Target space supergeometry of η and λ -deformed strings
aa r X i v : . [ h e p - t h ] O c t Imperial-TP-LW-2016-03
Target space supergeometry of η and λ -deformedstrings Riccardo Borsato and Linus Wulff
Blackett Laboratory, Imperial College, London SW7 2AZ, U.K. r.borsato, [email protected]
Abstract
We study the integrable η and λ -deformations of supercoset string sigma models, the basicexample being the deformation of the AdS × S superstring. We prove that the kappa symmetryvariations for these models are of the standard Green-Schwarz form, and we determine the targetspace supergeometry by computing the superspace torsion. We check that the λ -deformationgives rise to a standard (generically type II*) supergravity background; for the η -model therequirement that the target space is a supergravity solution translates into a simple conditionon the R -matrix which enters the definition of the deformation. We further construct all suchnon-abelian R -matrices of rank four which solve the homogeneous classical Yang-Baxter equationfor the algebra so (2 , ontents η and λ -deformed string sigma models 7 λ -model 94 Target superspace for the η -model 12 R -matrices and the unimodularity condition 146 Some examples of supergravity backgrounds 18 h ⊕ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 r ′ , ⊕ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.3 r , − ⊕ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4 n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Z -graded superisometry algebras 24B Useful results for the deformed models 26 B.1 λ -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26B.2 η -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A remarkable property of the
AdS × S superstring sigma model is its classical integrability [1],see [2] for a review. In fact, this property extends to several other symmetric space string back-grounds [3, 4]. Recently two interesting deformations of the AdS × S superstring sigma model were proposed which preserve the integrability. The η -model [5] and λ -model [6], named after thecorresponding deformation parameters. The former is based on the Yang-Baxter deformationof [7, 8, 9], it generalises the case of bosonic coset models [10], and its essential ingredient is an R -matrix which satisfies the modified classical Yang-Baxter equation (MCYBE). The λ -modelwas originally proposed by [11] and it extends the case of bosonic cosets [12] (see also [13]). Theconstruction is based on a G/G gauged Wess-Zumino-Witten (WZW) model, and it is morenaturally interpreted as a deformation of the non-abelian T-dual of the original string. The These deformations extend to any Z -symmetric supercoset sigma model, i.e. symmetric space RR stringbackground preserving supersymmetry. q -deformed [14, 15] (with q real and root of unity respectively), and the two models are related,at least at the classical level, by the Poisson-Lie T-duality of [16, 17], see [18, 19, 20].The attempt of interpreting these deformations as string theories has raised interesting ques-tions. In fact, both models possess a local fermionic symmetry believed to be the standard kappasymmetry—which was expected to guarantee a string theory interpretation. However, the targetspace of the η -model derived in [21, 22] does not solve the type IIB supergravity equations [22],but rather a generalisation of them as suggested in [25]. These generalised equations ensurescale invariance for the sigma model, but are not enough to have the full Weyl invariance, whichis present only when the target space satisfies the standard equations of supergravity. For the λ -model, on the other hand, it was shown that the target space does solve the supergravityequations, at least in the case of λ -deformed AdS × S × T [26] and AdS × S × T [27] stringsigma models .A possible resolution for the puzzle posed by the η -model could have been that, after all, thepossessed local fermionic symmetry was not the standard kappa symmetry of Green-Schwarz.However this state of affairs was clarified recently in [30] where it was shown that, contrary towhat was commonly believed, kappa symmetry of the type II Green-Schwarz superstring doesnot imply the full equations of motion of type II supergravity. Rather it implies a weaker(generalized) version of these equations, whose bosonic subsector coincides with the equationswritten down in [25]. These generalized supergravity equations involve a Killing vector field K a ,and reduce to the standard type II supergravity equations when this vector field is set to zero.This fact implies that kappa-symmetric backgrounds whose metric does not allow for isometriesmust in fact solve the standard type II equations. The λ -model falls into this class, which isconsistent with the fact that the corresponding target spaces were found to be supergravitybackgrounds. On the other hand, the η -model typically leads to a target-space metric whichpossesses isometries, so that a priori it is not possible to exclude the possibility that it solvesonly the generalized supergravity equations. It should be mentioned that, given a solution ofthe generalized supergravity equations and provided that K a is space-like, it is possible to finda genuine supergravity solution which is formally T-dual to it [32, 25] (i.e. only at the classicallevel of the sigma model, ignoring the fact that the dilaton is linear in the coordinate alongwhich T-duality is implemented). We will not consider this possibility here. Target space supergeometry
The procedure for the η -deformation can be generalised also to the case when the R -matrixsatisfies the classical Yang-Baxter equation (CYBE) [33, 34, 35]. Therefore several solutionsexist and the question is which choices lead to a string theory, i.e. a target space that solvesthe standard type II supergravity equations. Here we will answer this question and find asimple (necessary and sufficient) condition on R . We will also determine the form of the targetspace (super) fields for both the η and the λ -model in terms of the ingredients that define them See [23] for lower dimensional examples of bosonic truncations and [24] for a review These results differ from the ones proposed in [28]. The metric in target space of the λ -deformed AdS × S was obtained in [29] Earlier indications of this was seen in the pure spinor string in [31]. We will actually see that the kappa symmetry transformations of the λ -model take the standard form onlyafter inserting proper factors of i (see sec. 2.2). This leads to a target space geometry which is a solution of typeII* rather than type II supergravity. In the case of AdS × S × T [26] it was shown how one can get a standard(and real) type IIB background by analytic continuation, or equivalently by picking a different coordinate patch.The same should be true for the deformation of AdS × S × T [27], and probably AdS × S . We will use the names “ η -deformation” and “Yang-Baxter deformation” for both the homogeneous (CYBE)and inhomogeneous (MCYBE) cases, as we can treat them both at the same time. H abc = 3 M [ ab,c ] − i (cid:26) ˆ η − λ (cid:27) M ˆ α a ( γ b ) ˆ α ˆ β M ˆ β c ] , (1.1) S ˆ α β = 8 i (cid:26) [Ad h (1 + 2ˆ η − − O − )] ˆ α γ iλ [Ad h (1 + λ (1 − λ − ) O − )] ˆ α γ (cid:27) b K ˆ γ β , (1.2)where the upper (lower) expression in curly brackets refers to the η ( λ ) model and ˆ η = p − cη .The RR field strengths are encoded in the bispinor defined as [30, 36] S = − iσ γ a F a − σ γ abc F abc − · iσ γ abcde F abcde , (1.3)where for standard supergravity backgrounds F = e φ F contains the exponential of the dilaton.The remaining ingredients in these equations are defined in section 2, in particular the opera-tors O + , M and the group element h are defined in (2.5), (2.2), (B.2) and (2.12). From ourcomputation we obtain also the Killing vector of the generalised type II equations K a = − i
16 ( γ a ) ˆ α ˆ β ( ∇ ˆ α χ ˆ β − ∇ ˆ α χ ˆ β ) , (1.4)where χ I ( I = 1 ,
2) are the would be dilatino superfields χ α = i (cid:26) ˆ η − (cid:27) γ b ˆ α ˆ β [Ad h M ] ˆ β b , χ α = − i (cid:26) ˆ ηiλ (cid:27) γ a ˆ α ˆ β M ˆ β a . (1.5)When K a vanishes we have a standard supergravity solution and the dilaton is given by e − φ = sdet( O + ) . (1.6)For the λ -model K a automatically vanishes and the target space is always a supergravity solu-tion, consistently with the observation of [30] and the previous findings [26, 27]. The η -model as a string For the η -model the situation is more subtle. Let us review some details at this point and recallthat the η -deformation is defined by an antisymmetric R -matrix on the algebra R : g → g , R T = − R , satisfying the (M)CYBE[ R ( x ) , R ( y )] − R ([ R ( x ) , y ] + [ x, R ( y )]) = c [ x, y ] , ∀ x, y ∈ g , (cid:26) c = 0 CYBE c = ± . (1.7)In section 4.1 we prove that the condition K a = 0 for the η -model is equivalent to the followingalgebraic condition on the R -matrix STr( R ad x ) = 0 , ∀ x ∈ g (i.e. R BA f ABC = 0) . (1.8) Note that here we write the λ -model as a solution of type IIB supergravity, and the corresponding RR flux isimaginary. The background is real when written as a solution of type IIB*. The reason for this is a non-standardsign in the kappa symmetry transformations of the lambda model, see sec 2.2. For the λ -model this formula was argued in [6]. It is also consistent with the form of the bosonic dilatonsuggested in [37] for the η -model based on bosonic R -matrices. Essentially the same condition was argued to appear from the analysis of vertex operators of the β -deformed AdS × S superstring in [38], see equation (87) there. That discussion would correspond to the truncation of ourdeformed action at order O ( η ). We thank Arkady Tseytlin for pointing this reference out to us.
3e will refer to R -matrices satisfying this condition as “unimodular”, for reasons that will beclear in section 5. Therefore the η -model has an interpretation as a string sigma model preciselyfor the unimodular R -matrices.Let us consider the η -deformation based on an R -matrix which is a non-split ( c = 1 in(1.7)) solution of the MCYBE for the supercoset on AdS × S with superalgebra psu (2 , | R that multiplies by − i (+ i ) positive (negative) roots ofthe complexified algebra, and annihilates Cartan elements. Choices of different real forms of thesuperalgebra correspond to inequivalent R -matrices, but one can check that none of the examplesconsidered so far [5, 22, 14, 39] are unimodular, which is consistent with the findings of [22, 39].We are not aware of a complete classification of solutions of the MCYBE for psu (2 , | R -matrices that would lead to genuinestring deformations. We will not analyze this question further here.As first pointed out in [33], there is a rich set of solutions to the CYBE ( c = 0 in (1.7)) whichcan be used to define an η -deformation of the supercoset. These R -matrices can be divided intotwo classes: abelian and non-abelian. Writing the R -matrix as (sums over repeated indices areunderstood) R = 12 r ij b i ∧ b j , ( R ( x ) = r ij b i Str( b j x ) , x ∈ g ) , (1.9)abelian R -matrices are the ones for which [ b i , b j ] = 0 ∀ i, j while non-abelian ones have [ b i , b j ] =0 for some i, j . The unimodularity condition (1.8) takes the form r ij [ b i , b j ] = 0 . (1.10)This is trivially satisfied by any abelian R -matrix, which is consistent with observations in theliterature, see e.g [37, 40, 41]. This is also in line with the expectation that abelian R -matricesalways have an interpretation in terms of (commuting) TsT-transformations [35]. For non-abelian R -matrices the unimodularity condition (1.10) is non-trivial, and it is interesting tofind all the compatible ones. In fact, as explained in section 5 it rules out most of the R -matrices of the so-called Jordanian type, which is the only class considered in the literature sofar [33, 35, 37, 40, 41].Here we will focus on the problem of classifying all R -matrices which satisfy the CYBEon the bosonic subalgebra so (2 , ⊕ so (6) ⊂ psu (2 , |
4) and are unimodular. The question isnon-trivial only for non-abelian R -matrices, which we classify by the rank. From (1.10), anyunimodular R -matrix of rank two R = a ∧ b must be abelian, i.e. [ a, b ] = 0, so non-abelianunimodular R -matrices have at least rank four. In tables 1 and 2 we write down all non-abelianrank four R -matrices for so (2 ,
4) (the second table gives the inequvalent ones from the pointof view of the string sigma model), and in table 3 we provide the bosonic isometries and thenumber of supersymmetries that they preserve. These R -matrices are constructed in section 5,where we also show that the only other possibility is rank six. The extension to so (2 , ⊕ so (6)is essentially trivial as it turns out that they must be abelian . in so (6). Therefore there are nonew marginal deformations of the dual CFT. Notice that R , R and R can be embedded It is easy to see that this condition is compatible with the (M)CYBE. For the split case ( c = −
1) there exist no solution for the compact subalgebra su (4) ⊂ psu (2 , | TsT stands for T-duality – shift – T-duality [42, 43, 44]. Here we use it in the most general possible sense,e.g. including non-compact and fermionic T-dualities. This includes R -matrices mixing generators of AdS and S, e.g. as in the so-called dipole deformations of [45] This statement remains to be true also if we further allow the R -matrix to act non-trivially on supercharges:after imposing unimodularity, preservation of the so (2 ,
4) isometry, reality and CYBE, we find that the onlypossible R -matrices are abelian and they act just on so (6). so (2 ,
3) and can therefore be used to define deformations of
AdS . To have non-abeliandeformations of AdS , instead, one must involve also generators from the sphere.Because abelian R -matrices seem to generate backgrounds which can be equivalently ob-tained by doing (commuting) TsT-transformations on the undeformed model, one might suspectthat η -deformed strings always correspond to TsT-transformations. With the exception of thelast three R -matrices our results appear to be consistent with this expectation, see section 5 fora discussion.The outline of the rest of the paper is as follows. In section 2 we first review the definitions ofthe deformed models, we introduce a notation that highlights their similarities, and prove thatthe local fermionic symmetries of both deformed models are of the standard Green-Schwarz form.In section 3 we derive the target space supergeometry for the λ -model, and by comparing to theresults of [30] we extract the corresponding background fields. Section 4 achieves the same goalfor the η -model. Here we also show how the unimodularity condition for the R -matrix is derived.In section 5 we study this condition in detail. We discuss its compatibility with Jordanian R -matrices, and derive all rank-four non-abelian unimodular R -matrices for so (2 ,
4) which solvethe CYBE. In section 6 we consider the case of backgrounds generated by R -matrices which actonly on the bosonic subalgebra. We work out certain examples generated by the R -matricespreviously derived, and we check in some cases that they are equivalent to sequences of TsTtransformations on the original undeformed model. R = p ∧ p + ( p + p ) ∧ ( J − J ) R = p ∧ p + ( p + p ) ∧ ( p + J − J ) R = p ∧ ( J − J ) + ( p + p ) ∧ ( p + J − J ) R = ( p − J + J ) ∧ ( k + k + 2 p − J ) + 2( p + p ) ∧ ( p + J − J ) R = p ∧ ( J − J ) + ( p + p ) ∧ ( D + J ) R = p ∧ J + 2 p ∧ p R = J ∧ J + 2 p ∧ p R = p ∧ p + ( p + p ) ∧ J R = p ∧ p + ( p + p ) ∧ ( p + J ) R = p ∧ p + p ∧ ( p + J ) R = p ∧ p + p ∧ J R = p ∧ p + p ∧ ( p + J ) R = p ∧ p + p ∧ J R = p ∧ p + J ∧ J R = p ∧ p + ( J − J ) ∧ ( p + p ) R = p ∧ p + ( p + J − J ) ∧ ( p + p ) R = p ∧ ( p + J − J ) + ( p + p ) ∧ ( p + J − J )Table 1: All non-abelian unimodular rank-four R -matrices (CYBE) of so (2 ,
4) up to automor-phisms of the corresponding subalgebras (see section 5).5 = ( p + a ( J − J )) ∧ p + ( p + p ) ∧ ( J − J ) R = ( p + a ( p + J − J ) + b ( p + p )) ∧ p + ( p + p ) ∧ ( p + J − J ) R = ( p + a ( p + J − J )) ∧ ( p + J − J ) + ( p + p ) ∧ ( p + J − J ) R = (( p − J + J ) + 2 a ( p + J − J ) + 2 b ( p + p )) ∧ ( k + k + 2 p − J + c ( p + p ))+2 d ( p + p ) ∧ ( p + J − J ) R = p ∧ ( J − J ) + a ( p + p ) ∧ ( D + J ) R = p ∧ J + 2 p ∧ p R = J ∧ J + 2 p ∧ p R = p ∧ p + ( p + p ) ∧ J R = p ∧ p + a ( p + p ) ∧ ( p + J ) R = p ∧ p + a p ∧ ( p + J ) R = p ∧ p + p ∧ J R = p ∧ p + a p ∧ ( p + J ) R = p ∧ p + p ∧ J R = p ∧ p + J ∧ J R = ( p + a ( p + p )) ∧ p + ( J − J ) ∧ ( p + p ) R = ( p + a ( p + p )) ∧ p + ( p + J − J ) ∧ ( p + p ) R = ( p + a ( p + p )) ∧ ( p + p + J − J ) + ( p + p ) ∧ ( p + J − J )Table 2: All non-abelian unimodular rank four R -matrices (CYBE) of so (2 ,
4) up to innerautomorphisms.supercharges bosonic isometries R p + p , p , p , p − p − J − J ) , ( a = 0)8 p + p , p + a ( J − J ) , p , ( a = 0) R p + p , p , p , p − p − J − J + J + J , ( a = 0)8 p + p , p + a ( J − J ) , p , ( a = 0) R p + p , p , J − J , ( a = 0)8 p + p , p + ( J − J ) , J − J − a ( J − J + p ) , ( a = 0) R − J + J + p + 2 a ( J − J + p ) , p + p , J − p − k − k ,R D + J , p + p ,R J , p , p ,R J , J ,R p , p , J ,R p , p , J ,R p , p , J ,R p , p , J ,R p , p , J ,R p , p , J ,R J , J ,R p + p , p , p ,R p + p , p , p ,R p + p , p , J − J − p + p , Table 3: For each R -matrix of Table 2 we indicate the number of unbroken supercharges andwe list the unbroken bosonic isometries. 6 η and λ -deformed string sigma models The η and λ deformations are deformations of supercoset sigma models that preserve the classicalintegrability of the original models. In the string theory context the most studied example is thedeformation of the AdS × S string described by a P SU (2 , | SO (1 , × SO (5) supercoset sigma model [47].However, there are many other backgrounds where at least a subsector of the string worldsheettheory is described by a supercoset sigma model, e.g. AdS × CP [48, 49, 50], AdS × S × T [51], AdS × S × T [52] and several others [3].We start by reviewing the definitions of the deformed models. The relevant superalgebraconventions are collected in appendix A. The η -model Lagrangian takes the form [5, 33] L = − (1 + cη ) − cη ) ( γ ij − ε ij )Str( g − ∂ i g ˆ d O − − ( g − ∂ j g )) , (2.1)where g is a group element of G , i, j are worldsheet indices, γ ij is the (Weyl-invariant) worldsheetmetric and ε = +1. Here η is the deformation parameter, and setting η = 0 yields theLagrangian of the undeformed supercoset sigma model. The deformation involves the Lie algebraoperators O + = 1 + ηR g ˆ d T , O − = 1 − ηR g ˆ d , (2.2)where R g = Ad − g R Ad g , R T = − R and R satisfies the (M)CYBE (1.7). Our derivation is generaland we will not need to pick a particular solution of (1.7): we only need to assume the aboveproperties for R , and we will treat the homogeneous ( c = 0, CYBE) and the inhomogeneous( c = 1, MCYBE) cases at the same time. In the Lagrangian the following combinations ofprojection operators appearˆ d = P (1) + 2ˆ η − P (2) − P (3) , ˆ η = p − cη . ˆ d T = − P (1) + 2ˆ η − P (2) + P (3) , where ˆ d + ˆ d T = 4ˆ η − P (2) . (2.3)The λ -model is defined as a deformation of the G/G gauged WZW model. To get a standardstring sigma model one integrates out the gauge-field which leads to a Lagrangian somewhatsimilar to that of the η -model, namely [6] L = − k π ( γ ij − ε ij )Str( g − ∂ i g (1 + b B − O − − )( g − ∂ j g )) . (2.4)Here k is the level of the WZW model, and the Lie algebra operators O ± are now defined as O + = Ad − g − Ω T , O − = 1 − Ad − g Ω . (2.5)In this case things are written in terms of the combinations of projectorsΩ = P (0) + λ − P (1) + λ − P (2) + λP (3) , Ω T = P (0) + λP (1) + λ − P (2) + λ − P (3) , − ΩΩ T = 1 − Ω T Ω = (1 − λ − ) P (2) . (2.6) Another supercoset closely related to this is the pp-wave background of [46]. This is the classical Lagrangian. At the quantum level there is also a Fradkin-Tseytlin term R (2) φ present,where φ is the dilaton superfield, generated by integrating out the gauge-field, whose form will be discussed insection 3. b B = − b B T is related to the original WZ-term, see section 3. η and (2.4) of the λ -model can be formally written in thesame way L = − T γ ij Str( A (2) − i A (2) − j ) + T ε ij Str( A − i b BA − j ) , (2.7)in terms of the one-forms A ± = O − ± ( g − dg ) , (2.8)where the string tension T and the operator b B (responsible for the B -field) in the two cases are η − model : T = (cid:18) cη − cη (cid:19) , b B = ˆ η P (1) − P (3) + η ˆ d T R g ˆ d ) ,λ − model : T = kπ ( λ − − , b B = ( λ − − − ( O T − b B O − + Ω T Ad g − Ad − g Ω) . (2.9)An important role is played by the operator M = O − − O + (2.10)which relates A − to A + as A − = M A + . Using the expressions in (B.2) it is not hard to showthat M T P (2) M = P (2) , (2.11)which implies that the operator P (2) M P (2) implements a Lorentz transformation on the subspacewith grading-2 of the superisometry algebra. This implies that there exists an element h ∈ H = G (0) ⊂ G such that P (2) M P (2) = Ad − h P (2) = P (2) Ad − h . (2.12)The fact that Ad h is a Lorentz transformation implies the basic relation between the action onvectors and spinors [Ad h ] ˆ γ ˆ α γ a ˆ γ ˆ δ [Ad h ] ˆ δ ˆ β = [Ad h ] ab γ b ˆ α ˆ β . (2.13)We refer to appendix B for some useful identities satisfied by the operators entering the deformedmodels. Both the η and λ model have a local fermionic symmetry which removes 16 of the 32 fermions,and here we show that it takes the form of the standard kappa symmetry of the GS superstring.The transformations for the local fermionic symmetry take the form [5, 33, 6] O − ( g − δ κ g ) = P ij − { i ˜ κ (1) i , A (2) − j } + ζ s P ij + { i ˜ κ (3) i , A (2)+ j } , (2.14)where we denote the parameter by ˜ κ , which is related to the kappa symmetry parameter κ ofthe GS string as explained below. The above transformations are accompanied by the variationof the worldsheet metric δ κ γ ij = ζ (cid:0) Str( W [( P + i ˜ κ (1) ) i , ( P + A (1)+ ) j ]) + Str( W [( P − i ˜ κ (3) ) i , ( P − A (3) − ) j ]) (cid:1) , (2.15)where we have defined P ij ± = 12 ( γ ij ± ε ij ) , ζ = (cid:26) ˆ ηλ , s = (cid:26) η − model λ − model . (2.16) We have used (2.3), (2.6), Ad Tg = Ad − g and R Tg = − R g . A (2) − is related to A (2)+ by a gauge transformation, i.e. A (2) − = P (2) M A (2)+ = Ad − h A (2)+ , (2.17)we can write the kappa transformations as i δ κ E (2) = 0 , i δ κ E (1) = P ij − { iκ (1) i , E (2) j } , i δ κ E (3) = P ij + { iκ (3) i , E (2) j } δ κ γ ij = 12 Str( W [( P + iκ (1) ) i , ( P + E (1) ) j ]) + 12 Str( W [( P − iκ (3) ) i , ( P − E (3) ) j ]) , (2.18)where κ (1) = ζ Ad h ˜ κ (1) and κ (3) = ( − i ) s ζ ˜ κ (3) . This shows that the kappa symmetry variationshave the standard GS form, and at the same time it allows us to identify the supervielbeins withprojections of A ± as E (2) ≡ E a P a = A (2)+ , E (1) ≡ E ˆ α Q α = ζ Ad h A (1)+ , E (3) ≡ E ˆ α Q α = i s ζA (3) − . (2.19)In terms of these the Lagrangian (2.7) takes the standard form L = − T γ ij Str( E (2) i E (2) j ) + T ε ij B ij , (2.20)where the B -field can be read off from (2.9).Since the action and kappa symmetry transformations take the standard GS form, it followsfrom the analysis of [30] that the target superspace of these models solves the generalized typeII supergravity equations derived there. If the Killing vector K a appearing in these equationsvanishes, they reduce to the standard supergravity equations. In the next sections we will derivethe form of the target space supergeometry for the η and λ -deformed strings. Having identifiedthe supervielbeins of the background superspace we can find the supergeometry by calculatingthe torsion T a = dE a + E b ∧ Ω ba , T ˆ αI = dE ˆ αI −
14 ( γ ab E I ) ˆ α ∧ Ω ab ( I = 1 , , (2.21)and reading off the background superfields by comparing to the general expressions derivedin [30]. These are valid for a generalized type II supergravity background and reduce to those ofa standard supergravity background (see e.g. [36]) only when K a = 0. We will see that the λ -model background is a solution to standard (type II*) supergravity. For the η -model backgroundwe will derive the condition on the R -matrix of the η -model for it to give rise to a standard typeII background. λ -model In this section we present the derivation for the λ -model. We refer to appendix B.1 for moredetails. The supervielbeins are defined in terms of projections of A ± by (2.19). To calculate the In writing the transformations in this form we used (B.4). The explicit i in E (3) and κ (3) in the case of the λ -model is needed to put the transformations in the standardtype IIB form. The reason for having i can be traced to the relative sign between P (1) and P (3) in (2.6) comparedto (2.3). Alternatively, insisting on manifest reality of the model, the kappa symmetry transformations andsuperspace constraints become those of type IIB* rather than type IIB. This is rather natural since the λ -modelis a deformation of the non-abelian T-dual of the AdS × S string, which involves also a T-duality in the timedirection. Our conventions are the same as those of [30]. In particular d acts from the right and components of superformsare defined as ω n = n ! E A n ∧ · · · ∧ E A ω A ··· A n . A ± . Using A + = O − ( g − dg )where O ± are defined in (2.5) we find dA + = O − ( d O + ∧ A + ) + O − ( g − dg ∧ g − dg )= − O − { g − dg, Ad − g A + } + 12 O − { g − dg, g − dg } = − O − { Ad − g A + , Ad − g A + } + 12 O − { Ω T A + , Ω T A + } = − { A + , A + } − O − (Ω T { A + , A + } − { Ω T A + , Ω T A + } ) , (3.1)where we used the fact that g − dg = O + A + = (Ad − g − Ω T ) A + to write everything in terms of A + . An almost identical calculation gives dA − = 12 { A − , A − } + 12 O − − Ad − g (Ω { A − , A − } − { Ω A − , Ω A − } ) . (3.2)In the above equations it is useful to expand out the expressions inside parenthesis, see (B.5), (B.6).Projecting equation (B.5) with P (2) we find dE (2) = 12 { E (1) , E (1) } + 12 { E (3) , E (3) } − { A (0)+ , E (2) } − iλ { E (3) , P (3) M E (2) }− iλP (2) M T { E (2) , E (3) } − λ { P (3) M E (2) , P (3)
M E (2) } − P (2) M T { E (2) , E (2) }− λ P (2) M T { E (2) , P (3) M E (2) } . (3.3)where the result has been rewritten in terms of the supervielbeins (2.19), and we have used(B.4) and (2.12). Using the explicit form of the commutators in (A.1) and (A.2) we find thatthe component T a of the torsion takes the standard form (here and in the following we drop the ∧ ’s for readability) T a = dE a + E b Ω ba = − i E γ a E − i E γ a E , (3.4)if we identify the spin connection as Ω ab = − ( A + ) ab − λ ( E γ [ a ) ˆ α M ˆ α b ] − i λ E c M ˆ α a ( γ b ) ˆ α ˆ β M ˆ β c ] + 12 E c ( M ab,c − M c [ a,b ] ) . (3.5)To derive the other components of the torsion we first need to compute the exterior derivativeof the fermionic supervielbeins. Using (B.6) and (2.19) we find dE (3) = i λP (3) M { E (3) , E (3) } − { A (0)+ , E (3) } + { P (0) M E (2) , E (3) }− iλ (cid:2) λ (1 − λ − ) P (3) ( O T + ) − (cid:3) Ad − h (cid:0) { E (2) , E (1) } − { E (2) , Ad h P (1) M E (2) } (cid:1) + i λ (1 − λ − ) P (3) ( O T + ) − Ad − h { E (2) , E (2) } . (3.6)Since we have already identified the form of the spin connection (3.5) from the previous com-putation, we can now find the corresponding component of the torsion (2.21) and compare it tothe standard form given in [30], i.e. T ˆ α = E ˆ α E χ − E γ a E ( γ a χ ) ˆ α + 18 E a ( E γ bc ) ˆ α H abc − E a ( E γ a S ) ˆ α + 12 E b E a ψ ˆ α ab , (3.7) Here we rewrote A (0) ± = A ab ± J ab and used the relation between components of M and M T in (A.11). H is the NSNS three-form, S the RR bispinor, χ I ˆ α the dilatino and ψ ˆ αIab the gravitino fieldstrength superfields. We find that T ˆ α takes the above form if we identify H abc =3 M [ ab,c ] + 3 iλ M ˆ α a ( γ b ) ˆ α ˆ β M ˆ β c ] , (3.8) S ˆ α β = − λ (cid:2) Ad h (1 + λ (1 − λ − ) O − ) (cid:3) ˆ α γ b K ˆ γ β , (3.9) χ α = 12 λγ a ˆ α ˆ β M ˆ β a , (3.10) ψ ˆ α ab = i λ (1 − λ − )[( O T + ) − Ad − h ] ˆ α cd b K abcd −
14 [Ad h M ] ˆ β a ( γ b ] ) ˆ β ˆ γ S ˆ γ α . (3.11)As already remarked, the RR bispinor superfield is imaginary if we interpret the λ -model targetspace as a solution of type II supergravity, as here, rather than type II* supergravity. Thisdetermines the bosonic target space fields, with the exception of the dilaton which we will deter-mine shortly. First, let us calculate also the remaining components of the femionic superfields,which we will extract from the corresponding component of the torsion, T ˆ α . From (B.5) andusing (2.19) we find dE (1) = − { Ad h A (0)+ + dhh − , E (1) } + 12 λ (1 − λ − ) P (1) Ad h O − Ad − h { E (1) , E (1) }− iλ Ad h { E (2) , E (3) } − λ Ad h { E (2) , P (3) M E (2) } − iλ (1 − λ − ) P (1) Ad h O − { E (2) , E (3) }− λ (1 − λ − ) P (1) Ad h O − (cid:0) { E (2) , E (2) } + 2 λ { E (2) , P (3) M E (2) } (cid:1) . (3.12)Using this expression we find T ˆ α = E ˆ α E χ − E γ a E ( γ a χ ) ˆ α − E a ( E γ bc ) ˆ α H abc − E a ( E γ a S ) ˆ α + 12 E b E a ψ ˆ α ab , (3.13)is again of the standard form given in [30], where S ˆ β α = −S ˆ α β and χ α = − i γ b ˆ α ˆ β [Ad h M ] ˆ β b , ψ ˆ α ab = − λ (1 − λ − )[Ad h O − ] ˆ α cd b K abcd − i λ ( S γ [ a ) ˆ α ˆ β M ˆ β b ] . (3.14)We complete the set of background superfields for the λ -model by noting that the B -field canbe written in the two equivalent forms B = ( λ − − − (cid:2) B + Str( g − dg ∧ A − ) (cid:3) , dB = 13 Str( g − dg ∧ g − dg ∧ g − dg ) , = ( λ − − − (cid:2) B − Str( g − dg ∧ Ω T A + ) (cid:3) , (3.15)and that the dilaton is given by e − φ = sdet( O + ) = sdet(Ad g − Ω) . (3.16)This result for the dilaton arises from integrating out the gauge-fields in the deformed gaugedWZW model [6]. To verify that the λ -model gives rise to a standard supergravity background itis enough to verify that the dilatino’s found in (3.10) and (3.14) are indeed the spinor derivativesof φ ∇ ˆ α φ = i λ b K ˆ β γ STr( Q β M [ Q α , Q γ ]) = χ α , ∇ ˆ α φ = 12 (1 − λ − )[Ad − h ] ˆ β ˆ α STr( P a O − − [ Q β , P a ]) = χ α . (3.17) Let us recall that at least in some cases it is possible to define a real type II background, after analyticcontinuation or proper choice of coordinate patch [26, 27]. To calculate this component of the torsion we must first find the Lorentz-transformed spin connectionAd h A (0)+ + dhh − appearing in the first term, see equation (B.9) and the corresponding derivation. As pointed out in [30] this was clear from the fact that the metric of the λ -model does not admit any isometries,so that the Killing vector K a of the generalized supergravity equations vanishes. Target superspace for the η -model The calculations for the η -model proceed along the same lines as those for the λ -model withonly minor differences. We begin by calculating the derivative of A + dA + = O − ( d O + ∧ A + ) + O − ( g − dg ∧ g − dg )= η O − R g { g − dg, ˆ d T A + } − η O − { g − dg, R g ˆ d T A + } + 12 O − { g − dg, g − dg } = 12 O − { A + , A + } + η O − R g { A + , ˆ d T A + } + η O − R g { R g ˆ d T A + , ˆ d T A + }− η O − { R g ˆ d T A + , R g ˆ d T A + } = 12 O − { A + , A + } − cη O − { ˆ d T A + , ˆ d T A + } + η O − R g { A + , ˆ d T A + } , (4.1)where we used the fact that g − dg = O + A + and in the last step we used the fact that R (aswell as R g ) satisfies the (M)CYBE equation, so that { R g ˆ d T A + , R g ˆ d T A + } − R g { R g ˆ d T A + , ˆ d T A + } − c { ˆ d T A + , ˆ d T A + } = 0 . (4.2)The result for dA − is simply obtained by changing the sign of η and replacing ˆ d T → ˆ d in theabove expression dA − = 12 O − − { A − , A − } − cη O − − { ˆ dA − , ˆ dA − } − η O − − R g { A − , ˆ dA − } . (4.3)After rewriting dA + as in (B.13) and projecting with P (2) we find dE (2) = { A (0)+ , E (2) } + 12 { E (1) , E (1) } + 12 { E (3) , E (3) } − η { E (3) , P (3) O − − E (2) } + 4ˆ η − P (2) O − { E (2) , E (3) } − P (2) O − { E (2) , P (3) O − − E (2) } + 2ˆ η { P (3) O − − E (2) , P (3) O − − E (2) } + 2 η ˆ η − P (2) O − R g { E (2) , E (2) } , (4.4)where we have used (2.19) to write the result in terms of the supervielbeins, together with (B.4)and (2.12). We check again that the bosonic torsion T a takes the standard form (3.4), wherewe can now identify the spin connection for the η -model background asΩ ab = ( A + ) ab + 2 i ˆ η ( γ [ a E ) ˆ α M ˆ α b ] + 3 i η E c M ˆ α a ( γ b ) ˆ α ˆ β M ˆ β c ] − E c (2 M c [ a,b ] − M ab,c ) . (4.5)As before, we continue by computing the remaining components of the torsion. First, from(B.14) we get dE (3) = { A (0)+ , E (3) } + ˆ ηP (3) O − − { E (3) , E (3) } + 2 { P (0) O − − E (2) , E (3) } + P (3) (4 O − − − − η − )Ad − h { E (2) , E (1) } − η ˆ η − P (3) O − − R g Ad − h { E (2) , E (2) } + 2ˆ ηP (3) (4 O − − − − η − ) { Ad − h E (2) , P O − − E (2) } , (4.6)which we use to check that also T ˆ α is of the standard form (3.7). To do this we make use ofthe spin connection (4.5) and we identify the following superfields for the η -model H abc = 3 M [ ab,c ] − i ˆ η M ˆ α a ( γ b ) ˆ α ˆ β M ˆ β c ] , (4.7) S ˆ α β = 8 i [Ad h (1 + 2ˆ η − − O − )] ˆ α γ b K ˆ γ β , (4.8) χ α = − i ηγ a ˆ α ˆ β M ˆ β a , (4.9) ψ ˆ α ab = − η ˆ η − [ O − − R g Ad − h ] ˆ α cd b K abcd + 14 ˆ η [Ad h M ] ˆ β a ( γ b ] S ) ˆ β ˆ α . (4.10)12o identify the last component of the spinor superfields we must compute torsion T ˆ α . Startingfrom (B.13) we find dE (1) = { Ad h A (0)+ − dhh − , E (1) } + ˆ ηP (1) Ad h O − Ad − h { E (1) , E (1) } + P (1) Ad h (4 O − − − η − ) { E (2) , E (3) } + 2 η ˆ η − P (1) Ad h O − R g { E (2) , E (2) }− ηP (1) Ad h (4 O − − − η − ) { E (2) , P (3) O − − E (2) } . (4.11)Using this expression we can check that T ˆ α is standard, see (3.13), where S ˆ β α = −S ˆ α β and χ α = i ηγ b ˆ α ˆ β [Ad h M ] ˆ β b , ψ ˆ α ab = 2 η ˆ η − [Ad h O − R g ] ˆ α cd b K abcd −
14 ˆ η ( S γ [ a ) ˆ α ˆ β M ˆ β b ] . (4.12)Let us also note that in the case of the η -model the B -field can be written in the two ways B = ˆ η g − dg ∧ ˆ d T A + ) = − ˆ η g − dg ∧ ˆ dA − ) , (4.13)which are equivalent thanks to the properties of O ± under transposition. Unlike in the case of the λ -model, the η -model does not come with a natural candidate dilaton.Indeed, in general the target space geometry of the η -model is a solution of the generalized typeII supergravity equations of [25, 30] rather than the standard ones, and a dilaton does not exist.One of our goals is to determine precisely when a dilaton exists for the η -model. To do this, letus define a would-be dilaton in the same way as the dilaton is defined in the λ -model e − φ = sdet( O + ) = sdet(1 + ηR g ˆ d T ) . (4.14)For this to be the actual dilaton of the η -model its spinor derivatives must coincide with thedilatinos in (4.9) and (4.12). In (B.18) we write down the result for dφ . In particular we find ∇ ˆ α φ = − η − STr( P a O − [ Q α , P a ]) − η η − b K AB STr( T A R g [ T B , Q α ])= χ α − η η − b K AB STr([ T A , RT B ] gQ α g − ) , (4.15) ∇ ˆ α φ = − ˆ η [Ad − h ] ˆ β ˆ α ( b K ˆ γ δ STr( Q δ O − [ Q β , Q γ ]) − η b K AB STr( T A R g [ T B , Q β ]))= χ α + η η [Ad − h ] ˆ β ˆ α b K AB STr([ T A , RT B ] gQ β g − ) . (4.16)Therefore a sufficient condition for the η -model to lead to a standard supergravity backgroundis that b K AB STr([ T A , RT B ] gQ I ˆ α g − ) = 0 , (4.17)or, since g is an arbitrary group element (modulo gauge-transformations),STr( R ad x ) = 0 , ∀ x ∈ g (i.e. R BA f ABC = 0 , or R BC f ABC = 0) . (4.18)To see that this condition is also necessary we calculate the Killing vector superfield K a appearingin the generalized supergravity equations of [30], which in general is given by K a = − i
16 ( γ a ) ˆ α ˆ β ( ∇ ˆ α χ ˆ β − ∇ ˆ α χ ˆ β ) , (4.19) As in the previous section, we need to first find an expression for Ad h A (0)+ − dhh − , see (B.16). Here we used the fact that O − ± P (0) = P (0) . η -model has a standard type II supergravity solutionas target space if K a = 0. In fact, it must be that it vanishes order by order in the deformationparameter η . At linear order we find the equation b K AB STr([ T A , RT B ] gP a g − ) = 0 , (4.20)which, since g ∈ G is arbitrary implies (4.18). Therefore the condition (4.18) is both necessaryand sufficient, and also the higher order terms in η in (B.19) vanish when this condition isfulfilled. R -matrices and the unimodularity condition In this section we study the unimodularity condition (1.8) for the R -matrix. First we analyse itscompatibility with a class of non-abelian R -matrices—the Jordanian ones—and then we explainhow to classify all unimodular R -matrices solving the CYBE on the bosonic subalgebra of thesuperisometry algebra.Following [53] we define an “extended Jordanian” R -matrix for a Lie superalgebra g asfollows: we fix a Cartan element h (deg( h ) = 0) and a positive root e as well as a collection ofroots e γ ± i with i ∈ { , , . . . , N } such that deg( e ) = deg( e γ i ) + deg( e γ − i ) (mod 2) and satisfying[ h, e ] = e , [ h, e γ i ] = (1 − t γ i ) e γ i , [ h, e γ − i ] = t γ i e γ − i , ( t γ i ∈ C )[ e γ ± i , e ] = 0 , [ e γ k , e γ l ] = δ k, − l e , ( k > l ∈ {± , ± , . . . , ± N } ) . (5.1)The extended Jordanian R -matrix is then defined as R = h ∧ e + N X i =1 ( − deg( e γi ) deg( e γ − i ) e γ i ∧ e γ − i . (5.2)It is now easy to see that for a bosonic deformation, i.e. deg( e ) = 0, we have r ij [ b i , b j ] = ( N − N + 1) e , (5.3)with N = N + N , N ( N ) being the number of bosonic (fermionic) roots e γ i . For this tovanish we need precisely one more fermionic e γ i than bosonic. This is clearly a very strongrestriction on the allowed Jordanian R -matrices. Let us note that this result is compatible withthe findings of [37, 40, 41], where Jordanian R -matrices acting only on bosonic generators werefound to produce backgrounds which do not solve the standard supergravity equations. Wehave considered certain examples of bosonic Jordanian R -matrices (namely R = J ∧ ( P − P ), R = J ∧ ( J − J ) and R = D ∧ p i , i = 0 , . . . ,
3) and we have checked that it is not possible tofind a positive and a negative fermionic root satisfying (5.1) without spoiling the reality of theextended R -matrix. If possible, it would be interesting to find extended Jordanian unimodular R -matrices for psu (2 , | so (2 , ⊕ so (6) ⊂ psu (2 , | c = 0, for ordinary Lie algebras.The first important fact, due to Stolin [54, 55], is that there is a one-to-one correspondencebetween constant solutions of the CYBE for a Lie algebra g and quasi-Frobenius (or symplectic)subalgebras f ⊂ g (see also [56]). Notice that we do not need to assume anything about the Liealgebra g , in particular it does not need to be simple. A Lie algebra is quasi-Frobenius if it hasa non-degenerate 2-cocycle ω , i.e. ω ( x, y ) = − ω ( y, x ) , ω ([ x, y ] , z ) + ω ([ z, x ] , y ) + ω ([ y, z ] , x ) = 0 , ∀ x, y, z ∈ f . (5.4)14t is Frobenius if ω is a coboundary, i.e. ω ( x, y ) = f ([ x, y ]) for some linear function f . If R isa solution to the CYBE for g , then there is a subalgebra f on which R is non-degenerate. Thissubalgebra is necessarily quasi-Frobenius, and writing R in the form (1.9) the 2-cocycle is theinverse of the R -matrix, i.e. ω ( b i , b j ) = ( r − ) ij . The converse is also true, i.e. if f ⊂ g is quasi-Frobenius then the inverse of the 2-cocycle ω gives a solution to the CYBE, as is easily verified.Therefore, finding solutions to the CYBE for a given g reduces to finding all quasi-Frobeniussubalgebras of g . A fact with important consequences for our analysis is that if g is compactthen f must be abelian [58]. This leads to the conclusion that deformations involving only S (i.e. marginal deformations of the dual CFT) must necessarily have abelian R -matrices.We now show that the unimodularity condition (1.8) for the R -matrix adds a further propertyto the quasi-Frobenius subalgebra f . If we write the structure constants as f ijk in some basis,the 2-cocycle condition is ( r − ) i [ j f ikl ] = 0 . (5.5)Contracting this equation with r jk we get ( r − ) il f ijk r jk = − f iil , which together with theunimodularity condition for the R -matrix written as (1.10), i.e. f ijk r jk = 0, implies f iil = 0 ⇔ tr( ad x ) = 0 ∀ x ∈ f . (5.6)Therefore f is a unimodular Lie algebra. Clearly the converse is also true and we have establishedthat
Solutions of the CYBE for a Lie algebra g which satisfy the condition (1.8) are in one-to-onecorrespondence with unimodular quasi-Frobenius subalgebras of g . For this reason we refer also to the R -matrices which satisfy (1.8) as unimodular. A quasi-Frobenius Lie algebra must clearly have even dimension, and if the dimension is two the algebramust be abelian to respect unimodularity. To find a non-abelian R -matrix we must thereforeconsider at least the case of rank four. Luckily the real quasi-Frobenius Lie algebras of dimensionfour were classified in [59], and the five unimodular ones (Corollary 2.5 in [59]) are listed intable 4. The task of finding all R -matrices of rank four which solve the CYBE and lead to f Defining Lie brackets R – h ⊕ R [ e , e ] = e r , − ⊕ R [ e , e ] = e , [ e , e ] = − e r ′ , ⊕ R [ e , e ] = − e , [ e , e ] = e n [ e , e ] = − e , [ e , e ] = e Table 4: The four-dimensional real unimodular quasi-Frobenius Lie algebras. In all cases the2-cocycle can be taken as ω = e ∧ e + e ∧ e , where e i denotes the dual basis of f ∗ .a deformation of the AdS × S string with a proper supergravity background is thereforereduced to finding all inequivalent embeddings of these subalgebras in so (2 , ⊕ so (6). The mostinteresting problem is to find the embedding of the non-abelian algebras in so (2 , so (2 ,
4) must be embeddable inone of the maximal solvable subalgebras of so (2 , This was done for sl (2) and sl (3) in [57]. The extension to so (2 , ⊕ so (6) is essentially trivial and amounts to adding in commuting generators from so (6) in such a way that the commutation relations of the algebra are preserved. so (2 , s and s of dimension 9 and 8respectively. It is most convenient to write them using the conformal form of the so (2 ,
4) algebra,with dilatation generator D , translations and special conformal generators p i , k i ( i = 0 , . . . J ij . They are related to the form of so (2 ,
4) in (A.1)with b K ij kl = − δ k [ i δ lj ] by p i = P i + J i , k i = − P i + J i , D = P , (5.7)and the non-vanishing commutators are[ D, p i ] = p i , [ D, k i ] = − k i , [ p i , k j ] = − η ij D + 2 J ij , (5.8)[ J ij , p k ] = 2 η k [ i p j ] , [ J ij , k k ] = 2 η k [ i k j ] , [ J ij , J kl ] = η ik J jl − η jk J il − η il J jk + η jl J ik . The metric on the Lie algebra is given by tr( DD ) = 1, tr( p i k j ) = − η ij , tr( J ij J kl ) = − η i [ k η l ] j .The two non-abelian maximal solvable subalgebras of so (2 ,
4) then take the form s = span( p i , J − J , J − J , J , J , D ) , s = span( p + p , p , p , J − J , J − J , J , J − D, k + k + 2 p ) , (5.9)up to automorphisms. Our task is reduced to finding all embeddings of the non-abelian algebrasin table 4 in s and s . To simplify this problem further we will single out the element e inthis table and use automorphisms generated by elements of s ( s ) to simplify it as much aspossible. Using this freedom we can bring e to one of the following forms s : (1) e = p , (2) e = J − J , (3) e = p + J − J , (4) e = p , (5) e = p , (6) e = p + p , (7) e = p − p + J − J , (5.10) s : (1) e = p , (2) e = p + p , (3) e = ap + bp + J − J . (5.11)The rest is a straightforward if slightly tedious calculation. The results are summarized in tables5–8. Note that in writing these embeddings we have used automorphisms of the four-dimensionalsubalgebras which are not always inner automorphisms of so (2 , R -matrices. In Table 1 in the introduction we write thecorresponding R -matrices, R = e ∧ e + e ∧ e up to automorphisms. In Table 2 instead we listthe inequivalent, modulo inner automorphisms of so (2 , R -matrices. This is the result whichis interesting from the string sigma model perspective, since inner automorphisms correspond tofield redefinitions in the sigma model, i.e. coordinate transformations in target space. In Table3 we write down the bosonic isometries and the number of supercharges that each R -matrixpreserves. Given a generator t of the superalgebra g , the condition that it is preserved by the R -matrix is given by [ t, R ( x )] = R ([ t, x ]) , ∀ x ∈ g . (5.12)Most of these R -matrices all have a form which suggests that they should correspond tonon-commuting TsT-transformations , in the sense that they involve sequences of T-dualitiesalong non-commuting directions. All but the last three R -matrices in table 1 have the form R = a ∧ b + c ∧ d , (5.13)where [ a, b ] = [ c, d ] = 0 and c, d generate isometries of the corresponding background. It isnatural to conjecture that such R -matrices correspond to two successive TsT-transformations, The reason for picking e is that it always arises as a commutator of two other elements. Since the last threegenerators in s or s are never generated in commutators, they do not appear in e . Here we use TsT in a generalized sense, where we can involve also non-compact directions a, b and the second using isometries c, d . Note that unlike in standardapplications of TsT-tranformations, e.g. [62], the pairs of isometries a, b and c, d do not commutewith each other. This means that after the first TsT is implemented, it is necessary to makea change of coordinates in order to realize the isometries of the second TsT transformationas shift isometries. We will confirm this in section 6, when we will check in some examplesthat the deformed backgrounds are indeed equivalent to such sequences of TsT-transformations.These considerations suggest a very simple picture for how TsT-transformations are interpretedat the level of the R -matrix: the TsT-transformation involving isometries a, b should be simplyimplemented by adding a term a ∧ b to the R -matrix . Notice that the number of free parametersentering the definitions of the R -matrices (plus the overall deformation parameter) does notneed to be equal to the number of TsT-transformations implemented. In fact, the numberof parameters could be reduced in some cases, if they can be reabsorbed by means of fieldredefinitions. In other cases one might have more parameters than expected, which suggests thepossibility of applying TsT-transformations on linear combinations of the isometric coordinates.The structure of the last three R -matrices in table 1 is different, and one observes that now a, c generate isometries. However, one can check explicitly that the background correspondingto R , for example, is self-dual (up to field redefinitions) under a TsT-transformation involving a, c . This example is particularly instructive because it can be embedded in so (2 , a, c , whichsuggests that backgrounds corresponding to the algebra n are not of TsT-type. Note that n is the only algebra considered which is not the direct sum of a three-dimensional algebraand a commuting generator. One possibility is that non-abelian T-duality of the correspondingsubalgebra should instead play a role in the interpretation of these backgrounds. A hint towardsthis direction comes from the results of [63], where it was shown that a conformal anomaly isencountered when implementing non-abelian T-duality on a subalgebra, unless all generatorshave vanishing trace. In the case of the adjoint representation this condition is precisely thatof unimodularity of the corresponding subalgebra. h ⊕ R e e e e . p J − J p + p p . p p + J − J p + p p . p p + J − J p + p p + J − J . p − ( J − J ) p + J − J p + p k + k + 2 p − J Table 5: Embeddings of h ⊕ R in so (2 ,
4) up to automorphism.Let us now consider the case of higher ranks, which can only be six or eight. We havenot done a systematic study for the case of rank six R -matrices. One would first need toidentify all 6-dimensional subalgebras of s and s , and check which of them are unimodular andquasi-Frobenius. We have found that the subalgebra of s generated by { p i , J , J } has bothproperties. It is straightforward to find the 2-form ω that solves the cocycle condition (5.4), andinvert it to find the corresponding R -matrix. For particular choices of the free parameters thiscan be written e.g. as R = p ∧ p + p ∧ p + J ∧ J .We have also checked that there is no 8-dimensional subalgebra which is at the same timeunimodular and quasi-Frobenius. Therefore there is no rank eight R -matrix which produces a It is easy to check that this is compatible with the CYBE, since a, b are isometries and satisfy (5.12). Note that this is consistent with our above proposal on how to interpret the action of TsT at the level of the R -matrix; in fact, in this case the addition of the term a ∧ c to R can be removed by an inner automorphism of so (2 , a, c can be chosen to be p , p + p . We thank Arkady Tseytlin for pointing this reference out to us. , − ⊕ R e e e e . − D − J J − J p p + p . J p − p p + p p . J p − p p + p J (4 . ) D + 2 J p p + p − Table 6: Embeddings of r , − ⊕ R in so (2 ,
4) up to automorphism. The last case is an embeddingof r , − which does not extend to an embedding of r , − ⊕ R . It is the only case where thishappens and included only since it is relevant for constructing all non-abelian R -matrices of so (2 , ⊕ so (6). r ′ , ⊕ R e e e e . J p p p + p . p + J p p p + p . p + J p p p . J p p p . p + J p p p . J p p p . J p p J Table 7: Embeddings of r ′ , ⊕ R in so (2 ,
4) up to automorphism.background that solves the supergravity equations of motion. It is in fact easy to check that s (which is 8-dimensional) is quasi-Frobenius but not unimodular. To identify all 8-dimensionalsubalgebras of s (which is 9-dimensional), we first define e = P j =1 λ j e j to be the generatorwhich we want to remove, where e j are the generators of s . Then for a generic element X ∈ s we define its component perpendicular to e as X ⊥ = X − P ( X ), where P projects along e .Then the condition to have a subalgebra is P [ X ⊥ , Y ⊥ ] = 0 , ∀ X, Y ∈ s . These equations givetwo possible solutions, depending on some unconstrained parameters( a ) e = λ J + λ J + λ D , ( b ) e = λ ( p − p ) + λ ( J − D ) . (5.14)In the case ( a ) we find that the subalgebra is unimodular if λ = 0 and λ = 2 λ . However,for this choice it is not quasi-Frobenius—the cocycle condition gives a 2-form of rank six. In thecase ( b ) the subalgebra is not unimodular for any choice of λ , λ . In this section we give a brief discussion on the η -model backgrounds generated by solutions ofthe CYBE ( c = 0), when we restrict R to act only on the bosonic subalgebra. In most cases aconvenient parameterisation of the group element g = g a · g s ∈ SO (2 , × SO (6) is g a = exp (cid:0) x i p i (cid:1) · exp (log z D ) , (6.1) We define P ( X ) = e STr( Xe ∗ ), where e ∗ is a dual to e , STr( ee ∗ ) = 1. We can take it as e ∗ = P j =1 λ j || λ || e j ,where || λ || = P j =1 λ j and e j are the duals of the generators in the basis such that STr( e i e j ) = δ ji . In both cases ( a ) and ( b ) one needs to choose carefully a basis for the 8-dimensional subalgebra, in such away that the generators are linearly independent and non-degenerate for generic choices of the remaining λ j . Away to do it is to pick an orthogonal basis, and normalise the vectors such that they can be degenerate only if λ j = 0 ∀ j . e e e e . p J − J p + p p . p p + J − J p + p p . p + p + J − J p + J − J p + p p Table 8: Embeddings of n in so (2 ,
4) up to automorphism.where p i , D are the generators defined in (5.7). Here we will be interested only on deformationsof AdS, so we will not need to specify the parameterisation that we use for g s on the sphere. Inthis coordinate system the undeformed metric takes the familiar form ds η =0 = η ij dx i dx j + dz z + ds s . (6.2)Because of our restriction on R , it is enough to look at the action of the operators O ± on thebosonic subalgebra. They take a block form (cid:18) ( O ± ) bca ( O ± ) ba (cid:19) , (6.3)since O ± P (0) = P (0) . All the information about background fields of the deformed model canbe extracted by studying just the block ( O + ) ba —or in other words P (2) O + P (2) . Notice that theresults for ( O − ) ba are simply obtained by changing the sign of the deformation parameter η .The dilaton of the deformed model is easily obtained by computing the determinant of ( O + ) ba e φ = (det O + ) − / . (6.4)The rest of the background fields are written in terms of ( O − ) ba —the inverse of the block( O + ) ba . The vielbein components for the deformed model are E a = ( O − ) ab e b , (6.5)where e a is the bosonic vielbein of the undeformed background, related to the Maurer-Cartanform as g − dg = e a P a + ω ab J ab . (6.6)The spacetime metric of the deformed background is then straightforwardly obtained, ds = η ab E a E b . The B -field can be extracted immediately from the action of the bosonic σ -model,and it reads as B = dX n ∧ dX m B mn = ( O − − ) ab e a ∧ e b , (6.7)where it is assumed that indices are raised and lowered with η ab . To get the Ramond-Ramondfields we first need to consider the local Lorentz transformation given by M in (2.10) and writeits action on spinors (Ad h ) ˆ β ˆ α = exp[ − (log M ) ab Γ ab ] ˆ β ˆ α , (6.8)where here we have introduced a basis for 32 ×
32 Gamma-matrices . The RR fields are obtainedby solving the equation (note that (1.2) simplifies considerably for R -matrices of the bosonicsubalgebra) (Γ a F a + Γ abc F abc + · Γ abcde F abcde )Π = e − φ Ad h ( − )Π (6.9) We recall that P (0) and P (2) are projectors on the subspaces spanned by the generators J ab and P a respectively.A useful matrix realisation of the algebra generators can be found in [22]. Here we identify P a = P a , and J ab = − J ab , where P a , J ab are the generators used in [22]. For a convenient basis see [22]. (1 − Γ ) is a projector and ( − )Π encodes the 5-form flux of the undeformedmodel. The various components of F ’s are found by multiplying the above equation by therelevant Gamma-matrix Γ a ...a m +1 and then taking the trace. This computation yields the F ’s expressed with tangent indices, which are translated into form language by F (2 m +1) = m +1)! E a m +1 ∧ . . . ∧ E a F a ...a m +1 .In the rest of this section we present some backgrounds solving the standard supergravityequations which we have derived by using the above procedure. We work out one example foreach of the 4-dimensional non-abelian subalgebras in table 4.In section 5 we have argued that the R -matrices related to the subalgebras h ⊕ R , r ′ , ⊕ R , r , − ⊕ R should produce backgrounds which can be obtained by sequences of TsT-transformationsstarting from AdS × S . We check this explicitly for the backgrounds that we have derived,where we follow the conventions of [32] for the T-duality rules [64, 65, 66]. Because the isome-tries of the first TsT do not commute with those of the second one, we will see that before doingthe last step it is necessary to implement a coordinate transformation, which realizes the secondpair of isometries as shifts of the corresponding coordinates. Let us mention that since we havechosen to have just one overall deformation parameter η (i.e. we fix some free parameters in thedefinitions of the possible R -matrices), the shifts of the two TsT-transformations are related toeach other. This does not need to be true for generic cases. h ⊕ R Let us choose the R -matrix (this corresponds to R in table 1 with x ↔ x ) R = ( J + J ) ∧ ( p + p ) + p ∧ p , (6.10)which preserves 4 bosonic isometries p , p , p + p , p − p − J + J ) , (6.11)and 8 supercharges. Clearly, it is convenient to introduce lightcone coordinates x ± = x ± x ,since a shift of x + will correspond to an isometry. The spacetime metric that we obtain is ds = z − (cid:18) η z (cid:19) − (cid:0) η z − x − dx − (2 dx − x − dx − ) + dx + dx (cid:1) + − dx − dx + + dz z + ds s . (6.12)The dilaton depends only on the z -coordinate, while the B -field also on x − e φ = (cid:18) η z (cid:19) − / , B = 2 η ( dx − x − dx − ) ∧ dx (4 η + z ) . (6.13)The RR-fluxes turn out to be quite simple F (5) = (1 + ∗ ) 2 dx − ∧ dx + ∧ dx ∧ dx ∧ dzz ( z + 4 η ) , F (3) = 4 ηz (2 x − dx − dx + ) ∧ dx − ∧ dz. (6.14)In order to show that this background can be obtained by a sequence of TsT-transformations,we start from the deformed background and show that we can reach the undeformed AdS × S For F (5) it is enough to look at half of the components, e.g. F bcde , and construct the corresponding form f (5) . Then F (5) = (1 + ∗ ) f (5) , such that F (5) = ∗ F (5) .
20y TsT-transformations. We will write T ( x i ) to indicate that we apply T-duality along theisometric coordinate x i , and denote by ˜ x i the dual coordinate. In this case we need to do thesequence T ( x ) , x → x − η ˜ x , T (˜ x ) , T ( ψ ) , w + → w + − η ˜ ψ, T ( ˜ ψ ) , (6.15)where we need to redefine the coordinates in the 013 space x + = 2( ψ w − + w + ) , x − = 2 w − , x = − ψw − , (6.16)before applying the last TsT-transformation. Obviously, starting from AdS × S and applyingthese TsT-transformations backwards, we find the deformed background presented here. r ′ , ⊕ R In this case we can choose an R -matrix which involves generators along spacelike directions ( R in table 1) R = J ∧ p + p ∧ p . (6.17)It preserves 3 bosonic isometries J , p , p , (6.18)and no supercharges. It is more convenient to use the parameterisation g a = exp( ξJ ) · exp( rp + x p + x p ) · exp(log z D ) , (6.19)since ξ will be isometric. In the undeformed case ds η =0 = − ( dx ) + r dξ + dr + dx + dz z + ds s , (6.20)so that ( r, ξ ) are a radial and an angular coordinate in the 1 , ds = z − η (cid:0) r + 1 (cid:1) z ! − (cid:2) dr (cid:0) η r + z (cid:1) + r z dξ − η r drdx + dx (cid:0) η + z (cid:1)(cid:3) + dz − ( dx ) z + ds s (6.21)The dilaton and the B -field now depend on r and ze φ = η (cid:0) r + 1 (cid:1) z ! − / , B = 2 η r dξ ∧ ( dr + rdx ) z + 4 η ( r + 1) . (6.22)For the RR-fluxes we find F (5) = (1 + ∗ ) 4 r dx ∧ dr ∧ dξ ∧ dx ∧ dzz ( z + 4 η ( r + 1)) , F (3) = 8 ηz ( dx − rdr ) ∧ dx ∧ dz. (6.23)The sequence of TsT-transformations T ( x ) , ξ → ξ + 2 η ˜ x , T (˜ x ) , T ( x ) , x → x − η ˜ x , T (˜ x ) , (6.24)(where r = p x + x , ξ = arctan( x /x )) yields undeformed AdS × S .21 .3 r , − ⊕ R The R -matrix ( R in table 1 with x → x , x → x ) R = J ∧ p + 2 p ∧ p , (6.25)preserves 3 bosonic isometries J , p , p , (6.26)and no supercharges. As before, it is more convenient to parameterise the group element in adifferent way g a = exp( tJ ) · exp( ρp + x p + x p ) · exp(log z D ) , (6.27)so that t is an isometry. In the undeformed case we have the spacetime metric ds η =0 = − ρ dt + dρ + dx + dx + dz z + ds s , (6.28)while the defomation gives ds = z − − η (cid:0) ρ + 4 (cid:1) z ! − (cid:0) − ρ z dt − η ρdρdx + dx (cid:0) z − η (cid:1) + dρ (cid:0) z − η ρ (cid:1)(cid:1) + dx z + dz z + ds s . (6.29)The dilaton and the B -field depend on ρ and ze φ = (cid:18) − η (4 + ρ ) z (cid:19) − / , B = 2 η ρ dt ∧ (2 dρ − ρdx ) z − η (4 + ρ ) , (6.30)and the RR-fluxes are F (5) = − (1 + ∗ ) 4 ρ dt ∧ dρ ∧ dx ∧ dx ∧ dzz ( z − η (4 + ρ )) , F (3) = 8 η (2 dx + ρdρ ) ∧ dx ∧ dzz . (6.31)We can get back the undeformed AdS × S background by applying the sequence of TsT-transformations T ( x ) , t → t + 2 η ˜ x , T (˜ x ) , T ( x ) , x → x − η ˜ x , T (˜ x ) , (6.32)where x = ρ cosh t, x = ρ sinh t . n Let us consider the R -matrix ( R in table 1 with x ↔ x ) R = p ∧ p + ( p + p ) ∧ ( J + J ) (6.33)which preserves the 3 bosonic isometries p + p , p , p , (6.34)22nd 8 supercharges. The metric is given by ds = z − (cid:18) − η ξ − z (cid:19) − (cid:20) z dx − η ( dx + ) − dξ − (cid:0) η ξ − dξ − + 2 dx + (cid:0) z − η ξ − (cid:1)(cid:1)(cid:21) + dx + dz z + ds s , (6.35)where we preferred to redefine ξ − = 2 x − −
1. The dilaton and the B -field depend on ξ − and ze φ = (cid:18) − η ξ − z (cid:19) − / , B = η ( ξ − dξ − + 2 dx + ) ∧ dx z − η ξ − ) . (6.36)The RR-fluxes are F (5) = (1 + ∗ ) dξ − ∧ dx + ∧ dx ∧ dx ∧ dzz ( z − η ξ − ) , F (3) = 2 ηz (cid:0) ξ − dξ − − dx + (cid:1) ∧ dx ∧ dz. (6.37)We have checked that this background is self-dual (after field redefinitions) under a TsT-transformation involving p + p and p . If we view it as a deformation of AdS there areno other bosonic isometries at our disposal, so it appears that this background cannot be gen-erated by (bosonic) TsT-transformations. As remarked earlier, it would be very interesting tounderstand if it can be generated by applying non-abelian T-duality. We have derived the target space geometry of the η and λ -deformed type IIB supercoset stringsigma models. With this result we have checked that the λ -deformation leads to a (type II*)supergravity background, while in general the η -deformation only to a “generalized” one in thesense of [25, 30]. When this is the case, the sigma model is expected to be scale invariant butnot Weyl invariant, and therefore does not seem to define a consistent string theory. We haveidentified the (necessary and sufficient) condition for the η -model to have a standard supergravitybackground as target space. This is translated into an algebraic condition on the R -matrix,which we refer to as the unimodularity condition . It imposes strong restrictions on non-abelian R -matrices, and in fact all non-abelian R -matrices considered in previous works do not lead tosupergravity solutions.We have also analyzed the problem of finding all unimodular R -matrices which solve theCYBE for the bosonic subalgebra so (2 , ⊕ so (6) ⊂ psu (2 , | R -matrices for so (2 ,
4) has been given and we have showed that the onlyother non-abelian R -matrices in this case have rank six. We have argued that most of theseexamples should correspond to a sequence of non-commuting TsT-transformations and haveverified this explicitly in some cases. It should be possible to understand these deformationsin terms of twisted boundary conditions for the string just as in the standard TsT case [44].There are many similarities between the backgrounds we construct and that of Hashimoto-Itzhaki/Maldacena-Russo [67, 68] and the dual field theories are expected to be certain non-commutative deformations of N = 4 super Yang-Mills, see [69] and in particular [70].Many interesting open questions remain. It would be important to find all possible unimod-ular R -matrices of psu (2 , |
4) to have a complete list of Yang-Baxter deformations of
AdS × S with a string theory interpretation. A question is whether any of them are of the Jordanian type.It is particularly interesting to investigate whether it is possible to have unimodular R -matricesthat solve the MCYBE rather than the CYBE, to solve one of the puzzles of [22]. One could also23ry to give an interpretation to backgrounds generated by non-unimodular R -matrices; in manycases one can associate to them a formally T-dual model which does describe a string sigmamodel, so it is natural to wonder what these backgrounds correspond to. See [41] for some inves-tigations along these lines. It would be also interesting to clarify if these deformed models havea connection to non-abelian T-duality, in view of the similarities between our unimodularitycondition and the tracelessness condition of [63].Our results are also useful to make further progress in the case of the λ -model. In fact, wehave written the NSNS and RR background fields in terms of the Lie algebra operators whichare used to define the deformation procedure, and after picking a certain parameterisation forthe group element this enables to obtain their explicit form. This method is more efficient,albeit equivalent, to the ones used so far e.g. in [22, 26, 41]. One could then check the proposalof [27] for the background of the λ -deformed AdS × S × T string, and finally derive the onefor the AdS × S case. It would be interesting to understand whether there is room to modifythe definition of the λ -model, hence realising other possible deformations of the string. In fact,in the current status the λ -model is related through Poisson-Lie T-duality to the η -model basedon the MCYBE, but there is no known counterpart for deformations based on the CYBE. Acknowledgements
We would like to thank Arkady Tseytlin for useful discussions and helpful comments on a firstdraft of this manuscript. This work was supported by the ERC Advanced grant No. 290456. A Z -graded superisometry algebras In this appendix we review some facts about the relevant superalgebras and explain our notationand conventions. In [4] it was shown that for all cases of interest here the superisometryalgebra—which admits a Z -grading that extends the Z -grading of the bosonic subalgebra—can be written in the same form. The bosonic subalgebra is of the standard symmetric spaceform [ J ab , P c ] = 2 η c [ a P b ] , [ P a , P b ] = 12 b K abcd J cd , [ J ab , J cd ] = η ac J bd − η bc J ad − η ad J bc + η bd J ac . (A.1)Here a, b, c = 0 , . . . , J ab generate Lorentz-transformations and rotations while P a generatetranslations. Note that since the space is typically a product of factors J ab is block-diagonal withcomponents mixing different factors absent and this should be taken into account in interpretingthe last commutator above. In the case of RR backgrounds, i.e. no NSNS three-form flux, thecommutators involving the supercharges take the form (here and in the rest of the paper wespecialize to the type IIB case, but the type IIA case works in the same way)[ P a , Q I ˆ α ] = − i ( Q J b K JI γ a ) ˆ α , [ J ab , Q I ˆ α ] = −
12 ( Q I γ ab ) ˆ α , ( I, J = 1 , { Q α , Q β } = { Q α , Q β } = iγ a ˆ α ˆ β P a , { Q α , Q β } = ( γ a b K γ b ) ˆ α ˆ β J ab . (A.2)Here ˆ α = 1 , . . . , N where 2 N is the number of supersymmetries preserved by the background.For AdS × S ( psu (2 , | N = 16 and γ a ˆ α ˆ β are the standard 16 ×
16 symmetric Weyl blocks or We restrict our attention to models with only RR flux since these have certain simplifying features like Z -symmetry. AdS × S × T ( psu (1 , | ) N = 8 and for AdS × S × T ( psu (1 , | N = 4 and the gamma-matrices γ a ˆ α ˆ β involve an extraprojector to make them 8 × × Z automorphism acts as J ab → J ab , P a → − P a , Q → iQ , Q → − iQ . (A.3)We introduce projectors that split the generators T A = { P a , J ab , Q I ˆ α } according to their Z -grading as follows P (0) ( T A ) = J ab , P (1) ( T A ) = Q α , P (2) ( T A ) = P a , P (3) ( T A ) = Q α . (A.4)Finally b K AB appearing on the right-hand-side in (A.1) and (A.2) is the inverse of the Lie algebrametric defined by the supertrace Str( T A T B ) = K AB , T A = { P a , J ab , Q I ˆ α } , (A.5)e.g. 12 b K abef K ef,cd = 2 η a [ c η d ] b . (A.6)It can be expressed in terms of the geometry and fluxes of the corresponding symmetric spacesupergravity background as b K ab = η ab , b K abcd = − R abcd , b K ˆ αI ˆ βJ = i S ˆ αI ˆ βJ , (A.7)where R abcd and S IJ are the Riemann curvature and RR field strength bispinor respectively. Let us also note the relation b K abcd ( K γ cd ) ˆ α ˆ β = 8( γ [ a b K γ b ] ) ˆ α ˆ β . (A.8)Finally for operators acting on the Lie algebra (i.e. endomorphisms) M : g → g we defineits components in the following way M ( T C ) = T D M DC . (A.9)The transpose operator is defined with respect to the supertrace byStr( T A M ( T B )) = Str( M T ( T A ) T B ) , (A.10)or M AB = ( − AB ( M T ) BA M AB = K AC M C B (A.11)e.g. ( M T ) a ˆ β = K ˆ β γ M ˆ γ a , ( M T ) a,bc = 12 K bc,de M dea . (A.12)The supertrace of the Lie algebra operator M is given byStr( M ) = ( − A M AA = b K AB Str( T A M T B ) . (A.13)When we need to raise indices with b K AB we use the convention M A = M B b K BA . (A.14)To conclude, when writing generic commutation relations we write[ T A , T B ] = f CAB T C . (A.15) Note that our definition of K differs by a factor of i compared to the definition used in [4]. The curvature of
AdS is R abcd = 2 δ c [ a δ db ] while that of the sphere is R abcd = − δ c [ a δ db ] in our conventions. TheRR flux takes the form AdS n × S n × T − n : S ˆ αI ˆ βJ = − i ( σ ) IJ ( P γ ) ˆ α ˆ β , where the projector P , with Q I = Q I P , is given by 1 for n = 5, (1+ γ ) for n = 3 and (1+ γ ) (1+ γ )for n = 2. Useful results for the deformed models
In this appendix we collect some useful identities and expressions to obtain the results presentedin the main text. In the two deformed models, we can relate O T ± and O ± by λ − model : O T − = Ad − g O + , η − model : O T − ˆ d T = ˆ d T O + . (B.1)Using the definitions of O ± , we can express M defined in (2.10) in terms of O ± and projectorsonly λ − model : M = − Ω T + ( O T + ) − (1 − ΩΩ T ) = − Ω T + (1 − λ − )( O T + ) − P (2) ,η − model : M = O − − ( O − + 2 ηR g ˆ dP (2) ) = 1 − P (2) + 2 O − − P (2) , (B.2)which is useful to prove λ − model : Ad − h P (2) = O + (1 + Ω( O T + ) − ) P (2) = P (2) (1 + ( O T + ) − Ω) O + ,η − model : Ad − h P (2) = O + (2 P (2) − O − − P (2) . (B.3)Note that using the expression for M we can express A − in terms of A + as A − = M A + = ( A + + ( M − A (2)+ − Ω T A + + ( M + λ − ) A (2)+ . (B.4)The rest of this appendix is devoted to the two deformed models separately. B.1 λ -model The expressions for dA ± in (3.1),(3.2) can be rewritten as dA + = − { A + , A + } −
12 (1 − λ − ) O − ( { A (2)+ , A (2)+ } − λ { A (1)+ , A (1)+ } + 2 λ { A (2)+ , A (3)+ } ) , (B.5) dA − = 12 { A − , A − } + 12 (1 − λ − )( O T + ) − ( { A (2) − , A (2) − } − λ { A (3) − , A (3) − } + 2 λ { A (2) − , A (1) − } ) , (B.6)if we useΩ T { X, X } − { Ω T X, Ω T X } = (1 − λ − )( { X (2) , X (2) } − λ { X (1) , X (1) } + 2 λ { X (2) , X (3) } ) , (B.7)for X ∈ g , and the same for Ω but with X (1) and X (3) interchanged.To calculate the component T ˆ α of the torsion, we first need to compute the Lorentz-transformed spin-connection Ad h A (0)+ + dhh − . We do this by taking the exterior derivativeof both sides of the relation E (2) = Ad h A (2) − , from which we find the equation0 = { Ad h A (0)+ + dhh − − A (0)+ , E (2) } + λ (1 − λ − ) P (2) Ad h ( O T + ) − Ad − h { E (2) , E (1) } + { E (1) , Ad h P (1) M E (2) } − iλ { E (3) , P (3) M E (2) } − iλP (2) M T { E (2) , E (3) }− { Ad h P (0) M E (2) , E (2) } −
12 Ad h { P (1) M E (2) , P (1)
M E (2) } − λ { P (3) M E (2) , P (3)
M E (2) }−
12 (1 − λ − ) P (2) Ad h ( O T + ) − Ad − h ( { E (2) , E (2) } + 2 λ { E (2) , Ad h P (1) M E (2) } ) − P (2) M T { E (2) , E (2) } − λ P (2) M T { E (2) , P (3) M E (2) } , (B.8)26here we used (3.3) and (B.6). This equation determines Ad h A (0)+ + dhh − completely: this isobvious for the terms involving fermionic vielbeins, while for the terms involving E a it followsfrom symmetry in the same way that the condition T abc = 0 determines the spin connectionΩ abc . Using the algebra (A.1), (A.2) as well as (B.3) the result is[ dhh − + Ad h A + ] ab = − Ω ab + 12 E c H abc + 2 i ( E γ [ a ) ˆ α (Ad h M ) ˆ α b ] . (B.9)Here we have used the fact, which will be proven below, that the expression that we find H abc = 3[Ad h M ] [ ab,c ] + 3 iM ˆ α a [Ad h ] b | d | γ d ˆ α ˆ β M ˆ β c ] , (B.10)is equivalent to the one in (3.8). In fact, if we calculate H = dB using the first definition for B in (3.15) we find H = dB = 13 (1 − λ − ) − (cid:0) Str(Ω A − ∧ Ω A − ∧ Ω A − ) − Str( A − ∧ A − ∧ A − ) (cid:1) −
12 (1 − λ − ) − Str( A + ∧ (Ω { A − , A − } − { Ω A − , Ω A − } ))= − Str(( A (0)+ + A (0) − ) ∧ A (2) − ∧ A (2) − ) + λ Str( A (2)+ ∧ A (3) − ∧ A (3) − ) −
12 Str( A (2) − ∧ { A (1) − , A (1) − + 2 λA (1)+ } )=Str( E (2) ∧ E (1) ∧ E (1) ) − Str( E (2) ∧ E (3) ∧ E (3) ) − Str( P (0) Ad h M E (2) ∧ E (2) ∧ E (2) ) − Str( E (2) ∧ P (1) Ad h M E (2) ∧ P (1) Ad h M E (2) )= − i E a E γ a E + i E a E γ a E + 13! E c E b E a H abc , (B.11)with H abc given by (B.10). On the other hand, if we start from B given in the second lineof (3.15), we find a result which is mapped to the previous one by the replacements A − ↔ A + ,Ω ↔ Ω T and A (3) ↔ A (1) . This leads to the same form of H except now with H abc given by (3.8),which proves the equivalence of the two expressions. Let us also remark that this computationshows that the NSNS three-form superfield H = dB satisfies the correct superspace constraints.In order to check that the dilatinos in (3.10),(3.14) are in fact the spinor derivatives of thedilaton φ , we start from (3.16) and compute dφ = −
12 STr( O − − Ad − g d Ad g ) = − b K AB STr( T A O − − [ g − dg, T B ])= − b K AB STr( T A O − − [ O − A − , T B ])= − b K AB STr( T A O − − [ A − , T B ]) + 12 b K AB STr( T A O − − Ad − g [Ω A − , Ad g T B ])= − b K AB STr( T A O − − [ A − , T B ]) + 12 b K AB STr( T A Ad g O − − Ad − g [Ω A − , T B ])= − b K AB STr( T A O − − [ A − , T B ]) + 12 b K AB STr( T A Ad g ( O T + ) − [Ω A − , T B ])= − b K AB STr( T A O − − [ A − , T B ]) + 12 b K AB STr( T A Ω O − − Ad − g [Ω A − , T B ])= − b K AB STr( T A O − − [ A − , T B ]) + 12 b K AB STr( T A Ω O − − Ω T [Ω A − , T B ])+ 12 (1 − λ − ) λ b K AB STr( T A Ω O − − P (2) [Ω A − , T B ]) , (B.12)where we used (A.13) and in the last step we inserted 1 = 1 − ΩΩ T +ΩΩ T = (1 − λ − ) P (2) +ΩΩ T .It is easy to see that the A (0) − -terms cancel, as they must since they transform as a connection.27 .2 η -model The expressions for dA ± in (4.1), (4.3) can be rewritten as dA + = 12 { A + , A + } − cη { ˆ d T A + , ˆ d T A + } + ( O − − (cid:16) { A (2)+ , A (3)+ } + ˆ η { A (1)+ , A (1)+ } (cid:17) + η O − R g { A (2)+ , ˆ d T A (2)+ } , (B.13) dA − = 12 { A − , A − } − cη { ˆ dA − , ˆ dA − } + ( O − − − (cid:16) { A (2) − , A (1) − } + ˆ η { A (3) − , A (3) − } (cid:17) − η O − − R g { A (2) − , ˆ dA (2) − } , (B.14)where we have rewritten e.g. the last term in the expression for dA + as η O − R g { A (2)+ , ˆ d T A (2)+ } + (1 − O − ) (cid:16) { A + , A + } − cη { ˆ d T A + , ˆ d T A + } − { A (2)+ , A (3)+ } − ˆ η { A (1)+ , A (1)+ } (cid:17) . (B.15)As in the case of the λ -model, to calculate the component T ˆ α of the torsion we must first findthe Lorentz-transformed spin connection Ad h A (0)+ − dhh − (note the difference in sign betweenthe two models). We use the same method explained in the previous subsection and we find[Ad h A (0)+ − dhh − ] ab = Ω ab − E c H abc + 2 i ˆ η ( γ [ a E ) ˆ α [Ad h M ] ˆ α b ] , (B.16)where we write the components of H abc as H abc = 3[Ad h M ] [ ab,c ] − i ˆ η [Ad h ] [ a | d | M ˆ α b γ d ˆ α ˆ β M ˆ β c ] . (B.17)This expression is equivalent to the one in (4.7), which is easy to verify by a calculation similarto the one performed for the λ -model: the B -field written as in the first way of (4.13) leads to H abc of the form (4.7), while the second way leads to the form in (B.17). The same calculationalso shows that the remaining components of the superform H satisfy the standard supergravityconstraints.If we take (4.14) as the definition of the dilaton in the case of the η -model we find dφ = − η b K AB STr( T A ˆ d T O − R g [ g − dg, T B ]) + 12 η b K AB STr( T A R g ˆ d T O − [ g − dg, T B ])= − η b K AB STr( T A ˆ d T O − R g [ A + , T B ]) − b K AB STr( T A O − [ A + , T B ]) − η b K AB STr( T A O − [ R g ˆ d T A + , T B ]) − η b K AB STr( T A ˆ d T O − R g [ R g ˆ d T A + , T B ])= − b K AB STr( T A O − [ A + , T B ]) + 12 cη b K AB STr( T A ˆ d T O − [ ˆ d T A + , T B ]) − η b K AB STr( T A ˆ d T O − R g [ A + , T B ]) − η b K AB STr( T A O − R g [ ˆ d T A + , T B ])+ 12 η b K AB STr( T A R g [ ˆ d T A + , T B ]) , (B.18)where we used the (M)CYBE (1.7) in the last step. 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