Supergravity solution-generating techniques and canonical transformations of σ-models from O(D,D)
aa r X i v : . [ h e p - t h ] F e b Supergravity solution-generating techniquesand canonical transformations of σ -modelsfrom O ( D, D ) Riccardo Borsato and Sibylle Driezen
Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela, Spain [email protected], [email protected]
Abstract
Within the framework of the flux formulation of Double Field Theory (DFT) we employ a gen-eralised Scherk-Schwarz ansatz and discuss the classification of the twists that in the presence ofthe strong constraint give rise to constant generalised fluxes interpreted as gaugings. We analysethe various possibilities of turning on the fluxes H ijk , F ij k , Q ijk and R ijk , and the solutions forthe twists allowed in each case. While we do not impose the DFT (or equivalently supergravity)equations of motion, our results provide solution-generating techniques in supergravity whenapplied to a background that does solve the DFT equations. At the same time, our resultsgive rise also to canonical transformations of 2-dimensional σ -models, a fact which is inter-esting especially because these are integrability-preserving transformations on the worldsheet.Both the solution-generating techniques of supergravity and the canonical transformations of2-dimensional σ -models arise as maps that leave the generalised fluxes of DFT and their flatderivatives invariant. These maps include the known abelian/non-abelian/Poisson-Lie T-dualitytransformations, Yang-Baxter deformations, as well as novel generalisations of them. ontents O ( d, d ) parametrisation of the twist U . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Twist ansatz for orbits with H -flux . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ∅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 F, Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6
F, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.7
Q, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8
F, Q, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.9 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.10 H, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.11
F, H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.12
H, Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.13
F, H, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.14
H, Q, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.15
F, H, Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
B.1 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
C On the O ( d, d ) parametrisation of the twist 45D Details on RR fields and type II 46 DFT equations of motion 52F Other ansatze used for orbits with H -flux 52 An important concept in physics is that of symmetry. Symmetries are powerful tools to constrainthe form of solutions of physical theories, and they may be used as a guiding principle to constructnew physical models. Certain symmetries are particularly useful because they connect physicalsolutions as well as different theories that are at first sight unrelated. A prominent example isT-duality in string theory, which provides such a connection between strings on backgroundswith very different geometry [1, 2]. In this paper we will employ generalised notions of T-duality that go beyond the case of reduction on a torus implemented by Buscher’s rules. Whennecessary we will refer to the latter case as “abelian” T-duality, to distinguish it from the “non-abelian” T-duality (NATD) transformation of [4, 5, 6]. NATD may be viewed as a generalisationwhere a set of non-abelian isometries of the string background are dualised to give rise to a newsolution of the low-energy (super)gravity equations of motion, modulo a known anomalous-freecondition [7, 8]. The transformation can be implemented by a Busher-like gauging procedureof the initial isometries, similar to the abelian case. Global issues of NATD still need to beunderstood [6] and currently the transformation is viewed as a solution-generating techniquerather than a fully-fledged symmetry of string theory [9, 10]. Similar comments apply to a furthergeneralisation of NATD that goes under the name of “Poisson-Lie” T-duality (PLTD). In thiscase there is still a notion of dualisation of a symmetry algebra, but the symmetry does not needto be realised as an isometry of the initial background [11, 12]. This advantage puts the originaland dual models on an equal footing at the expense of losing the interpretation of the gaugingprocedure. All these generalised notions of T-duality transformations were recently shown toadmit α ′ -corrections that promote them to solution-generating techniques in the bosonic andheterotic strings at least to 2 loops (first order in α ′ ) [13, 14, 15].Double Field Theory (DFT) [16, 17, 18, 19] is an attempt to make (abelian) T-dualitymanifest at the level of the low-energy action, at the expense of doubling the coordinates x m ofthe D -dimensional target-space by pairing them with a set of “dual” coordinates ˜ x m in X M =(˜ x m , x m ). Diffeomorphisms, B-field gauge transformations and T-duality maps are combinedinto the O ( D, D ) group, defined by the matrices O M N that leave the O ( D, D ) metric η MN invariant η MN = (cid:18) δ mn δ mn (cid:19) , (1.1)meaning that O M P O N Q η P Q = η MN . The action of DFT is manifestly invariant under constant O ( D, D ) transformations of the coordinates X M and of the dynamical fields of the theory, i.e. thegeneralised metric H MN and the generalised dilaton d , where the latter is an O ( D, D ) scalar.The DFT action reduces to the low-energy effective action of the string when constraining thefields to depend only on the physical coordinates x m , also known as the “strong constraint”. Byrelaxing the strong constraint DFT may also be viewed as a way to go beyond supergravity toprovide a description of backgrounds that are locally non-geometric. Nevertheless, in this paperwe will simply use DFT as a convenient O ( D, D )-covariant rewriting of (super)gravity, and thusthe strong constraint is always assumed. We will use in particular the so-called “flux formulation”of DFT [20] where the dynamical fields are the generalised dilaton d and a generalised vielbein Recently a notion of “T-duality” for point particles was introduced in [3]. AM for the generalised metric, and where the equations of motion are written exclusively interms of the generalised fluxes F ABC and F A and their (flat) derivatives. (Super)gravity solution-generating techniques — One motivation of our work is thestudy and possibly the classification of solution-generating techniques in (super)gravity, wherewe focus on those that admit a description in terms of O ( D, D ). In this case we assume thatthe starting point is a (super)gravity solution, say with Neveu-Schwarz-Neveu-Schwarz (NSNS)fields G mn , B mn , φ for the metric, Kalb-Ramond field and dilaton. The question is whether itis possible to construct a map from these fields to a new set of G ′ mn , B ′ mn , φ ′ that also give riseto a (super)gravity solution. Our strategy is to rewrite the D -dimensional fields in terms of thedoubled fields of DFT, and demand that G mn , B mn , φ and G ′ mn , B ′ mn , φ ′ give rise to the samegeneralised fluxes and flat derivatives F = F ′ , ∂ F = ∂ F ′ . This is a sufficient condition to haveagain a (super)gravity solution, and it is the mechanism that applies also for the generalisedT-duality transformations. As remarked, we are interested in classifying solution-generatingtechniques in (super)gravity rather than the actual (super)gravity solutions, and for this reasonat no point we need to impose the DFT equations of motion. Canonical transformations of σ -models — Another motivation of our work is theclassification of canonical transformations of 2-dimensional σ -models, see [21] for a similardiscussion. Consider a σ -model S = R d σ L whose Lagrangian can be put in the form L = G mn ˙ x m ˙ x n − G mn x ′ m x ′ n − B mn ˙ x m x ′ n , which is the Polyakov action in conformal gauge.Introducing momenta p m conjugate to x m and going to the first-order formalism, the Lagrangianis L = p m ˙ x m − H with the Hamiltonian H = Ψ M H MN Ψ N , where one has Ψ M ≡ ( p m , x ′ m ) and H MN ≡ (cid:18) G mn − ( G − B ) mn ( BG − ) mn G mn − ( BG − B ) mn (cid:19) , (1.2)which coincides with the known parametrisation commonly used for the generalised metric ofDFT. Canonical Poisson brackets for x m and p m translate into Poisson brackets for Ψ M , andredefining the phase-space variables as Ψ A = E AM Ψ M via a generalised vielbein E AM for thegeneralised metric H MN , one has { Ψ A ( σ ) , Ψ B ( σ ) } = −F ABC ( σ )Ψ C ( σ ) δ + η AB ∂ σ δ ( σ − σ ) , (1.3)where the generalised fluxes F ABC appear again. Therefore when two different σ -models admita rewriting of the phase-space variables in terms of Ψ A and Ψ ′ A respectively, and when they giverise to the same generalised fluxes F ABC when computing the Poisson brackets, the two σ -modelsare related by a canonical transformation — possibly up to zero modes . This last remark is a We are ignoring the overall string tension. A dot denotes derivatives with respect to the worldsheet time τ , and a prime with respect to the worldsheet spatial coordinate σ . In this case we limit the discussion to theclassical level, and we therefore ignore the dilaton. If we further identify the momenta with the spatial derivatives of a dual set of coordinates p m = ˜ x ′ m then X M = (˜ x m , x m ) and we have the action of Tseytlin’s double σ -model [22] S = R d σ (cid:0) ∂ X M ( η MN + Ω MN ) ∂ X N − ∂ X M H MN ∂ X N (cid:1) , where the matrix Ω MN ≡ (cid:18) − (cid:19) denotes thetopological term used also in [23]. In this discussion we want to look at the standard σ -model in the Hamiltonianformalism and we therefore refrain from going to the doubled σ -model. The Poisson brackets are { Ψ M ( σ ) , Ψ N ( σ ) } = η MN ( ∂ − ∂ ) δ − Ω MN ( ∂ + ∂ ) δ where we use theshorthand notation ∂ , = ∂ σ , , δ = δ ( σ − σ ), and Ω MN was defined in footnote 3. With appropriateboundary conditions ( ∂ + ∂ ) δ vanishes in the sense of distributions and the Poisson brackets can be rewrittenas { Ψ M ( σ ) , Ψ N ( σ ) } = η MN ∂ σ δ ( σ − σ ), which is the expression that is commonly used and the one we usein the text. On the line ( σ ∈ R ) test functions should be smooth functions with compact support, and on thecircle ( σ ∈ S ) they should be periodic. In the case of the circle one cannot drop terms with ( ∂ + ∂ ) δ if theboundary conditions for test functions allow for non-trivial winding. We refer to [21] for a discussion that usesthe original form of the Poisson brackets. spatial derivative of the coordinates x m appear in Ψ M , andtherefore with the above argument one is not able to claim that the zero-mode contribution tothe Poisson brackets remains invariant under the map. To use a uniform terminology throughoutthe paper we will use the term “solution-generating techniques” notwithstanding that we havein mind canonical transformations of σ -models as well. Integrable σ -models — Worldsheet (classical) integrability is a statement about the on-shell 2-dimensional σ -model, whose equations of motion can be put in the form of a flatnesscondition for an object known as the “Lax connection”. Canonical transformations provide on-shell identifications of the two σ -models, so that if one of the two models admits a formulation interms of a Lax connection, one can use the canonical transformation to construct the Lax connec-tion for the other σ -model, and argue integrability also in that case. Integrability has played animportant role in the understanding of the AdS /CF T correspondence [24], and much progresswas reached also in lower dimensional holographic examples. Being able to generate new super-gravity solutions (for example starting from AdS × S ) that retain worldsheet integrability giveshope of applying exact methods to non-maximally supersymmetric backgrounds. Additionally,at least when the transformations can be understood as deformations of the original model, thismotivates also the search of the corresponding transformation of the holographically-dual con-formal field theory (in this example N = 4 super Yang-Mills). Yang-Baxter (YB) deformationsare integrability-preserving solution-generating techniques that recently have been extensivelystudied from various points of view, and that started to be relevant for the superstring in [25, 26].At least the so-called “homogeneous” YB deformations can be interpreted as solution-generatingtechniques in the sense of this paper, since they leave the generalised fluxes invariant. The orig-inal formulation, the so-called “ η -model” or “inhomogeneous” YB-model constructed in [27, 28]will also be discussed later. The homogeneous YB deformations are based on solutions of theclassical YB equation on the algebra of isometries of the original background, and they can beapplied to generic isometric backgrounds [29] including the integrable AdS × S background,giving rise to generalisations of TsT transformations [30]. Preliminary proposals for the corre-sponding deformations of N = 4 super Yang-Mills were put forward in [31, 32]. As remarked, weare interested in classifying integrability-preserving transformations of 2-dimensional σ -models,rather than the actual integrable models, and for this reason at no point we need to impose theexistence of a Lax connection. α ′ -corrections — The low-energy effective action of the string has higher-derivative α ′ -corrections that admit an O ( D, D )-covariant formulation at least to 2-loop order. In particular,as at leading order in α ′ , the α ′ -corrected equations of motion of DFT may still be written inthe flux formulation in terms of the generalised fluxes and their flat derivatives exclusively.This observation can be used to extend the solution-generating techniques that we classify tohigher orders in α ′ . This strategy was first employed in [34] to obtain the first α ′ -correctionfor homogeneous YB deformations, and later for NATD and PLTD in [13, 14, 15]. As arguedin [15] the same methods can be applied to more general O ( D, D )-covariant solution-generatingtechniques, as the ones that we consider in this paper.
Summary of the paper —
Starting with section 2, we will employ an ansatz for thegeneralised vielbeins that is known in the literature as “generalised Scherk-Schwarz reduc-tions” [35, 36, 37]. Under this ansatz we identify a d -dimensional subspace of the full D -dimensional spacetime, so that we can discuss the more general case of solution-generatingtechniques acting non-trivially in d ≤ D dimensions. Our discussion will be local and we will This is a weak notion of integrability, since in general one should argue that the full tower of conserved chargesobtained from the monodromy matrix are in involution. Recently a tension was identified at 4 loops for the quartic-Riemann terms multiplied by ζ (3) that appear forthe (super)string [33]. constant generalised fluxes in d dimensions. This set-up encompasses the generalised T-dualities and theprominent integrable deformations so far considered in the literature. Subsequently, we discussthe O ( d, d ) parametrisation of the “twist” used in the reduction, as well as the constraints andthe redundancies that arise. We finally explain methods that will turn out to be useful to treatin particular cases with non-vanishing H -flux. In section 3 we present the classification of the“orbits”, namely the possibilities of turning on the different components of the generalised flux F IJK (usually denoted as H ijk , F ij k , Q ijk , R ijk ), and their “representatives”, i.e. the solutionsfor the twists that they allow. While we initially focus on the fields of the NSNS sector, insection 4 we discuss the Ramond-Ramond (RR) fields of the type II superstring as well. Wefinish in section 5 with conclusions and an outlook. In appendix A we collect our conventionson notation, in appendix B we give a brief recap on some aspects of DFT and gauged DFT thatare relevant for our discussion, in appendix C we discuss how to obtain the parametrisation ofthe twist that we use, in appendix D we give more details on the formulations to include the RRfields of type II, in appendix E we review the DFT equations of motion in the flux formulation,and in appendix F we report on other attempts we made to treat orbits with non-vanishing H -flux. We start by discussing a specific ansatz for the generalised vielbein of DFT. In turn this impliesa specific ansatz for the NSNS fields (metric, B-field and dilaton) of the class of backgroundsthat we consider. We refer to appendix B for some definitions and details that may be helpfulfor readers not familiar with DFT.
To be more general, we assume that our backgrounds are parametrised by coordinates x m =( ˙ x ˙ µ , y µ ), where we take m = 0 , . . . , D − µ takes d ≤ D values. In principle y µ may includethe time direction. In this splitting of coordinates ˙ x ˙ µ will play the role of “spectators” — wewill never specify the ˙ x ˙ µ dependence of the solution and ˙ x ˙ µ will not participate in the solution-generating technique. The interesting discussion will therefore involve only the coordinates y µ .From now on, we will use a boldface notation for coordinates and fields of the full D -dimensionalspacetime, to distinguish them from the coordinates and fields of the d -dimensional spaces. Wedo not make any assumptions regarding their global properties, in fact our discussion will bevalid only in local patches. We refer to [38] for a discussion on how to construct generalisedLeibniz parallelisable spaces from generalised Scherk–Schwarz uplifts of gauged supergravities.The ansatz we take for the generalised vielbein and generalised dilaton is [35, 36, 37] E AM ( x ) = ˙ E AI ( ˙ x ) U I M ( y ) , d ( x ) = ˙ d ( ˙ x ) + λ ( y ) . (2.1)The factorisation of the dependence on the coordinates is what plays a crucial role. We will usea dot for fields depending only on spectator coordinates. In the following we will distinguishbetween M, N, . . . indices and
I, J, . . . indices, and similarly for the boldface version, and thereason will be clarified in section 2.3. See also appendix A for a recap of our conventions onnotation. The matrix U is in general an element of O ( D, D ), and U and λ are usually called twists . Because of the role of spectators of the coordinates ˙ x ˙ µ , it is natural to take U of block4orm, not mixing ˙ µ and µ directions, and acting as the identity in the spectator block, so that U I M = (cid:18) a bc d (cid:19) , U I M = (cid:18) a bc d (cid:19) , (2.3)where a = (cid:18) D − d a (cid:19) , b = (cid:18) b (cid:19) , c = (cid:18) c (cid:19) , d = (cid:18) D − d d (cid:19) . (2.4)In the rest of this paper we will therefore work with the twist matrix U I M ∈ O ( d, d ) ⊂ O ( D, D ).This set-up is in fact known as “generalised Scherk-Schwarz compactifications” and used alsoin gauged DFT, see appendix B for a short recap of certain aspects. Let us also mention thatif we define the generalised metric H MN = E AM H AB E BN and parametrise it as in (1.2) interms of G mn and B mn (and similarly for the dotted fields) then equation (2.1) implies that M ≡ G − B is of the form M = ( ˙ M b + d ) − ( ˙ M a + c ) , (2.5)where ˙ M ≡ ˙ G − ˙ B . This is the known transformation of M under O ( D, D ) transformations.In other words the generalised Scherk-Schwarz ansatz is selecting a class of backgrounds withmetric, B-field (and dilaton) of a specific form.Because of the above ansatz, the Weitzenb¨ock connection constructed out of U I M is non-vanishing only when the
IJ K indices are of the type
IJ K , i.e. they are vector indices of the O ( d, d ) ⊂ O ( D, D ) subgroup. The generalised fluxes then become F ABC = ˙ F ABC + ˙ E A I ˙ E B J ˙ E C K F IJK , F A = ˙ F A + ˙ E A I F I , (2.6)where ˙ F ABC = 3 ˙ Ω [ ABC ] , ˙ Ω ABC = ˙ E A ˙ µ ∂ ˙ µ ˙ E BJ ˙ E CJ , ˙ F A = ˙ Ω BBA + 2 ˙ E A ˙ µ ∂ ˙ µ ˙ d , and F IJK = 3Ω [ IJK ] , F I = Ω J JI + 2 U I µ ∂ µ λ, Ω IJK = U I µ ∂ µ U J N U KN . (2.7)After defining the flat derivative ∂ A ≡ E AM ∂ M one finds that ∂ A F BCD = ˙ E A ˙ µ ∂ ˙ µ F BCD + ˙ E A I ˙ E B J ˙ E C K ˙ E D L U I µ ∂ µ F JKL ,∂ A F B = ˙ E A ˙ µ ∂ ˙ µ F B + ˙ E A I ˙ E B J U I µ ∂ µ F J . (2.8)We will now further restrict the assumptions behind our calculations, and present the constraintscoming from Bianchi identities. In general F IJK , F I may be y -dependent but in this paper we assume them to be constant.One motivation comes from interpreting the components of the 3-form generalised flux as the Equation (2.1) is rather natural and it is hard to imagine a more general ansatz if we want to exploit O ( D, D ). A possible generalisation of (2.3) is to replace the identity matrix in the spectator block with another O ( D − d, D − d ) element. The simplest example is U ˙ I ˙ M = (cid:18) e − γ ( y ) δ ˙ µ ˙ ν e γ ( y ) δ ˙ µ ˙ ν (cid:19) , (2.2)where γ playes the role of a generically y -dependent warping factor in front of G ˙ µ ˙ ν and B ˙ µ ˙ ν , see [36, 39]. O ( d, d )-covariant way by calculating the Jacobiator of the C-bracket (see appendix B for itsdefinition) of the generalised vielbeins (here the twists U ) [20]. In fact[[ U I , U J ] (C) , U K ] M (C) = [ F IJ L U L , U K ] M (C) = ( F IJ H F KLH − ∂ K F IJL + ∂ L F IJK ) U LM , (2.9)and the Jacobiator is Jac( U I , U J , U K ) M = ( −Z IJKL + ∂ L F IJK ) U LM . (2.10)On the strong constraint the Bianchi identities for the generalised fluxes imply Z IJKL = 0 (seeappendix B). If we in addition want to deal with the structure of a Lie algebra rather than the oneof a Courant algebroid, it is natural to take the generalised fluxes constant, ∂ L F IJK = 0, suchthat the Jacobiator vanishes. The assumption of constant fluxes (both F IJK and F I ) will simplifyconsiderably our calculations to identify the solution-generating techniques, and ultimately thisis the main motivation for this assumption. In particular it will be easier to find different twists( U, λ ) and ( U ′ , λ ′ ) that give rise to the same generalised fluxes F IJK = F ′ IJK , F I = F ′ I , and theirconstancy immediately implies that not only F ABC , F A but also their flat derivatives remainthe same, see (2.8). Let us remark that demanding the invariance of F IJK , F I is a strongercondition compared to the invariance of F ABC , F A , and relaxing this assumption may lead tointeresting generalisations on which we will comment in the conclusions.Let us now discuss the constraints coming from Bianchi identities for the generalised fluxes F IJK and F I . When these are constant the Bianchi identities (on the strong constraint) reduceto F [ IJ H F K ] LH = 0 , F K F IJK = 0 , F I F I − F IJK F IJK = 0 . (2.11)The first can be understood as the Jacobi identity for a Lie algebra, while the rest are constraintsinvolving also F I . We introduce Lie algebra generators T I = ( ˜ T i , T i ) so that [ T I , T J ] = F IJ K T K ,where indices are raised and lowered with the ad-invarant metric, η IJ = (cid:18) (cid:19) , (2.12)that corresponds to the pairing ⟪ T I , T J ⟫ = η IJ . We adopt the usual notation for the componentsof the 3-form flux F IJK when choosing upper or lower indices F ijk = H ijk , F ij k = F ij k , F ijk = Q ijk , F ijk = R ijk . Notice that such an algebra is unimodular F IJ J = 0. The commutationrelations of the “pre-Roytenberg algebra” spanned by T I , and that we denote by r , read[ T i , T j ] = F ij k T k + H ijk ˜ T k , [ T i , ˜ T j ] = Q ijk T k − F ikj ˜ T k , [ ˜ T i , ˜ T j ] = Q kij ˜ T k + R ijk T k . (2.13)The Lie algebra of the Drinfel’d double is obtained by setting H = R = 0 and keeping F and Q . When only R = 0 the above Lie algebra is known as quasi-Manin triple. In terms of all the In the context of gauged supergravities the constancy of the generalised fluxes is imposed to make sure thatthe ( D − d )-dimensional theory does not depend on the coordinates y . Our motivation is different and at least inprinciple there would be nothing wrong with taking F IJK , F I to be y -dependent. We follow the terminology usually employed in the literature, so that “Roytenberg algebra” is used in thecase of the Courant algebroid where F IJK are generically non-constant, while “pre-Roytenberg algebra” is just aLie algebra with a pairing η IJ that splits generators as T I = ( ˜ T i , T i ). ∅ )( F ) ↔ ( Q ) ( H ) ↔ ( R )( F, Q ) (
F, H ) ↔ ( Q, R ) (
F, R ) ↔ ( H, Q ) (
H, R )( F, Q, R ) ↔ ( F, H, Q ) (
F, H, R ) ↔ ( H, Q, R )( F, H, Q, R )Figure 1: Diamond representing the possible (sub)orbits. Arrows relate orbits connected byrigid T -transformations. The orbits at the four corners of the diamond are self-dual under rigid T -transformations.fluxes the Bianchi identities read F [ ijk H lm ] k = 0 , (2.14) F [ ijk F l ] km − H k [ ij Q l ] mk = 0 , (2.15) F ij k Q klm + H ijk R klm − Q [ ik [ l F j ] km ] = 0 , (2.16) Q k [ jl Q im ] k − F ik [ j R lm ] k = 0 , (2.17) Q k [ ij R lm ] k = 0 . (2.18) H ijk F k + F ij k F k = 0 , Q ijk F k − F ikj F k = 0 , R ijk F k + Q kij F k = 0 , (2.19) H ijk R ijk + 3 F ij k Q kij = 6 F i F i . (2.20)The more generic case is the one when all fluxes F ij k , H ijk , Q ijk , R ijk are non-zero. Neverthelessit is interesting to consider simpler cases in which only some of the fluxes are turned on. Wewill call “orbits” the classes that have a definite set of fluxes turned on. The elements of eachorbit (namely the possible solutions for the twist U ) will be called “representatives” of theorbits. Turning on fewer fluxes essentially corresponds to studying sub-orbits. The possiblecases are organised in the structure of a diamond, see Figure 1, where each node of the diamondcorresponds to a possible orbit and its dual orbit under a rigid T -transformation, connected byan arrow. In particular under a rigid T -transformation F ↔ Q and H ↔ R , see also the nextsection. The four corners of the diamond correspond to orbits that are self-dual under rigid T -transformations. The “empty” orbit ( ∅ ) with all fluxes vanishing contains as representativethe d -dimensional torus with no flux turned on. One possible representative of the ( F )-orbitis any background with isometries whose algebra has structure constants F ij k , so that in the( Q )-orbit one finds for example their non-abelian T-dual backgrounds. Other examples worthmentioning are the torus with H -flux in the ( H )-orbit, Wess-Zumino-Witten (WZW) models inthe ( F, H ) orbit, and Poisson-Lie (PL) symmetric models in the (
F, Q ) orbit. In section 3 wewill focus on each orbit and discuss the representatives that appear in each of them. Self-duality is in general only at the level of the orbit, but it may be that even some representatives areself-dual. .3 O ( d, d ) parametrisation of the twist U Under mild assumptions (see [45] and appendix C) the most general O ( d, d ) parametrisation ofthe twist U is U I M = (cid:18) β ij (cid:19) (cid:18) b jk (cid:19) (cid:18) ( ρ t ) kµ
00 ( ρ − ) kµ (cid:19) , (2.21)where β ij and b ij are antisymmetric ( β ij = − β ji , b ij = − b ji ) and ρ µj are components of thematrix ρ ∈ GL ( d ). From now on we will use the shorthand notation ρ iµ ≡ ( ρ − ) iµ , so that theplacing of indices will indicate if we use ρ or its inverse. Our motivation is to find different U, U ′ and λ, λ ′ that give rise to the same generalised fluxes F IJK , F I . Notice however that twists thatare related by the group of allowed gauge transformations — i.e. GL ( d ) diffeomorphisms andgauge transformations of the B-field in d dimensions — should not be considered as genuinelydifferent. We will call H geom this group. Therefore, rather than O ( d, d ), we should take U tobelong to the coset O ( d, d ) \ H geom . Under a GL ( d ) diffeomorphism one has the transformation U I M → U I N R N M where [46] R N M = (cid:18) r t r − (cid:19) with ( r − ) µν = ∂ µ V ν . (2.22)Therefore, we see that the action of GL ( d ) diffeomorphisms is all reabsorbed in the transforma-tion of ρ in (2.21), meaning ρ iµ → ρ iν ∂ ν V µ . This also implies that β and b in (2.21), while theymay be y -dependent, they must transform as scalars under GL ( d ) diffeomorphisms. This is thereason why we prefer to distinguish between i, j and µ, ν indices, as well as I, J and
M, N forthe double indices. Gauge transformations of the B-field in d dimensions are obtained by thetransformations U I M → U I N J N M where [46] J N M = (cid:18) x (cid:19) , with x µν = ∂ [ µ ξ ν ] . (2.23)It is easy to see that this transformation is completely reabsorbed just by the redefinition of b in (2.21) as b ij → b ij + ρ iµ ρ jν ∂ [ µ ξ ν ] . To conclude, when we say that U and U ′ must be differentwe mean that they must be different elements of the coset O ( d, d ) \ H geom , or in other wordsthat there exists no element h ∈ H geom such that U = U ′ h .An important feature in our discussion is that the ansatz (2.1) is invariant under the redefi-nition ˙ E AI → ˙ E AJ V J I , U I M → V J I U J M , (2.24)where V I J = (cid:18) D − d V I J (cid:19) , (2.25)and V J I is a constant O ( d, d ) matrix. This redefinition by V J I does not give rise to new back-grounds, because it signals only the redundancy of the freedom in the decomposition between˙ E and U in (2.1). However, under the rotation by V the generalised fluxes will be rotatedas F ′ IJK = V I U V J V V K W F UV W and F ′ I = V I U F U . This shows that different forms of thecommutation relations of the pre-Roytenberg algebra r may be in fact related by rigid O ( d, d )transformations, and therefore they need to be considered physically equivalent. Hence, orbitsare classified by equivalent classes , where the equivalence is given by a rigid O ( d, d ) change ofbasis. Nevertheless, in section 3 we will list them in the intuitive way of Figure 1, since thegrouping into equivalent classes may require additional knowledge on the pre-Roytenberg alge-bra. On the other hand, given a background with ˙ E AI and U I M , if we are able to find a U ′ I M up to a rigid O ( d, d ) transformation, then we can applya compensating transformation on ˙ E AI to make sure that F ABC , F A remain invariant.Another interesting rigid O ( d, d ) transformation, that now can give rise to new inequivalentrepresentatives for the twist U , is U I M → ˜ U I M = W I J U J M , W I J = (cid:18) D − d W I J (cid:19) , (2.26)where W I J is a constant O ( d, d ) matrix entailing an automorphism of the pre-Roytenberg al-gebra r . Notice that here we are not transforming ˙ E AI . In particular, the fact that W is anautomorphism, i.e. it satisfies W I I ′ W J J ′ W K K ′ F I ′ J ′ K ′ = F IJK , (2.27)guarantees that the transformation (2.26) leaves the generalised fluxes F IJK invariant. Whenstudying (super)gravity solution-generating techniques we additionally require the automor-phism to satisfy W I J F J = F I . In this case, if U I M is a representative, than the transforma-tion (2.26) generates a new representative ˜ U I M = W I J U J M . Let us point out that — since W ∈ O ( d, d ) — after this transformation the twist can still be parametrized as in (2.21). Whenviewing the transformation as a solution-generating technique, it will be important to under-stand in which cases W gives rise to gauge transformations in H geom . This may be complicatedto discuss in general, but in certain explicit examples particular conclusions can be made. For in-stance in the ( F )-orbit, modding out by GL ( d ) transformations requires the automorphism W tobe outer . Interestingly, when r is not semisimple outer automorphisms may also involve continu-ous parameters. For example, given a 2-cocyle ω : r → r (see section 3.2 for the definition), then W = exp( ζω ) is an automorphism of the algebra with ζ ∈ R a continuous parameter. In orderto avoid having W inner we have to impose that ω is not coboundary. Notice finally that using(2.24), the transformation by W in (2.26) may equally be seen as leaving the twist U invariantbut transforming the spectator contribution ˙ E AI of the background as ˙ E AI → ˙ E AJ W I J .In general O ( d, d ) transformations with “off-diagonal” components will reshuffle the typesof fluxes in complicated ways. Given the commutation relations (2.13), an O ( d, d ) redefinition T ′ I = h I J T J with V I J = (cid:18) δ ij + ( βb ) ij β ij b ij δ ij (cid:19) , (2.28)gives rise to the rotated structure constants [ T ′ i , T ′ j ] = F ′ ij k T ′ k + H ′ ijk ˜ T ′ k , [ T ′ i , ˜ T ′ j ] = Q ′ ijk T ′ k − F ′ ikj ˜ T ′ k , [ ˜ T ′ i , ˜ T ′ j ] = Q ′ kij ˜ T ′ k + R ′ ijk T ′ k , where H ′ ijk = H ijk + 3 F [ ij l b k ] l − b l [ i b j | m Q | k ] lm + b il b jm b kn R lmn ,F ′ ij k = F ij k − F [ ij l b m ] l β mk − H ijl β lk + 2 b l [ i Q j ] lk + 3 b l [ i b j | m Q | n ] lm β nk − b il b jm b pn R lmn β pk + b il b jm R lmk ,Q ′ ijk = Q ijk + 4 b l [ i Q m ] l [ k β j ] m + 3 b l [ i b m | p Q | n ] lp β nj β mk − F il [ j β k ] l + 3 F [ inl b m ] l β nj β mk + H imn β jm β kn + b il R ljk − b il b mn R ln [ j β k ] m + β l [ j β k ] n b lm b np R mpq b qi ,R ′ ijk = R ijk − b lm β l [ i R jk ] m − β l [ i R j | mp β | k ] q b lm b pq − β il β jn β kr b lm b np b qr R mpq + 3 Q n [ ij β k ] n − Q nl [ i β j | n β | k ] m b lm − β l [ i β j | n β | k ] q b lm b pq Q nmp + 3 β l [ i F lmj β k ] m + 3 β l [ i β j | m β | k ] p b np F lmn + β il β jm β kn H lmn . (2.29) This is (2.21) where we take ρ = 1 since it does not rotate the fluxes. However here b and β are constant antisymmetric matrices. O ( d, d ) trans-formation to see if it can be rotated to a different orbit with less (or just different) types offluxes turned on. In section 3 we will often use this possibility to go from one orbit to anotherby means of rigid O ( d, d ) transformations. Alternatively, the above expressions may be used toclassify the automorphisms of r implemented by b - and β -shifts. Another important case is totake U ′ = T U related by the rigid T -transformation T I J = (cid:18) δ ij δ ij (cid:19) , (2.30)which leads to the usual T-duality-like relations among the fluxes F ′ i = δ ij F j , H ′ ijk = δ il δ jm δ kn R lmn , Q ′ ijk = δ il δ jm δ kn F mnl , F ′ i = δ ij F j , R ′ ijk = δ il δ jm δ kn H lmn , F ′ ijk = δ il δ jm δ kn Q nlm . (2.31)This mechanism is what allows us to describe solution-generating techniques that involve alsorigid T -transformations, like abelian/non-abelian/Poisson-Lie T-duality.Let us remark that the concept of “Poisson-Lie plurality” — namely the possibility of decom-posing the same Drinfel’d double d in terms of different choices of subalgebras d = g ⊕ ˜ g = g ′ ⊕ ˜ g ′ — can be understood as the transformation of U by a constant O ( d, d ) matrix V as in (2.24).What we have here is a generalisation of the traditional definition of PL plurality in the sensethat we do not require r to have the structure of a Drinfel’d double, and any O ( d, d ) transfor-mation may in principle be considered, even those relating different orbits. Let us now look at the explicit expressions for the components of the fluxes in the parametri-sation (2.21) used for the twist U . One has F IJK = δ [ I i δ J j δ K ] k H ijk + 3 δ [ I i δ J j δ K ] k F ij k + 3 δ [ I i δ Jj δ K ] k Q ijk + δ [ Ii δ Jj δ K ] k R ijk , F I = δ Ii F i + δ I i F i , (2.32)where H ijk ≡ F ijk = F [ ijk ] , F ij k ≡ F ij k = F [ ij ] k , Q ijk ≡ F ijk = F i [ jk ] , R ijk ≡ F ijk = F [ ijk ] , (2.33)are H ijk = 3( ∂ [ i b jk ] + w [ ijl b k ] l ) ,F ij k = w ij k + β kl H ijl Q ijk = ∂ i β jk − β l [ j F ilk ] + β l [ j β k ] m H ilm R ijk = − β l [ i Q ljk ] + 3 β l [ i β j | m F lm | k ] − β l [ i β j | m β | k ] n H lmn . (2.35)Here we defined ∂ i ≡ ρ iµ ∂ µ , w ijk ≡ − ρ [ iµ ρ j ] ν ∂ µ ρ νk . (2.36)To make w ijk appear also in the generalised flux with one index, we rewrite λ = ¯ λ − log det ρ so that using d log det ρ = Tr( dρρ − ) we find F i = w ij j + 2 ∂ i ¯ λ, F i = ∂ j β ji + β ik ( w kjj + 2 ∂ k ¯ λ ) = ∂ j β ji + β ik F k . (2.37) The matrix form of T IJ and η IJ coincide, but notice the different position of indices in the two definitions. For explicit examples on how to relate representatives of the (
F, H ) and of the (
F, Q ) orbits see e.g. [47]. See for example [20] for similar expressions. The last two equations could be rewritten also as Q ijk = ∂ i β jk − β l [ j w ilk ] − β m [ j β k ] n H imn R ijk = − β l [ i ∂ l β jk ] − β l [ i β j | m w lm | k ] − β l [ i β j | m β | k ] n H lmn . (2.34) if w ijk can be interpreted as structure constants of a Lie algebra, then its definitionis simply the Maurer-Cartan (MC) identity for ρ , which can then be taken of MC form ρ = g − dg = dy µ ρ µi t i , where t i ∈ Lie ( G ) are the generators of the Lie algebra with structureconstants w ij k and g ∈ G . This happens for example for all orbits that have H ijk = 0, since inthis case the Bianchi identities imply that F ij k solves the Jacobi identity and the above equationsgive F ij k = w ijk . We will see that the discussion for orbits with non-vanishing H -flux is morecomplicated, and in the next section we present some methods that we will use in this case. H -flux The methods explained in this section may be useful in general, but later we will actuallyuse them only for orbits which have non-vanishing H ijk , so this section is not necessary forreading the discussion of the other orbits. The particular issue with non-vanishing H -fluxis that w ij k may not be interpreted as structure constants of a Lie algebra (it may not evenbe constant) and therefore ρ can not be taken of MC form. Nevertheless, we can maintain ageometrical interpretation by exploiting the fact that the doubled manifold — spanned locally bythe coordinates y µ and their T-duals ˜ y ˜ µ — can be interpreted as a group manifold R associatedto the pre-Roytenberg algebra r , equipped with the bilinear ad-invariant form η of split signature.This is in fact the set-up also known as DFT WZW of [48] and employed also in [49]. We decomposethe group elements g ( Y M ) ∈ R specifically as g ( Y ) = ˜ m (˜ y ) m ( y ), where ˜ m = e ˜ m i (˜ y ) ˜ T i and m = e m i ( y ) T i , which is always possible locally. Notice that the subspaces spanned by T i and ˜ T i are not necessarily Lie subalgebras of r . Let us introduce in r the adjoint action by m and theright-invariant one-form dmm − as m T I m − = M I J T J = e ad m T I , dmm − = dy M V M I T I = dy µ V µi T i + dy µ V µi ˜ T i . (2.38)We will denote the inverse of V µi as V iµ , i.e. V µi V iν = δ νµ and V iµ V µj = δ ji . The MC identity forthe one-form dmm − in r projected to the subspaces is2 ∂ [ µ V ν ] i = V µj V ν k F jki + 2 V [ µj V ν ] k Q jki + V µj V νk R ijk , ∂ [ µ V ν ] i = V µj V νk Q ijk + 2 V [ µj V ν ] k F ij k + V µj V νk H ijk . (2.39)In addition, to first order in the expansion m = e m i T i , we have M ij = δ ij + m l F lij + O ( m ), M ij = δ ij − m l F lj i + O ( m ), M ij = m l H lij + O ( m ) and M ij = m l Q lij + O ( m ), so that we cantake M ij and M ij invertible at least in a certain neighborhood around the identity, while M ij and M ij may not be invertible.When the H -flux is non-vanishing, depending on the orbit in consideration, we will considertwo possible parametrisations for the twist U I M that are equivalent to further specifications ofthe more general parametrisation given in (2.21). Essentially they encompass an ansatz for the For a discussion on dealing with orbits with H = 0 without appealing to the methodology of this section, seeappendix F. The interpretation given in [49] is that the physical manifold is understood as a coset R \ ˜ M , where ˜ M isparametrised by the ˜ y µ . This is possible only when the R -flux vanishes, since in this case the generators ˜ T i spana subalgebra of r . We will not need to appeal to this coset interpretation. The parametrisations for the twiststhat we use manifestly satisfy the strong constraint. -twist and a rewriting of the β - and b -twist of (2.21). In particular, we will consider U (1) I M = (cid:18) δ ij + ξ il ω lj ξ ij ω ij δ ij (cid:19) I J M J K (cid:18) ( V t ) kµ V kµ (cid:19) K M , (2.40)or U (2) I M = M I J (cid:18) δ jk + ξ jl ω lk ξ jk ω jk δ j k (cid:19) J K (cid:18) ( V t ) kµ V kµ (cid:19) K M , (2.41)where the ξ ij , ω ij that we use in U (1) and U (2) are different because of the different positionof the adjoint action M I J in the above expressions. In general a twist given by U (1) may notbe rewritten in the form of U (2) , and viceversa, so generically speaking the two twists shouldnot be viewed as equivalent. When, respectively, M ij = 0 ( M ij = 0) we will consider U (1) ( U (2) ) since the inclusion of the adjoint action realises a simple shift of ω ij ( ξ ij ) in terms of M ij ( M ij ). Notice that both twists are automatically elements of O ( d, d ) since ξ ij and ω ij areantisymmetric matrices and the bilinear form η is ad-invariant. Furthermore ξ ij and ω ij dependonly on the coordinates y µ such that the twists satisfy manifestly the strong constraint.In the case that the R -flux R ijk is vanishing and ξ ij = 0 the twist U (2) I M coincides withthe generalised frame fields of [49] in which a particular solution for ω was found for general( F, H, Q )-flux. In fact, in the parametrisation U (2) I M we can generalise this solution to includenon-vanishing R ijk and ξ ij as well. In order to do so we follow [49] and observe that the fluxequations can be written as F IJK = M I I ′ M J J ′ M K K ′ ( T I ′ J ′ K ′ + S I ′ J ′ K ′ ) = T IJK + S IJK , (2.42)where, T IJK = 3 ˆ U [ I M ∂ M ( ˆ U J N ) ˆ U K ] N , ˆ U I M = (cid:18) ( δ ij + ξ il ω lj )( V t ) j µ ξ ij V j µ ω ij ( V t ) j µ V iµ (cid:19) ,S IJK = 3Λ [ I L F JK ] L , Λ I J = ˆ U I M V M J = (cid:18) ξ ik V kµ V µj ξ ij V iµ V µj δ ij (cid:19) , (2.43)and we have used the identity ∂ M M I J = V M K M I L F KLJ . (2.44)as well as the fact that the adjoint action is an automorphism of r , F IJK = M I I ′ M J J ′ M K K ′ F I ′ J ′ K ′ . (2.45)While the equations for F ij k , Q ijk and R ijk become involved due to the presence of ξ ij , theequation for H ijk is independent of ξ ij (and the inclusion of R ijk ). In particular it reads3 V iµ V j ν V kρ ∂ [ µ ω νρ ] = − H ijk − V [ iµ F jk ] l V µl , (2.46) If we relaxed the strong constraint we could simply take for the twist the components of g − d g , since theMC identity would ensure that they satisfy the correct algebra relations. When imposing the strong constraintthis is not possible because of the non-invertibility of the would-be frame field. Therefore one has to look forother solutions for the twists. Notice that U (2) is a generalisation of the twist used in [49], which is necessary ifone wants to be able to describe, for example, the so-called Yang-Baxter deformations. By equating the four block components of U (1) (˜ ξ, ˜ ω ) and U (2) ( ξ, ω ) we will have for M = (cid:18) M M M M (cid:19) the relations ˜ ω = ( M + M ξ ) ωM − and ˜ ξ = ( M + M ξ ) − M ξ constraint with the consistency relations ωM − M = ( M + M ξ ) − M ξ and M ξ ( M + M ξ ) − M M − M = M ( M + M ξ ) − M ξ . As we will see theseare consistent when M or M are vanishing. In general it may not be possible to solve these constraints since M or M may not be invertible. If we considered instead U (2) ( U (1) ), the ρ -twist would receive a contribution from the (unknown) ω ij ( ξ ij )which would further complicate the discussion. ω µν = V µi ω ij V νj . One can verify using (2.39) that a particular solutionfor this (inhomogeneous) differential equation is given by ω inhom. = ¯ ω (2) − Ω (2) where,¯ ω (2) = 12 V µi V νi dy µ ∧ dy ν , (2.47)which satisfies ∂ [ µ ¯ ω (2) νρ ] = 12 (cid:16) − V µi V νj V ρk H ijk − V [ µi V νj V ρ ] k F ij k + V [ µi V νj V ρ ] k Q ijk + V µi V νj V ρk R ijk (cid:17) (2.48)and Ω (2) chosen such that d Ω (2) = 112 F IJK V I ∧ V J ∧ V K + 16 R ijk V i ∧ V j ∧ V k . (2.50)Notice that Ω (2) may not exist globally and an explicit expression can only be found by choosingparticular coordinates in a local patch. The most general solution to (2.46) is then given by ω ij = ω inhom. ij + ω hom. ij where ω hom. ij is a closed two-form. Let us point out that this is a solutionfor the ω form of U (2) , and it is not a solution for the ω of U (1) in (2.40). Furthermore, withinparametrisation U (2) , it is still necessary to solve the other flux equations for F ij k , Q ijk and R ijk in terms of the unknown ξ ij . Using the equations (2.42) and (2.43) we find in particular S ijk = 2 F ij k − Q [ ikl Λ j ] l + ξ kl H ijl + ξ kn Λ nl F ij l ,S ijk = Q ijk + Λ il R ljk − F il [ j ξ k ] l − ml Q il [ j ξ k ] m ,S ijk = 3 Q l [ ij ξ k ] l + 3Λ ml R l [ ij ξ k ] m , (2.51)with Λ ij = V iµ V µj , and T ij k = ˆ w ijk + ξ kl H ijl ,T ijk = ˆ ∂ i ξ jk − ξ l [ j ˆ w ilk ] − ξ m [ j ξ k ] n H imn ,T ijk = − ξ l [ i ˆ ∂ l ξ jk ] − ξ l [ i ξ j | m ˆ w lm | k ] − ξ l [ i ξ j | m ξ | k ] n H lmn , (2.52)where we have used eqs. (2.35) (by replacing β ij with ξ ij and ρ iµ with V iµ ) and defined ˆ ∂ i = V iµ ∂ µ and ˆ w ijk = − V [ iµ V j ] ν ∂ µ V νk . Using the MC identity (2.39) we haveˆ w ijk = − F ij k + 2 Q [ ikl Λ j ] l − Λ il Λ jm R lmk . (2.53)The equations for ξ ij describing the ( F, H, Q, R )-orbit can then be rewritten as2 ξ kl H ijl + ξ km Λ ml F ij l − Λ il Λ jm R lmk = 0 , (2.54)ˆ ∂ i ξ jk + 2Λ il ξ m [ j Q mk ] l + Λ il R ljk + 2Λ il Λ mn ξ m [ j R k ] ln − ξ m [ j ξ k ] n H imn = 0 , (2.55)3 Q l [ ij ξ k ] l + 3 ξ l [ i ξ j | m F lm | k ] − ξ l [ i ξ j | m R | k ] pq Λ lp Λ mq + 2 ξ l [ i ξ j | m ξ | k ] n H lmn = R ijk , (2.56)where in the latter equation we have used (2.55). Using (2.54) we can rewrite (2.56) also as3 Q l [ ij ξ k ] l + 3 ξ l [ i ξ j | m F lm | k ] − lp Λ mq ξ l [ i ξ j | m R | k ] pq − ξ l [ i ξ j | m ξ | k ] n Λ np F lmp = R ijk . (2.57) When R ijk = 0, the three-form d Ω (2) takes a nice expression in terms of the group elements m , in particular[49] d Ω (2) = 112 ⟪ dmm − , [ dmm − , dmm − ] ⟫ . (2.49)
13e will not solve these equations in general, but rather use them when studying sub-orbits with H = 0. Notice however that for a non-vanishing R -flux they do not allow a trivial solution ξ = 0.While we will use these methods to look at orbits with non-vanishing H -flux, let us pointout that the framework described here can cover the most general representatives of the ( F ),( Q ), ( F, Q ) and (
F, R ) orbits derived later.
Comments on other possible ansatze — The methodology described above to find asuitable twist when H = 0 is far from the most general parametrisation considered in (2.21):there are still several possibilities to generalise the two ansatze considered in (2.40) and (2.41).(i) The most obvious generalisation is to replace V µi in (2.40) and (2.41) as V µi → A µν V νi forsome matrix A µν ( y ) which, to exclude diffeomorphisms, is not of the form ∂ µ a ν for some(vector) a ν ( y ). Up to a rewriting, this is now equivalent to the general parametrisation(2.21), which will be more convenient to use in practice.(ii) Another possibility is to consider a different parametrisation of the group elements g ∈ R .In particular we can take instead g ( Y ) = m (˜ y ) ˜ m ( y ), with ˜ m = e ˜ m i ( y ) ˜ T i and m = e m i (˜ y ) T i ,and define all the quantities of the twist U in terms of the group elements ˜ m (so that,again, U manifestly satisfies the strong constraint). We define ˜ M I J T J = ˜ m T I ˜ m − and˜ V = d ˜ m ˜ m − = dy µ ˜ V µi ˜ T i + dy µ ˜ V µi T i . When trying to find non-trivial solutions for theequivalent of ξ ij , this parametrisation could be convenient when the generators ˜ T i span asubalgebra of r such that ˜ V µi = 0 and ˜ M ij = 0. In that case, we can consider the followingtwists ˜ U (3) I M = (cid:18) δ jk + ξ jl ω lk ξ jk ω jk δ j k (cid:19) I J ˜ M J K (cid:18) δ kl ( ˜ V t ) lµ δ kl ˜ V lµ (cid:19) K M , (2.58)˜ U (4) I M = ˜ M I J (cid:18) δ jk + ξ jl ω lk ξ jk ω jk δ j k (cid:19) J K (cid:18) δ kl ( ˜ V t ) lµ δ kl ˜ V lµ (cid:19) K M , (2.59)where we have included a rigid T -transformation, and where we have denoted the inverseof ˜ V µi as ˜ V iµ . Because ˜ M ij = 0 these two parametrisations are actually equivalent. Whenspan( ˜ T i ) does not form a subalgebra, however, we expect that not much is gained comparedto the parametrisations of (2.40) and (2.41).Although we will not consider these generalisations further, it would certainly be interesting tosystematically understand whether non-trivial and inequivalent solutions for ξ can be found inspecific orbits within these other ansatze. See also appendix F for some further comments. In the following we will discuss the orbits found by turning on all combinations of the fluxes
F, H, Q and R as in Figure 1. Let us remark that when looking at solution-generating techniquesin supergravity we want to impose that both F IJK = F ′ IJK and F I = F ′ I , while when lookingat canonical transformations of σ -models it is enough to impose F IJK = F ′ IJK . That meansthat there may be certain (classical) canonical transformations of the σ -model that cannotbe interpreted as supergravity solution-generating techniques. Examples of this kind are non-unimodular homogeneous Yang-Baxter deformations, that will be discussed in the ( F )-orbit(see [50] for some exceptions in this class). 14 .1 ∅ The simplest orbit is the one where all fluxes are zero. From the equation for F ij k one gets w ij k = 0, which can be solved up to GL ( d ) transformations by ρ µi = δ iµ . The equation for H ijk can be solved at least locally by taking b ij constant up to gauge transformations. Finally, theequation for Q ijk is solved by taking also β ij constant, and the equation for R ijk is automaticallysatisfied. If the space is compact, this is the example of the d -dimensional torus with no flux,decorated with constant b - and β -shifts. F In this orbit we assume that the only non-vanishing flux is the F -flux. First let us discuss therestrictions imposed by the Bianchi identities. They imply that F ij k satisfies the Jacobi identity.We will call g the corresponding Lie algebra with generators T i ∈ g . The remaining generators ˜ T i of the pre-Roytenberg algebra r form an abelian algebra ˜ g , so that we have r = g ⊕ ˜ g . Moreover F ij k F k = 0 and F ikj F k = 0 imply that F k = 0 if T k ∈ [ g , g ] , F k = 0 if T k / ∈ Z ( g ) , (3.1)where [ g , g ] is the derived algebra and Z ( g ) is the center of the algebra. Notice that for semisimple algebras this means that F i = 0 and F i = 0. Finally we also have the orthogonality condition F i F i = 0.Let us now turn to solving the equations (2.35) for the fluxes in terms of the functions ρ, b, β appearing in the parametrisation of the twist U . First we have F ij k = w ij k and, therefore, thedefinition of w ijk in (2.36) reads like the (left) MC identity. Although we can take ρ of MCform, this solution can be generalised by taking ρ iµ = W ij ¯ ρ jµ with ¯ ρ = ¯ g − d ¯ g of MC form fora ¯ g ∈ G and W an automorphism of the Lie algebra, W il W jm F lmn = F ij k W kn . To mod outby GL ( d ) diffeomorphisms, we have to take W to be an outer automorphism. The dressing bythe automorphism W will burden the following expressions, but we prefer to keep W explicitlybecause different outer automorphisms W will correspond to inequivalent representatives.Solving for H ijk = 0 globally is a question about the second de Rham cohomology of themanifold. It is well known that if g is the Lie algebra of the compact and connected Lie group G , then the n -th cohomology group H n ( G ) with real coefficients is isomorphic to H n ( g , R ), the n -th Chevalley-Eilenberg Lie algebra cohomology with coefficients in R [51]. In other words,one has to impose H ijk = 0 with b ij constant, which leads to the equation F [ ijl b k ] l = 0 implyingthat b ij is a constant 2-cocycle. It is useful to rewrite this in operatorial form. Given a 2-cocycle ω ij solving F [ ij l ω k ] l = 0 one can construct a linear operator ω : g → g such that ωT i = ω ij T j ,where algebra indices are raised and lowered with a symmetric invariant bilinear form on thealgebra, κ ij = h T i , T j i . Then the 2-cocycle condition in operatorial form reads ω [ x, y ] = [ ωx, y ] + [ x, ωy ] , x, y ∈ g . (3.2)Because we want to mod out by gauge transformations of the B-field, we have to impose that b ij is not coboundary, in other words b ij = F ij k x k for some constants x k , or equivalently inoperatorial form b = ad x for some constant x ∈ g . If g is semisimple, all 2-cocycles are 2-coboundaries — the second Lie algebra cohomology is trivial. Therefore, in the compact caseinteresting solutions to H ijk = 0 are possible only for g non-semisimple. In the non-compact Remember that for R d the second de Rham cohomology is trivial, and the exact b ij can be gauged away. Notice that when we write T i we mean T i = κ ij T j , which is different from the other generators ˜ T i of r . Puttingglobal issues aside, the equation H ijk = 0 may be solved for example by b ij = ˆ b ij + ˆˆ b ij withˆ b ij = h ˆ ωT i , T j i and ˆˆ b ij = h ˆˆ ω g T i , T j i , where ˆˆ ω g = W − ◦ ˆˆ ω ¯ g ◦ W , ˆˆ ω ¯ g = Ad − g ◦ ˆˆ ω ◦ Ad ¯ g and both ˆ ω and ˆˆ ω are constant 2-cocycles that are not 2-coboundaries.The equation for β ij coming from Q ijk = 0 is ∂ i β jk − β l [ j F ilk ] = 0 , (3.3)and its most generic solution is β ij = h r g ( T i ) , ( T j ) i , (3.4)with r g = W − ◦ r ¯ g ◦ W , r ¯ g = Ad − g ◦ r ◦ Ad ¯ g and r a constant and antisymmetric r t = − r linearoperator on the algebra. Transposition is understood with respect to the ad-invariant symmetricbilinear form of g , i.e. h x, ry i = h r t x, y i . It is easy to argue that the above is the most generalsolution. In fact, we may view r ≡ Ad ¯ g W ◦ β ◦ W − Ad − g just as a redefinition of the variablesfor which we want to solve the equation. But then, by using d Ad g = Ad g ad g − dg , one finds thatthe differential equation for β is equivalent to ∂ µ r ij = 0, and therefore r must be constant.The equation R ijk = 0 reads β l [ i β j | m F lm | k ] = 0 , (3.5)which is known as the classical Yang-Baxter equation (CYBE) on g . Therefore both β ij and r ij must solve the CYBE. To rewrite it in operatorial form we may introduce the linear operator r : g → g such that rT i = r ij T j . Then the CYBE is[ rx, ry ] − r ([ rx, y ] + [ x, ry ]) = 0 , ∀ x, y ∈ g . (3.6)Notice that r may come with an overall continuous parameter — in fact its entries may depend onseveral independent parameters — so that we can interpret it as a deformation of a representativewith β = 0.It is simple to discuss the rigid O ( d, d ) automorphisms that leave this orbit invariant.From (2.29) one sees that the algebra is invariant if and only if F [ ijl b k ] l = 0 and F il [ j β k ] l = 0. Theformer is a constant b -shift by a 2-cocycle, while the latter a constant β -shift where β commuteswith the adjoint action on the algebra. Both are included in the above solutions. One also finds F i = F ij j + 2 ∂ i ¯ λ, F i = − β jk F jki + 2 β ij ∂ j ¯ λ. (3.7)Notice that for consistency ∂ i ¯ λ must be constant which means ∂ µ ( ρ iν ∂ ν ¯ λ ) = 0. This in turnimplies ∂ µ ∂ ν ¯ λ = ∂ µ ρ νi ρ iǫ ∂ ǫ ¯ λ. Antisymmetrising in µ, ν and using the invertibility of ρ one getsthe condition F ij k ρ kµ ∂ µ ¯ λ = 0 . (3.8)If an algebra g is such that its derived algebra is strictly a subalgebra [ g , g ] ⊂ g , i.e. there arecertain generators of g not contained in [ g , g ], then for those generators ∂ i ¯ λ is not set to zeroby (3.8). Solvable algebras are examples of this kind of algebras, and in this case one may have ∂ i ¯ λ = constant but ¯ λ not constant. If the algebra is semisimple, then the above equation (andthe invertibility of ρ ) imply ¯ λ constant. Notice in addition that β must be such that F i isconstant. For example R is de Rham trivial, but the 2-dimensional abelian algebra has non-trivial Lie algebra coho-mology. In fact it admits a non-trivial 2-cocycle giving rise to the three-dimensional Heisenberg-Weyl algebra. Notice that for such β one has r ¯ g = r because the adjoint action commutes with β .
16o summarise, the ( F )-orbit contains representatives that are invariant under a group G ofisometries. They are found by taking ρ of MC form and setting β = 0. From these representativesone may construct the so-called Yang-Baxter (YB) deformations. They are found by switchingon β of the above form, in general multiplied by a deformation parameter. We refer to thesection on the ( F, R )-orbit for a discussion on the relation of YB-deformations to PL-plurality.Finally we have also included the possibility of outer automorphisms in this ( F )-orbit. Comments on Yang-Baxter deformations —
Notice that if we start from an isometricbackground ( β = 0) then F i = 0, and if we turn on a YB deformation ( β = 0) then to haveinvariance of F we must have β jk f jki = 2 β ij ∂ j ¯ λ . If the original dilaton is isometric (¯ λ isconstant), then this implies the unimodularity condition for β and r [52]. Otherwise it lookslike a generalisation of this condition. We can use YB deformations to give an example of thediscussion on the geometric interpretation of the backgrounds as in appendix B.1, and see howthe fluxes in curved indices change under a YB deformation of an isometric background. As thestarting point ( β = b = 0) we take a background that has only non-vanishing F µν ρ . From (B.15)with h M N = (cid:18) δ µν − β ′ µν − b ′ µν δ µν + b ′ µρ β ′ ρν (cid:19) , where β ′ µν = ρ iµ ρ jν β ′ ij , b ′ µν = ρ µi ρ νj b ′ ij , (3.9)one finds that the fluxes in curved indices after the transformation are H ′ µνρ = 6 b ′ δ [ µ | β ′ δα F α | νγ b ′ ρ ] γ ,F ′ µνρ = F µν ρ + 4 b ′ δ [ µ β ′ α [ ρ F ν ] αδ ] ,Q ′ µνρ = 2 β ′ β [ ν F µβ ρ ] ,R ′ µνρ = 0 , (3.10)where we used the 2-cocycle condition for b and the CYBE for β . Notice that we have therelations F ′ µνρ = F µν ρ + 2 b ′ δ [ µ Q ′ ν ] ρδ , H ′ µνρ = b ′ α [ ρ F ′ µν ] α . (3.11)To conclude, R µνρ remains vanishing, but it is possible to shift F µν ρ and generate H µνρ and Q µνρ . In general YB backgrounds have therefore an interpretation as T-folds, see also [53]. Q The ( Q )-orbit is related to the previous one by a rigid T -transformation. Because the ( F )-orbitcontains isometric backgrounds, the ( Q )-orbit will contain their non-abelian T-duals. Similarlyto the previous case, Bianchi identities imply the Jacobi identity for Q ijk . Now the generators ˜ T i span a non-abelian algebra ˜ g , while the generators T i span an abelian algebra g so that r = g ⊕ ˜ g .The conditions F ij k = w ijk = 0 and H ijk = 0 may be solved as in the ( ∅ )-orbit by taking ρ iµ = δ iµ and, at least locally, b constant up to gauge transformations. The equation Q ijk = ∂ i β jk is the interesting one in this orbit and it is easily integrated to β ij = y µ δ µk Q kij + ˜ ω ij , (3.12)with ˜ ω ij the constants of integration. Finally, imposing R ijk = 0 at each order in y and usingthe Jacobi identity for Q ijk , we find that ˜ ω must be a 2-cocycle of ˜ g Q l [ ij ˜ ω k ] l = 0 . (3.13)Notice that in this case the role of upper and lower indices is exchanged compared to theprevious ( F )-orbit. The 2-cocycle ˜ ω may be multiplied by an overall continuous parameter — in17act its entries may depend on several independent parameters — so that we can think of it as adeformation of the ˜ ω = 0 case. Also in this discussion the above solutions are already includingthe rigid O ( d, d ) transformations of the pre-Roytenberg algebra r under consideration.For the generalised flux with one index we have F i = 2 ∂ i ¯ λ, F i = Q jji + 2˜ ω ik ∂ k ¯ λ + 2 y j Q jik ∂ k ¯ λ. (3.14)Constancy of the fluxes implies ¯ λ = α i y i + ¯ λ and Q jik α k = 0, for all i, j , for some constants α i and ¯ λ . Therefore the fluxes become F i = 2 α i , and F i = Q jji + 2˜ ω ik α k . Bianchi identitiesimply Q ijk F k = 0 , Q kij F k = 0 , F i F i = 0 , (3.15)which again imply that F k = 0 if ˜ T k / ∈ Z (˜ g ) and F k = 0 if ˜ T k ∈ [˜ g , ˜ g ]. Notice that the first andthird conditions above hold thanks to Q jik α k = 0 and antisymmetry of ˜ ω . The second conditionreads Q kij Q llk + 2 Q kij ˜ ω kl α l = 0. Recall that in the case of ˜ g semisimple we have the strongerconditions F i = F i = 0.The fact that the ( F )- and ( Q )-orbits are related by a rigid T -transformation means thatstarting with an F ij k we can relate it to a Q ′ ijk as F ij k = δ il δ jm δ kn Q ′ nlm . Similarly, F i = δ ij F ′ j and F i = δ ij F ′ j , and the generators are identified as T i = δ ij ˜ T ′ j , T i = δ ij ˜ T ′ j so that the roles of g and ˜ g are exchanged. Comments on non-abelian T-duality —
If the starting point is an isometric backgroundwith only F ij k = 0, the above β with ˜ ω ij = 0 is in fact the β -twist of non-abelian T-duality(NATD), as can be easily seen by comparing to (2.5). Starting from F i = F ij j + 2 ∂ i ¯ λ and F i = 0in the ( F )-orbit, then invariance of the fluxes implies F ′ i = 2 α i = 0 in the Q ′ -orbit, and also Q ′ jji = δ ik ( F kj j + 2 ∂ k ¯ λ ) which means ∂ i ¯ λ = − F ij j . For an isometric dilaton ¯ λ is constant, andone gets also in this case a unimodularity condition F ij j = 0 which is the anomaly-free conditionof [7, 8]. Let us use the example of NATD to see how the fluxes in curved indices can transform,following the discussion of appendix B.1. We start again with an isometric background whereonly F µν ρ = 0. From (B.15) we have h M N = (cid:18) − δ ˜ µi ˜ β ij δ jk ρ νk δ ˜ µi δ ij ρ j ν δ ˜ µi (1 + ˜ b ˜ β ) ij δ jk ρ νk − δ ˜ µi ˜ b ij δ jk ρ kν (cid:19) = − ˜ β ˜ µν ρ ˜ µν ρ t ˜ µν + (˜ b ˜ β ) ˜ µν − ˜ b ˜ µν ! , (3.16)where ˜ β ˜ µν = δ ˜ µi ˜ β ij δ jk ρ νk and ˜ b ˜ µν = δ ˜ µi ˜ b ij δ jk ρ kν . We are being explicit in writing all tensorsand indices, and we prefer to use a tilde (rather than a prime) both on the functions and onthe coordinate indices when they refer to the NATD representative of the ( Q )-orbit. After theNATD transformation the fluxes are˜ H ˜ µ ˜ ν ˜ ρ = 3 δ [˜ µi δ ˜ νj δ ˜ ρ ] k ˜ b il ˜ b jm F lmk ˜ F ˜ µ ˜ ν ˜ ρ = 2 δ [˜ µi δ ˜ ν ] j δ ˜ ρk ˜ b il F lkj ˜ Q ˜ µ ˜ ν ˜ ρ = δ ˜ µi δ ˜ νj δ ˜ ρk F jki , ˜ R ˜ µ ˜ ν ˜ ρ = 0 . (3.17)where we used that ˜ β is a 2-cocycle for the Lie algebra with structure constants Q ijk . Noticethat ˜ Q ˜ µ ˜ ν ˜ ρ is always non-zero when we dualise a non-abelian algebra, and the background hasthe interpretation of a T-fold. If we turn on ˜ b we can also generate geometric fluxes. Finally wehave the relations ˜ F µν ρ = 2˜ b δ [ µ ˜ Q ν ] ρδ , ˜ H µνρ = − ˜ b α [ ρ ˜ F µν ] α , notice the different sign in the lastequation compared to the YB case. Comments on “deformed T-duals” —
When in the ( Q )-orbit we turn on ˜ ω ij (which mayinclude an overall deformation parameter) we generate representatives that can be understood as18 eformations of NATD, and that in [54] were called “deformed T-duals” (DTD). See also [55, 29].It turns our that YB-deformations are actually related to DTD models [56, 54], and here werephrase this fact in O ( d, d ) language. In order to do that we will take the point of view of theYB-deformation, and for this reason we will have structure constants F ij k for the non-abelian Liealgebra g . This also means that we will need to lower the indices of ˜ ω with deltas ˜ ω ij = δ ik δ jl ˜ ω kl to take care of the duality relation. First let us point out that in the above discussion g does notneed to be semisimple. Actually, if we want ˜ ω ij to generate a non-trivial deformation from thepoint of view of DTD, then we must take g non semisimple. Therefore in the following we willnot assume semisimplicity. Nevertheless, we will assume that g is a subalgebra of an algebra f that admits a non-degenerate symmetric invariant bilinear form κ . If T i are generators of g , wegenerate the subalgebra g ∗ of f by T i , where we use κ − to raise the indices of T i . We can thinkof ˜ ω as a linear operator in f if we restrict it to the subalgebras ˜ ω : g → g ∗ . Let us now assumethat ˜ ω is invertible in this restriction and let us call r : g ∗ → g the inverse of ˜ ω . This means that r ˜ ω = P and ˜ ωr = P T , where P, P T project on g , g ∗ respectively. Because ˜ ω is a 2-cocycle in g , r satisfies the CYBE (on the whole f ). Now, given the coordinates y used in the DTD model,let us consider the following change of coordinates y = ζP T − Ad − g log Ad ¯ g ˜ ω log ¯ g, ¯ g ∈ G. (3.18)Here y = y µ δ µi T i ∈ g ∗ . One can prove that this change of coordinates implies [55] P T (ad y + ζ ˜ ω ) P = ζP T (˜ ω ¯ g ) P, dy = ζP T (˜ ω ¯ g ¯ g − d ¯ g ) , (3.19)where ˜ ω ¯ g = Ad − g ˜ ω Ad ¯ g , which in components read y µ δ µk f ijk + ζ ˜ ω ij = ζ (˜ ω ¯ g ) ij , dy µ = ζd ¯ y ν ¯ ρ νi (˜ ω ¯ g ) ij δ jµ , (3.20)where we define ¯ ρ such that d ¯ y ν ¯ ρ νi T i = ¯ g − d ¯ g . We can take into account the above change ofcoordinates in the twist ˜ U under consideration by using the above substitution and Jacobian,so that˜ U = (cid:18) δδ (cid:19) (cid:18) β (cid:19) → ¯ U = (cid:18) δδ (cid:19) (cid:18) ζδω ¯ g δ (cid:19) (cid:18) ζ (¯ ρω ¯ g δ ) t ζ − (¯ ρω ¯ g δ ) − (cid:19) (3.21)where the last matrix implements the Jacobian, and we are writing explicitly the δ ij or δ ij whileomitting the indices. A straightforward calculation gives¯ U = (cid:18) ηr ¯ g ¯ ρ − − ζω ¯ g ¯ ρ t ¯ ρ − (cid:19) = (cid:18) ηr ¯ g (cid:19) (cid:18) − ζω ¯ g (cid:19) (cid:18) ¯ ρ t
00 ¯ ρ − (cid:19) (3.22)where we used that r is the inverse of ω and η = ζ − . Thanks to a diffeomorphism, we weretherefore able to rewrite the twist of DTD in terms of the one of a YB-deformation, plus a shiftof b so that H ijk = 0. Interestingly, this ¯ U is not included in the discussion of appendix C. In fact, given that the second Lie algebra cohomology of semisimple algebras is trivial, any ˜ ω ij would be acoboundary (i.e. ˜ ω ij = F ijk c k for some c k ). Therefore we could remove ˜ ω by redefining the coordinates. It maystill be interesting to consider ˜ ω coboundary because we can still relate to it YB deformations, as we are about tosee. These will be equivalent to NATD only when the deformation parameter takes finite values, the limit η → When writing the equations in components we are automatically implementing the projectors, because theindices i, j were restricted to g from the beginning. .4 R Having H = F = Q = 0 immediately implies that R = 0. Indeed, it is well-known that the( R )-orbit is not realisable when imposing the strong constraint. Notice that therefore we cannotinclude the rigid T -transformation of the ( H )-orbit as a solution-generating technique. F, Q
In the (
F, Q )-orbit the Bianchi identities imply that both the F - and Q -flux satisfy the Jacobiidentity on their own, and in addition there is an identity mixing them F ij k Q klm − Q [ ik [ l F j ] km ] = 0 . (3.23)Hence, F and Q can be interpreted as the structure constants of Lie algebras g and ˜ g respectively,with generators T i and ˜ T i . Together with the ad-invariant pairing η the structure r = g ⊕ ˜ g isknown as a Drinfel’d double. Furthermore the Bianchi’s for F I read F ij k F k = 0 , Q kij F k = 0 , Q ijk F k − F ikj F k = 0 , F k F k = 12 F ij k Q kij . (3.24)The first two identities imply F i = 0 if T i ∈ [ g , g ] , F i = 0 if T i ∈ [˜ g , ˜ g ] . (3.25)Finally notice that by tracing the mixed Jacobi identity (3.23) over i = l and j = m we can alsowrite the last identity as F i F i = Q iji F jkk .Let us now solve the flux equations (2.35) for the twist U ( ρ, b, β ). The solution for the ρ -and b -twist found from the F - and H -flux equation, respectively, are equivalent to the solutionsderived and explained in the ( F )-orbit, see section 3.2. Recall that, up to diffeomorphisms, wehave ρ iµ = W ij ¯ ρ jµ with W an outer automorphism of g and ¯ ρ = ¯ g − d ¯ g of (left-invariant) MCform. The β -twist, however, must now solve Q ijk = ∂ i β jk + 2 β [ j | l F ilk ] . (3.26)Notice that this is a linear inhomogeneous partial differential equations (PDE). The most generalsolution to such equations is found by adding the most general homogeneous solution to aparticular solution β inhom. of (3.26). The β -twist of Poisson-Lie symmetric backgrounds [11, 12,58], appropriately dressed by the automorphism W , is an example of such a particular solution.It is given by β ij inhom. = ⟪ Ad ¯ g − · P · Ad ¯ g f W ( ˜ T i ) , f W ( ˜ T j ) ⟫ , (3.27)where f W = W − t is an automorphism of ˜ g , P projects on g , and recall that we have definedthe bracket ⟪ T I , T J ⟫ = η IJ . The most general homogeneous solution is known from thediscussion in the ( F )-orbit, i.e. given by (3.4), which we denote here as β ij hom. = η h r ¯ g ( T i ) , T j i , (3.28)where, recall, r ¯ g ≡ W − ◦ Ad − g ◦ r ◦ Ad ¯ g ◦ W , r t = − r and T i = κ ij T j with κ ij = h T i , T j i .Notice also that here we explicitly introduce a deformation parameter η . At this point this isnot necessary, but we can do it because we are solving a homogeneous equation, and it will be The property f W = W − t follows from requiring that W and f W form an automorphism in the Drinfel’d doubleas W ( r ) = f W (˜ g ) ⊕ W ( g ) which preserves the bilinear form η IJ . R -flux equation. Therefore, turning on β hom. can be seen as a deformationof the PL-symmetric background at η = 0. Concluding, the most general solution to (3.26) is β ij = β ij inhom. + β ij hom. . Solving for R ijk = 0 gives additional algebraic conditions on the operator r . In particular, by expanding order by order in η , it must satisfy the equations β l [ i inhom. β j | m hom. F lm | k ] − β l [ i hom. Q ljk ] = 0 , (3.29) β l [ i hom. β j | m hom. F lm | k ] = 0 , (3.30)where we have used the identity β l [ i inhom. β j | m inhom. F lm | k ] − β l [ i inhom. Q ljk ] = 0 (see e.g. [58]). Therefore,from the second condition we find again the requirement that r must satisfy the CYBE (3.6). The first condition, on the other hand, should be viewed as a compatibility condition between β inhom. and β hom. . In order to interpret it in a field-independent way it will be convenient to writeout the components of the adjoint action by ¯ g in the Drinfel’d double r ,¯ gT i ¯ g − = (Ad ¯ g ) ij T j , ¯ g ˜ T i ¯ g − = (Ad ¯ g ) ij T j + (Ad ¯ g ) ij ˜ T j . (3.31)From Ad ¯ g ∈ O ( d, d ) and Ad ¯ g ◦ Ad ¯ g − = 1 we can derive the relations (Ad ¯ g ) ij = (Ad ¯ g − ) j i ,(Ad ¯ g − ) ij = (Ad ¯ g ) ji and (Ad ¯ g ) ij = − (Ad ¯ g − ) li (Ad ¯ g ) ml (Ad ¯ g ) mj . The solution for β ij can nowbe written as β ij = ( W − ) li ¯ β lm ( W − ) mj , ¯ β ij = (Ad ¯ g ) il (Ad ¯ g − ) lj + η (Ad ¯ g − ) li r lm (Ad ¯ g − ) mj . (3.32)Using the previous relations, the automorphism properties for W and those for Ad ¯ g in r , inparticular using Q ijk = (Ad ¯ g ) il (Ad ¯ g − ) mj (Ad ¯ g − ) nk Q lmn − ¯ g ) il (Ad ¯ g − ) m [ j (Ad ¯ g ) k ] n F lnm , (3.33)we find that the compatibility condition (3.29) takes the following simple form r l [ i Q ljk ] = 0 , (3.34)which is the condition of r ij being a 2-cocycle of ˜ g .It is interesting to see in which cases we are genuinely in the ( F, Q )-orbit modulo rigid O ( d, d )transformations. From (2.29) we find that we can turn off the Q -flux — and thus describe onlythe ( F )-orbit — when it is of the form Q ijk = 2 F il [ j β k ] l for some constant antisymmetric β ij which satisfies the CYBE β l [ i F lmj β k ] m = 0 on g . Equivalently, the F -flux can be turned off bya rigid O ( d, d ) when it is of the form F kij = − b l [ i Q j ] lk for some constant antisymmetric b thatsatisfies the CYBE b l [ i b jm Q k ] lm = 0 on ˜ g .For the generalised flux F I we have F i = F ij j + 2 ∂ i ¯ λ, F i = Q jji − β jl F jli + 2 β ij ∂ j ¯ λ. (3.35)Similarly to the ( F )-orbit we must have that F ij k ∂ k ¯ λ = 0 such that ∂ i ¯ λ = α i , and thus F i , isconstant. Additionally, notice that the β -twist must be such that F i is constant. One can verifythat this is already implied by using the ordinary Jacobi identities, the trace of the mixed Jacobiidentity (3.23) over j, m and the mixed Bianchi identity for F i and F i . This is where the explicit parameter η becomes useful. Because of the presence of η , the R -flux equation givestwo equations for β inhom. and β hom. which have a nice interpretation. However this splitting is not necessary, andin general one gets one single algebraic constraint on β . Let us point out that when ˜ g is abelian, and thus Q ijk is vanishing, then β ij inhom. = 0, and we safely reduceto the most general representative of the ( F )-orbit in which r must satisfy the CYBE. Q -flux — which deforms the ordinaryPL symmetric backgrounds [11, 12] described previously in the literature. Interestingly thesedeformations can be implemented even when the initial background has no isometry, in contrastto the usual homogeneous YB-deformations described in the ( F )-orbit. F, R
In the (
F, R )-orbit the Bianchi identities imply that F ij k are the structure constants of the Liealgebra g generated by T i and that the R -flux satisfies F nm [ k R ij ] n = 0. In turn the latter impliesthe (weaker) condition that R ijk = κ il κ jm κ kn R lmn with κ ij = h T i , T j i is a 3-cocycle of g . Forthe fluxes with one-index we must impose that F ij k F k = F ikj F k = R ijk F k = 0 as well as theorthogonality F i F i = 0. Hence we must have the same conditions as given in (3.1) togetherwith R ijk F k = 0.The solutions for ( ρ, b, β ) to the flux equations (2.35) are equivalent to the solutions of the( F )-orbit apart from the condition that follows from the R -flux equation. Hence, the expressionfor the β -twist is given in (3.4) and the solution for ρ is given by ρ iµ = W ij ¯ ρ j µ , where W is anouter automorphism and ¯ ρ = ¯ g − d ¯ g is a left-invariant MC of g . The R -flux equation now readsby using W, Ad ¯ g ∈ Aut( g )( W ¯ g ) li ( W ¯ g ) mj ( W ¯ g ) nk R lmn = 3 r l [ i r j | m F lm | k ] , W ¯ g ≡ W · Ad ¯ g . (3.36)Using the fact that in r the adjoint action by ¯ g is of block-diagonal form, i.e. (Ad ¯ g ) ij =(Ad ¯ g ) ij = 0, and that Ad ¯ g is an automorphism of r , one can derive the following identity(Ad ¯ g ) li (Ad ¯ g ) mj (Ad ¯ g ) nk R lmn = R ijk , so that the condition (3.36) is in reality field-independentand can be written as R ijk = 3¯ r l [ i ¯ r j | m F lm | k ] , ¯ r ij ≡ ( W − ) li r lm ( W − ) mj . (3.37)A natural choice is to take the R -flux of the form R ijk = ακ il κ jm F lmk , (3.38)which is a 3-cocycle of g , and where α is a real constant. This is of course always a possiblechoice. Assuming it from now on, (3.37) requires that r solves the modified CYBE (mCYBE),that is [ rx, ry ] − r ([ rx, y ] + [ x, ry ]) = − c [ x, y ] , ∀ x, y ∈ g , (3.39)with, up to redefinitions of the r operator, c = α = {− , +1 } . Here c = 1 is known as a split r -matrix while c = − r -matrix. Notice that when r satisfies the mCYBE thenso does ¯ r . Turning on an outer automorphism W can therefore be seen as mapping differentsolutions of the mCYBE to each other. When g is a semisimple (and bosonic) Lie algebra, thecanonical solution of the mCYBE is known as the Drinfel’d-Jimbo r -matrix [59, 60] which isunique [61] up to a GL ( l, C ) freedom on the Cartan subalgebra (CSA) directions of g with l therank of g . In particular the canonical Drinfel’d-Jimbo r -matrix is given in a Cartan-Weyl basisof the complexified algebra g C and annihilates the Cartan generators while multiplying positiveand negative roots with ∓ c respectively. When the real form g C is compact, no split solutionsexist and one can only consider a non-split r -matrix. When the real form is non-compact, how-ever, both possibilities can exist. Since an outer automorphism of semisimple algebras maps the As in the case of the standard homogeneous YB-deformation of the ( F )-orbit, it should be possible to under-stand also this case as a version of PL plurality, and it would be interesting to see this explicitly. W will not affect the canonical r -matrix.To truly sit in the ( F, R )-orbit we must take c = 0 such that the R -flux does not vanishes.This means that “turning on” r (equivalently, β ), e.g. by means of a deformation parameter, cannot be seen as a solution-generating technique in the ( F, R )-orbit. Hence, in the (
F, R )-orbit wecan describe only the split ( c = 1) and the non-split ( c = −
1) inhomogeneous Yang-Baxtermodels (see e.g. [27, 28, 62]). One can verify that in these cases we are genuinely in (
F, R ) underrigid O ( d, d ) equivalence relations. In other words, we cannot turn off the R -flux and F -flux ascan be seen by rewriting (2.29) for H = Q = H ′ = Q ′ = 0 such that R ′ ijk = 3¯ r l [ i ¯ r j | m F lm | k ] + 3 β l [ i β j | m F lm | k ] (3.40)for some constant antisymmetric matrix β ij . In general there may be no real solution for β ij that sets R ′ ijk to zero. When we cannot turn off R ijk , we can also conclude that we cannotturn off F ij k . Otherwise, we would be in the ( R )-orbit and recall that this orbit cannot berealised under the strong constraint.However, an interesting observation is that these ( F, R )-representatives can be mapped, usinga particular rigid O ( d, d ) transformation, to the self-dual ( F, Q )-orbit of Poisson-Lie symmetricbackgrounds. From (2.29) and taking b ij = 0 and β ij = h ¯ rT i , T j i we find that H ′ ijk = 0 , F ′ ij k = F ij k , Q ′ ijk = − F il [ j β k ] l , R ′ ijk = 0 , (3.41)for generic c , so that this discussion applies also to the homogeneous YB-deformations of the( F )-orbit. Here Q ′ ijk are the structure constants of g r whose Lie bracket is defined as[ x, y ] r = [ rx, y ] + [ x, ry ] , ∀ x, y ∈ g , (3.42)and which precisely underlies the bi-algebra structure corresponding to the Drinfel’d double r = g ⊕ g r that in the c = − λ ⋆ -deformation [63, 64, 65]. Therefore, when assuming the expression for R ijk given in (3.38), the discussion of the solution-generating techniques in this orbit (as wellas the discussion on the F I -flux equations) is implicitly captured by the discussion given insection 3.5. As already remarked, the above rigid O ( d, d ) transformation that sends us to the( F, Q )-orbit can be understood as a notion of a PL-plurality transformation.
Q, R
In the (
Q, R )-orbit the Bianchi identities imply that Q ijk can be interpreted as structure con-stants of a Lie algebra and that R ijk is a 3-cocycle. Solving the flux equations (2.35) gives, forthe same reasons as in section 3.3 for the ( Q )-orbit, that ρ µi = δ iµ and that β ij is given by (3.12).Using the Bianchi identity the R -flux equation becomes R ijk = − ω l [ i Q ljk ] , (3.43)where recall ˜ ω ij is an antisymmetric constant matrix. Importantly, a rigid O ( d, d ) transformationas in (2.29) can undo this R -flux contribution at no other expense, and therefore we are effectively Interestingly, it would be possible to remove the R -flux if the algebra g admitted both split and non-splitsolutions of the mCYBE. It would also be interesting to understand the consequences of removing R by relaxingthe reality of β . Notice that here we do not require Q ′ = 0 so that we do not have the expression (3.40) for R ′ . Q )-orbit. Hence, the genuine (
Q, R )-orbit with non-vanishing R -flux can notbe realised when imposing the strong constraint, and therefore we do not expect to be ableto apply a rigid T -transformation from the ( F, H )-orbit when the H -flux is non-vanishing.This is reminiscent of the known anomaly obstructions to gauging global symmetries in WZWmodels [66], which are part of the ( F, H )-orbit. F, Q, R
In the (
F, Q, R )-orbit the F ij k flux can still be interpreted as the structure constants of a Liealgebra g spanned by the generators T i , while in general the Q ijk flux satisfies Bianchi identitiesmixed with R ijk and F ijk , see (2.16)—(2.18).To solve for the twist functions ( ρ, b, β ) of U parametrised as in (2.21) we can observe thatthe flux equations for H ijk , F ij k and Q ijk are identical to the equations in the ( F, Q )-orbit ofsection 3.5. Therefore, the b -twist is found from the second de Rham cohomology of the manifoldwhile the ρ - and β -twist take the form ρ = g − dg, β ij = ⟪ Ad g − · P · Ad g ˜ T i , ˜ T j ⟫ + η h r g ( T i ) , T j i . (3.44)with g ∈ exp g and r g = Ad g − ◦ r ◦ Ad g . For simplicity we have dropped here the possibility ofautomorphisms of r but recall from the general discussion around eq. (2.26) that this is alwayspossible. What does change in this orbit, however, is the algebraic R -flux equation. We find R ijk = 3 β l [ i hom. β j | m hom. F lm | k ] − β l [ i hom. Q ljk ] + 6 β l [ i inhom. β j | m hom. F lm | k ] , (3.45)or in terms of the adjoint action by g in r ,(Ad g ) li (Ad g ) mj (Ad g ) nk R lmn = 3 η r l [ i r j | m F lmk ] − ηr l [ i Q ljk ] (3.46)where we have used the identity (3.33) (which we note is not affected by the presence of R -flux). In the ( F, Q ) orbit, in order to have vanishing R -flux, r was required to be a solution ofthe CYBE for F ij k and to satisfy the 2-cocycle condition with Q ijk . These conditions are nowrelaxed by the presence of the non-trivial R -flux. Notice that the right-hand-side of the aboveequation is constant (i.e. g -independent), which puts a strong constraint on the left-hand-side.Taking g = e x with x ∈ g and expanding order by order in x we have respectively at leadingand first order R ijk = 3 η r l [ i r j | m F lmk ] − ηr l [ i Q ljk ] , F lm [ i R jk ] m = 0 . (3.47)The first condition then implies that we can remove the R -flux using a rigid O ( d, d ) transfor-mation. Indeed taking in (2.29) b = 0 and β = − ηr we can turn off the R -flux at the expenseof shifting the Q -flux and, therefore, we are effectively describing here the ( F, Q )-orbit. There-fore we do not expect to be able to apply a rigid T -transformation from the ( F, H, Q )-orbit ofquasi-Manin triples to the (
F, Q, R )-orbit. In section 3.3 for the ( Q )-orbit, ˜ ω ij was constrained to be a 2-cocycle in order to have vanishing R -flux. Inthat discussion it was assumed that the symmetry in (2.24) had been fixed, or in other words that ˙ E is not allowedto transform. Here we do not need to impose any constraint on ˜ ω ij because the R -flux is not required to vanishat the start, it is rather removed by the transformation (2.24) under which ˙ E is allowed to transform. Notice that in our framework we can only deform/dualise one copy of the symmetry group of the WZWmodel, e.g. the left one. See section 3.5 for the definitions of the various objects, which are not affected by the presence of the R -flux. For example, recall that ρ iµ = W ij ¯ ρ jµ with W ij ∈ Out( g ) and ¯ ρ = ¯ g − d ¯ g , ¯ g ∈ exp g , is also a solution for ρ .When we take for instance the automorphism W IL ∈ Aut( r ) to be block-diagonal, i.e. W ij = W ij = 0 then we willhave f W ij ≡ W ij = ( W − ) ji as well as the relations Q ijk = W il W jm W kn Q lmn and R ijk = W il W jm W kn R lmn ,so that the solution of the Q -flux equation for β ij given in (3.27) still holds. .9 H We now start the discussion of orbits with non-vanishing H -flux, following the framework ex-plained in section 2.4. The first example is the ( H )-orbit. Notice that the Bianchi identitiesfor F IJK are trivially satisfied while those for F I require H ijk F k = 0 and the orthogonality F i F i = 0.Using the notation of section 2.4, for the adjoint action by m and the one-form V I we simplyfind in this orbit that M I J = (cid:18) δ ij m k H ijk δ ij (cid:19) , V µi = ∂ µ m i , V µi = − V µj m k H ijk . (3.48)As argued, since M ij = 0, we continue with the twist U (1) I M to solve the flux equations (2.35).In terms of the functions ρ, b, β of (2.21) we have ρ µi = V µi , β ij = ξ ij and b ij = ω ij + 2 V [ iµ V µj ] .In fact this ρ µi can be gauged away by a diffeomorphism to get ρ µi = δ iµ , which we understandas a consequence of the ansatz U (1) I M , and thus we have w ijk = 0. Consequently, solving for F ij k = 0 constrains β ij to satisfy β kl H ijl = 0 . (3.49)Solving for Q ijk = 0 gives that β ij must be constant, and R ijk = 0 is solved automatically.From (2.29) one sees that this constant β -transformation satisfying (3.49) is precisely an au-tomorphism of the pre-Roytenberg algebra, and therefore it can be removed by a rigid O ( d, d )transformation as in (2.24). Then we are describing the example of the torus with H -flux, seee.g. [67]. To complete the discussion we can also solve for the b -twist, or equivalently ω ij . Bydefining b µν = ρ µi b ij ρ νj and observing that b µν = ω µν − ω µν , with ¯ ω defined as in (2.47), wefind that 3 ∂ [ µ ω νρ ] = − V µi V ν j V ρk H ijk (3.50)which has a particular solution ω inhom. µν = V µi V νi , so that we have an explicit expression at leastlocally. The most general solution is ω µν = ω inhom. µν + ω hom. µν with ω hom. a closed two-form.For the generalised flux F I we have F i = 2 ∂ i ¯ λ and F i = 0 (after setting β = 0 by the rigid O ( d, d ) transformation) so that the above Bianchi identities are automatically satisfied. Another class of representatives.
Let us now present another class of representativeswithin the ( H )-orbit that are not captured by the above discussion and the methods of section2.4. We will try to look for representatives with non-vanishing w ij k . In particular we willrestrict to the case of constant w ij k and we will want to interpret them as structure constantsof a Lie algebra g , so that we will prefer to use the notation f ijk = w ijk . This can be achievedsimply by taking ρ = g − dg of MC form, with g a group element of G such that g = Lie ( G ).Then from F ij k = 0 we have f ijk = − β kl H ijl . (3.51)For g non-abelian this is possible only if we turn on a certain β , and in general it is consistentonly if β kl H ijl is constant. The Jacobi identity for the structure constants f ijk now implies β nl H n [ ij H k ] lp β mp = 0 . (3.52)While a non-constant β may be still possible, let us take a constant β to simplify the discussionfurther. Then from Q ijk = 0 it follows β l [ j f ilk ] = 0 , = ⇒ β jl β km H ilm = 0 , (3.53) For more comments on solving the ( H )-orbit without relying on section 2.4 see appendix F. R ijk = 0 and the Jacobi identity for f ijk are automatically satisfied. For the fluxes F I we havethat F i = − H ijl β jl + 2 ∂ i ¯ λ and F i = 2 β ik ∂ k ¯ λ , where (3.53) was used, and the Bianchi identitiesinvolving F I are automatically satisfied. Constancy of the fluxes imposes also the condition f ijk ∂ k ¯ λ = 0, which is equivalent to the Bianchi identity H ijk F k = 0.One can check that in d = 3 there is no solution to (3.53) but already in d = 4 there areseveral. Given a basis e i we first write the 3-form H -flux as H = 4 h [ l e i ∧ e j ∧ e k ] = h e ∧ e ∧ e − h e ∧ e ∧ e + h e ∧ e ∧ e − h e ∧ e ∧ e . (3.54)Then one possible solution to (3.53) is found by setting for example all β ij = 0 except β , and h = h = 0. The only non-vanishing components of the structure constants are then f = − h β , f = − h β , (3.55)which define a Heisenberg algebra[ t , t ] = − β ( h t + h t ) . (3.56)Interestingly we may view this class of representatives as a deformation of the representativesdescribed previously. It would be nice to extend the methods of section 2.4 to include these also. H, R
In the (
H, R )-orbit the Bianchi identities read H ijk R klm = 0 , H ijk F k = 0 , R ijk F k = 0 , F i F i = 0 . (3.57)Following section 2.4, one can verify that, compared to the ( H )-orbit, the adjoint action by m and the one-form dmm − do not change with the presence of R -flux and thus are given in(3.48). Using the Bianchi identities notice that we therefore have that V µl R ijl = 0. Continuingwith the parametrisation U (1) I M and solving for the flux equations with ρ µi = V µi , β ij = ξ ij and b ij = ω ij + 2 V iµ V µj we ultimately find that R ijk must be vanishing and, therefore, that wereduce to the ( H )-orbit. Hence, we must conclude that the ( H, R )-orbit (with non-vanishing R -flux) can not be realised within the framework of section 2.4. F, H
In the (
F, H )-orbit the Bianchi identities imply that F ij k satisfies the Jacobi identity and canbe interpreted as the structure constants of a Lie algebra that we will call f . We should howeverpoint out that the generators are not T i (since their commutation relations in r are of the form[ T, T ] ∼ F T + H ˜ T ), and we will denote them by t i ∈ f . Moreover, the Bianchi identities implythat H ijk is a 3-cocycle of f and that H ijk F k + F ij k F k = 0, F ikj F k = 0 and F i F i = 0, and inparticular F k = 0 if t k / ∈ Z ( f ). When f is semisimple the Bianchi identities imply F i = F i = 0.Let us now ask when we are genuinely inside this orbit, and not in simpler sub-orbits, or inother words when there exists a transformation of the pre-Roytenberg algebra that sets either F or H (or both) to zero. From (2.29) one sees that H can be removed by a rigid O ( d, d )if it is of the form H ijk = − F [ ij l b k ] l for some constant b , i.e. if H is a coboundary for f . This is consistent with the MC identity (2.39) for V µi .
26e therefore want to restrict ourselves to H in the third Lie algebra cohomology of f . For f semisimple H ( f , R ) = R n [68], where n labels the number of simple factors in f , such thatone can take H ijk = P na =1 α a F ij l κ lk with κ ij = h t i , t j i , and α a , a = 1 , . . . , n real constants.Viceversa, starting from ( F, H ) and using (2.29) one sees that F can be removed if it is of theform F ij k = H ijl β lk for some constant β which additionally satisfies H ilm β jl β km = 0 such thatthe Q and R -fluxes are to remain vanishing.Following section 2.4, the adjoint action by m now takes the form M I J = (cid:18) M ij M ij M ij (cid:19) , (3.58)where, using ad-invariance of η IJ , we have M ij = ( M − ) j i and M ij = − M il M kl ( M − ) j k .Furthermore, using that M ∈ Aut( r ) we have the following identities F ij l M lk = M il M jm F lmk ,F ij l M lk + H ijl ( M − ) kl = 2 M [ i | l M | j ] m F mkl + M il M j m H lmk , (3.59)of which the last identity can be rewritten as M il M jm M kn H lmn = H ijk + 3 F [ ij l M k ] n M ln . (3.60)Notice in addition that the components M ij give an automorphism of f .Since M ij = 0 we prefer the parametrisation U (1) I M for the twist, that corresponds to ρ iµ = M ij V jµ , β ij = ξ ij , b ij = ω ij + M ik M jk . (3.61)Using the MC identity (2.39), the expression for the derivatives of M I J (2.44) and the automor-phism properties, we find w ijk = F ij k . Hence we might as well take the ρ -twist as ρ iµ = W ij ¯ ρ j µ with ¯ ρ = ¯ g − d ¯ g , ¯ g ∈ exp f and W ∈ Out( f ). Solving for the F -flux equation now implies that β kl H ijl = 0 . (3.62)Solving for Q ijk = 0 and R ijk = 0 then simply gives the homogeneous YB solution for β ij = h r g ( t i ) , t j i — as in (3.4) but defined with different generators — in which the constant antisym-metric operator r solves the CYBE on f . Notice that β is, however, constrained by the condition(3.62). Finally, to solve for the b -twist we rewrite the H -flux equation as 3 ∂ [ µ b νρ ] = ρ µi ρ νj ρ ρk H ijk with b µν = ρ µi ( ω ij + M ik M j k ) ρ νj , and observe that for ˜ b µν ≡ ρ µi M ik M jk ρ νj we find3 ∂ [ µ ˜ b νρ ] = ρ µi ρ νj ρ ρk H ijk + 2 V µi V νj V ρk H ijk + 3 V [ µi V νj V ρ ] k F ij k , (3.63)where we have used (2.39), (2.44), (3.59) and (3.60). From the definitions for ¯ ω (2) and Ω (2) givenin (2.47) and (2.50) we can solve the H -flux equation for b µν by b µν = ¯ ω (2) µν − Ω (2) µν + ˜ b µν = ¯ ω (2) µν − Ω (2) µν + ρ µi M ik M jk ρ νj . (3.64)Notice that the most general solution adds closed two-forms admitted by the manifold to b µν but also that this solution might not exist globally.Finally for the equations for the generalised flux F I in this orbit we have F i = F ij j + 2 α i , F i = − β jl F jli + 2 β il α l (3.65)27here we have used β kl H lij = 0 and where, as before, α i = ∂ i ¯ λ is a constant, which implies F ij k α k = 0. The Bianchi identities for F I therefore read F ij l F lmm − H ijl β mn F mnl = 0 , F ilj β lm α m − F ilj β mn F mnl = 0 . (3.66)One can verify that the second identity guarantees also the orthogonality condition F i F i = 0.Together with the second identity and the Jacobi identity for F ij k one can verify that all theseconditions guarantee also the constancy of F i . Notice that if the algebra f is semisimple andthe dilaton is isometric ( α i = 0) these condition reduce to the requirement that f as well as β and the r -operator are unimodular [52].An example of a σ -model in the ( F, H )-orbit is the Principal Chiral Model (PCM) with theaddition of a Wess-Zumino (WZ) term (in the absence of spectators in fact we describe thewhole σ -model). When fixing the correct normalisation between the coupling in front of thePCM and the WZ actions, this becomes the WZW model [69], which is in addition a conformalfield theory. The pure NSNS bosonic-string background AdS × S × T can be realised as an SL (2 , R ) × SU (2) WZW model plus four free bosons parametrising the torus.Inhomogeneous and homogeneous YB-deformations of PCM + WZ models were discussedin [70]. The homogenous-type deformations leave the generalised fluxes invariant and thereforecan be included in our discussion. However we would like to point out that the homogeneousYB-deformations of [70] are always such that the image of the r -matrix (which is a subalgebra f of the full Lie algebra of the group used to construct the models) is a solvable algebra. As aconsequence of Cartan’s criterion, the components of the H -flux computed from the WZ term(e.g. H ijk = αf ij l κ kl for some constant α ) are zero when all three legs are along f . Selectinga solvable subalgebra is equivalent to splitting the coordinates x m of the full target-space intospectator coordinates ˙ x ˙ µ and coordinates y µ such that H µνρ = 0. Therefore in our languagethe homogeneous deformations of [70] should be viewed as a deformation of the simpler ( F )-orbit. Notice that the solvability condition can be seen as a possible solution to (3.62). Newdeformations of the PCM + WZ model correspond to solving (3.62) whilst relaxing the solvabilitycondition.Let us now turn to the generically inhomogeneous YB-deformation of the PCM + WZ modelof [71, 72, 70]. To compare to those results let us consider the O ( d, d ) matrix¯ U I J = (cid:18) a bc d (cid:19) = (cid:18) γ − cosh( ζrκ ) γ − κ − sinh( ζκr ) γκ sinh( ζrκ ) γ cosh( ζκr ) (cid:19) , (3.67)where γ, ζ ∈ R are parameters, κ ij is a symmetric matrix that we will interpret as the Killingmetric of an algebra f , and r ij is an antisymmetric matrix. No further restriction on r is requiredfor ¯ U to be in O ( d, d ). However, to have an integrable σ -model further conditions are necessary,see [71, 72, 70]. If we take ¯ H IJ = ¯ U K I ˙ H KL ¯ U LJ , where simply˙ H IJ = (cid:18) κ ij κ ij (cid:19) , (3.68)and in the absence of spectators, then using (2.5) it is easy to check that the combination ofmetric and B-field ¯ M = ¯ G − ¯ B parametrising ¯ H will be¯ M = (cid:18) γ − γ − e − ζκr (cid:19) − (cid:18) γ − γ + e − ζκr (cid:19) κ. (3.69)Comparing to equation (1.2) of [70] we see that in order to rewrite the σ -model action in an28 ( d, d ) covariant form it is sufficient to take for example the twist U = (cid:18) b (cid:19) ¯ U (cid:18) v t v − (cid:19) , (3.70)with v = dgg − and g ∈ exp f , and where the ˆ b -shift of the B-field takes care of the WZ term.Let us mention that ¯ U is of the form of (2.21) with¯ ρ = γ − (cosh( ζκr )) − , ¯ β = γ − κ − sinh( ζκr )(cosh( ζκr )) − , ¯ b = γ κ sinh( ζrκ ) cosh( ζrκ ) . (3.71)Let us now ignore the WZ term for a moment (ˆ b = 0). The v -twist is very simple and whentaken alone it gives rise to F -flux only, in particular F ij k = − f ijk in terms of the structureconstants of f . The multiplication by ¯ U from the left can be seen as a rigid O ( d, d ) rotation, andfor this reason (when ˆ b = 0) we can still interpret the model as being in the ( F )-orbit. On theother hand, if we include ¯ U when computing the fluxes we obtain H ijk = − γ (cosh ζκr ) [ il (cosh ζκr ) jm (sinh ζκr ) k ] n f lmn ,F ij k = − γ [(cosh ζκr ) il (cosh ζκr ) jm (cosh ζrκ ) kn + 2(cosh ζκr ) [ il (sinh ζκr ) j ] m (sinh ζrκ ) kn ] f lmn ,Q ijk = − γ − [(sinh ζκr ) il (sinh ζrκ ) jm (sinh ζrκ ) kn + 2(cosh ζκr ) il (sinh ζrκr ) [ jm (cosh ζrκ ) k ] n ] f lmn ,R ijk = − γ − (sinh ζrκ ) [ il (sinh ζrκ ) jm (cosh ζrκ ) k ] n f lmn , (3.72)where the indices of f ijk are raised and lowered with κ − and κ . Notice that all the fluxes aboveare non-zero. While, as already mentioned, at ˆ b = 0 this is just a rigid O ( d, d ) rotation of arepresentative of the ( F )-orbit, when including now ˆ b the new fluxes written in terms of theabove ones are H ′ ijk = H ijk + 3 F [ ij l ˆ b k ] l − b l [ i ˆ b j | m Q | k ] lm + ˆ b il ˆ b jm ˆ b kn R lmn + ˆ H ijk ,F ′ ij k = F ij k + 2ˆ b l [ i Q j ] lk + ˆ b il ˆ b jm R lmk ,Q ′ ijk = Q ijk + ˆ b il R ljk ,R ′ ijk = R ijk , (3.73)where ˆ H ijk is the shift of the H -flux produced by ˆ b . The requirement is that in the unde-formed case ( ζ = 0 , γ = 1) the H -flux is proportional to the structure constants of f , namely − f [ ijl ˆ b k ] l + ˆ H ijk = αf ijk for some coefficient α . It would be interesting to see if there is adifferent parametrisation of the twist of the inhomogeneous YB-deformation of PCM + WZthat gives rise to simpler expressions for the fluxes, and if there is a parametrisation in whichthe fluxes are manifestly constant. This would help understand what is the simplest orbit inwhich one can describe this model. Notice that we have not used the mCYBE for r above, andit will likely play a role in that computation. H, Q
In the (
H, Q )-orbit the Bianchi identities imply that the Q -flux represents the structure constantsof a Lie algebra ˜ g with generators ˜ T i and that the H -flux satisfies H k [ ij Q l ] mk = 0. The latterin turn implies that H ijk = ˜ κ il ˜ κ jm ˜ κ kn H lmn , with ˜ κ ij = h ˜ T i , ˜ T j i the Killing form, must be a3-cocycle of ˜ g . Moreover Q ijk F k = 0 and Q kij F k = 0 imply that F k = 0 if ˜ T k Z (˜ g ) and F k = 0 if ˜ T k ∈ [˜ g , ˜ g ]. We also have the orthogonality condition F i F i = 0 as well as H ijk F k = 0. To relate to the parameters in [70] we have to identify γ − γ = e χ and ζ = − ρ/ H -flux is non-vanishing we turn to a particular parametrisation ofthe twist U explained in section 2.4. As all of the components of the adjoint action M I J = ⟪ mT I m − , T J ⟫ are turned on, we prefer to continue with U (2) I M given in (2.41), where we knowin general the solution for ω ij . In terms of the functions ( ρ, b, β ) of (2.21) we have in general ρ iµ = ( M ij ξ jk + M ij δ kj ) V kµ ,β ij ρ jµ = ( M ij ξ jk + M ij δ kj ) V kµ ,b ij ρ µj = ( M ij ( δ + ξω ) jk + M ij ω jk ) V µk . (3.74)Recall from section 2.4 that the H -flux equation is solved, up to closed two-forms, by ω =¯ ω − Ω (2) . Instead of trying to solve the other flux equations given in eq. (2.35) directly, it willbe more convenient to use the expressions given in (2.54)–(2.57). In this orbit they read ξ kl H ijl = 0 , (3.75) ∂ µ ξ ij + 2 V µl ξ m [ i Q mj ] l = 0 , (3.76) Q l [ ij ξ k ] l = 0 . (3.77)Notice that ξ = 0 is a trivial possible solution. To solve these equations in general, it will beimportant to consider the series expansion of the one-form dmm − . We find ∂ µ mm − = V µi T i + V µi ˜ T i = ∞ X N =0 N + 1)! ad Nm ∂ µ m, (3.78)with V µi = ∞ X N =0 N + 1)! ∂ µ m j (( mH · mQ ) N ) j i ,V µi = ∞ X N =0 N + 2)! ∂ µ m j ( mH ) jl (( mQ · mH ) N ) li , (3.79)where we have defined ( mH ) ij ≡ m l H lij and ( mQ ) ij ≡ m l Q lij , and which can be proven byinduction. Notice that we can always write V µi = A jkµ H jki for some tensor A which carries allthe field dependence. With this observation, it is now easy to show that using eq. (3.75) andthe Bianchi identity H k [ ij Q l ] mk = 0 that V µl ξ m [ i Q mj ] l = 0 . (3.80)Hence eq. (3.76) simply requires that ξ ij must be a constant matrix. Then the remainingconditions are algebraic: eq. (3.77) implies that ξ must be a 2-cocycle for ˜ g while eq. (3.75) canbe seen as a compatibility condition requiring that ξ and H must be orthogonal. Therefore,turning on such a particular ξ from ξ = 0 can be interpreted as a deformation which is a (novel)solution-generating technique.To have the expression of the ( ρ, b, β )-twists it will again be convenient to consider a seriesexpansion, now of M ij . One finds in particular M ij = P ∞ N =0 1(2 N +1)! ( mH ) il (( mQ · mH ) N ) lj sothat upon (3.75) we have M ij ξ jk = 0. Hence ρ iµ = M ij V jµ , β ij = ( M − ) li ξ lk ( M − ) kj + M ik ( M − ) kj , b ij = M il ω lm M j m , (3.81)where we have used M il M j l + M il M jl = δ ij and M il M jl + M il M j l = 0, from M I L ∈ O ( d, d ), aswell as M ij ξ jk = 0. We can now easily calculate the expression for w ijk resulting in w ijk = H ijl M ln ( M − ) nk . (3.82)30et us now consider the conditions in which case a rigid O ( d, d ) transformation by (2.29)describes a simpler orbit in disguise (i.e. H or Q can be turned off at no other cost). We findthat we can not turn off the H -flux with a constant b and β transformation since the conditionfound from H ′ ijk = 0 together with the consistency condition from F ′ ij k = 0 requires H ijk to bevanishing, which is of course not possible. The same holds for the Q -flux where the conditionrequired from Q ′ ijk = 0 does not match with the conditions found from R ′ ijk = 0 as well as F ′ i jk = 0.To complete the discussion of this orbit in general we have for the generalised fluxes F I F i = w ij j + 2 ∂ i ¯ λ, F i = Q jji + 2 β ij ∂ j ¯ λ, (3.83)with w ij k ∂ i ¯ λ = 0 such that F i is constant. From the requirement that F i is constant, ∂ i F j = 0,we therefore also must have that Q ijk ∂ k ¯ λ = 0. From the Bianchi Q ijk F k = 0 we then findthe condition Q ijk w kll = Q ijk H klm β ml = 0. Furthermore H ijk F k = 0 implies H ijk Q llk = 0,and Q kij F k = 0 implies Q kij Q llk + 2 Q kij β kl α l = 0. Notice that the orthogonality condition F i F i = 0 will be automatically satisfied because of the previous conditions. (Asymmetrical) λ -deformations on group manifolds — It is known from [49], whenignoring spectators, that the WZW model and its λ -deformation [73] can be described by theparticular twist U (2) with ξ = 0 and by a particular choice of fluxes (or, equivalently, a particularpre-Roytenberg algebra). In the following we will show that, generalising [49], also the asym-metrical λ -deformation on group manifolds of [74] fits within the same pre-Roytenberg algebra.We take r = k ⊕ k where k is a Lie algebra with generators t i and structure constants f ijk . Thebilinear form η IJ is taken to be [65, 49] ⟪ { x , y } , { x , y } ⟫ = h x , x i − h y , y i , (3.84)where x i , y i ∈ k and κ ij = h t i , t j i is the Killing form on k . The generators ˜ T i with structureconstants Q ijk generate the diagonal embedding ˜ g = k diag in r by the map x
7→ { x, x } / √
2. Thissubgroup is maximally isotropic. The complementary isotropic subspace, spanned by T i , is theanti-diagonal embedding by the map x
7→ { x, − x } / √
2. Hence, in this case we have H ijk = 1 √ f ijl κ lk , Q ijk = 1 √ κ il κ jm κ kn f mnl , F ij k = 0 , R ijk = 0 . (3.85)Notice that the Bianchi identities for F IJK are automatically satisfied upon the Jacobi identityfor f ijk and the ad-invariance of κ ij . To connect to the λ -deformed background in terms of themetric and B-field there is a subtlety in choosing a good parametrisation of the group element m = e m i ( y ) T i . Following [49] we take m = { ¯ g, ¯ g − } with ¯ g ∈ exp( k ) and define a group element˜ g ≡ ¯ g ∈ exp( k ). Using this identification in the twist U (2) I M we find in terms of the generalparametrisation (2.21) that ρ iµ = 1 √ ˜ g ) ij ˜ v jµ , β = − κ − − Ad ˜ g ˜ g , b = b + ρ − ωρ − t , (3.86)with b = 14 (Ad ˜ g − Ad − g ) κ, (3.87)and ˜ v = d ˜ g ˜ g − . Deviating from [49] we now consider a different parametrisation denoted by m = { g, g − } with g ∈ exp k and which is related to the previous parametrisation as Ad ˜ g = For details of this calculation see [49]. Following the logic of [49] we find that to find the same deformedgeometry, we should replace e = g − dg with v = dgg − in their frame fields of eq. (5.63), an overall minus sign ismissing in entry (1 ,
2) and (2 , λ → − λ in eq. (5.66). This likely corresponds to a knownsymmetry of these backgrounds [75] and indeed we agree on the final result. g ◦ W and ˜ v = v = dgg − in which W is a constant outer automorphism of k which preservesthe metric κ . In particular this means that the group elements ˜ g and g are not related by atrivial field redefinition. The twist functions simply become ρ − = 1 √ g W ) v − , β = − κ − − Ad g W g W , (3.88) b = 14 (Ad g W − W − Ad − g ) κ + ρ − ωρ − t . (3.89)A final subtlety in calculating the background from the twist functions is the choice of the matrix˙ E AI , which is constant when we turn off the spectator fields. To obtain the (asymmetrical) λ -deformed background we must take ˙ E AI such that [65, 49]˙ H IJ = ˙ E AI H AB ˙ E BJ = (cid:18) λ − λ κ − λ λ κ − (cid:19) , (3.90)in which λ is the eponymous deformation parameter. The metric and the B-field can thenbe extracted from the generalised metric H MN = U (2) I M ˙ H IJ U (2) J N resulting, in the coordinateframe, in G = 1 − λ vO g κO tg v t , B = 12 v ( O g κ − κO tg ) v t − ω, (3.91)with O g = (1 − λ Ad g W ) − and where dω = − f ijk v i ∧ v j ∧ v k gives rise to the well-knownWZ term. This coincides precisely with the metric and B-field of the asymmetrical λ -model onthe group manifold G = exp k [74] and with the original λ -model when W = 1 [73]. Let us pointout that here W should not be mistaken with an automorphism of the pre-Roytenberg algebraas given in (2.26).Curiously this implies that the WZW model background, which can be found by taking thelimit λ →
0, is described in the (
H, Q )-orbit as well as the (
F, H )-orbit. This then begs thequestion if there is a rigid O ( d, d ) transformation (2.29) that relates the ( H, Q )-orbit (with the H and Q fluxes proportional to each other via the Killing form as in (3.85)) to the ( F, H )-orbit (with F and H again proportional). However, in general there is no such rigid O ( d, d )transformation, as can be checked for example in the su (2) case.On the other hand, a rigid O ( d, d ) transformation does allow to describe the (asymmetrical) λ -models in the self-dual ( F, Q )-orbit of Poisson-Lie symmetric backgrounds. By taking β ij = 0and b ij = h ˜ rt i , t j i , with ˜ r a constant antisymmetric operator that satisfies the mCYBE (3.39)for c = 1 on k , we find H ′ ijk = 0 , F ′ ij k = √ b l [ i f j ] mk κ lm , Q ′ ijk = 1 √ κ il κ jm κ kn f mnl , R ′ ijk = 0 , (3.92)where we have used the ad-invariance of κ . This is now up to analytic continuations (essentiallysending c = 1 to c = − η -deformation[63, 64, 65, 49] of which the ( F, Q )-fluxes were given in eq. (3.41).Let us emphasise that the generalised fluxes F IJK remain invariant when turning on a non-trivial automorphism W from the (original) λ -model representative with W = 1. Hence this mapcan be understood as a solution-generating technique — recall that in the case of supergravities A useful trick to get this generalisation Ad ˜ g = Ad g ◦ W and ˜ v = v = dgg − is to consider a constant w ∈ exp g such that ˜ g = gw and define W ( t i ) = wT i w − . This W would be an inner automorphism and it would be removedby a trivial field redefinition, but the dressing by W in the formulas above is the same also in the case of outer automorphisms.
32e have to require that also F I stays invariant. Furthermore starting e.g. with the λ -modelrepresentative ( ξ = 0), and turning on a constant ξ which is orthogonal to H as well as a 2-cocycle for Q , is a novel solution-generating technique. In fact, if we transform the twist bya constant β -shift from the left, the fluxes transform as in (2.29), with β = ξ and b = 0 inthis case; then demanding that the fluxes remain invariant produces the two conditions (3.75)and (3.77). Therefore turning on a constant ξ can be understood as the implementation of anautomorphism of the pre-Roytenberg algebra as in (2.26). Interestingly this deformation maybe viewed as a generalisation of the DTD models (which on its own generalise ordinary NATD)in the ( Q )-orbit, but now with the addition of H -flux. Comments on finding other representatives — Let us illustrate that outside of theansatz considered in this section, the (
H, Q )-orbit may still capture different non-trivial repre-sentatives. Since the generators ˜ T i span a subalgebra ˜ g , another interesting ansatz allowing forsystematic progress is the twist considered in eq. (2.58) or eq. (2.59). Taking the latter, andassuming that δ ij is an ad-invariant bilinear form of ˜ g , we find for the equivalent of the F -, Q -and R -flux equations of (2.42) and (2.43) that ξ kl H ijl = δ il Q jlk , Q ijk = ˆ ∂ i ξ jk − δ il ξ m [ j Q mk ] l , ξ l [ i Q ljk ] − ξ l [ i ξ j | m Q m | k ] n δ ln = 0 , (3.93)with ˆ ∂ i ≡ δ ij ˜ V jµ ∂ µ and ˆ w ij k = δ il Q jlk . In this case notice that the expansion of ˜ V µi is in termsof Q instead of H , namely ˜ V µi = P N =0 1 N +1) ∂ µ ˜ m j ( ˜ mQ ) j i . It is now indeed clear that theseequations may hold a genuinely different solution. We leave this problem, and the possibility ofother ansatze, open. F, H, R
In the (
F, H, R )-orbit the Bianchi identities imply that the F -flux represents the structure con-stants of a Lie algebra f whose generators we denote by t i and which — as in section 3.11 — donot coincide with T i . Then H ijk and R ijk = κ il κ jm κ kn R lmn , with κ ij = h t i , t j i , are 3-cocyclesof f . In fact the R -flux satisfies the stronger condition F ij [ k R lm ] j = 0 Additionally we have H ijk R klm = 0 as well as the Bianchi identities (2.19) and (2.20) for F I .Since the H -flux is non-vanishing we turn again to methodology of section 2.4. Comparedto the ( F, H )-orbit of section 3.11, notice that the presence of R -flux does not affect the adjointaction M I J given in (3.58). Therefore taking the ansatz U (1) I M of eq. (2.40) we have in terms ofthe ( ρ, b, β )-twists again that ρ iµ = M ij V jµ , β ij = ξ ij , b ij = ω ij + M ik M jk . (3.94)Now, however, the equation for w ijk = − ρ iµ ρ jν ∂ [ µ ρ ν ] k will in principle receive contributionsfrom V µi R ijk by using the MC identity (2.39) and the expressions for the derivatives of M I J (2.44). Before calculating w ijk let us first derive several useful properties. First notice from theautomorphism identities that we have from the vanishing Q -flux that M il R lmn = 0 , (3.95)where we have used the fact that M il has an inverse. Hence the automorphism property for F ij k becomes simply F ij k = M il M jm ( M − ) nk F lmn . Additionally it will be useful to calculate Notice that in this parametrisation λ enters ˙ E , and for this reason turning on λ is not necessarily a solution-generating technique. It is in fact known that the λ -deformation is not a marginal deformation of the WZWmodel [73], and in general one has to add RR fluxes to get a supergravity solution [76, 52, 77]. V = dmm − in terms of m = exp( m i T i ). In general we have ∂ µ mm − = V µi T i + V µi T i = ∞ X N =0 V ( M ) µ , V ( N ) µ ≡ N + 1)! ad Nm ∂ µ m. (3.96)Using the commutation relations in r we have the following expressions V µi = ∞ X N =0 N + 1)! ∂ µ m j ( mF N ) ji V µi = ∞ X N =1 N + 1)! ∂ µ m j N − X K =0 ( − ) K ( mF N − − K · mH · ( mF t ) K ) ji (3.97)where we have defined ( mF ) ij ≡ m l F lij ,( mF t ) ij ≡ m l F lj i and ( mH ) ij ≡ m l H lij , and which canbe proved by induction using the relation V ( N +1) = N +2 ad m V ( N ) . An important consequenceis that using the Bianchi identities F ij [ k R lm ] j = 0 and H ijk R klm = 0 we have V µi R ijk = 0 . (3.98)In particular with H ijk R klm = 0 this is easily seen for the case K = 0 in (3.97). When K = 0 itis sufficient to verify that, upon using F ij [ k R lm ] j = 0 enough times, eventually an index of R ijk will be contracted with an index of H ijk . Using all of the above in the calculation for w ijk wefind that the contributions of the R -flux will vanish w ij k = F ij k − ρ iµ ρ j ν V µl V νm R lmn M kn , = F ij k . (3.99)The same is true when trying to solve the H -flux equation in terms of the b -twist: in thecalculation of d ˜ b with ˜ b µν = ρ µi M ik M jk ρ νj the contributions received from R ijk are all of theform V µi R ijk . Hence, also in this orbit the solution for the b -twist is given in eq. (3.64).From solving the F -flux equation of (2.35) we now have the condition β kl H ijl = 0, againlike in the ( F, H )-orbit. Instead of the parametrisation ρ iµ = M ij V j µ we might now as well take ρ = g − dg with g ∈ exp f some group element of the Lie group associated to f . Then solving for Q ijk = 0 gives β ij = h r g t i , t j i , with r g = Ad − g ◦ r ◦ Ad g , t i = κ ij t j and r a constant antisymmetricoperator. Notice only the difference in generators compared to eq. (3.4). Furthermore r mustadditionally satisfy r kl H ijl = 0 . (3.100)The R -flux equation, on the other hand, gives(Ad g ) li (Ad g ) mj (Ad g ) nk R lmn = 3 r l [ i r j | m F lm | k ] , (3.101)where we have used that Ad g is an automorphism of f . Notice that the matrices (Ad g ) ij and M ij coincide if we parametrise g ∈ exp g as g = e m i t i , which is most easily seen when writingboth in a series expansion. Indeed in that case M ij = (Ad g ) ij = ∞ X N =0 N ! ( mF N ) ij . (3.102) We ignore again here the possibility of (outer) automorphisms of which the general discussion is given aroundeq. (2.26). M ij = 0 we also have from (2.45) that M il M j m M kn R lmn = R ijk where M ij = ( M − ) j i . These observations combined ensure that eq. (3.101) becomes simply R ijk = 3 r l [ i r j | m F lm | k ] . (3.103)When we consider R ijk = ακ il κ jm F lmk , with α ∈ R , eq. (3.103) becomes ακ il κ jm F lmk =3 r l [ i r j | m F lm | k ] which is precisely the mCYBE (3.39) on f for c = α as in the ( F, R ) orbit,cf. section 3.6.Interestingly, using a rigid O ( d, d ) transformation the R -flux can be traded for Q -flux whenwe take b = 0 and β = r in (2.29), so that this representative can be equally described in the( F, H, Q )-orbit. In particular, in that case we will have H ′ ijk = H ijk , F ′ ij k = F ij k , Q ′ ijk = − F il [ j r k ] l , R ′ ijk = 0 , (3.104)so that, as in the ( F, R )-orbit of section 3.6, the Q -flux are the structure constants of g r whoseLie bracket was defined in (3.42).On a different note, the ( F, H, R )-orbit that we are describing is a simpler orbit in disguiseif we can turn off at least one of the three types of fluxes. The possibilities that we haveare: (i) we can turn off the H -flux if it is of the form H ijk = − F [ ij l b k ] l − b il b jm b kn R lmn forsome constant antisymmetric matrix b satisfying 2 F il [ j β k ] l − b il R ljk + 2 b il b mn R ln [ j β k ] m = 0 forsome constant antisymmetric matrix β . The F - and R -flux will obtain in that case a shift, F ′ ij k = F ij k + b il b jm R lmk and R ′ ijk = R ijk + β l [ i R jk ] m b ml , which vanishes when we take β = 0(since then we must have b il R ljk = 0); (ii) to turn off the R -flux we find from Q ′ ijk = 0 andtaking b = 0 for simplicity in (2.29) the condition H imn β mj β nk − F il [ j β k ] l = 0, so that the R -flux should be of the form R ijk = β l [ i β j | m F lm | k ] . In that case H stays invariant while F receives a shift F ′ ij k = F ij k − H ijl β lk . Notice that while in the previous discussion we did takethe R -flux of this form with β = √ r , the condition (3.100) also implies F il [ j r k ] l = 0 which isinconsistent with the mCYBE unless α = 0 from the very beginning, which is actually not thecase that we want to consider. Finally (iii) we do not need to discuss the possibility of turningoff the F -flux since in that case we would end up in the ( H, R )-orbit which, as discussed before,can not be realised on the strong constraint with the methods of section 2.4, while here we dofind possible representatives.Finally let us close the discussion of this orbit by briefly commenting on the generalisedfluxes F I . We have F i = F ij j + 2 ∂ i ¯ λ, F i = − β jl F jli + 2 β il ∂ l ¯ λ . (3.105)Constancy of F i implies F ij k ∂ k ¯ λ = 0 while one can verify that constancy of F i is guaranteedby the Bianchi F ikj F k = 0. In addition, this condition as well as H ijk R klm = 0 implies also theBianchi F i F i = 0. At last we point out that when writing down all the Bianchi identities for F I explicitly one finds immediately that they are satisfied when f is semisimple, the dilaton isisometric, and r as well as f are unimodular. H, Q, R
In the (
H, Q, R )-orbit, the Bianchi identities imply that Q ijk are structure constants for ˜ g =span( ˜ T i ) and that the R -flux is a 3 cocycle of ˜ g . Furthermore we have H k [ ij Q l ] mk = 0 and H ijk R klm = 0. Within the ansatz of 2.4 this again implies several simplifications. First noticethat the R -flux will not alter the series expansion of the one-form dmm − given in (3.79) as wellas the expansion of M ij . Hence, upon the Bianchi H ijk R klm = 0 we have V µl R ljk = 0 , M il R ljk = 0 . (3.106)35et us now use the parametrisation U (2) of (2.41) where, recall, we have the general solution ofthe H -flux equation in terms of the two-form ω . For the expressions of the other flux equationsin terms of the yet unknown ξ ij we refer to the most general orbit discussed in section 2.4 bysetting F = 0 in the equations (2.54)–(2.57). Let us point out here that due to the presence of R -flux, ξ = 0 will not be a solution. Using (3.106) the expressions simplify to ξ kl H ijl = 0 , (3.107) ∂ µ ξ ij + 2 V µl ξ m [ i Q mj ] l = 0 , (3.108)3 Q l [ ij ξ k ] l = R ijk (3.109)Similarly as in section 3.12 the Bianchi identity H k [ ij Q l ] mk = 0 implies that V µl ξ m [ i Q mj ] l = 0 sothat ξ ij must simply be a constant matrix. However, given a rigid O ( d, d ) transformation it ispossible to turn off the R -flux for this particular constant ξ ij . In order to do so one should takein (2.29) the constant matrices b ij = 0 and β ij = − ξ ij such that H ijk β kl = 0. This rigid O ( d, d )will leave the other fluxes invariant and thus we are in fact describing the ( H, Q )-orbit (which,recall, is a genuine orbit modulo rigid O ( d, d )). Concluding, within the ansatz of section 2.4we cannot describe a particular non-trivial representative in the ( H, Q, R )-orbit and, therefore,we cannot describe a rigid T -transformation from the ( F, H, R )-orbit as a solution-generatingtechnique. It would be interesting to explore other ansatze for this purpose.
F, H, Q
The (
F, H, Q )-orbit describes what is known as a quasi-Manin triple. The generators ˜ T i spana subalgebra ˜ g of r with structure constants Q ijk , while the generators T i do not. Furthermorewe have the Bianchi F [ ij k H lm ] k = 0 as well as (2.15) and (2.16). To discuss this orbit we use theansatz of section 2.4 and the parametrisation U (2) of (2.41) for the twist. The general solutionof the H -flux equation was given in terms of ω ij in (2.46). The equations to be solved for theunknown ξ ij can be found in section 2.4 by setting R = 0 in (2.54)–(2.57). They are2 ξ kl H ijl + ξ km Λ ml F ij l = 0 , (3.110)ˆ ∂ i ξ jk + 2Λ il ξ m [ j Q mk ] l − ξ m [ j ξ k ] n H imn = 0 , (3.111)3 Q l [ ij ξ k ] l + 3 ξ l [ i ξ j | m F lm | k ] − ξ l [ i ξ j | m ξ | k ] n Λ np F lmp = 0 (3.112)In the first place, we always have the trivial solution ξ = 0. Recall that no genuine representativesexist in the ( F, Q, R )-orbit, so that we cannot employ a rigid T -transformation as a solution-generating technique in this case. On the other hand, it is possible to consider a rigid β -transformation relating this representative and the one of the ( F, H, R )-orbit. It would beinteresting to find also non-trivial solutions for ξ ij within this orbit in the hope of havingsolution-generating techniques mapping cases with different ξ ’s. We leave this problem open. Let us now include also the RR fields in the discussion of the solution-generating techniques,which in general may be relevant for type II backgrounds. RR fields in the doubled formulationwere discussed in various works, see for example [78, 79, 80]. Here we will employ the spinorialformulation of [78] following the rewriting of [20]. In this section we will review only the essen-tial ingredients of the construction, and we refer to appendix D for more details. One uses ademocratic formulation [81, 82] where all even (odd) forms from 1 to D = 10 are used for the36R fields strengths of type IIA (IIB). The RR potentials are encoded in a spinor | c i and theRR field strengths in a spinor | F i . Given Gamma matrices Γ A in 2 D -dimensions satisfyingthe Clifford algebra relations { Γ A , Γ B } = 2 η AB (notice the flat indices) we write Γ A = √ ψ A so that { ψ a , ψ b } = δ ba , { ψ a , ψ b } = 0, { ψ a , ψ b } = 0 are anticommutation relations for thefermionic oscillators ψ a , ψ a . Starting from the Clifford vacuum | i such that ψ a | i = 0 for all a , we rewrite the spinor | F i as | F i = D X p =0 e φ p ! ˆ F m ··· m p e a m · · · e a p m p ψ a · · · ψ a p | i . (4.1)This rewriting relies on the one-to-one map between spinors | F i and polyforms ˆ F , where ˆ F = P Dp =0 1 p ! ˆ F m ··· m p d x m · · · d x m p on R ,D − . Here we are using e am which is the (inverse) vielbeinfor the metric G mn . Importantly, ˆ F m ··· m p are the RR field strengths that are commonly used intype II supergravity (they are the ˆ F of [78]), and they are the ones that appear in the quadraticcouplings of the fermions in the Green-Schwarz formulation of the superstring (they are the F of [83]). The two spinors | F i and | c i are related by | F i = ( ψ A ∂ A − F ABC ψ ABC − F A ψ A ) | c i = / ∇ | c i , / ∇ ≡ /∂ − / F (3) − / F (1) , (4.2)where / F ( n ) includes the 1 /n ! factor. Gauge transformations of RR potentials read as δ λ | c i = / ∇ | λ i and Bianchi identities as / ∇ | F i = 0. Notice that these Bianchi identities are a consequenceof / ∇ = 0 which holds on the strong constraint. After defining Ψ − = ( ψ + ψ )( ψ − ψ ) · · · ( ψ − ψ ) we also impose the self-duality condition Ψ − | F i = | F i . Notice that this differs from [20],see appendix D. The self-duality condition translates into the duality conditions for the p -formsas ˆ F ( p ) = − ( − p ( p +1) ∗ ˆ F (10 − p ) .The transformation rules of RR fields ˆ F m ··· m p under the O ( D, D ) solution-generating tech-niques are found by the observation that | F i is in fact invariant under these transformations.Notice that knowing that the generalised fluxes F are invariant under the O ( D, D ) solution-generating techniques, it is obvious that keeping also | F i invariant ensures that even whenincluding the RR sector we still have a solution of the supergravity equations, and that wecorrectly satisfy the Bianchi identities and the constraints. The fact that we can identify adynamical field (in this case | F i ) that remains invariant under the transformation is again aconfirmation of the usefulness of the DFT formulation that we are employing.The fact that | F i is invariant does not mean that the RR fields ˆ F m ··· m p remain invariant.This is similar to what we saw in the NSNS sector: while the generalised fluxes are invariant, thebackground metric G mn , the field B mn and the dilaton φ do transform. To present the trans-formation rules it is useful to define | F i ≡ e − d S − E ] | F i where S [ E ] is a Spin ( D, D ) element corre-sponding to the O ( D, D ) element E . See the appendix for more details. Another useful rewritingof the spinor for the RR field strenghts is | ˆ F i ≡ S [ B ] | F i , where we use the Spin ( D, D ) element S [ B ] ≡ exp( − B mn ψ m ψ n ) with B mn the B-field of the supergravity background. We willthen write | F i = P Dp =1 1 p ! F m ··· m p ψ m · · · ψ m p | i and | ˆ F i = P Dp =1 1 p ! ˆ F m ··· m p ψ m · · · ψ m p | i ,which is compatible with (4.1). From the invariance of | F i and using that E = ˙ EU and d = ˙ d + λ ,it follows that | ˙ F i ≡ e λ S [ U ] | F i , (4.3) Our | F i is G of [20]. Our | F i and | ˆ F i appearing later are respectively | F i and | ˆ F i of [78]. The Hodge dual is defined as ( ∗ A ) m ··· m p = − p )! G m n · · · G m p n p ε k p +1 ··· k n ··· n p A k p +1 ··· k where ǫ ··· = 1, ǫ ··· = − ε m ··· m = √− G ǫ m ··· m , ε m ··· m = √− Gǫ m ··· m . We also have ∗ ∗ ω ( p ) = − ( − p (10 − p ) sω ( p ) where the additional minus sign is due to the Lorentz signature.
37s also invariant under the O ( D, D ) transformations. If we use primes to denote the new back-ground related by the O ( D, D ) transformation, this fact can be exploited to compute the RRfields of the new solution just by identifying | ˙ F i = e λ ′ S [ U ′ ] | F ′ i .From an operational point of view, starting from the polyform ˆ F = P Dp =1 1 p ! ˆ F m ··· m p d x m · · · d x x p of a supergravity solution, we can get F = exp( B (2) ) ∧ ˆ F with B (2) ≡ B mn d x m ∧ d x n , whichis a consequence of the relation between the corresponding spinors. To obtain ˙ F we write the Spin ( D, D ) element S [ U ] = S [ ρ ] S [ ¯ β ] S − b ] which follows from the rewriting of the twist U as U = (cid:18) β (cid:19) (cid:18) b (cid:19) (cid:18) ρ t ρ − (cid:19) = (cid:18) ρ t ρ − (cid:19) (cid:18) β (cid:19) (cid:18) b (cid:19) (4.4)where ¯ β = ρ − t βρ − and ¯ b = ρbρ t have curved indices. We prefer this rewriting because itsimplifies the translation of the action on polyforms. In fact | F (¯ b ) i = S − b ] | F i = e ¯ b µν ψ µ ψ ν | F i = ⇒ F (¯ b ) = e ¯ b ∧ F = F + ¯ b ∧ F + 12!¯ b ∧ ¯ b ∧ F + . . . , (4.5)where ¯ b ≡ ¯ b µν dy µ ∧ dy ν and | F ( ¯ β ) i = S [ ¯ β ] | F i = e ¯ β µν ψ µ ψ ν | F i = ⇒ F ( ¯ β ) = e ¯ β ∨ F = F + ¯ β ∨ F + 12! ¯ β ∨ ¯ β ∨ F + . . . , (4.6)where ¯ β ∨ F = ¯ β µν ι µ ι ν F and ι m dx n = δ nm , ι m ( ω ( p ) ∧ χ ( q ) ) = ι m ω ( p ) ∧ χ ( q ) + ( − p ω ( p ) ∧ ι m χ ( q ) .Finally one has S [ ρ ] | F i = (det ρ ) / exp( − ψ m R mn ψ n ) X p p ! F m ··· m p ψ m · · · ψ m p | i = (det ρ ) / X p p ! F m ··· m p ρ i m · · · ρ i p m p ψ i · · · ψ i p | i (4.7)where ρ = exp R acts as the identity on spectator coordinates ˙ x ˙ µ and coincides with ρ on the y -block. In other words the only action of S [ ρ ] is to translate curved indices µ, ν into algebra-likeindices i, j , and multiply by (det ρ ) / . Notice that this factor cancels in | ˙ F i once we rewrite λ = ¯ λ − log det ρ . With the above formulas and starting from ˇ F , U, λ one can obtain ˙ F ,and given another representative with U ′ , λ ′ one can similarly compute F ′ and ˆ F ′ by the sameformulas.As we argued in the previous sections, the solution-generating techniques that we are study-ing cover not only those that leave the generalised fluxes invariant. They also cover those thatrelate different sets of generalised fluxes by constant O ( D, D ) transformations, the prominentexample being the rigid T -transformation implemented by the matrix in (2.30). Let us show howthe RR fields transform when this T -transformation is involved. When the fluxes are related as in F ′ A ′ B ′ C ′ = T A ′ A T B ′ B T C ′ C F ABC and F ′ A ′ = T A ′ A F A , from (4.2) and using S [ T ] ψ A S − T ] = ψ B T BA one sees that we have the relations | F i ′ = S [ T ] | F i and | c i ′ = S [ T ] | c i , provided that | c i is takento be independent of the coordinates that are being dualised. Notice that the chirality of thespinors remains the same if we dualise in even d dimensions, and changes in odd d . In addi-tion notice that this transformation gives Ψ ′− = S T Ψ − S − T = ( − d Ψ − , therefore in d -evendimensions it preserves the self-duality condition, while in d odd it changes the sign, see also [78]. Notice that the ansatz for the spinor taken in [84] is equivalent to this condition, and a similar condition istaken also in [49, 85]. In the presence of RR fields we assume that we do not dualise the time direction.
38o obtain the transformation rules of RR fields one uses the fact that the relation to thedualised model is via a twist˜ U = (cid:18) (cid:19) (cid:18) β (cid:19) (cid:18) b (cid:19) (cid:18) ρ t ρ − (cid:19) = (cid:18) ρ − ρ t (cid:19) (cid:18) (cid:19) (cid:18) β (cid:19) (cid:18) b (cid:19) (4.8)that now has an additional matrix implementing the T -transformation. Also in this case it ispreferable to pull the GL ( d ) block with ρ to the left, because this rewriting permits to implementthe action of the T-duality matrix (the second block in the last equation) on polyforms in a simpleway. Notice that because of the above rewriting we can think of the operation as a dualisation of d coordinates not just in the case of abelian T-dualities, but also in their generalisations. Whendualising more than one coordinate we can think of it as a factorised product of single T-dualitiesalong the m direction implemented by S [ µ ] = ( ψ µ − ψ µ )( − N F . Under such transformation onehas in polyform notation | F ( µ ) i = S [ µ ] | F i = ⇒ F ( µ ) = F ∧ dy µ + F ∨ dy µ , (4.9)where if A is a p -form A ∨ dy µ = ( − p − dy µ ∨ A = ( − p − ι µ A . A similar point of view toobtain the transformation rules of RR fields under PL duality was used in [85]. We have discussed an ansatz for the generalised vielbein of DFT by demanding that it takesa “twisted” form, as in generalised Scherk-Schwarz reductions of D -dimensional backgroundson d -dimensional spaces, and that the twist U gives rise to constant generalised fluxes in the d -dimensional space when imposing the strong constraint of DFT. The results are organised asin Figure 1 into orbits depending on which of the different fluxes F, H, Q and R are turned on,and in general they can be related by rigid T -transformations by F ↔ Q and H ↔ R , or moregeneric O ( d, d ) transformations. Our classification of the representatives is complete when the H -flux is vanishing, while we have employed particular methods for the cases with non-trivial H -flux which may not cover all possible representatives.When an orbit contains more than one representative, or when by means of a rigid O ( d, d )transformation (including a rigid T -transformation) we can relate it to another orbit that admitsrepresentatives, we can view the maps relating two of them as a solution-generating techniquein (super)gravity, as well as a canonical transformation at the level of the σ -models. Thereforewhen one of the two σ -models is classically integrable it follows that the other enjoys the sameproperty. Our results add new possibilities to the known zoo of generalised T-dualities andYang-Baxter deformations. Generic O ( d, d ) transformations relating possibly different orbitsmay be seen as generalisations of the so called PL- plurality , which is traditionally defined as thepossibility of decomposing in different ways the same Drinfel’d double. Let us remark that in thispaper we have only imposed the O ( d, d ) symmetry of the background (i.e. the fact that it takesthe form of a generalised Scherk-Schwarz ansatz and that it gives rise to constant F IJK , F I ). Atno point we have imposed the DFT (or supergravity) equations of motion, and this is actuallynot relevant if one is only interested in canonical transformations of σ -models. In general, onemay have to choose appropriate RR fields and/or spectator background fields in order to solvethe DFT/supergravity equations. Similar comments apply to the integrability of the σ -modelsgiving rise to the backgrounds that we describe. At no point we imposed the existence of a Laxconnection, and in general this may introduce additional conditions.Let us now summarise our classification. In the ( F )-orbit one can independently switchon a twist β satisfying the CYBE, which corresponds to possible (homogeneous) YB deforma-39ions [27, 28, 26] of isometric backgrounds. This orbit is related by rigid T -transformations tothe ( Q )-orbit, which contains backgrounds that arise from applying NATD [4] to the ones in the( F )-orbit, and more generally include the DTD models of [54, 29]. The ( F, Q ) orbit containsrepresentatives that are PL-symmetric [11, 12], and it allows also for novel deformations that canbe understood as the natural generalisation of the homogeneous YB-deformations, now withoutthe need of having isometries in the initial background. The ( R )-orbit is empty, it contains nonon-trivial representative when demanding the strong constraint. The ( F, R ) orbit contains theso-called inhomogeneous YB-model (or η -model) [27, 28], slightly generalised in our treatmentby the presence of spectators. The ( Q, R ) and (
F, Q, R ) orbits are trivial in the sense that onecan turn off the R -flux by a rigid O ( d, d ) transformation.When the H -flux is non-zero our classification is not exhaustive, because the methods ofsection 2.4 that we employ in these cases do not guarantee that we are covering all possiblerepresentatives. Nevertheless we identify interesting possibilities for the ( H ) , ( F, H ) , ( H, Q ) and(
F, H, R ) orbits. The (
H, R ) and (
H, Q, R ) orbits seem trivial within these methods since the R -flux can again be turned off by a rigid O ( d, d ). The ( F, H )-orbit contains the PCM+WZ modelas well as a novel generalisation of the homogeneous YB-deformation in which the YB-operatoris constrained to be compatible with H . The ( H, Q ) orbit contains the integrable (asymmetrical) λ -model [73, 74] as well as additional novel deformations by 2-cocycles compatible with the H -flux, which are reminiscent of the deformations of NATD in the ( Q )-orbit that were called DTD.In the ( F, H, R )-orbit we can describe a generalisation of the inhomogeneous YB-model as wellas the fact that a rigid O ( d, d ) relates it to the ( F, H, Q )-orbit. Within the methods that we use,the (
F, H, Q )-orbit admits at least the solutions of [49]. The question of having more generalrepresentatives in this orbit, as well as representatives in the most general (
F, H, Q, R )-orbit,remains open. Finally in the ( H )-orbit we identified possible deformations of the torus with H -flux without relying on the methods of section 2.4. This confirms that it should be possibleto go beyond these methods, and it would certainly be interesting to do so in a systematic wayin the other orbits with non-vanishing H -flux.A general observation that we made, which generalises previously known results, is that aconstant automorphism W of the pre-Roytenberg algebra r (modulo gauge transformations) maygenerate a new representative from a known representative for the twist. They must not be mis-taken with the rigid O ( d, d ) transformations of (2.24), since these automorphisms do not involvea compensating transformation of the spectator background. For instance, in the ( F )-orbit theautomorphism reduced to the subalgebra generated by F ij k must be an outer automorphism.An interesting possibility is that for non-semisimple algebras outer automorphisms may involvecontinuous parameters and can therefore be seen as deforming the background.While we have focused on the NSNS sector in most of the paper, in section 4 we haveexplained how to obtain the transformation rules of RR fluxes by demanding that the mapsunder consideration are in fact mapping type II solutions to type II solutions.Let us now comment on several interesting open questions.While we have discussed in our classification of representatives various known integrable2-dimensional σ -models, it would be interesting to rewrite in this language also others such asthe bi-YB-deformation of [86, 71, 72] and the deformations of [87].In this paper we have not analysed the special case of solution-generating techniques involv-ing (super)cosets, and this would be a very interesting future direction. In the (super)coset casethe spectator background fields are expected to project on the coset part of the algebra, and We stress that we cannot view it as a solution-generating technique from the PCM+WZ model since we aregoing from the (
F, H ) to the (
F, H, R ) orbit by turning on R . O ( d, d ) generalised fluxes F IJK , F I , see [88, 50] for an ob-servation along these lines. In this paper we cannot view the inhomogeneous YB-deformations(the η -deformation) as a solution-generating technique from η = 0 to η = 0, because in generalit entails going from the ( F )-orbit to the ( F, R )-orbit, but it is possible that the (unimodular) η -deformation of the superstring [89] turns out to be a solution-generating technique becauseof these additional features of the supercoset. The transformation rules of the NSNS and RRfields [52] are in fact strongly suggesting the underlying O ( D, D ) structure also in this case. Sim-ilar comments apply to the λ -deformation of the superstring [76], as well as to the constructionof [77].Another possible future direction, which is also necessary for the point above, is to discusssolution-generating techniques that involve super algebras. It is likely that for this purposethe formulation of [80] for type II superstrings in DFT language will be more useful than theformulation of [78].In our discussion we focused only on backgrounds defined on local patches and we ignoredpossible global issues. It would be interesting to employ the methods of [38] where generalisedLeibniz parallelisable spaces are constructed for generalised Scherk–Schwarz uplifts of gaugedsupergravities, since this may shed some light for example on the global issues of NATD.It would be interesting to relax some assumptions that we have made. In particular, thesolution-generating techniques discussed in this paper arise by demanding that the generalisedfluxes (and their flat derivatives) remain invariant under the map. To find more general solution-generating techniques in supergravity, one may try to look for more complicated symmetries ofthe DFT equations of motion that do not necessarily leave the generalised fluxes invariant,see [50] for a step in this direction. Additionally we were interested in backgrounds that satisfythe strong constraint of DFT, namely those whose fields depend only on the physical coordinates y and not on the dual ˜ y . In the context of gauged DFT it is possible to relax the strong constraintas done in [45], and it would be interesting to look at this generalisation of the classificationas well. One may also relax our analysis by looking for solution-generating techniques in thecontext of the generalised supergravity equations of [90, 91]. Acknowledgements
We thank S. Demulder, B. Hoare, S. Lacroix, D. Osten and L. Wulff for useful discussions,and we are very grateful especially to D. Marqu´es for discussions and for inspiring this project.We thank D. Marqu´es, D. Thompson and L. Wulff for comments on the manuscript. Thiswork was supported by the fellowship of “la Caixa Foundation” (ID 100010434) with codeLCF/BQ/PI19/11690019, by AEI-Spain (FPA2017-84436-P and Unidad de Excelencia Mar´ıade Maetzu MDM-2016-0692), by Xunta de Galicia-Conseller´ıa de Educaci´on (Centro singularde investigaci´on de Galicia accreditation 2019-2022), and by the European Union FEDER.
A Notation
We follow the convention of using a boldface notation for the objects (and their indices) of the full D -dimensional space, both before and after the doubling (e.g. we have the metric G mn and thegeneralised fluxes F ABC ), and the same notation but without boldface for the correspondingquantities in d dimensions, (e.g. G mn and F ABC ). A recap on our indices conventions is as41ollows: m, n , . . .
Curved D -dimensional indices of coordinates x m µ, ν, . . . , α, β, . . . Curved d -dimensional indices of y µ ˙ µ, ˙ ν, . . . Curved ( D − d )-dimensional indices of spectator coordinates ˙ x ˙ µ M, N , . . .
Curved O ( D, D ) indices
M, N , . . .
Curved O ( d, d ) indices A, B , . . .
Flat O (1 , D − × O ( D − , I, J , . . .
Algebra indices in 2 d dimensions i, j, . . . Algebra indices in d dimensions B Brief recap on DFT and gDFT
The generalised vielbein may be parametrised as E AM = 1 √ e (+) an ( G − B ) nm e (+) am − e ( − ) a n ( G + B ) nm e ( − ) a m ! . (B.1)Here e ( ± ) are two vielbeins for the metric G mn and B mn is the Kalb-Ramond field. The gener-alised vielbein satisfies the following relations with the O ( D, D ) metric (1.1) and the generalisedmetric (1.2) η MN = E AM η AB E BN , H MN = E AM H AB E BN , (B.2)where η AB = (cid:18) ¯ η ab − ¯ η ab (cid:19) , H AB = (cid:18) ¯ η ab
00 ¯ η ab (cid:19) , (B.3)Curved indices M, N, . . . are raised and lowered with η MN and η MN , while flat indices A, B, . . . with η AB and η AB . The generalised dilaton is d = φ − log( − det G ) with φ the usual dilaton. The generalised fluxes are defined as F ABC = 3 Ω [ ABC ] , F A = Ω BBA + 2 E AM ∂ M d (B.4)in terms of the generalized Weitzenb¨ock connection Ω ABC = E AM ∂ M E BN E CN , (B.5)and they satisfy the following Bianchi identities ∂ [ A F BCD ] − F [ ABE F CD ] E = Z ABCD , ∂ [ A F B ] + ( ∂ C − F C ) F ABC = Z AB ,∂ A F A − F A F A + F ABC F ABC = Z (B.6)where Z ABCD ≡ − Ω E [ AB Ω ECD ] , Z AB ≡ ( ∂ M ∂ M E [ AN ) E B ] N − Ω C AB ∂ C d , Z ≡ − ∂ A d ∂ A d + 2 ∂ A ∂ A d + Ω ABC Ω ABC . (B.7)42n the strong constraint one has Z ABCD = 0, Z AB = 0 and Z = 0.A generalised diffeomorphism is implemented on tensors V M N and on the generalised dilaton(transforming as a density) as δ ξ V M N = ˆ L ξ V M N = ξ P ∂ P V M N + ( ∂ N ξ P − ∂ P ξ N ) V M P + ( ∂ M ξ P − ∂ P ξ M ) V P N ,δ ξ e − d = ˆ L ξ e − d = ∂ M ( ξ M e − d ) , ⇐⇒ δ ξ d = ˆ L ξ d = ξ M ∂ M d − ∂ M ξ M , (B.8)where ˆ L ξ is the generalised Lie derivative and ξ M the parameter of the transformation. On thestrong constraint the generalised fluxes transform as scalars under generalised diffeomorphisms.In terms of the generalised Lie derivative they may be written also as F ABC = ˆ L E A E BM E CM , F A = 2 ˆ L E A d . (B.9)The strong constraint is again a sufficient condition also for the closure of the algebra of gener-alised diffeomorphisms, so that [ ˆ L ξ , ˆ L ξ ] = ˆ L ξ with ξ = [ ξ , ξ ] M (C) = ( δ ξ ξ M − δ ξ ξ M ) =2 ξ P [1 ∂ P ξ M + ∂ M ξ [1 P ξ P given by the C-bracket.The point of view of this paper is similar to the setup of gauged DFT [92, 37]. Ratherthan the original interpretation of [92], where the O ( D, D ) theory is gauged by shifting thegeneralised fluxes by some constant gaugings, we are closer to the interpretation of [37] where anappropriate generalised Scherk-Schwarz (gSS) reduction from D to ( D − d ) dimensions essentiallygives the same construction, i.e. generalised fluxes for the ( D − d )-dimensional theory that aregauged by the fluxes of the d -dimensional space. If tensors are decomposed as in the gSSansatz V M N ( x ) = ˙ V I J ( ˙ x ) U I M ( y ) U J N ( y ) then generalised diffeomorphisms with parameter ξ M = ˙ ξ I U I M respect the ansatz in the sense that δ ξ V M N = ˙ δ ˙ ξ ˙ V I J U I M U J N where in the caseof a vector ˙ δ ˙ ξ ˙ V I = δ ˙ ξ ˙ V I + F I JK ˙ ξ J ˙ V K . Imposing the strong constraint both on the “externalspace” (with coordinates ˙ x ) and “internal space” (with coordinates y ) in a sufficient condition forclosure of the algebra of the ˙ δ ˙ ξ transformations, since [ ˙ δ ˙ ξ , ˙ δ ˙ ξ ] = ˙ δ ˙ ξ where ˙ ξ I = [ ˙ ξ , ˙ ξ ] I ( F ) ≡ [ ˙ ξ , ˙ ξ ] I (C) + F I JK ˙ ξ J ˙ ξ K . Notice the relation to the C-bracket [ ξ , ξ ] M (C) = [ ˙ ξ , ˙ ξ ] I ( F ) U I M . Whilethe strong constraint on the internal space is a sufficient condition to define consistently gaugedDFT, it may be relaxed, as long as Z IJKL = Z IJ = Z = 0. B.1 Geometric interpretation
The fact that the solution-generating techniques that we discuss do not modify the generalisedfluxes when written in flat indices, still allows for the possibility of finding maps between ge-ometric and (globally) non-geometric backgrounds. (Non)geometry is in fact captured by thefluxes written in curved indices — otherwise their definition in flat indices is dependent on thechosen double-Lorentz gauge, see [40] for a discussion. After going from flat to curved indices,the independent components of F MNP are usually divided into geometric fluxes F mnp , H mnp and non-geometric fluxes Q mnp , R mnp [40], see also [20, 93]. These fluxes are often used todistinguish between • geometric backgrounds , i.e. backgrounds that are well defined globally when using diffeo-morphisms and B-field gauge transformations as transition functions in different patches.They have Q mnp = R mnp = 0. • globally non-geometric but locally geometric backgrounds , e.g. T-folds, that require T-duality transformations to glue different patches. They have Q mnp = 0 and R mnp = 0.43 locally non-geometric backgrounds , i.e. depending on dual coordinates. They have Q mnp =0 and R mnp = 0.We do not consider locally non-geometric backgrounds because we always impose the strongconstraint. If F ′ ABC = F ABC then the generalised fluxes in curved indices are related as F ′ MNP = h M Q h N R h P S F QRS , where h M Q = E ′ M A E AQ = U ′ M I U I Q . (B.10)Notice that because of the assumptions we made on the twists U , U ′ , the matrix h M Q is ofblock diagonal form as U in (2.3), i.e. it acts as the identity in the block for the (doubled) ˙ x coordinates, while it is h M Q = U ′ M I U I Q , (B.11)in the block for the y -coordinates. If h M Q has off-diagonal components h µρ , h µρ then it is possiblethat the type of fluxes (in curved indices) change after the transformation. To show examples wesay how they change in solution-generating techniques involving the ( F ) and ( Q )-orbits. Noticethat F MNP = U M I U N J U P K ( ˙ F IJK + F IJK ) , (B.12)where ˙ F IJK = ˙ E I A ˙ E J B ˙ E KC ˙ F ABC and ˙ F IJK = 3 δ [ I ˙ µ ˙ E J | B ∂ ˙ µ ˙ E B | K ] . Because of the deriva-tive at least one leg is dotted in ˙ F IJK , and the corresponding U and h act as the identity onthat leg. The other two legs are not necessarily in the dotted directions, so that U and h can actnon-trivially on them. Therefore let us stress that in general it is important to include the con-tribution of ˙ F IJK to understand how the fluxes change under the transformation. However, inorder to have a more concrete discussion, we will look in the following only at the piece comingfrom F IJK , and we remind that this is non-zero only for
IJ K = IJ K . We are therefore lookingat flux components with all legs along y -directions, and these do not mix with the contributionsfrom ˙ F that have at least one leg along dotted coordinates. Hence we will analyse only thesecontributions to the fluxes F MNP = U M I U N J U P K F IJK , F ′ MNP = U ′ M I U ′ N J U ′ P K F IJK , (B.13)that are related as F ′ MNP = h M Q h N R h P S F QRS . (B.14)Separating the components we have F ′ µνρ = h µα h νβ h ργ F αβγ + 3 h [ µα h ν β h ρ ] γ F αβ γ + 3 h [ µα h ν | β h | ρ ] γ F αβγ + h µα h νβ h ργ F αβγ , F ′ µν ρ = h µα h νβ h ργ F αβγ + ( h µα h ν β h ργ + 2 h [ µβ h ν ] γ h ρα ) F αβ γ + ( h µβ h νγ h ρα + 2 h [ µα h ν ] β h ργ ) F αβγ + h µα h νβ h ργ F αβγ , F ′ µνρ = h µα h νβ h ργ F αβγ + ( h µγ h να h ρβ + 2 h µα h [ ν | β h | ρ ] γ ) F αβ γ + ( h µα h νβ h ργ + 2 h µβ h [ νγ h ρ ] α ) F αβγ + h µα h ν β h ργ F αβγ , F ′ µνρ = h µα h νβ h ργ F αβγ + 3 h [ µ | α h | ν | β h | ρ ] γ F αβ γ + 3 h [ µ | α h | νβ h ρ ] γ F αβγ + h µα h ν β h ργ F αβγ (B.15) In the standard example of the T-duality chain on the 3-torus with H -flux done for example in [67] onedualises one leg at a time, and therefore the contribution to the flux that is discussed in each step is indeedcoming from ˙ F IJK , with two legs undotted. On the O ( d, d ) parametrisation of the twist Here we explain why we take (2.21) as a parametrisation of the twist U . The O ( d, d ) group isgenerated by the matrices [93] R M N = (cid:18) ( ρ t ) µν
00 ( ρ − ) µν (cid:19) , J M N = (cid:18) δ µν b µν δ µν (cid:19) , T [ σ ] NM = δ µν − u µ ( σ ) ν u µν ( σ ) u ( σ ) µν δ µν − u ( σ ) νµ ! , (C.1)where ρ ∈ GL ( d ), b t = − b and u ( σ ) = u ( σ ) = diag(0 , . . . , , , , . . . , σ ∈ { , . . . , d } . The matrices R and J generate two separate subgroups, whichtogether generate a larger subgroup of O ( d, d ) that we call G geom . Notice that RJ = J ′ R, with b ′ = ρ − bρ − t . Hence we can always use this move to write any element of G geom e.g. in the order J R . Each T [ σ ] can be understood as a T-duality along the direction σ . When we involve thematrices T [ σ ] we generate also different types of matrices. A distinguished one is given by themultiplication of all the T [ σ ] T M N = ( T [1] T [2] . . . T [ d ] ) M N = (cid:18) δ µν δ µν (cid:19) , (C.2)which we will call a rigid T -transformation since it is constant. Notice that T = 1 and that T is not equivalent to the O ( d, d ) metric η , because of the position of the indices in the definitions.We also have matrices S ≡ T R S
M N = (cid:18) σ − ) µν ( σ t ) µν (cid:19) , (C.3)with σ ∈ GL ( d ), which form a subgroup on their own. Notice that T RT = R ′ where ρ ′ = ρ − t .Finally, we also have matrices K ≡ T J TK
M N = (cid:18) δ µν β µν δ µν (cid:19) , (C.4)with β t = − β , which again form a subgroup on their own. More complicated elements occur ifinstead of T we multiply by single T [ σ ] matrices. However, we will not consider this case heresince we will assume that we have already reduced from D to d dimensions, so that T-duality isimplemented on all the d coordinates.One way to classify the possibilities in the parametrisation of the most general O ( d, d ) elementis to observe whether they involve an even or an odd number of T ’s, which we consequently call“even” and “odd” elements respectively. Let us first analyse the most generic parametrisation foran even twist and argue that they are generated by the matrices R, J, K . As we have remarked
T RT = R ′ and we can pull any R to the right of any J . Notice that the same considerationsapply for K . Therefore, any even twist can be written as a generic product of J ’s and K ’s timesa single R . Finally notice that we can always write J K = K ′ J ′ R ′ where ρ ′ = (1 + bβ ) − , b ′ = b (1 + βb ) , β ′ = β (1 + bβ ) − , (C.5) if we assume that 1+ bβ is invertible. Therefore, under this assumption we can always re-arrangethe order of the matrices J and K in the product, at the cost of introducing a matrix of type R . We can therefore conclude that any twist U of even type can be written as the product U even = KJ R, (C.6)45or some
K, J, R . Notice, however, because of the previous assumption we may be excludingtwists involving products of the type
J K with 1 + bβ not invertible. Let us finally discuss themost generic parametrisation for odd twists. As they must be generated by an odd numberof T ’s we can write them as a generic product of U even ’s and T ’s (odd). By noticing that T KJ R = (
T KT )( T J T )( T RT ) T = J ′ K ′ R ′ T = K ′′ J ′′ R ′′ T where we used T = 1, we concludethat we can always pull T to the left, for example. Then the most generic parametrisation foran odd twist is U odd = T KJ R. (C.7)The only effect of the rigid T matrix in the odd twist is to map an orbit to its dual. D Details on RR fields and type II
In this appendix we collect some additional details that are useful for the DFT formulation oftype II supergravity. We closely follow [78] and [20] and highlight the differences when present.In this appendix we will always assume that D = 10. We will also remove the boldface fromall objects and indices, since the reduction from D to d dimensions is not relevant here, and weprefer to have a simpler notation.Given the Clifford algebra C ( D, D ) generated by the Gamma matrices Γ M satisfying { Γ M , Γ N } =2 η MN (notice the curved indices as in [78]) we can define ψ M = Γ M / √ ψ m , ψ m satisfying { ψ m , ψ n } = δ nm , { ψ m , ψ n } = 0, { ψ m , ψ n } = 0. The Clifford vacuum | i is defined such that ψ m | i = 0 for all m . A generic spinorcan be rewritten as | χ i = P Dp =0 1 p ! C m ··· m p ψ m · · · ψ m p | i which is consequence of the one-to-onemap between spinors χ and polyforms C = P Dp =0 1 p ! C m ··· m p dx m · · · dx m p on R ,D − . One fixesthe normalisation h | i = 1 and defines conjugation such as ( ψ m · · · ψ m p | i ) † = h | ψ m p · · · ψ m . P in ( D, D ) is a subgroup of C ( D, D ), see [78] for its definition. The algebras of O ( D, D ) and
P in ( D, D ) are isomorphic and if we call J MN the generators of the algebra of O ( D, D ) satisfying[ J MN , J P Q ] = η MP J QN − η NP J QM − η MQ J P N + η NQ J P M then in the spinor representation wecan identify J MN ≡ Γ MN , where Γ M ··· M p is the totally antisymmetric product of p Gammamatrices, with the factor 1 /p ! included, e.g. Γ MN = [Γ M , Γ N ]. We denote by ρ the grouphomomorphisms ρ : P in ( D, D ) → O ( D, D ). P in ( D, D ) is in fact the double cover of O ( D, D )because ρ ( S ) = ρ ( − S ) for S ∈ P in ( D, D ). We have S Γ M S − = Γ N h N M , where ρ ( S ) = h •• . (D.1)This is a change of conventions compared to [78], where they define ρ such that ρ ( S ) = h •• .Notice that this amounts to swapping the two diagonal blocks of h , and the two off-diagonalblocks. Spin ( D, D ) is the subgroup generated by the S made out of only an even number ofGamma matrices and under the group homomorphism ρ it is mapped to SO ( D, D ).There are some distinguished elements of
Spin + ( D, D ) that are important for the rest of thesection. Their definition and the corresponding SO + ( D, D ) elements are S [ b ] = exp( − b mn ψ m ψ n ), S [ β ] = exp( β mn ψ m ψ n ), S [ r ] = (det r ) − / exp( ψ m R mn ψ n ), h [ b ] = (cid:18) − b (cid:19) , h [ β ] = (cid:18) β (cid:19) , h [ r ] = (cid:18) r − T r (cid:19) .Above we have r mn = (exp R ) mn ∈ GL + ( D ), where the plus stands for elements with a positivedeterminant. Other useful elements of P in ( D, D ) that are not in
Spin ( D, D ) are S ± m = ψ m ± ψ m .46hey satisfy ( S ± m ) = ± ρ ( S + m ) = − (cid:18) − u m − u m − u m − u m (cid:19) , ρ ( S − m ) = − (cid:18) − u m u m u m − u m (cid:19) , (D.2)where u m = diag(0 , . . . , , , , . . . ,
0) has 1 only at position m . It is nevertheless preferable toimplement one factorised T-duality along the direction m as in [85] by S [ m ] = ( ψ m − ψ m )( − N F with ( − N F to be defined below. In fact this corresponds to S [ m ] ψ M S − m ] = ψ N T [ m ] N M where T [ m ] implements T-duality along direction m , and acts as the identity on other coordinates,without other unwanted signs.The charge conjugation matrix is defined as C = ( ψ − ψ )( ψ − ψ ) · · · ( ψ − ψ ) and it satisfies C − = − C . One has the relations Cψ m C − = ψ m , Cψ m C − = ψ m and C Γ M C − = (Γ M ) † , C Γ M C − = (Γ M ) † . Defining the T-duality matrix as in (2.30) one sees that C Γ M C − = T M N Γ N and therefore ρ ( C ) = T . Under conjugation C † = C − , and we have that S † = CS − C − if S ∈ Spin + ( D, D ), while S † = − CS − C − if S ∈ Spin − ( D, D ). Moreover ( S + m ) † = CS + m C − .One can define chiral spinors from the number operator N F = P m ψ m ψ m that gives N F | χ i p = p | χ i p and ( − N F | χ i = P Dp =0 ( − p p ! C m ··· m p ψ m · · · ψ m p | i . Chiral projections of spinors aretherefore defined as | χ ± i = (1 ± ( − N F ) | χ i so that ( − N F | χ ± i = ± | χ ± i . We have that χ + is mapped to even forms and χ − to odd forms. Notice that the action of Spin ( D, D ) preservesthe chirality.The generalised metric H is an element of SO − ( D, D ) because it has determinant 1 andwe are in Lorentz signature. One therefore takes an S ∈ Spin − ( D, D ) such that ρ ( S ) = H •• .Notice again the change in conventions, in [78] one has ρ ( S ) = H •• . Here it is S rather than H that is considered the fundamental field. We have S = S † = − C S − C − . Let us furtherdefine S [¯ η ] = ψ ψ − ψ ψ , then S [¯ η ] = S † [¯ η ] = S − η ] ∈ Spin − ( D, D ). If G mn is the NSNSmetric of the background and G = e ¯ ηe T in terms of a standard vielbein e and the Minkowskimetric ¯ η , then we can define S [ G ] ≡ S [ e ] S [¯ η ] S † [ e ] where S [ e ] = S [ r = e ] , and where we used thedefinition of S [ r ] above. We then have ρ ( S − G ] ) = diag( G, G − ). Given the NSNS field B mn ofthe background we can further define S [ B ] ≡ S [ b = B ] using the definition of S [ b ] above. Then ifwe define S H ≡ S † [ B ] S − G ] S [ B ] we have ρ ( S [ H ] ) = H •• . Notice that because of our identification ρ ( S ) = h •• , if we consider an O ( D, D ) transformation H ′ = h T H h on the generalised metric,then it corresponds to S ′ = S † S S which again differs from [78]. For later convenience it will beuseful to define also K = C − S , so that ρ ( K ) = ρ ( C − ) ρ ( S ) = T H •• = H •• .To write the contribution of RR fields to the DFT action according to [78] one uses a spinor | χ i that encodes the RR potentials. The contribution of RR fields to the Lagrangian is L RR = 14 ( /∂χ ) † S /∂χ = 18 ∂ M ¯ χ Γ M K Γ N ∂ N χ = 14 ( /∂χ ) K /∂χ, (D.3)where we dropped the bra/ket notation and we defined ¯ χ = χ † C and /∂ = √ Γ M ∂ M = ψ m ∂ m + ψ m ˜ ∂ m . Using the C ( D, D ) relations and the weak constraint it is easy to see that /∂ = 0.Reality of the action follows from S † = S and the fact that χ is Grassmann even. Under theconstant Spin ( D, D ) transformation χ → S − χ and X M → X N h N M (where ρ ( S ) = h so that ∂ M → ( h − ) M N ∂ N ) one has /∂χ → S − /∂χ , and the Spin ( D, D ) invariance of the Lagrangian ismanifest. It is not invariant under the full
P in ( D, D ) because this would break the chirality of χ . In addition to the above Lagrangian, and after deriving the equations of motion, one imposesthe duality relation /∂χ = −K /∂χ, (D.4)47hich are invariant under χ → S − χ, S → S † S S for S ∈ Spin + ( D, D ). The restriction toonly
Spin + ( D, D ) (and not
Spin ( D, D )) invariance has to do with the Lorentz signature, sothat timelike T-dualities would spoil the relations. Notice that the duality relation is consistentwith K = 1. The equations of motion for χ that one obtains from the above Lagrangian are /∂ ( K /∂χ ) = 0. They are consistent with the duality relation and the identity /∂ = 0 valid on theweak constraint. The abelian gauge symmetries acting on RR potentials are here implementedas δ λ χ = /∂λ , and because δ λ /∂χ = /∂ λ = 0, the invariance of the Lagrangian is obvious.The above Lagrangian can be rewritten as a Lagrangian for the RR fields in the usual way. We recall that we are taking the spinor χ rewritten as | χ i = P Dp =0 1 p ! C m ··· m p ψ m · · · ψ m p | i which we associate to a polyform C = P Dp =0 1 p ! C m ··· m p dx m · · · dx m p . Let us now define F m ··· m p ≡ p∂ [ m C m ··· m p ] so that on the strong constraint (i.e. taking ˜ ∂ m = 0) we have | F i = | /∂χ i = P Dp =1 1( p − ∂ m C m ··· m p ψ m · · · ψ m p | i = P Dp =1 1 p ! F m ··· m p ψ m · · · ψ m p | i and in poly-form notation F = dC . Using the NSNS B-field B mn we also define | ˆ F i ≡ exp( − B mn ψ m ψ n ) | F i so that ˆ F = exp( − B (2) ) ∧ F where B (2) = B mn dx m ∧ dx n . The conjugate is obtained by h ˆ F | = P Dp =1 1 p ! h | ψ m p · · · ψ m ˆ F m ··· m p . With this rewriting the RR Lagrangian becomes L RR = 14 ( /∂χ ) † S /∂χ = 14 h F | S [ H ] | F i = 14 h ˆ F | S − G ] | ˆ F i = 14 D X p =1 D X q =1 p ! q ! h | ψ m p · · · ψ m S − G ] ψ n · · · ψ n q | i ˆ F m ··· m p ˆ F n ··· n q = − √− G D X p =1 | ˆ F p | , | ˆ F p | ≡ p ! ˆ F m ··· m p G m n · · · G m p n p ˆ F n ··· n p , (D.6)where we used h | ψ m p · · · ψ m ψ n · · · ψ n q | i = p ! δ pq δ [ n m · · · δ n p ] m p . This is the Lagrangian in thedemocratic formulation, see [78]. In fact ˆ F m ··· m p are the RR field strengths that are commonlyused in type II supergravity (they are the ˆ F of [78]), and they are the ones that appear in thequadratic couplings of the fermions in the Green-Schwarz formulation of the superstring (theyare the F of [83]).In terms of | ˆ F i the self duality relation reads as | ˆ F i = − S [ G ] C | ˆ F i and in terms of p -formsˆ F ( p ) = − ( − p ( p +1) ∗ ˆ F (10 − p ) . The Hodge dual is defined as( ∗ A ) m ··· m p = 1(10 − p )! G m n · · · G m p n p ε k p +1 ··· k n ··· n p A k p +1 ··· k , (D.7)where ǫ ··· = 1, ǫ ··· = − ε m ··· m = √− G ǫ m ··· m , ε m ··· m = √− Gǫ m ··· m . We also have ∗ ∗ ω ( p ) = − ( − p (10 − p ) ω ( p ) where the additional minus sign is due to the Lorentz signature.Under the abelian RR gauge transformations the spinor transforms as δ λ | χ i = /∂ | λ i whichimplies δ λ C = dλ . Under D -dimensional diffeomorphisms δ ξ C m ··· m p = L ξ C m ··· m p , i.e. it trans-forms with the standard Lie derivative as expected. Under the gauge transformation of theB-field with gauge parameter ˜ ξ the transformation is δ ˜ ξ C m ··· m p = p ( p − ∂ [ m ˜ ξ m C m ··· m p ] , To do so it is useful to notice that exp( ψ m R mn ψ n ) ψ p | i = (exp R ) mp ψ m | i and S [¯ η ] ψ m | i = − δ mn ¯ η np ψ p | i .It follows that S [ G ] ψ m · · · ψ m p | i = − √− G δ m n · · · δ m p n p G n q · · · G n p q p ψ q · · · ψ q p | i ,S − G ] ψ m · · · ψ m p | i = −√− GG m n · · · G m p n p δ n q · · · δ n p q p ψ q · · · ψ q p | i , (D.5)where G is the determinant of the metric G mn . δ ˜ ξ C = d ˜ ξ ∧ C where δ ˜ ξ B (2) = d ˜ ξ . Therefore F is not invariant under gauge transfor-mations of the B-field, but ˆ F is invariant. Defining ˆ A = e − B (2) ∧ C (i.e. C = e B (2) ∧ ˆ A ) then δ ˜ ξ ˆ A = 0. Notice that in terms of these new potentials ˆ F = d ˆ A + H ∧ ˆ A , with H = dB . If wefurther define A ( p ) = ˆ A ( p ) for p = 4 and A (4) = ˆ A (4) + B (2) ∧ ˆ A (2) then one findsˆ F (1) = dA (0) , ˆ F (2) = dA (1) , ˆ F (3) = dA (2) + H ∧ A (0) , ˆ F (4) = dA (3) + H ∧ A (1) , ˆ F (5) = dA (4) + 12 H ∧ A (2) − B (2) ∧ dA (2) , (D.8)where on the left we have IIB and on the right IIA. These are the more familiar parametrisationsof the RR field strenghts in terms of potentials [78].To rewrite the above results as in [20] one has to pass from a basis of Gamma matrices Γ M in terms of curved indices to a basis Γ A in terms of flat indices. This can be done by using S [ E ] ,the P in ( D, D ) representative of the generalised vielbein, as S [ E ] Γ A S − E ] = Γ M E M A . It is alsonecessary to change vacuum, since we have to go from a vacuum defined as Γ m | i = 0 to a newone such that Γ a | ′ i = 0. In the following we will omit the prime but always think in terms ofthe new vacuum. Notice also that we can still take the charge conjugation matrix C and S [¯ η ] tobe defined as before, while we interpret the ψ as being defined in terms of flat indices. We firstdefine | F i = e d S [ E ] | F i . (D.9)Because we can decompose S [ E ] = S − e ] S [ B ] and we get | F i = X n e φ n ! ˆ F m ··· m n e a m · · · e a n m n ψ a · · · ψ a n | i . (D.10)From the definition it follows that | F i = /∂ | c i + S E ψ M ∂ M S − E | c i + e d ∂ M e − d ψ M | c i in terms of | χ i = e − d S − E ] | c i . This may be rewritten (see also [84]) using E AM ∂ M S [ E ] S − E ] = Ω ABC ψ BC as well as Ω ABC ψ A ψ B ψ C = F ABC ψ ABC + Ω
BBA ψ A , which is a consequence of the C ( D, D )relations. To conclude | F i = ( ψ A ∂ A − F ABC ψ ABC − F A ψ A ) | c i = / ∇ | c i , / ∇ ≡ /∂ − / F (3) − / F (1) , (D.11)where / F ( n ) includes the 1 /n ! factor. Notice that / ∇ = 0 on the strong constraint. Now gaugetransformations read as δ λ | c i = / ∇ | λ i , and the Bianchi identities as / ∇ | F i = 0. The Lagrangianof [78] is readily rewritten as L RR = 14 e − d h F | S − η | F i = − e − d h ¯ F | CS − η | F i . (D.12)Now we have CS − η = ( ψ − ψ )( ψ − ψ ) · · · ( ψ − ψ )( ψ ψ − ψ ψ )= ( ψ − ψ )( ψ ψ − ψ ψ )( ψ − ψ ) · · · ( ψ − ψ )= − ( ψ + ψ )( ψ − ψ ) · · · ( ψ − ψ ) = Ψ − , (D.13) Given the Lie algebra isomorphism betweem O ( D, D ) and
P in ( D, D ) one can identify the generators J MN → ψ MN and use them to construct group elements E AM = exp(Λ PQ J PQ ) AM and S [ E ] = exp(Λ PQ ψ PQ )with the same Λ PQ . Therefore one has ( ∂ M EE − ) BC = Ξ MPQ ( J PQ ) BC and ∂ M S [ E ] S − E ] = Ξ MPQ ψ PQ with the same Ξ MPQ . It follows that E AM ∂ M S [ E ] S − E ] = E AM Ξ MPQ ψ PQ = E AM Ξ MPQ (2 δ [ PB δ Q ] C ) ψ BC = E AM Ξ MPQ ( J PQ ) BC ψ BC = E AM ( ∂ M EE − ) BC ψ BC = Ω ABC ψ BC , where we used the matrix realisation( J MN ) P Q = 2 δ [ MP η N ] Q . In (4.32) of [84] only the leading term in the expansion is taken into account. L RR = − e − d h ¯ F | Ψ − | F i . (D.14)Varying the Lagrangian with respect to | c i one finds the equations of motion / ∇ Ψ − | F i = 0.The above Lagrangian is different when comparing to [20], since they have Ψ + instead ofΨ − , where Ψ ± = ( ψ ∓ ψ )( ψ ± ψ ) · · · ( ψ ± ψ ). To further test the mismatch we first moveto prove (A.9) of [20]. Using Ψ ± ψ A = ∓H AB ψ B Ψ ± one findsΨ + | ω i = X n n ! ω a ··· a n Ψ + ψ a · · · ψ a n | i = X n ( − n n ! ω a ··· a n H a B · · · H a n B n ψ B · · · ψ B n Ψ + | i = − X n ( − n n ! ω a ··· a n ¯ η a b · · · ¯ η a n b n ψ b · · · ψ b n ψ ψ · · · ψ | i = X n ( − n ( − n ( n − n !(10 − n )! ω a ··· a n ¯ η a b · · · ¯ η a n b n ǫ b ··· b n b n +1 ··· b ψ b n +1 · · · ψ b | i = X n n !(10 − n )! ǫ b n +1 ··· b a ··· a n ω a n ··· a ψ b n +1 · · · ψ b | i . (D.15)This agrees with (A.9) of [20]. In the first step we used the commutation relations between Ψ + and the ψ ’s. In the second step we used Ψ + | i = − ψ ψ · · · ψ | i , in the third one we used(5.39) of [78] (notice that that is written with an ǫ with upper indices, so there is an overallminus sign when written for an epsilon with lower indices). In the last step we got a factor of( − n ( n − by rearranging the indices of ω and a factor of ( − n (10 − n ) by swapping the positionof the b · · · b n and b n +1 · · · b D indices in ǫ . In the last line, indices in the epsilon tensor areraised with the Minkowski ¯ η .The computation for Ψ − is identical, since also Ψ − | i = − ψ ψ · · · ψ | i , but because of thedifferent sign in the commutation relations with the ψ ’s one gets an additional factor of ( − n Ψ − | ω i = X n ( − n n !(10 − n )! ǫ b n +1 ··· b a ··· a n ω a n ··· a ψ b n +1 · · · ψ b | i . (D.16)Let us now compare to (A.11) of [20]. First we compute C | χ i = X n n ! χ a ··· a n Cψ a · · · ψ a n | i = X n n ! χ a ··· a n ψ a · · · ψ a n C | i = X n n ! χ a ··· a n ψ a · · · ψ a n ψ · · · ψ | i = X n ( − n ( n − n !(10 − n )! χ a ··· a n ǫ a ··· a n b n +1 ··· b ψ b n +1 · · · ψ b | i , (D.17)where in the last step we used again (5.39) of [78]. Now using ( C | χ i ) † = h χ | C † = − h χ | C we50nd h χ | C Ψ + | ω i = − X n,m ( − n ( n − n !(10 − n )! 1 m !(10 − m )! χ a ··· a n ǫ a ··· a n b n +1 ··· b ǫ d m +1 ··· d c ··· c m ω c m ··· c × h | ψ b · · · ψ b n +1 ψ d m +1 · · · ψ d | i = − X n ( − n ( n − n ! n !(10 − n )! χ a ··· a n ǫ a ··· a n b n +1 ··· b ǫ b n +1 ··· b c ··· c n ω c n ··· c = − X n ( − n ( n − ( − n (10 − n ) n ! n !(10 − n )! χ a ··· a n ǫ b n +1 ··· b a ··· a n ǫ b n +1 ··· b c ··· c n ω c n ··· c = X n ( − n ( n − ( − n (10 − n ) n ! χ a ··· a n ¯ η a b · · · ¯ η a n b n ω b n ··· b = X n ( − n ( n − ( − n (10 − n ) ( − n ( n − n ! χ a ··· a n ¯ η a b · · · ¯ η a n b n ω b ··· b n = X n ( − n n ! χ a ··· a n ¯ η a b · · · ¯ η a n b n ω b ··· b n . (D.18)Compared to (A.11) of [20] we have an additional factor of ( − n . In the computation abovewe used also ǫ b n +1 ··· b a ··· a n ǫ b n +1 ··· b c ··· c n = − n !(10 − n )!¯ η [ a | c · · · ¯ η a n ] c n . The computation forΨ − works in the same way, and because of the previous factor we have in fact h χ | C Ψ − | ω i = X n n ! χ a ··· a n ¯ η a b · · · ¯ η a n b n ω b ··· b n . (D.19)This computation is independent from the one that was giving us Ψ − in the Lagrangian, andit is a further confirmation that Ψ − rather than Ψ + should be used there, since with Ψ + therewould be an additional factor of ( − n when matching the DFT action to the one of standardsupergravity. In the IIB case this would affect the sign of the RR Lagrangian (because for n odd ( − n = −
1) while for IIA using Ψ − or Ψ + is inconsequential.Let us also remark that the self-duality condition Ψ − F = F (rather than Ψ + F = F ) is in factconsistent with the conventions of [78]. The left-hand-side is X n ( − n n !(10 − n )! ǫ b n +1 ··· b a ··· a n ˆ F a n ··· a ψ b n +1 · · · ψ b | i = X n ( − (10 − n ) n !(10 − n )! ǫ b ··· b n a n +1 ··· a ˆ F a ··· a n +1 ψ b · · · ψ b n | i ( n → − n )= X n ( − (10 − n ) ( − n (10 − n ) n !(10 − n )! ǫ a n +1 ··· a b ··· b n ˆ F a ··· a n +1 ψ b · · · ψ b n | i ( ǫ ······ → ǫ ······ )= X n ( − (10 − n ) ( − n (10 − n ) ( − (10 − n )(10 − n − / n !(10 − n )! ǫ a n +1 ··· a b ··· b n ˆ F a n +1 ··· a ψ b · · · ψ b n | i ( ˆ F a ··· a n +1 → ˆ F a n +1 ··· a )= − X n ( − n ( n +1)2 n !(10 − n )! ǫ a n +1 ··· a b ··· b n ˆ F a n +1 ··· a ψ b · · · ψ b n | i , (D.20)51hich implies the self-duality relation − ( − n ( n +1)2 (10 − n )! ǫ a n +1 ··· a b ··· b n ˆ F a n +1 ··· a = ˆ F b ··· b n , (D.21)in agreement with the conventions of [78]. E DFT equations of motion
In this appendix we continue to omit the boldface notation, since the dimensional reductionplays no role. The action in the NSNS sector is S NSNS = R dXe − d R where we prefer to use therewriting of [50] R = − ∂ A F ( − ) A + 2 F A F ( − ) A − F ( − ) ABC F ( − ) ABC − F ( −− ) ABC F ( −− ) ABC . (E.1)Here P ± = ( η ± H ) and F ( ± ) A = ( P ± ) AB F B , F ( ± ) ABC = ( P ∓ ) AD ( P ± ) BE ( P ± ) C F F DEF , F ( ±± ) ABC =( P ± ) AD ( P ± ) BE ( P ± ) C F F DEF . To compute the equations of motion one finds that δ E S NSNS = R dXe − d Ξ AB NSNS ∆ AB where ∆ AB = − ∆ BA = δE AM E BM and δ d S NSNS = R dXe − d Ξ NSNS δd where [20] Ξ NSNS[ AB ] = 4 ∂ [ A F ( − ) B ] + ( F C − ∂ C ) ˇ F C [ AB ] + ˇ F CD [ A F B ] CD , Ξ NSNS = − R , (E.2)where we defined ˇ F ABC = ˇ S ABC A ′ B ′ C ′ F A ′ B ′ C ′ andˇ S ABCA ′ B ′ C ′ = H AA ′ η BB ′ η CC ′ + η AA ′ H BB ′ η CC ′ + η AA ′ η BB ′ H CC ′ − H AA ′ H BB ′ H CC ′ − η AA ′ η BB ′ η CC ′ = − (cid:16) P ( − ) AA ′ P ( − ) BB ′ P ( − ) CC ′ + P (+) AA ′ P ( − ) BB ′ P ( − ) CC ′ + P ( − ) AA ′ P (+) BB ′ P ( − ) CC ′ + P ( − ) AA ′ P ( − ) BB ′ P (+) CC ′ (cid:17) . (E.3)It is convenient to write ˇ S in the second way, which is the unique way to write it as a linearcombination of products of projectors. In fact it is then easy to check that P (+) AA ′ P ( − ) BB ′ Ξ [ A ′ B ′ ]NSNS = 2Ξ ′ NSNS AB , Ξ ′ NSNS AB ≡ ∂ (+) A F ( − ) B + ( ∂ C − F C ) F ( − ) ABC − F (+)
CDA F ( − ) DC B . (E.4)Notice the transposition of CD indices in the last term. One also has P (+) AA ′ P (+) BB ′ Ξ [ A ′ B ′ ]NSNS = 0 and P ( − ) AA ′ P ( − ) BB ′ Ξ [ A ′ B ′ ]NSNS = 2 P ( − ) AA ′ P ( − ) BB ′ Z A ′ B ′ ∼ P ( − ) AA ′ P (+) BB ′ Ξ [ A ′ B ′ ]NSNS = − ′ NSNS BA we can conclude that Ξ NSNS[ AB ] ∼ ′ NSNS[ AB ] (i.e. on the constraint ofDFT). Hence because we can project with P ( ± ) , Ξ NSNS[ AB ] = 0 is equivalent to Ξ ′ NSNS AB = 0.When considering the RR contribution to the action one has [20] δ E S RR = R dXe − d Ξ AB RR ∆ AB and δ d S RR = R dXe − d Ξ RR δd where Ξ [ AB ]RR = − ¯ G ψ AB G and Ξ RR = 0 after imposing the self-duality condition. The full equations of motion in the type II case are then Ξ [ AB ] = Ξ NSNS[ AB ] +Ξ RR[ AB ] = 0 and Ξ = Ξ NSNS = 0.
F Other ansatze used for orbits with H -flux In this appendix we will report on other attempts to deal with orbits with non-vanishing H -fluxin the standard parametrisation of (2.21) rather than appealing to the methodology of section52.4. In particular, by preferring an ordinary Lie group G action on the coordinates y , we willassume here that ρ µi of (2.21) are the components of a (left-invariant) Maurer-Cartan form ρ = g − dg = dy µ ρ µi t i where t i ∈ g = Lie( G ) are the generators of the Lie algebra and g ∈ G is aLie group element. In this case the components w ij k defined in (2.36) are precisely the structureconstants of g so that we can write them in the more standard way w ijk = f ij k . Furthermore,we can use the well-known expansion of the MC form ρ = g − dg = ∞ X N =0 ( − ) N ( N + 1)! ad Nx dx, x ∈ g . (F.1)The main difficulty with pursuing in this way is finding the most general solution to thedifferential equation obtained from the Q -flux equation of (2.35). Focusing here on the simplesthomogeneous equation ( Q = 0) with F = 0 we must solve ∂ µ β ij + ρ µn β l [ i β j ] m H nlm = 0 , (F.2)or equivalently upon the F -flux equation ∂ µ β ij − ρ µn β l [ i f nlj ] = 0 . (F.3)Let us remark here that by lowering all indices in eq. (F.3) with the Killing form and antisym-metrising in i, j, k we find that a necessary (but not sufficient) condition is that as a two-form β is closed, and possibly exact. For semisimple algebras h it is known that there exists no non-trivial closed two-forms which are not exact. An exact form such as β ij = ρ µi ρ νj ∂ [ µ λ ν ] for someone-form λ does not, however, solve eq. (F.3). We can therefore conclude that for semisimplealgebras no non-trivial solutions to eq. (F.3) exist.To find a non-trivial solution of (F.3) we have considered for β ij the following ansatze. (i) In the ( H )-orbit of section 3.9 we found a non-trivial representative, where ρ is of MCform, by considering β ij to be a constant matrix.(ii) Take now a non-constant ββ ij = h O t ROt i , t j i , O = ∞ X N =0 a N ad Nx , = ∞ X N =0 ∞ X M =0 a N a M (ad Nx ) il R lk (ad Mx ) j k , (F.5)where R is an antisymmetric matrix and where t i = κ ij t j and κ ij = h t i , t j i is the Killingform of g . In this case one can observe that fixing the unknown coefficients a N becomesinconsistent at order O ( ∂ µ x, x ). When generalising (F.5) as β ij = ∞ X N =0 ∞ X M =0 a NM (ad Nx ) [ il E lk (ad Mx ) j ] k , (F.6)where E ij = S ij + R ij with S a symmetric and R an antisymmetric matrix, and a NM unknown coefficients, one will find that the symmetrical part does not contribute and thisis effectively equivalent to the ansatz (F.5). In the calculations that follow it is useful to use the identity[ad Nx dx, y ] = N X K =0 ( − ) K N ! K !( N − K )! ad N − Kx ad dx ad Kx y, (F.4)which holds for any x, y ∈ Lie( G ) and which can be proved by induction and the Jacobi identity. β = O − O t with O a series in ad x , as defined above, which is important toresonate with the MC form. Hence β ij = 2 ∞ X N =0 a N (ad Nx ) [ ik κ k | j ] . (F.7)This ansatz turns out to be empty: for every N ∈ N we find a N = 0.We have also considered the possibility that ρ is not of MC form and therefore tried to solve thePDE (F.2) instead. Assuming nevertheless an underlying Lie algebra g we take H ijk = α f ijl κ lk for some real constant α , and we expand ρ as ρ µi = ∞ X N =0 b N (ad Nx ) lm ∂ µ x l κ mi , x ∈ g , (F.8)for some undetermined coefficients b N . 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