We build string backgrounds for Yang-Baxter deformations of the $AdS_4\times\mathbb{CP}^3$ superstring generated by $r$-matrices satisfying the classical Yang-Baxter equation. We obtain the metric and the NS-NS two-form of the gravity dual corresponding to noncommutative and dipole deformations of ABJM theory, as well as a deformed background with Schrödinger symmetry. The first two backgrounds may also be found by TsT transformations while for the last background we get a new family of non-relativistic ABJM theories with Schrödinger symmetry.
We consider three-parameter Yang-Baxter deformations of the $AdS_5\times T^{1,1}$ superstring for abelian $r$-matrices which are solutions of the classical Yang-Baxter equation. We find two new backgrounds which are dual to the dipole deformed Klebanov-Witten gauge theory and to the nonrelativistic Klebanov-Witten gauge theory with Schrödinger symmetry
We present an integrable deformation of the $AdS_{5}\times S^{5}$ pure spinor action based on homological perturbation theory. Its equations of motion, Lax connection and BRST symmetry are discussed. The resulting model describes a pure spinor superstring in a generalized supergravity background.
The gravity dual of $β$-deformed ABJM theory can be obtained by a TsT transformation of $AdS_4\times\mathbb{CP}^3$. We present a supercoset construction of $\mathbb{CP}^3$ to obtain this gravity dual theory as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to get an Abelian $r$-matrix satisfying the classical Yang-Baxter equation. Our results provide another illustration of the relation between Abelian $r$-matrices and TsT transformations.
Interesting deformations of AdS_5 x S^5 such as the gravity dual of noncommutative SYM and Schödinger spacetimes have recently been shown to be integrable. We clarify questions regarding the reality and integrability properties of the associated construction based on R matrices that solve the classical Yang-Baxter equation, and present an overview of manifestly real R matrices associated to the various deformations. We also discuss when these R matrices should correspond to TsT transformations, which not all do, and briefly analyze the symmetries preserved by these deformations, for example fi…
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 consider so-called Yang-Baxter deformations of bosonic string sigma-models, based on an $R$-matrix solving the (modified) classical Yang-Baxter equation. It is known that a unimodularity condition on $R$ is sufficient for Weyl invariance at least to two loops (first order in $α'$). Here we ask what the necessary condition is. We find that in cases where the matrix $(G+B)_{mn}$, constructed from the metric and $B$-field of the undeformed background, is degenerate the unimodularity condition arising at one loop can be replaced by weaker conditions. We further show that for non-unimodular defo…
Based on the formulation of Yang-Baxter sigma models developed by Klimcik and Delduc-Magro-Vicedo, we explain that various deformations of type IIB superstring on AdS$_5\times$S$^5$ can be characterized by classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). The relation may be referred to as `the gravity/CYBE correspondence.' We present non-trivial examples of the correspondence including Lunin-Maldacena backgrounds for $β$-deformations of the ${\cal N} = 4$ super Yang-Mills theory and the gravity duals for non-commutative gauge theories. We also discuss non-integrable…
The so-called homogeneous Yang-Baxter (YB) deformations can be considered a non-abelian generalization of T-duality--shift--T-duality (TsT) transformations. TsT transformations are known to preserve conformal symmetry to all orders in $α'$. Here we argue that (unimodular) YB deformations of a bosonic string also preserve conformal symmetry, at least to two-loop order. We do this by showing that, starting from a background with no NSNS-flux, the deformed background solves the $α'$-corrected supergravity equations to second order in the deformation parameter. At the same time we determine the re…
We investigate Poisson-Lie duals of the $η$-deformed $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ superstring. The $η$-deformed background satisfies a generalisation of the type II supergravity equations. We discuss three Poisson-Lie duals, with respect to (i) the full $\mathfrak{psu}(1,1|2)$ superalgebra, (ii) the full bosonic subalgebra and (iii) the Cartan subalgebra, for which the corresponding backgrounds are expected to satisfy the standard type II supergravity equations. The metrics and B-fields for the first two cases are the same and given by an analytic continuation of th…
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 study Baxter's T-Q equation of XXX spin-chain models under the semiclassical limit where an intriguing SU(N)/SU(2)^{N-3} correspondence emerges. That is, two kinds of 4D \mathcal{N}=2 superconformal field theories having the above different gauge groups are encoded simultaneously in one Baxter's T-Q equation which captures their spectral curves. For example, while one is SU(N_c) with N_f=2N_c flavors the other turns out to be SU(2)^{N_c-3} with N_c hyper-multiplets (N_c > 3). It is seen that the corresponding Seiberg-Witten differential supports our proposal.
We show that new BRST charges in RNS superstring theory with nonstandard ghost numbers, constructed in our recent work, can be mapped to deformed pure spinor (PS) superstring theories, with the nilpotent pure spinor BRST charge $Q_{PS}=\oint{λ^α}d_α$ still retaining its form but with singular operator products between commuting spinor variables $λ^α$. Despite the OPE singularities, the pure spinor condition $λγ^mλ=0$ is still fulfilled in a weak sense, explained in the paper. The operator product singularities correspond to introducing interactions between the pure spinors. We conjecture that …
In this paper, based on the notion of plane wave string/gauge theory duality, we explore the pp wave limit associated with the bosonic sector of $ η$ deformed superstrings propagating in $ (AdS_5 \times S^{5})_η $. Our analysis reveals that in the presence of NS-NS and RR fluxes, the pp wave limit associated to full ABF background satisfies type IIB equations in its standard form. However, the beta functions as well as the string Hamiltonian start receiving non trivial curvature corrections as one starts probing beyond pp wave limit which thereby takes solutions away from the standard type IIB…
We recently proposed an integrable q-deformation of the AdS_5 x S^5 superstring action. Here we give details on the hamiltonian origin and construction of this deformation. The procedure is a generalization of the one previously developed for deforming principal chiral and symmetric space sigma-models. We also show that the original psu(2,2|4) symmetry is replaced in the deformed theory by a classical analog of the quantum group U_q(psu(2,2|4)) with q real. The relation between q and the deformation parameter entering the action is given. The framework used to derive the deformation also enabl…
We find an exact type IIB supergravity solution that represents a one-parameter deformation of the T-dual of the AdS_5 x S^5 background (with T-duality applied in all 6 abelian bosonic isometric directions). The non-trivial fields are the metric, dilaton and RR 5-form only. The latter has remarkably simple "undeformed" form when written in terms of a "deformation-rotated" vielbein basis. An unusual feature of this solution is that the dilaton contains a linear dependence on the isometric coordinates of the metric precluding a straightforward reversal of T-duality. If we still formally dualize …
Homogeneous Yang-Baxter (YB) deformation of AdS$_5\times$S$^5$ superstring is revisited. We calculate the YB sigma model action up to quadratic order in fermions and show that homogeneous YB deformations are equivalent to $β$-deformations of the AdS$_5\times$S$^5$ background when the classical $r$-matrices consist of bosonic generators. In order to make our discussion clearer, we discuss YB deformations in terms of the double-vielbein formalism of double field theory. We further provide an O(10,10)-invariant string action that reproduces the Green-Schwarz type II superstring action up to quadr…
Yang-Baxter type models are integrable deformations of integrable field theories, such as the principal chiral model on a Lie group $G$ or $σ$-models on (semi-)symmetric spaces $G/F$. The deformation has the effect of breaking the global $G$-symmetry of the original model, replacing the associated set of conserved charges by ones whose Poisson brackets are those of the $q$-deformed Poisson-Hopf algebra $\mathscr U_q(\mathfrak g)$. Working at the Hamiltonian level, we show how this $q$-deformed Poisson algebra originates from a Poisson-Lie $G$-symmetry. The theory of Poisson-Lie groups and thei…
Baxter operators are constructed for quantum spin chains with deformed $s\ell_2$ symmetry. The parallel treatment of Yang-Baxter operators for the cases of undeformed, trigonometrically and elliptically deformed symmetries presented earlier and relying on the factorization regarding parameter permutations is extended to the global chain operators following the scheme worked out recently in the undeformed case.
We construct supergravity backgrounds for the integrable eta-deformations of the AdS2 x S2 x T6 and AdS5 x S5 superstring sigma models. The eta-deformation is governed by an R-matrix that solves the non-split modified classical Yang-Baxter equation on the superisometry algebra of the model. Such R-matrices include those of Drinfel'd-Jimbo type, which are constructed from a Dynkin diagram and the associated Cartan-Weyl basis. Drinfel'd-Jimbo R-matrices associated with inequivalent bases will typically lead to different deformed backgrounds. For the two models under consideration we find that th…
The Yang-Baxter $σ$-model is an integrable deformation of the principal chiral model on a Lie group $G$. The deformation breaks the $G \times G$ symmetry to $U(1)^{\textrm{rank}(G)} \times G$. It is known that there exist non-local conserved charges which, together with the unbroken $U(1)^{\textrm{rank}(G)}$ local charges, form a Poisson algebra $\mathscr U_q(\mathfrak{g})$, which is the semiclassical limit of the quantum group $U_q(\mathfrak{g})$, with $\mathfrak{g}$ the Lie algebra of $G$. For a general Lie group $G$ with rank$(G)>1$, we extend the previous result by constructing local and n…
String backgrounds of the form $\mathbb{M}_3 \times {\cal M}_7$ where $\mathbb{M}_3$ denotes $3$-dimensional Minkowski space while ${\cal M}_7$ is a $7$-dimensional G$_2$-manifold, are characterised by the property that the world-sheet theory has a Shatashvili-Vafa (SV) chiral algebra. We study the generalisation of this statement to backgrounds where the Minkowski factor $\mathbb{M}_3$ is replaced by ${\rm AdS}_3$. We argue that in this case the world-sheet theory is characterised by a certain ${\cal N}=1$ superconformal ${\cal W}$-algebra that has the same spin spectrum as the SV algebra and…
Yang-Baxter deformations of superstring sigma-models have recently inspired a supergravity solution generating technique. Using the open/closed string map and a Killing bi-vector as a deformation parameter, new solutions can be built, such that the (generalised) supergravity field equations were conjectured to always reduce to the classical Yang-Baxter equation (CYBE). In this work we provide a proof of this conjecture, using a systematic approach based on the so-called $β$-supergravity, which is a dynamical theory for the field $β^{mn}$ instead of the NSNS 2-form $b_{mn}$.
We investigate the worldsheet S matrix of string theory in $η$-deformed $AdS_5\times S^5$. By computing the six-point tree-level S matrix we explicitly show that there is no particle production at this level, as required by the classical integrability of the theory. At one and two loops we show that integrability requires that the classical two-particle states be redefined in a non-local and $η$-dependent way. This is a significant departure from the undeformed theory which is probably related to the quantum group symmetry of the worldsheet theory. We use generalized unitarity to carry out the…
Superstrings on NS5 backgrounds, deformed AdS3 and holography
我们研究了D1/D5-膜系统的一个非标准退耦极限，它介于D1/D5背景的近视界几何和纯D5-膜几何的近视界极限之间。这个背景的S-对偶描述实际上是一个精确可解的二维（世界面）共形场论：{零变形SL（2，R）} x SU（2）x T^4或K3。该模型不存在强耦合奇点。通过对SL（2，R）的细致处理，基于更好理解的SL（2，R）/ U（1）陪集，我们得到了SL（2，R）x SU（2）x K 3上超弦的完全配分函数.这使我们能够计算J^3和J^2电流-电流变形的配分函数，以及连接SL（2，R）共形场论和线性声子理论的超对称零变形的全线。这种设置的全息解释是一个重整化群流之间的解耦NS 5膜世界体积理论在紫外线（小弦论），和低能量动力学的超杨-米尔斯弦状瞬子在六个维度。
A classical exchange algebra of the superstring on S^5 with the AdS-time is shown on the light-like plane. To this end we use the geometrical method of which consistency is guaranteed by the classical Yang-Baxter equation. The Dirac method does not work, there being constraints which contain first-class and second-class and one can disentangle with each other keeping the isometry hardly.
Using an elementary perturbative open string field theory solution involving a twistor-like parameter, we study the cohomology of new nilpotent BRST charge corresponding to the space-time background defined by this solution. The BRST cohomology of the deformed background automatically cuts off the GSO-odd spectrum in RNS superstring theory and keeps the GSO-even spectrum intact, without a need of GSO-projection. The on-shell constraints in the GSO-even sector get deformed in the new background, corresponding to BRST type transformations of the low-energy effective action with the ghost-like co…
We prove that abelian Yang-Baxter deformations of superstring coset sigma models are equivalent to sequences of commuting TsT transformations, meaning T dualities and coordinate shifts. Our results extend also to fermionic deformations and fermionic T duality, and naturally lead to a TsT subgroup of the superduality group OSp(d_b,d_b|2d_f). In cases like AdS_5 x S^5, fermionic deformations necessarily lead to complex models. As an illustration of inequivalent deformations, we give all six abelian deformations of AdS_3. We comment on the possible dual field theory interpretation of these (super…
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…
The investigation of the q deformation of the S-matrix for excitations on the string world sheet in AdS5 x S5 is continued. We argue that due to the lack of Lorentz invariance the situation is more subtle than in a relativistic theory in that the nature of bound states depends on their momentum. At low enough momentum |p|<E the bound states transform in the anti-symmetric representation of the super-algebra symmetry and become the solitons of the Pohlmeyer reduced theory in the relativistic limit. At a critical momentum |p|=E they become marginally unstable, and at higher momenta the stable bo…
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 consider γ-deformations of the AdS_5xS^5 superstring as Yang-Baxter sigma models with classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). An essential point is that the classical r-matrices are composed of Cartan generators only and then generate abelian twists. We present examples of the r-matrices that lead to real γ-deformations of the AdS_5xS^5 superstring. Finally we discuss a possible classification of integrable deformations and the corresponding gravity solution in terms of solutions of CYBE. This classification may be called the gravity/CYBE correspondence.
Integrable deformations of type IIB superstring theory on $\mathrm{AdS}_5\times S^5$ have played an important role over the last years. The Yang-Baxter deformation is a systematic way of generating such integrable deformations. Since its introduction, this topic has seen important conceptual progress and has among others led to the intriguing discovery generalized supergravity, a new low-energy effective theory. This review endeavors to not only introduce the historical development of the Yang-Baxter deformation, but also its relation to generalized supergravity, non-geometric backgrounds, non…
We explicitly derive Lax pairs for string theories on Yang-Baxter deformed backgrounds, 1) gravity duals for noncommutative gauge theories, 2) $γ$-deformations of S$^5$, 3) Schrödinger spacetimes and 4) abelian twists of the global AdS$_5$\,. Then we can find out a concise derivation of Lax pairs based on simple replacement rules. Furthermore, each of the above deformations can be reinterpreted as a twisted periodic boundary conditions with the undeformed background by using the rules. As another derivation, the Lax pair for gravity duals for noncommutative gauge theories is reproduced from th…
In this thesis we discuss how one can derive the quantum spectral curve for the $η$-deformed AdS$_5 \times S^5$ superstring, an integrable deformation of the AdS$_5 \times $S$^5$ superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the AdS$_5 \times $S$^5$ superstring, like the Heisenberg xxz spin chain is a trigonometric version of the xxx spin chain. Our derivation starts from the ground-state thermodynamic Bethe ansatz equations and discusses the construction of both the undeformed and the $η$-deformed quantum spectral curve. We reformulate it firs…
Yang-Baxter (YB) deformations of type IIB string theory have been well studied from the viewpoint of classical integrability. Most of the works, however, are focused upon the local structure of the deformed geometries and the global structure still remains unclear. In this work, we reveal a non-geometric aspect of YB-deformed backgrounds as $T$-fold by explicitly showing the associated $\mathrm{O}(D,D;\mathbb{Z})$ $T$-duality monodromy. In particular, the appearance of an extra vector field in the generalized supergravity equations (GSE) leads to the non-geometric $Q$-flux. In addition, we stu…
The spectral problem for the ${\rm AdS}_5\times {\rm S}^5$ superstring and its dual planar maximally supersymmetric Yang-Mills theory can be efficiently solved through a set of functional equations known as the quantum spectral curve. We discuss how the same concepts apply to the $η$-deformed ${\rm AdS}_5\times {\rm S}^5$ superstring, an integrable deformation of the ${\rm AdS}_5\times {\rm S}^5$ superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the ${\rm AdS}_5\times {\rm S}^5$ superstring, like the relation between the XXZ and XXX spin chains, or…
Two distinct $η$-deformations of strings on AdS$_5\times$S$^5$ can be defined; both amount to integrable quantum deformations of the string non-linear sigma model, but only one is itself a superstring background. In this paper we compare their conjectured all-loop worldsheet S matrices and derive the corresponding Bethe equations. We find that, while the S matrices are apparently different, they lead to the same Bethe equations. Moreover, in either case the eigenvalues of the transfer matrix, which encode the conserved charges of each system, also coincide. We conclude that the integrable stru…
We consider bosonic subsectors of the q-deformed AdS_5 x S^5 superstring action and study the classical integrable structure of anisotropic Landau-Lifshitz sigma models (LLSMs) derived by taking fast-moving limits. The subsectors are 1) deformed AdS_3 x S^1 and 2) R x deformed S^3. The cases 1) and 2) lead to a time-like warped SL(2) LLSM and a squashed S^3 LLSM, respectively. For each of them, we construct an infinite number of non-local conserved charges and show a quantum affine algebra at the classical level. Furthermore, a pp-wave like limit is applied for the case 1). The resulting syste…
In this Letter, based on the notion of gauge/gravity duality, we compute $ q $ SYK spectra in the presence of Yang-Baxter (YB) deformations. The gravitational counterpart of this duality turns out to be the YB deformed Almheiri-Polchinski (AP) model embodied with hyperbolic potential for the dilaton. Following Kaluza-Klein reduction on $ (AdS_2)_η\times (S^1/ Z_2) $, we compute the deformed Green's function that reproduces the original $ q $ SYK spectrum in the limit of vanishing deformations.
We determine the bosonic part of the superstring sigma model Lagrangian on $η$-deformed AdS5 x S5, and use it to compute the perturbative world-sheet scattering matrix of bosonic particles of the model. We then compare it with the large string tension limit of the q-deformed S-matrix and find exact agreement.
We build the bosonic $η$-deformed $AdS_4\times\mathbb{CP}^3$ background generated by an $r$-matrix that satisfies the modified classical Yang-Baxter equation. In a special limit we find that it is the gravity dual of the noncommutative ABJM theory.
We study the worldsheet scattering theory of the $η$ deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring corresponding to the purely fermionic Dynkin diagram. This theory is a Weyl-invariant integrable deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring, with trigonometric quantum-deformed symmetry. We compute the two-body worldsheet S matrix of this string in the light-cone gauge at tree level to quadratic order in fermions. The result factorizes into two elementary blocks, and solves the classical Yang-Baxter equation. We also determine the corresponding e…
Yang-Baxter (YB) deformations of string sigma model provide deformed target spaces. We propose that homogeneous YB deformations always lead to a certain class of $β$-twisted backgrounds and represent the bosonic part of the supergravity fields in terms of the classical r-matrix associated with the YB deformation. We then show that various $β$-twisted backgrounds can be realized by considering generalized diffeomorphisms in the undeformed background. Our result extends the notable relation between the YB deformations and (non-commuting) TsT transformations. We also discuss more general deformat…
Yang-Baxter deformed sigma models on $\mathrm{SU}(N)/\mathrm{SO}(N)$
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；HSE University（高等经济学院）；Institute for Theoretical and Mathematical Physics MSU（莫斯科大学理论与数学物理研究所）
