Type II DFT solutions from Poisson–Lie $T$-duality/plurality

Abstract

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 technique, the Poisson-Lie (PL)T -duality or T -plurality. We describe the PL T -plurality as an O(n, n) transformation and clearly show the covariance of the DFT equations of motion by using the gauged DFT.We further discuss the PL T -plurality with spectator fields, and study an application to the AdS 5 × S 5 solution. The dilaton puzzle known in the context of the PL T -plurality is resolved with the help of DFT. IntroductionThe T -duality was discovered in [1] as a symmetry of string theory compactified on a torus.The mass spectrum or the partition function of string theory on a D-dimensional torus was studied for example in [2][3][4][5][6] and the T -duality was identified as an O(D, D; Z) symmetry.It was further studied from a different approach [7,8], and the transformation rules for the background fields (i.e. metric, the Kalb-Ramond B-field, and the dilaton) under the T -duality were determined. In [9,10], the T -duality was understood as an O(D, D) symmetry of the classical equations of motion of string theory. The classical symmetry was clarified in [11] by using the gauged sigma model, and this approach has proved quite useful, for example when we discuss the global structure of the T -dualized background [12]. The transformation rules for the Ramond-Ramond (R-R) fields and spacetime fermions were determined in [13][14][15][16].This well-established symmetry of string theory is called the Abelian T -duality since it relies on the existence of Killing vectors which commute with each other (see [17,18] for reviews).An extension of the T -duality to the case of non-commuting Killing vectors was explored in [19] (see [20,21] for earlier works), and this is known as the non-Abelian T -duality (NATD).Various aspects have been studied in [12,[22][23][24][25][26][27][28][29][30][31][32][33][34][35], but unlike the Abelian T -duality, there are still many things to be clarified. For example, the partition function in the dual model is not the same as that of the original model (see [36] for a recent study), and NATD may rather be regarded as a map between two string theories. The global structure of the dual geometry is also not clearly understood [12]. However, NATD at least generates many new solutions of supergravity, and it can be utilized as a useful solution-generating technique.Under NATD, the isometries are generally broken, and naively we cannot recover the original model from the dual model. However, this issue was re...

U -duality extension of Drinfel’d double

“…In this case, by using a supergravity solution obtained in [60] (which has the symmetry of the Bianchi type 6 0 ), we can perform a U -duality extension of the PL T -duality. Namely, in 7 The flux f1 234 = a corresponds to the Q-flux Q1 23 = −a in type IIA theory and the double T -duality transforms it to the H-flux H123 = −a . The 11D uplift corresponds to f ′ 1234 = −a .…”

“…The familiar T -duality is a symmetry of string theory when the target space has commuting (or Abelian) Killing vectors. An extension of the T -duality, where the Killing vectors do not commute with each other, has been proposed in [1][2][3][4][5][6], and it is known as non-Abelian T -duality (see [7] for a list of references). Subsequently, a further extension, called the Poisson-Lie (PL)…”

“…A new formulation of DFT on group manifolds (called DFT WZW ) has been developed in [20][21][22], and in the recent works [23,24], the PL T -duality has been studied in the framework of DFT WZW . In more recent papers [7,25], the non-Abelian T -duality and the PL T -duality have been discussed by using another approach, called the gauged DFT [26][27][28][29][30][31] (see also [32,33] for recent discussion on the Drinfel'd double and related aspects in DFT). Thus, DFT is now not restricted to Abelian T -duality but can be applied also to the non-Abelian extensions.…”

“…On the other hand, if we eliminate the gauge fields first, the string sigma model on the dual geometry is recovered [3][4][5][6], and in this sense, the gauged sigma model connects the original geometry and the dual geometry. If we try to apply the same procedure to the membrane sigma model, we face a difficulty (see [7]). In order to formulate the non-Abelian extension of U -duality, the approach of the PL T -duality will be more useful.…”

Tri-vector deformations in d = 11 supergravity

“…Supergravity formulations that are natural to look at in this context are Double [26] and Exceptional [27] Field Theories (DFT and ExFT, respectively). Specifically designed to render supergravities in various dimensions covariant under T-and U-duality groups at the expense of extending the spacetime dimension, they are useful in describing Yang-Baxter deformations [15,[28][29][30][31] and Poisson-Lie T-duality [32][33][34][35][36][37]. The proof of [23] that (1.1), (1.2) is a supergravity symmetry relied upon DFT techniques, in particular the β-supergravity formalism [38][39][40].…”