On non-abelian T-duality and deformations of supercoset string sigma-models

Journal of High Energy Physics, Oct 2017

We elaborate on the class of deformed T-dual (DTD) models obtained by first adding a topological term to the action of a supercoset sigma model and then performing (non-abelian) T-duality on a subalgebra \( \tilde{\mathfrak{g}} \) of the superisometry algebra. These models inherit the classical integrability of the parent one, and they include as special cases the so-called homogeneous Yang-Baxter sigma models as well as their non-abelian T-duals. Many properties of DTD models have simple algebraic interpretations. For example we show that their (non-abelian) T-duals — including certain deformations — are again in the same class, where \( \tilde{\mathfrak{g}} \) gets enlarged or shrinks by adding or removing generators corresponding to the dualised isometries. Moreover, we show that Weyl invariance of these models is equivalent to \( \tilde{\mathfrak{g}} \) being unimodular; when this property is not satisfied one can always remove one generator to obtain a unimodular \( \tilde{\mathfrak{g}} \), which is equivalent to (formal) T-duality. We also work out the target space superfields and, as a by-product, we prove the conjectured transformation law for Ramond-Ramond (RR) fields under bosonic non-abelian T-duality of supercosets, generalising it to cases involving also fermionic T-dualities.

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On non-abelian T-duality and deformations of supercoset string sigma-models

HJE On non-abelian T-duality and deformations of supercoset string sigma-models Riccardo Borsato 0 1 3 Linus Wul 0 1 2 0 611 37 Brno , Czech Republic 1 Roslagstullsbacken 23 , SE-106 91 Stockholm , Sweden 2 Department of Theoretical Physics and Astrophysics, Masaryk University 3 Nordita, Stockholm University and KTH Royal Institute of Technology We elaborate on the class of deformed T-dual (DTD) models obtained by rst adding a topological term to the action of a supercoset sigma model and then performing (non-abelian) T-duality on a subalgebra g~ of the superisometry algebra. These models inherit the classical integrability of the parent one, and they include as special cases the socalled homogeneous Yang-Baxter sigma models as well as their non-abelian T-duals. Many properties of DTD models have simple algebraic interpretations. For example we show that their (non-abelian) T-duals | including certain deformations | are again in the same class, where g~ gets enlarged or shrinks by adding or removing generators corresponding to the dualised isometries. Moreover, we show that Weyl invariance of these models is equivalent to g~ being unimodular; when this property is not satis ed one can always remove one generator to obtain a unimodular g~, which is equivalent to (formal) T-duality. We also work out the target space super elds and, as a by-product, we prove the conjectured transformation law for Ramond-Ramond (RR) elds under bosonic non-abelian T-duality of supercosets, generalising it to cases involving also fermionic T-dualities. Integrable Field Theories; String Duality 1 Introduction The deformed T-dual models 2.1 Relation to Yang-Baxter sigma models Global symmetries 3.1 3.2 DTD of DTD models DTD models not related to YB models by NATD Kappa symmetry and Green-Schwarz form Target space super elds 5.1 Supergravity condition and dilaton 2 3 4 5 6 7 Some explicit examples 6.1 6.2 A TsT example A new example Conclusions A Useful identities B Derivation of the action C Classical integrability 1 Introduction In this paper we investigate further the deformed T-dual (DTD) supercoset sigma models introduced in [1], and we nd results that are of interest also when considering the undeformed case, i.e. when applying just non-abelian T-duality (NATD). The construction of DTD models is equivalent to applying NATD on a centrally extended subalgebra as rst suggested in [2].1 subalgebra of the (super)isometry algebra g~ The models are constructed by picking a g | the canonical example is the AdS5 superstring where g = psu(2; 2j4) | and a 2-cocycle, i.e. an anti-symmetric linear map ! : g~ g~ ! R satisfying !(X; [Y; Z]) + !(Z; [X; Y ]) + !(Y; [Z; X]) = 0 ; 8X; Y; Z 2 g~ : 1The rst hint of the relation of YB models to NATD appeared in [3] for the case of Jordanian deformations. { 1 { Together with an element of the corresponding group g~ 2 G~, the 2-cocycle de nes a 2-form B = !(g~ 1dg~; g~ 1dg~) which is closed, i.e. dB = 0, thanks to the 2-cocycle condition. The idea behind the construction is to add this topological term to the supercoset sigma model Lagrangian and then perform NATD on G~. If B is added to the Lagrangian, with a parameter, the resulting model can be thought of as a deformation of the non-abelian T-dual of the original model with deformation parameter . The classical integrability of the original sigma model is preserved by the deformation, since both adding a topological term and performing NATD preserve integrability. We refer to [1] for more details on how this procedure relates to the construction of [2]. Let us remark that DTD models may be constructed starting from a generic -model, for example the principal chiral model as in [1], and the starting model does not have to be (classically) integrable. In this paper we will only consider the supercoset case. It was proven in [1] that the so-called Yang-Baxter (YB) sigma models [4{7], de ned by an R-matrix solving the classical Yang-Baxter equation (CYBE), are equivalent to DTD models with invertible !. This relation was rst conjectured and checked for many examples | in the language of T-duality on a centrally extended subalgebra | in [2]. See also [8] for a more detailed discussion of some of the examples. In [1] we used the fact that when ! is invertible its inverse R = ! 1 solves the CYBE, and therefore de nes a corresponding YB model; by means of a eld rede nition and relating the deformation parameters as = 1 we could prove the equivalence of the two sigma model actions [1]. Note that simply by setting the deformation parameter to zero, DTD models include all non-abelian and abelian T-duals of the original supercoset model, including fermionic T-dualities. Therefore all the statements we prove for DTD models apply also to (nonabelian) T-duals of supercoset models. They are also easily seen to describe all so-called TsT-transformations of the underlying supercoset model. In fact we will argue here that the class of (...truncated)


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Riccardo Borsato, Linus Wulff. On non-abelian T-duality and deformations of supercoset string sigma-models, Journal of High Energy Physics, 2017, pp. 24, Volume 2017, Issue 10, DOI: 10.1007/JHEP10(2017)024