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)