Non-trivial outer algebra automorphisms may be utilized in $λ$-deformations of (gauged) WZW models thus providing an efficient way to construct new integrable models. We provide two such integrable deformations of the exact coset CFT $SU(2)_k\times U(1)/U(1)_q$ with a vector and axial residual gauge. Besides the integer level $k$ and the deformation parameter $λ$, these models are characterized by the embedding parameter $q$ of the $U(1)$ factor. We show that an axial-vector T-duality persists along the deformations and, therefore, the models are canonically equivalent. We demonstrate integrability even though the space is non-symmetric and compute the $RG$-flow equations for the parameters $λ$ and $q$. Our example provides an integrable deformation of the gravitational solution representing a Euclidean three-dimensional black string.
We review some recent developments in the construction of integrable $η$- and $λ$-deformations of the $AdS_5 \times S^5$ superstring. We highlight their link with Poisson-Lie T-duality.
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.
We study integrable deformations of sine-Liouville conformal field theory. Every integrable perturbation of this model is related to the series of quantum integrals of motion (hierarchy). We construct the factorized scattering matrices for different integrable perturbed conformal field theories. The perturbation theory, Bethe ansatz technique, renormalization group and methods of perturbed conformal field theory are applied to show that all integrable deformations of sine-Liouville model possess non-trivial duality properties.
This contribution is based on a talk given by the author at the "Dualities and Generalized Geometries" session of the Corfu Summer Institute 2018 workshops. We overview the results of [1], focusing our attention on integrable $λ$-deformations of WZW models with boundaries.
We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti--self--dual Yang--Mills equations with a gauge group Diff$(S^1)$.
We present new integrable models of interacting spin-1/2 chains, which can be interpreted as hard rod deformations of the XXZ Heisenberg chains. The models support multiple particle types: dynamical hard rods of length $\ell$ and particles with lengths $\ell'<\ell$ that are immobile except for the interaction with the hard rods. We encounter a remarkable phenomenon in these interacting models: exact spectral degeneracies across different deformations and volumes. The algebraic integrability of these systems is also treated using a recently developed formalism for medium range integrable spin c…
We consider the non-integrable bosonic backgrounds $W_{2,4}\times T^{1,1}$ and $AdS_5\times T^{1,1}$ and derive their bosonic $η$-deformed versions using an $r$-matrix that solves the modified Yang-Baxter equation obtaining new integrable deformed backgrounds.
An integrable deformation of the type IIB AdS_5 x S^5 superstring action is presented. The deformed field equations, Lax connection, and kappa-symmetry transformations are given. The original psu(2,2|4) symmetry is expected to become q-deformed.
We define a two-parameter family of integrable deformations of the principal chiral model on an arbitrary compact group. The Yang-Baxter sigma-model and the principal chiral model with a Wess-Zumino term both correspond to limits in which one of the two parameters vanishes.
We study d=2 0A string theory perturbed by tachyon momentum modes in backgrounds with non-trivial tachyon condensate and Ramond-Ramond (RR) flux. In the matrix model description, we uncover a complexified Toda lattice hierarchy constrained by a pair of novel holomorphic string equations. We solve these constraints in the classical limit for general RR flux and tachyon condensate. Due to the non-holomorphic nature of the tachyon perturbations, the transcendental equations which we derive for the string susceptibility are manifestly non-holomorphic. We explore the phase structure and critical be…
We consider current-current deformations that generalise $T\bar{T}$ ones, and show that they may be also introduced for integrable spin chains. In analogy with the integrable QFT setup, we define the deformation as a modification of the S matrix in the Bethe equations. Using results by Bargheer, Beisert and Loebbert we show that the deforming operator is composite and constructed out of two currents on the lattice; its expectation value factorises like for $T\bar{T}$. Such a deformation may be considered for any combination of charges that preserve the model's integrable structure.
A multi-parameter integrable deformation of the principal chiral model is presented. The Yang-Baxter and bi-Yang-Baxter sigma-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.
For supercosets with isometry group of the form $\hat{G} \times \hat{G}$, the eta-deformation can be generalised to a two-parameter integrable deformation with independent $q$-deformations of the two copies. We study its kappa-symmetry and write down a formula for the Ramond-Ramond fluxes. We then focus on $\hat{G}= PSU(1,1|2)$ and construct two supergravity backgrounds for the two-parameter integrable deformation of the $AdS_3 \times S^3 \times T^4$ superstring, as well as explore their limits. We also construct backgrounds that are solutions of the weaker generalised supergravity equations o…
We consider the three-parameter integrable deformation of the $AdS_3\times S^3$ superstring background constructed in arXiv:1811.00453. Working on the string worldsheet in uniform lightcone gauge, we find the tree-level bosonic S matrix of the model and study some of its limits.
We construct integrable deformations of the $λ$-type for asymmetrically gauged WZW models. This is achieved by a modification of the Sfetsos gauging procedure to account for a possible automorphism that is allowed in $G/G$ models. We verify classical integrability, derive the one-loop beta function for the deformation parameter and give the construction of integrable D-brane configurations in these models. As an application, we detail the case of the $λ$-deformation of the cigar geometry corresponding to the axial gauged $SL(2,R)/U(1)$ theory at large $k$. Here we also exhibit a range of both …
In this pedagogical review we introduce systematic approaches to deforming integrable 2-dimensional sigma models. We use the integrable principal chiral model and the conformal Wess-Zumino-Witten model as our starting points and explore their Yang-Baxter and current-current deformations. There is an intricate web of relations between these models based on underlying algebraic structures and worldsheet dualities, which is highlighted throughout. We finish with a discussion of the generalisation to other symmetric integrable models, including some original results related to Z T cosets and their…
We provide a new construction of the dressing cosets sigma-models which is based on an isotropic gauging of the E-models. As an application of this new approach, we show that the recently constructed multi-parametric integrable deformations of the principal chiral model are the dressing cosets, they are therefore automatically renormalizable and their dynamics can be completely characterized in terms of current algebras.
Lie algebra valued equations translating the integrability of a general two-dimensional Wess-Zumino-Witten model are given. We found simple solutions to these equations and identified three types of new integrable non-linear sigma models. One of them is a modified Yang-Baxter sigma model supplemented with a Wess-Zumino-Witten term.
In this article we study an integrable deformation of the Kapustin-Witten equations. Using the Weyl-Wigner-Moyal-Groenewold description an integrable $\star$-deformation of a Kapustin-Witten system is obtained. Starting from known solutions of the original equations, some solutions to these deformed equations are obtained.
A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg-Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg-Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.
After a short review of the algebraic setting of N=2 superconformal field theories, their chiral ring and their perturbations, I present some recent results on curious relations between the integrability of their perturbations and algebraic properties of their deformed chiral ring. (Lecture given at Hang-zhou, China, Sept 1993)
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 construct integrability-preserving deformations of the integrable $σ$-model coupling together $N$ copies of the Principal Chiral Model. These deformed theories are obtained using the formalism of affine Gaudin models, by applying various combinations of Yang-Baxter and $λ$-deformations to the different copies of the undeformed model. We describe these models both in the Hamiltonian and Lagrangian formulation and give explicit expressions of their action and Lax pair. In particular, we recover through this construction various integrable $λ$-deformed models previously introduced in the liter…
The exact effective field equations of motion, corresponding to the perturbative mixed theory of open and closed (2,2) world-sheet supersymmetric strings, are investigated. It is shown that they are only integrable in the case of an abelian gauge group. The gravitational equations are then stationary with respect to the Born-Infeld-type effective action.
A generalization of the notion of a (pseudo-) Riemannian space is proposed in a framework of noncommutative geometry. In particular, there are parametrized families of generalized Riemannian spaces which are deformations of classical geometries. We also introduce harmonic maps on generalized Riemannian spaces into Hopf algebras and make contact with integrable models in two dimensions.
We describe a unifying framework for the systematic construction of integrable deformations of integrable $σ$-models within the Hamiltonian formalism. It applies equally to both the `Yang-Baxter' type as well as `gauged WZW' type deformations which were considered recently in the literature. As a byproduct, these two families of integrable deformations are shown to be Poisson-Lie T-dual of one another.
A three-parameter integrable deformation of $\mathbb{Z}_4$ permutation supercosets is constructed. These supercosets are of the form $ {\widehat{F}} / {F_0}$ where $F_0$ is the bosonic diagonal subgroup of the product supergroup $\widehat{F}=\widehat{G} \times \widehat{G}$. They include the AdS$_3$ $\times$ S$^3$ and AdS$_3$ $\times$ S$^3$ $\times$ S$^3$ supercosets. This deformation encompasses both the bi-Yang-Baxter deformation of the semi-symmetric space $σ$-model on $\mathbb{Z}_4$ permutation supercosets and the mixed flux model. Truncating the action at the bosonic level, we show that on…
A procedure is developed for constructing deformations of integrable sigma-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F, one recovers the Yang-Baxter sigma-model introduced a few years ago by C. Klimcik. In the case of the symmetric space sigma-model on F/G we obtain a new one-parameter family of integrable sigma-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresp…
We propose a scheme for nonlinearizing linear equations to generate integrable nonlinear systems of both the AKNS and the KN classes, based on the simple idea of dimensional analysis and detecting the building blocks of the Lax pair. Along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the AKNS family, e.g. for the KdV, the mKdV, the NLS and the SG equation, showing thus a two-fold universality of the recently found deformation for the KdV equation.
This is the text of the Hermann Weyl Prize lecture given by the author at the XXIV Colloquium on Group Theoretical Methods in Physics, Paris, July 2002 (to appear in the Proceedings of the Colloquium).
We search for integrable boundary conditions and their geometric interpretation as $D$-branes, in models constructed as generalized $λ$-deformations of products of group- and coset-spaces. Using the sigma-model approach, we find that all the conformal brane geometries known in the literature for a product of WZW models solve the corresponding boundary conditions, thus persisting as integrable branes along the RG flows of our sigma-models. They consist of the well known $G$-conjugacy classes, twisted $G$-conjugacy classes by a permutation automorphism (permutation branes) and generalized permut…
We consider the finite volume mean values of current operators in integrable spin chains with local interactions, and provide an alternative derivation of the exact result found recently by the author and two collaborators. We use a certain type of long range deformation of the local spin chains, which was discovered and explored earlier in the context of the AdS/CFT correspondence. This method is immediately applicable also to higher rank models: as a concrete example we derive the current mean values in the SU(3)-symmetric fundamental model, solvable by the nested Bethe Ansatz. The exact res…
We prove that the description of pencils of compatible (N x N)-metrics of constant Riemannian curvature is equivalent to a special class of integrable N-parametric deformations of quasi-Frobenius (in general, noncommutative) algebras.
We construct two-parameter families of integrable $λ$-deformations of two-dimensional field theories. These interpolate between a CFT (a WZW/gauged WZW model) and the non-Abelian T-dual of a principal chiral model on a group/symmetric coset space. In examples based on the $SU(2)$ WZW model and the $SU(2)/U(1)$ exact coset CFT, we show that these deformations are related to bi-Yang-Baxter generalisations of $η$-deformations via Poisson-Lie T-duality and analytic continuation. We illustrate the quantum behaviour of our models under RG flow. As a byproduct we demonstrate that the bi-Yang-Baxter $…
We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the $T\bar T$-deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of $\mathcal{N} = 4$ super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefor…
We consider the worldsheet S matrix of superstrings on an AdS3xS3xT4 NS-NS background in uniform light-cone gauge. We argue that scattering is given by a CDD factor that is non-trivial only between opposite-chirality particles, yielding a spin-chain-like Bethe Ansatz. Our proposal reproduces the spectrum of non-protected states computed from the Wess-Zumino-Witten description and the perturbative tree-level S matrix. This suggests that the model is an integrable deformation of a free theory similar to those arising from the T-Tbar composite operator.
A proposal for constructing a universal nonlinear ${\hat W}_{\infty}$ algebra is made as the symmetry algebra of a rotational Killing-symmetry reduction of the nonlinear perturbations of Moyal-Integrable deformations of $D=4$ Self Dual Gravity (IDSDG). This is attained upon the construction of a nonlinear bracket based on nonlinear gauge theories associated with infinite dimensional Lie algebras. A Quantization and supersymmetrization program can also be carried out. The relevance to the Kadomtsev-Petviashvili hierarchy, $2D$ dilaton gravity, quantum gravity and black hole physics is discussed…
Integrable deformations of principal chiral model from solutions of associative Yang-Baxter equation
Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科国立大学理论物理与数学物理研究所）；National Research University Higher School of Economics（国立高等经济大学）；National Research Centre ”Kurchatov Institute”（库尔恰托夫国家研究中心）；Institute for Information Transmission Problems RAS (Kharkevich Institute)（俄罗斯科学院信息传输问题研究所（哈尔凯维奇研究所））；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
It has been shown recently that deformations of some integrable field theories in (1+1)-dimensions possess an infinite number of charges that are asymptotically conserved in the scattering of soliton like solutions. Such charges are not conserved in time and they do vary considerably during the scattering process, however they all return in the remote future (after the scattering) to the values they had in the remote past (before the scattering). Such non-linear phenomenon was named quasi-integrability, and it seems to be related to special properties of the solutions under a space-time parity…
We study the approaches to two-dimensional integrable field theories via a six-dimensional(6D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces to a four-dimensional Chern-Simons theory, while under solving along fibres it leads to four-dimensional(4D) integrable theory, the anti-self-dual Yang-Mills or its generalizations. From both four-dimensional theories, various two-dimensional integrable field theories can be obtained. In this work, we try to investigate several two-dimensional integrable deformations in this framework. We find that the $λ$-…
Relativistic integrable field theories like the sine-Gordon equation have an infinite set of conserved charges. In a light-front formalism these conserved charges are closely related to the integrable modified KdV hierarchy at the classical level. The latter hierarchy admits a family of symplectic structures which we argue can be viewed as deformations of the relativistic sine-Gordon symplectic structure. These deformed theories are integrable but no longer relativistic and the basic excitations of the theory, the solitons, have an interesting non-relativistic dispersion relation that in a cer…
We consider integrable deformations of the XXZ spin chain for periodic and open boundary conditions. In particular, we classify all long-range deformations and study their impact on the spectrum. As compared to the XXX case, we have the z-spin at our disposal, which induces two additional deformations: the short-range magnetic twist and a new long-range momentum-dependent twist.
In this paper we study in detail the deformations introduced in [1] of the integrable structures of the AdS$_{2,3}$ integrable models. We do this by embedding the corresponding scattering matrices into the most general solutions of the Yang-Baxter equation. We show that there are several non-trivial embeddings and corresponding deformations. We work out crossing symmetry for these models and study their symmetry algebras and representations. In particular, we identify a new elliptic deformation of the $\rm AdS_3 \times S^3 \times M^4$ string sigma model.
A group of volume-preserving diffeomorphisms in 3D turns out to play a key role in an Einstein-Maxwell theory whose Weyl tensor is selfdual and whose Maxwell tensor has algebraically general anti-selfdual part. This model was first introduced by Flaherty and recently studied by Park as an integrable deformation of selfdual gravity. A twisted volume form on the corresponding twistor space is shown to be the origin of volume-preserving diffeomorphisms. An immediate consequence is the existence of an infinite number of symmetries as a generalization of $w_{1+\infty}$ symmetries in selfdual gravit…
We find new homogeneous r matrices containing supercharges, and use them to find new backgrounds of Yang-Baxter deformed superstrings. We obtain these as limits of unimodular inhomogeneous r matrices and associated deformations of AdS2 x S2 x T6 and AdS5 x S5. Our r 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 transformati…
We study integrable deformations of AdS/CFT by focusing upon three kinds of warped AdS_3 geometries, 1) space-like warped AdS_3, 2) time-like warped AdS_3 and 3) null warped AdS_3. These geometries are embedded into type IIB supergravity solutions and are regarded as consistent string backgrounds. By restricting the classical motion of strings on the warped AdS_3xS^1 subspace, the Landau-Lifshitz sigma models are derived by taking the fast-moving limit. The first two warped AdS_3 spaces correspond to anisotropic deformations of the sl(2) spin chain and the last one to Jordanian deformations. A…
We study deformations of 2D Integrable Quantum Field Theories (IQFT) which preserve integrability (the existence of infinitely many local integrals of motion). The IQFT are understood as "effective field theories", with finite ultraviolet cutoff. We show that for any such IQFT there are infinitely many integrable deformations generated by scalar local fields $X_s$, which are in one-to-one correspondence with the local integrals of motion; moreover, the scalars $X_s$ are built from the components of the associated conserved currents in a universal way. The first of these scalars, $X_1$, coincid…
We present a method to deform (generically non-abelian) T duals of two-dimensional $σ$ models, which preserves classical integrability. The deformed models are identified by a linear operator $ω$ on the dualised subalgebra, which satisfies the 2-cocycle condition. We prove that the so-called homogeneous Yang-Baxter deformations are equivalent, via a field redefinition, to our deformed models when $ω$ is invertible. We explain the details for deformations of T duals of Principal Chiral Models, and present the corresponding generalisation to the case of supercoset models.
We show that the anisotropic Heisenberg-Ising chains with higher spin allow, for special values of the anisotropy, integrable deformations intimately related to the theory of quantum groups at roots of unity. For the spin one case we construct and study the symmetries of the hamiltonian which depends on a spectral variable belonging to an elliptic curve. One of the points of this curve yields the Fateev-Zamolodchikov hamiltonian of spin one and anisotropy $Δ= \frac{ q^2 + q^{-2}}{2} $ with $q$ a cubic root of unity. In some other special points the spin degrees of freedom as well as the hamilt…
We address the question of whether integrable models allow for PT-symmetric deformations which preserve their intgrability. For this purpose we carry out the Painleve test for PT-symmetric deformations of Burgers and the Korteweg-De Vries equation. We find that the former equation allows for infinitely many deformations which pass the Painleve test. For a specific deformation we prove the convergence of the Painleve expansion and thus establish the Painleve property for these models, which are therefore thought to be integrable. The Korteweg-De Vries equation does not allow for deformations wh…
In a recent paper it was shown that the response of an integrable QFT under variation of the Unruh temperature can be computed from a S-matrix preserving deformation of the form factor approach. We give explicit expressions for the deformed two-particle formfactors for various integrable models: The Sine-Gordon and SU(2) Thirring model, several perturbed minimal CFTs and the real coupling affine Toda series. A uniform pattern is found to emerge when both the S-matrix and the deformed form factors are expressed in terms Barnes' multi-periodic functions.
Marginal beta deformations of N=4 super-Yang-Mills theory are known to correspond to a certain class of deformations of the S^5 background subspace of type IIB string theory in AdS_5 x S^5. An analogous set of deformations of the AdS_5 subspace is reviewed here. String energy spectra computed in the near-pp-wave limit of these backgrounds match predictions encoded by discrete, asymptotic Bethe equations, suggesting that the twisted string theory is classically integrable in this regime. These Bethe equations can be derived algorithmically by relying on the existence of Lax representations, and…
We consider two integrable deformations of 2d sigma models on supercosets associated with AdS_n x S^n. The first, the "eta-deformation" (based on the Yang-Baxter sigma model), is a one-parameter generalization of the standard superstring action on AdS_n x S^n, while the second, the "lambda-deformation" (based on the deformed gauged WZW model), is a generalization of the non-abelian T-dual of the AdS_n x S^n superstring. We show that the eta-deformed model may be obtained from the lambda-deformed one by a special scaling limit and analytic continuation in coordinates combined with a particular …
We solve the discrete Hirota equations (Kirillov-Reshetikhin Q-systems) for $A_r$, and their analogue for $D_r$, for the cases where the second variable ranges over either a finite set or over all integers. Until now only special solutions were known. We find all solutions for which no component vanishes, as required in the known applications. As an introduction we present the known solution where the second variable ranges over the natural numbers.
We calculate the $\mathcal{S}$-multiplets for two-dimensional Euclidean $\mathcal{N}=(0,2)$ and $\mathcal{N} = (2,2)$ superconformal field theories under the $T\overline{T}$ deformation at leading order of perturbation theory in the deformation coupling. Then, from these $\mathcal{N} = (0, 2)$ deformed multiplets, we calculate two- and three-point correlators. We show the $\mathcal{N} = (0,2)$ chiral ring's elements do not flow under the $T\overline{T}$ deformation. Specializing to integrable supersymmetric seed theories, such as $\mathcal{N} = (2,2)$ Landau-Ginzburg models, we use the thermod…
We build on the results of arXiv:1912.11036 for generalised frame fields on generalised quotient spaces and study integrable deformations for $\mathbb{CP}^n$. In particular we show how, when the target space of the Principal Chiral Model is a complex projective space, a two-parameter deformation can be introduced in principle. The second parameter can however be removed via a diffeomorphism, which we construct explicitly, in accordance with the results stemming from a thorough integrability analysis we carry out. We also elucidate how the deformed target space can be seen as an instance of gen…
