving the statement made in [1]. These results bring us closer to the proof of the “YB solution generating technique” for (generalized) supergravity advertised in [1] and in particular provide an economical way to perform TsT transformations.

Light-cone reduction vs. TsT transformations: a fluid dynamics perspective

We compute constitutive relations for a charged (2+1) dimensional Schrödinger fluid up to first order in derivative expansion, using holographic techniques. Starting with a locally boosted, asymptotically AdS, 4 + 1 dimensional charged black brane geometry, we uplift that to ten dimensions and perform TsT transformations to obtain an effective five dimensional local black brane solution with asymptotically Schrödinger isometries. By suitably implementing the holographic techniques, we compute the constitutive relations for the effective fluid living on the boundary of this space-time and extract first order transport coefficients from these relations. Schrödinger fluid can also be obtained by reducing a charged relativistic conformal fluid over light-cone. It turns out that both the approaches result the same system at the end. Fluid obtained by light-cone reduction satisfies a restricted class of thermodynamics. Here, we see that the charged fluid obtained holographically also belongs to the same restricted class.

Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable \u03c3-model

A multi-parameter integrable deformation of the principal chiral model is presented. The Yang-Baxter and bi-Yang-Baxter σ-models, the principal chiral model plus a Wess-Zumino term and the TsT transformation of the principal chiral model are all recovered when the appropriate deformation parameters vanish. When the Lie group is SU(2), we show that this four-parameter integrable deformation of the SU(2) principal chiral model corresponds to the Lukyanov model.