We continue our study of $λ$-deformed $σ$-models by setting up a $1/k$ perturbative expansion around the free field point for cosets, in particular for the $λ$-deformed $SU(2)/U(1)$ coset CFT. We construct an interacting field theory in which all deformation effects are manifestly encoded in the interaction vertices. Using this we reproduce the known $β$-function and the anomalous dimension of the composite operator perturbing away from the conformal point. We introduce the $λ$-dressed parafermions which have an essential Wilson-like phase in their expressions. Subsequently, we compute their anomalous dimension, as well as their four-point functions, as exact functions of the deformation and to leading order in the $k$ expansion. Correlation functions with an odd number of these parafermions vanish as in the conformal case.
We study the integrable asymmetric $λ$-deformations of the $SO(n+1)/SO(n)$ coset models, following the prescription proposed in \cite{AsyLambda}. We construct all corresponding deformed geometries in an inductive way. Remarkably we find a $Z_2$ transformation which maps the asymmetric $λ$--deformed models to the symmetric $λ$--deformed models.
We examine integrable $λ$-deformations of $SO(n+1)/SO(n)$ coset CFTs and their analytic continuations. We provide an interpretation of the deformation as a squashing of the corresponding coset $σ$-model's target space. We realise the $λ$-deformation for $n=5$ case as a solution to supergravity supported by non-vanishing five-form and dilaton. This interpolates between the coset CFT $SO(4,2)/SO(4,1)\times SO(6)/SO(5)$ constructed as a gauged WZW model and the non-Abelian T-dual of the $AdS_5\times S^5$ spacetime.
We study dynamics of scalar fields on a large class of geometries described by integrable sigma models. Although equations of motion are not separable due to absence of isometries and Killing tensors, we completely determine the spectra using algebraic and group-theoretic methods.
A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term $Tr(gTg^{-1}\Tb )$, where algebra elements $T, \Tb $ belong to the center of the algebra {\bf h} associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equ…
We provide a new class of exactly solvable superconformal field theories that corresponds to type II compactification on manifolds with exceptional holonomies. We combine N=1 Liouville field and N=1 coset models and construct modular invariant partition functions of strings moving on these manifolds. The resulting theories preserve spacetime supersymmetry. Also we explicitly construct chiral currents in these models to realize consistent string theories.
Fusion rules for Wess-Zumino-Witten (WZW) models at fractional level can be defined in two ways, with distinct results. The Verlinde formula yields fusion coefficients that can be negative. These signs cancel in coset fusion rules, however. On the other hand, the fusion coefficients calculated from decoupling of singular vectors are non-negative. They produce incorrect coset fusion rules, however, when factorisation is assumed. Here we give two prescriptions that yield the correct coset fusion rules from those found for the WZW models by the decoupling method. We restrict to the Virasoro minim…
We compute the flat coordinates on the Frobenius manifolds arising on the deformation space of Gepner $\hat{SU}(3)_k$ chiral rings. The explicit form of the flat coordinates is important for exact solutions of models of topological CFT and 2d Liouville gravity. We describe the case k=3, which is of particular interest because apart from the relevant chiral fields it contains a marginal one. Whereas marginal perturbations are relevant in different contexts, their analysis requires additional care compared to the relevant perturbations.
We obtain classical solutions of $ł$-deformed $\s$-models based on $SL(2,\mathbb{R})/U(1)$ and $SU(2)/U(1)$ coset manifolds. Using two different sets of coordinates, we derive two distinct classes of solutions. The first class is expressed in terms of hyperbolic and trigonometric functions, whereas the second one in terms of elliptic functions. We analyze their properties along with the boundary conditions and discuss string systems that they describe. It turns out that there is an apparent similarity between the solutions of the second class and the motion of a pendulum.
We present an algorithm for the construction of the branching functions in the vacuum sector for affine Lie algebras based on the string hypothesis solution to a system of Bethe equations for generalized RSOS models. We also mention how the ground state structure and features of the excitation spectra like the Brillouin zone schemes of these models (and those in the same universality classes) can be extracted from combinatoric arguments and encoded in Lie algebraic terms.
We study integrable structure of the coset conformal field theory and define the system of Integrals of Motion which depends on external parameters. This system can be viewed as a quantization of the ILW type hierarchy. We propose a set of Bethe anzatz equations for its spectrum.
We study the effective action for the integrable $λ$-deformation of the $G_{k_1} \times G_{k_2}/G_{k_1+k_2}$ coset CFTs. For unequal levels theses models do not fall into the general discussion of $λ$-deformations of CFTs corresponding to symmetric spaces and have many attractive features. We show that the perturbation is driven by parafermion bilinears and we revisit the derivation of their algebra. We uncover a non-trivial symmetry of these models parametric space, which has not encountered before in the literature. Using field theoretical methods and the effective action we compute the exac…
The largest known class of gravitational backgrounds with an exact string theoretical description is based on coset G/H CFTs and the corresponding gauged WZW models. These backgrounds generically lack isometries and are quite complicated. Thus the corresponding field equations seem impossible to solve and their use in physical applications becomes problematic. We develop a systematic general method enabling us to overcome this problem using group theory. The method is inspired by observations made in some elementary geometric coset and coset CFTs, but its full power is apparent in non-abelian …
In hep-th/9506151 we started a programme devoted to the systematic study of the conformal field theories derived from WZW models based on nonreductive Lie groups. In this, the second part, we continue this programme with a look at the N=1 and N=2 superconformal field theories which arise from both gauged and ungauged supersymmetric WZW models. We extend the supersymmetric (affine) Sugawara and coset constructions, as well as the N=2 Kazama--Suzuki construction to general self-dual Lie algebras.
We construct Cardy states, defect lines and chiral operators for rational coset conformal field theories on the lattice. The bulk theory is obtained by taking the overlap between tensor network representations of different string-nets, while the primary fields emerge from using the topological superselection sectors of the anyons in the original topological theory. This mapping provides an explicit manifestation of the equivalence between conformal field theories in two dimensions and topological field theories in three dimensions: their groundstates and elementary excitations are represented …
We investigate the thermodynamical features of two Lorentzian signature backgrounds that arise in string theory as exact CFTs and possess more than two disconnected asymptotic regions: the 2-d charged black hole and the Nappi-Witten cosmological model. We find multiple smooth disconnected Euclidean versions of the charged black hole background. They are characterized by different temperatures and electro-chemical potentials. We show that there is no straightforward analog of the Hartle-Hawking state that would express these thermodynamical features. We also obtain multiple Euclidean versions o…
We examine rectangular W-algebras with $so(M)$ or $sp(2M)$ symmetry, which can be realized as the asymptotic symmetry of higher spin gravities with restricted matrix extensions. We compute the central charges of the algebras and the levels of $so(M)$ or $sp(2M)$ affine subalgebras by applying the Hamiltonian reductions of $so$ or $sp$ type Lie algebras. For simple cases with generators of spin up to two, we obtain their operator product expansions by requiring the associativity. We further claim that the W-algebras can be realized as the symmetry algebras of dual coset CFTs and provide several…
We study the $SU(2)_k/U(1)$-parafermion model perturbed by its first thermal operator. By formulating the theory in terms of a (perturbed) fermionic coset model we show that the model is equivalent to interacting WZW fields modulo free fields. In this scheme, the order and disorder operators of the $Z_k$ parafermion theory are constructed as gauge invariant composites. We find that the theory presents a duality symmetry that interchanges the roles of the spin and dual spin operators. For two particular values of the coupling constant we find that the theory recovers conformal invariance and th…
We give a proof that the expected counting of strings attached to giant graviton branes in AdS_5 x S^5, as constrained by the Gauss Law, matches the dimension spanned by the expected dual operators in the gauge theory. The counting of string-brane configurations is formulated as a graph counting problem, which can be expressed as the number of points on a double coset involving permutation groups. Fourier transformation on the double coset suggests an ansatz for the diagonalization of the one-loop dilatation operator in this sector of strings attached to giant graviton branes. The ansatz agree…
The relationship between the nonlinear Schrodinger hierarchy and the parafermion and SL(2,R)/U(1) coset models, analogous to the relationship between the KdV hierarchy and the minimal models, is explained. To do this I first present an in depth study of a series of integrable hierarchies related to NLS, and write down an action from which any of these hierarchies, and the associated second Poisson bracket structures, can be obtained. In quantizing the free part of this action we find many features in common with the bosonized parafermion and SL(2,R)/U(1) models, and particularly it is clear th…
We present the construction of the $λ$-deformation of $AdS_5\times S^5$ superstring from the four dimensional Chern-Simons-type gauge theory. The procedure is applicable to all the semi-symmetric coset models and generalizes the previous construction of $λ$-deformation of the symmetric coset models.
We study $T\bar T$ deformations of 2d CFTs with periodic boundary conditions. We relate these systems to string models on $\mathbb{R}\times {S}^1\times{\cal M}$, where $\cal M$ is the target space of a 2d CFT. The string model in the light cone gauge is identified with the corresponding 2d CFT and in the static gauge it reproduces its $T\bar T$ deformed system. This relates the deformed system and the initial one by a worldsheet coordinate transformation, which becomes a time dependent canonical map in the Hamiltonian treatment. The deformed Hamiltonian defines the string energy and we express…
$J\bar T$-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 …
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 or…
We consider $λ$-deformed current algebra CFTs at level $k$, interpolating between an exact CFT in the UV and a PCM in the IR. By employing gravitational techniques, we derive the two-loop, in the large $k$ expansion, $β$-function. We find that this is covariant under a remarkable exact symmetry involving the coupling $λ$, the level $k$ and the adjoint quadratic Casimir of the group. Using this symmetry and CFT techniques, we are able to compute the Zamolodchikov metric, the anomalous dimension of the bilinear operator and the Zamolodchikov $C$-function at two-loops in the large $k$ expansion, …
Scharmm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT we consider CFT correlation functions which are martingales with respect to SLE. We establish connection between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset…
The duality symmetries of WZW and coset models are discussed. The exact underlying symmetry responsible for semiclassical duality is identified with the symmetry under affine Weyl transformations. This identification unifies the treatement of duality symmetries and shows that in the compact and unitary case they are exact symmetries of string theory to all orders in $α'$ and in the string coupling constant. Non-compact WZW models and cosets are also discussed. A toy model is analyzed suggesting that duality will not generically be a symmetry.
In this work, we study the holographic entanglement entropy of two dimensional $T\bar{T}$-deformed conformal field theory. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of New massive gravity, General minimal massive gravity, and Exotic general massive gravity. We also apply Song's prescription to obtain the entanglement entropy for deformed theories. In each cases, we find agreement between the results.
We construct the spectral curve of gamma-deformed AdS/CFT from the strong coupling scaling limit of the T-system. As we interpret the twisted T-functions in the classical limit as characters of the highest weight representations of the psu(2,2|4) symmetry group, we compute the twisted quasimomenta which characterize classical integrability and analyze their analytic and asymptotic properties. These twisted quasimomenta are compared to Beisert-Roiban Bethe ansatz equations and classical string solutions.
In the framework of the AdS/CFT correspondence, imposing a scalar field in the bulk space-time leads to deform the corresponding CFT in the boundary, which may produce corrections to entanglement entropy, as well as the so-called subregion complexity. We have computed such corrections for a set of singular subregions including kink, cones and creases in different dimensions. Our calculations shows new singular terms including universal logarithmic corrections for entanglement entropy and subregion complexity for some distinct values of conformal weight.
In previous works, we have developed an approach to fix the leading behaviors of the pure AdS$_3$ and BTZ black hole from the entanglement entropies of the free CFT$_2$ and finite temperature CFT$_2$, respectively. We exclusively use holographic principle only and make no restriction about the bulk geometry, not only the kinematics but also the dynamics. In order to verify the universality and correctness of our method, in this paper, we apply it to the $T\bar{T}$ deformed CFT$_2$, which breaks the conformal symmetry. In terms of the physical arguments of the $T\bar{T}$ deformed CFT$_2$, the d…
The $T\bar{T}+Λ_2$ deformation of a Holographic CFT on a cylinder (or torus in Euclidean signature) inhabits the stretched cosmological horizon in dS$_3$ when the deformation parameter is tuned to a particular value. I will describe how this insight allows us to compute the entropy of the stretched cosmic horizon including the logarithmic correction.
We compute both classical and quantum finite-size corrections at leading order in the strong coupling limit for the (dyonic) giant magnon in the Lunin-Maldacena background. Based on the exact S-martix conjectured for the deformed theory, we generalize the Luscher formula to include twisted boundary conditions and show that the results match with those derived both by finite-size classical solutions of the giant magnon and by algebraic curve analysis.
We study the embedding of spacetime filling D7-branes in beta-deformed backgrounds which, according to the AdS/CFT dictionary, corresponds to flavoring beta-deformed N=4 super Yang-Mills. We consider supersymmetric and more general non-supersymmetric three parameter deformations. The equations of motion for quadratic fluctuations of a probe D7-brane wrapped on a deformed three-sphere exhibit a non-trivial coupling between scalar and vector modes induced by the deformation. Nevertheless, we manage to solve them analytically and find that the mesonic mass spectrum is discrete, with a mass gap an…
We investigate generic n-point correlation functions of conformal field theories (CFTs), with $T\bar{T}$ and $J\bar{T}$ deformations, in terms of the perturbative CFT approach. We systematically obtain the first order correction to the generic correlation functions of CFTs with $T\bar{T}$ or $J\bar{T}$ deformation. We compute the out-of-time ordered correlation function (OTOC) in the Ising model with $T\bar{T}$ or $J\bar{T}$ deformation, which confirms that these deformations do not change the integrable property up to the first order level.
In this paper we study string theory in the background $\mathcal{M}_3$ that interpolates between $AdS_3$ in the IR and linear dilaton spacetime $\mathbb{R}^{1,1}\times\mathbb{R}_ϕ$ in the UV. Via holographic duality this background corresponds to $\rm{CFT}_2$ deformed by a dimension $(2,2)$ operator. Here we discuss the holographic Wilson loop in such a model and shed more light in support of the non-local structure of the theory (Little String Theory (LST)) in the UV. We also discuss quantum and thermal phase transitions of the boundary theory.
We introduce a set of gauge invariant fermion fields in fermionic coset models and show that they play a very central role in the description of several Conformal Field Theories (CFT's). In particular we discuss the explicit realization of primaries and their OPE in unitary minimal models, parafermion fields in $Z_k$ CFT's and that of spinon fields in $SU(N)_k, k=1$ Wess-Zumino-Witten models (WZW) theories. The higher level case ($k>1$) will be briefly discussed. Possible applications to QHE systems and spin-ladder systems are addressed.
We study the fuzzy structure of the general Kaehler coset space G/S\otimes{U(1)}^k deformed by the Fedosov formalism. It is shown that the Killing potentials satisfy the fuzzy algebrae working in the Darboux coordinates.
In the AdS/CFT correspondence motion in the radial direction of the AdS space is identified with renormalization group flow in the field theory. For the N=4 Yang-Mills theory this motion is trivial. More interesting examples of renormalization group flow occur when the N=4 theory is deformed. Aspects of the flows are discussed for the N=4 theory on its moduli space, and deformed to N=2 in the infra-red within the context of 5d SUGRA. 10d lifts and brane probing are crucial tools for linking the spacetime backgrounds to the dual field theory.
In this paper, we regard the $T\bar{T}$/$J\bar{T}$-deformed CFTs as perturbation theories and calculate the first order correction of the correlation functions due to the $T\bar{T}$/$J\bar{T}$-deformation. As applications, we study the Rényi entanglement entropy of excited state in the $T\bar{T}$/$J\bar{T}$-deformed two-dimensional CFTs. We find, up to the perturbation first order of the deformation, the Rényi entanglement entropy of locally excited states will acquire a non-trivial time dependence. The excess of the Rényi entanglement entropy of locally excited state will also be dramatically…
We show that the $T \bar T$ deformation of conformal field theories whose entropy grows as $S(E) \sim E^γ$ for $γ> 1/2$ exhibits negative specific heat in its microcanonical thermodynamic function $S({\cal E})$. We analyze the large $N$ symmetric product CFT as a concrete example of a CFT with this property and compute the thermodynamic functions such as $S({\cal E})$ and ${\cal E}(T)$. The negative specific heat in the microcanonical data is interpreted as signaling the first order phase transition when the system is coupled to a heat bath.
We study two dimensional conformal field theory with a left-moving conserved current $J$, perturbed by an irrelevant, Lorentz symmetry breaking operator with the quantum numbers of $J\bar{T}$, using a combination of field and string theoretic techniques. We show that the spectrum of the theory has some interesting features, which may shed light on systems of interest for holography and black hole physics.
We present a new method to calculate the spectrum of (slow-roll) inflationary perturbations, inspired by the conjectured dS/CFT correspondence. We show how the standard result for the spectrum of inflationary perturbations can be obtained from deformed CFT correlators, whose behavior is determined by the Callan-Symanzik equation. We discuss the possible advantages of this approach and end with some comments on the role of holography in dS/CFT and its relation to the universal nature of the spectrum of inflationary perturbations.
We propose a symmetry of $T\bar T$ deformed 2D CFT, which preserves the trace relation. The deformed conformal killing equation is obtained. Once we consider the background metric runs with the deformation parameter $μ$, the deformation contributes an additional term in conformal killing equation, which plays the role of renormalization group flow of metric. The conformal symmetry coincides with the fixed point. On the gravity side, this deformed conformal killing equation can be described by a new boundary condition of AdS$_3$. In addition, based on the deformed conformal killing equation, we…
We construct semi-classical string solutions of the Schrödinger $Sch_5 \times S^5$ spacetime, which is conjectured to be the gravity dual of a non-local dipole-deformed CFT. They are the counterparts of the giant magnon and spiky string solutions of the undeformed $AdS_5 \times S^5$ to which they flow when the deformation parameter is turned off. They live in an $S^3$ subspace of the five-sphere along the directions of which the $B$-field has non-zero components having also extent in the $Sch_5$ part of the metric. Finally, we speculate on the form of the dual field theory operators.
We consider non-perturbative effects in the beta-deformed N=4 supersymmetric gauge theory in the context of the AdS/CFT correspondence. We concentrate on the correlators of the Yang-Mills operators which correspond to the lowest Kaluza-Klein modes propagating on the dual supergravity background found by Lunin and Maldacena in hep-th/0502086. In particular, we calculate all multi-instanton contributions to these correlators in the beta-deformed SYM and find a compelling agreement with the results expected in supergravity.
In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional $T\overline{T}$-deformed conformal field theory. For such case, the (Rényi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large $c$ CFT, which is propos…
In this lecture we discuss `beyond CFT' from symmetry point of view. After reviewing the Virasoro algebra, we introduce deformed Virasoro algebras and elliptic algebras. These algebras appear in solvable lattice models and we study them by free field approach.
We present a number of D=4 bosonic and heterotic string solutions with a covariantly constant null Killing vector which, like the solution of Nappi and Witten (NW), correspond to (gauged) WZW models and thus have a direct conformal field theory interpretation. A class of exact plane wave solutions (which includes the NW solution) is constructed by `boosting' the twisted products of two D=2 `cosmological' or `black-hole' cosets of SL(2,R) and SU(2). We describe a general limiting procedure by which one can construct new solutions with a covariantly constant null Killing vector starting with kno…
We consider the relation between affine Toda field theories (ATFT) and Landau-Ginzburg Lagrangians as alternative descriptions of deformed 2d CFT. First, we show that the two concrete implementations of the deformation are consistent once quantum corrections to the Landau-Ginzburg Lagrangian are taken into account. Second, inspired by Gepner's fusion potentials, we explore the possibility of a direct connection between both types of Lagrangians; namely, whether they can be transformed one into another by a change of variables. This direct connection exists in the one-variable case, namely, for…
We present a new exact treatment of $T\bar{T}$ deformed 2D CFT in terms of the worldsheet theory of a non-critical string. The transverse dimensions of the non-critical string are represented by the undeformed CFT, while the two longitudinal light-cone directions are described by two scalar fields $X^+$ and $X^-$ with free field OPE's but with a modified stress tensor, arranged so that the total central charge adds up to 26. The relation between our $X^\pm$ field variables and 2D dilaton gravity is indicated. We compute the physical spectrum and the partition function and find a match with kno…
The coset construction is the most important tool to construct rational conformal field theories with known chiral data. For some cosets at small level, so-called maverick cosets, the familiar analysis using selection and identification rules breaks down. Intriguingly, this phenomenon is linked to the existence of exceptional modular invariants. Recent progress in CFT, based on studying algebras in tensor categories, allows for a universal construction of the chiral data of coset theories which in particular also applies to maverick cosets.
We perform some simple computations for the first order deformation of the D1D5 CFT off its orbifold point. It had been shown earlier that under this deformation the vacuum state changes to a squeezed state (with the further action of a supercharge). We now start with states containing one or two initial quanta and write down the corresponding states obtained under the action of deformation operator. The result is relevant to the evolution of an initial excitation in the CFT dual to the near extremal D1D5 black hole: when a left and a right moving excitation collide in the CFT, the deformation…
