Yang-Baxter Deformations Research Articles

Regularising spectral curves for homogeneous Yang-Baxter strings

In this Letter, we study the semi-classical spectrum of integrable worldsheet σ-models using the Spectral Curve. We consider a Homogeneous Yang-Baxter deformation of the AdS5×S5 superstring, understood as the composition of a Jordanian with a “non-diagonal” TsT deformation. We derive its type IIB supergravity solution, whose isometry algebra features zero supercharges and a non-relativistic conformal algebra in 0+1 dimensions. While the Spectral Curves of non-diagonal TsT models are ill-defined, we demonstrate that the composition with a Jordanian model regularises this issue. From the regularised Curve, we derive the one-loop shift of the classical energy and the semi-classical spectrum of excitations of a point-like string. In the TsT limit, the one-loop shift vanishes despite the loss of supersymmetry. Our results suggest that it may be possible to use standard Bethe Ansatze on spin chain pictures of deformed N=4 Super-Yang-Mills theory dual to non-diagonal TsT models.

Yang-Baxter deformed wedge holography

In this paper, we construct the wedge holography in the presence of homogeneous Yang-Baxter deformation. We observe that the DGP term is the reason for the existence of non-zero tension of the Karch-Randall branes in Yang-Baxter deformed wedge holography. The homogeneous Yang-Baxter deformation introduces non-trivial island surfaces inside the black hole horizon whose entanglement entropy is lower than the twice of thermal entropy of the black hole. Therefore, we obtain the Page curve even without the DGP term on the Karch-Randall branes due to the homogeneous Yang-Baxter deformation in the context of wedge holography. Finally, we compute the holographic complexity in homogeneous Yang-Baxter deformed AdS2 background.

Yang-Baxter deformations of the GL(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GL(2,{\\mathbb {R}})$$\\end{document} WZW model and non-Abelian T-duality

By calculating inequivalent classical r-matrices for the gl(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$gl(2,{\\mathbb {R}})$$\\end{document} Lie algebra as solutions of (modified) classical Yang-Baxter equation ((m)CYBE), we classify the YB deformations of Wess-Zumino-Witten (WZW) model on the GL(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GL(2,{\\mathbb {R}})$$\\end{document} Lie group in twelve inequivalent families. Most importantly, it is shown that each of these models can be obtained from a Poisson-Lie T-dual σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-model in the presence of the spectator fields when the dual Lie group is considered to be Abelian, i.e. all deformed models have Poisson-Lie symmetry just as undeformed WZW model on the GL(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GL(2,{\\mathbb {R}})$$\\end{document}. In this way, all deformed models are specified via spectator-dependent background matrices. For one case, the dual background is clearly found.

On 10-dimensional Exceptional Drinfeld algebras

Abstract Based on Mubarakzyanov’s classification of four-dimensional real Lie algebras, we classify ten-dimensional Exceptional Drinfeld algebras (EDAs). The classification is restricted to EDAs whose maximal isotropic (geometric) subalgebras cannot be represented as a product of a 3D Lie algebra and a 1D abelian factor. We collect the obtained algebras into families depending on the dualities found between them. Despite algebras related by a generalized Yang–Baxter deformation we find two algebras related by a different Nambu–Lie U-duality transformation. We show that this duality relates two Type IIA backgrounds.

All Jordanian deformations of the $AdS_5 \times S^5$ superstring

We explicitly construct and classify all Jordanian solutions of the classical Yang-Baxter equation on \mathfrak{psu}(2,2|4)𝔭𝔰𝔲(2,2|4), corresponding to Jordanian Yang-Baxter deformations of the AdS_5\times S^5AdS5 × S5 superstring. Such deformations preserve the classical integrability of the underlying sigma-model and thus are a subclass of all possible integrable deformations. The deformations that we consider are divided into two families, unimodular and non-unimodular ones. The former ensure that the deformed backgrounds are still solutions of the type IIB supergravity equations. For the simplest unimodular solutions, we find that the corresponding backgrounds preserve a number N < 32 of supercharges that can be N=12,8,6,4,0.

Topological gauging and double current deformations

We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved U(1) currents Ja. We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the S-matrix of the system, similarly as what happens in the case of a textrm{T}overline{textrm{T}} deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents Ja are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the textrm{T}overline{textrm{T}} deformation to the central extension of the two-dimensional Poincaré algebra.

Yang-Baxter deformation of WZW model based on Lie supergroups: The cases of GL(1|1) and (C3 + A)

We proceed to generalize the Yang-Baxter (YB) deformation of Wess-Zumino-Witten (WZW) model to the Lie supergroups case. This generalization enables us to utilize various kinds of solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE) to classify the YB deformations of WZW models based on the Lie supergroups. We obtain the inequivalent solutions (classical r-matrices) of the (m)GCYBE for the gl(1|1) and (C3+A) Lie superalgebras in the non-standard basis, in such a way that the corresponding automorphism transformations are employed. Then, the YB deformations of the WZW models based on the GL(1|1) and (C3+A) Lie supergroups are specified by skew-supersymmetric classical r-matrices satisfying (m)GCYBE. In some cases for both families of deformed models, the metrics remain invariant under the deformation, while the components of B-fields are changed. After checking the conformal invariance of the models up to one-loop order, it is concluded that the GL(1|1) and (C3+A) WZW models are conformal theories within the classes of the YB deformations preserving the conformal invariance. However, our results are interesting in themselves, but at a constructive level, may prompt many new insights into (generalized) supergravity solutions.

Generalized 11D supergravity equations from tri-vector deformations

In Bakhmatov et al. (Phys. Rev. D 105(8): L081904, 2022) we presented a modification of 11-dimensional supergravity field equations which upon dimensional reduction yields generalized supergravity equations in 10-dimensions. In this paper we provide full technical details of that result which is based on SL(5) exceptional field theory. The equations are obtained by making a non-unimodular tri-vector Yang-Baxter deformation which breaks the initial local diffeomorphism symmetry in 11 dimensions down to separate coordinate transformations in external 7 and internal 4 directions. We also give some non-trivial solutions to these equations.

Yang-Baxter deformations of the flat space string

We define integrability preserving Yang-Baxter deformations of symmetric space sigma models with non-semi-simple symmetry group, in particular the flat space string, using only the essential structures of a symmetric space sigma model. For homogeneous deformations, the Lax connection is of the same form as the semi-simple case, although the R operator no longer satisfies a freestanding operator equation. For inhomogeneous deformations, the form of the Lax connection needs to be relaxed, by modifying the underlying algebra. We illustrate the construction by discussing nonabelian deformations of three dimensional Minkowski space.

Non-isometric U-dualities

I study generalisations of U-duality transformations which do not rely on the existence of isometries. I start by providing more details of a recently proposed generalised U-duality map between solutions of type IIA supergravity of the form M7× S3, with NSNS flux, and solutions of 11-dimensional supergravity, in which the three-sphere is replaced by a four-dimensional geometry which encodes three-algebra structure constants. I then show that when M7 admits two abelian isometries, TsT deformations on the IIA side become six-vector deformations in the 11-dimensional setting. These six-vector deformations involve an action of E6(6) on both isometric and non-isometric directions. I discuss the algebraic interpretation of these deformations, and compare and contrast them with (generalised) Yang-Baxter deformations in supergravity.

Semiclassical spectrum of a Jordanian deformation of AdS5×S5

We study a Jordanian deformation of the $AdS_5 \times S^5$ superstring that preserves 12 superisometries. It is an example of homogeneous Yang-Baxter deformations, a class that generalises TsT deformations to the non-abelian case. Many of the attractive features of TsT carry over to this more general class, from the possibility of generating new supergravity solutions to the preservation of worldsheet integrability. In this paper, we exploit the fact that the deformed $\sigma$-model with periodic boundary conditions can be reformulated as an undeformed one with twisted boundary conditions, to discuss the construction of the classical spectral curve and its semi-classical quantisation. First, we find global coordinates for the deformed background, and identify the global time corresponding to the energy that should be computed in the spectral problem. Using the curve of the twisted model, we obtain the one-loop correction to the energy of a particular solution, and we find that the charge encoding the twisted boundary conditions does not receive an anomalous correction. Finally, we give evidence suggesting that the unimodular version of the deformation (giving rise to a supergravity background) and the non-unimodular one (whose background does not solve the supergravity equations) have the same spectrum at least to one-loop.

Integrable supersymmetric deformations of AdS3× S3× T4

We construct a family of type IIB string backgrounds that are deformations of AdS3× S3× T4 with a “squashed” AdS3× S3 metric supported by a combination of NSNS and RR fluxes. They have global SU(1, 1) × SU(2) symmetry, regular curvature, constant dilaton and preserve 8 supercharges. Upon compactification to 4 dimensions they reduce to mathcal{N} = 2 supersymmetric AdS2× S2 solutions with electric and magnetic Maxwell fluxes. These type IIB supergravity solutions can be found from the undeformed AdS3× S3× T4 background by a combination of T-dualities and S-duality. In contrast to T-duality, S-duality transformations of a type IIB supergravity background do not generally preserve the classical integrability of the corresponding Green-Schwarz superstring sigma model. Nevertheless, we show that integrability is preserved in the present case. Indeed, we find that these backgrounds can be obtained, up to T-dualities, from an integrable inhomogeneous Yang-Baxter deformation (with unimodular Drinfel’d-Jimbo R-matrix) of the original AdS3× S3 supercoset model.

Lax pairs for new [formula omitted]-symmetric coset σ-models and their Yang-Baxter deformations

Two-dimensional σ-models with ZN-symmetric homogeneous target spaces have been shown to be classically integrable when introducing WZ-terms in a particular way. This article continues the search for new models of this type now allowing some kinetic terms to be absent – analogously to the Green-Schwarz superstring σ-model on Z4-symmetric homogeneous spaces. A list of such integrable ZN-symmetric (super)coset σ-models for N≤6 and their Lax pairs is presented. For arbitrary N a big class of integrable models is constructed that includes both the known pure spinor and Green-Schwarz superstring on Z4-symmetric cosets.Integrable Yang-Baxter deformations of this class of ZN-symmetric (super)coset σ-models can be constructed in same way as in the known Z2- or Z4-cases. Deformations based on solutions of the modified classical Yang-Baxter equation, the so-called η-deformation, require deformation of the constants defining the Lagrangian and the corresponding Lax pair. Homogeneous Yang-Baxter deformations (i.e. those based on solutions to the classical Yang-Baxter equation) leave the equations of motion and consequently the Lax pair invariant and are expected to be classically equivalent to the undeformed model.As an example, the relationship between Z3-symmetric homogeneous spaces and nearly (para-)Kähler geometries is revisited. Confirming existing literature it is shown that the integrable choice of WZ-term in the Z3-symmetric coset σ-model associated to a nearly Kähler background gives an imaginary contribution to the action.

Do Drinfeld twists of $AdS_5 \times S^5$ survive light-cone quantization?

We study how a wide class of Abelian Yang-Baxter deformations of the AdS_5 \times S^5AdS5×S5 string behave at the quantum level. These deformations are equivalent to TsT transformations and conjectured to be dual to beta, dipole, and noncommutative deformations of SYM. Classically they correspond to Drinfeld twists of the original theory. To verify this expectation at the quantum level we compute and match (1) the bosonic two-body tree-level worldsheet scattering matrix of these deformations in the uniform light-cone gauge, and (2) the Bethe equations of the equivalent model with twisted boundary conditions. We find that for a generalization of gamma deformations of the BMN string the we are able to express the S matrix either through a Drinfeld twist or a shift of momenta. For deformations of the GKP string around the null-cusp solution we encounter calculational obstacles that prevent us from calculating the scattering matrix.

Homogeneous Yang-Baxter deformations as undeformed yet twisted models

The homogeneous Yang-Baxter deformation is part of a larger web of integrable deformations and dualities that recently have been studied with motivations in integrable σ-models, solution-generating techniques in supergravity and Double Field Theory, and possible generalisations of the AdS/CFT correspondence. The σ-models obtained by the homogeneous Yang-Baxter deformation with periodic boundary conditions on the worldsheet are on-shell equivalent to undeformed models, yet with twisted boundary conditions. While this has been known for some time, the expression provided so far for the twist features non-localities (in terms of the degrees of freedom of the deformed model) that prevent practical calculations, and in particular the construction of the classical spectral curve. We solve this problem by rewriting the equation defining the twist in terms of the degrees of freedom of the undeformed yet twisted model, and we show that we are able to solve it in full generality. Remarkably, this solution is a local expression. We discuss the consequences of the twist at the level of the monodromy matrix and of the classical spectral curve, analysing in particular the concrete examples of abelian, almost abelian and Jordanian deformations of the Yang-Baxter class.

Exceptional Algebroids and Type IIB Superstrings

AbstractIn this note we study exceptional algebroids, focusing on their relation to type IIB superstring theory. We show that a IIB‐exact exceptional algebroid (corresponding to the group , for ) locally has a standard form given by the exceptional tangent bundle. We derive possible twists, given by a flat ‐connection, a covariantly closed pair of 3‐forms, and a 5‐form, and comment on their physical interpretation. Using this analysis we reduce the search for Leibniz parallelisable spaces, and hence maximally supersymmetric consistent truncations, to a simple algebraic problem. We show that the exceptional algebroid perspective also gives a simple description of Poisson–Lie U‐duality without spectators and hence of generalised Yang–Baxter deformations.

Brief lectures on duality, integrability and deformations

We provide a pedagogical introduction to some aspects of integrability, dualities and deformations of physical systems in [Formula: see text] and in [Formula: see text] dimensions. In particular, we concentrate on the T-duality of point particles and strings as well as on the Ruijsenaars duality of finite many-body integrable models, we review the concept of the integrability and, in particular, of the Lax integrability and we analyze the basic examples of the Yang–Baxter deformations of nonlinear [Formula: see text]-models. The central mathematical structure which we describe in detail is the [Formula: see text]-model which is the dynamical system exhibiting all those three phenomena simultaneously. The last part of the paper contains original results, in particular, a formulation of sufficient conditions for strong integrability of non-degenerate [Formula: see text]-models.

Supergravity solution-generating techniques and canonical transformations of \u03c3-models from O(D, D)

Within the framework of the flux formulation of Double Field Theory (DFT) we employ a generalised Scherk-Schwarz ansatz and discuss the classification of the twists that in the presence of the strong constraint give rise to constant generalised fluxes interpreted as gaugings. We analyse the various possibilities of turning on the fluxes Hijk, Fijk, Qijk and Rijk, and the solutions for the twists allowed in each case. While we do not impose the DFT (or equivalently supergravity) equations of motion, our results provide solution-generating techniques in supergravity when applied to a background that does solve the DFT equations. At the same time, our results give rise also to canonical transformations of 2-dimensional σ-models, a fact which is interesting especially because these are integrability-preserving transformations on the worldsheet. Both the solution-generating techniques of supergravity and the canonical transformations of 2-dimensional σ-models arise as maps that leave the generalised fluxes of DFT and their flat derivatives invariant. These maps include the known abelian/non-abelian/Poisson-Lie T-duality transformations, Yang-Baxter deformations, as well as novel generalisations of them.

Yang-Baxter deformations of WZW model on the Heisenberg Lie group

The Yang-Baxter (YB) deformations of Wess-Zumino-Witten (WZW) model on the Heisenberg Lie group (H4) are examined. We proceed to obtain the nonequivalent solutions of (modified) classical Yang-Baxter equation ((m)CYBE) for the h4 Lie algebra by using its corresponding automorphism transformation. Then we show that YB deformations of H4 WZW model are split into ten nonequivalent backgrounds including metric and B-field such that some of the metrics of these backgrounds can be transformed to the metric of H4 WZW model while the antisymmetric B-fields are changed. The rest of the deformed metrics have a different isometric group structure than the H4 WZW model metric. As an interesting result, it is shown that all new integrable backgrounds of the YB deformed H4 WZW model are conformally invariant up to two-loop order. In this way, we obtain the general form of the dilaton fields satisfying the vanishing beta-function equations of the corresponding σ-models.

Yang-Baxter deformations of the AdS5 \xd7 T1,1 superstring and their backgrounds

We consider three-parameter Yang-Baxter deformations of the AdS5× T1,1 superstring for abelian r-matrices which are solutions of the classical Yang-Baxter equation. We find the NSNS fields of 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.