Yang-Baxter Deformations Research Articles

String backgrounds of the Yang-Baxter deformed AdS4 \xd7 \u2102\u21193 superstring

We build string backgrounds for Yang-Baxter deformations of the AdS4 × ℂℙ3 superstring generated by r-matrices satisfying the classical Yang-Baxter equation. We obtain the metric and the NSNS 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.

Extended Drinfel’d algebras and non-Abelian duality

Abstract The Drinfel’d algebra provides a method to construct generalized parallelizable spaces and this allows us to study an extended $T$-duality, known as the Poisson–Lie $T$-duality. Recently, in order to find a generalized $U$-duality, an extended Drinfel’d algebra (ExDA), called the exceptional Drinfel’d algebra (EDA), was proposed and a natural extension of Abelian $U$-duality was studied both in the context of supergravity and membrane theory. In this paper, we clarify the general structure of ExDAs and show that an ExDA always gives a generalized parallelizable space, which may be regarded as a group manifold with generalized Nambu–Lie structures. We then discuss the non-Abelian duality that is based on a general ExDA. For a coboundary ExDA, this non-Abelian duality reduces to a generalized Yang–Baxter deformation and we find a general formula for the twist matrix. In order to study the non-Abelian $U$-duality, we particularly focus on the $E_{n(n)}$ EDA for $n\leq 8$ and study various aspects, both in terms of M-theory and type IIB theory.

Yang–Baxter deformations of the principal chiral model plus Wess–Zumino term

A large class of integrable deformations of the principal chiral model, known as the Yang–Baxter deformations, are governed by skew-symmetric R-matrices solving the (modified) classical Yang–Baxter equation. We carry out a systematic investigation of these deformations in the presence of the Wess–Zumino term for simple Lie groups, working in a framework that treats both inhomogeneous and homogeneous deformations on the same footing. After analysing the cohomological conditions under which such a deformation is admissible, we consider an action for the general Yang–Baxter deformation of the principal chiral model plus Wess–Zumino term and prove its classical integrability. We also show how the model is found from a number of alternative formulations: affine Gaudin models, -models, four-dimensional Chern–Simons theory and, for homogeneous deformations, non-abelian T-duality.

Relaxing unimodularity for Yang-Baxter deformed strings

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 deformations satisfying the one-loop conditions the Weyl invariance extends at least to two loops (first order in α′). The calculations are simplified by working in an O(D, D)-covariant doubled formulation.

Yang-Baxter deformations of the AdS5\xd7S5 supercoset sigma model from 4D Chern-Simons theory

We present homogeneous Yang-Baxter deformations of the AdS5×S5 supercoset sigma model as boundary conditions of a 4D Chern-Simons theory. We first generalize the procedure for the 2D principal chiral model developed by Delduc et al. [5] so as to reproduce the 2D symmetric coset sigma model, and specify boundary conditions governing homogeneous Yang-Baxter deformations. Then the conditions are applicable for the AdS5×S5 supercoset sigma model case as well. In addition, homogeneous bi-Yang-Baxter deformation is also discussed.

The first \u03b1\u2032-correction to homogeneous Yang-Baxter deformations using O(d, d)

We use the O(d, d)-covariant formulation of supergravity familiar from Double Field Theory to find the first α′-correction to (unimodular) homogeneous Yang-Baxter (YB) deformations of the bosonic string. A special case of this result gives the α′-correction to TsT transformations. In a suitable scheme the correction comes entirely from an induced anomalous double Lorentz transformation, which is needed to make the two vielbeins obtained upon the YB deformation equal. This should hold more generally, in particular for abelian and non-abelian T-duality, as we discuss.

Comments on η-deformed principal chiral model from 4D Chern-Simons theory

We study η-deformations of principal chiral model (PCM) from the viewpoint of a 4D Chern-Simons (CS) theory. The η-deformed PCM has originally been derived from the 4D CS theory by Delduc, Lacroix, Magro and Vicedo [arXiv:1909.13824]. The derivation is based on a twist function in the rational description. On the other hand, we start with a twist function in the trigonometric description and discuss possible boundary conditions. We show that a certain boundary condition reproduces the usual η-deformed PCM and another one leads to a new kind of Yang-Baxter deformation.

Constrained affine Gaudin models and diagonal Yang–Baxter deformations

We review and pursue further the study of constrained realisations of affine Gaudin models, which form a large class of two-dimensional integrable field theories with gauge symmetries. In particular, we develop a systematic gauging procedure which allows to reformulate the non-constrained realisations of affine Gaudin models considered recently in Delduc et al (2019 J. High Energy Phys. JHEP06(2019)017) as equivalent models with a gauge symmetry. This reformulation is then used to construct integrable deformations of these models breaking their diagonal symmetry. In the second part of the article, we apply these general methods to the integrable coupled σ-model introduced recently, whose target space is the N-fold Cartesian product of a real semi-simple Lie group G0. We present its gauged formulation as a model on with a gauge symmetry acting as the right multiplication by the diagonal subgroup and construct its diagonal homogeneous Yang–Baxter deformation.

Double Yang-Baxter deformation of spinning strings

We study the reduction of classical strings rotating in the deformed three- sphere truncation of the double Yang-Baxter deformation of the AdS3×S3×T4 background to an integrable mechanical model. The use of the generalized spinning-string ansatz leads to an integrable deformation of the Neumann-Rosochatius system. Integrability of this system follows from the fact that the usual constraints for the Uhlenbeck constants apply to any deformation that respects the isometric coordinates of the three-sphere. We construct solutions to the system in terms of the underlying ellipsoidal coordinate. The solutions depend on the domain of the deformation parameters and the reality conditions of the roots of a fourth order polynomial. We obtain constant-radii, giant-magnon and trigonometric solutions when the roots degenerate, and analyze the possible solutions in the undeformed limit. In the case where the deformation parameters are purely imaginary and the polynomial involves two complex-conjugated roots, we find a new class of solutions. The new class is connected with twofold giant-magnon solutions in the degenerate limit of infinite period.

Non-abelian tri-vector deformations in d = 11 supergravity

A truncation of the SL(5) Exceptional Field Theory that allows to describe spacetimes of the form M4 × M7 with the 4-form flux on M4 is constructed. The resulting theory is used to test the recently proposed tri-vector generalisation of Yang-Baxter deformations applied to the AdS4 × mathbbm{S} 7 solution of d = 11 supergravity. We present two new supergravity solutions corresponding to non-abelian non-unimodular tri-vector deformations of AdS4 × mathbbm{S} 7.

Poisson-Lie U-duality in exceptional field theory

Poisson-Lie duality provides an algebraic extension of conventional Abelian and non-Abelian target space dualities of string theory and has seen recent applications in constructing quantum group deformations of holography. Here we demonstrate a natural upgrading of Poisson-Lie to the context of M-theory using the tools of exceptional field theory. In particular, we propose how the underlying idea of a Drinfeld double can be generalised to an algebra we call an exceptional Drinfeld algebra. These admit a notion of “maximally isotropic subalgebras” and we show how to define a generalised Scherk-Schwarz truncation on the associated group manifold to such a subalgebra. This allows us to define a notion of Poisson-Lie U-duality. Moreover, the closure conditions of the exceptional Drinfeld algebra define natural analogues of the cocycle and co-Jacobi conditions arising in Drinfeld double. We show that upon making a further coboundary restriction to the cocycle that an M-theoretic extension of Yang-Baxter deformations arise. We remark on the application of this construction as a solution-generating technique within supergravity.

Two-loop conformal invariance for Yang-Baxter deformed strings

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 required α′-corrections of the deformed background, which take a relatively simple form. In examples that can be constructed using, possibly non-commuting sequences of, TsT transformations we show how to obtain the first α′-correction to all orders in the deformation parameter by making use of the α′-corrected T-duality rules. We demonstrate this on the specific example of YB deformations of a Bianchi type II background.

Yang–Baxter deformation as an transformation

The rules for Yang–Baxter (YB) deformation for a generic Green–Schwarz string sigma model has been obtained recently. We show that the deformation can be described through the action of a coordinate dependent matrix on the target space fields both in the NS–NS and the RR sectors, generalizing previous results. This enables us to show that the YB deformed fields can be regarded as duality twisted fields in the context of gauged double field theory (GDFT). We compute the fluxes associated with the twist and show that the conditions on the R-matrix determining the YB deformation give rise to conditions for the fluxes on the GDFT side. More precisely, we show that YB deformation is a process which takes a solution of DFT with geometric flux associated with the isometry group G and deforms it to another solution of DFT with vanishing R-flux and non-vanishing Q-flux given by the structure constants of the dual Lie algebra determined by the R-matrix. We also show that the non-unimodularity of the R-matrix forces the generalized dilaton field to pick up a linear dependence on the winding type coordinates of DFT, implying that the corresponding target space fields satisfy the field equations of DFT in the generalized supergravity frame. This provides a new perspective on the relation between the non-unimodularity of the R-matrix, the trace of the Q-flux and the generalized supergravity equations.

A unifying 2D action for integrable sigma -models from 4D Chern\u2013Simons theory

In the approach recently proposed by K. Costello and M. Yamazaki, which is based on a four-dimensional variant of Chern–Simons theory, we derive a simple and unifying two-dimensional form for the action of many integrable sigma -models which are known to admit descriptions as affine Gaudin models. This includes both the Yang–Baxter deformation and the lambda -deformation of the principal chiral model. We also give an interpretation of Poisson–Lie T-duality in this setting and derive the action of the textsf {E} -model.

U -duality extension of Drinfel’d double

Abstract A family of algebras $\mathcal{E}_n$ that extends the Lie algebra of the Drinfel’d double is proposed. This allows us to systematically construct the generalized frame fields $E_A{}^I$ which realize the proposed algebra by means of the generalized Lie derivative, i.e., . By construction, the generalized frame fields include a twist by a Nambu–Poisson tensor. A possible application to the non-Abelian extension of $U$-duality and a generalization of the Yang–Baxter deformation are also discussed.

Nonlocal charges from marginal deformations of 2D CFTs: Holographic TT¯ & TJ¯ , and Yang-Baxter deformations

In this paper we study generic features of nonlocal charges obtained from marginal deformations of WZNW models. Using free-fields representations of CFTs based on simply laced Lie algebras, one can use simple arguments to build the nonlocal charges; but for more general Lie algebras these methods are not strong enough to be generally used. We propose a brute force calculation where the nonlocality is associated to a new Lie algebra valued field, and from this prescription we impose several constraints on the algebra of nonlocal charges. Possible applications for Yang-Baxter and holographic \(T\bar{T}\) and \(T\bar{J}\) deformations are also discussed.

Tri-vector deformations in d = 11 supergravity

We construct a d = 11 supergravity analogue of the open-closed string map in the context of SL(5) Exceptional Field Theory (ExFT). The deformation parameter tri-vector Ω generalizes the non-commutativity bi-vector parameter Θ of the open string. When applied to solutions in d = 11, this map provides an economical way of performing TsT deformations, and may be used to recover d = 10 Yang-Baxter deformations after dimensional reduction. We present a generalization of the Classical Yang-Baxter Equation (CYBE) for rank 3 objects, which emerges from d = 11 supergravity and the SL(5) ExFT. This equation is shown to reduce to the d = 10 CYBE upon dimensional reduction.

Holographic derivation of q SYK spectrum with Yang-Baxter shift

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 (AdS2)η×(S1/Z2), we compute the deformed Green's function that reproduces the original q SYK spectrum in the limit of vanishing deformations.

SUSY and the bi-vector

In this note we give an explicit formula for the preserved Killing spinors in deformed string theory backgrounds corresponding to integrable Yang–Baxter deformations realized via (sequences of) TsT transformations. The Killing spinors can be expressed only in terms of the bi-vector Θ which encodes the deformation. This formula is applicable to deformed backgrounds related to r-matrices of various ranks, including those that do not satisfy the unimodularity condition and give rise to backgrounds in generalized supergravity. We conjecture that our formula also remains valid for integrable deformations which are not realized via TsT transformations and motivate this conjecture by explicit examples.

Assembling integrable \u03c3-models as affine Gaudin models

We explain how to obtain new classical integrable field theories by assembling two affine Gaudin models into a single one. We show that the resulting affine Gaudin model depends on a parameter γ in such a way that the limit γ → 0 corresponds to the decoupling limit. Simple conditions ensuring Lorentz invariance are also presented. A first application of this method for σ-models leads to the action announced in [1] and which couples an arbitrary number N of principal chiral model fields on the same Lie group, each with a Wess-Zumino term. The affine Gaudin model descriptions of various integrable σ-models that can be used as elementary building blocks in the assembling construction are then given. This is in particular used in a second application of the method which consists in assembling N − 1 copies of the principal chiral model each with a Wess-Zumino term and one homogeneous Yang-Baxter deformation of the principal chiral model.