Lie T-duality Research Articles

A hierarchy of WZW models related to super Poisson–Lie T-duality

Motivated by super Poisson–Lie (PL) symmetry of the Wess–Zumino–Witten (WZW) model based on the (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C^3+A)$$\\end{document} Lie supergroup of our previous work (Eghbali et al., in J High Energy Phys 07:134, 2013. arXiv:1303.4069 [hep-th]), we first obtain and classify all Drinfeld superdoubles (DSDs) generated by the Lie superbialgebra structures on the (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\mathscr {C}}^3+ {\\mathscr {A}})$$\\end{document} Lie superalgebra as a theorem. Then, introducing a general formulation we find the conditions under which a two-dimensional 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 may be equivalent to a WZW model. With the help of this formulation and starting the super PL symmetric (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C^3+A)$$\\end{document} WZW model, we get a hierarchy of WZW models related to super PL T-duality, in such a way that it is different from the super PL T-plurality, because the DSDs are, in this process, non-isomorphic. The most interesting indication of this work is that the (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C^3+A)$$\\end{document} WZW model does remain invariant under the super PL T-duality transformation, that is, the model is super PL self-dual.

A QP perspective on topology change in Poisson–Lie T-duality

We describe topological T-duality and Poisson–Lie T-duality in terms of QP (differential graded symplectic) manifolds and their canonical transformations. Duality is mediated by a QP-manifold on doubled non-abelian ‘correspondence’ space, from which we can perform mutually dual symplectic reductions, where certain canonical transformations play a vital role. In the presence of spectator coordinates, we show how the introduction of bibundle structure on correspondence space realises changes in the global fibration structure under Poisson–Lie duality. Our approach can be directly translated to the worldsheet to derive dual string current algebras. Finally, the canonical transformations appearing in our reduction procedure naturally suggest a Fourier–Mukai integral transformation for Poisson–Lie T-duality.

Gauged sigma models and exceptional dressing cosets

Abstract The Poisson–Lie (PL) T-duality is a generalized T-duality based on the Lie algebra of the Drinfel’d double. In particular, when we consider the PL T-duality of a coset space, the dual space is found to be a generalized coset space, which is called the dressing coset. We investigate an extension of the dressing cosets to the U-duality setup. We propose the gauged actions for various branes in M-theory and type IIB theory, where the generalized metric is constructed by using the exceptional Drinfel’d algebra (EDA) and the gauge algebra is a certain isotropic subalgebra of the EDA. By eliminating the gauge fields, the gauged action reduces to the standard brane action on a certain reduced background, which we call the exceptional dressing coset. We also propose an alternative definition of the exceptional dressing cosets based on Sfetsos’s approach and reproduce a known example of non-Abelian T-duality in the U-duality-covariant formulation.

Non-Abelian U duality at work

Non-abelian U-duality originates from the construction of exceptional Drinfel'd algebra (EDA), which extends the constriction of the classical Drinfel'd double. This symmetry is a natural extension of Poisson--Lie T-duality and is believed to be a symmetry of Type II string/M-theory or their low-energy effective theories. In this paper, we consider non-abelian U-dualities of 11- or 10-dimensional backgrounds starting with E${}_{n(n)}$ EDA with $n\leq 6$ with vanishing trombone gauging. The latter guarantees that all dual backgrounds satisfy the standard supergravity equations of motion. In particular, when the duality includes a timelike T-duality, we obtain solutions of M$^*$-theory or Type II$^*$ background equations, as expected. Also starting with coboundary EDA's we provide examples of generalised Yang--Baxter deformations of M-theory and Type IIB backgrounds. The obtained results provide explicit examples when non-abelian U-duality works well as a solution generating transformation.

T-dualization of G\xf6del string cosmologies via Poisson\u2013Lie T-duality approach

Using the homogeneous Gödel spacetimes we find some new solutions for the field equations of bosonic string effective action up to first order in alpha ' including both dilaton and axion fields. We then discuss in detail the (non-)Abelian T-dualization of Gödel string cosmologies via the Poisson–Lie (PL) T-duality approach. In studying Abelian T-duality of the models we get seven dual models in such a way that they are constructed by one-, two- and three-dimensional Abelian Lie groups acting freely on the target space manifold. The results of our study show that the Abelian T-dual models are, under some of the special conditions, self-dual; moreover, by applying the usual rules of Abelian T-duality without further corrections, we are still able to obtain two-loop solutions. We also study the Abelian T-duality of Gödel string cosmologies up to alpha '-corrections by using the T-duality rules at two-loop order derived by Kaloper and Meissner. Afterwards, non-Abelian duals of the Gödel spacetimes are constructed by two- and three-dimensional non-Abelian Lie groups such as A_2, A_2 oplus A_1 and SL(2, mathbb {R}). In this way, the PL self-duality of AdS_3 times mathbb {R} space is discussed.

T-folds as Poisson\u2013Lie plurals

In previous papers we have presented many purely bosonic solutions of Generalized Supergravity Equations obtained by Poisson–Lie T-duality and plurality of flat and Bianchi cosmologies. In this paper we focus on their compactifications and identify solutions that can be interpreted as T-folds. To recognize T-folds we adopt the language of Double Field Theory and discuss how Poisson–Lie T-duality/plurality fits into this framework. As a special case we confirm that all non-Abelian T-duals can be compactified as T-folds.

Current algebras, generalised fluxes and non-geometry

A Hamiltonian formulation of the classical world-sheet theory in a generic, geometric or non-geometric, NSNS background is proposed. The essence of this formulation is a deformed current algebra, which is solely characterised by the generalised fluxes describing such a background. The construction extends to backgrounds for which there is no Lagrangian description—namely magnetically charged backgrounds or those violating the strong constraint of double field theory—at the cost of violating the Jacobi identity of the current algebra. The known non-commutative and non-associative interpretation of non-geometric flux backgrounds is reproduced by means of the deformed current algebra. Furthermore, the provided framework is used to suggest a generalisation of Poisson–Lie T-duality to generic models with constant generalised fluxes. As a side note, the relation between Lie and Courant algebroid structures of the string current algebra is clarified.

Courant Algebroids, Poisson–Lie T-Duality, and Type II Supergravities

We reexamine the notions of generalized Ricci tensor and scalar curvature on a general Courant algebroid, reformulate them using objects natural w.r.t. pull-backs and reductions, and obtain them from the variation of a natural action functional. This allows us to prove, in a very general setup, the compatibility of the Poisson–Lie T-duality with the renormalization group flow and with string background equations. We thus extend the known results to a much wider class of dualities, including the cases with gauging (so called dressing cosets, or equivariant Poisson–Lie T-duality). As an illustration, we use the formalism to provide new classes of solutions of modified supergravity equations on symmetric spaces.

Hitchhiker’s guide to Courant algebroid relations

Courant algebroids provide a useful mathematical tool (not only) in string theory. It is thus important to define and examine their morphisms. To some extent, this was done before using an analogue of canonical relations known from symplectic geometry. However, it turns out that applications in physics require a more general notion. We aim to provide a self-contained and detailed treatment of Courant algebroid relations and morphisms. A particular emphasis is placed on providing enough motivating examples. In particular, we show how Poisson–Lie T-duality and Kaluza–Klein reduction of supergravity can be interpreted as Courant algebroid relations compatible with generalized metrics (generalized isometries).

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.

Affine Poisson and affine quasi-Poisson T-duality

We generalize the Poisson–Lie T-duality by making use of the structure of the affine Poisson group which is the concept introduced some time ago in Poisson geometry as a generalization of the Poisson–Lie group. We also introduce a new notion of an affine quasi-Poisson group and show that it gives rise to a still more general T-duality framework. We establish for a class of examples that this new T-duality is compatible with the renormalization group flow.

Poisson–Lie T-duality of string effective actions: A new approach to the dilaton puzzle

For a particular class of backgrounds, equations of motion for string sigma models targeted in mutually dual Poisson-Lie groups are equivalent. This phenomenon is called the Poisson-Lie T-duality. On the level of the corresponding string effective actions, the situation becomes more complicated due to the presence of the dilaton field. A novel approach to this problem using Levi-Civita connections on Courant algebroids is presented. After the introduction of necessary geometrical tools, formulas for the Poisson-Lie T-dual dilaton fields are derived. This provides a version of Poisson-Lie T-duality for string effective actions.

Jacobi–Lie symmetry and Jacobi–Lie T-dual sigma models on group manifolds

Using the concept of Jacobi–Lie group and Jacobi–Lie bialgebra, we generalize the definition of Poisson–Lie symmetry to Jacobi–Lie symmetry. In this regard, we generalize the concept of Poisson–Lie T-duality to Jacobi–Lie T-duality and present Jacobi–Lie T-dual sigma models on Lie groups, which have Jacobi–Lie symmetry. Using this symmetry, new cases of duality appear and some examples are given. This generalization may provide insights to understand the quantum features of Poisson–Lie T-duality, in a more satisfactory way.

BTZ black hole from Poisson–Lie T-dualizable sigma models with spectators

The non-Abelian T-dualization of the BTZ black hole is discussed in detail by using the Poisson–Lie T-duality in the presence of spectators. We explicitly construct a dual pair of sigma models related by Poisson–Lie symmetry. The original model is built on a 2+1-dimensional manifold M≈O×G, where G as a two-dimensional real non-Abelian Lie group acts freely on M, while O is the orbit of G in M. The findings of our study show that the original model indeed is canonically equivalent to the SL(2,R) Wess–Zumino–Witten (WZW) model for a given value of the background parameters. Moreover, by a convenient coordinate transformation we show that this model describes a string propagating in a spacetime with the BTZ black hole metric in such a way that a new family of the solutions to low energy string theory with the BTZ black hole vacuum metric, constant dilaton field and a new torsion potential is found. The dual model is built on a 2+1-dimensional target manifold M˜ with two-dimensional real Abelian Lie group G˜ acting freely on it. We further show that the dual model yields a three-dimensional charged black string for which the mass M and axion charge Q per unit length are calculated. After that, the structure and asymptotic nature of the dual space–time including the horizon and singularity are determined.

Yang–Baxter σ-model with WZNW term as [formula omitted]-model

It turns out that many integrable σ-models on group manifolds belong to the class of the so-called E-models which are relevant in the context of the Poisson–Lie T-duality. We show that this is the case also for the Yang–Baxter σ-model with WZNW term introduced by Delduc, Magro and Vicedo in [5].

Ricci flow, Courant algebroids, and renormalization of Poisson–Lie T-duality

We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson-Lie T-duality is compatible with the 1-loop renormalization group.

Generalised integrable λ- and η-deformations and their relation

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 σ-model for a general group is one-loop renormalisable.

Poisson–Lie T-Duality and Courant Algebroids

Poisson–Lie T-duality is explained using the language of Courant algebroids.

Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group H4

We show that the WZW model on the Heisenberg Lie group H4 has Poisson–Lie symmetry only when the dual Lie group is A2⊕2A1. In this way, we construct the mutual T-dual sigma models on Drinfel'd double generated by the Heisenberg Lie group H4 and its dual pair, A2⊕2A1, as the target space in such a way that the original model is the same as the H4 WZW model. Furthermore, we show that the dual model is conformal up to two-loop order. Finally, we discuss D-branes and the worldsheet boundary conditions defined by a gluing matrix on the H4 WZW model. Using the duality map obtained from the canonical transformation description of the Poisson–Lie T-duality transformations for the gluing matrix which locally defines the properties of the D-brane, we find two different cases of the gluing matrices for the WZW model based on the Heisenberg Lie group H4 and its dual model.

Super Poisson–Lie symmetry of the [formula omitted] WZNW model and worldsheet boundary conditions

We show that the WZNW model on the Lie supergroup GL(1|1) has super Poisson–Lie symmetry with the dual Lie supergroup B⊕A⊕A1,1|.i. Then, we discuss about D-branes and worldsheet boundary conditions on supermanifolds, in general, and obtain the algebraic relations on the gluing supermatrix for the Lie supergroup case. Finally, using the supercanonical transformation description of the super Poisson–Lie T-duality transformation, we obtain formulae for the description of the dual gluing supermatrix, then, we find the gluing supermatrix for the WZNW model on GL(1|1) and its dual model. We also discuss about different boundary conditions.