aa r X i v : . [ h e p - t h ] A ug Poisson-Lie T-Duality in Double Field Theory
Falk Hassler
1, 2, ∗ University of North Carolina, Department of Physics and Astronomy, Chapel Hill, NC 27599-3255, USA University of Pennsylvania, Department of Physics and Astronomy, Philadelphia, PA 19104-6396, USA
We present a formulation of Double Field Theory with a Drinfeld double as extended spacetime. Itmakes Poisson-Lie T-duality (including abelian and non-abelian T-duality as special cases) manifest.This extends the scope of possible applications of the theory, which so far captured abelian T-dualityonly, considerably. The full massless bosonic subsector (NS/NS and R/R) of type II string theoriesis covered.
PACS numbers: 11.25.-w, 11.30.Ly
Superstring theory is a very successful framework toexplore quantum field theories and quantum gravity. Du-alities lie at its heart by connecting the five perturbativeformulations and M-theory. Especially T-duality is inher-ently linked to the extended nature of the string. Unfor-tunately, this term is a bit ambiguous because it is usedin different contexts. Normally, it refers to abelian T-duality [1] which requires the two target spaces it links tohave abelian isometries. But there is also a non-abeliancounterpart (NATD) [2]. It only holds at the classi-cal level but still admits to construct new string back-grounds, for example [3]. This provides an important toolin studying the underlying structure of the AdS/CFTcorrespondence. Like dualities are essential for stringtheory, integrability plays a similar distinguished role inthis endeavor. Considering the close relation betweenstring theory and AdS/CFT, it is not surprising thatnon-abelian T-duality also gives access to deformationsof integrable models [4]. Based on these three pillars,NATD, AdS/CFT and integrability, many interesting re-sults have been presented recently.T-duality is however not directly manifest in the low-energy effective target space description, 10D supergrav-ity (SUGRA). Both variants, abelian and non-abelian,are meditated by involved, non-linear transformations ofthe fields. Double Field Theory (DFT) [5, 6] (see also[7]) is an approach to make abelian T-duality manifestby formally doubling the complete target space. Phys-ically, this doubling is only relevant for directions withcapture the abelian isometries of the background. Theyform a flat d -torus, while the additional coordinates aredual to string modes winding along non-contractible cy-cles of this torus. Abelian T-duality exchanges momen-tum with winding modes and therefore is a simple O( d , d )rotation of the coordinates in the doubled space. For abackground independent, consistent formulation of thetheory [6] the section condition has to be imposed. Itguarantees that the theory can be reduced to the com-mon SUGRA description (at least in one duality frame).DFT has become an indispensable tool for studying con-sistent truncations, non-geometric backgrounds, α ′ cor-rection and many other questions related to abelian T-duality. Nevertheless, it does not help with NATD and thus didnot enter the line of research outlined above (a notableexception is [8]). Hence, objective of this letter is topresent a version of DFT with manifest abelian and non-abelian T-duality. In fact it implements the even moregeneral Poisson-Lie T-duality [9, 10] which includes theformer as special cases. We discuss the full bosonic partof type IIA/B SUGRA, also including the R/R sector. Poisson-Lie T-duality:
As it is the starting point forall following derivations, let us review the salient featuresof Poisson-Lie T-duality. We start from a bosonic σ -model with target space M which admits a free action ofthe Lie-group G . It is governed by the action S = Z dzd ¯ z E ij ∂x i ¯ ∂x j (1)where E ij = g ij + B ij combines the metric on M withthe anti-symmetric B -field. If we vary this action withrespect to infinitesimal changes of the target space co-ordinates δx i = v ai δǫ a , where v ai is the left-invariantvector field on G , we find δS = Z dzd ¯ z L v a E ij ∂x i ¯ ∂x j δǫ a − Z dJ a δǫ a (2)with the Noether currents J a = − v ai E ji ∂x j dz + v ai E ij ¯ ∂x j d ¯ z . (3)In order to permit Poisson-Lie T-duality, these currentshave to satisfy the on-shell integrability condition dJ a − F bca J b ∧ J c = 0 (4)where F bca denotes the structure constants of the Lie-algebra ˜ g corresponding to the dual Lie-group ˜ G . On-shell, the variation (2) vanishes. In combination withthe integrability condition for J a , this gives rise to theconstraint L v a E ij = − F bca v bk v cl E ik E lj (5)on the background. This equations reveals a remark-able feature of Poisson-Lie T-duality: In contrast toits abelian/non-abelian descendants, it does not requireisometries. Of course, it still works with them present. Inthis case, the dual Lie-group is abelian and dJ a =0 holds.But in general the σ -model admits a non-commutativeconservation law [9].As v ai is the left-invariant vector field on G , itsLie derivative generates the corresponding Lie-algebra[ L v a , L v b ] = L F bac v c , denoted as g , with structure con-stants F abc . By combining (5) with this commutator,we obtain a compatibility relation between the structureconstants of g and ˜ g . It requires that these two groups arethe maximally isotropic subgroups of a Drinfeld double D [11] with the bilinear, invariant pairing h t a , t b i = δ ba , h t a , t b i = δ ab , h t a , t b i = h t b , t a i = 0 (6)where t a and t a are the generators of G and ˜ G , respec-tively. This double is the central object behind the wholeconstruction. The dual σ -model˜ S = Z dzd ¯ z ˜ E ij ∂x ′ i ¯ ∂x ′ j (7)arises after exchanging its two subgroups G and ˜ G . Dueto the integrability condition (4), solutions g ( z, ¯ z ) ∈ G of the field equations for (1) can be lifted to the Drinfelddouble D ∋ d ( z, ¯ z ) = g ( z, ¯ z )˜ g ( z, ¯ z ) by J a = d ˜ g ( z, ¯ z )˜ g − ( z, ¯ z ) . (8)Using the alternative decomposition d = ˜ h ( z, ¯ z ) h ( z, ¯ z )provides a solution ˜ h ( z, ¯ z ) ∈ ˜ G for the dual model [9].After changing from the Lagrangian description to theHamiltonian, the two models are connected by a canoni-cal transformation on their phase space variables [12]. Soat the classical level their dynamics is indistinguishable.It is convenient to exploit the underlying structure ofthe Drinfeld double to find an explicit expression for themetric and the B -field by integrating (5). At the unitelement of the double, and therewith also of G , we use aconstant, invertable E ab as initial conditions. What isleft to do, is to transport it by the adjoint action to allother points g ∈ G [9]. More explicitly, if we denote theadjoint action as g t a g − = M ab t b , g t a g − = M ab t b + M ab t b , (9)one obtains E ij = v ci M ac ( M ae M be + E ab ) M bd v dj (10)where E ij is the inverse of E ij . For the dual model,the same procedure applies with the adjoint action of ˜ g instead of g and a map E ab : ˜ g → g (the inverse of E ab )resulting in˜ E ij = ˜ v ci ˜ M ac ( ˜ M ae ˜ M be + E ab ) ˜ M bd ˜ v dj . (11)A decomposition of the Drinfeld double in two maxi-mally isotropic subgroups refers to a Manin triple. We are interested in the set M ( D ) of these triples for agiven double D . It plays the role of the moduli spaceof dual σ -models and has at least two points originatingfrom G / ˜ G and swapping them [10]. Abelian and non-abelian T-duality are special cases of this construction.While the former arises if the double is abelian and has M ( D ) = O( D , D , Z ), where D denotes the dimension ofthe target space M , the latter results from a non-abelian G and an abelian ˜ G . Double Field Theory on Drinfeld doubles:
So far, weapproached Poisson-Lie T-duality from the world-sheetperspective. Main objective of this letter is to show howthis symmetry can be made manifest in the correspond-ing target space low-energy effective theory. As the Drin-feld double is essential to construct all dual backgrounds,it should be a fundamental part of this theory. Remem-ber that D is a Lie-group equipped with a bi-invariantlinear form (6) of split signature. Hence, the frameworkof DFT on group manifolds naturally applies. It wasoriginally derived via Closed String Field Theory for aWess-Zumino-Witten model [13] and afterward general-ized to arbitrary, 2 D -dimensional group manifolds [14]which admit an embedded into O( D , D ). Its action S NS = Z d D Xe − d (cid:16) H CD ∇ C H AB ∇ D H AB − H AB ∇ B H CD ∇ D H AC − ∇ A d ∇ B H AB + 4 H AB ∇ A d ∇ B d + 16 F ACD F BCD H AB (cid:17) (12)governs the dynamics of the generalized metric H AB , asymmetric, O( D , D ) valued matrix, and the generalizeddilation d on the extended space D . The indices A to F label the adjoint representation of the correspondingLie-algebra d . They are lowered/raised with the O( D , D )invariant metric η AB and it inverse, which directly resultsfrom the pairing (6) as h t A , t B i = η AB . Furthermore, weneed the η -compatible, covariant derivative ∇ A V B = D A V B + 13 F AC B V C − wF A V B , (13)here for a vector density V A = ( V a V a ) of weight w , inorder to evaluate the action. It incorporates the structureconstants F ABC of d and the flat derivative D A = E AI ∂ I , with [ D A , D B ] = F ABC D C , (14)expressed in terms of a partial derivative and the right-invariant vector field E AI on D whose coordinates are X I = (˜ x i x i ). Furthermore, there is the density partwith F A = D A log E and E = det E AI . The action (12)is invariant under a local O( D , D ) symmetry mediated bythe generalized Lie derivative L ξ V A = ξ B ∇ B V A + (cid:0) ∇ A ξ B − ∇ B ξ A (cid:1) V B + w ∇ B ξ B V A (15)which extends in the usual way to higher rank tensorslike H AB ( w = 0), and e − d ( w = 1). Closure of theseinfinitesimal transformations requires to impose the sec-tion condition D A · D A · = 0 (16)on arbitrary combinations of physical fields and gaugeparameters, formally denoted by · .A technique to solve (16) was introduced in [15]. Itselects D physical coordinates x i on the group manifoldusing a maximally isotropic subgroup. If we apply it toa Drinfeld double D , group elements are parameterizedby d = g ( x i )˜ g (˜ x i ) and the target space M is identifiedwith the coset D / ˜ G which is nothing else than G . Allphysical fields and parameters of gauge transformationsare restricted to depend on x i only. Thus, the sectioncondition (16) is solved. A refinement of this statement isrequired for tensor densities, like the generalized dilaton.They split into a background and a fluctuation part. Herethe section condition (16) only applies for the latter [14].Finally, we need to connect the generalized metric H AB and the GL( D ) element E ij used in the σ -model (1).Latter decomposes into the metric g ij and the B -field B ij which are embedded in the generalized metric b H ˆ I ˆ J = (cid:18) g ij − B ik g kl B lk − B ik g kl g ik B kj g ij (cid:19) (17)on the generalized tangent bundle T M ⊕ T ∗ M over M .In order to clearly distinguish this bundle from T D , thecorresponding indices and quantities are decorated witha hat. b H ˆ I ˆ J is linked to it flat counterpart by the gener-alized frame field b E A ˆ I = M AB (cid:18) v bi v bi (cid:19) B ˆ I , M AB t B = g t A g − , (18)as b H ˆ I ˆ J = b E A ˆ I H AB b E B ˆ J . Note that the components of M AB were already introduced in (9). Taking furthermore E ab = g ab + B ab and writing H AB like (17) but now withindices a , b instead of i , j , we indeed exactly reproducethe background (10) derived from the σ -model.It is instructive to rewrite the action (12) and it gaugetransformations (15) in terms of these hatted quantities.For the generalized dilaton d , we also have to take intoaccount the volume form on ˜ G to introduce b d = d + 12 log ˜ v , ˜ v = det(˜ v ai ) . (19)A crucial part in the rewriting is to transform the covari-ant derivatives. It is straightforward to show that ∇ A V B → ∂ ˆ I b V ˆ J + ( b Ω [ˆ I ˆ K ˆ L ] − b Ω ˆ I ˆ K ˆ L ) η ˆ L ˆ J b V ˆ K (20) ∇ A d → ∂ ˆ I b d + 12 b Ω ˆ J ˆ J ˆ I with b Ω ˆ I ˆ J ˆ K = ∂ ˆ I b E A ˆ K b E A ˆ J holds if we assume that both Lie-groups G and ˜ G areunimodular. As result we get the action S NS = V ˜ G Z d D xe − b d (cid:16) b H ˆ K ˆ L ∂ ˆ K b H ˆ I ˆ J ∂ ˆ L b H ˆ I ˆ J − (21)2 ∂ ˆ I b d∂ ˆ J b H ˆ I ˆ J − b H ˆ I ˆ J ∂ ˆ J b H ˆ K ˆ L ∂ ˆ L b H ˆ I ˆ K + 4 b H ˆ I ˆ J ∂ ˆ I b d∂ ˆ J b d (cid:17) of conventional DFT and the corresponding generalizedLie derivative after neglecting boundary terms and takinginto account the section condition ∂ ˆ I · ∂ ˆ I · . Evaluatingthe integral in (12) over the unphysical directions ˜ x i con-tributes the volume factor V ˜ G . After suppressing it, asit is common practice in DFT, and following [6], we findthe action S NS = Z d D x √ ge − φ (cid:0) R +4 ∂ i φ∂ i φ − H ijk H ijk (cid:1) (22)for the NS/NS sector of type II SUGRA with the cur-vature scalar R , the dilaton φ = b d − / g ) and thethree-form flux H ijk = 3 ∂ [ i B jk ] . Finally note that b E A ˆ I is a globally defined, O( D , D ) valued generalized framefield which renders M a generalized parallelizable space.This property is essential to obtain (21), because it givesrise to F ABC = 3 b Ω [ ABC ] .For the R/R sector the action on D , S R = 14 Z d D X ( / ∇ χ ) † S H / ∇ χ , (23)is a straightforward generalization of [16]. It incorpo-rates the O( D , D ) Majorana-Weyl spinor χ and the spinorversion S H of the generalized metric. We exclude time-like T-dualities, so this map is well-defined. Our Γ-matrices satisfy the canonical anti-commutator relation { Γ A , Γ B } = 2 η AB . Additionally, the spin connection ofthe covariant derivative has to be fixed. We do so byrequiring ∇ A Γ B = 0, resulting in ∇ A χ = D A χ − F ABC Γ BC χ − F A χ (24)under the assumption D A Γ B = 0 and taking into ac-count that χ is a density of weight 1/2. Generalized dif-feomorphisms leave (23) invariant and are mediated bythe generalized Lie derivative L ξ = ξ A ∇ A χ + 12 ∇ A ξ B Γ AB χ + 12 ∇ A ξ A χ . (25)Depending on its chirality, χ either captures the typeIIA or IIB R/R gauge potentials. A convenient way toparameterize it is in terms of p -forms C ( p ) with even orodd degree as χ = D X p =0 p/ p ! C ( p ) a ...a p Γ a . . . Γ a p | i . (26)Moreover, one has to impose the self-duality constraint / ∇ χ = − C − S H / ∇ χ [16], where C is the charge conjuga-tion matrix. As for the generalized metric in the NS/NSsector, we finally employ the spinor version S b E of thegeneralized frame field (18) to make contact with theconventional DFT R/R action S R = V ˜ G Z d D x
14 ( /∂ b χ ) † S b H /∂ b χ (27)where b χ = ˜ v − / S b E χ . Again the transformation of thecovariant derivatives (20) plays the central role in thiscalculation. Here, it has to be adapted to spinors. Aftersome Γ-matrix algebra ones find the simple identification / ∇ χ → √ ˜ vS − b E /∂ b χ (28)which makes the identity between (23) and (27) mani-fest. Note that /∂ denotes the contraction with the Γ-matrices b Γ ˆ I , while / ∇ takes Γ A . They are related by b Γ ˆ I = b E A ˆ I S b E Γ A S − b E . Assuming that the section con-dition holds, which is the case if b χ depends on the physi-cal coordinates only, (27) reduces to the democratic R/Raction of type IIA/B SUGRA [16]. Dual backgrounds:
At least in the NS/NS sector, allessential parts of DFT on D have a natural analog onthe world-sheet. This connection extends to the dualbackground which arises after swapping G and ˜ G . Werequire that each duality frame (with coordinates X I and X ′ I ) parameterizes the same group elements D ∋ d = g ( x i )˜ g (˜ x i ) = ˜ g ( x ′ i ) g (˜ x ′ i ) (29)in the Drinfeld double. This equation describes the 2 D -diffeomorphism X I → X ′ I which is a manifest symmetryof our formulation [14]. It does not exists in conventionalDFT. Requiring that in both duality frames H AB andthe fluctuations of χ and d depend on the physical coor-dinates only, we have to choose them constant. This is inagreement with the world-sheet perspective, where E ab is constant, too. Furthermore, we adapt the generalizedframe field eb E A ˆ I = ˜ M AB (cid:18) ˜ v bi
00 ˜ v bi (cid:19) BI , ˜ M AB t B = ˜ g t A ˜ g − , (30)because ˜ E A ˆ I would depend on unphysical coordinates ˜ x ′ i after applying (29). However, (30) still allows to reducethe action to (21) and (27). Now, we find eb H ˆ I ˆ J = b O ˆ I ˆ M b H ˆ M ˆ N b O ˆ J ˆ N with b O ˆ I ˆ J = eb E A ˆ I b E B ˆ J (31)for the generalized metric which specifies the metric ˜ g ij and the B -field ˜ B ij of the dual background (11). Forthe generalized dilaton e ˆ d = d + 1 / v holds. Finally,assuming a constant fluctuation part of χ , the R/R fieldstrengths spinor transforms covariantly e /∂ eb χ = S b O /∂ b χ , (32) like χ . Clearly the two dual backgrounds share the sameaction on the Drinfeld double D . Hence, the DFT versionpresented in this letter makes Poisson-Lie T-duality man-ifest like the conventional formulation does for abelianT-duality.In order to compare the R/R sector results with theliterature, we take ˜ G to be abelian. Now, b O admits thedecomposition b O ˆ I ˆ J = 12 (cid:18) ˜ e T − ˜ B ˜ e − e − (cid:19) (cid:18) − Λ 1 + Λ1 + Λ 1 − Λ (cid:19) (cid:18) e − T e (cid:19) ˆ I ˆ J (33)where ˜ e T ˜ e = ˜ g and e is the left-invariant Maurer-Cartanform on G . Note that the Λ appearing here is the onefound in (2.8) of [17] by comparing the chiral and anti-chiral components of the world-sheet current (3). Start-ing for this observation, it is straightforward to show that(32) matches their transformation prescription.In summary it is evident that DFT on a Drinfeld dou-ble is a natural proposal for a low-energy effective targetspace theory with manifest Poisson-Lie (and therewithnon-abelian) T-duality. It provides a powerful frameworkto address the questions outlined in the first paragraph ofthis letter. Merging these two lines of research seems nowfeasible and hopefully will produce many new insights onboth sides. Acknowledgements:
I would like to thank the organiz-ers and participants of the workshop “Recent Advancesin T/U-dualities and Generalized Geometries” 6-9 June2017, Zagreb, Croatia for the inspiring atmosphere andfor highlighting the significance of Poisson-Lie and non-abelian T-duality to me. This work is supported by theNSF CAREER grant PHY-1452037 and NSF grant PHY-1620311. ∗ [email protected][1] T. Buscher, Phys.Lett. B194 , 59 (1987).[2] X. C. de la Ossa and F. Quevedo,Nucl. Phys.
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