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.
Poisson-Lie T-duality and plurality are important solution generating techniques in string theory and (generalized) supergravity. Since duality/plurality does not preserve conformal invariance, the usual beta function equations are replaced by Generalized Supergravity Equations containing vector $\mathcal{J}$. In this paper we apply Poisson-Lie T-plurality on Bianchi cosmologies. We present a formula for the vector $\mathcal{J}$ as well as transformation rule for dilaton, and show that plural backgrounds together with this dilaton and $\mathcal{J}$ satisfy the Generalized Supergravity Equation…
We generalize the prescription realizing classical Poisson-Lie T-duality as canonical transformation to Poisson-Lie T-plurality. The key ingredient is the transformation of left-invariant fields under Poisson-Lie T-plurality. Explicit formulae realizing canonical transformation are presented and the preservation of canonical Poisson brackets and Hamiltonian density is shown.
Poisson-Lie T-plurality constructs a chain of supergravity solutions from a Poisson-Lie symmetric solution. We study the Poisson-Lie T-plurality for supergravity solutions with $H$-flux, which are not Poisson-Lie symmetric but admit non-Abelian isometries, $\mathcal{L}_{v_a}g_{mn}=0$ and $\mathcal{L}_{v_a}H_3=0$ with $\mathcal{L}_{v_a}B_2\neq 0$. After introducing the general procedure, we study the Poisson-Lie T-plurality for two WZW backgrounds, the AdS$_3$ with $H$-flux and the Nappi-Witten background.
Covariance of the one-loop renormalization group equations with respect to Poisson-Lie T-plurality of sigma models is discussed. The role of ambiguities in renormalization group equations of Poisson-Lie sigma models with truncated matrices of parameters is investigated.
We investigate (non-)Abelian T-duality from the perspective of Poisson-Lie T-plurality. We show that sigma models related by duality/plurality are given not only by Manin triples obtained from decompositions of Drinfel'd double, but also by their particular embeddings, i.e. maps that relate bases of these decompositions. This allows us to get richer set of dual or plural sigma models than previously thought. That's why we ask how T-duality is defined and what should be the `canonical' duality or plurality transformation.
We write the conditions for open strings with charged endpoints in the language of gluing matrices. We identify constraints imposed on the gluing matrices that are essential in this setup and investigate the question of their invariance under the Poisson-Lie T-plurality transformations. We show that the chosen set of constraints is equivalent to the statement that the lifts of D-branes into the Drinfel'd double are right cosets with respect to a maximally isotropic subgroup and therefore it is invariant under the Poisson-Lie T-plurality transformations.
Conditions for the gluing matrix defining consistent boundary conditions of two-dimensional nonlinear sigma-models are analyzed and reformulated. Transformation properties of the right-invariant fields under Poisson-Lie T-plurality are used to derive a formula for the transformation of the boundary conditions. Examples of transformation of D-branes in two and three dimensions are presented. We investigate obstacles arising in this procedure and propose possible solutions.
The Poisson-Lie T-plurality is an equivalence of string theories on various cosets $\mathcal{D}/\tilde{G}$, $\mathcal{D}/\tilde{G}'$, $\cdots$, where $\mathcal{D}$ is a Drinfel'd double and $\tilde{G}$, $\tilde{G}'$, $\cdots$ are maximal isotropic subgroups. This can be extended to the equivalence for dressing cosets, i.e., $F\backslash\mathcal{D}/\tilde{G}$, $F\backslash\mathcal{D}/\tilde{G}'$, $\cdots$, where $F$ is an isotropic subgroup of $\mathcal{D}$. We explore this extended Poisson-Lie T-plurality, clarifying the relation between several previous approaches. We propose a gauged sigma m…
We look for 3-dimensional Poisson-Lie dualizable sigma models that satisfy the vanishing beta-function equations with constant dilaton field. Using the Poisson-Lie T-plurality we then construct 3-dimensional sigma models that correspond to various decompositions of Drinfeld double. Models with nontrivial dilaton field may appear. It turns out that for ``traceless'' dual algebras they satisfy the vanishing beta-function equations as well. In certain cases the dilaton cannot be defined in some of the dual models. We provide an explanation why this happens and give criteria predicting when it hap…
Classical equations of motion for three-dimensional sigma-models in curved background are solved by a transformation that follows from the Poisson-Lie T-plurality and transform them into the equations in the flat background. Transformations of coordinates that make the metric constant are found and used for solving the flat model. The Poisson-Lie transformation is explicitly performed by solving the PDE's for auxiliary functions and finding the relevant transformation of coordinates in the Drinfel'd double. String conditions for the solutions are preserved by the Poisson-Lie transformations. T…
Starting from the classification of 6-dimensional Drinfeld doubles and their decomposition into Manin triples we construct 3-dimensional Poisson-Lie T-dual or more precisely T-plural sigma models. Of special interest are those that are conformally invariant. Examples of models that satisfy vanishing beta-function equations with zero dilaton are presented and their duals are calculated. It turns out that for "traceless" dual algebras they satisfy the beta-function equations as well but usually with rather nontrivial dilaton. We also present explicit examples of several kinds of obstacles and di…
We generalize the formulation of Poisson-Lie (PL) T-plurality proposed by R. von Unge [JHEP 07 (2002) 014] from Lie groups to Lie supergroups. By taking a convenient ansatz for metric of the $σ$-model in terms of the left-invariant one-forms of the isometry Lie supergroups $(C^3 +A)$ and $GL(1|1)$ we construct cosmological string backgrounds, including $(2+1|2)$-dimensional metric, time-dependent dilaton and vanishing torsion, in a way that they satisfy the one-loop beta-function equations. Starting from the decompositions of semi-Abelian Drinfeld superdoubles (DSDs) generated by the $({\C}^3 …
String theory has the T-duality symmetry when the target space has Abelian isometries. A generalization of the T-duality, where the isometry group is non-Abelian, is known as the non-Abelian T-duality, which works well as a solution-generating technique in supergravity. In this paper, we describe the non-Abelian T-duality as a kind of O(D,D) transformation when the isometry group acts without isotropy. We then provide a duality transformation rule for the Ramond-Ramond fields by using the technique of double field theory (DFT). We also study a more general class of solution-generating techniqu…
We extend the path-integral formalism for Poisson-Lie T-duality to include the case of Drinfeld doubles which can be decomposed into bi-algebras in more than one way. We give the correct shift of the dilaton, correcting a mistake in the litterature. We then use the fact that the six dimensional Drinfeld doubles have been classified to write down all possible conformal Poisson-Lie T-duals of three dimensional space times and we explicitly work out two duals to the constant dilaton and zero anti-symmetric tensor Bianchi type V space time and show that they satisfy the string equations of motion.…
We investigate a special class of Poisson--Lie T-plurality transformations of Bianchi cosmologies invariant with respect to non-semisimple Bianchi groups. For six-dimensional semi-Abelian Manin triples $\mathfrak{b}\bowtie \mathfrak{a}$ containing Bianchi algebras $\mathfrak{b}$ we identify general forms of Poisson--Lie identities and dualities. We show that these can be decomposed into simple factors, namely automorphisms of Manin triples, B-shifts, $β$-shifts, and "full" or "factorized" dualities. Further, we study effects of these transformations and utilize the decompositions to obtain new…
Various approaches to T-duality with NSNS three-form flux are reconciled. Non-commutative torus fibrations are shown to be the open-string version of T-folds. The non-geometric T-dual of a three-torus with uniform flux is embedded into a generalized complex six-torus, and the non-geometry is probed by D0-branes regarded as generalized complex submanifolds. The non-commutativity scale, which is present in these compactifications, is given by a holomorphic Poisson bivector that also encodes the variation of the dimension of the world-volume of D-branes under monodromy. This bivector is shown to …
We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of A-series. For the simply-laced case this is just a direct generalization of the well-known relativistic Toda chains, and we construct explicitly the set of the Ad-invariant integrals of motion on symplectic leaves, which can be described by the Poisson quivers being just the blown up Dynkin diagrams. We also demonstrate how to get the set of "minimal" integrals of motion, using the co-multiplication rules for the corresponding Lie algebras. In the non si…
Geometric structures and dualities arise naturally in quantum field theories and string theory. In fact, these tools become very useful when studying strong coupling effects, where standard perturbative techniques can no longer be used. In this thesis we look at several conformal field theories in various dimensions. We first discuss the structure of the nilpotent networks stemming from T-brane deformations in 4D $\mathcal{N}=1$ theories and then go to the stringy origins of 6D superconformal field theories to realize deformations associated with T-branes in terms of simple combinatorial data.…
