JHEP05(2019)209 Springer May 19, 2019 May 31, 2019 : : February 28, 2019 : Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP05(2019)209 b,c,d [email protected] , . 3 1812.00026 and Christian S¨amann The Authors. a c Differential and Algebraic Geometry, Flux compactifications, String Duality

We describe the global geometry, symmetries and tensors for Double Field , [email protected] E-mail: Department of Mathematics, Heriot-WattColin University, Maclaurin Building, Riccarton, EdinburghMaxwell EH14 Institute 4AS, for U.K. MathematicalEdinburgh, Sciences, U.K. Higgs Centre for Theoretical Physics, Edinburgh, U.K. Faculty of Mathematics andSokolovsk´a83, Physics, 186 Charles 75 University, Praha 8, Czech Republic b c d a Open Access Article funded by SCOAP ArXiv ePrint: solution of this condition,fluxes. we We also recover discuss theand the explain Courant finite, how algebroids global this symmetries of specializes of both to general the nilmanifolds local case Double with ofKeywords: Field T-duality Theory between nilmanifolds. straightforward application of astructs formalism the we analogue developed of previously. ausing This Courant the algebroid formalism language over con- the ofnilmanifolds correspondence in graded space terms manifolds, of of derived ation periodicity T-duality, brackets condition conditions and arises rather we purely than use local algebraically, and patches. the we description The show strong of that sec- for a particularly symmetric Abstract: Theory over pairs of nilmanifolds with fluxes or gerbes. This is achieved by a rather Andreas Deser Extended Riemannian geometry III:Field global Theory Double with nilmanifolds JHEP05(2019)209 25 25 31 12 15 16 19 6 23 9 27 21 14 8 ] and references therein. Currently, 3 26 – 1 7 ] for alternative approaches to ours. 9 – 1 – 10 6 ] or also [ 8 k,j j,k – 18 C 6 j,k C 18 K G ], [ E 5 and , 1 4 18 j,k -manifold on the correspondence space C K 29 ) T-duality symmetry of string theory and governs the dynamics E Q -structures and the section condition Z ; ∞ L d, d ( O 31 -algebra structures: solutions to the section condition ∞ L and it is suggested that some form of gluing has to be performed to recover the 1 With minor exceptions; see e.g. [ 5.2 Finite symmetries 5.3 Specialization to 4.4 4.5 T-duality between 4.6 Differential geometry on 5.1 Generalized Lie derivative and infinitesimal symmetries 4.1 Topological T-duality 4.2 Symplectic pre-N 4.3 Symmetries, 2.6 The Courant algebroid of 3.1 Lie 2-algebra3.2 associated to Action of3.3 the associated Lie Finite algebra symmetries and of its a action gerbe Lie algebra 2.1 Three-dimensional nilmanifolds 2.2 Gerbes2.3 on three-dimensional nilmanifolds Exact Courant2.4 algebroids and generalized Exact metrics Courant2.5 algebroids as symplectic The Lie Courant 2-algebroids algebroid of a gerbe 1 under the hidden of the massless sectorthe of available Lagrangian superstring and theory, itsspace see gauge [ structure are given only on a local patch of target 1 Introduction and results The goal of Double Field Theory is to provide a Lagrangian which is manifestly symmetric 6 Discussion 5 Finite symmetries of Double Field Theory 4 Global Double Field Theory 3 Finite global symmetries of Generalized Geometry Contents 1 Introduction and results 2 Geometry of gerbes on three-dimensional nilmanifolds JHEP05(2019)209 . 2 ]. of B 15 , = d ]. The 14 12 H actions generalized , cf. e.g. [ and we equip them 2 T ] should be straightforward. 11 carrying an abelian gerbe M and curvature (i.e. flux) ]. The infinitesimal symmetry struc- B 13 of symmetries of a gerbe is then the as- Generalized Geometry 3 ]. Recall that in the case of Generalized -algebra, which is essentially the same. 18 ∞ – 2 – L as pioneered in [ ] and extend the local picture developed there to a global ]. 10 17 , 16 -manifolds Q , which has some non-trivial gluing properties if the gerbe is topologically One approach to a global picture of Double Field Theory is to use the finite symmetry The aim is now to lift this rather well-understood global picture from Generalized The perspective of the generalized tangent bundle can be further improved by con- In physics terms, we are studying the NS-NS sector of supergravity without dilaton, In this paper, we study this problem in the case of the prominent example of 3- A generalization to heterotic NS-NSBy gravity this along term, the we lines shall discussed always in mean [ a 2-term 2 3 -field together with the gauge-for-gauge transformations of the latter. At the infinitesimal the infinitesimal symmetries by theto Dorfman the bracket semidirect form product a ofthe Lie case diffeomorphisms algebra, of and which Double closed integrates Field 2-forms Theory, on the the infinitesimal base symmetries are manifold. governed In by an analogue freedom but globally, we have topological features complicating the picture. actions to glueGeometry, together the local infinitesimal descriptions symmetriessymmetries are [ themselves governed form by a the Lie Dorfman 2-algebra bracket. which While is the awkward to integrate, the bracket construction of [ Geometry to Double Fieldfreedom to Theory. the The compactas circle latter an fibers associates exchange of of additional the the winding nilmanifold. coordinates degrees Fourier-dual Locally, to of T-duality the is momentum understood and winding degrees of From a more mathematicalor perspective, principal abelian 2-bundles gerbes for arecorresponding categorified short. class principal of For bundles Courant eachsymplectic algebroids Lie topological (and 2-algebroids). class Courant of Thesociated algebroids Lie abelian Lie are 2-algebra 2-algebra gerbes, best of there regarded the is corresponding as a Courant algebroid, obtained from the derived non-trivial. sidering the correspondinggraded Courant manifolds algebroid or inture the of language NS-NS of gravity symplectic is differential then simply obtained in terms of derived brackets, cf. [ with connective structure (i.e.The Kalb-Ramond symmetries of field) this theoryB consist of diffeomorphisms and gaugelevel, transformations of these the are elegantlyvector described field and by the Hitchin’s 1-formtangent parameterizing bundle these symmetries are sections of the terms of twisted periodicity conditions onwe the presented local previously data. in Weone will [ use by the demanding general compatibility formalism with the periodicity conditions. consisting of a vielbein (or metric) on a topological manifold a clean understanding of this gluing procedure is important. dimensional nilmanifolds. These arewith a circle three bundles form over flux,advantage the which of mathematically torus this corresponds to class the of curvature examples of is a gerbe. that the The non-trivial patchings can be understood in global picture. Since T-duality is fundamentally linked to compact target space directions, JHEP05(2019)209 ]. of 22 -flux ]. Q H 18 -actions 3 Z ] for further 20 -manifold picture of Q . This modification has ], see also [ to the Hamiltonian −} 19 , H , the Chern class of the bundle. {Q Z = ∈ j Q ]. . On these, we recover the generalized 15 are principal circle bundles over the 2- j N -manifolds underlying the structure of Double ]. In this paper, we show that this sketch can – 3 – Q -manifolds 15 Q is obtained by adding H it is not surprising that the above mentioned integrated ]. The key observation was that the 4 ], we presented a formalism in which the necessary modifica- 23 15 symplectic pre-N ] for a slightly different perspective. These finite transformations were 21 globally) to a non-trivial one therefore requires either to modify the Dorfman B , and thus they are characterized by an integer 2 = d . Therefore, we do not have to use cumbersome local patches, but we can work with T 3 H Recall that three dimensional nilmanifolds The above was already claimed in [ The modification of the Dorfman bracket of Generalized Geometry for topologically In our previous paper [ There is, however, a problem with this construction, which is already seen in the con- = 0, leading us to We are not aware of any discussion of this point before [ R 4 2 realm of smooth manifoldswhich and we generalize plan to to noncommutative do and in future nonassociative work. spaces, torus is still simple enough toparticular, we allow can for use the a fact transparenton that and nilmanifolds are detailed the discussion orbit spaces inflat of coordinates. particular space and In a numberof of the periodicity various conditions bundles. to describe Note functions also and that global for sections more general examples, one needs to leave the a clear analogue forField the Theory. symplectic pre-N indeed be completed to abetween 3-dimensional full nilmanifolds coherent carrying picture. topologically non-trivial We gerbes. study the This example simple example of T-duality tion (or rather athe slight derived weakening brackets form thereof) a is Lie recovered 2-algebra algebraically of symmetries, bynon-trivial as demanding on gerbes that the with Courantthe curvature algebroid. differential given by the homological vector field discussed in detail, seethe also Courant [ algebroid overtarget space-time space can of be Double extendedQ Field to Theory. an We analogous showedLie picture that derivative over this of the requires Double to Field drop Theory the as condition a derived bracket, and the strong section condi- this modification into account, infinitesimal symmetries, as well aslogically the trivial gluing gerbes. transformations, The are same only conclusion suitable was for reached topo- bytions explicit of computations in the [ D-bracket due to topologically non-trivial gerbes become natural and can be text of Generalized Geometry.with its As own stated class(i.e. above, of each Courant topological algebroids.bracket class or of Switching to gerbes from modifybundle. a comes the Both topologically changes gluing trivial are prescriptions relatedalized by Lie for a derivative the in coordinate Double corresponding change, Field as generalized Theory we is shall tangent almost see. exclusively Since discussed without the taking gener- the symmetries form aLie Lie algebra 2-algebra, which and can the bediscussions actions integrated and by to [ finite the transformations D-bracketthen [ form used as an gluing ordinary transformations for local descriptions of Double Field Theory [ of the Dorfman bracket, often called the generalized Lie derivative or D-bracket. Again, JHEP05(2019)209 , , k to j,k C (1.2) (1.1) k,j ] = C Locally H ]), which ]. 13 and 25 ]. The resulting 24 j,k 6 , -algebra of Double C ) ∞ j L is the fiber product k ] = O  k,j k,j N -algebra. G H C [ ∞ , k L . Tensoring the gerbes, we and N j K o } N . The space relevant for Double k = ( K K N identical to the and we show that the ingredients C ] further and derive the conditions -manifolds as given in [ ' ( C k,j Q . Altogether, we have the following 15 K G E not K k C between π Both gerbes and both associated Courant 2 k endowed with a gerbe of Dixmier-Douady of pr 5 N K k 2 K N in terms of coordinates. Explicitly, we lift w T E come with associated Courant algebroids o w × – 4 – K 2  K j E k,j T E T-duality ' . Under T-duality along the fiber direction of the G N ' plays the same role in Double Field Theory as the Z ←−−−−−−−−→ := K ∼ = 1 and ) E K pr ) k j π ] which arises from the symmetry Lie 2-algebra via the classical BV j,k Z , G 26 j ] = v N x H ( [ -algebra gives rise to, but is j 3 , K are also given by an integer, their Dixmier-Douady class [ ]. ∞ j N G ! -manifold H j / L 27 N Q N , we can follow our formalism of [ with associated Courant algebroid = ( K K E G j,k . The gerbes O  j,k j,k G k G C N 2 T (by which we always mean the corresponding × j : k,j N j C Having found A classical way to understand T-duality is to use correspondence spaces, which are A picture of T-dualityNote as an that isomorphism this of invariant 2-term Courant algebroids was given in [ = 5 6 for the antisymmetrized derived brackets toalgebra form of a symmetries Lie will 2-algebra always appear of in symmetries. the form This of Lie a 2- 2-term Field Theory as partiallyformalism, given as in explained [ in [ the periodicity conditions ofperiodicity the conditions coordinates of the onto various the construct coordinates Courant the on algebroids C-brackettherefore via they a are derived indeed bracket global. are invariant under this periodicity and The symplectic pre-N Courant algebroid for Generalizedwhose structures Geometry. and actions Its areis sections governed therefore by parameterize derived the the brackets. detailed Our symmetries first description important of result diagram: and encode the relevant Lie 2-algebra ofalgebroids symmetries. can be pulledobtain back the gerbe to theField correspondence Theory space is now a graded submanifold fibered products of originalwe associate and the T-dual original- and manifold dual fiberIn over coordinates a the to case momentum common and at base winding coordinates.K hand, manifold. the correspondence space which is an elementprincipal in bundle, one obtains theclass nilmanifold which follows from the Buscher rules and which was explained in detail in [ Similarly, gerbes over JHEP05(2019)209 , = j,k K but E and C F (1.3) (1.4) con- 2 Z K . Both Q 5 2 E ∈ ˆ pr → , k j,k with a brief C 6 : , we have 1 which encodes the Q e K along the lifts of the E . −} , k,j -manifold of even degree). 4 C p Q { → -manifold = K Q 4 E ∂ , we arrive at two integrable distri- : . We close in section ∂x 4 to the Courant algebroids of the two and the second one defines , we review nilmanifolds, gerbes over 2 V 4 , 1 ˆ 2 K 4 pr ˆ , we present the Double Field Theory lift E p pr and 3 and 4 p 3 −} as well as the symplectic form are pullbacks V = -manifolds. That is, they are smooth maps of , 3 Q Q – 5 – and p Q} { , and corresponding Hamiltonian = (actually on any pre-N Q {Q . In section 3 K j,k ∂ 3 ∂x E . C ] and the overlapping results agree. 2 reduce nicely to those of the two Courant algebroids e 28 → . The first defines on i K 3 E K F and E ,V : 1 3 e 1 ∂ to the Courant algebroids, ∂x now factorizes as ˆ pr h 2 pr Q} , along and are the momentum coordinates in the T-duality directions. They form i ], we had introduced the concepts of extended tensors, extended covariant K {Q 4 and 4 E ] before we construct the symplectic pre-N for functions p given by morphisms of pre-N 15 1 ,V } 24 4 pr K ∂ E ∂x and ,F for the corresponding solutions of the strong section condition. Section h → 3 , p Q} k,j , k,j C This paper is structured as follows. In section We note that some aspects of the specializationAltogether, of we our conclude example that in there which is no geometric obstacle to a global description Solutions to this section conditions are found in a very natural way. Given a Poisson C : = 0 were discussed earlier in [ {{Q 2 1 2 and the extended tensorsand of tains a discussion of thebefore infinitesimal and we finite specialize symmetries to ofdiscussion general the of Double case Field generalizations discussed Theory of in our section constructions. as found in [ geometry and symmetries of Doubleas Field well Theory. as We the discuss the mostnilmanifolds strong relevant with section solutions, gerbes. condition restricting Wegeometry. also In make [ some detailedderivatives, comments extended on torsion extended and differential Riemann tensors. All this is readily specialized to these and the correspondingfinitesimal Courant and finite algebroids. symmetries) of these Theterms Courant symmetry algebroids of are derived structures then brackets explained (both in in section of for detail this in in- picture. We start with a very brief review of topological T-duality for nilmanifolds, j of Double Field Theory, atsmooth least manifolds carrying for gerbes. those cases in which the manifest T-duality connects butions, projections admit particularly simplee sections, which aregraded embeddings manifolds and theof Hamiltonian the of data on projections Complementing these by global vector fields The expression where Hamiltonians of the vector fields condition for Double Fieldgerbes manifest. Theory making the T-duality between nilmanifoldsstructure carrying with compatible vector field conditions are the appropriate global form of (a slight weakening of) the strong section JHEP05(2019)209 . 3 T (2.4) (2.1) (2.2) , and = (2.3a) (2.3b) 2 1 , which 2 jx S x − × d 1 2 . = T ) jx 2 h = is connected and = jx ) used mostly in the Recall that up to is a nilpotent Lie 0 i − A N 7 , ζ N i 3 ξ of the principal circle , x 3 F 2 2 ∂ 2 . x ∂x , x T d 2 R 1 ) = 2 ), encoding a principal circle + 1 Γ, where jx = 3 x ¯ ζ 1 d j . The resulting curvature reads + 1 x N/ h, A 3-matrices. The cocompact dis- h ( 3 , d . We shall denote the nilmanifold jx x = ,... × ∼ 3 nilpotent Lie group − ] N + ∂ g , and we shall describe it in terms of 3 , = d ∂x M ∈ 2 1 x 1 3 T ) = g + 1) j ¯ ξ 1 8 3 jx is the coordinate along the fiber. Clearly, we x = [ ( comes with the natural metric + (d 3 − , x , 2 A 2 x j 2 2 2 g ) – 6 – x N 2 ∂ , x − 1 ∂x x , x ] = d ( can be described using a single transition function . Recall that a = g , while 2 + 1. In its additive form, it reads as + 1) , 2 2 + (d N ∼ ¯ 2 ξ ¯ g ζ 1 1 T 2 ) T x ) x 3 ( 1 , = [ . , → x is the first Chern number , x k 1 A → 1 1 g x j g j 1 ∂ N + 1 ∂x x (3) of upper-triangular 3 = (d . One compatible choice is the connection 2 : = d H is a circle bundle over g = A 1 j , x ¯ 1 ξ j π 1 ¯ ζ = 0, for which the circle bundle becomes trivial: . with the following periodicity conditions imposed: x N i j ( ˇ j Cech cocycle to a Deligne cocycle ( are classified by an integer 3 δ and thus is a compact quotient manifold j ∼ R such that the sequence 2 = ) x g i are coordinates on 3 d M on j . 2 ¯ ζ j i , x ∧ x , 2 i x N 1 ¯ ξ x . h , x j 1 d by and j x N 1 j ( x Γ = / Finally, we note that the nilmanifold The circle bundle We have a basis of globally defined 1-forms The nilmanifold We shall be interested in examples of 3-dimensional nilmanifolds. These examples play a prominent role in Thurston’s geometrization conjecture:The every bar closed is 3-manifold merely distinguishing these tensors from a slightly different set ( F 7 8 nilmanifold specifying the transition (3) can be decomposed into pieces with one oflater eight discussion. geometric structures, one being that of a nilmanifold. bundle with connection satisfies the cocycle condition as bundle satisfying h we can extend this together with dual vector fields Here, can include the case is the Heisenbergcrete group subgroups Γ H coordinates becomes trivial at some term isomorphisms, there is a unique, connected, simply connected nilpotent Lie group, which 2.1 Three-dimensional nilmanifolds A group and Γ a discretehas subgroup Lie in algebra 2 Geometry of gerbes on three-dimensional nilmanifolds JHEP05(2019)209 , b U (2.7) (2.5) (2.6) ] will , is a ∩ . There . . a 29 encodes [ M on triple U ab and we can is comple- ) G (1)-bundle U abcd g X U U := abc ( on h on on the patches ab ) ) X U a U ( ab are given with respect ( , the analogue of a B of a gerbe A , M = 0 on a H ) U . Let = d ˇ ) bcd Cech cocycle encoding the of a principal b ( M ( ˇ h Cech cocycle B F Hitchin-Chatterjee gerbe −  ) − ) a and 2-forms is connected, compact, orientable a acd . The Dixmier-Douady classes of ( U , or Hitchin-Chatterjee gerbes ( j j a ab B h , the cocycles on t N N U + M , ) connective structure ) -gerbe a ( abd , this cover induces a cover ( (1) B h M and U − → ) = d – 7 – X, a ], so the curvature 3-from , or abc U ( | F h abc H ]. U of some manifold gerbe H are described by cocycles with respect to two covers of ) = [ M on P M ( → ) are then completed by ) 1 ) = [ c X, with together with a cover 11 G on the double overlaps abc 2.5 9 ( ( (1)-gerbes and the relation between the various definitions is ) h M U dd d ab ( − R A (1)-bundles, gerbes can be tensored and restricted to submanifolds. by pullback. which is part of the connective structure of a gerbe. For our purposes, U = , etc. Then a → c M ) B c U bc ( U ]. , that is a collection of functions ∩ A ∩ h . Together with the embedding 30 b b + U M , 1-forms U ) U ∩ ∩ ac abc A restriction to some submanifold is also readily performed by restricting the ( a a U A U U : 10 − ) ) := Let us now come to gerbes on the nilmanifolds Just as principal Geometrically, gerbes should be regarded as central groupoid extensions. That is, they Consider a manifold ab I.e. up to torsionMore elements. explicitly, the two gerbes on Explicitly, given a submanifold abc ( . The cocycle conditions ( 9 ( 10 11 abc a A h ˇ is a common refinement of these two covers, overto which some we cover can addrestrict the the corresponding cocycles cocycles. on gerbes. cocycles to that submanifold. these are encoded in globaland closed 3-dimensional, 3-forms, these and forms since are unique up to rescaling. The cocycle conditions impose found, e.g., in [ Geometrically, this amounts to tensoring andcocycle restricting perspective, their tensoring categorified amounts fibers. towhose adding From Dixmier-Douady the the class underlying cocycles, is yielding the a sum gerbe of the Dixmier-Douady classes of the individual form indeed a globalcaptures 3-form. the first Chern Just class asits the Dixmier-Douady class curvature 2-form are particular categories in whichrather both detailed the review morphisms of and the objects form manifolds. A The second cocycle condition implies that the curvatures on each gerbe to a Deligneconnection cocycle for encoding a gerbe. aoverlaps gerbe Such a with Deligne cocycle consistsU of the U Cech 2-cocycle Just as in the case of principal circle bundles, we can extend the At the heart of Generalized Geometrymented and by a T-duality 2-form is the factthe that rather the metric simplistic definitionbe of sufficient. gerbes in the form of 2.2 Gerbes on three-dimensional nilmanifolds JHEP05(2019)209 . E B (2.8) (2.9) = id, (2.11) (2.12) (2.13) (2.14) (2.15) (2.10) and σ Courant ◦ g ρ called 6= 0. The cocycle E Γ 3 x such that d → ∧ E E 2 , X. , Γ x → ) } d is the global 3-form ) × k , X , k ) = giving rise to this curvature is E 0 )( = j,k ) (2.16) TM, α M. are characterized by an integer ∗ G TM : B B X ) = + H j T ( 1 σ ):Γ Γ( j + −−→ N β x . X N ⊕ ( g − ( ∈ . Z 3 , B x 3 ) + − X + ( TM d x ( − | TM Y , 2 . is usually defined as a vector bundle d ( ) ∧ X 3 µ ρ ∼ = α ∧ j,k 2 x X −−→ +1) M ) x G 2 d = )( 1 T M , ρ d ) = x , and which is endowed with a fiberwise inner P E ∧ x 1 i d B ( X – 8 – 2 β → ( k B + x k x ker( ω + −−→ d g = M ⊕ ∗ + ∧ =  A ) ). Note that a 2-form M, , i.e. a bundle map T 1 into its symmetric and antisymmetric parts d ∗ X ρ α, Y P x + ( B and a bracket ⊕ T ω 2.2 d on which the inner product + anchor map k X 7→ R and thus an isomorphism { E X −−→ × TM ρ h over a manifold = im( = = X ◦ 0 ∼ = : E M  σ H E C σ → : := ρ E P (1)-bundles. Altogether, gerbes on is called the × U E of the anchor map : TM ) then implies that −i → 2.6 , E h− : , subject to a number of axioms. splitting ρ A The 3-form representing the Dixmier-Douady class of exact Courant algebroid , and we denote the corresponding gerbe by which span the subbundles of A general splitting is then of the form where we split theIt rank also 2 leads tensor to the two graphs Explicitly, consider the anchor map product bracket yields the projector An which fits into the short exact sequence of vector bundles, where conditions ( 2.3 Exact Courant algebroids and generalized metrics where we used again coordinates ( Clearly, this 2-form is not global because class for principal k a quantization condition on the Dixmier-Douady class, just as it is the case for the Chern JHEP05(2019)209 G E of µ of p (2.22) (2.20) (2.21) (2.17) (2.18) (2.19) , ]. + . With ]. That µ C ζ 10 -degree 2 13 , N µ ξ , , . . µ  x g ! ! µ induces a Poisson 1 ∂ β Y 1 − ∂ζ ω µ −

. There is a canonical ∂ η µ ∂ζ . More explicitly, let 1   x T µ generalized metric f B Bg p = ! 1 µ are polynomials in the fiber . B g − + α ∂ T X 1 ∂ξ µ C µ −

p  Bg g  ∂ µ GG g | ∂x ξ − f − µ -manifold is essentially the vector | = as explained in detail in [ µ = ∂ g ξ = Q 1) 2 ∂p i

β = − G Q ( = +   − f generalized metric is also called the H µ  ,Q . This N g ∂ -algebroids in our conventions, see [ α, Y µ ∂x µ ηG ζ n M , which carries a symplectic form of with ∂ + d  – 9 – ∂ξ = 0. The symplectic form = [1] M ∧ X − of degree 1 generating a symplectomorphism. For h ω T and H µ ) and  Q ξ   Q 1 [2] g f L ∗ −} µ µ ∓ -manifold of degree 2 , + d T ∂ ∂ Q ∂x ∂ζ µ ! G ) with = {Q -manifold underlying an exact Courant algebroid over a p ∗ ), it follows that 1  with 1 d | − Q E C = − f   | ∧ such that f Γ( 2.17 = 0 and Q µ 1) = ker( µ x 2 E ∂ − → B Bg  ! ( ∂p Q 1 ) )-metric 0 C 1 B g −  = d − ], the N 1 E symplectic N ) and ( 0 1 − ω d, d 13 Bg g ( :=

or : Γ( O − } − 2.15 = C H g ] for details and our conventions. , but regarded as a graded manifold with local coordinates η f, g is isomorphic to 10 -graded manifold with body

{ N TM = be the 12 M ∗ η G T satisfies indeed is Hamiltonian: is an Q Q As shown in [ as given in ( See again [ C 12  and and structure, coordinates with coefficients which aresymplectic arbitrary form functions and a of canonical the vector field of degree 1 in the above coordinates, a very detailed discussion of symplectic Lie manifold bundle degrees 0, 1, 1 and 2, respectively. Locally, the functions on 2.4 Exact CourantA algebroids more as powerful symplectic perspective Lie isLie obtained 2-algebroids 2-algebroid by regarding ais, Courant algebroid as a symplectic as well as a nilquadratic vector field Then the map C with positive definite inducedbe metric an is endomorphism called on a and let is positive and negative definite, respectively. Such a half-dimensional subbundle JHEP05(2019)209 - is Q E with (2.29) (2.25) (2.27) (2.28) (2.23) (2.26) (2.24) , which M , . Q ) ) } , given over 1 in M ). Functions α ∗ . µ . p T 3 + TM ) 2.11 1 a ⊕ ν ( . → ξ ) ,X in ( ν a } x ( µν from the homological [1] TM 2 ). In terms of sections E d . The pairing on B α E µ Γ( C ∧ : ∂ and the Courant bracket, 2.6 + + ∂x ) with respect to a cover µ ∈ ρ ) . ) 2 µ , x a C a µ ( ( µ d ν ab X x ζ ,X x , cf. ( µν A ,B ) )d ) d )d := B x ab ) ab ) = ( ( b 1 2 , ( ( yields a vector bundle over µν µ µ ! ν A X ζ ) A B ξ α = ι a C µ ) µ ( ) } − {{Q ∂ + ] + a ab + X 2 = d ( ( ν ( ) ∂x ) α µ ) translates to maps b µν ρ A b ( and in terms of our local coordinates, we ∂ ( g

µ + [ α ∂x B ∂ = 0 on 2 ) E ,B ) be a representative of its Dixmier-Douady 2.25 of the anchor map + + ( x µ − ν + , ( ) p M ) ,X x µ , σ σ b ) µ – 10 – ( ( a } b d µν ) ∂ ( 3 ( µ 1 g a X ∂x X ζ B µν α H U [1] is the grade-shifted bundle = B = + := ∈ [1] ) E ) 1 + a a T (  ( µ , ) on functions of degree 1 on B µ ←→ α ζ µ ,X [2] ∂ ∂ ∗ + ∂x ∂x = d 2.21 ). T ) µ ( a  {{Q . Here, ξ ( ( := ) ∞ H σ 1 with connective structure ( X 1 2 -picture, equation ( M x 2.25  C ( Q µ µ G → ∂ α → ) = ∂x , this relation corresponds to the well-known patching relation for the 2 a + [1] α  and let M µ E ∗ σ ζ TU + ) : T 2 M x ( σ . We shall discuss this Lie 2-algebra in more detail in section ⊕ by µ of ,X Q a 1 X ). This data induces local splits U α . Explicitly, the anchor map reads as M TM µ G = + ( ξ of α 1  ) of degree 1 correspond to sections of dd α X In the symplectic N The relation between the classical definition of an exact Courant algebroid and the N a C + ( ( U 2 + a X ∞ µ because of the Deligne cocycleX relation twisted generalized tangent bundle On the intersection of patches, we then have the relation we can capture anyto splitting, splittings as of discussed the above. form For ( now, however, we restrict ourselves If we further insert a symmetric tensor, Consider now a gerbe t class the patch is part of anvector associated field Lie 2-algebra that canonically arises2.5 on The Courant algebroid of a gerbe The kernel of the anchoris map here are the sections spannedgiven by by the the Poisson coefficient bracket of ( manifold picture is now as follows.fibers of Putting degree 1, C identify JHEP05(2019)209 ]. E = 0 10 of 2 (2.31) (2.30) (2.34) (2.32) (2.33) + = 0, is Q C 2 -field can . and gerbes Q B , ! G . Q 1 C G − C 1 ˇ Severa class. As g ) − a g ( B -field contribution to is exact, see e.g. [ ) B ], the Dixmier-Douady a H ( , , H ) } B a ) T 1 ( T a − ) by a symplectomorphism. ! B g ! ) 1 ) a TU a − ( ab 1 ( . 2.25 g ]. 1 B Γ( A B κ − d ξ 0 1 ∈ , we still have 32 − ν , a 0 1 ξ if and only if g X U µ 31 |

= 0, which is equivalent to ) ξ

! C ) b 1 X ( = twisted Courant algebroid 0 − µνκ are now simply glued together in a different Q} )( ) H g , a H is still a half-dimensional subbundle ( ω B 0 g ! comes with an associated Lie 2-algebra which 1 3! induced by the splitting is still compatible with ) H {Q + ν G – 11 – ab . We shall see an explicit example of this trans- + ξ g ( and C . In this sense, the Courant algebroid generalizes κ 1 ! , µ  A x ) µν G p Q a d d µ ! ( B 1 ξ + ( 1 0 1 ∧ B + (1)-bundles, cf. [ − µ ν = 1 ) and the corresponding characteristic class is called the U X 0

g 1 ζ x ) − { a d Q g M

will have non-trivial transition functions, in general. This = ( and the generalized metric to be defined only locally: ( → = ) 3 ∧ B =  a µ a G ( µ ) ζ H ) U C and we obtain the transition function for the generalized metric. a | generalized metric a x H ( ) can be brought to the form ( ( . We conclude that the isomorphism classes of Courant algebroids ) d  ) B ab G H a C 1 ( ), the condition for ( µνκ − 2.26 A B g 1 H ) as a 2.30 − 1 3! +d ( ) g ) a b B ( ( − ], the Courant algebroid = − H B , which is best expressed by a modification of its Hamiltonian 10 of the Courant algebroid. B g Q = ) of the gerbe

) = 0 are local conditions, and a = d ( G are determined by = ( ω and its subbundles B ) H Q a dd M ( E L In the twisted case, a We thus obtain a one-to-one correspondence between Courant algebroids Note that in ( The coordinate change = 0. Moreover, one can show that the additional contributions from the G , up to isomorphism classes, mapping the Dixmier-Douady class to the H ˇ we readily verify that due to Decomposing but leads to the endomorphism the Atiyah algebroid of principal with positive definite induced metric. Over patches Severa class G discussed in [ describes the symmetries of the gerbe d be transformed away by aSimilarly, symplectomorphism splittings on ( Thus, relevant for theclass twist is indeedover the cohomology class of [ Here, formation later. In both cases, we arrive at the the original symplectic form and homologicaland vector field. (The necessaryway.) relations, One can, however, performthe a coordinate coordinate transformations change is in removedvector which but the field instead one has a change in the homological JHEP05(2019)209 ) i 3, ˜ p , and , 2 i , i ˜ ζ (2.40) (2.41) (2.42) (2.37) (2.43) (2.38) (2.39) (2.35) (2.36) ¯ ξ , i ˜ ξ = 1 . , i i ) , 4 x i ˜ p , . x , j . ) . 3 2 ), where we use ∂ i ) ˜ over the nilman- i p ∂x 3 ¯ x j p jx ¯ j p i , d j,k i τ j i , + identifies ¯ − ζ ) G 3 , 2 . 1 3 i ¯ p ˜ 7→ p E i − ¯ ξ j i ˜ , p , x , τ ¯ ζ i → i ( 2 , 1 . ˜ + ξ x ˜ p 3 3 2 , x M, , ↔ ¯ = p ∂ x ∗ or 3 ∂x j , d 0 ˜ ζ T ¯ ξ + 1 1 , Q ¯ p 1 3 . 7→ ↔ 3 , x ˜ for the gerbe ζ ∂ i 3 3 ( 3 or j ˜ ζ ˜ ∂x ζ ξ ¯ ζ ) d , j,k ∼ 3 + , G j 3 ¯ 2 p ∧ 7→ 2 2 δ C ¯ x i ˜ , ζ ζ 2 i and 3 , ξ ¯ d , δ ζ 2 , , are the canonical ones, 1 ) + 1) 1 2 ¯ p := 2 1 1 3 0 ¯ ˜ 3 ∂ ζ , ζ x ˜ p jx + d , jx ∂x d Q , 1 , ). The anchor map thus reads as j,k i , i , x ¯ 3 ˜ p 3 ζ 2 + p − C 2 2 ¯ ˜ ξ ∂ ξ , d ˜ 7→ ↔ d p 3 2.3 j i – 12 – j ∂x , 3 , x δ x , 2 2 ∧ ¯ 1 ζ ∧ 1 2 ˜ ˜ ζ ξ d 1 i − ¯ i ξ , x = ˜ ˜ p ξ jx ( 3 x j,k 2 , . We use again the periodic coordinates j ˜ , i ξ ¯ ↔ ζ G 1 k τ − 3 ∼ , ¯ , 3 ξ ˜ = 0. We have much freedom in choosing the coordinates + d , we can use the exact Courant algebroid as described , ζ ¯ , = d 2 ) ξ 2 j 1 , i , ˜ 1 3 ¯ ξ k ζ 1 ∂ p 2 N ω 2 ∂ , x , ∂x d˜ ˜ ζ , , x ∂x d 1 3 jx , 2 ˜ ¯ ∧ ξ ξ 7→ x 1 i − , , + 1 7→ ↔ ˜ ζ d x 2 2 [1] and it is sensible to consider Darboux coordinates, for which 2 3 2 ¯ , 1 1 ζ ¯ ], we then deduce that the identification of coordinates is ξ ˜ ˜ ζ ξ E 3 jx ↔ and the Hamiltonian , , x , x ˜ ξ = d 13 1 1 , 2 2 , we arrive at the following identification of coordinates: , − ω ¯ ¯ ξ x ξ i ω 1 2 ( 3 p , ˜ , x ξ ∂ 3 ∂x , ∼ , = ˜ , x 1 1 ) i 2 , x ˜ + 1 ξ 3 x p 7→ 2 1 , d , x 1 , x x 3 ¯ ζ 2 , x ↔ 1 , x , x of degree 1 on + 1 = ( 1 x 1 2 with Dixmier-Douady class ¯ ξ i ( 1 x ζ j x ( , x N 1 Much more convenient, however, is the coordinate system ( By considering patches on Let us first consider the case = ( x and are supposed to be Darboux coordinates, the embedding ( i i ¯ Together with ¯ not making any distinctionζ between coordinate and non-coordinate indices. Since the global vector fields and dual 1-forms ( where we introduced the vielbein The symplectic form As necessary, both expressions are indeed global. Using equation 3.3 of [ above and glue it togetherwith over the patches. anchor map This and leads embedding to the coordinate system ( ξ the symplectic form reads as We now give the explicit Courantifold algebroid with the identification 2.6 The Courant algebroid of JHEP05(2019)209 ) 2.10 -field (2.44) (2.47) (2.45) (2.46) B (2.48c) remain (2.48b) ]. This . 0 ) and the ). 33 . Q          ) 2.4 ) 2 2 2.34 ¯ ξ ) ). From ( , 2 1 ) (2.48a) and , k j ) ¯ ζ x 2 ¯ 2 1 p , we have to mod- ( ¯ − ω ) ξ j 1 i x ¯ 2 2.29 ξ, 1 1 G ( 3 τ . ( j i ¯ x 2 , cf. e.g. [ ζ jx ¯ ( ξ ) j kx  , 2 ). We certainly prefer, − -manifold, we have M 3 ¯ 3 ξ jk = ¯ ∗ ζ + ¯ ∂ ξ 3 (1 + Q ∂ 3 1 + 3 1 T k are then ¯ ¯ 2.29 X ζ p 2 3 ¯ 2 ξ + 0 ⊕ kx ¯ p ξ 1 − = of a gerbe 2 1 Q 2 ¯ 3 ζ 3 ) = ¯ k ¯ kx ξ , 1 1 1 jx 2 3 TM x 1 + ¯ ( ζ − jx kx X , ζ ) = 2 ( i , + 1 should read as ¯ − − 3 ζ jk ¯ k ∂ p G 3 , p ¯ 1 ξ i ∂ ( ¯ ξ − ¯ 1 2 ξ x 3 + , p ¯ ξ 0 00 1 i dd 2 = kx 1 ¯ ξ to ¯ ) 0 i ξ = ¯ 0 2 ) transformations. Hence, the expressions ), where the degree 1 members are global. 1 − , α kx 2 ) i Q x 2 2 1 1 ) ¯ ¯ + ζ are no longer global, as a non-trivial ξ − , p x 1 2 i i ( 2.45 , ) i 1 1 and Hamiltonian ¯ x ζ 2 , p = ¯ – 13 – 1 ζ ( i , ζ = 0 as before, they complete the structure of a k  i 3 2 x 2 ω kx ¯ ( ξ X k + 1 } , ξ 2 − 0 i ¯ 1 and ξ jk and ) = x x k Q (1 + , ζ ¯ ζ ) and ( i ( 1 + , 1 1 ¯ ξ 0 ¯ ¯ − ξ, ζ ( −→ jx 1 ∼ i } 2.43 ¯ = p ζ {Q ) 2 ). The result is d ) ,B 1 i ) 3 = ¯ ¯ 1 ¯ ξ , taking into account the gluing condition ( ζ ∧ {− i i 1 2.9 x i ¯ , ζ ) p ( α + 1, this generalized metric transforms indeed as ( ¯ 2 ξ , ζ 2 2 2 ) ,( ) ¯ 1 i ζ 1 k + 1 ¯ ξ x , 2 x j,k i x ) := e + d and ) ( ¯ ζ 1 ( G ¯ ζ i = i 1 2 ¯ i ζ 2 → p ¯ k i x ξ j ¯ , ξ, X ( 1 d¯ )(1 + ( 3 x 2 ¯ ξ B ∧ jk ) -field transformation on the sections of i , 1 , ξ -manifold and therefore the structure of a Courant algebroid. Note that we (1 + − (1 + B i x x 2 1 1 ( ¯ Q ξ x 2 ) = e , jx j kx ), but this is due to the fact that the transformation does not depend on the = ¯ = d 1 ), we glean that the transition from ¯ i ξ ξ, ζ ω x , ( 2.45 1 2.28 x Our coordinates of degree 1, To include a non-trivial Dixmier-Douady class 0 (1 + 0 00 0 0 0 1 0 0 1 0 0 0 0 0 (          -field for the gerbe on the basis vectors. In terms of coordinates on the symplectic N which is complemented by performing a amounts to In the transition induces the non-trivial gluing of thehowever, generalized to tangent work bundle ( withIt coordinates is ( well-known in Generalized Geometry, that the non-trivial gluing can be absorbed by are globally defined andsymplectic since N can compute a generalized metricB from the natural metric on the nilmanifold ( We first observefrom that ( therebody is coordinates. no Inity additional the condition general periodicity due case, to alsoinvariant condition the under the topological for the momenta non-triviality combined receive the of ( an the momenta additional gerbe. periodic- Both, Therefore, we have the identification ify the coordinates and ( The appropriate global symplectic form JHEP05(2019)209 } k ], ,B 17 . As {− , (2.49) (2.52) (2.50) (2.51) Q 16 := e ). , before we i B e j,k , p i ˇ Severa class . C is given by the , ζ ) i 3 j,k , ξ i , p G . o x 3 } j,k -algebra structure, see jp n C , and the The Courant algebroid is + -closed lifts of generalized k X,B 2 Q { -closed functions which rep- by acting with 13 , 0 ]. 0 0 , p Q 1 3 Q Q 10 ξ . n 2 , p i -manifold ξ 3 ζ + 1 ) Q ]: d , cf. [ -closed, 3 o o , ζ kξ ∧ 13 Q 2 j,k i , p ,B + C ,X ξ is 2 , ζ } } 3 1 p , p X 2 + d ,B ,X ]. , ζ 1 ξ B 0 0 i 1 3 e p 33 , p d – 14 – jx {Q {Q is obtained from } , ξ 3 . ∧ n n 2 − X Q i , ζ } i + + B x 2 , ξ p i } } 1 , e ,X ξ , ζ 0 0 = d 1 , ξ ,X ,X Q = 2 0 0 -gerbes. ω . This can be used to relate {Q B , ζ n 0 -closed, then Q e B 3 jx 0 {Q { {Q Q Q − , ξ B = = e = = e 2 3 } = ) is , ξ , x 1 2 ,X Q j,k : C , ξ , x ( {Q ν 3 ξ ∞ 1 -algebroids always come with an associated Lie µ , x n + 1 ξ 2 1 ∈ C µν x , x B ]. We start with a description of the Lie 2-algebra associated to X 1 2 1 = ( x ( 32 ] for more details. The symmetry Lie 2-algebra of the gerbe = , 10 B 31 As a side remark, we observe that The Hamiltonian in the new coordinates reads as This relation readily extends to 13 associated Lie 2-algebra of theto Courant the algebroid abelian gerbee.g. roughly [ what the Atiyah algebroidinterpret is it to as a the principal appropriate circle symmetry bundle, Lie see 2-algebra. One of the majorit advantages makes of the the relevantsymplectic above symmetry Lie formulation structures of veryalso Generalized obvious. [ Geometry is As that first observed in [ This result is well-knownmology in on the the classical generalized language: tangent bundle it [ relates ordinary and3 twisted coho- Finite global symmetries of Generalized Geometry with resent lifts of generalized vectorsvectors: (i.e. functions if of degree 1) to and we see that the dependencein on the the Dixmier-Douady transition class which function wasone previously of readily encoded the shows, coordinates theredescribes is is the no now isomorphism symplectomorphism moved classes of into eliminating the the symplectic Hamiltonian N In the following, we shall use essentially exclusively the coordinates ( For the new coordinate system, we have the identification to a symplectomorphism, cf. again equation 3.3 of [ JHEP05(2019)209 , 0 L
) (3.5) (3.3) (3.4) (3.1) (3.2) ⊕ β 1 + − L and forms Y H, ( i = H, Y ζ , so we get a ι i Y L 3 ↔ ι , X Y ξ ) ι ) H. X 2 ι H. α ξ Z = E ι 1 Z ι -flux written in the ) + + Y Y Γ( ) + ι kξ Y α , H
. ι : α i X X ∼ = Y i ( ζ + ι ι X
Y i ) -algebra) if we endow it ι ι j ... − + − p X ∞ j,k + − j + } − i ,
β L C τ ν
= k β } i ( ξ X ζ ξ γ ι ν + cycl. X ) ∞ 1 , ( X ι ], i.e. k β } C τ + k = perm. ) i d( d 10 X + perm. γ = j 2 1 2 1  Q ν ,Z ∂ 0  of degree 2 + ζ j } i γ . . . ξ − − L ν τ i γ β Y 0 i i α α µ ι Y together with the ξ + τ ι Y ,Y Y τ Y β, Z } d d k τ i ] , − · X X ξ + ,Y ...i , } i ι ι k · 1 − L − L } i 3 2 3 2 α Y Y α , f β β ω ( j } j 2 + + + ∂ X τ X – 15 – + ∂ α of degree 0 and 1, respectively. These correspond i j i γ γ and [ L L ζ 0 τ j τ i i d ) and i j,k + α, µ τ τ j j,k d ), the expressions look formally the same if we use + ,X Y C i ! given for vector fields X ] + ι has an underlying graded vector space C Y , 1 τ N + k ,X of the Courant algebroid as well as generalized vectors. ] ι ( , ζ ,X X τ } j f , i ι X = ( = j,k X f , ∞ ξ ι X N f , f , , f {{{Q { C C τ {{Q X,Y 1 τ X ω 3! τ ] L {{Q X,Y 1 τ 3! L 1 1 3! 3! 1 2 1 2 {Q d ) = ) = ) = d ) = [ ) = becomes a Lie 2-algebra (a 2-term = and a totally antisymmetrized sum is taken over the indices X,Y ) = d ) = [ ) = ) = γ f [ β j,k ) = ) = ) = ) = ) now depends on the chosen coordinate system since ordinary L ( γ f β i C γ f β 1 ( + + ∂ α, f ( ( 1 µ ∂x + + 3.2 α, f 1 by + + α, f µ ∞ 0 + µ C k + i + β, Z α, Y X β, Z and differential d = ( α, Y X β, Z α, Y X + 2 ( + 1 τ ( + 2 . . . ξ ] + µ + 2 − · + we arrive at the expressions µ 1 X , L µ i · i ( X ξ X 2 ξ ( α, Y ), the expressions are just the ones given in [ k ( 2 i α, Y µ 2 α, Y ˜ ζ is short for + µ ...i , µ + 1 i i + i in the last expression. Using d ˜ ξ X ∂ ω ( X k ! X 3 ( 1 k ( 3 In the nonholonomic frame ( µ 3 µ µ . . . i = 0 where i coordinates the vielbein: the homologicalnew bracket function [ takes theω form The explicit form of1-forms and ( vector fields areframe no ( longer global objects. We first note, that in the holonomic to ordinary functions on the base The graded vector space with the following totally anti-symmetric brackets The Lie 2-algebra associatedwhere to the homogeneous parts are the vector spaces of functions on 3.1 Lie 2-algebra associated to JHEP05(2019)209 on k (3.7) (3.8) (3.9) (3.6) loc ]. The α j L 34 ∂ j i τ ) i -algebra on ¯ ξ β ]. . ∞ k 27 ) L + Y a U Y | ( ) + 1 E α α -algebroids naturally + Γ( 1 + a ∞ X t L ˆ X L describes the action of gauge ( , 2 ∼ = i . Moreover, we can restrict 1 α ∂

) µ ) β i + k a β 2 β τ + U , this corresponds to extending | k X + j + α , ˆ Y L Y τ j,k N 14 Y d C − ,Y ( − ( Y ) }  j ι 2 Y ∞ α 1 β α 2 A ) − C + α j + a U 2 ) ) is simply the action of a Hamiltonian + a t 2 β X ,X j t 1 X X 3.8 ,X act on the functions and the local functions ∂ = ˆ α ι L ( ( + 1 j ) with “structure constants” induced by the i τ 1 {Q

2 α loc 0 loc τ i  X β – 16 – L L -algebras of symplectic + on the function ¯ ( ξ , ν 1 2 + d ∞ } + X µ loc ˆ ˆ L α β ) = L L L ) + and τ β ( ,Y + β d = L } + X ) = Lie + α ι ,X β Y ). This action is readily extended to an action of Y + ( + are not affected by this extension. Since the brackets are 15 ( via the Dorfman bracket, + α {Q j,k j τ i ] + ∂ C ) and µ Y ,X j,k j ( i a X ]( τ ˆ C i U ∞ L 2 1 {Q ( α X,Y C  X + ∞ with 2 C = [ = ∈ X a . In particular, the action of the Jacobiator of the Lie 2-algebra is ) = ˆ loc 0 β L 2 t ], there is a well-defined general notion of an action of an β , L µ 1 + 10 = + α ⊕ 1 + 1 1 loc − loc − L X α, Y L α, Y ˆ L [ + = + , since the generalized Lie derivative ( X 16 X loc ( j,k L 2 We can readily identify this Lie algebra as a Lie subalgebra of the symplectomorphisms In our case, the Lie 2-algebras For our purposes, it will be useful to generalize the underlying graded vector space of We note that the Courant and DorfmanThe brackets Jacobi on identity is nilmanifoldsThis trivially were satisfied is also for obvious given, commutators. from e.g., a in more [ abstract point of view: the image of C ν -manifold picture goes beyond this since it provides also the full Lie 2-algebra of symmetries. to from global to local functions. Given a cover 14 15 16 Q N transformations between gauge transformationsalways (level leave the 1) gauge on fields gauge invariant, see transformations e.g. (level the 0). explanation of The the former BRST complex in [ Courant bracket trivial. on vector field with Hamiltonian action together with the commutator forms an ordinary Lie algebra local functions. We can, in fact, regard the Dorfman bracket as a generalized Lie derivative as is customary in Generalized Geometry and Double Field Theory. We note that this where a graded vector spacecome and with associated such an action. on the Courant algebroid tives), the higher products given by tensorial expressions, the brackets will be3.2 compatible with the gluing Action conditions. of theAs explained associated in Lie [ algebra and its action Lie algebra L L Since the brackets only involve local expressions (i.e. expressions with finitely many deriva- JHEP05(2019)209 . (3.16) (3.11) (3.14) (3.15) (3.17) (3.18) (3.10) (3.12) (3.13) ). ) where j,k C 3.13 2.48a . − β ] 2 } = 0 ,X 1 ij ,Y X we have } [ H ι . } of degree 1 on , ˜ τ Q β j ,B } 2 , ζ 1 i , + Q . . Such symplectomorphisms ζ + d ) α } }} j ^ . } {{Q ij Y Y,B j,k β ,B , { β N ) τ -field corrected expression H C and + } ( 1 B d therefore removes the additional 1 2 {− ] } } + + B α Y,B α Y 2 ) ι 2 }} { β + Y β + ,X ,B τ X = e j 2 closed 1 + B + j } + ζ + } α, Y X i Ω [ β β Y,B X ^ ,X ξ ι Y i − L ,Y Y { + j n ,B ( ∂ + i , + 2 } } ) {Q α X ] + H j β Y { = ) = {Q 1 ,B { {Q {{Q N ,Y j β + τ X + i ( = ,Y + {− -field is merely its covariantization with respect + + } ] = j L e ) = + 2 } H ( ξ – 17 – } } } B } Q i , β τ ,Y diff β ,B β β β ξ β Y d ,X ( Q = , + ij + 1 } Y -closed forms are locally exact and give rise to the + + + } {Q + i {Q ι by the appropriate τ } ) = H X B Y α ,B ,B 2 ^ 1 2 ( Y j + , corresponds to ,Y − ,B β to the action on e ,Y ,Y ,Y ^ ∂ } B {− } {− , ν is thus clearly -field modified Hamiltonian by + α, + d e {− + α α Y − i {Q {Q { {Q {Q ], and they have general Hamiltonian of degree 2, B loc 6= 0, we recall the observation made around ( ) = [[ X + p = e in the Poisson brackets. Note that also Q j ^ + Y + L ˆ i := e β L 13 ( = = e = = = = α ν ^ H i = X H β + ξ } ∂ { ,X µ Lie ˜ β Q + = ξ Y = 0: Q, X = ( + {Q ^ ) Y µν H 2 {{ ij H due to a non-trivial ^ α B Y = H , 1 2 + in the Lie 2-algebra and its action. We thus recover again ( Q 2 ˜ H Q = B ,X { , 1 i α B X + 1 ), first for X = ( i 2 action. Denoting the 3.9 µ H ˆ B L To understand the case To be more specific about the structure of the Lie algebra, let us consider the explicit and similarly Translating from the actionterms on containing to the e as well as where we used The modification of local 1-forms which correspond to part of the localwe replaced functions the local sections The Lie algebra of actions of where the semidirect product isphisms given on by 2-forms. the natural Note action that of the the infinitesimal d diffeomor- form of ( have been discussed, e.g., in [ The case at hand, ourselves to the degree-preserving symplectomorphisms on JHEP05(2019)209 j N (4.1) , but of the (3.19) (3.20) (3.21) α k ]. maps the 12 j N ] together with F . . } ) = [ } P , α ( , f 1 has been given long ago c , the automorphism group {Q {Q ˇ Cech 1-cochains. j,k = = C α H f τ ) τ ]. Then we have a closed integral . x ( H ) 1 = d j − = d . That is, the T-duality interchanges π N k according to ( I with the Dixmier-Douady class δB δα ) = [ N j α ]). Together with the commutator, these G N ( ) := 10 2 closed was discussed in full detail. Let us briefly x dd Ω )( topological T-duality ˇ j,k Severa class is the Dixmier-Douady class of the n and first Chern class – 18 – with G H with ) ( j ∗ M N π ( are given by diffeomorphisms on the base manifold whose → ), which is readily integrated to the group ) = δB δα Diff P j,k j,k x : ( + + G C ˆ 3.13 F π α B 17 over the nilmanifold = = k,j ˜ α ˜ B G → → , and changing the 1-forms j α B N . of the principal circle bundle j with Dixmier-Douady class from the integral along the fiber, j N . on P loc M f L to the gerbe on over on G ˆ j,k j,k F G G ], and the example of the gerbes A mathematical picture behind this Consider a circle bundle On the Courant algebroid 24 More precisely, the symmetries of the gerbe over local patches, which is all that is needed for T-duality. . This is relevant, as a subset of these functions can be interpreted as extended vector 17 2 2-form On double overlaps, we also have gerbe isomorphisms parameterized by in [ summarize the essential points important for our purposes. a gerbe 4.1 Topological T-duality Performing a T-duality along thegerbe fiber direction of the principalthe circle Chern bundle class gerbe the symmetries of athe general base Courant manifold of bracket the are corresponding diffeomorphisms Courant and bracket closed is4 2-forms well-known, cf. on e.g. [ Global Double Field Theory actions form the Lie algebra ( This is also theof symmetry the group underlying of bundle the endowed Courant with algebroid the Courant bracket. We note that the fact that Lie algebra gerbe, this associated Lie 2-algebraν also acts on gradedand functions polyvector via fields the Dorfman (as bracket formalized, e.g., in [ Therefore, there are gauge transformationsby between functions gauge transformations, parameterized The graded vector space of both these transformations arranges now into the categorified The symmetries of the gerbe as well as gaugeonly transformations. their differential acts The non-trivial, latter shifting are the parameterized 2-form connective by structure local 1-forms 3.3 Finite symmetries of a gerbe JHEP05(2019)209 ), ⊗ ˆ H G (4.5) (4.6) (4.2) (4.3) (4.4) ∗ ˆ P, pr ) = 1, we ˆ A ) to ( ) and ( ∗ 4.3 π ). It was shown with Chern class P,H 4.1 P ) = ˆ with the first Chern . One can now argue A O  )). Given normalized j ( k,j k,j M ∗ . P G C N π ( ˆ } 1 → , G ) c ] to be the homomorphism ( ˆ x P F . Altogether, we obtain the ∗ (ˆ 24 over : o ˆ p π } P k π o K j,k N C on G ˆ ) = ˆ ' ( ) =: d P # x ˆ ˆ A ( G ∗ p , | ˆ ˆ ˆ ) π pr pr k ˆ π P P 2 k ∧ ( ˆ pr P 1 o N × { A c ∗ 2 M w P T k,j M pr w is defined analogously to ( $ , by which we mean × E of the gerbe ∈ × ) = 2 – 19 – ˆ  with Chern class ˆ P ˆ j ) ˆ P P k = T H ˆ map was defined in [ ' x ( N = d( ˆ & P ∗ . Note that the latter implies that the gerbe K π pr x, → ˆ π E and ( P H := ∗ { { ˆ 1 P M P pr K pr j P := × / M π − with ˆ on × P P T-duality ˆ ˆ H . We thus specialize and extend the diagram ( P ˆ -manifold on the correspondence space v P x A j ∗ M : Q j ˆ N pr × K on G N ∗ , the G ! / P ˆ and ˆ H F pr correspondence space A 2-form on O  j,k where j,k is the G C ∗ global ˆ P pr M ◦ is a × )) or as a circle bundle over of the nilmanifold ] is the first Chern class of a principal circle bundle ˆ ˆ F F∧ P P is trivial. Using ˆ j e ( F 1 ˆ ◦ c G ( ∗ ∗ ∗ ˆ ˆ 4.2 Symplectic pre-N Let us now returninterchanges to the Dixmier-Douady our class exampleclass of pairs of nilmanifolds. Here, topological T-duality where pr pr that this map descends torespectively. an One of isomorphism the of aims twistedin of cohomologies the order on following to sections ( have is a to starting reformulate this point in to our investigate language T-duality in later works. π connection 1-forms find that Here, The correspondence space can either be seen as a circle bundle over that there is a 3-form which defines the Dixmier-Douadyfollowing fibration class of of spaces: a gerbe and [ JHEP05(2019)209 , . K G (4.8) (4.9) (4.7) . We (4.11) (4.10) . k,j -algebra. , the ap- 4 C -manifold . -structure ξ ∞ K ) Q 2 ∞ . E L 2 ξ captures the L 1 ) , respectively. and 4 kx K ), such that the k,j j ξ E − , p j,k G E + 3 4 ( C 3 ∞ . Its symmetries are ξ on a symplectic pre- , p , x 2 and -manifold of degree 2 . 4 2 K ξ L Q ) K 1 ⊂ C 4 kp G j,k jx , G . C L k ξ + − , x  k 3 3 + 3 ζ to the picture which describe 4 jp + 1 p , x  and 2 3 closes and forms an 2 k,j + 4 ξ . ) j -structure over this space for the gerbe ξ 1 C 2 4 E , x , x ˇ Severa class is the Dixmier-Douady  ∞ k,j K 2 kx 1 2 , p L , p G G 1 3 ∗ 2 − and C + 1 , x symplectic pre-N := 3 1 1 pr , p , p p of the homological vector field are then x x j,k 4 whose 4  2 2 ( ( ⊗ yields the gerbes ξ . The latter is a four-dimensional manifold C Q 1 , ζ k , p K ∼ ∼ j,k 3 4 subject to the identification K 1 N , θ ) jx G . Note that the latter two gerbes can be pulled , we now linearly combine the fiber coordinates -algebra on 4 – 20 – 2 ∗ 1 on , ζ  → n , , p K − T 4 k,j , we readily derive an equally explicit description for Double Field Theory. Since 2 k pr + 1) E 4 ζ , x j µ K G × ,..., 4 p G N 3 = 0. In particular, a symplectic pre-N , ζ N j should be a  , ζ K µ := C 1 ξ 3 3 , x N , x = 1 ω K ξ and K 3 , ζ -manifold. By an Q and = E , ζ  µ G -graded manifold of degree 2 endowed with a vector field 4 := 4, subject to the identification L Q 2 1 2 + 1 , x j,k N Q ), K , ξ 2 2 G µ , ζ , we expect it to be a subspace of -manifold K 3 := → 1 , , Q , p , x , x j k,j ,..., 3  µ µ , ξ 1 1 , ζ G N θ ζ 2 x x 4 to the Courant algebroid , ζ d and the Hamiltonian ( ( = 1 µ , ξ ∧ , ξ ω µ ∼ ∼ K and , ξ 1 3 µ , µ ) ξ µ 4 x , ξ j,k , ξ x G 2 , x + d , we mean a subset of the algebra of functions is thus indeed a suitable correspondence object on 3 + 1 µ , ξ E p 1 1 K , x d x G . and tensored there to the gerbe ( 4: 2 , ξ ∧ , K = 0 is a symplectic N 1 µ K G ∼ , x x 2 x ( 1 = 3 ], which also suggests that Q x To construct the pre-N We now extend From our explicit description of This point of view is further supported by our general discussion of the local case As explained in the introduction, our goal is to characterize the space ( α -manifold -structure is a solution to the strong section condition. 10 = d of degree 1, which satisfies , Q ∞ ω α  We note the expected symmetry under the exchange of of degree 1 correspondingθ to the directions involved in the T-duality to new coordinates The symplectic form which is also straightforward,have coordinates as ( it combines the Courant algebroids L of the correspondence space with coordinates with N usual construction of theTranslated associated to the Lie physics nomenclature, theamounts algebraic essentially condition to of (a having slight an weakening) of the strong section condition and an explicit duality between in [ By this we mean aQ symplectic described by the Courantclass algebroid of propriate correspondence objectsymmetries on and the differential geometry of the Double Field Theory relevant for the T- the symmetries of theback gerbes to Restricting this gerbe to The gerbe where we also added the Courant algebroids JHEP05(2019)209 - - of Q ∞ L loc (4.16) (4.17) (4.15) (4.12) (4.13) (4.14) L 4. More -manifold , Q = 3 . 4 and used the . α ]. This pre-N , , α −  is a subset β + 4 θ + 10 g θ θ  K d 2 4 + g = 3 . = 0, ξ E ∂ a ∧ 1 by pull-back along the ) ∂θ ∂ α 4 + 2 α ]. The pre-N − endowed with gerbes of ∂ζ α + jξ θ ξ θ K ) (4.18a) are captured by an θ = 0 for the corresponding 1 k d 10 E 2, K d ξ +   , k N ω ) ∧ E   f j 3 ( + αβ to Q N f 3 0 + θ 3 + η , L 2 a 2 θ − ∂ 1 2 = 1 . Q ξ T ∂ ∞ ∂θ loc n 1 ∂ξ β a + ×  + d p   | a -structure on kξ j α g 2 ⊕ C + ( | ζ f and and | p ζ ∞ f µ N ) 4 d + | d j p ∂ L 1) K 4 αβ αβ ∧ 1) ∂p N ∧ = p E η )( − a ! ( 2 2 ( ), etc., − induces the Poisson bracket 2 1 ξ ξ ( ξ , α = ξ 1 K 1 −   ω 0 1 1 0 4 ξ − ∞ loc , x f ) p + d kx  a

+ d µ 3  -manifold, since g is the ideal generated by the x m α p – 21 – 1 ∂ g − = ⊂ C p Q 3 + ζ ∂x a I 3 − d ∂ d ∂ p  -N = ( ∂θ = 2 αβ loc 1 k ∂ξ 2 ∧ ∧ η L ξ − µ 1 α 1 pre + ( x ξ Q} ⊕ x      , 3 a jx ]. Recall that an f g f p , where loc 0 µ 4 + d + a − + d K 10 {Q I L ) between which there is no T-duality. The factorization of , respectively, as we shall see in the following. ∂ ∂ ∂ E µ / a α j ∂ζ ∂θ ∂x ) p p p = = 2( d d K m,n   α + G | | θ C loc ∧ ∧ f f   and C , -structures and the section condition | | Q} L ( µ a f + , . The symplectic form render k ∞ 1) 1) x x j,k µ a ∞ L ∂ C − − p C αβ {Q Q ( ( a ∂p η = d = d ξ  − − = as defined in [ ) = ω = and K := K αβ ω Q E E } η ( ∞ on C f, g { would here have read as ) (we restrict immediately to local functions), K 1)-metric E We also note that the above constructions are readily extended to general pairs of Restricting the symplectic form and the Hamiltonian Q} in which we are interested is then obtained as the locus of , ( , (1 ∞ K loc 4.3 Symmetries, The symmetries ofstructure Double Field TheoryC on {Q We see that this factorization simplifies for T-dual pairs. manifold describes indeedunder the the target T-duality space connectingDixmier-Douady the for class nilmanifolds double field theory manifestly invariant Courant algebroids ( We note that hamiltonian vector field, and This is precisely the relation also found in the local case discussed in [ O with inverse and where we split all coordinates as E formally, implied embedding, we obtain closely following the construction in the local case as outlined in [ JHEP05(2019)209 as , K ) (4.20) (4.19e) (4.19c) A (4.19a) (4.19d) (4.18b) (4.19b) Y ): τ d A 2.41 -structure are X ∞ . − L ν , A , cf. ( A j ∂ i Ξ ], we showed that in ν X τ µ ! f , A τ τ 10 d µ τ X X . ξ 0 1 1 0 A L 1 2 = A

Y = ( . a X ). We also used the index τ 1 2 ) to be an = = ζ ν a K ∂ f      + , E αβ ν ν X µ ( 4.18 τ ν , , 4 ∂ τ 0 ] , 0 0 δ ν + µ µ 2 and that further µ -structure translate to the section τ 2 µ ∞ Z loc δ , η 0 0 0 µ = 0 = 0 = 0 ξ 1 1 ∞ A A β ] loc and d µ X,Y } } X Y L 2 x L kx 0 00 0 1 0 1 0 0 1 2 X ⊂ C 1 + = [ αβ − g, f      = = η f, X A
X,Y,Z ) = Ξ 2  [ ν 3 loc 1 2 } a µ } δ Z X = L = Q ζ }} Q 2 ν µ ν { , ζ µ α ,Z δ { ∂ ∂ µ ,Z ∈ -field, which is part of the connective structure , f 1 } ν ν µ + ξ µ } X – 22 – + perm. τ τ B are higher gauge transformations, that is gauge , η } f jx µ µ f , {Q } + ,Y , x τ B Y a Y . − ) loc 1 }  ) = ( X,Y f, g ξ X, A parameterize infinitesimal diffeomorphisms on α X ν µ L ], the conditions for ( 2 µ 2 a − loc 0 { X, f δ ) for having an . = d X ,Z 1 2 L Q 2 , x Q X , ζ 10 ) and AB } Y loc 0 ν a { a = loc 0 η Q + ν L {{ ∂ K x ξ = = } − {{Q L { ν ∂ 4.20 ν µ µ E ,Y A = } ν τ ( µ τ α + } ∈ 1 τ Y ,Y = ( µ θ , A ν , f µ } ξ α ) = ( µ } ∂ ,X ∞ X loc α + X x ν X µ = ,X τ , θ ,X } = + µ a ⊂ C X,Y,Z a τ {{{Q , f X ] ζ , ζ {{Q  A a a {{Q loc 0 1 ξ 1 2 3! 1 2 Z {Q ( X L and . The elements of is the four-dimensional generalisation of the vielbein X,Y = [ + K ∈ ν µ loc 1 ) = ) = ) = ) = a G τ L f ξ ( a 1 ∈ X, f X µ X,Y ( ( 2 2 X,Y,Z f, g µ X,Y,Z ( µ 3 The algebraic conditions ( Explicitly, the elements of As shown in Theorem 4.7 of [ µ transformations parameterized by condition in the usualan Double appropriate Field symmetry Theory algebra nomenclature. onthe They the local correspondence are case, space. necessary they for In correspond having [ in fact to a weakening of the section condition. for all well as the gauge transformationsof of the the gerbe transformations between gauge transformations which encode the redundancy in the gauge that the brackets and the Poisson bracket close on The vielbein and gives rise to the modified Lie and total derivatives, and so that Here, notation which becomes a Lie 2-algebra with the following higher brackets: JHEP05(2019)209 to K (4.21) (4.22) (4.24) (4.26) (4.27) (4.23) E . Both into the k,j k,j C C , , ) (4.25) completing the , ) . 2 ) and 3 gives a clear hint and 4 4 x ) V d 0) 3 4 ) ) , , , p 1 ˜ j,k , p p 4 4 p 4 , j,k 2 ) ) 4 3 3 p C , 3 C 4 4 p ∂ p kx p ˜ , p , p p 0 1 ∂p 1 and ) span integrable distri- 3 3 , . , , p , p 1 + 4 . . 2 3 ) ) 2 kx jx and = 2 3 3 4 ˜ , p , p ˜ p 3 3 p 2 V kx ,V x 3 ˜ ˜ p p −} 2 2 , − − 4 , x k,j , p , p p , , , + ∂ 1 Q} d 2 2 (d 1 2 2 C 4 ˜ , p , p 1 , ˜ p 2 2 p p 4 − ˜ ˜ p p 1 1 , , p , p , { , p , p ∂ 2 kx , , ) ) 3 , respectively. Explicitly, we have 1 1 3 {Q ∂p 1 1 = ˜ ˆ 3 3 ζ , p , p ˜ 1 pr 1 1 = ξ ˜ ˜ + p p ˜ ˜ p p , −−−→ ω ˆ , p , p , 4 4 + + , p , p = pr 4 ) and ( , , 4 4 , , 3 3 3 4 3 + + 4 V ∂ ˜ 4 4 K + + x 2 2 ˜ ξ ζ 3 3 , θ , θ ι ∂x ˜ ˜ ˜ ˜ V p p E ζ ζ , , ,V , θ , θ 3 3 + + , θ , θ , , 3 , , and 2 2 2 2 ˜ ˜ ζ ∂ 1 1 ζ 2 2 3 3 + + 2 ˜ ˜ , θ , θ ˜ ˜ ζ ζ p p , , ˆ , ζ , ζ 2 2 pr , , , , 1 1 , θ , θ 1 1 ˜ ˜ ζ ζ 1 1 3 3 2 2 ˜ ˜ , ζ , ζ ˜ ˜ ζ ζ ζ ζ , , and and and and d , ζ , ζ – 23 – 1 1 , , , , 2 2 , ζ , ζ ˜ ˜ ξ 3 4 3 3 + + 2 2 ξ 1 1 , ζ , ζ ˜ ˜ ˜ ˜ ζ ζ ξ ξ , , 2 2 , , , , θ , , θ 1 ) and , ζ , ζ 1 . The factorization ˜ 2 2 ˜ 2 ξ 1 1 2 2 2 2 ξ 2 2 ˜ ˜ , ξ , ξ ˜ ˜ j,k ζ ζ x ξ ξ ∂ x , , −} k,j 1 1 ∂p d , , C d , , ξ , , ξ , 2 , ξ , ξ 3 1 1 1 3 C 3 3 ˜ 1 1 1 1 x ˜ 1 1 x ˜ ˜ p , ξ , ξ ˜ ˜ ξ ξ 1 ξ ξ 1 , jx { jx jx 4 4 ˜ , , x , , ξ , , ξ , ξ , ξ ˆ 2 pr k 2 2 3 3 3 4 + 4 4 and + ˜ + = x −−−→ ˜ ˜ ξ ξ , x , x We note that both ( ˜ ˜ x x 1 − 3 3 3 3 , , 3 3 ˜ , , x , , x x K x , x , , x x j,k ∂ 1 1 ∂ j 18 2 2 2 2 d E 3 3 3 ˜ ˜ ∂p ξ ξ C ∂x , x , x ˜ ˜ (d x x ˜ x − , , 2 2 , , x , , x , x , , x , − = 3 3 : 1 1 1 1 2 2 2 2 ˜ ˜ 3 x x , x , x x x x x = ˜ ˜ K x x V , , 1 1 E , x , , x , ) are the coordinates on the Courant algebroids ω 2 2 x x 3 1 1 1 1 3 ( ( = (˜ = ( = (˜ = ( ˜ ˜ x x ˜ p V 2 1 x x x x , , ι ˆ ˆ pr pr 1 1 x x = ( = (˜ = ( = (˜ (˜ (˜ ,..., 1 2 -algebra structures: solutions to the section condition 1 e e x ∞ -manifold L Q The symplectic form is global and invertible and so are the two 1-forms. 18 projections allow for sections,pre-N which are the evident embeddings of where (˜ the obvious projections which satisfy and are therefore global. butions, giving indeed rise to the projections distributions. It is natural to choose the symplectic dual vector fields to theUp global 1-forms to a sign, we then arrive at We want to regardfoliations whose these leaves vector are the fields fibers as to part the projections of integrable distributions whichFor this, encode we clearly need an additional pair of vector fields Let us now comethe to Courant discussing algebroids the solutionsthat to we the want to section quotient condition by which the reduce Hamiltonian vector fields of 4.4 JHEP05(2019)209 ) ) K } and E ( 4.31 (4.28) (4.29) (4.32) (4.30) } ∞ = 0 loc (4.31b) C F = 0 3 ∈ V F . = 4 F V ) between which F 3 −} = , ∂ m,n . 2 . Altogether, ( , | ξ C 1 2 K F 1 ) , ˆ ˆ 4 pr pr E ξ = 0 for K ) ∂ L L j,k j E follows fully analogously. | C ( ∈ ∈ F 2 -manifolds, but we refrain from − 4 ) = 0 (4.31c) ˆ ∞ loc } pr Q F F V ). For this, we rewrite n L K } , E ,F ∈ C ( for for + ( ,Z yields two different vector fields, 4.20 } -manifolds: the symplectic forms 4 = 0 = 0 ∞ F loc Q} } } = 0 (4.31a) = 0. Second, the above discussion p , { Q } } } { ,Y Q} F ,F ,F } = , 3 3 4 ∈ C , f , g = ,X p p } 2 V {{Q } ; the case } 4 { { ˆ 1 2 1 ,X pr {Q F 4 3 3 , f ˆ L pr D { ,X , f p p 3 p = 3 3 L 2 2 p p p ) of ( }} {{{ {{ {{ {{ and 4 4 4 4 = p p – 24 – ,F p p 4.17 } } and = = is a function of degree 0 on = = {Q ,F 19 } } , } } } . 4 = 0 are functions of degree 0 and the Poisson bracket on 2 p e ,Z 3 g F {Q ,Y f, g determining the projected Courant algebroids also form } p 4 2 } X, f f, X { −} V = } 2 2 ) to be satisfied. Note, however, that they are by no means Q , and 2 { ,X Q Q and = F ξ 3 X,Y ) and analogously for 1 { { = 2 since the Poisson bracket also vanishes between a function of 2 = 0 1 } 2 p e 4.20 F ξ } 0 Q , 4 ) Q F {{ , f 1 ∂ 4.22 3 3 ˆ ,F m pr {{ | p V L { ) − along Q} of ( = ∈ K , k = 0 seems to be merely necessary for reducing to a symplectic space 1 K E , since F ( X E ˆ 0 3 pr F , + ( ∂ 4 1 {{Q ∞ loc -algebra structures, verifying conditions ( 3 ). We also note that , since ˆ V | pr p 1 ∞ K , and L ) { 1 L E ∈ C 1 ˆ ( K ∈ pr , = 1 E L F ∞ ( loc ˆ 3 pr { ∈ L ) vanishes separately. D ∞ loc = ∈ C ∈ 1 algebras, as it is to be expected from the fact that they represent Courant algebroids. A few remarks are in order. First, we have not used We now restrict our discussion to the case It is now easy to show that the subsets between two such functions vanishes. Similarly, ˆ ∈ C f, g 4.20 X,Y,Z f F Mathematically, we can thus encode T-duality as a cospan of pre-N pr L 19 K loc ∞ F The projection map is correspondingly more complicated. pushing this thought any further in the current paper. { L The condition in the projection straightforwardly extends to general pairs ofthere Courant algebroids is ( no T-duality. The factorization ( are sufficient for equations ( necessary. They arein rather ( special, in that each term inat the any symmetrizations point contained of the proof. Thus the subsets for degree 0 and a function of degree 1. Finally, we have for for E We have indeed form for and the Hamiltonians ofbacks the of homological those vector on fields on the Courant algebroids are pull- These embeddings are morphisms of symplectic pre-N JHEP05(2019)209 , ) 3 K ˜ ζ E 2 ) in ˜ ξ 4.16 ˆ 1 H (4.36) (4.33) (4.34) (4.35) ˜ , ξ k k N . = H k,j , lifted to ∗ 1 ˆ Q A ˆ ∗ 2 pr ∗ ) and ( ◦ ) in our language. ˆ ) and embeddings ˆ pr pr ∗ 2 e ,H ∧ 4.5 j 4.26 = A N ∗ 3 + 4 + θ , respectively. Evaluating θ of ( pr 2 2 i ξ ξ k,j 1 1 ˆ pr . . The very general framework C . ) . kξ jξ ) according the untwisted part K 4 2 + to the respective Courant alge- E θ 4 ξ + and 2 1 ) takes the form θ K ) + ) + . We also have ξ 2 4.13 3 3 E 1 jξ ξ , j,k 4.5 k,j 1 jp jp C jξ + Q j,k jξ 4 + + + . Q p = + 4 4 ∗ 1 ( 3 + H 4 ˆ + θ 3 + pr kp kp j,k θ 2 θ ( ( ◦ ξ 2 1 1 -manifold Q 1 ξ ∗ 2 ∗ 1 x x 1 e 2 2 ) = – 25 – ˆ kξ pr ξ ξ 2 NQ kξ ◦ = ξ − using the projections k,j ∗ 2 ) allows to relate the homological functions on the 1 − − e + C 3 4 = k,j ), we first rewrite the condition ( ) is the untwisted de Rham differential on the corre- k,j kξ 0 K p p 4.5 Q E C 3 4 + + Q + . Hence, equation ( 4.6 4.5 θ θ F} that 3 4 + , and = 2 θ p 0 + + e by itself describes the symmetries of Double Field Theory, ( 3 + 2 2 Q j,k 3 + θ p p {Q K θ C 2 2 − from the pre- E ξ ξ ) gives the relation for the twists: i = + + ˆ pr 1 1 F 4.34 p p 1 1 are the homological functions on ξ ξ , we split the homological function ( = = k,j K -manifold ] is readily applied to the case at hand. -flux” T-dual to the 3-form flux Q Q j,k f 10 Q ∗ 1 and ˆ pr j,k ) in composition: Q to its corresponding object on The pre-N We see that there are two steps for T-duality here: first, the factorization of ( 4.27 -manifolds corresponding to the projected Courant algebroids. This is the reformula- Q j,k 4.6 Differential geometry on Let us briefly comment also on thedeveloped differential in geometry on [ which act on extended tensors. For these, we need to extend the algebra of functions on N tion of the T-duality mapthe in two nilmanifolds our with language. interchanged Chern- Ofthis and course to Dixmier-Douady we hold classes, restricted true but to we whenever expect the geometric very T-duality example is of possible. It follows from the definition of which is the “ determines projections broids. And second, the condition ( This enables the exchangetions: of the parts differing in the pullbacks of the homological func- where the left hand side of ( Using this condition, it turnsC out that we are ablein to transport ( the homological function on We further note thattakes the the product simple of form the connection forms, i.e. We finally want to investigateour language. the meaning Recalling diagram of ( Noting T-duality that between the ( differentialspondence d space in ( plus the flux part: 4.5 T-duality between JHEP05(2019)209 = are , F on a 3 R ) (4.40) (4.41) (4.37) (4.39) V K }} E (and the = and , = 0 (4.38) ) such that F F on  3 K T o } ) ∂ E , t QX, Y ( Y loc ,Y {

L ∞ X ↔ C Z, ) , QZ X {∇ = 0 or . } f ) to K W F + ( E 4 K + }} − { E V ↔ o Y,X ( -structures. An extended Y , 2 -structure } = Z ) Q ,Y Q X ∞ K

F , L QZ, Y + ( 4 E ) ( { ∂ ,X o K T QX, f } -manifold } − { E { X, ( ,Z Q { ,W → ( = W

, Theorem 4.11]. This condition is the ) X ). For this notion of extended covariant X,Y 1 2 } 2 K 10 K -structures. ,Z E → + ) Q E ( }−{∇ ) of degree 1. Note that both ] ∞ ( { ) , fY T K L  T K ,W X,Y X E , cf. [ ( n E Z × ( ∈ 2 ( ) X,Y,Z µ reduce to the extended tensors on the Courant T loc 0 [ ) for the parameters of infinitesimal symmetries. – 26 – {∇ {∇ Y L K X K E : o ∈ ( E } 4.20 } − ∇ ∞ loc } − ∇ ) generated by the local coordinate functions over the −} ,Z C Y , K W : Y and ∇ X E , ∇ ( , X,Y,Z {∇ X −} T ) gives rise to a map {∇ , , is defined as a linear map from Z , = 0 and ) ) K } X, {∇ K E K K

), the subset of {∇ {− n (  E E E  ,Y K , t ( ( − . } E n X X n ). In order for this action to be a reasonable action of a 2-term  ] also knows extended notions of covariant derivative, torsion and on ∞ ∞ ( 0 0 1 2 k,j K + ∈ 10 ∇ {∇ E X C Y,X ( f X → C → C ) and extended tensors 2 ∈ T ) ) := 3 ) ) := K = Q ) as well as all and E K K for ⊂ } ( K E E ). The Poisson bracket now uniquely extends to a map ( ( E ) X ∞ j,k 0 ( K ,Y } − { K C ∇ E C T X X E X,Y,Z ( 3 4 fX ( ( 0 ∈ , ∈ ⊗ ⊗ T T t X,Y,Z,W X,Y X,Y,Z,W f : : ∞ {∇ loc ( 2 C Our formalism [ to the free tensor algebra T Q R R -algebra on a vector space, we require that { ∞ K  indeed tensorial up to terms which vanish on 5 Finite symmetries ofWe Double shall first Field discuss the Theory finitebefore symmetries of specializing local to Double Field the Theory case in of a general the context symplectic pre-N where which readily generalizes to extended tensors and satisfies for all derivative, we readily write down the extended torsion and extended Riemann tensors, Riemann tensor.covariant These derivative merely relythe on image the available pre-N We note that bothobvious extension conditions to are elements trivially of0. the satisfied tensor So if algebra) the with wealgebroids above consider extended functions tensors on L for all analogue for tensors of the condition ( ring which encodes in particularsubset the action of elements of an E JHEP05(2019)209 . M T loc + L , p , (5.4) (5.5) (5.7) (5.1) (5.2) of an M  loc 0 , θ Y L M M x ∂ , ∂x  } , which is the ]. That is, we -structure 2 , Z ν ,Y 10 ∞ M M   L N p f , ∂ X . ). We now want to M } M 2 θ ∂ M E ∂  ( ∂x = ∂x X,Y ∞  1 Q N = 0 C M θ = ∂ F } M = 0 (5.3) ⊂ µ θ {{ and set of extended tensors ∂ } } ,Y ∂p + + loc 0 } loc and ,Z ,Y } L = 0 (5.6) L = M } , f } p }} ,Z in F M ) ,Y M p , f ,Z µ } Y } M X ∂ N M ), which has local coordinates Y θ ∂x M p {Q  d ,X ( d ,Y , ∂ . The symplectic form and the Hamiltonian

M } R d M = θ )( ∧ {{ ( ), trivially satisfied, for example, for the trivial 2 p -structure )) 2 1 2 } K X d {{Q ,X – 27 – E M = M M ∞ ( θ X p R = M ,Y L 2 ( d ∂ Q} } Y ,..., } 2 ], this also requires that E M ( ) = , -structures have to satisfy the constraint (or section . Note that ∂ E {{{ { , ( 10 ∞ T ,X MN ∂x ,Y ,Y = 1 M η = = } ∞ L ) K ∈ 1 2 X f } C , f ( M Y {{Q -manifold {{{{Q Y 1 M + ∈ ,Z M θ Q µ )-structure. The Lie 2-algebra structure is given by the same to form a Leibniz algebra on the degree 0 elements Q} } ( θ M , F 2 2 ), and ) = p = ν ν d, d  d ,Y ( ] + {Q ) and } ∧ 1 2 = X O K X,Y 4.18b E ( ,X M {{ ( 2 loc + X,Y x . Explicitly, ν L ∞ = 0 Q} loc 0 , . As explained in [ = = [ C } L = d ). loc Y ⊂ ∈ ,Y ω L X ˆ }} loc 1 is the usual 4.20 L {{{{Q L , f ∈ MN {Q X,Y,Z f η , Second, we would like The generalized Lie derivative of Double Field Theory is simply the action of the -structure -structure read as {{Q ∞ ∞ again an analogueintroduced of above. the section condition trivially satisfied for the L for all which is the usual section conditionL on for all for generalized vectors impose two further restrictionsFirst on of our all, weThis want implies that that symmetries between symmetries have no effect on the action. extended tensors. On extendedfollowing vector derived fields, bracket: it is given by the D-bracket where derived bracket as in ( condition) ( extended vector fields in the Lie 2-algebra of symmetries on extended vector fields and We start from theconsider description the of symplectic local pre-N Doubleof Field degrees Theory 0, as 1 andQ given 2 in and [ index 5.1 Generalized Lie derivative and infinitesimal symmetries JHEP05(2019)209 (5.8) (5.9) (5.11) (5.10) (5.12) ) as again d } R ( ,F f , 2 ) E X K 1 L X. X } = Y, }} M M ) f ∂ ,F 2 } ∂ ,F ∂x M K ,X 2 }} } K ∂ 1 2 1 N X X X ∂ M ( X,Y ( X, 2 ,X ∂x ,X N M X M µ } ∂ ∂ ˆ ∂ N M 1 L ∂ = θ N K M {{Q θ ∂x Y = } ,X θ , − M } M − } Y + 2 θ N θ 1 . of degree 1, however, we have an ex- , {{Q X, f X M X − M ,F 2 , + ,X ˆ . ] p M } L K µ Y X 2 M 2 2 ∂ X ∂ K M E X , which is homomorphic to the Lie algebra N X 1 {{Q M ∂x 2 M ˆ , ∂ ,X X L M θ X 1 E ∂ { ∂x N ], where also a finite form was given to which ∂ = M − p M X + M N K [ } = 13 ∂ F, X – 28 – − θ }} − {{Q Y Y Y ) M M } K 2 2 2 ,F p ∂ M = = = − X ,F X θ ,Y M ˆ } } }} M } } ,X L Y } 2 2 M θ 1 1 M X M M θ ] + M X { X + ˆ ,X ,X ,X ∂ ( , x , θ , p L } } } = ] + R M M 2 = {Q p } X,Y , {{Q {{Q X + ,X ,X ,X M , } Y ,X , f ] ] 1 } F = = = [ 1 2 } 2 ] 1 : 2 X X {{Q {{Q {{Q X Y ,X [ ˆ ,X ,X ,X L ,X , L 1 X 1 , = = = ˆ 1 X L X [ ), which is easily verified using the derived bracket formulation of this [ X M M M 2 {{Q {{Q {{{Q {{Q { L ˆ θ p x L E [ ( = X X X = = = = = ˆ ˆ ˆ L L ∞ L f 1 F ] C X 2 ˆ of degree 0: L X ⊂ ˆ L f , loc 0 1 L X ˆ ∈ L To find the Lie algebra of general actions, we consider the Leibniz algebra formed by On generalized vectors, which are functions We note that the generalized Lie derivative acts as the ordinary Lie derivative on Now the action of the generalized Lie derivative, together with the commutator, forms [ 2 , , which is fully determined by its action on the coordinate functions. We have 1 2 A similar action was discussedwe already shall in return [ later.an The commutator interesting of form, two actions of the generalized Lie derivative has ν infinitesimal diffeomorphisms on the bodyof of infinitesimal diffeomorphisms on the body of pression differing from functions and the Lie algebra of actions on these functions is simply the Lie algebra of actions of equation. Since theto action be is a Hamiltonian Lie andfamiliar subalgebra degree from of preserving, the the we caselittle degree know of bit preserving that the harder symplectomorphisms this to Courant on has identify algebroid. this Lie Contrary subalgebra. to that case, however, it is a a Lie algebra, justencoded as in in the C-bracket, the i.e. case the of higher the product Courant algebroid. The structure constants are X JHEP05(2019)209 MN η (5.17) (5.16) (5.13) (5.14) (5.15) , = 0 )-transformation, )-transformation. } , . d, d d, d P ( ( ,F . θ } d via the identification o ) O K 2 } X 1 x ] and the finite transfor- ι -field are in the transition ( X ,X ,B P 13 + B } 1 } L α . M β B X α X d ∂ T ι α { Y + K 2 + + ι 1. Here, we used the variants of ]. KL = X X = d η − ˆ ,Y 2 L 18 ) ,X M } ) is a local | ≤ X β 0 x p X + x α ( L 1 F { X ( | M α N is the Hamiltonian of the homological X L + N ˜ ι x {{Q K 0 ∂ M B + d + ] + Q ,X T ∂T X 0 } B L N with a local and antisymmetric matrix. So we θ , α = , X,Y + 0 1 2 and {{Q – 29 – F ). In this picture, the last term describes the N α )-transformation. A clear description of the action + θ = [ {Q δB , where ) , ). N x ) := 5.16 + d + + ) ( d, d p } = 0 and β ( x α, β 5.1 N B B B ( } N M O + ,X ˜ M x M = = = -structure and thus ∂x ,F ∂ Y M x T 0 ( } p ∞ 2 ) has already been integrated in [ α B { = ˜ = = L ) due to the gauge transformation + M β ,X 0 X M M M θ } ˜ ˜ 5.11 ˆ -field corrections (i.e. the effects of the θ p ˜ x 7→ L + 1 = B is a contraction of and 1-forms Y 7→ 7→ 7→ ,X ( X B F B ) M M d M any 2 Q} θ p x , X,Y ,X , induces the local 1 X {{{Q ( MN without η R is an element of an F From the infinitesimal case, we know that the choice of diffeomorphism, together with additional term in e for vector fields vector field functions). The generalized LieLie derivative gives algebra an of action ofGeometry. diffeomorphisms the There and semidirect is, product local however, a ofis 1-forms second the action given on within by generalized ordinary differential the vectors geometry, in which first Generalized two terms in ( bracket from Generalized Geometry, because we are integrating an infinitesimalis degree-preserving part symplectomorphism of and the symplectic form, cf. ( the metric of the generalized Lie derivative is found by recalling the interpretation of the Dorfman where the first map is a diffeomorphism and 5.2 Finite symmetries A more general formmations of are ( given by Note that conclude that the actionmorphisms of whose the action generalized on Lie extendedNote tensors derivative that is forms the modified the same by Lie conclusion a algebra was local already of reached diffeo- in [ the section condition as well as the definition of the ordinary Lie derivative where JHEP05(2019)209 , + } }} X,Y (5.22) (5.23) (5.18) (5.21) (5.19) (5.20) ,B } β, B β ): + + Y 5.16 , and it is the { ) as follows: 0 ,Y , } } β δB α , α α d + 0 = + Y + ι ) generates the diffeo- 0 Y 2 ( − K 0 E B ,X , namely with the action ,X ( ) 0 B 0 β X Y. β M ∞ 0 X M ∗ M + L ∂ X, ∂ {{Q T {{Q = 0 on arbitrary tensors d ∈ C K M Y ⊕ + Y + ( Y ] + Y ι f X α } β = 0 and M M + . − β induces a subgroup of the symplec- + θ }} + TM X 2 B X,Y N Y ˆ + X∂ Y L + E Y θ ,B X { } M + + [ = M L X ,Y ∂ α 0 θ + } ) is mapped to e β β M + N β Y } + ∂ 0 + + β ,X X Y B + M 0 e Y Y + M ,X . Second, we also know that the action of the = Y Y – 30 – 0 ∂ 2 ( 0 = = 1 2 E − {{Q B 0 ,Y Y } β = Y {{Q + X for . regarded as functions and the second and third term α ˆ L } + M ) + 0 + Y ∂ Y δB 7−−−−→ β } X Y M ˆ δB L + β, B , α ,X X 0 Y 0 7−−−−→ Y + ( = = {Q B Y 0 X ˆ Y {{Q { L Y Y )e β, 0 X B 7−−−−→ + + X ˆ e + L L β β Y β + + . The only remaining candidate is thus the group of diffeomorphisms. { ). Here, the first term describes the action of infinitesimal diffeomor- 2 2 + -transformed doubled vectors as scalars on the doubled space may be E Y Y + E ( ) transforms as an ordinary section of B Y = = (1 + β ∞ 1 -field coordinate change, e ) = 0 + C B β Y ∈ ( + 0 B Y ( Regarding In Double Field Theory, we have a similar split of contributions as in ( 0 X,Y B e Third, this fits with thewhich strong section always condition, seemed incompatibleview. with covariance Fourth, from doubled a vectorsposite differential combine ways geometric vector under fields point diffeomorphisms. and of consistent 1-forms, “compromise.” The which scalar transform transformation in law op- can be regarded as a unusual, but our interpretationthe is action supported of by the themorphisms generalized following group Lie points. derivative on on thegeneralized First, functions base Lie we derivative space saw on of that arbitrarytomorphisms functions on on This transformation isanalogue readily of checked the in above situation the in case Generalized Geometry. We thus interpret the generalized Lie derivative as a transformation which amounts to the diffeomorphism for phisms on the generalized vectors describe the terms of the form e Thus, e of diffeomorphisms. but after a Explicitly, we have JHEP05(2019)209 ). in not B 2 . The 4.28 (5.27) (5.26) (5.24) (5.25) x Y , and
2 and carrying , 1 ξ ). .
) x 2 K 3 M ξ defined in ( E ) X ( 4 = 0 ) = 0 2 1 1 , ˆ
loc pr } X Y Y 1 L ∞ loc α Z , ... − ∂ Y α
3 )( + ∈ C − Y , ∂ is more involved. First of and f 1 } ... 4 M α 1 B Y ∂ ˆ X loc pr N + 1 ( L X,Y B k X X,Y M N α + K ) = ( ..., ∂ K + ( 1 f B ξ 3 B + 1 {{ ) and ) K ∂ ξ 3 K } } ) ; the necessary adaptions are rather N + )( Y 4 Y K 2 ) translate to B } Y 4.19 ,B } Y 1 2 ,Z E 1 2 2 4 -structures } } X ∂ 5.6 ,Z + ,Y X ∞ } ) + + − } ,Y L A with connective structure, we always have a M Y − 3 } Y,B M ,Y , f α 4 N 2 ) + ( X } X – 31 – N {{ ∂ τ -field modification from the transition function to 2 G ,X B X Y 1 2 d B 4 ) and ( , f Q} )( 2 4 B Y N 4 , p A ( ∂ Y + 3 p + X Y Y k 5.3 3 )( p α and } {Q p f M ∂ N + + + 1 2 + ( 2 1 ( 3 δ fibered over a common base manifold and closed 2-forms along the directions τ {{{ { ∂ G 3 + 4 ] + M {{ {{ Y,B θ θ 2 N ) = = { ) Y K 2 = = ( = 2 X Y } + Y } X,Y Y M M 1 K 1 ,Z θ θ Y E ,Y } X = [ and X = = = } }} 1 − − ,Y 2 Y } 2 X , f ,Y B X } X e 1 ,X 1 {Q , Y Y ,X ( ( Q} , j k {{Q {{Q + + = {{{{Q does not truncate, Y X B ˆ The generalized Lie derivative now acts as the semi-direct product of diffeomorphisms The relevant section conditions ( The situation for an arbitrary initial Kalb-Ramond field L -fields or, equivalently, gerbes 6 Discussion We would like to stressField that Theory nilmanifolds were as chosen base merelybetween spaces for two for pedagogical spaces a reasons. discussionB If of we global have Double a T-duality relation Both conditions are trivially satisfied for the on the correspondenceinvolved space in the T-duality. and where we used again the notation introduced in ( our framework. 5.3 Specialization to Let us brieflystraightforward. specify The the generalized Lie above derivative reads discussion as to but this expression converges: itproblem is simply is the rather usual matrix that exponentialGeneralized in multiplying Geometry Double which moves Field the Theory,the the homological vector analogue field of (and thus thenoncommutative into or transformation the nonassociative generalized e spaces, Lie which derivative) may cannot now be produce described without generalizing all, e JHEP05(2019)209 566 (6.1) Class. -field are , C agree. K Phys. Rept. ] is still missing. , 10 2 2 O  G C ]. The situation over the , which can be restricted to 36 ]. K , o } C 2 35 K X C ' ( SPIRE IN ][ 2 π 2 2 pr X which is the tensor product of the pullbacks w M o w K Double field theory: A pedagogical review × G – 32 –  K 1 M E ' X ]. is readily captured in our framework as the special ' Duality symmetric string and M-theory 20 := 027/0008495, International Mobility of Researchers at 1 arXiv:1305.1907 K SPIRE pr [ 1 π ), which permits any use, distribution and reproduction in IN ][ v w 1 K carrying a gerbe X G ! / capturing the global geometry of Double Field Theory: , not even in the local case. K CC-BY 4.0 K (2013) 163001 K E E This article is distributed under the terms of the Creative Commons 30 1 1 O  G C arXiv:1306.2643 . This gerbe comes with a Courant algebroid [ 2 G -manifold Q and Quant. Grav. (2014) 1 The extension to Exceptional Field Theory (EFT), however, is more problematic. The 1 G. Aldazabal, D. Marques and C. N´u˜nez, D.S. Berman and D.C. Thompson, Not involving noncommutative or nonassociative spaces. G 20 [1] [2] Attribution License ( any medium, provided the original author(s) and source are credited. References nology (COST) Action MP1405 QSPACE. Theproject research No. of CZ.02.2.69/0.0/0.0/16 AD wasCharles supported University. by OP RDE Open Access. higher analogue of Acknowledgments This work has been partially supported by the European Cooperation in Science and Tech- issue here is thatNote that a the clear analogues analogue ofmostly Courant of clear. algebroids the capturing They local the areto symmetries picture higher of accommodate versions developed the of both in Vinogradov M2-correspondence [ algebroids space, and containing however, shifted M5-brane is copies windings, much less cf. clear. 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