We consider several classes of $σ$-models (on groups and symmetric spaces, $η$-models, $λ$-models) with local couplings that may depend on the 2d coordinates, e.g. on time $τ$. We observe that (i) starting with a classically integrable 2d $σ$-model, (ii) formally promoting its couplings $h_α$ to functions $h_α(τ)$ of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that $h_α(τ)$ must solve the 1-loop RG equations of the original theory with $τ$ interpreted as RG time. This provides a novel example of an 'integrability - RG flow' connection. The existence of a Lax connection suggests that these time-dependent $σ$-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such $σ$-models with $D$-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a $(D+2)$-dimensional conformal $σ$-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.
We show that SU(2)_L Yangian and q-deformed SU(2)_R symmetries are realized in a two-dimensional sigma model defined on a three-dimensional squashed sphere. These symmetries enable us to develop the two descriptions to describe its classical dynamics, 1) rational and 2) trigonometric descriptions. The former 1) is based on the SU(2)_L symmetry and the latter 2) comes from the broken SU(2)_R symmetry. Each of the Lax pairs constructed in both ways leads to the same equations of motion. The two descriptions are related one another through a non-local map.
We consider two dimensional non linear sigma models on few symmetric superspaces, which are supergroup manifolds of coset type. For those spaces where one loop beta function vanishes, two loop beta function is calculated and is shown to be zero. Vanishing of beta function in all orders of perturbation theory is shown for the principal chiral models on group supermanifolds with zero Killing form. Sigma models on symmetric (super) spaces on supergroup manifold $G/H$ are known to be classically integrable. We investigate a possibility to extend an argument of absence of quantum anomalies in non l…
The conditions under which a general two-dimensional non-linear sigma model is classically integrable are given. These requirements are found by demanding that the equations of motion of the theory are expressible as a zero curvature relation. Some new integrable two-dimensional sigma models are then presented.
This is a write-up of lectures on integrable sigma-models, which covers the following topics: (1) Homogeneous spaces, (2) Classical integrability of sigma-models in two dimensions, (3) Topological terms, (4) Background-field method and beta-function, (5) S-matrix bootstrap in the O(N) model, (6) Supersymmetric cosets and strings on AdS(d)xX.
The classical integrability the O(N) nonlinear sigma model on a half-line is examined, and the existence of an infinity of conserved charges in involution is established for the free boundary condition. For the case N=3 other possible boundary conditions are considered briefly.
We discuss classical integrable structure of two-dimensional sigma models which have three-dimensional Schrodinger spacetimes as target spaces. The Schrodinger spacetimes are regarded as null-like deformations of AdS_3. The original AdS_3 isometry SL(2,R)_L x SL(2,R)_R is broken to SL(2,R)_L x U(1)_R due to the deformation. According to this symmetry, there are two descriptions to describe the classical dynamics of the system, 1) the SL(2,R)_L description and 2) the enhanced U(1)_R description. In the former 1), we show that the Yangian symmetry is realized by improving the SL(2,R)_L Noether c…
Lie algebra expansion is a technique to generate new Lie algebras from a given one. In this paper, we apply the method of Lie algebra expansion to superstring $σ$-models with a $\mathbb{Z}_4$ coset target space. By applying the Lie algebra expansion to the isometry algebra, we obtain different $σ$-models, where the number of dynamical fields can change. We reproduce and extend in a systematic way actions of some known string regimes (flat space, BMN and non-relativistic in AdS$_5 \times$S$^5$). We define a criterion for the algebra truncation such that the equations of motion of the expanded a…
In this paper we study the integrability of a family of models with U(1)xSU(N) symmetry. They admit fermionic and bosonic formulations related through bosonization and subsequent T-duality. The fermionic theory is just the CP^(N-1) sigma model coupled to a self-interacting massless fermion, while the bosonic one defines a one-parameter deformation of the O(2N) sigma model. For N=2 the latter model is equivalent to the integrable deformation of the O(4) sigma model discovered by Wiegmann. At higher values of N we find that integrability is more sporadic and requires a fine-tuning of the paramet…
We construct integrability preserving boundary conditions for Green-Schwarz sigma-models on semi-symmetric spaces. The boundary conditions are expressed as gluing conditions of the flat-connection, using an involutive metric preserving automorphism. We show that the boundary conditions preserve half of the space-time supersymmetry and an infinite set of conserved charges. We find integrable D-brane configurations for AdS_5 x S^5 and AdS_4 x CP^3 backgrounds.
This review is devoted to the classical integrability of the AdS5xS5 superstring theory. It starts with a reminder of the corresponding action as a coset model. The symmetries of this action are then reviewed. The classical integrability is then considered from the lagrangian and hamiltonian points of view. The second part of this review deals with the gauge fixing of this theory. Finally, some aspects of the pure spinor formulation are also briefly reviewed.
Recently, a gauged linear sigma model was proposed by Berkovits and Vafa which can be used to describe the AdS_5xS^5 superstring at finite and zero radius. In this paper we show that the model is classically integrable by constructing its first non-local conserved charge and a superspace Lax "quartet". Quantum conservation of the non-local charge follows easily from superspace rules.
Exact massive S-matrices for two dimensional sigma models on symmetric spaces SU(2N)/Sp(N) and Sp(2P)/Sp(P)*Sp(P) are conjectured. They are checked by comparison of perturbative and non perturbative TBA calculations of free energy in a strong external field. We find the mass spectrum of the models and calculate their exact mass gap.
Non-local conserved charges in two-dimensional sigma models with target spaces $SO(2n)/SO(n){\times}SO(n)$ and $Sp(2n)/Sp(n){\times}Sp(n)$ are shown to survive quantization, unspoiled by anomalies; these theories are therefore integrable at the quantum level. Local, higher-spin, conserved charges are also shown to survive quantization in the $SO(2n)/SO(n){\times}SO(n)$ models.
We construct recursively an infinite number of Poisson structures for the supersymmetric integrable hierarchy governing the Pohlmeyer reduction of superstring sigma models on the target spaces AdS_{n}\times S^n, n=2,3,5. These Poisson structures are all non-local and not relativistic except one, which is the canonical Poisson structure of the semi-symmetric space sine-Gordon model (SSSSG). We verify that the superposition of the first three Poisson structures corresponds to the canonical Poisson structure of the reduced sigma model. Using the recursion relations we construct commuting charges …
We introduce and study conformal field theories specified by $W-$algebras commuting with certain set of screening charges. These CFT's possess perturbations which define integrable QFT's. We establish that these QFT's have local and non-local Integrals of Motion and admit the perturbation theory in the weak coupling region. We construct factorized scattering theory which is consistent with non-local Integrals of Motion and perturbation theory. In the strong coupling limit the $S-$matrix of this QFT tends to the scattering matrix of the $O(N)$ sigma model. The perturbation theory, Bethe anzatz …
A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The Sinh-Gordon model, the affine sl(3) Toda mo…
We construct topological Wess-Zumino term for supercoset sigma-models on various AdS(3) backgrounds. For appropriately chosen set of parameters, the sigma-model remains integrable when the Wess-Zumino term is added to the action. Moreover, the conditions for integrability, kappa-symmetry and conformal invariance are equivalent to each other.
This short note is devoted to the study of the Hamiltonian formalism and the integrability of the bosonic model introduced in [hep-th/0612079]. We calculate Poisson bracket of spatial components of Lax connection and we argue that its structure implies classical integrability of the theory.
It is shown how derived brackets naturally arise in sigma-models via Poisson- or antibracket, generalizing a recent observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression for the derived bracket is obtained. The generalized Nijenhuis tensor of generalized complex geometry is shown to coincide up to a de-Rham closed term with the derived bracket of the structure with itself, and a new coordinate expression for this tensor is presented. The insight is applied to two known two-dimensional sigma models in a background with genera…
We first review the result that the noncommutative principal chiral model has an infinite tower of conserved currents, and discuss the special case of the noncommutative CP^1 model in some detail. Next, we focus our attention to a submodel of the CP^1 model in the noncommutative spacetime A_θ(R^2+1). By extending a generalized zero curvature representation to A_θ(R^2+1) we discuss its integrability and construct its infinitely many conserved currents. Supersymmetric principal chiral model with and without the WZW term and a supersymmetric extension of the CP^1 submodel in noncommutative spacet…
A master equation expressing the classical integrability of two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. In particular, a closer connection between integrability and T-duality transformations is emphasised. Finally, a whole new class of integrable non-linear sigma models is found and all their corresponding Lax pairs depend on a spectral parameter.
We probe warped BTZ $ \times S^3 $ geometry with various string solitons and explore the classical integrability criteria of the associated phase space configurations using Kovacic's algorithm. We consider consistent truncation of the parent sigma model into one dimension and obtain the corresponding normal variational equations (NVE). Two specific examples have been considered where the sigma model is reduced over the subspace of the full target space geometry. In both examples, NVEs are found to possess the Liouvillian form of solutions which ensures the classical integrability of the associ…
The bi-Yang-Baxter sigma-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimcik who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset sigma-model on G x G / G_diag, we show that it also admits a Lax matrix whose Poisson bracket is of…
We consider a class of 2d $σ$-models on products of group spaces that provide new examples of a close connection between integrability and stability under the RG flow. We first study the integrable $G \times G$ model derived from the affine Gaudin construction (for which the 1-loop $β$-functions were found in arXiv:2010.07879) and show that its condition of integrability is preserved also by the 2-loop RG flow. We then investigate the RG flow in the gauged $G \times G /H$ model, in particular the integrable $T^{1,1}$ model found in arXiv:2010.05573. We also construct a new class of integrable …
Yang-Baxter sigma models, proposed by Klimcik and Delduc-Magro-Vicedo, have been recognized as a powerful framework for studying integrable deformations of two-dimensional non-linear sigma models. In this short article, as an important generalization, we review a non-integrable sigma model in the Yang-Baxter sigma model approach based on [arXiv:1406.2249]. In particular, we discuss a family of deformations of the 5D Sasaki-Einstein manifold $T^{1,1}$, instead of the standard deformations of the $5$-sphere S$^5$. For this purpose, we first describe a novel construction of $T^{1,1}$ as a superco…
We use the notion of the gauge/string duality and discuss the Liouvillian (non) integrability criteria for string sigma models in the context of recently proposed Arutyunov-Bassi-Lacroix (ABL) model [JHEP \textbf{03} (2021), 062]. Our analysis complements those previous results due to numerical analysis as well as Lax pair formulation. We consider a winding string ansatz for the deformed torus $T^{\qty(λ_{1},λ_{2},λ)}_{k}$ which can be interpreted as a system of coupled pendulums. Our analysis reveals the Liouvillian nonintegrablity of the associated sigma model. We also obtain the \emph{gener…
The existence of higher-spin quantum conserved currents in two dimensions guarantees quantum integrability. We revisit the question of whether classically-conserved local higher-spin currents in two-dimensional sigma models survive quantization. We define an integrability index $\mathcal{I}(J)$ for each spin $J$, with the property that $\mathcal{I}(J)$ is a lower bound on the number of quantum conserved currents of spin $J$. In particular, a positive value for the index establishes the existence of quantum conserved currents. For a general coset model, with or without extra discrete symmetries…
We discuss an interplay between T-duality and integrability for certain classical non-linear sigma models. In particular, we consider strings on the AdS_5 x S^5 background and perform T-duality along the four isometry directions of AdS_5 in the Poincare patch. The T-dual of the AdS_5 sigma model is again a sigma model on an AdS_5 space. This classical T-duality relation was used in the recently uncovered connection between light-like Wilson loops and MHV gluon scattering amplitudes in the strong coupling limit of the AdS/CFT duality. We show that the explicit coordinate dependence along the T-…
We show that the integrability of the $SO(N)/SO(N-1)$ Principal Chiral Model (PCM) originates from the Pohlmeyer reduction of the $O(N)$ Non Linear Sigma Model (NLSM). In particular, we show that the Lax pair of the PCM is related upon redefinitions and identification of parameters to the zero curvature condition, which is a consequence of the flatness of the enhanced space used in the Pohlmeyer reduction. This identification provides the solution of the auxiliary system that corresponds to an arbitrary NLSM/PCM solution.
Skyrme theory on S^2 (Faddeev coset proposal), is analyzed with a generalization of 0-curvature integrability, based on gauge techniques. New expressions valid for models in the sphere are given. The relation of the minimum energy configurations to gauge vacua is clarified. Consequences of adding a potential term to break the SO(3) symmetry are discussed.
We prove that the field equations of supergravity for purely time-dependent backgrounds, which reduce to those of a one--dimensional sigma model, admit a Lax pair representation and are fully integrable. In the case where the effective sigma model is on a maximally split non--compact coset U/H (maximal supergravity or subsectors of lower supersymmetry supergravities) we are also able to construct a completely explicit analytic integration algorithm, adapting a method introduced by Kodama et al in a recent paper. The properties of the general integral are particularly suggestive. Initial data a…
This paper is devoted to the analysis of the integrability of D1-brane on group manifold. We consider D1-brane as principal chiral model, determine corresponding equations of motions and find Lax connection. Then we calculate the Poisson brackets of Lax connection and we find that it has similar structure as in case of principal chiral model. As the second example we consider more general background with non-zero NS-NS two form. We again show that D1-brane theory is integrable on this background and determine Poisson brackets of Lax connection.
We obtained the formal solution of the auxiliary system of Non Linear Sigma Models, whose target space is a rank 1 symmetric space based on the indefinite orthogonal group O(p,q), corresponding to an arbitrary solution of the NLSM. This class includes Anti-de Sitter, de Sitter and Hyperbolic spaces, which are of interest in view of the AdS/CFT correspondence. The formal solution is related to the Pohlmeyer reduction of the NLSM, constituting another link between the NLSM and the reduced theory. Besides deriving the solution, we also review the Pohlmeyer reduction of such models. Finally, we co…
We discuss the integrability of 2d non-linear sigma models with target space being the squashed three-sphere, warped anti-de Sitter space and the Schroedinger spacetime. These models can be obtained via T-duality from integrable models. We construct an infinite family of non-local conserved charges from the T-dual Lax currents, enhancing the symmetry of warped anti-de Sitter space and the Schroedinger spacetime to sl2(R)+sl2(R).
We consider D1-brane as a natural probe of the group manifold with mixed three form fluxes. We determine Lax connection for given theory. Then we switch to the canonical analysis and calculate the Poisson brackets between spatial components of Lax connections and we argue for integrability of given theory.
Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d $σ$-models. We include non-conformal $σ$-models, which have historically been a good arena for discovering integrable models that were later generalized to Weyl-invariant ones. General $σ$-models feature a quantum RG flow, given by a 'generalized Ricci flow' of the target-space geometry. This thesis is based on the conjecture that integrable $σ$-models are renormalizable, or stable under the RG flow. It is widely understood that classically integrable theories are stable at th…
The integrability of N=(2,2) dilaton supergravity in two dimensions is studied by the use of the graded Poisson Sigma model approach. Though important differences compared to the purely bosonic models are found, the general analytic solutions are obtained. The latter include minimally gauged models as well as an ungauged version. BPS solutions are an especially interesting subclass.
Quantum Flat Connections, KZ equations, and Integrability
Institut des Hautes Études Scientifiques (IHES)（高等科学研究所）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）；Qom University of Technology (QUT)（库姆理工大学）
In the absence of NSNS three-form flux the bosonic string on a symmetric space is described by a symmetric space coset sigma-model. Such models are known to be classically integrable. We show that the integrability extends also to cases with non-zero NSNS flux (respecting the isometries) provided that the flux satisfies a condition of the form H_{abc}H^{cde}~R_{ab}^{de}. We then turn our attention to the type II Green-Schwarz superstring on a symmetric space. We prove that if the space preserves some supersymmetry there exists a truncation of the full superspace to a supercoset space and deriv…
For type II supergravity backgrounds with superisometries the corresponding transformations and conserved currents for the superstring are constructed up to fourth order in $Θ$. It is then shown how, for certain backgrounds related to near horizon geometries of intersecting branes, the components of the superisometry current can be used to construct a Lax connection demonstrating the classical integrability of the string in these backgrounds. This includes examples of $AdS_2$ and $AdS_3$ backgrounds with a $D(2,1;α)$ isometry group which have not previously been studied from an integrability p…
We review the deformed instanton equations making connection with Hilbert schemes and integrable systems. A single U(1) instanton is shown to be \asd\ with respect to the Burns metric.
It is well-known that sigma-models with symmetric target spaces are classically integrable. At the example of the model with target space the flag manifold U(3)/U(1)^3 -- a non-symmetric space -- we show that the introduction of torsion allows to cast the equations of motion in the form of a zero-curvature condition for a one-parametric family of connections, which can be a sign of integrability of the theory. We also elaborate on geometric aspects of the proposed model.
We investigate boundary conditions for the nonlinear sigma model on the compact symmetric space $G/H$, where $H \subset G$ is the subgroup fixed by an involution $σ$ of $G$. The Poisson brackets and the classical local conserved charges necessary for integrability are preserved by boundary conditions in correspondence with involutions which commute with $σ$. Applied to $SO(3)/SO(2)$, the nonlinear sigma model on $S^2$, these yield the great circles as boundary submanifolds. Applied to $G \times G/G$, they reproduce known results for the principal chiral model.
The Gauge/YBE correspondence states a surprising connection between solutions to the Yang-Baxter equation with spectral parameters and partition functions of supersymmetric quiver gauge theories. This correspondence has lead to systematic discoveries of new integrable models based on quantum-field-theory methods. We provide pedagogical introduction to the subject and summarizes many recent developments. This is a write-up of the lecture at the String-Math 2018 conference.
We construct a zero-curvature Lax connection in a sub-sector of the superstring theory on AdS(4) x CP(3) which is not described by the OSp(6|4)/U(3) x SO(1,3) supercoset sigma-model. In this sub-sector worldsheet fermions associated to eight broken supersymmetries of the type IIA background are physical fields. As such, the prescription for the construction of the Lax connection based on the Z_4-automorphism of the isometry superalgebra OSp(6|4) does not do the job. So, to construct the Lax connection we have used an alternative method which nevertheless relies on the isometry of the target su…
We derive the AdS${}_5\times$S${}^5$ Green-Schwarz superstring from four-dimensional Beltrami-Chern-Simons theory reduced on a manifold with singular boundary conditions. In this construction, the Lax connection and spectral parameter of the integrable superstring have a simple geometric origin in four dimensions as gauge connection and reduction coordinate. Kappa symmetry arises as a certain class of singular gauge transformations, while the worldsheet metric comes from complex-structure-changing Beltrami differentials. Our approach offers the possibility of investigating integrable holograph…
In this thesis we investigate quantum aspects of the Green-Schwarz superstring in various AdS backgrounds relevant for the AdS/CFT correspondence, providing several examples of perturbative computations in the corresponding integrable sigma-models. We start by reviewing in details the supercoset construction of the superstring action in $AdS_5 \times S^5$, pointing out the limits of this procedure for $AdS_4$ and $AdS_3$ backgrounds. For the $AdS_4 \times CP^3$ case we give a thorough derivation of an alternative action, based on the double-dimensional reduction of eleven-dimensional super-mem…
We find new integrable boundary conditions, depending on a free parameter $g$, for the O(N) nonlinear $σ$ model, which are of nondiagonal type, that is, particles can change their ``flavor'' through scattering off the boundary. These boundary conditions are derived from a microscopic boundary lagrangian, which is used to establish their integrability, and exhibit integrable flows between diagonal boundary conditions investigated earlier. We solve the boundary Yang-Baxter equation, connect these solutions to the boundary conditions, and examine the corresponding integrable flows.
Integrable Sigma Models and Universal Root $T\bar{T}$ Deformation via Courant-Hilbert Approach
Ningbo University（宁波大学）；Peking University（北京大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））
We consider superstring sigma models that are based on coset superspaces G/H in which H arises as the fixed point set of an order-4 automorphism of G. We show by means of twistor theory that the corresponding first-order system, consisting of the Maurer-Cartan equations and the equations of motion, arises from a dimensional reduction of some generalised self-dual Yang-Mills equations in eight dimensions. Such a relationship might help shed light on the explicit construction of solutions to the superstring equations including their hidden symmetry structures and thus on the properties of their …
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…
In this paper, we motivate how the Hodge dual related with S-duality gives the hidden symmetry in the moduli space of IIB string. Utilizing the static $% κ$-symmetric Killing gauge, if we take the Hodge dual of the vierbeins keeping the connection invariant, the duality of Maure-Cartan equations and the equations of motion becomes manifest. Thus by twistly transforming the vierbein, we can express the BPR currents as the Lax connections by a unique spectral parameter. Then we construct the generators of the infinitesimal dressing symmetry, the related symmetric algebra becomes the affine $% gl…
The short-distance singularity of the product of a composite scalar field that deforms a field theory and an arbitrary composite field can be expressed geometrically by the beta functions, anomalous dimensions, and a connection on the theory space. Using this relation, we compute the connection perturbatively for the O(N) non-linear sigma model in two dimensions. We show that the connection becomes free of singularities at zero temperature only if we normalize the composite fields so that their correlation functions have well-defined limits at zero temperature.
Various recently developed connections between supersymmetric Yang-Mills theories in four dimensions and two dimensional integrable systems serve as crucial ingredients in improving our understanding of the AdS/CFT correspondence. In this review, we highlight some connections between superconformal four dimensional Yang-Mills theory and various integrable systems. In particular, we focus on the role of Yangian symmetries in studying the gauge theory dual of closed string excitations. We also briefly review how the gauge theory connects to Calogero models and open quantum spin chains through th…
We review how Geroch's reduction method is extended from Ricci-flat to Einstein spacetimes. The Ehlers-Geroch SL(2,R) group is still present in the three-dimensional sigma-model that captures the dynamics, but only a subgroup of it is solution-generating. Holography provides an alternative three-dimensional perspective to integrability properties of Einstein's equations in asymptotically anti-de Sitter spacetimes. These properties emerge as conditions on the boundary data (metric and energy-momentum tensor) ensuring that the hydrodynamic derivative expansion be resummed into an exact four-dime…
