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In contradistinction to the case of massive excitations, the connection between integrability and the tree-level massless scattering matrix of integrable $\sigma$-models is lost. Namely, in well-known 2-d integrable models the tree-level massless S-matrix exhibits particle production and fails to factorise. This is conjectured to happen due to IR ambiguities in the massless tree-level amplitudes. We present a definition of the massless S-matrix which has all the nice properties of integrable theories, there is no particle production and the S-matrix factorises. As an example, we present in detail the case of the $SU(2)$ principal chiral model (PCM)., Comment: 32 pages

We calculate holographically three-point functions of scalar operators with large dimensions at finite density and finite temperature. To achieve this, we construct new solutions that involve two isometries of the deformed internal space. The novel feature of these solutions is that the corresponding two-point function depends not only on the conformal dimension but also on the difference between the two angular momenta. After identifying the dual operators, we systematically calculate three-point correlators as an expansion in powers of the temperature and the chemical potential. Our analytic perturbative results are in agreement with the exact numerical computation. The three point correlator (when the background contains either temperature or density but not both) is always a monotonic function of the temperature or the chemical potential. However, when both parameters are present the three point correlator is no longer a monotonic function. For fixed finite temperature and small values of the chemical potential a minimum of the three-point function appears. Surprisingly, contributions from the internal space do not depend on the chemical potential or the temperature, as long as those are treated as perturbations., Comment: 36 pages, 5 figures

We set up the computation of correlation functions for operators that are dual to semiclassical string states in strongly coupled defect conformal field theories (dCFTs). In the dCFT that is dual to the D3-D5 probe-brane system, we calculate the correlation function of two heavy operators perturbatively, in powers of the conformal ratio. We find that the leading term agrees with the prediction of the operator product expansion (OPE). In the case of two heavy BMN operators, we find agreement in subleading orders as well., Comment: 10 pages, 1 table, 3 figures; typos corrected, references added. Matches published version

We calculate holographically one and two-point functions of scalar operators at finite density and/or finite temperature. In the case of finite density and zero temperature we argue that only scalar operators can have non-zero VEVs. In the case in which both the chemical potential and the temperature are finite, we present a systematic expansion of the two-point correlators in powers of the temperature T and the chemical potential $\Omega$. The holographic result is in agreement with the general form of the OPE which dictates that the two-point function may be written as a linear combination of the Gegenbauer polynomials $C_J^{(1)}(\xi)$ but with the coefficients depending now on both the temperature and the chemical potential, as well as on the CFT data. The leading terms in this expansion originate from the expectation values of the scalar operator $\phi^2$, the R-current ${\cal J}^\mu_{\phi_3}$ and the energy-momentum tensor $T^{\mu\nu}$. By employing the Ward identity for the R-current and by comparing the appropriate term of the holographic result for the two-point correlator to the corresponding term in the OPE, we derive the value of the R-charge density of the background. Compelling agreement with the analysis of the thermodynamics of the black hole is found. Finally, we determine the behaviour of the two-point correlators, in the case of finite temperature, and in the limit of large temporal or spatial distance of the operators., Comment: 1+31 pages

Particle production in integrable field theories may exist depending on the vacuum around which excitations are defined. To tackle this and analogous issues with conventional field theoretical tools, we consider the integrable $\lambda$-deformed model for $SU(2)$ together with a timelike coordinate. We construct the corresponding four-dimensional plane wave background keeping also post-plane wave corrections, as well as all the non-trivial $\lambda$-dependence. After imposing the light-cone gauge and the Virasoro constraints, we obtain an interacting field theory for the transverse physical modes which are massive. We explicitly demonstrate the absence of particle production to leading order in the large $k$-expansion. This is based crucially on the form of the interaction vertices and their dependence on the $\lambda$-deformation parameter. In addition, we compute the $S$-matrix for the two-particle elastic scattering exactly in $\lambda$ and to leading order in the large $k$-expansion. Our method can be applied to any integrable theory with at least one isometry., Comment: 31 pages, 1 figure

We construct and thoroughly study a new integrable example of the AdS/CFT correspondence with Schr\"{o}dinger symmetry. On the gravity side, the supergravity solution depends on two parameters and is obtained by marginally deforming the internal space of the Schr\"{o}dinger 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. A non-trivial test of the correspondence is provided by using the Landau-Lifshitz coherent state approach to reproduce the leading, in the deformation parameters, terms of that relation. 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 Schr\"{o}dinger parameter. When the separation length is much greater than the Schr\"{o}dinger 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., Comment: 1+36 pages, 3 figures

We present the Hamiltonian formulation of the recently constructed integrable theories of arXiv:2006.12525. These theories turn out to be canonically equivalent to the sum of an asymmetrically gauged CFT and of the most general $\lambda$-deformed model of arXiv:1812.04033. Using the Hamiltonian formalism, we prove that the full set of conserved charges of the models of arXiv:2006.12525 are in involution, ensuring their Hamiltonian integrability. Finally, we show that the equations of motion of these theories can be put in the form of zero curvature Lax connections., Comment: 32 pages, minor changes, version accepted in Nucl. Phys. B

We construct and study new giant graviton configurations in the framework of the non-supersymmetric Schrodinger holography. We confirm in the original Schrodinger spacetime, the picture discovered previously in the pp-wave limit of the geometry, namely that it is the giant graviton that becomes the energetically favored stable configuration compared to the point graviton one. Furthermore, there is a critical value of the deformation above which the point graviton disappears from the spectrum. The former fact leads also to the possibility of tunnelling from the point graviton to the giant graviton configuration. We calculate, explicitly, the instanton solution and its corresponding action which gives a measure of the tunnelling probability. Finally, we evaluate holographically the three-point correlation function of two giant gravitons and one dilaton mode as a function of the Schrodinger invariant., Comment: 20 pages, 8 figures

We present an intuitive diagrammatic representation of a new class of integrable $\s$-models. It is shown that to any given diagram corresponds an integrable theory that couples $N$ WZW models with a certain number of each of the following four fundamental integrable models, the PCM, the YB model, both based on a group $G$, the isotropic $\s$-model on the symmetric space $G/H$ and the YB model on the symmetric space $G/H$. To each vertex of a diagram we assign the matrix of one of the aforementioned fundamental integrable theories. Any two vertices may be connected with a number of lines having an orientation and carrying an integer level $k_i$. Each of these lines is associated with an asymmetrically gauged WZW model at an arbitrary level $k_i$. Gauge invariance of the full action is translated to level conservation at the vertices. We also show how to immediately read from the diagrams the corresponding $\s$-model actions. The most generic of these models depends on at least $n^2+1$ parameters, where $n$ is the total number of vertices/fundamental integrable models. Finally, we discuss the case where the level conservation at the vertices is relaxed and the case where the deformation matrix is not diagonal in the space of integrable models., Comment: 28 pages, 3 figures,v2: extended discussion on the integrability of the theories is added, v3: version accepted in Nucl. Phys. B

We construct a generalization of the cyclic $\lambda$-deformed models of \cite{Georgiou:2017oly} by relaxing the requirement that all the WZW models should have the same level $k$. Our theories are integrable and flow from a single UV point to different IR fixed points depending on the different orderings of the WZW levels $k_i$. First we calculate the Zamolodchikov's C-function for these models as exact functions of the deformation parameters. Subsequently, we fully characterize each of the IR conformal field theories. Although the corresponding left and right sectors have different symmetries, realized as products of current and coset-type symmetries, the associated central charges are precisely equal, in agreement with the valuesobtained from the C-function., Comment: 41 pages, v2: minor typos corrected to match the published version

We continue our study of $\lambda$-deformed $\sigma$-models by setting up a $1/k$ perturbative expansion around the free field point for cosets, in particular for the $\lambda$-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 $\beta$-function and the anomalous dimension of the composite operator perturbing away from the conformal point. We introduce the $\lambda$-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., Comment: v1: 1+36 pages, Latex, v2: JHEP version

We construct a new giant graviton solution on the recently constructed pp-wave geometry of the non-supersymmetric Schrodinger background. That solution exhibits an intriguing behavior as the deformation parameter of the spacetime varies. Firstly, the degeneracy between the giant and the point graviton is lifted for the benefit of the giant graviton as soon as the deformation is turned on. Secondly, when the deformation parameter exceeds a critical value the barrier separating the point from the giant graviton disappears. This suggests that the mere presence of a D3-brane leads to the spontaneous breaking of conformal invariance. We perform a detailed analysis of the full bosonic spectrum, which reveals that the deformation induces a coupling between the scalar and the gauge field fluctuations. It is exactly this coupling that keeps the giant graviton free of tachyonic instabilities. Furthermore, the giant graviton configuration completely breaks the supersymmetry of the pp-wave background, as the Kappa-symmetry analysis suggests., Comment: 19 pages, 1 figure

We study the notion of strong integrability for classically integrable $\lambda$-deformed CFTs and coset CFTs. To achieve this goal we employ the Poisson brackets of the spatial Lax matrix which we prove that it assumes the Maillet $r/s$-matrix algebra. As a consequence the system in question are integrable in the strong sense. Furthermore, we show that the derived Maillet $r/s$-matrix algebras can be realized in terms of twist functions, at the poles of which we recover the underlying symmetry algebras., Comment: v1:1+22 pages, Latex, v2: NPB version

We consider $\lambda$-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, $\beta$-function. We find that this is covariant under a remarkable exact symmetry involving the coupling $\lambda$, 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, as exact functions of the deformation parameter. Finally, we extend the above results to $\lambda$-deformed parafermionic algebra coset CFTs which interpolate between exact coset CFTs in the UV and a symmetric coset space in the IR., Comment: v1: 1+27 pages, Latex, v2: JHEP version

We explore the structure of the $\lambda$-deformed $\sigma$-model action by setting up a perturbative expansion around the free field point corresponding to the identity group element. We include all field interaction terms up to sixth order. We compute the two- and three-point functions of current and primary filed operators, their anomalous dimensions as well as the $\beta$-function for the $\lambda$-parameter. Our results are in complete agreement with those obtained previously using gravitational and/or CFT perturbative methods in conjunction with the non-perturbative symmetry, as well as with those obtained using methods exploiting the geometry defined in the space of couplings. The advantage of this approach is that all deformation effects are already encoded in the couplings of the interaction vertices and in the $\lambda$-dressed operators., Comment: 29 pages, minor improvements, published version

We study string theory on the pp-wave geometry obtained by taking the Penrose limit around a certain null geodesic of the non-supersymmetric Schrodinger background. We solve for the spectrum of bosonic excitations and find compelling agreement with the dispersion relation of the giant magnons in the Schrodinger background obtained previously in arXiv:1712.03091. Inspired by the pp-wave spectrum we conjecture an exact in the t'Hooft coupling dispersion relation for the magnons in the original Schrodinger background. We show that the pp-wave background admits exactly 16 Killing spinors. We use the explicit form of the latter in order to derive the supersymmetry algebra of the background which explicitly depends on the deformation parameter. Its bosonic subalgebra is of the Newton-Hooke type., Comment: 35 pages; v2: Minor changes, JHEP version

We invent a method that exploits the geometry in the space of couplings and the known all-loop effective action, in order to calculate the exact in the couplings anomalous dimensions of composite operators for a wide class of integrable $\sigma$-models. These involve both self and mutually interacting current algebra theories. We work out the details for important classes of such operators. In particular, we consider the operators built solely from an arbitrary number of currents of the same chirality, the composite operators which factorize into a chiral and an anti-chiral part, as well as those made up of three currents of mixed chirality. Remarkably enough, the anomalous dimensions of the former two sets of operators turn out to vanish. In our approach, loop computations are completely avoided., Comment: 1+60 pages, minor improvements, published version

We construct the all loop effective action for WZW models perturbed by current-bilinear terms of the type $J_+J_- $, $J_+J_+ $ and $J_-J_- $, the last two of which explicitly break Lorentz invariance. For isotropic couplings we prove integrability. For the case in which only the first two terms are present we identify a non-perturbative symmetry of the effective action and we compute the exact beta-functions. These become identically zero outside a bounded region in the parametric space., Comment: 21 pages, 1 figure. some presentation improvements

We show that the CFT with symmetry group $G_{k_1}\times G_{k_2}\times \cdots \times G_{k_n}$ consisting of WZW models based on the same group $G$, but at arbitrary integer levels, admits an integrable deformation depending on $2(n-1)$ continuous parameters. We derive the all-loop effective action of the deformed theory and prove integrability. We also calculate the exact in the deformation parameters RG flow equations which can be put in a particularly simple compact form. This allows a full determination and classification of the fixed points of the RG flow, in particular those that are IR stable. The models under consideration provide concrete realizations of integrable flows between CFTs. We also consider non-Abelian T-duality type limits., Comment: 27 pages

The Collatz conjecture is one of the easiest mathematical problems to state and yet it remains unsolved. For each $n\ge 2$ the Collatz iteration is mapped to a binary sequence and a corresponding unique integer which can recreate the iteration. The binary sequence is used to produce the Collatz curve, a 2-D visualization of the iteration on a grid, which, besides the aesthetics, provides a qualitative way for comparing iterations. Two variants of the curves are explored, the r-curves and on-change-turn-right curves. There is a scarcity of acyclic r-curves and only three r-curves were found having a cycle of minimum length greater than 4., Comment: 12 pages, 16 figures. Added reference to A304715, added new sections on reverse curves and miracle curves. Minor corrections

We construct the all loop effective action representing, for small couplings, simultaneously self and mutually interacting current algebra CFTs realized by WZW models. This non-trivially generalizes our previous works where such interactions were, at the linear level, not simultaneously present. For the two coupling case we prove integrability and calculate the coupled RG flow equations. We also consider non-Abelian T-duality type limits. Our models provide concrete realisations of integrable flows between exact CFTs and exhibit several new features which we discuss in detail., Comment: 33 pages, 4 figures, typos corrected in version 2, version published in Nucl. Phys. B

We calculate, for the first time, three-point correlation functions involving "heavy" operators in the Schrodinger/null-dipole CFT correspondence at strong coupling. In particular, we focus on the three-point functions of the dilaton modes and two "heavy" operators. The heavy states are dual to the single spin and dyonic magnon, the single spin and dyonic spike solutions or to two novel string solutions which do not have an undeformed counterpart. Our results provide the leading term of the correlators in the large $\lambda$ expansion and are in perfect agreement with the form of the correlator dictated by non-relativistic conformal invariance. We also specify the scaling function which can not be fixed by using conformal invariance., Comment: 26 pages, 1 figure

By employing CFT techniques, we show how to compute in the context of \lambda-deformations of current algebras and coset CFTs the exact in the deformation parameters C-function for a wide class of integrable theories that interpolate between a UV and an IR point. We explicitly consider RG flows for integrable deformations of left-right asymmetric current algebras and coset CFTs. In all cases, the derived exact C-functions obey all the properties asserted by Zamolodchikov's c-theorem in two-dimensions., Comment: v1: 1+15 pages, Latex, v2: PLB version, v3: Correcting a typo in footnote 6

We construct semi-classical string solutions of the Schr\"odinger $Sch_5 \times S^5$ 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., Comment: 16 pages; v2: references added

We explicitly construct families of integrable $\sigma$-model actions smoothly interpolating between exact CFTs. In the ultraviolet the theory is the direct product of two current algebras at levels $k_1$ and $k_2$. In the infrared and for the case of two deformation matrices the CFT involves a coset CFT, whereas for a single matrix deformation it is given by the ultraviolet direct product theories but at levels $k_1$ and $k_2-k_1$. For isotropic deformations we demonstrate integrability. In this case we also compute the exact beta-function for the deformation parameters using gravitational methods. This is shown to coincide with previous results obtained using perturbation theory and non-perturbative symmetries., Comment: 1+27 pages, text improvements, version published in JHEP

We prove that the doubly lambda-deformed sigma-models, which include integrable cases, are canonically equivalent to the sum of two single lambda-deformed models. This explains the equality of the exact beta-functions and current anomalous dimensions of the doubly lambda-deformed sigma-models to those of two single lambda-deformed models. Our proof is based upon agreement of their Hamiltonian densities and of their canonical structure. Subsequently, we show that it is possible to take a well defined non-Abelian type limit of the doubly-deformed action. Last, but not least, by extending the above, we construct multi-matrix integrable deformations of an arbitrary number of WZW models., Comment: 1+17 pages, Latex; v2: PLB version, v3: Correcting a typo in Eq.(3.7)

We study quantum aspects of the recently constructed doubly lambda-deformed sigma-models representing the effective action of two WZW models interacting via current bilinears. We show that although the exact beta-functions and current anomalous dimensions are identical to those of the lambda-deformed models, this is not true for the anomalous dimensions of generic primary field operators in accordance with the fact that the two models differ drastically. Our proofs involve CFT arguments, as well as effective sigma-model action and gravity calculations., Comment: 1+26 pages, Latex

We construct a new class of integrable $\sigma$-models based on current algebra theories for a general semisimple group $G$ by utilizing a left-right asymmetric gauging. Their action can be thought of as the all-loop effective action of two independent WZW models for $G$ both at level $k$, perturbed by current bilinears mixing the different WZW models. A non-perturbative symmetry in the couplings parametric space is revealed. We perform the Hamiltonian analysis of the action and demonstrate integrability in several cases. We extend our construction to deformations of $G/H$ CFTs and show integrability when $G/H$ is a symmetric space. Our method resembles that used for constructing the $\lambda$-deformed integrable $\sigma$-models, but the results are distinct and novel., Comment: 25 pages, minor typos corrected, published version

We compute the all-loop anomalous dimensions of current and primary field operators in deformed current algebra theories based on a general semi-simple group, but with different (large) levels for the left and right sectors. These theories, unlike their equal level counterparts, possess a new non-trivial fixed point in the IR. By computing the exact in $\lambda$ two- and three-point functions for these operators we deduce their OPEs and their equal-time commutators. Using these we argue on the nature of the CFT at the IR fixed point. The associated to the currents Poisson brackets are a two-parameter deformation of the canonical structure of the isotropic PCM., Comment: 1+27 pages, Latex, v2: Correcting a typo in Eqs. (3.1), (3.2), v3: Correcting a typo in Eq.(2.20)

We compute the 2- and 3-point functions of currents and primary fields of $\lambda$-deformed integrable $\sigma$-models characterized also by an integer $k$. Our results apply for any semisimple group $G$, for all values of the deformation parameter $\lambda$ and up to order $1/k$. We deduce the OPEs and equal-time commutators of all currents and primaries. We derive the currents' Poisson brackets which assume Rajeev's deformation of the canonical structure of the isotropic PCM, the underlying structure of the integrable $\lambda$-deformed $\sigma$-models. We also present analogous results in two limiting cases of special interest, namely for the non-Abelian T-dual of the PCM and for the pseudodual model., Comment: 30 pages plus appendices; v2: few minor changes, NPB version

We present a detailed study of the Entanglement Entropy (EE) of excited states in all closed rank one subsectors of N=4 SYM, namely SU(2), SU(1|1) and SL(2). Exploiting the techniques of the Coordinate and the Algebraic Bethe Ansatz we obtain the EE for spin chains with up to seven magnons, at leading order in the coupling expansion but exact in the length of the spin chain and of the part of it that we cut. Focusing on the superconformal primary operator with two magnons in the BMN limit, we derive analytic and exact, in the coupling $\lambda'$, expressions for the Renyi and the EE. The interpolating functions for the Renyi and the EE monotonically increase as the coupling increases from the weak coupling $\lambda' \rightarrow 0$ regime to the strong coupling $\lambda' \rightarrow \infty$ regime. This results to a violation of a certain bound for the EE that is present at weak coupling and confirms the physical intuition that entanglement increases when the coupling increases., Comment: 33 pages, 12 figures; v2: Typos corrected and a new appendix with the Bethe roots added; JHEP published version