Jacobi–Lie Models and Supergravity Equations

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Introduction
Dualities play an important role in the study of string theory.Whereas Abelian Tduality [1] (the T stands for target-space) is recognized as a symmetry of the theory, non-Abelian T-duality [2] is rather understood as a solution-generating technique.
In both cases we rely on the presence of symmetries of non-linear sigma-model characterizing dynamics of a string propagating in generally curved background.These symmetries have to be gauged in order to obtain "dual" background.With the introduction of Poisson-Lie T-duality [3,4] and plurality [5] it became clear that isometries of sigma-model backgrounds are not strictly necessary and mutually dual models can be constructed on Lie groups G and G * of Drinfel'd double D with corresponding Lie bialgebra (g, g * ).Such models are said to be Poisson-Lie symmetric.
Revived interest in Poisson-Lie T-duality appeared recently with its extension to U-duality [6,7,8] which is based on the exceptional Drinfel'd algebra (EDA) [9,10].While the algebra of the Drinfel'd double is a Lie algebra, meaning that the structure constants satisfy X AB C = −X BA C , this doesn't have to hold for EDA.An extension of Lie bialgebra (g, g * ) was also introduced in [11], where the authors considered Jacobi structures on Lie groups [12] and Jacobi-Lie bialgebras ((g, φ 0 ) , (g * , X 0 )).Carefully repeating the process carried out in the derivation of Poisson-Lie T-duality in [3] authors of [11] found conditions for cocycles φ 0 , X 0 that allowed them to generalize Poisson-Lie symmetry to Jacobi-Lie symmetry replacing Lie bialgebra by Jacobi-Lie bialgebra as the underlying algebraic structure.Lowdimensional Jacobi-Lie bialgebras and their Jacobi structures were further studied and classified in [13,14].
It is natural to ask, whether models obtained this way satisfy Generalized Supergravity Equations [15,16] that for bosonic fields in the NS-NS sector read Here R µν and R are Ricci tensor and scalar curvature of metric G specifying sigmamodel background F = G + B together with Kalb-Ramond field B giving rise to torsion Components of one-form X are calculated from dilaton Φ and Killing vector field J as The condition that the field J is Killing vector field of the background F is not necessary for solution of eqns.( 1)-( 3), but it is required for their full version containing the R-R fields [16].
Note that X is invariant with respect to gauge transformation where Λ is arbitrary differentiable function.It means that if dX = 0, we can always find dilaton Φ Λ such that X µ = ∂ µ Φ Λ , J Λ vanishes, and the equations ( 1)-( 3) transform to usual Supergravity Equations, i.e. vanishing beta function equations.Moreover, it immediately follows from the equation ( 2) that dX = 0 whenever torsion H ρµν vanishes.If dX = 0, we cannot trivialize Killing vector field J by gauge transformation and the equations ( 1)-( 3) are called Generalized Supergravity Equations.More general conditions for trivialization of the Killing vector were considered in [18].
Authors of [17] approach Jacobi-Lie transformation from the perspective of Double Field Theory (DFT) [19] that has proven useful earlier in the study of Poisson-Lie T-duality [20,21,22].Similarly to EDA they break the antisymmetry of structure constants X AB C and propose 2D-dimensional Leibniz algebra DD + composed of D-dimensional subalgebras g (generated by T a ) and g * (generated by T a ) in the form where a = 1, . . ., D. Since g and g * are Lie algebras, antisymmetry holds for their structure constants f ab c = −f ba c and f c ab = −f c ba .A symmetric bilinear form can be introduced on DD + , such that the subalgebras g and g * are maximally isotropic with respect to it, i.e. it holds that As opposed to the case of Lie bialgebra, the form , on DD + is no longer adinvariant.Under certain conditions on f ab c , f c ab , Z a and Z a following from Leibniz identities there is one-to-one correspondence between Leibniz algebra DD + of [17] and Jacobi-Lie bialgebra of [11].In the following we prefer the approach based on DD + and DFT since it not only allows us to investigate Jacobi-Lie transformation of background fields G and B, but also enables us to study transformation of dilaton and vector field J that appear in generalized fluxes of DFT.Moreover, the construction of DFT fields and corresponding sigma-model backgrounds is not limited to coboundary Jacobi-Lie bialgebras.
When performing Poisson-Lie T-duality transformation with respect to a group whose structure constants are non-unimodular, f b ba = 0, one finds that resulting background does not always satisfy vanishing beta function equations [23,24] but Generalized Supergravity Equations ( 1)-( 3) instead [25].In such cases the transformed dilaton may depend on dual coordinates that have to be eliminated at cost of introducing Killing vector field J [22,26].
In Jacobi-Lie symmetric models the presence of non-vanishing Z a in DD + gives rise to a scale factor that brings the dependence on dual coordinates even in the metric and B-field.Even if this is consistent from the point of view of DFT, we can hardly consider such fields as solutions to (Generalized) Supergravity Equations.
There are three types of models considered in Section 3 of [17] depending on values of f b ba and Z a .Examples with f b ba = Z a = 0 belong to the first case and satisfy usual Supergravity Equations, i.e. ( 1)-( 3) with vanishing J .Examples with Z a = 0 belong to the third case and give backgrounds depending on dual coordinates.In this paper we want to investigate models of type 2, i.e. those obtained from Leibniz algebras with f b ba = 0 and Z a = 0. Our primary goal is to find solutions of Generalized Supergravity Equations predicted in [17].We discuss construction of J and give examples where Generalized Supergravity Equations reduce to usual Supergravity Equations even in this case.
In the language of DFT it is natural to consider not only duality, but also plurality.When viewing DD + as a vector space, plurality allows us to pass to different pair of subalgebras g and g * instead of just exchanging them.We shall show that three-dimensional Jacobi-Lie bialgebras with f b ba = 0, Z a = 0 classified in Table 7 of [13] are isomorphic to type 1 algebra ((g, φ 0 ) , (g * , X 0 )) = (III, −T 2 + T 3 ), (I, 0) and use this plurality to construct our examples.
Plan of the paper is the following.In Section 2 we summarize notation used to describe the algebra DD + and results concerning plurality of Jacobi-Lie bialgebras.The particular isomorphisms are given in the Appendix.In Section 3 we recapitulate the construction of Jacobi-Lie sigma model backgrounds and find sigma model satisfying the Supergravity Equations.Section 4 sums up Jacobi-Lie T-plurality and appearance of vector field J .Examples of plural backgrounds without spectators are presented in Section 5 and examples of Jacobi-Lie T-plurality with spectators are in Section 6.
2 Leibniz algebras with f b ba = 0, Z a = 0 The 2D-dimensional Leibniz algebra DD + was introduced in [17].Denoting T A = (T a , T a ), Z A = (Z a , Z a ), a = 1, . . ., D; A = 1, . . ., 2D, the structure constants of DD + can be written as while equations (8) read There is one-to-one correspondence between Leibniz algebras (7) and Jacobi-Lie bialgebras studied in [11,13] and we can use the classification of three-dimensional Jacobi-Lie bialgebras given in [13].Investigating Table 7 of [13] one can find that g g * Product definitions of g Table 1: Real three-dimensional type 2 Leibniz algebras with X 0 = 2Z a T a = 0, f b ba = 0.
all algebras with f b ba = 0 and Z a = 0 appear as duals of the last seven bialgebras.We give their list1 in Table 1, where algebra g * is always defined as Maybe surprisingly, all algebras in Table 1 are isomorphic in the sense that there are matrices C that transform algebraic relations of algebra generated by T A to those of algebra generated by TA as Since (10) has to hold as well, the conditions on C are Moreover, all algebras in Table 1 are also isomorphic to algebra ((g, φ 0 ), (g * , X 0 )) = (III, −T2 + T 3 ), (I, 0) defined by and For better readability and similarity to the Manin triples we shall denote the Leibniz algebras using curly brackets as The existence of isomorphisms then can be written as In each case there are several possible solutions to algebraic relations (13) that can be used for Jacobi-Lie plurality transformation and different solutions may give models with different properties.The transformation matrices C are given in the Appendix.
Our strategy for getting Jacobi-Lie models satisfying Generalized Supergravity Equations is following.We start with flat torsionless model with vanishing dilaton on the group corresponding to the algebra with f b ba = Z a = 0 and then use Jacobi-Lie T-plurality to get models with f b ba = 0 and Z a = 0.
3 Jacobi-Lie models The construction of Jacobi-Lie symmetric models resembles construction of Poisson-Lie symmetric models [17].In the absence of spectator fields the background fields are given by constant matrix E 0 and algebraic structure of Leibniz algebra DD + .The metric G and the B-field of Jacobi-Lie model on Lie group G corresponding to Lie algebra g can be expressed as where r a m are components of right-invariant one-form dgg −1 = r a m dx m T a and g ∈ G .The anti-symmetric matrix π ab is calculated as follows.For group action is of the form from which we can read π ab , ∆ and define Jacobi-Lie structure on G .
For the factor e −2ω we have to use DFT.Let us denote the doubled coordinates x M = (x m , xm ) and corresponding derivatives ∂ M = (∂ m , ∂m ).The generalized frame fields satisfy condition for generalized Lie derivative £ in DFT hold, where v m a are components of left-invariant fields on G .If Z a = 0, we can choose σ = 1 and ω = ∆.For Z a = 0 the backgrounds in ( 16) may be dependent on "dual" coordinates xm .
The standard dilaton Φ can be found as Knowledge of function ϕ is complementary to knowledge of dilaton.G, B and Φ constructed above define sigma model on G and should satisfy equations ( 1)-(3).

Three-dimensional flat models corresponding to the algebra {3|1}
As mentioned in the previous section, for getting models satisfying Generalized Supergravity Equations we are going to look for models corresponding to the algebra with f b ba = Z a = 0 and then apply the Jacobi-Lie T-plurality to construct models with f b ba = 0, Z a = 0.The algebra denoted in [13] as (III, −T2 + T 3 ), (I, 0) =: {3|1} is given by relations ( 14) and (15).It is of the type 1, so the models should satisfy usual Supergravity Equations.We parametrize elements g ∈ G as Given E 0 , it is then easy to calculate F.More difficult is to find dilaton that solves Supergravity Equations.However, one can proceed reversely and look for matrices E 0 that together with the algebra {3|1} produce sigma model satisfying usual Supergravity Equations with vanishing (or constant) dilaton Φ.This will enable us to find the function ϕ(x) in ( 23) necessary for Jacobi-Lie T-plurality transformation of dilaton.
For vanishing Φ and J equation ( 2) implies H µνρ = 0. Then from (1) and (3) one can see that the Jacobi-Lie background has to be Ricci flat.In three dimensions that means that the metric is also flat.Altogether there are two families of matrices E 0 depending on real constants λ i that produce flat torsionless backgrounds via (16), namely For simplicity we choose λ 3 = λ 4 = λ 6 = λ 7 = 0 and λ 5 = 1 in (24) to work with diagonal E 0 and get background of Jacobi-Lie sigma model 2 Choosing 25) becomes symmetric and Jacobi-Lie model is given by background In both cases we get constant dilaton Φ from the expression (23) e.g. for Even though these two flat models must be equivalent by transformation of coordinates, they give different results after the Jacobi-Lie transformation 3 .Let us note that the structure of algebra {3|1} leads to vanishing π ab in (18).We choose σ = 1 to have ω = ∆ = − x 2 2 + x 3 2 and get the overall factor e x 2 −x 3 in ( 26) and (27).This is similar to construction of Poisson-Lie symmetric backgrounds (where Z a = 0) on semi-Abelian Lie bialgebra (3|1) where the Poisson bivector also vanishes.However, then we have ∆ = 0, e −2∆ = 1, and we get backgrounds with constant scalar curvature (the sign of R depends of the choice of λ i in E 0 ) that do not satisfy Generalized Supergravity Equations for any Φ or J .

Jacobi-Lie T-plurality
Jacobi-Lie T-plurality was described in [17] in the DFT language.Transformation of metric and B-field is very similar to the Poisson-Lie T-plurality of models constructed from Drinfel'd doubles.Namely, we first transform the constant matrix E 0 by formula where the matrices P, Q, R, S are D × D blocks of the transformation matrix C −1 and then apply formulas for construction of Jacobi-Lie models given in Section 3.More difficult is the transformation of dilaton.Before we can apply the formula (23) we have to transform function ϕ.For that we have to compute generalized fluxes F A associated with generalized vielbein E A M ≡ e ω(x) E A M (x) and DFT dilaton d through where Finally we find for f b ba = 0, Z a = 0, Transformed function φ is then obtained from the transformation of the flux However, for the algebras of type 2, where fb ba = 0, Ẑa = 0, it may happen that the r.h.s. of (32) is not gradient of a function of x m .Instead, we may define form where vm a are components of left-invariant vector fields of the group associated with ĝ.It turns out that Ĵ m (x) then are components of Killing vector field that occurs in the Generalized Supergravity Equations for the Jacobi-Lie plural model.Values of functions φ and Ĵ m for various type 2 algebras are given in the Table 2.It is evident that if Ĵ m vanishes then corresponding plurality produces Jacobi-Lie model that satisfies usual Supergravity Equations.On the other hand, there are cases with non-vanishing Ĵ m where d X = 0 and we can get rid of the Killing vector field by the gauge transformation (6).The model then satisfies usual Supergravity Equations -see e.g.5.2.1.
Let us note in the end that Jacobi-Lie T-duality T a ↔ T a for Z a = 0 is not useful for construction of new models as it transforms type 1 and type 2 algebras to the type 3 where X0 = 0, and these produce Jacobi-Lie models dependent on dual DFT coordinates.

Jacobi-Lie models plural to {3|1}
When we have transformation matrices C we can construct supergravity fields of Jacobi-Lie models plural to flat backgrounds found in 3.1.However, not all matrices C generate plural models.Some of them may give singular matrix Ê0 .Beside that it turns out that even for the type 2 algebras some Jacobi-Lie plural models satisfy usual and others Generalized Supergravity Equations.Models satisfying Generalized Supergravity Equations are produced only by Jacobi-Lie transformation of the background (26).Unfortunately the backgrounds are usually so extensive that they are difficult to display.
Plural algebra C-matrix (2 e x 2 −x 3 , 0, 0) Table 2: Values of functions ϕ and components of Killing vector fields for type 2 algebras and corresponding C-matrices from Appendix.

Plural models exist only for
because matrix (P + E 0 • R) in the formula ( 29) is singular for C α , C γ and the matrices Ê0 do not exist in these cases.
For (36) we get with nontrivial torsion.This background is obtained from ( 16)-( 19) since from the matrix M A B we obtain and Formulas ( 33) and ( 34) with σ = 1 then give The background (37) together with dilaton given by ( 23) satisfy Generalized Supergravity Equations.
Using the matrix we get from (33),(34) Jacobi-Lie transformation of the model ( 26) then yields torsionless curved background satisfies usual Supergravity Equations.

Models obtained by Jacobi-Lie T-plurality of the background (27)
Below we present examples obtained by Jacobi-Lie transformation of the flat model (27) given by {3|1} and It turns out that nearly all models obtained by Jacobi-Lie T-plurality of the background (27) produce either singular Ê0 or satisfy usual Supergravity Equations.
Ostensible exception is model corresponding to the Leibniz algebra

Jacobi-Lie models corresponding to {3.v|3}
For the C β matrix (36) Jacobi-Lie transformation of the model ( 27) yields that together with or satisfy Generalized Supergravity Equations.However, in this case we can use gauge transformation (6) to get rid of the Killing vector Ĵ because d X = 0, and the above given background together with dilaton satisfy usual Supergravity Equations, i.e. vanishing beta function equations.Analogous situation occurs for the Jacobi-Lie transformation given by the matrix C γ .Beside that, the background (45) is torsionless and flat.Vanishing dilaton and (47) are related by the so called χ-symmetry [29]   where χ = − X satisfies Other type of reduction to usual Supergravity Equations was presented in [28] where J = 0 but J • F = 0.

Jacobi-Lie models corresponding to {3.x|3}
Jacobi-Lie transformation of the model (27) given by (42) yields torsionless curved background This background satisfies usual Supergravity Equations with the dilaton 6 Jacobi-Lie models and Jacobi-Lie T-plurality with spectators Extension of construction of Jacobi-Lie models and Jacobi-Lie T-plurality by spectators can be done similarly as in the case of Poisson-Lie T-plurality.Namely, first we transform the spectator dependent matrix E 0 (y), y = (y 1 , . . ., y n ) by formula where the matrices P, Q, R, S are obtained by extension of the D × D matrices P, Q, R, S to (n + D) × (n + D) matrices to accommodate the spectator fields.Then we apply formulas for construction of Jacobi-Lie models from Section 3. Meantime we have to extend the formula for dilaton to e −2Φ(x,y) = e −2 d(y) |det ĝab (y)| e −2ϕ(x) e (D−1) ∆(x) σ D−3 where ĝab (y) is symmetric part of Ê0 (y).On the other hand, at least for Z a = 0, it turns out that Supergravity equations are satisfied for |det ĝab (y)| = e 2 d(y) so that formula for dilaton reads e −2Φ(x,y) = e −2ϕ(x)+(D−1) ∆(x) |det E 0 (y) −1 ab + π ab (x) det a a b (x) |. (52) 6.1 Four-dimensional flat models corresponding to the algebra {3|1} As mentioned in Section 3.1, good strategy for getting models satisfying Generalized Supergravity Equations is finding models corresponding to the algebra of the type 1 with f b ba = Z a = 0 and then apply the Jacobi-Lie T-plurality to the algebras of type 2. Similarly as in Section 3.1, we will start with flat torsionless model and vanishing dilaton given by the algebra {3|1} = III, −T 2 + T 3 , (I, 0) and some E 0 (t) 4 .
After rather tedious calculations we were able to find E 0 (t) for flat Jacobi-Lie model in four dimensions as Jacobi-Lie model then is and ϕ(x) = x 1 + 1 2 (x 3 − x 2 ).
6.1.1Jacobi-Lie models corresponding to {3.v|3} and spectator t Formula (50) for the matrix From the algebra {3.v|3} we get and very complicated torsionless background that together with dilaton satisfy usual Supergravity Equations for d(t) = − ln t + const.
to zero, so that in fact the model satisfies usual Supergravity Equations (see e.g.

5.2.1).
Appendix -C-matrices transforming {3|1} to type 2 algebras In general we get much greater set of solutions of equations ( 13), but they can be reduced to those given below by choice of free parameters or by change of basis of subalgebras g and ĝ.

Table 3 :
(26)lts of Jacobi-Lie T-plurality of {3|1} to type 2 algebras without spectators.GSUGRA means that models satisfy Generalized Supergravity Equations.β-equationsmeansthat models satisfy usual Supergravity Equations, i.e. vanishing beta function equations.We omit results of plurality to {6 a± |3} given by C α , since for(26)we were not able to check SUGRA equations in reasonable computer time while for (27) Ê0 does not exist.