JHEP11(2017)175 -dimen- d Springer August 10, 2017 : November 21, 2017 November 27, 2017 : : , is supplemented by an Received η into two Lagrangian sub- Accepted ˜ Published L ⊕ L a ) metric = , even if they are not flat and con- d, d ω P ( T Published for SISSA by O https://doi.org/10.1007/JHEP11(2017)175 define an almost bi-Lagrangian structure and [email protected] η ω , and η ) where the and David Svoboda b , η, ω are constructed. We find integrability conditions under P P . Together T ω . 3 Felix J. Rudolph 1706.07089 a The Authors. c Differential and Algebraic Geometry, String Duality, Non-Commutative Ge-

We formulate a kinematical extension of Double Field Theory on a 2 , [email protected] which provides a splitting of the tangent bundle Arnold Sommerfeld Center forLudwig-Maximilians-Universit¨atM¨unchen,Theresienstr. Theoretical 37, Physics, 80333 Department M¨unchen,Germany f¨urPhysik, E-mail: [email protected] Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON, N2L 2Y5, Canada b a Open Access Article funded by SCOAP Keywords: ometry ArXiv ePrint: which the symmetry algebra closesstant. for This formalism general thus providesField a Theory. generalisation of We the also kinematicalField show structure that Theory of Double with this Generalised formalism Geometry allows which one is to thoroughly reconcile discussed. and unify Double sional para-Hermitian manifold ( almost symplectic two-form K spaces. In this paper afor canonical the connection and Leibniz a algebroid corresponding on generalised Lie derivative Abstract: Laurent Freidel, Generalised kinematics for double field theory JHEP11(2017)175 ); 28 29 TM Γ( → )) ) will be referred TM ( ∇ 2 g, ] : Γ(Λ · ) which defines a notion of , · M ∗ T 9 6 ⊗ 26 TM 17 6 23 equipped with a differentiable structure, Γ( 12 7 19 → M ) 9 21 18 – 1 – TM D J : Γ( 15 ]) which encodes the underlying structure of spacetime · ∇ , · [ 16 M, ), which gives us a notion of distance between points; and a -para-Hermitian ): a set of points M L ∗ ∇ T , g, D ] ⊗ component; in the Einsteinian picture, this component is subject to · 28 J , · [ M 1 ∗ component of the geometry, since in the usual formulation of gravity it is 25 T M, Γ( ∈ -para-Hermitian manifolds dynamical g L kinematical B.1 Computing B.2 Anchoring and tensorality of 4.2 Generalized lie4.3 derivative and bracket Para-K¨ahlerand 3.2 3.3 The para-K¨ahlercase 4.1 Projected and partial connections 2.1 Review of2.2 the generalised geometry Review setup of2.3 the double field Relationship theory between setup GG and DFT 3.1 Integrability, closure and a key example parallel transport. The data ( and the symmetry ofthe the theory bygiven infinitesimal from diffeomorphisms, the onset will andto be is as referred not the to subject to as any dynamics. The data ( The notion of spacetimefour elements geometry ( used inwhich is the encoded context into of thea Lie general metric bracket relativity on is thecompatible tangent based torsionless bundle connection on [ 1 Introduction A Properties of para-Hermitian connections B The Jacobiator 5 Relationship with generalized geometry 6 Conclusion 4 connections on para-Hermitian manifolds 3 Para-Hermitian, para-K¨ahlerand integrability Contents 1 Introduction 2 Lightning reviews of GG and DFT kinematics JHEP11(2017)175 ) ). ). is H 1) d ˆ ( flat H − H (1.1) O η, ω η, , d × its kine- (1 and ) O d η ( ] are a × O both 1) 20 , − 19 , d ]. (1 . At the infinitesimal 18 which is locally given O P ). P is an element of the coset H -field gauge transformations − · · · − B , + which provides an . This is a generalisation of the X η . ··· H ] focuses on the generalisation of the L = ˆ 1 ), the compatibility between ˆ ηJ , in which case we denote them by (ˆ T P metric P . This is the doubled space of Double Field T ∗ M T – 2 – End( ,J is unified with the 1 P → encodes the kinematical structure of DFT while M T ˆ η H ∈ ). The purpose of this paper is to focus on the gener- of DFT is in fact a phase space carrying a symplectic = + 1 generalised ∇ − 2 P η ]. This has been used to show that at the quantum level J g, 17 := ˆ ] and it has also quite remarkably led us to an understanding and a J ). In order to write this relation, let us recall that we can think 10 η ]. Recently, the correspondence between the string target and the d – ( 7 13 O – × ], it has twice the number of dimensions of spacetime and it is seen as 11 6 ) – d denoted with spacetime denoted by ]. The presence of this structure radically modifies the interpretation of ( 2 0) tensors or as maps ]) and dynamics ( 1 ˜ , 16 · M /O – , ) · ) metric is a neutral metric with split signature (+ [ chiral structure × 14 d, d d, d ( M, M ( ) as (2 O O ) metric H One of the central ingredients of DFT is the idea that the notion of diffeomorphism The key geometrical elements entering the construction of DFT [ The construction presented here is profoundly motivated by the study of effective string By the generalisation of geometry we mean a structure that generalises An η, P' d, d 1 ( symmetries acting on theinto spacetime a generalisation of diffeomorphisms actinglevel on the this doubled action space is given by the Dorfman derivative of ( Defining a expressed as the condition structure on the generalised tangent bundlein in the the Euclidean Lorentzian case, or case.encodes The its metric dynamics. Thesewhich follows two from geometrical the structures T-dualityspace satisfy symmetry a and imposes compatibility condition that structure [ the effective string geometrythe [ string target can be thought of as aO non-commutative manifold [ of the constraints of supergravityof purely supersymmetry from [ geometrical argumenteffective and without description the has need beenof revisited string in theory the context calledestablish of metastring that a theory. the duality doubled A symmetric space key formulation element of this description has been to by Theory (DFT) [ a target space forcoordinates. the string, This its new coordinatesflux picture are compactifications has a [ combination in of particular spacetime allowed and to winding deepen our understanding of itself when properly generalisedThis is opens encoded up into amight geometrical be radically elements subject new denoted to possibility: a by dynamical ( the selection rule. kinematicalgeometries. structure It of is spacetime now well itself theory established and that quantum the gravity arena is of a several novel mathematical formulations space of denoted string by fixes the torsion to be zero and determines thematics Ricci ( component of thealisation curvature. of the kinematics whiledynamics. a companion As paper we [ will see, one of the main components of our results is that the kinematics the dynamics encoded into the choice of an action invariant under diffeomorphisms that JHEP11(2017)175 ] ]. H 24 21 [ (1.2) (1.3) . The P = 0. M η ∗ ] for more X T . L 29 ) , ⊕ of DFT. This . When acting 28 M B η ∗ ∂ TM T is not doubled and A ∂ Γ( := M AB ∈ η M T = η . , α, β  ) ) α d ]. One of the main challenges of y TM i ) — which acts on tensors in 13 ], who introduced a bracket, which ] to unify the Dirac formalism [ – Γ( P − 25 23 T 11 β ∈ ]. This led to the construction of the x Γ( L 26 , ∈ ] ]] is the Courant bracket obtain by skew- · , · X x, y – 3 – , x, y ) x ( ) = ([ β ] and Dorfman [ is generalised. In the simplest case (see [ y, β ( 22 ) + ]. ) y M ( . In the simplest cases the section condition imposes that 27 [ x,α α ( 2 L := which descends from the canonical pairing of forms and vectors . + + i M ·i ) , T Φ (where the bracket [[ h· ]] y, β ( , ) = 0 for all fields and their products, where X,Y ) 0 [[ L section condition also carries a Dorfman derivative that generalises the Lie derivative and is x, α (ΦΦ ( η h M  ]Φ = T Y L In the approach of generalised geometry, the spacetime manifold A separate but related development — also aiming at describing geometrical entities Unlike the usual Lie derivative, the Dorfman derivative defines an infinitesimal trans- , Equivalently known as Courant algebroid. (Φ) = 2 X η L This derivative generates astraints. group of This transformations kinematical withoutwhich the structure is need a can of pairing then on additional be con- augmented by a generalised metric This symmetric pairing ofbundle GG plays adefined role by analogous to the flat metric general examples) it isadvantage promoted of to this the approach is doublebase that tangent manifold there bundle is is unchanged to no begin needa with. symmetric to The pairing impose generalised a tangent section bundle naturally condition carries since the is the skew-symmetrisation ofquently the generalised Dorfman by Lie Liu,notion derivative. Weinstein of and a The Leibniz Xu construction algebroid [ was subse- instead the tangent bundle over relevant to the geometryvious of issue; string this theory is —attempts the of is field of Courant, an Generalised Weinstein approachof Geometry [ that symplectic (GG). does reduction This not with fieldconstraints. the have originates This the formalism from was of pre- the later Poisson also reduction studied by by first Courant and [ second class symmetry. This symmetry cantional in actions without turn the bethis need used approach of is to supersymmetry to constrain find [ thetranslates a directly consistent form mathematical into of description an stringy of underlying the gravita- choice section of condition a that geometrical structure. the so-called  on tensor fields that satisfysection the condition, section it condition, can and be[ for shown vector that fields the that Dorfman also derivativesymmetrisation satisfies satisfy of the the the Jacobi Dorfman identity derivative) and in this case it therefore defines a notion of By construction, this generalised Lie derivative preserves the neutralformation metric: that does notlimitation, integrate the to strategy a used groupand in action double tensors in field general. on theory which In is order the to to generalised restrict overcome Lie the this space derivative of acts, vector and fields demand that they satisfy notion of Lie derivative — labeled by a vector JHEP11(2017)175 ]. At 31 , in lieu of 17 M T ]. ]. 30 , 18 18 is viewed as a subspace in addition to the DFT ω M for the generalised geometry limit of compactified strings) M R limit of compactified strings), i.e. R ] that constitutes the classical geometrical 14 [ ) transformation on the fibers – 4 – 3 of string probes combining spacetime and energy- d, d serves as the base ( O P Born geometry phase space (which appears in the large ] and in the corresponding modular spacetime that represents ) completes the picture and allows one to connect the generalised , but it cannot be accommodated in GG since there the base is M (which appears in the small 16 ]. The metastring theory is a formulation of string theory in which P H , ˜ 17 η, 15 M . This freedom of choice makes DFT in a sense undetermined. The main point — which is in total agreement with DFT — is that the string target One of the key ingredients in the resolution of this puzzle has been revealed in the study To summarise, we see that DFT can accommodate some aspects of T-duality symmetry This would suggest that GG is therefore the proper mathematical expression of string Despite the many successes of both the DFT and GG approaches, there are still some We now understand that this change requires non-commutative geometry [ 3 M the doubled space ofclassical DFT level as is a the dynamical factmomentum space. that this The presence target ofgeometrical space elements a ( should dynamical be symplectic form interpretedgeometrical at concepts the with concepts in non-commutative geometry [ the classical level, this translatesnon-trivial into line the bundle fact over that a modularexpression spacetimes of can the be quantum viewed string as geometry. a space is a doubled space. The key element that differentiates the Born geometry from its effective geometry [ T-duality symmetry is manifest.works The is central what question type that ofphenomena. generalisation has of It been target has investigated space been in geometry foundof these allows that spacetime to the is encapsulate the proper stringy idea geometrical of concept modular generalising spacetime the which notion is fundamentally quantum [ to understand what extrainside geometrical double data enables field generalised theoryT-duality geometry in in to order the be to context embedded eventually of allow GG. a mathematical characterisationof of metastring theory [ a T-duality transformation, butstructure this inherent does to not T-duality express symmetry. the profound change of geometrical but is mathematically undetermined, whilevide GG a is geometrically satisfying mathematically description sound of but T-duality. does It not is therefore pro- of utmost importance onto a dual manifold a symmetry that exchangesthe the base manifold base can of eventually(or be the a implemented generalised quotient in bundle. space) DFT since fixed. of The At action best of we changing can implement an T geometry since it does notof contain string any undeterminations. theory, the However, fromsymmetry. main the reason point The to of stringy view expectT-duality expression a is of generalisation a T-duality of map geometry in from is terms the of T-duality sigma models suggests that unsatisfactory aspects that motivateDFT, us it to has been go noted beyond thathalf locally the the the coordinates. state section condition However, of there forcesno the the is geometrical fields art. not information to a that depend unique Starting reveals on solutionIn from only which to other half this words, of constraint, which the i.e. submanifold coordinates there of the is fields depend on. JHEP11(2017)175 ] ˜ ∈ L to 18 ) as ω = 0. ω (1.4) d 1 and L − ]). We | η ω L 34 , ) and sat- ]. := ˆ 33 is a distribu- , η, ω (cf. [ K . On the other is curved. The 32 L is exactly what η L η is encoded in the ω P Lagrangian is the complementary where ˜ ˜ L L , i.e. ⊕ η in DFT based on a canonical L = 0 and η L = | ˆ and we also do not assume η. η P − 4 , T = null ˆ and the associated Lagrangians ηK T are ω is a non-degenerate two-form. These satisfy to be flat compatible with ˜ L η ω to a para-Hermitian manifold is what is needed – 5 – P and with eigenvalue +1 and ,K L 1 K = + 2 ) is an almost para-Hermitian manifold. As shown in [ — which gives the more reduced structure group GL( K 1. Both ω − , η, ω P almost product structure is a neutral metric and η ), where ) defines an P DFT and GG can be identified. para-Hermitian connection that generalisesthat the it Levi-Civita satisfies the connection Jacobi and identity a underAn proof our extension closure of and integrability thepara-Hermitian conditions. realm geometry of that generalised allows geometryDFT to us and GG foliations to frameworks. associated construct Under with an this the isomorphism isomorphism the between generalised the Lie derivatives of isfying a new typestructure of that closure allows and toalso integrability formulate include conditions DFT key is examples on along the a theon proper manifold way a and with geometrical flat show curved para-K¨ahlermanifold. that the usual DFT naturallyA lives generalisation of the Dorfman derivative for curved The proposition that para-Hermitian geometry encoded into the pair ( η, ω T Most of the works on GG and DFT have so far not considered the presence of the extra Accordingly, the goal of this paper is also twofold: on one hand, we are going to show The key difference between the metastring target space and the DFT realm is twofold. This means that the kinematics of the metastring target space For a different approach to DFT on curved spaces, namely on group manifolds, see [ • • • 4 almost symplectic structure is needed in ordergeneralised to geometry. connect In theGG. other kinematical In words, part summary, we the of key are Born results going geometry of to this to describe paper an are: how extension to of unify DFT with in order to generalise the kinematicalThis structure is of what DFT we and formalise intend the to section prove. condition. that we can generaliseonly the condition notion that of we the willhand, Dorfman impose we derivative is are even the when integrability going of to the show distribution that the extra para-Hermitian structure First, in Born geometrybe we closed. do Moreover, not the assume provide compatible exactly two-form the extra geometricalgeometrically. structure In needed other to words, implement promoting the section condition In other words, the datathis ( para-Hermitian structure is essentialThis in implies order to that construct the thetion tangent string defined bundle vertex to operators. decomposes be as distribution the with eigenvalue eigenspace of pair ( a compatibility condition whichEnd( can be stated as follows. The operator JHEP11(2017)175 ⊕ (2.2) (2.1) TM := M ] who was the T 39 , ]. 38 is also equipped with 42 M -para-Hermitian manifold. T L , , it is given by ) α M d T y ι , the so-called Dorfman bracket or − α, One notable exception is in the work ) of y β ι 5 x TM + y, β L , β ] x = ( ι x, y Y ]. – 6 – ) = 16 structures will be given in [ ) = ([ ) and X,Y ( y, β η ( -para-K¨ahlermanifolds are introduced. In section 4 the x, α ) L algebroid x,α = ( ( L X . This bundle is equipped with a symmetric pairing described as M is introduced to give a formulation in terms of almost para-Hermitian or ] on target space duality in the context of symplectic manifolds. Our ω 41 , is curved, we will review the main geometrical elements of both GG and DFT, 40 ] who formalised the geometry of non-geometric vacua in terms of a flat Born η -para-Hermitian and denotes the interior product. The doubled bundle L 37 ]. x ι over a manifold 36 , The remainder of this paper is organised as follows: section 2 provides a concise review More general approaches to graded symplectic manifolds in the context of doubled spaces can be found 35 M 5 ∗ generalized Lie derivative in [ follows: given two sections where a natural generalisation of the Lie derivative of different) setups can be relatedformulated in in the the proposed language framework. of More details2.1 on this interplay Review ofIn the generalised generalised geometry geometry the setup T central object of study is the doubled bundle 2 Lightning reviews ofBefore GG embarking and onto an DFT extension kinematics case of when generalised geometry and doublerestricting field our theory summary to to the the kinematical aspects. We then outline how the two (a priori In section 5 theconstructing relation an of explicit this isomorphismidentifies construction between the to them generalised two and geometry notionstechnical showing is details of and that made generalised computations precise this to Lie by isomorphism supplement derivatives. the main Lastly, text. the appendix contains cepts of generalisation of the kinematics ofalisation the of doubled the space Dorfman is derivativemanifold. presented and by finding constructing This suitable a connection section gener- of on contains the the para-Hermitian our Dorfman main derivative satisfies theorem the which Jacobi states identity that on an our generalisation work can be viewedmetastring as formulation an of extension string of theory these [ pioneering works profoundlyof inspired the by generalised the geometrysymplectic and form double fieldalmost theory manifolds. para-K¨ahler approaches. A In discussion section on integrability 3 is the also almost included and the con- of Hull [ geometry. The otherfirst important to exception recognise is thator the the relationship work a between of para-K¨ahlerstructure. GG Vaismanof and [ DFT Alvarez We hinges [ would on also a para-Hermitian like to draw attention to the earlier work we will see — and the key geometrical role it plays. JHEP11(2017)175 ) ˜ L M × (2.7) (2.8) (2.9) (2.3) (2.4) (2.5) (2.6) (2.10) . by (cf. η, π, M M L = . . T , such that  P )) P D X y, β C ( ) ) Y B , The tangent bundle z,γ ( . ∂ ) , ( L 6 )) , − ))] π ). ) . ) algebroid on such that the distribution D Y Z,W )] Y η x, α B Y X,Y x, α C ( ( ) X , (( . θ ) L y, β ) X η ( . X, ∂ . L B − µ ( ) through ( ∂ can be written as Courant ) , π η x ( η x, α η )  y,β d Y,X η )) + ( (( ( − ) AB η (or θ ⊗ . There is also a natural projection ) η CD Y,X x, α )[ ) are local coordinates of x x,α η ( µ y, β ( ( + . µ ( θ x L π ( ˜ D , x AB ) L X y, β , η 1 2 Y Z,W ) := ( ] µ 2 1 Y = [ + d˜ B π x Y ∂ + ∂ Leibniz y 1 2 µ ] = + x, α C L ) x ( x − – 7 – , ) A X X,Y X d˜ L ) ( y,β Y X Y , respectively, ( . non-degenerate metric . If ( z,γ X A ∂ ⊗ ( CD η )) = L η ˜ ˜ X θ L B L does not, however, satisfy the Jacobi identity. The M L µ ([ , ∂ ∂ − ( )) = ( AB ) x η . Using this, we find that the three structures ( ⊕ is compatible with flat η y, β B Y Y = x Y ( x,α L Y L , ∂ X ( and X y, β ) = d Y ) = ( ∂ L − L ) = = ) [ : ) = ∂ X η A ))] = M = ) L : ˜ x, α x,α ) ( M ( Y X,Y ∂ ) x, α ( B L y, β T ]] ( B A ( ( ∂ π ( θ , x,α π ⊕ ( ) B ]: L X,Y,Z,W X X,Y ( ( 43 [[ η x, α = TM J (( A satisfy the anchoring property is equipped with a ) = η ). The generalised Lie derivative, whose skew-symmetrization gives the ) captures the difference to the ordinary Lie bracket and is therefore expressible in terms ∂ given by ]] )[ π satisfies the Jacobi identity P P ) index notation used in DFT, the C-bracket reads d, d T L X,Y ( z, γ and appendix ( d, d X,Y TM ( are coordinates on ( and denotes the usual Lie derivative along O A 8 ([[ π θ is normalised with respect to are isotropic with respect to O µ x L → x ˜ L L L M The space splits as In the T and ˜ 6 and : P µ The vector of the Y-tensor of DFT [ This generalized Lie derivative Jacobiator capturing thisDefinition failure is defined for any generalised Lie derivative The transition functions betweenbelong coordinate to patches that preserveC-bracket, is these given flat by coordinates where we have introducedT the vector L x 2.2 Review ofIn the double double field field theory we theory start setup with a C-bracket defined on the doubled space In addition, The four properties above define a so-called Third, π interplay in a natural way. First, Second, where JHEP11(2017)175 ) 2.9 (2.13) (2.11) (2.12) restrict . in ( )) corresponds ∂ µ L x . Accordingly, = 0 and hence W, Z ) , this is a result ( X B ) to any metrical D ( ∂ , θ ˜ J P ) ). Given two com- A 2.9 ∂ ∇ X or only on ˜ Z µ Y,X AB := are maximally isotropic ( x ∇ η θ . We can then decompose )] ˜ X P ( ˜ Y, P ˜ Y η D ( ( , η − ) Im ,P )) and ) ⊕ ) and Im Y,X X P ( X )) := ( ( ] in double field theory is to P Y. ∇ W, X P P ( θ ˜ D X 13 ∇ , Y,X , θ L + = Im , which implies ( 6 ) ) = [ identically vanish. For := ∇ + ], X P ) = 0. Y , one can check that the Jacobiators for the ˜ D Y T X D Y ∂ 38 D X Z,Y J θ Z, θ ( D X ( L ( = θ ( – 8 – L η ( P − ∇ P η and ∇ ) corresponds in this setup to imposing the sec- = Y , such that Im − D X P Y J , i.e. )) 2.13 ∇ T , i.e. ∇ X ˜ D where D P D L , respectively. = section condition L M W, Y ∗ Y ( . P → T ˜ D ∇ X = D+ T , θ J ) L : ∇ and ˜ P vanishes = 0. This construction is not canonical and from a geometrical Z,X P, ∂ ( TM ∂ θ ( belongs to Ker J ) is defined such that η D is an anchor of , the tangent bundle splits as θ η ) = Y,X P ( as first developed by Vaisman [ ∇ . θ ∇ P projecting to = 0 and T ˜ P B of θ the curvature of the map the vector A 7 X,Y,Z,W and We can see that the splitting ( A more fruitful approach is to impose the restriction on the type of generalised Lie The usual way to satisfy the Jacobi identity [ θ ( The standard solution of the section condition where fields depend only on • • • ∂ 7 P J AB geometrically appealing becauseferent of solutions the of fact the thattings section the condition can fields still remain be unrestricted recovered and by choosing dif- different split- to Similar statements hold for tion condition. However,Lie the derivatives only fields “see” are the still dependence on unrestricted, half but the the coordinates. projected This approach generalized is more Coming back to the DFTprojected setting, generalized where Lie derivatives of the following facts: with respect to the covariant derivative as this gives a decompositionderivatives of the generalised Lie derivative in terms of the projected Lie connection where now plementary projectors the vector fields by imposingη the perspective rather ad-hoc. derivative one allows. Let us first adapt our notation by generalising ( we have and an explicit computation shows that for the generalized Lie derivative JHEP11(2017)175 , ˜ P M 15 T and and × µ (2.14) L . This ∂ ω ∂x M transverse = . Working = µ F P structure and ∂ T the , is to demand ]. It is given by bundle over the µ , without loss of η x 46 L d – distributions P → , the tangent bundle 44 d T [ and ˜ P i T P M and is a central object that . Even though the bundles , which is the approach we is spanned by as described in the previous ˜ ], that a natural framework P para-Hermitian as a rank 2 M of DFT — a two-form P T L 39 η × T P is the integrable subdistribution. T ), , and the splitting is only done on i M L ˜ M ). However, in the general setting of M = = µ ) forms a × ˜ x F P ) metric . , T µ µ M x x d, d , η, ω d ( . Intuitively, this could be seen as one of the P O . To recover the distinguished local coordinates ↔ M – 9 – ˜ L µ ˜ ∂ ⊕ L para-Hermitian geometry , such that . We can then construct an isomorphism between µ , which provides us with a map P . We can now view = is a product of affine manifolds. That is ∂ ] F , have the same rank, the dimensions of the underlying p of P [ P P are coordinates along the leaves T M F T µ ] p x [ split globally into ( ` . If one of the distributions is integrable (say, and P ˜ L = M ⊕ F is spanned by T L are vector spaces or flat tori. The situation is considerably simplified F = ˜ T M P is not a flat metric. The triple ( T is not globally a product manifold to get a half-dimensional base , in DFT the base is extended to give η and while P P M , which can be expressed in local coordinates as µ is closed and that its Lagrangian submanifolds are integrable. This corresponds ˜ ), we therefore have to impose the integrability of x ∂ M -dimensional manifold, we introduce a splitting into two rank T F ∂ µ d ω ˜ T x As we will see the simplest generalisation that allows for a curved In the usual DFT setting, Let us now consider the following construction. Starting with , = is a d-dimensional manifold which is bijectively identified with such that ] and to reunite the central ideas behind DFT and GG, we need to include in our µ µ -dimensional foliation manifold x ˜ ˜ allows one to pass fromeven the when local constructions of DFTthe to goal a of globally this well-defined geometry, section is to review andthat discuss its relevant properties. 3 Para-Hermitian, para-K¨ahlerand integrability It is now clear that to16 accomodate the string geometry thatgeometrical follows framework from — the in metastring addition [ totwo-form the can be thought of as giving a preferred splitting of ( paragraph. It turnsfor these out, considerations as isdescribe first given in observed the by by following sections. Vaisman [ where since the coordinates of the Born geometry, weexample, have to consider issuesthe stemming level from of the global tangent considerations. bundle For ∂ and generality), we get a foliation F the union of leaves d in local coordinates, where coordinates (i.e. coordinates that are constant along reasons why the section conditionin of the DFT previous needs discussion, to this has be to imposed, be but spelledof as a out we 2 more have explained carefully. L We have seen aboveto that get while inof generalized interest, geometry namely one extendsmanifolds the do fibre not directions match.the base To relate the two setups, one needs to find a way to “reduce” 2.3 Relationship between GG and DFT JHEP11(2017)175 . . ]. ), L η 42 and = with (3.5) (3.2) (3.4) (3.1) ⊥ . The ˜ 8 , η, ω M L P K × M equipped with ) as ( such that P ] that a natural L Lagrangian P' , η, K 39 have the same rank , P are shows up in the non- ˜ L 38 ˜ η L 1 and the tangent bundle is an important theme of ) to . 1 eigenbundles of as a space  and ) (3.3) ) η R − and ω . K is a manifold L ) ,Y d, ) L ) R − is locally split η. ) with a compatible real struc- X L. such that 1 d, ( ( − ) ∈ , ω P 1 2 K , η, K P ( = P η P T ] for some standard results on para- ( := is neutral, X,Y from GL(2 ˜ ηK is maximally isotropic with respect to 48 P . This is a natural generalisation of the ) = GL( ) geometry of ) = ∀ T η ˜ End( R P L M ]. The curvature of are the +1 and T d, d ∈ d, T ( X,Y 47 ˜ L ] for the relationship with bi-Lagrangian struc- ( , O K – 10 – ω also imposes that 49 Sp(2 ,K and ) and ω 1 and ∩ K ) = 0 or ) L + R TM = + and as and the 1 2 X,Y ( η ˜ L ( ω ηK 1 2 K d, d, η := O( := , where and ˜ L ω P L ⊕ are affine manifolds [ L ˜ and an endomorphism = M η P An almost para-Hermitian manifold . T and is also maximally isotropic. Since . The axioms further imply that . We see that an almost para-Hermitian structure could have also been equiv- is not closed we can still defined a Lagrangian subspace of ηK M ω ˜ P L ω = is a real (or para-complex) structure, it has eigenvalues is symplectic or that the distributions are integrable. The interplay between the . When no confusion is possible, we will sometimes refer to ( ω K Dim splits as ω K 1 2 More conceptually, an almost para-Hermitian structure on a manifold is a reduction We refer the reader to the review paper [ Even if P 8 = of the structure group of the tangent bundle Now the compatibility between respect to alently defined asture an almost symplecticwhere manifold ( is an almostnatural symplectic projections structure, onto i.e. a non-degenerate two-form. We can also define Similarly, d Since of second condition implies thatIn the other distribution words Let’s start by the definition of aDefinition para-Hermitian 1. geometry: a neutral metric our work. Hermitian manifolds and totures. the article [ We alsoframework acknowledge for the that geometrical it understandinggeometry. of was DFT Our first geometry work is and noted given results by by can para-Hermitian Vaisman be [ viewed as a continuation of his work, also see [ the leaves constancy of the pairingDFT between geometry setting. Ifthat we want to goalmost beyond symplectic this geometry case, of we have to relax the condition to a para-K¨ahlergeometry. In this case the manifold JHEP11(2017)175 . ˚ ∇ 1) K , can (3.7) (3.8) (3.6) η (3.10) and )). The ω Y,Z can be split in ˆ Ω( ] for connections η ˆ X, Ω which is a (2 51 ( η is closed. Therefore, ω . ) . . ) := A ) (3.9) )) and therefore is not a para- , . The Levi-Civita connection ], also see [ ) ω of the metric Y,Z X,Y,Z K Y,Z ( 50 ( X,Y,Z , i.e. ω ∇ ˆ Ω( Y,Z ω ) X and imposes that contorsion tensor ˚ X, ∇ K η is of diagonal support with respect to ( ) ω X η . and ω ˚ ∇ η ˚ ∇ X (( , = 0 ) + X,Y,Z , η ( . A general discussion and classification of and is torsionless. ω ) condition for the Levi-Civita connection ω 0) tensor, Ω( A η 2 ∇ Y,Z , ) = ) = preserves )) = 0 X – 11 – = Z ˚ ∇ ˚ ∇ η ( ( Y,Z and a so-called ˜ η ( ∇ P is central for para-Hermitian geometry. In order to , ω ˚ ∇ X,Y,Z ) satisfies the following properties ( ω X ) = Y ω ( ω ˚ ∇ d . The second property relates the derivatives of is symplectic. This follows from the following lemma: P ˜ ( and L Y,Z ω ω which preserves both X η ⊕ X ∇ be an almost para-Hermitian manifold with compatible two-form L ∇ ( ˚ ∇ ) η ) geometry. = P d, d ( T , η, K O P ( is the sum over all cyclic permutations of A para-Hermitian connection on an almost para-Hermitian manifold ) Let is a connection X,Y,Z ( ) P It is a well-known fact that any metrical connection The first property emphasises that unlike Since we have a metric, the Levi-Civita connection is a natural connection to consider. Given an almost para-Hermitian manifold, we can define compatible connections: is the unique connection which preserves , η, K . The Levi-Civita derivative of P ˚ We also denote byrelationship Ω between a the para-Hermitian corresponding connectionthen (3 and be the expressed Levi-Civita purely connection of in terms of this tensor: The third one isthe standard, connection it is simply torsionless. relates the This lemma differential and is proven covariant derivative interms when of appendix the Levi-Civita connection tensor. By definition it is given by where the decomposition Hermitian connection. Demanding that a necessary (but not sufficient,to see be Corollary para-Hermitian is that Lemma 1. ω Alternatively, a para-Hermitian connection preserves ∇ However, in general the Levi-Civita is not compatible with Definition 2. ( understand this geometry one needspatible to with understand both the structures. propertiesgiven of statements We the can review connections be these com- compatible found properties connections and in here, classifications appendix can details bein and found the proofs in context [ of of the The compatibility between JHEP11(2017)175 I ) = (3.12) (3.13) (3.11) X,Y,Z and we see ˜ L Ω( . . ) for which the are conjugate to ˜ )] x )]) Y Y C x, ( ( 1 and with structure ˜ P , − ) P ⊗ . = X,K X T , )) ( 2 Z ˜ )) I P ( [ , ] + [ Z is an Euclidean metric and ˜ ( P ]. P H , ,Y ) H the integrability of 52 ) = ,P : = ) Y L ( ) X I . Y ( ˜ ( P ˜ ∈ x H K ∂ ˜ T P are independent from one another, i.e. ([ X, X,Y − I is encoded into the real analogue of the ( N ˜ ), where X,P K L ˜ P has a skew contorsion tensor, 2Ω( ,I K ) = − and decompose as ˜ x ] − ) as a sum associated with each Lagrangian H ∇ ∂ and , I ( ,N ˜ , while P – 12 – L )] K 3.12 L X,Y )) = 2Ω( )) = Y ( Z Z ( ( and ˜ ,P P )] + [ ) ) where x ,P , ∂ Y ) ) X ( ( Y Y ( ( ] for definitions and examples. When the manifold is para- P X,Y ) = [ ,K ˜ ( P P ) x ˜ ( ( ˜ P P is closed, the manifold is said to be almost K¨ahler.When both 53 ∂ , ω ω X ( N ( ω = X X : 50 . K K ) ˚ ˚ ∇ ∇ ω ) + X,Y ) := [ ( X,Y ( P P N X,Y governs the integrability of N ( , whose diagonal components are determined by the Levi-Civita derivative of Any para-Hermitian connection is closed, the manifold is said to be almost para-K¨ahler.When both conditions ) ˜ L K ). In the almost Hermitian case, there are two levels of of integrability one usually ω ) = N ∈ d 4 P N X,Y X,Z,Y Despite the similarities, the geometry of the para-Hermitian structure is fundamentally Therefore, the same characterisation of almost para-Hermitian manifolds can be im- These properties imply that only the off-diagonal components of the contorsion tensor ( as follows K Ω( Now that one can vanish evenof if “half-integrability”, the we introduce other some one new does terminology. not. In order to account for the possibility each other, so one isthe integrable integrability if of and the onlyone two if distributions Lagrangian the distribution other one canseen is. be by In rewriting integrable the while theN para-Hermitian Nijenhuis the case tensor other ( is not. This can be best Hermitian it is locallyproduct split, structure reads i.e. it admits a setdifferent of from local its coordinates complex ( case analogue. the two There distributions are that two diagonalise reasons behind this: in the complex When this tensor vanishes,two-form the manifold isare said realised to the be manifoldstudied para-Hermitian. is extensively, said see Moreover, [ to if be the para-K¨ahler. These different situations have been conditions are realised the manifold is said to beplemented. [ K¨ahler Integrability of theNijenhuis real tensor structure notion of an almostis Hermitian an manifold almost ( complex structuregroup with U( compatibility studies: if themanifold; complex if the structure two-form is integrable, the manifold is said to be a Hermitian are not determined by 3.1 Integrability, closure andOn a the key surface, example the notion of an almost para-Hermitian manifold is very similar to the − ω Lemma 2. JHEP11(2017)175 9 Z ∈ X ) TM D X is the (3.15) (3.14) ( ∈ v f ) )) = d denotes its , ) X ( x, v P L Z is integrable. T -para-Hermi- its differential Γ( (d L ˜ = ( L , → P . Here ∈ p ) ) TM to , we can define the M is the unique vector . These are defined TM ( X -para-hermitian then ( )) = 0 ˜ M : L . ∞ M ˜ , h L x Z )-para-K¨ahlermanifold x ) ˜ L Z C ( T P → ˜ -para-Hermitian, almost T L → TM Y ˜ P ˜ T ∈ L ∈ , ∈ , X is defined by and ] L, : ∈ ) is said to be f and v v ˜ L D and d ) and a compatible connection ) L π X Y ( ) TM ⊕ ( L v g : for ˜ L of P x, v = , ) -Lie bracket (i.e. a Lie bracket on v with ) -para-Hermitian if X,Y = -para-K¨ahlerthen it is said to be → P ) , η, ω f ( ˜ P P L L X ( P X = ( P D ( )] M ( ( T v X and R p TM : Y [ P ]) that illustrates the usefulness of the ( ( v is integrable, the manifold is said to be L v ∈ Z to ω ] = , − ) L ) 54 d , whose kernel defines the horizontal bundle. df → ]) M can be viewed as an Ehresmann connection, X ]). )[ ( 39 is the total space of the tangent bundle of a . Before doing so, we present a key example. x x, v h T ω X [ D TM 56 ( – 13 – , , TM X,Y ] v = ( ∈ : ([ , TM 55 p 57 h P → X h is integrable). or T = )) = 0 = 0 = : ˜ L . The vertical lift (see [ Z P of the tangent bundle in terms of a sum of a horizontal D P P ( X -para-Hermitian and ) equipped with a metric ˜ )] )] L is viewed as a map ˜ L L such that . the canonical projection, by d ,P Y Y ) ) p ⊕ ( ( f )) = ˜ v h L Y ( TM M L , , ( and a point v ) X ) ∈ ∈ ( M, g, D = ) X associated with the decompostion h ,P X ) to obtain its horizontal and vertical lifts M ( ( ) ( Z ) = P X v h D are integrable the manifold is para-Hermitian and similarly for the π P → [ [ ( X ∈ pr-¨he,by-para-K¨ahler, echanging the role of T d : ( v -para-K¨ahlerif TM ˜ L L ) the vertical bundle. In physical terms the vertical bundle is the space x, v x ˜ L P π ( π ( Γ( . The affine connection f . Here ω An almost para-Hermitian manifold The horizontal lift of d ∈ M and d x a canonical example (see also [ TM can be viewed as a map T L ), satisfying the relations [ X -integrability is when such that ∈ D ˜ is integrable. Also we say that an almost para-Hermitian manifold is almost L and a vertical bundle = Ker(d p -para-K¨ahler. Similarly we can define the notion of ) = L ˜ L L L L x TTM ( X,Z ∈ 1 We can then promote the point-wise construction of the previous definition to any Given a point From this definition it is clear that if a manifold is both ) = ). These vector fields can then be equipped with a − The affine connection 9 ˜ π L X . We denote by P -para-K¨ahlerand -para-K¨ahler(or -para-K¨ahlerif the conditions -dimensional manifold ( ( ˜ where differential. vector field Γ( T as follows: Definition 4. h is the unique vector linear function is i.e. a decomposition bundle notion of horizontal and vertical lift notion of d D and by of velocities and it is canonically identified with the tangent bundle since the vertical fiber When both para-K¨ahlerand almost para-K¨ahlercases. The factis that para-K¨ahleris needed an in ( theintegrability conditions next and section properties where of we d study theExample: relationship between the almost L it is simply para-hermitian.L When in addition tian if L are satisfied. If a manifold is Definition 3. JHEP11(2017)175 b v ) = ) = c ab t t ( ( (3.21) (3.17) (3.19) (3.18) (3.20) -para- p p ˜ L : ˙ := Γ h = 0. This v c a v . ˙ ) x a D ∂ are well defined. ( 4 v if c ) (3.16) x − ) can be decomposed . The projection map c is torsionless. d ˜ a ∂ M P ∂ vanishes. It is v ∧ D . ∂v c a T . We see that the set of b ∈ )) = ] Γ x a D is integrable if and only if D d ) d := ∂ t x − ( ( L ] = 0 a d of Γ( v ∧ a b ˜ x ( ∂ v ∂ v a b d ( and the almost symplectic form X -para-Hermitian structure x , and we denoted Γ a [d c ˜ d a c L . and . ∂ ˜ v X ∂ + Γ and corresponding local coordinates b b ,K v b a c ab ) x Dv given in Definition c a abc ∂ v a d M ∂x Γ − Dv ω v ) = ∂ R v , where , ( a b 1 2 )[d , we need to consider half para-Hermitian ∧ − = Γ h h c P on := ] ˜ P a x, v ∗ ∂ ) = b + d a a x + Γ v a T )( ∂ c x x c a ∂ a a )d vanishes. . An element ) is horizontal we recall that a vector Γ ∂ ∈ X [d is the curvature of v , where the dot expresses the time derivative. Dv a ( x a x Dv )) = + ) v – 14 – ( ( D h ∂ − v ∂ D c a a a ] ( ∧ v ab . We also see that the set of horizontal vectors is ∂ a K Γ h can also be described as the almost bi-Lagrangian b d := d ( g b a ∂ ) = , , P is integrable and that h ˙ x . The vertical and horizontal maps associated with X,Y a x a a [ ( , Dv b = T d ˜ x ∂ a ˜ ˜ − ∂ L = Γ D a x, v x ) ω ∧ TM Dv ( ] = ( = X b a − π ,K x c x, v v ) ) = d and we denote ] d ( v Dv a x b ) = Y d a ( ab x )[ ∂ V a T ab a (d ∂ ,D g x, v cd v ( + is canonically equipped with an g X K h )( a ) are then explicitly given by = ) D ∂ = X ) is given by (d v ( )) = vanishes which establishes the first claim. The pair dual to the basis b is transported parallelly along a path ω v = [ TM forms an ideal of ∂ d ( x, v D v ( Γ( h ˜ L TM ) a ) where , ). The result then follows by direct computation. ) TM ∈ a ∈ X a T M, g, D , where ∂ a is torsionless. It is almost para-K¨ahlerif and only if ( ∂ is para-Hermitian if and only if the curvature of given by a ∈ )) ∂ ( are the connection symbols: X,Y ω, K ) x a ) ) v ( D ( ˙ ( x ∂ ) = TM )) D ab c p h ( ( t η a We have already seen that v ( R , x, v ) X , v ( ) = ) on We have seen that to curve the geometry of We will now also show that the lifting maps a ) ˙ v t X ∂ = ( ( h T M, η, K T M, η, K x x, x, v is given by It is straightforward to compute the differential of The para-Hermitian structure on structure ( the curvature of ( This can be checked by direct evaluation, for instance ( K¨ahlerif Proof. manifolds. The above example is anLemma instance 3. of such a( geometry: for simplicity. In( order to check that translates into the conditionTherefore, ˙ the tangent components( ˙ to a parallel transport are in the image of in these coordinates isas given by X where Γ vertical vectors integrable if and only if the curvature of For this purpose, we choose local( coordinates where JHEP11(2017)175 . ) d )) as v Z ( L 3.23 (3.22) (3.23) (3.25) (3.26) -para- abcd ,P ˜ L we have R ) )) ) is Y Z = ( ( v ,Z (3.24) . ) ,P ,P ) abc TM )) ) )) R X Z )) Y Y and properties of ( ( ( ( play an equivalent X,Y Z ˜ ( P ( L ( ˜ ˆ ,P must vanish. These ,P L ,P T ) ω ) ( )) ,P . η ) F X X Y )) ( ( ( and X . ( Z P . )), means that there exists P on ( ( ( = 0 implies that the torsion L ) = )) ,P P )) ) T ( ω ω Z ω ) Z ,P ( ω 3.13 ( ) X Z − ) ( ( Y Y P ,P ( )) P ( ) ( P X,Y,Z ˚ ∇ )) = 0, and d Z ( ˆ Y ( T ,P ˚ ∇ ( Z ) T − is Lagrangian. This establishes ( ( X,Y,K − ,P − ˜ ( X is tangent to the leaves, in other words P ,P )) ) ) L = 0 (see ( ( ) , T ) in the appendix, which establishes that )) Z ) and the curvature tensor L X X ) P ( P X ) ( ] ( Z as being “tangent to the momentum fibre” Y ( ( Y N ( ω A.4 ab P ( c [ ,P ˜ P ) ˜ L ) P X ( P ,P Y X,Y,Z , ( ) ( Y T ) ∇ – 15 – ( P is integrable this establishes that = Γ ( ). Y − ) = d − X ( with torsion P ( ˜ c L ab is encoded in the following formula ,P ,Z such that P ˜ ) P T ) ( − ∇ − ∇ ) = ( ω ) = ω ) ) X F ω ) ( ˚ ∇ Y Y X,Y,Z X ) P ( we have that ( X,Y ( ( ( X P P P ( ω ) and P N X,Y,Z ˚ P ∇ X,Y,Z N X ( ( N ( ) P ∇ ( P = d ω P ( η N , which is insured if d ) = N ∇ P L ( ( X X,Y,Z is integrable then the pull back of d η η ( ) := L = ) = X,Y,Z ( P -dimensional leaves X,Y,Z d N ( 2 The cyclic permutation of the projected Nijenhuis tensor is given by P is torsionless. It is also clear that the condition d X,Y,Z ( N D P . There is a profound interplay between the integrability of -para-Hermitian manifolds From the definition of L N L = The integrability of F . For instance, if if and only if the torsion vanishes. From this expression we seevanishes that if d the torsion vanishes.K¨ahlerif Since vanishes. On the other hand,vanishes if by the the torsion Bianchi vanishes the identity. second This term concludes in that this the expression also structure is almost para-K¨ahler where we have introduced the torsion first two terms in theand last the equality first vanish part because of the lemma follows from ( Proof. The second equality follows from the fact that the connection is para-Hermitian, and the Given a para-Hermitian connection the relationship relationships are investigateddefinition in the following lemmas.Lemma 4. In the following, one uses the as in the previous example. a foliation by T ω 3.2 In the almost para-Hermitian geometry,role. the two For distributions the purposebeing “tangent of to our the presentation space-time it base” will and be useful to think from now on of The relationship between JHEP11(2017)175 ). )), and Z van- ( )) = 3.14 a (3.27) . The ∂ Z P vanish. a ( ∂x ˜ ∂ N ˜ Y,P P ˚ ∇ ) F = , a ) a K x . ) Y ∂ b implies that ( on X x ( ˜ P = 0, are an im- 1 d˜ establishes, this and similarly for ). P , := d˜ ω ) 2 . ∧ ω ˚ ∇ then one sees di- ˜ a ˚ X D (( η, ω ( , and that x on a K¨ahlermanifold. η L P )d 1 ( ˚ ∇ ˜ x and ω instead. Since the torsion = const )) = a x, a ∂ ( ˚ ∇ Z b x ( ˚ ∇ a a η . x ,P ) = Y = 0 pr-¨hei and-para-K¨ahlerif only if the Levi- ( -para-K¨ahler. On the other hand := d L ω P ) L ) ˚ are integrable and d , where we have denoted is X D X ) and using ( a ( )) = 0. Now, Lemma ) , ω ˜ ˜ P ∂ L P ) are given by ˜ and the pull-back of d Y )-para-K¨ahler. Using the previous lemma a ) is ˚ ( − ∇ ˚ ∇ is symplectic along 3.24 ˜ x take off-diagonal form and can be expressed L F η, ω ( : P ( d η ,P and ω η, ω η L, ) N ) = L ⊗ – 16 – a L X b ∂ ( x ( ). The second equality follows directly from the and P ( )) also vanish and that d K η 3.8 + d˜ ω Y ) b ( Z x ˜ . In practice, this means that there exist local coordinates ( ) are satisfied thanks to Lemma and P d˜ ˜ , L P ) is a para-K¨ahlermanifold. As Corollary a ) ˚ ⊗ ∇ ∂ = ) as X 3.14 a ˜ ( x ˜ x F , η, ω P x, ( T P ( )(d ) = + ω b is symplectic along ) a ˜ a x Z ∂ η η ( and ( x, )). This last component also vanishes by the second condition of ( P ( ), which means that ( L K b Z ˚ ∇ a ( = η ˜ The Levi-Civita connection is a para-Hermitian connection if and only if P 3.26 The para-Hermitian structure is symplectic. We now explore further properties of ) is para-K¨ahlerthen it is ( , = F ) T ω η ) it follows that Y ( . This in turn implies that , η, ω ˜ is para-K¨ahler. . In these coordinates, the leaves of P -para-K¨ahler implies that both a ( a P ) 3.26 If the Levi-Civita connection of x ˜ L x ∂ ) such that ω ∂ ) with a ) ) while ˜ ˜ First, the condition of being para-Hermitian implies that there exists a double foliation This corollary is of central importance to us. The para-K¨ahlercase is the closest If ( x X F ( , = , , η, ω and a P a F η, ω P x ˜ ˜ ˚ From ( the components ∇ This establishes the corollary. Proof. rectly that theishes conditions due ( tobeing ( of the Levi-Civita connection vanisheswe and 2 obtain thedefinition first of equality the from exterior derivative. ( This leads toCorollary the 1. following corollary: Civita connection of This can be seen starting from the definition ( in terms of functions It is convenient toimportance define of the these partial differentials lies differentials in the following lemma: ( ( ∂ F The para-K¨ahlermanifolds, where both portant subset of examples straightforwardlysection, generalising we the assume that flat ( DFTmeans geometry. that For this the Levi-Civita( connection is the natural connection to use since it preserves generalisation of DFT geometry.Civita connection The is previous the results natural connection shows to that use in3.3 since this it preserves case ( The the para-K¨ahlercase Levi- twice we can then conclude that Corollary 2. ( JHEP11(2017)175 ω are (4.1) a . The (3.28) (3.29) η . Their F K and ˜ are flat. In = a ˜ ˜ can locally be ˚ η D P b and the isotropy − a η η ), we can construct and P − . This can be true if ˚ ˚ D D = , and , η, K is not flat the leaf geom- )) P 1 ηK η Y T ( = 0. The same conclusion is ]. = ˜ P 2 K , ˜ 52 P ) ˚ D + X ( ) (3.30) . ˜ x P form a basis of one-forms on P ( . a = 0. If one introduces the one-forms x, η ) mean that the forms η = 0 ( φ 2 = 0 b ˜ ˜ ˚ ˚ ˜ D Dω ∂ a 3.29 )) = a ˜ η ]. This result of central importance means ∂ Y = ˜ ˚ ( D 60 ˜ – P – 17 – , = ) = ˚ Dω 58 ˜ a x X, vanishes, that is if ( = 0 ), which satisfy x, η ˚ L Dη ( 2 P b gives rise to para-Hermitian geometry. We will now a D T η η ) = φ along ,Y , one can easily see that the closure condition implies . The compatibility condition ) b ˜ ˚ L D x X is invertible means that in End( d˜ ( b b a P a ˜ is para-K¨ahlerthen the partial differential P and ( η η η ) L := and a , η, ω η P P ( . This proves the lemma. and ˜ If ˜ ˚ D translate into the following property b a ˜ η L is the para-K¨ahleranalog of the K¨ahlerpotential [ a First, since the Levi-Civita connection is torsionless, it means that the condition x φ and As we have seen, given an almost para-Hermitian manifold ( Let’s finally remark that the conditions ( This lemma expresses that in the para-K¨ahlercase even if = 0 is equivalent to the conditions := d L ω b previous condition implies that thisand basis only is if parallel the with curvature respectreached of for to η Now the condition that other words, they satisfy Proof. d Lemma 5. of which will be needed frequently. In the previous sectionwhich we together have with seen theexplore the canonical metric constructions introduction related of to the connections in almost this symplectic setting. two form projections, respective images are where 4 connections on para-Hermitian manifolds closed. Using thewritten Poincar´elemma in twice, terms we of conclude a that potential the tensor etry, i.e. the geometryin induced on symplectic the geometry leaf, thatconnection is the called flat. leaves the This of Bottthat is a connection leaves related symplectic of [ to a foliation thedirect symplectic admits well-known derivation foliation fact of a are this canonical fact affine flat in manifolds. the para-K¨ahlercase. The previous lemma is a more JHEP11(2017)175 , P are , i.e T (4.5) (4.6) (4.7) (4.8) (4.9) (4.2) (4.3) (4.4) ˜ L P (4.10) N defines a P → T X : , the projected P P is a vector field ) P , ), then D . T . P is an element of )]) f T Γ( f Y ( ] = 0 ∈ → Y. f ,P . the complimentary projector. f ) ) L, Z X ))[ and a projector )] and a projector D P Y X Y ( X ( ∇ X − ( ∇ ) P , := D and we introduce the twist of the ˜ [ ,P , D 1 ) P ] ) ( P ( ) f Y Y − X f T ( P on a function x . := ] )[ Y. D P ) + [ Γ( ˜ ). Moreover, D X X )]) P X X ( D ( Z,W X, is defined by L Y Y Y ) ( )) = ] P P and the projected Nijenhuis tensor ( ) D − η Γ( f → 7→ ∇ X ∇ P with P ( )[ − Z ,P . ∈ τ ] T (where of course ˜ ) ) ) X,Y P ) ˜ ) := X := P – 18 – ( Y Y ) := x ( Y X f Γ( L P P f, ( D 4.1 X Y Z P D ˆ T , R ) P X → D ( (D X ) = X,Y ([D ([ Γ( η = Im η ) D X, ( ) = ˜ P → ( X P P X, P × D ( x , Y η ( ]: T θ ) ( T η ) = P fY := [D L , a compatible connection er P ( , a compatible connection , the projected differential ) := 60 ) := ) := η ) η , X Z ,X : Γ( ) ) P : Γ( )) = D 46 ( = K f [ X ) above): ∇ ∞ X,Y X,Y ˜ L L ( ( (D X,Y C P P X,Y ( P τ ( 3.13 ( P N ∈ D and η X,Y,Z,W ) such that ˆ , the projected torsion R ( P along P f P P ˆ Z, θ R T given by R ( η Given a metric Given a metric satisfies the Leibniz rule Γ( P = Im ∈ X T denotes the action of the vector field as the surjective map L ] X f P [ X ) = 0. This can be seen by using ( projected derivative D f Using this projected connection, we can define the analog of curvature, torsion and the (D We also define projected derivative which satisfies elements of Nijenhuis tensor (cf. ( Definition 6. Riemann tensor for some P Viewing partial connection where Note that D Given a function such that In order to definethe a curved generalisation case, of we the need Lie the derivative following andDefinition objects. the 5. corresponding bracket for we denote The 4.1 Projected and partial connections JHEP11(2017)175 is → ) ) P (4.11) (4.12) (4.13) (4.16) (4.14) (4.15) P T T is related Γ( , the asso- P 1) Riemann : Γ( P → . , R ) 2 ) . . Z ) ) = 0. X P ) Z is not necessarily T D = D η X,Y Γ( X,Y ) = 0 ) = 0 X,Y ( ( Y, P D ( ( Y × P P θ ) R P η ( ) Z N N Z ˆ ) R ( P P ˜ f P + T ).
)) = X,Y D and a projector = and it is easy to check that ( , i.e. , X P , i.e. ) L is related to the (3 : Γ( ) ). This curvature Z . X, L ∇ N ] ˜ L ( Z L X,Y Y,X P f ) K D η )) ( ∇ ( ˆ )[ Y R L D Y,X Y + ( ( to form a generalisation of the Lie D Z, θ fθ ( D X,Y Z )) = ( η θ ( ,P η X,Y + ) − )) P ( + ˆ f Y P X R Y ( ( X,Y ) f X . N ( )D P Y K Z D ,P ( = ) is valued in + ) 1) tensor as can be seen from ˆ D X R , θ , Z Z ) – 19 – X D − X,Y ) ) ( ( ( Z X,Y ( P ) = η Y ( 1) tensor valued in is such that ( 1 2 ˜ P , X D ˆ ) ( R ). It can be expressed in terms of the ordinary torsion X,Y θ − ( η ) = X, fY X X,Y = ( P ( )) ] as (Dorfman derivative) D ˆ does not vanish. Then , a compatible connection P := D R P Z X,Y Y η ˆ , forms an integrable distribution if and only if the projected ) R f is a (2 ( ˆ ( , via P Y, R Y is the dual vector bundle of ))) = ( P P is a proper (3 X, fY N ∇ X,Y = P η ∗ D X ( ,P [ τ fτ ) ) = P L L D ) vanishes. Since D can be viewed as a partial connection D : X,Y Z ), we can define the curvature of D as θ ˆ − ( ) R X,Y X P ( ( fZ ) := P ) = X D ˆ T P )( R X,Y θ Y ( ( ( Γ( T while Y,X P fX, Y ( ( ) (where × N X,Y − ∇ X, fY D X P ( Z,P ) ( P , the image of θ ) = Given a metric ˆ ( ∗ R Y P via P L T η ˆ τ L ]. X R ∇ ∇ Γ( 38 X,Y ( End ) = of P , the projected twist vector → ⊗ τ ˆ ) = R f generalised Lie derivative ∗ ) L , the usual curvature of The definition given for the projected torsion and curvature is still meaningful even if The projected torsion Note that P 2 fX, Y R X,Y ( T ∧ is not integrable, i.e. when ( P Definition 7. ciated given by where the twist The main ingredient enteringfield the theory construction is of theThe both purpose Dorfman generalised of derivative geometry our that workflat. and generalises is double to the We generalise can notion thisderivative now [ notion of to use the the any Lie case metrical where derivative. connection of which in general is of course not tensorial4.2 in Generalized lie derivative and bracket It is tensorial in L tensor Like D where for the last equality to hold we needτ integrability of T The projected curvature Nijenhuis tensor Γ( Γ( to JHEP11(2017)175 . is Y (4.20) (4.19) (4.17) (4.18) and . If they X B Jacobiator gives the Courant , the , , ) ) 7 ) Z Z X . Y ) D Y X D L D , ) − Y, X, Z,W Z ( ( Y, Y . η Y η D X holds. ] ) D X L D X f D L D . L ) + ) + )[ Y, L ( L )] is a Killing vector field. It can be ( 1 2 Z X as in Definition X,Y η − ( ( Y X η D Z,Y P D Z := ] L X,X D X ) + ( D Y 1 2 D X, η ) + L ]] (D ( )[ , η η + Y Y,Z D X Y D X D D X − − ( ]] L X,Y L ) ) L . Moreover, unlike the usual Lie derivative, a P – 20 – [[ ( ([ ( η Y 1 2 η f η does not satisfy the Jacobi identity in general. In Z X,Y X,Y D D ) = ) = Z ) := )] = L with = [[ X, X (D ( fY . The second property — which implies the preservation Y D X ∇ η η ( Y,Z L D X Y ( D X η L are not skew-symmetric under the interchange of L Y, ) = ) = )[ ( Y Y η X X,Y,Z,W D Z ( D X ( X,Y L L P D under which this object is tensorial are given in appendix D Z J states that the Jacobi identity for L X, D ( ( L η η = 0 Given a generalized Lie derivative D and a normalisation condition which are respectively given by J η ], which is obviously skew but does not satisfy Jacobi identity. We will mainly In generalised geometry and double field theory, the Dorfman derivative satisfies 25 contains a derivative of for any vector field — distinguishes the generalised Lie derivative from the usual Lie We will take a detailed look at the Jacobiator for para-K¨ahlerand para-Hermitian The generalised Lie derivative This generalised Lie derivative satisfies three key properties: the Leibniz rule, compat- D L η Taking the skew-symmetric combination bracket [ work with the generalisedlated derivative by instead of the bracket as they are equivalent and re- geometries in the next subsection. Remark. the Jacobi identity (undergeneralised certain derivatives constraints). Unlike the ordinary Lie derivative, these The conditions on are satisfied, order to measure itscalled Jacobiator. failure of the Jacobi identityDefinition to 8. be satisfied,defined we as introduce the so- and using thegeneralised compatibility Lie of derivative ofthe a third vector property. field along itself does not necessarily vanish due to derivative for which suchsimply a proven condition by summing implies the that following two terms The first property followsof directly from the factof that only the first term in the definition ibility with JHEP11(2017)175 ) ), )). )) is a 4.21 , and ) 2.11 (4.24) (4.22) (4.23) (4.21) 4.1 4.13 X ( . P ). ), and the ) ) ˚ ∇ P be the Levi- Y . We recall X ) T = ) ˚ ∇ ∇ ) )) = 0 vanishes. This ) is said to be Γ( X Z,X Y ) W,Z ( P ( ( ˚ D ∈ P and corresponding ˚ P R τ τ )) ,P , η, K Z, W, X, Y ∇ ) -para-Hermitian ∇ ( ∇ P L with ˚ X,Y R X , as we have seen earlier Y, , we have (using ( ) W, W, Y, Z, X W, X ( is evaluated explicitly in ( ω are integrable Lagrangian Y ( ( η K η ) for the para-K¨ahlerand ) is 8 P ) D ) = Z,X,Y,W ˚ D X ˜ L ( − X R L Z,W , θ ( and P ) + ) ( ) P − R ˆ Z ˚ R X and , η, K ) ) ∇ ( ) P L ˜ P Z,Y η, K P ) + = ( . ( Y,Z Y,W preserves ( ( X η D X,Y,Z,W P P θ X ) and so the last two lines of ( vanishes (as can be seen from ( ˚ ∇ ˚ τ τ ∇ ( ( ˜ L ˜ η ˚ P R ∇ ∇ ) = 0 . P )) = τ Γ( from Definition − + W, X, Y, Z Y X, W, ( ( D ( ( )) ∈ P )) P Y,Z,X,W η , preserves η J (with D ( , the Nijenhuis tensor vanishes. Furthermore, X η R ,P is closed. In this case ( D − ) P − – 21 – ˚ ∇ θ K D X − ) ) ω W, Y R W, Z X is closed and we also relax the integrability condi- P L ( ) ( ( . Comparing this to the Jacobiator of DFT ( X,Y,Z,W Z Y ) ) ( D D ω P = 6 ) + P ) , θ , θ ) ˚ c X ) R X,Y X,W ( ( that the projected curvature tensor ∇ P P τ τ Z,W 5 ( Z,X Y,X ∇ of the metric ∇ ( ( ˆ ˚ be an almost para-Hermitian manifold and let W, Z, X, Y R ( D D ( -para-Hermitian ˚ ) ∇ W, Z, θ θ ) is a para-K¨ahlermanifold, then η P X,Y,Z,W L ( ( ( ( ( to be integrable. In this case, ( R η η η η are Lagrangian, P L − − − + − . Then the connection R , η, ω ˜ )) = from Definition L η P, η, K P Y D ( ( θ ) = and ,P Let ) L . Therefore, its projected torsion ˚ and ), imply that X canonical para-Hermitian connection no longer vanishes, the generalized Lie derivative associated to the Levi-Civita ( 1 . Because of integrability of P , we find τ ω . As we will show, in order to construct a Lie derivative that satisfies the Jacobi 4.11 , B ˜ L ˚ ∇ P X,Y,Z,W ( R D Z, W, P We now relax the condition that The simplest case is the case where we assume that ( J ˚ R is called the have to find a different connectionDefinition with 9. this property: Civita connection of tions on identity, we only need because connection no longer satisfies the Jacobi identity as in the para-K¨ahlercase. We therefore Therefore, the symmetry ofidentity the ( Riemann tensor, This shows that ifgeneralised ( Lie derivative that satisfies the Jacobi identity for any the Levi-Civita connection in Corollary We already know fromcan Lemma also be seen from an explicit calculation; since vanish identically. We now analyse theJacobiator possibility in to two construct different a cases connection that with vanishing are deeper generalisationsdistributions of and double field we theory. alsopara-K¨ahler assume that using we can seethat extra because terms due to curvature and torsion of the connection Now that wecepts have and properly quantities, introduced we(projected) would and like generalized defined to Lie use the derivatives para-Hermitian them generalisations geometries. to of find The Jacobiator connections variousappendix con- 4.3 Para-K¨ahlerand JHEP11(2017)175 ) 4.24 ) and (4.28) (4.26) (4.27) . . Con- )] canonical 4.21 = 0 Y,Z the K ( c η c (4.25) [ ∇ L . )) . The canonical X ˜ P )]) Z )) = , which is given by ( ) Y Z η = 0 ( ( P c . Since the canonical ˜ c and X t P ))] = . ∇ , ,P 3.10 ∇ X ˚ ) ) Z ∇ ( )) P ( , ˚ ∇ ) X Z of , ( ( ) ,P Y , i.e. Y,X ) c ( P P )]) [ ( Y τ ˜ . We will call c ( Y P K Y ( ( D − Y,K ( P η ˜ ( θ ) ( P = 0 ( ω η ,P and + c X η ) X ( J . This establishes that the connection η )) + , and according to this terminology, the X ˚ ∇ X P ˜ c P ) ( Z . )) + 1 2 c Y t ( )) + , the ordinary torsion of the Levi-Civita Y ∇ P − follows trivially from the definition ( Z ( D [ ˚ ( Z T − P ( P ,P ˜ ) − P K − ) ˜ ˚ ∇ P c , ] ω )) = 0 = ) , Y Y ) ∇ ( − X Y preserves Y c X Y,Z ˜ – 22 – ( K ) ( Y P Y c vanishes, we will use the expression ( = c ( X Y X P D c c can be checked directly ,P ( P ∇ ˚ ∇ ) = D X ˚ ( ∇ J ( ∇ − η P ( with η η ) ˚ ∇ X [ and the canonical projectors -para-Hermitian manifold. Then the generalized Lie η Y K ( ( Y ˜ X c L P ( η + X P D X c X P ) = ( with ηK P, L = ˚ ˚ ∇ T c ([D ( ( ) = = be a := ∇ Y,Z P P P Z ω ) c X given by c X Y = = . ∇ c X ∇ ) = ( K , it is simply related to , satisfies the Jacobi identity: L η Y, , η, K K Y ( ) P η ( X,Y is a canonical para-Hermitian connection with X ( ( c ) P c P ], a class of para-Hermitian connections called “canonical” is given. These ) + τ can also be expressed using its contorsion tensor Let ∇ 50 c , which vanishes 4.24 The canonical connection = Y,Z ( ˚ ∇ ∇ In [ c X Y ∇ c X ( The compatibility of To show that the Jacobiator D η This canonical connection has all the desired properties. First, we show that it is Now we will show that the generalized Lie derivative associated to the canonical con- The compatibility with and the properties of projectors is para-Hermitian. Proof. compatible with the para-Hermitianjustified geometry. in the The next reason section. we call itLemma “canonical” 6. will be sequently, it also preserves connection generalized Lie derivative Proof. show that all thevanish identically. individual We terms now vanish.connection evaluate preserves the As projected was torsion discussed earlier, the last two lines derivative associated to the canonical connection where nection indeed satisfies the Jacobi identity: Theorem 1. connections are parametrized by aconnection real parameter connection the derivative of the almost symplectic form Remark. JHEP11(2017)175 p ). is F . )), q M y ∈ P ] F by its p [ 4.27 (4.29) (4.30) (4.31) ˜ p -para- Φ(˜ and P . Once , , i.e. L ` c ) . p L ˜ y ∇ to rewrite )) = y, ˚ W R ]. . ( F . This means is the equivalence (Φ( )] ˜ 42 L T ,P F ) )) as is simply 7→ ⊕ ∼ , connecting Z ) ( L ˜ F y F ˜ 4.11 P )) , ]. Given a point y, = where ) ]. In these coordinates, 44 W F Y W, Z, X, Y ), the projected curvature P [ such that ( 45 ( ( and the generalised bundle [ ∼ T Y c : the fact that the canonical P -dimensional manifold, P ( P / . ,P d ,P P (using ( R ) 1 maps ( P ) P , T can be written as a cyclic sum Z X ˚ − ∇ X ( ∞ ( ) P ˜ ˚ ∇ P C of is a 2 P R ) = 0 P ( = const ˚ R ˚ . The foliation R P p y vanish as shown above, we have = 0. X,Y, ) = ( c c ) for the canonical connection such that the intersection of any leaf with Y P J . Let us recall that by definition we can )) + ( J ˚ R X,Y,Z,W U P W /L ( 4.21 ( c X , i-e a path within one leaf of X,Y,Z,W c P P ˜ ( P ∇ – 23 – ), we can use the symmetries of R , )) + ˚ [ R ) ) ∈ F 10 ) W Z γ ( ( 4.29 ˜ P X,Y,Z , ,P X,Y,Z ( ) ( X ) X Z Y ) and apply the Bianchi identity, ( ( Cycl Cycl as a set of leaves. If ) on an open set ,P ˜ ) y P be the corresponding foliation of ) = X y, ) is mapped onto the canonical generalized Lie derivative ( the leaf passing through -para-Hermitian manifold such that ( F X,Y,P ( L ), the first two lines involving P we design an isomorphism between 2.2 ( P . Under this isomorphism we show that the Dorfman bracket of X,Y,Z,W if there exist a path ( ˚ ˚ R R P L q 4.21 ⊂ F ˚ R = F = − p is characterised by the condition ˜ ) = X,Y,Z,W ∼ ) be a ) along with the property ( p M c F U ) = J ). Since all the remaining parts of 4.1 T , η, ω , can be written in terms of the curvature is viewed as an infinitesimal diffeomorphism and the quotient refers to the quotient of P c P L is integrable; let R X,Y,Z,W X,Y,Z , ( L c Let ( , where c P Here -dimensional manifold which is bijectively identified with W, Z, X, Y ∇ d F R 10 ( ˚ to cancel all the terms and arrive at the result Rewriting both aboveR terms using ( In the expression ( over ( more using ( of Next, let us evaluate the first two lines in ( the foliation preserving diffeomorphismsmapping are leaves invertible onto leaves. exponentiated action. The leafrelation space given can by also be defined as a quotient represents the partition of a we denote by where the index spacechoose local is coordinates the ( leafthe space open set generalized Lie derivative satisfies the Jacobithat identity the follows Dorfman now from bracket thein satisfies well particular the known the fact Jacobi underlying identity. Lie More algebroid details structures, on will this bethat construction, discussed in [ framework, described in previousHermitian manifold sections, and generalisedT geometry. Given an generalized geometry ( As a corollary, this gives another proof of our main theorem 5 Relationship with generalizedIn geometry this section we give the precise relationship between the para-Hermitian geometrical JHEP11(2017)175 and (5.7) (5.4) (5.5) (5.6) (5.1) (5.2) (5.3) L are given ) = is a mor- F F ρ -dimensional T T d ( α. ρ y ı , we introduce the . That is we define − F . F . Then β T ) denote the integral fo- x T )]) ı ∈ . 5.1 F ( x, y )]) , such that (˜ y )) = it is possible to pull-back the P η ( . ). There is an isomorphism X,Y ) , it gives the usual generalised α T a . P p ) translates to a generalized Lie ) y y, β ). T . ( M y, β )d )] X. . ) + d[ 5.4 + d[ ( ˜ 5.1 , ρ y . We define y, β x ρ ) Y ( α ) (˜ = ( and Y ))) y, : ρ y ˆ 1 η ) ( ) y X a x,α − F x, α ( ( α ρ ( α x,α c ρ ( ) T ˜ − L ( ρ P 1 c ρ L ( ( − L X + β ( − ˆ L η ) 1 x η a , y − ) L ∂ correspond to the duality pairings (˜ ρ and the canonical generalized Lie derivative on ) := ) , + ˆ P ˆ η ] X L ˜ y , given by – 24 – )] = ω x [ ( T x α, ω F ρ between y P L y, -dimensional manifold, we can define its tangent P y, β ı ( T x, y ( ( ρ d T 7→ 1 . In local coordinates the elements of y, β ) can be expressed using Cartan’s formula as a -para-Hermitian manifold, := ; it is precisely the connection that reproduces the + ( p and we also introduce the generalised tangent bundle ) − x c ) and similarly for F → L η β x,α F the isomorphism defined in M Y L ) = ( ( F x and T x x,α ) ) = ([ ı ∗ T ρ X x, α ( T c : X , the Dorfman bracket ( ( ] + ˆ P ( T L L F L ρ ρ [ 1 be a ) = ( = T ( -para-Hermitian structure on ) = ˜ y, β ρ T − ∗ ) defined in the previous section onto ( ) p L )) = x, y ) on ρ ) ρ X and two-form and M ( x, α . For simplicity and given vectors 2.2 x,α = [ η viewed as a via the natural isomorphism ( P 4.27 y, β ( ˜ F F| L , η, ω ( F → L T Y T F F P ⊕ T , ρ ( ρ X bundles : ) T and : L p L d x . If this bundle is restricted to any leaf ρ M = Let x, α is given by ( F P ρ ∗ ρ ) = ( T T and η X . The metric ( ˜ . The relationship between these two Lie derivatives is contained in the following L L P F ⊕ F T Under the isomorphism ) = F := This proposition retrospectively justifies the denomination “canonical” for the con- Now that we have an isomorphism Given a foliation ∗ , i.e. T F P ( This can be explicitely written as nection entering the definitionDorfman bracket of on Proof. derivative on notations phism between the Dorfman bracketT on Here, the Lie bracket, Liefoliation derivative and the differentialproposition: are defined on the Proposition 1. liation of The Dorfman derivative ( The inverse of canonical Lie derivative ( This isomorphism defines an ρ and cotangent bundles T bundle with base by elements of thebetween the form rank ( 2 JHEP11(2017)175 ] - . f L [ )]) x (5.8) (5.9) Y (5.10) (5.11) ( ) = ,P x ) ( )) f X Y ( . ( ) P )] . [ , ,P )) ))) ) ) x, y Z Y,Z X Z (˜ ( X ( ( η ( c X [ P ˜ P P ˜ ) L z P ) ( ) and noticing various ( ( X )), we have η Y ( η η . ( ) . Using in particular that P 5.8 P ) + ]) ), using that d 4.1 ˜ L Z z ˚ ∇ x ˚ ∇ ( ) = )])+ ( )]( x, y P α Z − [ ) as )])) x ( . This finally proves our main and ) (˜ Y,P )] z, Y ( ˆ η X,Y (˜ ( Y 5.7 1 L ,P ) η y ( )) + η ) Z x, y L ( Z P ,P [ (˜ X ) ( ) c P η ( ) + [ X − ]) + X ∇ ( P z ,P ) ( ( [ ,Z P ) , z η P ) . The differential d is the horizontal dif- )) ˚ x, z ∇ X [ ( ([ )]( L ( )] + Y Y ( y ˜ ( ) + y, are isotropic (see ( P (˜ ˜ (˜ ˜ ( ⊂ P ˆ P η x, z ˜ ( η . ) η L Z,P Z,P ] x – 25 – (˜ ) η ( ( L X η L X,Z − ( η η [ Y ) − ( P y = x, y ]) Y and = ( ]), we can rewrite ( P ˚ )]) ∇ − F c P ( L ]) + [ Z y, z − ˜ )]) = )] T ( [ P x, y ( Y )) ([ η x, y x, ( − ∇ ,P [ y, z (˜ α Z ) between forms and vector as (˜ ( η Y z, ,P F → η Y ) [ (˜ ) α ( + )) = T ) + X ,P η x x ( ) ı X P y : Z [ ( c P ( = , ]( Y a is integrable, [ ) P ( ∇ ) = )) we have α [ ( ˜ ,P , x P X L Y η ) ) ( ( is curved. ( L η ˜ Z Y Y,Z P and the corresponding Lie derivative is given by the Cartan formula ) ( η ( ( ,P ρ X ) = ˜ ˜ η X ) P P F ( ( L ( . The differential d can be viewed as the Lie algebroid differential corre- )] = [ ( X x η + η P y ı ( η ) ( Y,Z ˜ P α X ρ X ( [ is the canonical connection from Proposition ( ) = η L x c ( P d + d η x ∇ ı Y,Z ) = Denoting the pairing Our works opens up several new avenues. First, we have not included the presence of ρ X = L x, y x ( (˜ η where proposition. use its metricity, we get Repeating a similarcancellations, procedure we arrive with at the the remarkably remaining simple terms expression in ( Now, using the property that both where we also used the fact that the Levi-Civita connection is torsionless. If we further η We can rewrite this identity in terms of the projections ferential along L sponding to the anchor and that Notice that because para-Hermitian. We haveassociated also with shown the its corresponding equivalenceto foliation. with the This the case opens where generalised up Lie the derivative possibility to extendfluxes DFT in our description of generalised geometry and it is a natural question to wonder how 6 Conclusion In this work weHermitian have geometry, studied we the have generalised relationnection the between and Dorfman DFT derivative showed and using that the GG canonical it in con- the satisfies context the of Jacobi para- identity in the case the geometry is JHEP11(2017)175 (A.1) ) Z X ˚ ∇ , ) Y ( ) K ( Z η X − . ˚ ∇ ) ) )) Y, ,Z Z ,Z ( ) ( ) ω Y Y − X X ) Y,K ˚ ∇ ˚ ∇ ( X ( ˚ K ∇ K Y,Z ( ( ( X η η η ˚ ∇ − − ( – 26 – ) ) + ω )] ,Z ,Z − ) ) ,Z that enters both the string action and the definition ) )] Y Y ( ( Y H ( K K Y,Z K ( X X ( ω η ˚ ˚ ∇ ∇ [ [ ( ( X X η η = = = For the first part of the lemma we use that ) = . Y,Z 1 ( ω X ˚ ∇ -integrability is necessary for the construction of the Dorfman derivative, or whether This research was supported in part by Perimeter Institute for Theoretical Physics. F.J.R. would like to thank the Perimeter Institute where part of this work was carried Finally, so far we have only described the generalisation of the kinematics of DFT; it is L through the Ministry of Research, Innovation and Science. A Properties of para-HermitianProof connections of lemma The work of F.J.R. iscurrently now a supported PhD by student DFGis grant at supported TRR33 Perimeter “The by institute Dark NSERC and Universe”. Discovery University Grants D.S. of 378721. is Waterloo. HisResearch research at Perimeter Institutenovation, is Science supported and by Economic the Government Development of Canada Canada and through by In- the Province of Ontario for additional mathematical guidance andpresented consultations matter. both related and unrelated to the out for its kind hospitality anda support. PhD While student a at resident at Queen Perimeter Mary Institute, University F.J.R. was of London supported by an STFC studentship. F.J.R. would like to thankZagreb the (June participants 2017), of the especiallyPark, workshops Chris David in Blair Banff Berman, and (January Chrisvated Charles 2017) us Hull, Strickland-Constable, and to for present Dan this many Waldram,many work sharp conversations, in Jeong-Hyuck questions discussions a that and better moti- clarifications. form, and D.S. additionally would Emanuel like Malek to for thank a Dylan great Butson L.F. and D.S.continuous would mathematical like support to and endlessL.F. greatly discussions would thank around like the Ruxandra to themelaborations Moraru thank of and collaborators and our many Djordje paper. discussions Shengda on Minic Hu related and subjects for Rob which the Leigh have for been truly an inspirational. ongoing col- of utmost importance to understandsetting the and interplay between the the dynamical para-Hermitianof geometrical metric Born geometry. We hope to come back toAcknowledgments these issues in the near future. the it can be relaxed modulowonder extra whether conditions the on geometrical thethe extension admissible exceptionally we vectors. extended have Also, generalised presentedand it geometries Exceptional is here Field associated natural can Theory. to with be flux generalised compactifications to these would enter in the para-Hermitian context. A related question is to explore whether JHEP11(2017)175 ) = ) = this (A.5) (A.6) (A.7) (A.8) (A.2) (A.3) (A.4) ω ,Y  ) = 0 we ) which is ) we get X,Y ( X η ( Z ω ,Z X ] A.1 − K ˚ ∇ ( )) η , Z = ) ( ) = X,Y Z ([ η Z ( X ω . X ) Y,K ). . K ∇ ∇ − Z . , X )) ) ) ) 3.7 X )) ˚ ∇ Z Y ( Z ,Z and ∇ Y ( ] η X ( X Y K Y, ∇ . ∇ ( ( ] ) + η η Y, X,Y Y, ( , ) = ( )). Subtracting a similar expres- ([ − − ,Z  ω ω ) ) Y X,Y ) ω Y X,Z,K [ gives the result of the lemma since ( )) ( Y ). Then, by expanding ( − − Z ,Z K − Ω( ) + Z X ( ) ) Y,Z )] 3.7 ˚ − ∇ X X X,K ( ( Y Y,Z )) Y,Z ∇ Y,K η K X,Y Y,Z ( X ) and imposing that ( ( Z X ( X η − ( η ˆ ) = 0 we obtain ∇ T ω − ∇ ∇ [ ∇ ( ( ( − ( , ω ω Y,Z Y X ω η and the compatibility condition ) = ) + ) ( )] – 27 – ( Y,Z ) X η ( η − ,Z ,Z ,Y ) + ) + X ω ∇ ) ) ) X,Y,K )] )] + Y,Z X ˚ ∇ ( X Y Y X X,Y,Z and similarly for η ). This proves ( ( ∇ ( ) = Y,Z Y,Z [ )( )( the torsion vanishes and the lemma follows. ( ( Y,Z Y,Z Z K Y ( ( ( X K K ω ω ( ) = 0. Given a para-Hermitian connection we evaluate ˜ ˚ ) = Ω( ω ω ∇ η P X X ) = X X [ [ X,Y preserves ( ˚ ˚ ∇ ∇ ∇ ∇ X X = ) = ˆ   ) for . It also gives an alternative proof of ( T ˚ Y,Z (( (( ∇ ) = ) ) ˜ ( Z = Y η η P ( ω X,Z,Y ) = Y,Z − ˜ X,Y,Z = P ( X Given a para-Hermitian connection we evaluate ( X X and using the definition of the covariant derivative acting on X,Y η X,Y,Z X,Y,Z ) = = ( ˚ ω ( ( ∇ X Y,Z ) and d ω ∇ ˜ ( P ) or = 2 ∇ ) + Ω( Y ) and using that ω Y,Z ( ). Y K ) = ( ( X P ω P 3.8 ∇ Y,Z X = ( ω )). The last expression clearly vanishes if ˚ ∇ X,Y,Z while ), the fact that Y X X,Y,Z Y ( ( P ˚ ∇ ,Y ω ) d X = + By definition the exterior derivative of a two-form is given by ( X,K ( K η ( sion for Now inserting KP where we have used Subtracting a similar expressionget for that Ω( For the Levi-Civita connection Proof of Lemma where we have used the definition of the torsion tensor Given a connection can be written as which proves ( η − the case if where we have used the definition of the covariant derivative, the definition of JHEP11(2017)175 is ) = P (B.5) (B.3) (B.4) = 0 it ) ˜ Y,X P X ( P D W ) θ . D X is any metric ) ) )) Y Y W Y ∇ ) = 0. Similarly D Y ) D W W D X X − ] ( D ). Now subtracting D Y, ˜ and since ) (B.2) Y Y P Z W, X W Z ( X, . ˜ X L W D D ) X Z (D ) Z , η − D η ) D ) and X Z, D Z ] (D X ( Y Y − X η Y ] η Z, D ) Y, X ) Z,W D [D X ( W Y ) (B.1) . Recall that Z η Y D ) X W Z, D ) + ) + Z, ( D D X (D − X ) = D X ( W η ( D L − η X has image η Y,X W W, L ) + P X Z D L ( ( ( Y W ( X η P η )+ ∇ D Y,W , ) + Z, and obeys a normalisation condition − D P [D Z, θ [D Z D Y − − )) Y X,Z D η Z := Y, D + D L ) ] 1 Y )) + ( X Z − Z,W (D X D Y Y X − η X,W X ] then = D η X Y L (D Z Y (D Y X (D , ] − η D ˜ X D η P P ] η is a projector but the calculation presented D 8 D W W, D X X . X )] ) + Y ( − Y D is an arbitrary anchor map. The image of − L W W, − Z − X ) Im P Y − ( ) – 28 – D , η ([ D (D ) D ) P T η η W Y ∈ − W − , η − Z,W Y ) X Z ) = X D Y,Z W D Y X Y Y D Y [D X Z,W X X Z,W ] L Z, exchanged yields ) := Z )] = ( [D D D ( [D Y Y Z, P → ˜ X D (D P η := D D ( D Y D D T η Z, Z, η X )[ , ( (D ( ( X Y : Z,W Z, W, η η η η X X (D ( ( ( X D ( Y ) = (D and η η η η P + − − L P η ([D (D − − − − X η η X,Y,Z,W = Y,Z ( )[ ) = ) = 0 and . Therefore if D X D D ) = X P X J L ( ( ( J η P P Z,W we have the projected Dorfman derivative (recalling that D Y Z,W 7 ] L = Ker D Y D X then . The complementary projector ˜ L L L , L ( P D X η )). Under contraction with another vector this reads L )) ([ Ker X η 4.17 ∈ D Y, X ( where we also used the same expression with We now want tovarious expand contributions. this We start expression by to using the check compatibility it property is to indeed consider the tensorial term and analyse the (cf. ( We also recall the Jacobiator from Definition From Definition η which satisfies the Leibniz rule, is compatible with is clear that if B.1 Computing In this appendix wealised fill Lie in derivative some for of acompatible the projected connection details derivative and for D computinghere the generalises Jacobiator easily to of thedenoted the case by gener- where B The Jacobiator JHEP11(2017)175 ). X Z (B.8) (B.9) (B.6) (B.7) W ) ) ) D Y ) X D ) . ) ) ) X,Y ) cancel out. Y, ] ( Z,X X ) W,Z ] ( Y η ( W X,W )) P ] P W τ W D − τ )) Z Y ) ) Z , ) D ∇ , ) ∇ D Z D D X X ) , ) X , − ) Z, X Z )]) Y, Z ] Z W, W Z Y Y ] W, Y, Z, X W, X ( W, ([D ( ] X , the projected torsion D Y ( ( Y ( D D W η η X η ( X P )) P D Z,X,Y,W W Z W ([D Y (D [D W Z , η ( Z,W R − D R η X ) D η D D , θ D ,P ) — we find P )] D D ) + 6 ) + ) − ) ) ) − D − X Y − R Z Y, Y, ( Y W Z X Y, Y Y, ,U ) X ) ) + ( ( D ) ( ( ) Y, X ( Y X ,P η η D X ( η η ) Z,Y X,Z Z ) + P [D [D ] [D Z, L ( [ Y,Z Y,W X − , η ( ( ( − ( D D )] (B.10) D D D ) W Y,W ) W ) + η )+ P P ) − P θ ] ( [ τ τ Y ( Z D ] D Z Z, X X η ( W, W, X , η D W, ( ∇ ∇ ( D ( Z ( Y X D Y η η Z, W η , . η − ,P − Y ( ( D W, X, Y, Z . D X,W ) X, W, ( W, D η η X )) + − ( Y )) D ( ( )) Z ([D W )) from Definition P Y, ) + ] X Y,Z,X,W η η ) η ) + η D Y, X . ( ( X D − ( R Y Y ) ([D W D Z ) + Z ] X − ) = P − η D X P D − ] θ η D ] Y Y − Z, ) ) X W, Y , R ) W, Z (D Y ( (D (D W ( ) ( D Y, − D X – 29 – Z Y η W W, Z η X η η Z ( ) + ) ) D ( D ) J D Y,W ) = [ D D Z,W ) + − X, − − , η Y,Z ([D − , θ , θ , η D X − U Y ] ([D ( ) ) ) ) ) + ) ) X,Y Z Z W X,W L η P η )) = ( X Z ( W Y Z,W Y Y (D X Z,W Z P X ] W P L ] X τ D η D τ X [D ( Z D X D [D [D , Y Y,X Z,X D ∇ Y,W Y D D ) := ) := ) := D P ∇ ( ( D D X W, Z, X, Y D − Z, W, D X ( D D , ( ( Y, W Y, − W, Z, θ θ ( Z, W, Y, η η P D Z ) is in general a differential operator just as the generalized X X,Y,Z,W ( ( ( ( D ( ( ( ( ( ([D η Y ( Z W Y η R η η η X,Y X,Y η η η η η η 8 X ( ( P W D X Z, ([D − − − − + D P (D ( (D L [D − − + − − + R D η τ θ η η η (D (D Z, X,Y,Z,W − + + η η ( ( ) = ) = η P = ˆ R and so it is not tensorial. If we wish to demand that the Jacobiator ) = L D Z,W J Y X,Y,Z,W X,Y,Z,W ( D X ( D D L D J L J ( η and the twist of the projected derivative P The Jacobiator (Definition Lie derivative vanishes, the tensoriality is ofthe course anchoring a property necessary condition. In the following, we show that B.2 Anchoring and tensorality of to express the Jacobiator as τ In particular we see thatWe all can the now non-tensorial use terms the of definition the of form the projected Riemann tensor where we have used Putting everything together andJacobiator canceling terms, we arrive at the full expression for the On the other hand — noting that JHEP11(2017)175 ) L = 0 Y,Z ( X (B.14) (B.11) (B.12) (B.13) (B.15) (B.16) (B.17) η f D = L f ). Then the to hold. . 7 ]) ) as f f. ). Then the Jaco- = 0 )[ M ( )D 7 D X ⇔ ∞ ( J holds. C P ])) Y,Z . f. ) ( f ∈ ⇔ η (Definition )[ ) ) )D f = 0 = D( f . Now, let us show that the X X 11 B.10 ( Y η ( D ( Z f ), we get ) X P P X X,Y D , ( L ) ) L ] X + D( η f X,Y f ( B.11 L f Y, η + L )[ Y, ) and the definition of the Courant ( )D ( D D η Z X,Y ⇒ η Y X L ( ( X ] ] η ]) ) and using the normalisation condition 1 2 f L P f B.11 Y,Z , ) + ) f ] + ] ( )[ Y )[ = 0, which consequently implies + f − η )[ f f ) X Y = D + D )[ )[ ( ( Z Z X to avoid unnecessary cluttering of the expressions. + Y,Z ( D Y Y ( Y P P X ]] X Y, X,Y L Y ( η P – 30 – ( X X L ] D Y η X ] + − Y f L L L + D ]]. This finishes the proof. f L X,Y ( ( + [[ )[ ( Y Y f L )[ + Y D X η P P L Z X X )[ ] ( X L ) = D( f L L Y ( X = P f simply as )[ f f ( ] = D X ) = P ]] = Z f D Z f D P − L ] f = ( + . Using this and again ( L )[ D Y )[ )][ ) = Z P X, ) is satisfied. Z Y )] along with the symmetry of Z Y L ) ( Y Y , because we can always write any Y Y, ( , ( ( Y η is a necessary condition for the equation ( f fY L X fX P X ( η L P ( ) f ) B.10 L and ,P ( L X )[ X P X,X ) [ = D ( X X L L X Y ( η X − ( L f B.10 ( ])) = X )[ P = 0. Rewriting this using ( be a projected generalized Lie derivative ( P f f P be a projected generalized Lie derivative (Definition = = Y [ D )[ ( D L L L Z J P X ( 1 2 + fY Let P Let f L is tensorial if and only if the anchoring property X D D( ) = and any function L X D X Y, Assume First, we note the “Leibniz-like” properties of L ( J X D X η L We will now show that the anchoring property is equivalent to the Jacobiator being vanishes only if the ( In the following, we will denote Y, 11 D ( since Finally, the statement ofand the using lemma the follows above from derived rewriting properties: the metric compatibility of bracket, we get The symmetric part of this equationfor is all for some vector fields J Lemma 7. anchoring property Proof. First, we note the following property which can be shownη by expanding is another necessary condition and in fact is equivalent to the tensoriality of the Jacobiator. Proof. Using this, we expand tensorial: Lemma 8. biator JHEP11(2017)175 W is an ] (B.22) (B.18) (B.19) (B.20) (B.21) (B.24) f ] P f ))[ )[ Z the expres- X . X ( ) = Im ˜ L L . P ( ) ) ]) L ) vanishes. This , we get P f ). Moreover, the W L )[ − ( . )]) (B.23) X P )] Y X,Y ( ( ( B.10 Z − P W, Y, Z, X ( P ] = 0 ( ) ) holds. The tensoriality τ ,P f ) . P f Z,X,Y,W ,P ( )D( ) R D X ))[ ])) P ( B.10 X f )]) f − ) and contracting the second R W ( P f reveals that it is equivalent to W, [ )[ Y,Z , . ) Y X ( P ( X )D . )D X L η − L ([ ) + L ( Z ( B.10 ) ( ,P + )]) P X − ) η P -entry is ensured if and only if ) = 0 ]] = 0 Y,Z Y L Y Z f Y X Z ( − − D( ] X ( ) Y,X )[ X ) f Y, ( )] L Y P − ( ,P f L [ )[ ( W, X, Y, Z X D X ) η ( ( ( η f θ D W L . X − ( P P X P + Y,Z,X,W ( D D ] L + + ( ( R W, X P − J ) in the ,P D[ Y P P X ( ) − Y ] [ ) X L − η )] ] – 31 – Y R f Z − X ) Y ) )( X ( − Y Z D )[ -entry if and only if ( ( L f D ( ) X ] L P Z ( Y ) + P , − f D Y − X Y,Z ,P − P )[ X X ( ) X Y L Y L η Z L Z ( = ([ L P X X ( X,Y,Z,W ( X ([ Y ( ( P + ), which permits any use, distribution and reproduction in P f ), and using the metric compatibility of P L D ])) = P ] ([D (D ( W, Z, X, Y [ − f J ( − = f P P P )[ P Z )[ X,Y,Z,W = Ker Z B.18 Z ( = = R X ) X Y X ˜ ) P L ( ) = ( L X − R P Y L P fY ( L L CC-BY 4.0 D( ) and using metric compatibility of X + is tensorial in the ) and ( f f · X,Y ) = L ( − . Second, it implies that the projected torsion This article is distributed under the terms of the Creative Commons D L = = P W, entries is shown by analogous calculation while the tensoriality in the L J τ f ( B.17 − Z Z η D ) ] ⊂ Z X X is valued in fY L )] fY L ( D L Y X θ and fY , ( ) simplifies to X,Y,Z,W L W, ( L X ( L X D ,P η L B.9 ) J ([ We have shown above that the tensoriality is equivalent to ( X ( P in the entry follows directly from the definition of This shows that The first equation isequation with exactly the anchoringthe anchoring property property ( as well Therefore, the tensoriality of By combining ( Similarly, we find out Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. 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