Canonical Formulation and Conserved Charges of Double Field Theory

Canonical formulation and conserved charges of double field theory

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Citation Naseer, Usman. “Canonical Formulation and Conserved Charges of Double Field Theory.” Journal of High Energy Physics 2015.10 (2015): n. pag.

As Published http://dx.doi.org/10.1007/JHEP10(2015)158

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/103389

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP10(2015)158 Springer August 8, 2015 October 2, 2015 : October 26, 2015 : It is shown that : Received Accepted 10.1007/JHEP10(2015)158 Published e obtained and applied to ssions for conserved energy doi: en. By including appropriate ts. The Poisson bracket algebra ng to the C-bracket. A systematic f Technology, Published for SISSA by . 3 1508.00844 The Authors. c Space-Time Symmetries, String Duality

We provide the canonical formulation of double field theory. , [email protected] Center for Theoretical Physics, MassachusettsCambridge, Institute MA o 02139, U.S.A. E-mail: double field theory Canonical formulation and conserved charges of Open Access Article funded by SCOAP Abstract: Usman Naseer ArXiv ePrint: and momentum of ana asymptotically number flat of doubled solutions. space-time ar Keywords: this dynamics is subject to primaryof and secondary secondary constraints constrain is shown toway close of on-shell writing accordi boundary integralsboundary in terms doubled in geometry the is giv double field theory Hamiltonian, expre JHEP10(2015)158 ) ) 6 8 9 1 6 29 32 38 39 12 13 15 16 22 37 37 12 15 22 d, d d, d ( ( O O rdinates, ‘winding’ co- written in terms of the ) T-duality symmetry in o-form are combined into so be written in an d, d tained after compactification ( int restricts the theory to live ed coordinate transformations’ s linearly under global O se transformations. Double field – 1 – -dimensional torus. In addition to the usual space-time coo d A.1 Conditions onA.2 gradient vector General result 5.1 Boundary terms 5.2 Boundary terms5.3 in the Hamiltonian Applications and conserved charges 3.2 Constraints 4.1 Generalities 4.2 Algebra of constraints 2.1 Generalized metric2.2 formulation Frame field2.3 formulation Space/time split of double field theory 3.1 Canonical momenta and the Hamiltonian ordinates are introduced. Thea metric ‘generalized and metric’. the Kalb-Ramond This tw generalized metric transform transformations. Gauge transformations of the fields can al covariant form and they canin be interpreted the as the doubled ‘generaliz generalized space-time. metric, is The thentheory manifestly action invariant is of under a the double restricted field theory. theory, The so-called strong constra Double field theory was developed to make manifest the 1 Introduction 6 Conclusions and outlook A Generalized Stokes’ theorem 5 Conserved charges and applications 4 Algebra of constraints 3 Canonical formulation of double field theory Contents 1 Introduction 2 Double field theory and its space/time split the low energy effectiveon field theory a limit of string theory ob JHEP10(2015)158 . - ]. ]. t d 48 10 (1.3) (1.2) (1.1) (1.5) (1.4) – , can 7 , µν g h. R is the Lie √ (3) N = N L g , denoted by Σ d + N t − e dilaton and the can be written as: i √ ) d kl ]. For reviews on this of double field theory. solutions of the strong h oordinates, this action terms of the generalized 47 N – ij er) formulation of general and in-Hilbert action, h 11 to boundary terms, on a 2 [ of bosonic string theory [ − L ] the Ricci scalar. The space- nction and the shift vector as N , s: . ) kl µν e the lapse function and shift vector ij 3 g h and dilaton , h [ t − L f constant time 2 , ∂ MN , ] R = ij tion before we discuss the canonical )( H MN h µν ij t ) takes the following form: ij g = 1 d, H [ ∂ h The metric on full space-time, ( i 1.2 N ) 1 R . gR ij i d ] and earlier ideas can be found in [ on the space-like hyper-surface by introduc- 2 − h , g 6 i N − L . − – i √ N ij N 1 ij x h – 2 – N where 4 h = t X e d − L d ∂ , 0 2 ( i ij Z d g i h kl t = h ), the action ( = Z ∂ t, x ( ij ij i 0 = h h EH , h 1 4 S − i = 1 µ N − DFT , x N S , g h 2 N √ x − 3 j ]. N dtd 50 i – N Z 48 denotes the Ricci scalar on the space-like hyper-surface an ij = h is the generalized scalar curvature which is a function of th R the metric on four dimensional space-time and = EH (3) R -dimensional subspace of the doubled space-time. Different µν S 00 d g g In the case of general relativity, one starts with the Einste The most geometrical formulation of double field theory is in Our aim in this paper is to provide the canonical formulation Double field theory was developed in [ By a slight abuse of notation, we will use same symbols to denot 1 Space-time coordinates are split into time and space parts a One can now define a purely spatial metric ing the lapse function N and the shift vector with generalized metric. Byreduces to demanding the no low energy dependence effective action on of the the NS-NS dual sector c where metric and the dilaton. The action for double field theory, up derivative with respect to the shift vector then be expressed in termsfollows: of the spatial metric, the lapse fu where dimensional doubled space with generalized metric In terms of the variables (N formulation for double field theory. Due to many similaritiesrelativity, it with is instructive the to ADM(Arnowitt-Deser-Misn briefly review this formula Further developments of double fieldsubject theory see are [ discussed in time manifold is foliated into space-like hyper-surfaces o for double field theory. on a constraint are then related by T-duality. JHEP10(2015)158 (1.8) (1.6) (1.7) (1.11) (1.10) i metric Σ of the ∂ , i re’ secondary ij π ) = 0. Precise j x n onserved charges ) (1.9) ( i ) is zero. However, i λ N j C 1.6 ρ∂ that can be computed + 2 . under time evolution is i ar the ADM energy and lead to a non-zero value − ij a conjugate to the lapse C e primary and secondary i h ρ kj procedure is the emergence j h ly flat. This is done by iden- s, i.e., their invariance under o in the on-shell Hamiltonian i constraints: x β λ∂ ∂ ltonian ( Σ, one has to add appropriate 3 ( cal constant, ADM energy and the primary constraints arising k ∂ d ) = 0 and ]. The Hamiltonian for general n ij x gebra of the ‘smeared’ constraints. h ij ( 52 Z h , , ≡ B ≡ i − is function of space-time coordinates. ]) ] 51 γ β r N kj h i ∂ ] and it is: , i n 53 ]] λ where kj i 2 – 3 – and C [ ∂ h i , β , β 1 ] N β − . In general γ h = (B [N] + C [ σ ∞ H x λB, √ 3 = C [[ = B = C [ x ]: d 2 } } } d 2 is the metric on the two dimensional boundary ] ] ] 54 . ADM momentum is obtained by setting the lapse function , 2 Z ρ is the unit normal vector to the boundary. β β Z i ≡ B[ n C[ ), one can compute canonical momenta and perform a Legendre = 1 ] = , C[ , ] , λ ] → ∞ approaches i ¯ ] 1.5 λ λ , 1 r r in the metric such that the curved metric reduces to Minkowsk bdy B[ j ¯ β i ] are the so called ‘smeared’ constraints built out of the ‘ba ¯ B[ r B[ H σ β { { C[ = { ). 1.5 is the canonical momentum conjugate to the metric ] and C [ on-shell ] denotes the Lie bracket of vector fields. λ ij H , π From the action ( Since any solution of general relativity has to satisfy thes Now, one can define notions of conserved charges, in particul It can be shown that the invariance of secondary constraints boundary terms to theof Hamiltonian. the These on-shell Hamiltonian boundary [ terms can where easily from ( if the space-like hyper-surface has a non trivial boundary relativity can then be expressed in terms of these secondary transform to compute the Hamiltonian.of The primary key aspect and offunction secondary this and constraints. the shiftin The vector the vanish canonical ADM identically. formalism. moment These The are time consistency evolution, of primary leads constraint to two secondary constraints, constraints, we conclude that the on-shell value of the Hami space-like hyper-surface. expressions for these constraints can be found in [ where [ where B [ This algebra is computed in great detail in [ equivalent to the on-shell closure of the Poisson bracket al constraints as follows: ADM energy is then obtained byand setting taking the shift the vector limit to zer tifying a parameter momentum. In general relativity, withmomentum are vanishing defined cosmologi for space-times which are asymptotical when the parameter to zero. The ADM energy and momentum can then be identified as c JHEP10(2015)158 . - ) d M M D , 1.5 X N . ) (1.12) (1.13) (1.15) (1.14) ˜ t al vari- MN H such that d, d ( 2 − R e oncepts. d 2 ) = 0. The precise 1). This split allows , − e X = N − ( der generalized diffeo- b d ur starting point is the . The ADM energy and kj D 2 eral relativity. We have M tion and the shift vector h + N menta and doing the Leg- into the following: − i C ), the action of the double refer the interested reader ual time coordinate = e ld theory. In that process, ∂ ˆ -dimensional hyper-surface. N k d d → ∞ ˆ ) upon proper truncation. n MN M ativity given in equation ( r alism. In particular the notion ij MN b ance of these constraints under H H 1.5 t h H t − , d, D -dimensional doubled hyper-surface. M kj ) = 0 and , d h i MN N X . We split the coordinates on the 2 ( , ∂ . i M H D t ij n 2 B π D , kj j 8 1 h ) reduces to ( ˜ t, t, X ··· σn + – 4 – , σ 2 √ 2 1.15 ) , = x √ d 2 t x ˆ d M 2 D -dimensional doubled space, with generalized metric = 1 d X Z D 4 ( ˆ Z M d →∞ denoting the generalized lie derivative with respect to 2 r →∞ − r N e b 1 L = 2 lim = lim − -dimensional doubled hyper-surface ( respectively) vanish. This puts constraints on the dynamic -dimensional manifold is redefined as , where d N b d D M − with i ADM ADM P E N X b L d and Π 2 of the 2 − d N t b d ∂ Z , the induced generalized metric on the 2 , the generalized shift vector. ≡ dt ] for a detailed exposition of the ADM formalism and related c t M MN 54 Z D and the dilaton – morphisms of the doubled hyper-surface. N, the generalized lapse function which behaves as scalar un = ˆ Following the usual procedure of computing the canonical mo For the case of double field theory, a similar story unfolds. O N 51 •H •N • ˆ S M b ables of the theory,time called primary evolution constraints. leads to The secondary invari constraints, This completes our quickglossed review over of a the lot ofof ADM the asymptotic formulation details flatness of and needs gen to intricacies to of [ be this handled form very carefully. We endre transform, we canwe obtain find the that Hamiltonian the for(denoted canonical double by momenta Π fie conjugate to the lapse func are obvious. Indeed one can show that ( momentum are given by: associated with time and space translations respectively a where Similarities between this action and the one for general rel H double field theory action on a 2 are coordinates on the 2 The dilaton behaves as a density with respect to diffeomorphisms of the 2 an ADM-like decomposition of the full generalized metric field theory reads: dimensional manifold into temporal and spatial parts as: In terms of this new set of dynamical variables (N and demand that fields and parameters are independent of the d JHEP10(2015)158 ) aint, 3.12 (1.20) (1.19) (1.18) (1.16) (1.21) (1.17) on the ). Let d 2 , MN H grals are well , ) λ , as follows: N ) does not contain red’ constraints by eneral relativity, we M on space-like hyper- ρ∂ ξ , ij 1.17 − ary then the expression h M . . The importance of the undary terms. The full dary terms, can then be ρ raints are also preserved e full generalized metric and C N 2) has to satisfy primary and bdy λ nts under time evolution is M tion ( he smeared constraints, on- ue of the Hamiltonian. The y flat. To make precise the λ∂ H y. These conserved quantities s is given in equations ( ↔ ( eneralized metric ) vanishes on-shell. However, if metric + a of double field theory and that (1 X ξ d MN H 2 − 1.17 d . = P 2 ] and it is shown that it closes on-shell, ξ Z ≡ H N 4 M [ -dimensional doubled hyper-surface and ≡ M DFT ∂ d ). ] C χ P ξ H in such a way that it assumes the flat form 1 [ ξ 5.52 1 2 C , P , – 5 – [N] + − ] λ , C B P where ] M 2 ∂ 2 ξ B = P P , ξ , ξ 1 ∂ ] H is the Minkowski-type metric of signature (2 ξ − P 1 χ [ [[ X λ ξ ˆ d N 2 C B C ˆ = M ) against suitable test functions d b δ = = = X M C Z ] ( . } } } 2 ] ] ] ≡ 2 M ξ ρ , ξ [ [ ξ ] 1 C [ λ 1 be the coordinates on its boundary. Due to the strong constr ξ C [ B [ → ∞ be the coordinates on the 2 C , , − , ] ] B d ] d λ λ P 1 2 [ [ 2 ) and ξ , are smooth functions of coordinates such that the above inte , [ B B X { { ( M C ··· ξ ··· { , denotes the C-bracket defined as: , B C ) respectively. We introduce the convenient notion of ‘smea ] = 1 and = 1 , λ -dimensional doubled hyper-surface has a non trivial bound depends on a function of coordinates ¯ M in the limit 3.20 M d ˆ Consistency of the theory requires that the secondary const Motivated by the constructions of ADM energy and momenta in g , , N ˆ N ˆ ¯ M M M ˆ M b b X δ doubled hyper-surface as expected. The Hamiltonian in equa under time evolution.equivalent to The the invariance closure ofshell. of secondary This the algebra constrai is Poisson computed bracket in algebra detail in of section t H where [ defined. The Hamiltonian ofwritten as: the double field theory, up to boun Y surface are replaced by the generalized Lie bracket and the g where Again we see the similaritiesof between the general constraint relativity. algebr In particular the Lie bracket and the notion of flatness for a doubled space-time, we assume that th introduce conserved energy and momenta inare double field defined theor for doubled space-times which are asymptoticall boundary terms. Since anysecondary constraint, solution the of bulk Hamiltonian the of doublethe equation ( field 2 theory as required. The constraint algebra is given by: and ( integrating boundary terms is evidentboundary because Hamiltonian they is give given in the equation on-shell ( val form of these constraints in terms of the dynamical variable Hamiltonian of the double field theory is then, for Hamiltonian needs to be modified by adding appropriate bo JHEP10(2015)158 M (1.24) (1.23) (1.22) can only oral and S -dimensional . it is equal to d , A , we compute the LP d 5 Π H momentum, one can P M ∂ N − 1 4 eld theory monopole and ection he frame field and obtain lowing form: erwards we briefly review e generalized metric and a s bosonic sector of closed ld. Finally we discuss how d a of secondary constraints y reviewing the formulation − It does so by introducing an P . Also, if the boundary of heory and re-write its action eory are also computed there. ∂ ly depend on a L o spatial and temporal parts. ns of double field theory. We . n and define conserved charges. M 6 N LP l in the appendix H we briefly review the formulation KL 4 Π 2 . L N MK ], and confirm the physical interpretation d S 2 H , is devoted to the discussion of constraints 2 56 − ¯ M 4 e ,

Y ′ ′ 55 – 6 – ∂X ∂X ∂X ∂X =

) = constant, then it can be shown that . With these considerations in mind, expressions for M 1 X ′ Y Y ( 1 1 X M − − S d d d d 1 1 we present the canonical formulation of double field theory M M ∂ ∂ 3 Z Z -dimensional doubled hyper-surface →∞ →∞ d P P = lim = lim are the coordinates adapted to the boundary, i.e., ) by splitting the space-time of double field theory into temp E M of the 2 M ′ P 1 1.15 is the gradient vector which characterizes the boundary and X M M ). N X ( This paper is organized as follows. In section Using the expressions obtained for the conserved energy and S M Properties of gradient vectors are discussed in great detai where ∂ re-definition of the dilaton. 2.1 Generalized metricDouble formulation field theorystring is theory an which makes effective the T-duality description symmetry of manifest. the massles In this section we reviewin important a facts form about better double suitedof field for double t our field later computations. theorythe formulation in We start of terms b double ofto field the theory split in generalized the terms metric. space-timeThis of of split the Aft is double frame accompanied fie field by theory an explicitly ADM-like decomposition int of th Conserved charges for some known solutionsFinally of we double conclude field th and summarize our results in section 2 Double field theory and its space/time split spatial parts. Inand compute section the bulkarising Hamiltonian. in Section the canonicalunder formulation. Poisson brackets We andboundary compute contribution show to the the that double algebr field it theory Hamiltonia closes on-shell. In s given there for various free parameters. of double field theorythe in action terms ( of the generalized metric and t depend on the ‘allowed’ sub-space for a particular solution of double field theory, fields can on sub-space the conserved energy and the conserved momentum take the fol is characterized by the constraint apply our formulae togeneralized pp-wave compute solutions conserved discussed in charges [ for double fi compute conserved charges associated with specific solutio JHEP10(2015)158 M . ] as and X 6 (2.5) (2.6) (2.2) (2.1) (2.9) (2.4) (2.7) (2.8) (2.3) d d ij vature g N N ∂ ∂ . ) and the M ∂ MN 2.5 d H N MN ∂ M MK H ∂ 2 H MN L + 4 − ∂ H . d 4 ace-time metric . KL N . MK − MN H H d∂ ! MN the generalized metric [ H j N L η , i M 0 elation and it reads, ∂ N ∂ MN δ ) , ∂ ∂ = H j ! i KL M 0 MN N δ MN lj MN ∂ H ∂ b QN H

H H − N 2 1 kl H d kj ∂ g ) invariant metric and its inverse de- 2 d, = d . b ( +4 − ) tensor given by: − φ ik N PQ ik . 2 e b η MN g d, d − ( d∂ ! MN KL d, d − of the eﬀective action by: i i − H DFT ( ge ˜ M M x x O MP H 2 1 ij φ L ∂ ∂ − O g

N H − ∂ √ -dimensional generalized coordinate vector – 7 – X = kj d MN d ) + ij g , η 2 = KL KL H g M d ik d ! , 4 H b H 2 MN j X i N Z − 0 N

M − M H δ e ∂ ∂ δ d = j = d, i 0 , conjugate to the winding modes of the string. The total ( N = KL δ x ∂ MN R

H MN d H DFT PN 2 M 8 1 S MN = H − into a symmetric ∂ H e H d ij 2 M b MN MP MN − ∂ η ) = e H H 1 8 +4 coordinates, ˜ MN ) = d H ) can also be expressed in terms of the generalized scalar cur ) = is raised and lowered with the is related to the scalar dilaton d, MN 2.3 ( d MN H M ) up to some boundary terms. Indeed the Lagrangian density ( H d, DFT ( d, MN L ( is called the generalized metric which combines the usual sp coordinates are combined into the 2 H R DFT d d, L ( MN The dilaton and it satisfies following constraints. fined as Now the action of double field theory can be written in terms of the Kalb-Ramond field The index as follows. H The action ( R generalized scalar curvature are related as in The generalized curvature scalar can be obtained from this r where additional set of of 2 JHEP10(2015)158 , ··· -field (2.10) (2.12) (2.13) (2.16) (2.15) (2.14) (2.11) b a mani- ressed as: M,N, indices and ations of , ] i.e., n was explained 5 P -dimensional sub- ) is an infinitesimal d N M H ζ M ( o be a proper element of ζ up. However, the failure to eory has a gauge symmetry P eomorphisms of the doubled ong constraint, which has its y. Such a formalism was first ∂ etric can then be defined as: ralized Lie derivative, i.e., e C-bracket [ , where . We call the indices − M M t generate a gauge transformation ry to live on a P A ζ ζ E . , , , − , M C ( ) covariant fashion. Under a general- ] AB ∂ 2 AB ! M η ,ξ H ab 1 ) = 0 0 X d, d ξ B h ( [ ). This algebra does not satisfy the Jacobi = N b + 2 L = ··· O E P ab ( 0 − ζ A h 1.21 MN M -duality symmetry made manifest in the double M P N ′ – 8 – MN η

∂ E ∂ T ] = N X H 2 1 2 = B M = ξ P ∂ E − ∂ → , δ AB P d M 1 ξ A ζ MN MN P H M δ [ η E ∂ H = X P ]. ‘flat’ indices. The frame field is subject to a tangent space ζ , 43 MN = H d ··· ζ ζ b b L L = = A, B, ) is written in terms of covariant quantities and hence it has dimensional doubled space. The strong constraint can be exp d ζ ) symmetry. This is the d δ MN 2.3 H ) metric with ‘flat’ indices takes the same form as with curved d, d ] and its connection with the generalized metric formulatio ζ ( δ 1 ]. Generalized diffeomorphisms combine the gauge transform d, d O 23 ( ’ contains any arbitrary field, parameter or their product. , O ··· 21 ) as has been done in [ is the ‘flat’ generalized metric given by: ]. In this formalism, one works with a frame field The action ( Double field theory is a restricted theory. The so-called str 11 d, d ( AB where ‘ space of the full 2 The algebra of gauge transformations is characterized by th ‘curved’ indices and where the C-bracket is defined in equation ( H it is used to raise and lower ‘flat’ indices. The generalized m O gauge group. Here, it is convenient to choose the frame field t parameter, the transformation of fields is generated by gene Here we review the frame fieldprovided formalism in for [ double field theor identity so generalized diffeomorphismssatisfy do the not Jacobi identity form is of a awhen Lie trivial acting type gro and on it fields. does no 2.2 Frame field formulation fest, global origins in the level matching condition, restricts the theo i.e., the and the diffeomorphisms of the metric in an field theory. Apart fromwhich this can global be symmetry, interpreted double asspace field a [ symmetry th under generalized diff ized coordinate transformation in [ JHEP10(2015)158 ˆ N ˆ M . b being H (2.19) (2.17) (2.18) (2.21) (2.20) (2.22) (2.23) B ab compact h Ω n A Ω , can be built − A AB D 1) with H -dimensional doubled , he frame field, we D + ··· , 1 BDF ations but its completely , is not doubled, i.e., fields le field theory can be ex- 1 Ω -dimensional manifold can eory by splitting the full, ial coordinates are doubled. , − , D ACE tly. The basic idea is to write 2 M A : Ω . , . , subsection. On 2 non-compact and 0 ¯ 0 EF NC , d = diag ( , n 2 0 ¯ 0 H E − ab . N e h B CD    = M E = , A H ] M ˆ ˆ ˜ t t M M A E ∂ AB X ABC M H [    M A – 9 – f is split as follows: ∂ 1 E = 12 d 2 ], which we follow closely here. We start with a ˆ ˆ A M e M = = 3 − M 1. Coordinates on the 2 20 , , − X ¯ 0 0 − = , , BCD ABC ABC ¯ 0 D 0 A f Ω Ω = Ω d = = CD A ˆ , transforms as a scalar. Another scalar object, Ω ˆ -dimensional doubled space with the generalized metric A M Ω D ABC AB , with H , d 1 4 -dimensional Minkowski metric, . The ‘hatted’ index d b ··· d d , 2 2 ¯ 0 to differentiate between ‘flat’ and ‘curved’ time. − , e is the = = 1 ab M h is not well behaved under generalized coordinate transform DFT In order to write the action of double field theory in terms of t L Note a slight departure from the notation used in the previous 2 ABC In terms of thesepressed as objects, the Lagrangian density of the doub with the help of the dilaton and the frame field as follows: anti-symmetric part, Ω f introduce generalized coefficients of anholonomy as follows its inverse. and the dilaton double field theory on 2 where doubled space-time into temporalan and action spatial for parts double explici are field independent of theory the wherein dualSuch the time a coordinate time re-writing and coordinate of only double the field spat theory action, with This concludes our review of double field2.3 theory. Space/time splitIn of double this field subsection, theory we re-write the action for double field th where then be expressed as: The ‘flat’ indices are also split in the similar fashion, space we use ‘hatted’ fields and indices. coordinates has been performed in [ where we use JHEP10(2015)158 , M N MN takes (2.30) (2.29) (2.24) (2.31) (2.25) (2.27) (2.26) (2.28) t as the ˆ N , ˆ    M M EE , b N η . K N ˆ B    N formalism of N = d b Ω M M 2 ˆ A − N NK N N b e Ω M 2 1 A ˆ ˆ MN H N B − − M ˆ ˆ 2 E ime coordinate, i.e., A A H N N + − b coordinates and b E H − y via a straightforward N − N + ˆ ry in the frame field for- N M ˆ 1 F in the Lorentz gauge fixed ∂ α MN − ˆ b 0 d D , − 2 ˆ ˆ H N B -hatted’ objects to this split. M ˆ e ) invariant metric ˆ B A b ˆ Ω A K b is frame field is indeed a proper − ˆ E b E η K ˆ N C , D,D ˆ = = , N A P and all other objects are defined in ( ˆ b K ˆ    K Ω A N ˆ    N O ˆ M has the following form: F K b ˆ ˆ N Ω A N M ˆ AB E N 0 0 1 ˆ b b η K B PK N E δ MN b A ˆ − ˆ H A N η ˆ H 1 0 E B ˆ b N D H b MK ˆ 1 2 − E C + ˆ b H 2 − 1 0 0 M H ˆ 0 0 0 α A 0 0 0 11 0 0 0 ˆ N – 10 – B −    + b ˆ E A       − = b M H = K = N    1 , ˆ 12 ˆ N N α B M M M ˆ ˆ K A    0 M ¯ 0 A − − b η b b b b E E H E , ˆ N N N ) invariant metric. Note that the ‘flat’ generalized metric D ] 0 0 ˆ MN ˆ C α C b b b ˆ H H ˆ 1 2 0 B 0 0 H N d, d ¯ 0 ¯ 0 A ( b b ], we now give the frame field Ω − E 0 b b , b 0 E E E ˆ ˆ D 0 ˆ 00 N O B 20 M ˆ ˆ B ˆ M C 2 b b A b H 1) invariant. H ˆ b H − 0 A E 0 0 b N ¯ 0 H ¯ 0 ], we identify N as the generalized lapse function and A α 2 d, ˆ b 0 N Ω 0 ˆ M b b ( B 0 00 b M E E − ˆ ˆ E B ∂ N − 51 b b b ˆ H H H N O b A ˆ    E ˆ M A b ˆ -dimensional identity matrix. The [ A       H ˆ × d M b = E 1 4 b E = = 1) ˆ is the frame field for the induced generalized metric, ) covariant vector. Due to obvious similarities with the ADM is the usual M ˆ is 2 b ˆ d A = = 3 d, N 2 ( b ˆ M E ˆ ˆ d, d M − C N A ( ˆ MN e O AB ) element. The generalized metric can be computed explicitl b B ˆ H E M dimensional doubled hyper-surface. N is scalar function of δ η ˆ ) = 0. Following [ A O is b = H b d Ω With this split, the Lagrangian density for double field theo To proceed, we demand that fields are independent of the dual t ˆ ··· B D,D ( ˆ L A ( ˜ t b ∂ ∂ is an calculation and one obtains, general relativity [ generalized shift vector. AO short calculation shows that th on 2 malism can be written as: form as follows: where all the ‘hatted’In objects particular are we have proper the adaptations frame of field ‘un denoted by where terms of it as before, i.e., where the form: The generalized metric with the ‘flat’ indices, where H JHEP10(2015)158 and (2.35) (2.37) (2.40) (2.36) (2.32) (2.33) (2.39) (2.38) , 1 ABC b N (2.34) − M -dimensional .Ω ) ∂ . d ) M M N A MN N N E on 2 ∂ MN H 1 N − H wing non-zero coeffi- d, M e field given in equa- ( N A M d, ∂ ( E R , − d is a total derivative term gging these coefficients of R 2 A can be computed easily by MN d − 1 2 d e b rms for the dilaton and the ). After a short computation H ry term. We ignore this term − N = Ω e ∂ + 2.5 MN A H b Ω + N MN , N = N MN ), and , H N + 1-dimensional space-time (up to a , H 4 ∂ t 2.9 d L M , . N D B − MN d . N 2 . We conclude this section by giving the M E X H log (N) b t L − t ∂ d MN 1 2 2 e MN D 5.1 D − H d, t H − t t MN MN dtd ∂ MC D N d = N – 11 – MN D H H ∂ 8 1 1 E b d , Z d ≡ H 2 1 2 − t t d − − + − = 2 D e e D DFT 2 − 1 1 8 ) S e − L d = N = 2N logN in the expression ( t N + log (N) -dimensional doubled hyper-surface. N ¯ 0 1 2 D d 2 1 2 b ) − Ω BC behaves as a scalar density with respect to the generalized − d M 4 ( ). After some algebra, one finds that: ¯ 0 t − ∂ d d b Ω D d − d 2 2.24 MN , 4 ( = − . We also re-define the dilaton according to: e H N with 1 K 1 , d b d DFT M − 2 N ∂ − L N K e M ) is given precisely by equation ( 1 + − , C N − 2 E log (N) = 1 − N MN 1 2 − ABC N − H 2 1 L is the generalized Lie derivative with respect to the vector − = d, is a differential operator defined as ). After a straightforward computation, we obtain the follo d ( = Ω = N = N t b L C L R D α ¯ 0 2.29 ¯ 0 Let us now evaluate the double field theory action for the fram ABC corresponds to coefficients of anholonomy for the frame field b Ω DFT b Ω A L tion ( where doubled hyper-surface. We cananholonomy in now equation evaluate ( the action by plu coordinate transformations on 2 where where In the above expression,generalized metric. first The two last terms term, provide ‘kinetic’ te Ω This definition is such that final form of the action of double field theory on 2 one finds that: cients of anholonomy: replacing the dilaton given by: boundary term). where where when included in the action,here this and will correspond will to come a back bounda to it in the section JHEP10(2015)158 , he ) ical (3.6) (3.7) (3.8) (3.9) (3.3) (3.5) (3.1) (3.4) (3.2) MN . These H . We de- d, ( MN R MN Π d 2 , H − d e Π N , , d, − M M , xercise to compute the Π . N , MN ensity can be written as: , N unction and the shift vector H orm the Legendre transform H nd the secondary constraints le field theory starting from MN -dimensional doubled hyper- N ints at the level of equations , Π d N − H t b L L D − d MN 2 d, MN ) are , t H − t . e D MN M ∂ 1 d + Π C − 2 H H 2.40 t are to be written in terms of the canon- d − N M ∂ MN e X 4 1 N 1 N d b L − 2 − MN MN , d + Π d + H = 8N d , – 12 – t Z B = 0 − + Π ∂ + Π d = d = 0 = and MN t M Π d = N ∂ L H N N d H d L d δ Π t t t t L δ L d δ H δ X δ∂ δ∂ δ∂ δ∂ Π d = Π 2 1 = = = = 32 d dtd H N + M Π Π MN Π Z ). We follow the usual procedure of computing the canonical ). A short computation yields the following expression for t Π MN = Π 3.4 2.40 S ) can be written in terms of the Hamiltonian density and canon ), ( MN 2.39 Π 3.3 d 2 e 2N − = The action ( H and the Hamiltonian can be obtained by integrating over 2 surface, i.e., It will be shown in the next subsection that the Hamiltonian d canonical momenta can be computed easily and one obtains: note their canonically conjugate momenta respectively as variables as follows: Hamiltonian density 3.1 Canonical momentaDynamical and the variables Hamiltonian in the Lagrangian density ( momenta corresponding to dynamical variablesto compute and the then Hamiltonian perf density.arising We also in derive the primary a canonical formalism. ical momenta ( where the time derivatives of fields In this section weLagrangian present density the ( canonical formulation of doub 3 Canonical formulation of double field theory of motions whichHamiltonian will density be by performing discussed the later. Legendre transform. It is now a trivial e Note that the canonical momentaare corresponding constrained to to the be lapse zero. f This will lead to further constra JHEP10(2015)158 M as and N d (3.12) (3.15) (3.10) (3.11) (3.13) (3.14) , MN H d Π stency of the d Π 1 32 . + d ) are evaluated at the Π t vector. Treating esponding to the lapse X MN onstraints arising in the ( M erms of the Hamiltonian, Π ], they are called ‘primary’ ) implies that: grange multipliers and are N traint. This happens when B , need to look at the part of tten as: cy condition implies that: 53 3.1 Lie derivative terms: MN ay be possible in special cases mposing primary constraints. M c [ t vector. The shift vector ··· Π ∂ n leads to further constraints on dary constraint: primary constraint do not change 2 1 + d 2 − -dimensional doubled hyper-surface, e d 2 MN d − H Π ) . N . . M b L ∂ . MN = 0 , the induced generalized metric = 0 2 1 = 0 d H MN N ) requires the following to hold. M + MN – 13 – N d, Π H H d ( t δ N 3.2 . This form of the action makes manifest the fact δ δ + Π ∂ δ M R d ∂ d d 2 MN N and Π − b Π L d e d M and Π ≡ − = Π N ) d X = H ( ) are no longer definitions of the canonical momenta but they d B N 3.4 b L d depend on the dilaton -dimensional doubled hyper-surface. Similarly, the consi Π d M C where ) and ( , 3.3 and on the 2 B ) = 0 X X Consistency of the first primary constraint in equation ( ( B point where fields on the right hand side of the defining equation of second primary constraint in equation ( the dynamical fields, known asthat ‘secondary this constraints’. consistency It condition m the does time not derivative lead of toHowever, any we primary new will constraints cons see vanishescanonical that after formulation this of i double is field not theory. the case for the primary c constraints. Consistency of the theoryunder requires that time these evolution. In general this consistency conditio become equations of motion for Π that the lapse functionnot and dynamical the fields. shift vectorequations appear Also, ( only in as this La formulation of action in t their canonical momenta, Π Using Hamilton equation of motion we see that this consisten 3.2 Constraints In the lastfunction subsection and the we shift saw vector vanish. that In the the language of canonical Dira momenta corr where A straightforward calculation leads to the following secon To derive the secondarythe constraint Hamilonian associated density with which involves this, the we generalized shif appears in the Hamiltonian density through the generalized a density under generalized diffeomorphisms on the 2 the term involving the dilaton and the shift vector can be wri where we have omitted the terms which do not depend on the shif JHEP10(2015)158 ) ) ), he )it , 2 3.4 ) are 3.14 3.18 b (3.16) (3.22) (3.19) (3.21) (3.17) (3.20) (3.18) , 3.20 + d 3.20 Π ues for the K ) and ( ∂ ) and ( 1 2 NK 3.3 NK Π + Π 3.12 . ndary term in the MN MN NK Π NK − H nclude this term in the . − H MN Π clude our discussion here ide of equation ( is zero. However, we will MN H nstraints must be satisfied NK n equations ( K NK MN ctions. Using the expression ty takes the following form: an density can be written as: H ∂ NK H − H . Π ts. However, we will see in the − H ey should not change under time MN MN ly satisfied and one does not need MN M NK Π , the second term in equation ( MN Π C , H NK + Π M K M MN H d ∂ M − H C N MN 2 ∂ Π 2 M Π − K + MN NK ∂ − N Π and Π 1 2 B M H + ∂ N K + 2 MN – 14 – MN MN B d N H − H Π K X K d ∂ d ∂ + 4 2 = N d ) and ignoring the total derivative term, it is easy to K which ensures that the total derivative term ( K d d ∂ Π d d N Π MN H 2 Z 3.13 K − Π M e = M + Π N ∂ K H ) and the defining equations for constraints ( −N = . These secondary constraints are also required to satisfy t +2 N MN ) and ( X 3.6 M H = ∂ MN K 1 4 3.18 ∂ where H MN N = b that inclusion of boundary terms provide finite non-zero val MN L 2 H , b Π N b L MN 5.1 ≡ ) = 0 ) MN + Π is a total derivative term which, again, corresponds to a bou X X Π ( ( d 2 b K K N b C C L d Using equations ( Π same consistency condition as primaryevolution. constraint, i.e., In th principle,next this section can that lead these to consistencyto conditions further impose are constrain any trivial moreby constraints writing on the the dynamical Hamiltonianfor fields. in Hamiltonian terms We density con of ( these constraints fun So, the shift vector dependent term in the Hamiltonian densi Note that for anyand solution hence of the the on-shellsee double value in field of section theory, the these Hamiltonian co as given here Hamiltonian. where, Hamiltonian. is easy to see that up to total derivative terms the Hamiltoni to be evaluated at point transforms as a density.Hamiltonian to This write down fact a will boundary be term. importantderive when the we following costraint: i there is an implicit factor of can be written as: so that up to boundary terms, the Hamiltonian becomes: and as before, fields and canonical momenta on the right hand s Notice that in the definitons of the canonical momenta given i Similarly, using the symmetric nature of JHEP10(2015)158 ) is and (4.7) (4.6) (4.5) (4.4) (4.3) (4.1) (4.2) X . ( } ) M , ξ Y } ( ) Y M ( C , M ) C , X ( ) N X ( , C nds here and secondary { ) B ore constraints. We start under time evolution, we is well defined. nstrate that they are zero enta. Any Poisson bracket Y M { nts under time evolution is ritten as: bing the general method of ute the Poisson brackets in ion between fields and their d two secondary constraints. at it closes on-shell. M . enta can be computed using N − ) as follows: + N )) involve Dirac deltas and their , X to handle under an integration. , } X ( ) g the Hamiltonian is of particular + . ) ( ) X gebra of these constraints. Finally δ } ( } ), is given by: Y Y ) X M ) ( H ( C X − Y L C N , ( , ( δ ) B ) ) B , X ) M F ( X B X ) K ( X ( , ( ( δ δ X X ) ( ( M F B = = X { N ( } } C ) ) X ξ X λ – 15 – B { d d Y Y ) = { 2 2 ) ( ( ) d d X d Y ( Y Π KL Z Z , N( F Π ) N( ≡ , d Y dt ) ] ] = X d Y ( ξ λ 2 d [ [ X d 2 d ( ) is an arbitrary function of coordinates such that the inte- { d C B Z X -dimensional Dirac delta distribution. The lapse function Z ( MN = d ) first and then impose the constraints. It is easy to see that λ = } } {H H 4.5 , H , N C B { { ) is the 2 ) and ( Y ) = ) = − 4.4 X X ( ( X ( N δ B C d dt d dt derivatives. Dirac deltas areTherefor, distributions we which introduce are the easy notion of ‘smeared’ constraints In the above definition, gration on the right hand side of the above defining equations equations ( Hence, to showneed that to the compute secondary Poissonwhen brackets constraints among the are constraints constraints preserved and themselves demo are satisfied, i.e., we comp So the time derivative of secondary constraints can then be w Poisson brackets involving ‘bare’ constraints, ( conjugate momenta: equivalent to the on-shell closure of thewe Poisson compute bracket al the algebra of constraints explicitly and4.1 show th Generalities Let us start by giving the fundamental Poisson bracket relat In the last section weThe saw purpose how two of primaryconstraints constraints this yielde arising section in is doubleby to field introducing demonstrate theory the do that notioncomputation. not the of lead story We smeared to argue e constraints any that m and the descri invariance of these constrai 4 Algebra of constraints where the shift vector commute withinvolving all arbitrary the functionals fields of andthese canonical fundamental fields relations. mom and The Poisson conjugateimportance bracket involvin mom as the time derivative of a functional JHEP10(2015)158 . ) in , we 2) (4.8) (4.9) P (4.12) (4.11) (4.14) (4.13) (4.10) 2) ξ ↔ 3.20 P ↔ ∂ ) is equiv- (1 1 2 X (1 − ( − ) − ) 2 M ) ξ 2 X X ( ( X d d ( nstraints associated P ∂ ) and . KL )Π , P , from equation ( 1 X ξ ) ( , . 1 X K 2) f the smeared constraints } } )Π λ ( ] ] X 1 C d ) = MN eful notations as we proceed following form: ↔ N N ared constraints are now just 1 X − er time evolution it suffices to [ [ secondary constraints actually ξ ( X H pressed by taking their Poisson forward and better defined. In 2 b ( ξ L (1 C C b } d L X , , ) ξ ( − MN ] ] 2 b ξ L δ ) λ . We replace the dummy integration [ H [ X , MN 2 1 P 2 ( X ] ξ ) finally becomes: C B ( X ∂ b d { L { N 2 d [ ∂X ξ 1 } + Π + ξ + b 4.12 ) P L 2 C b 2 } , L ξ } d ξ P ) X − b ( 1 ∂ [N] L [N] d ). Direct computation shows that: P = 2 X – 16 – 2 [N] + B ξ B ( Π KL } , , d ) X ] ) B ] H ( 2 ξ Π λ 2 [ [ X ξ { X = ] to vanish for all choices of M ( ). We use integration by parts to remove the derivative X 2 ) to vanish at all points on the doubled hyper-surface. ( b L ξ C d ξ B d [ , 2 d { H X { 2 2 Π d ( ) ξ C 4.12 1 X b K L Z d X , ] = ] = 2 C ( ) and ξ ) λ d [ 1 ) and doing an integration by parts, the smeared constraint [ X 1 ] and d 1 ] = X 2 C X B λ MN ( ξ 4.7 [ ( d X [ d d d Π dt d ) and dt M 1 2 B { C Z ξ Π d X { ( − . The first line in equation ( Using the definition of the bare constraint + Z B X = . } } ] ] 2 2 with ξ ξ [ 1 C[ C X , , ] ] 1 1 ξ ξ Let us now compute the Poisson bracket between two smeared co [ ] can be written in the following form: ξ [ C C[ { Let us focus on the term involving the dilaton. Since, Using smeared constraint makes calculationsterms of more the straight smeared constraints, the Hamiltonian takes the with vector functions In this subsection,explicitly we and compute show the thatwith it Poisson our closes. bracket calculations. We algebra will o introduce some{ us Hence, to show that secondarydemonstrate constraints that are the preserved Poisson bracket und algebracloses of the on smeared smeared secondary constraints. 4.2 Algebra of constraints and time derivatives of smeared constraintsbracket can with be the neatly Hamiltonian ex as follows. alent to requiring We use this relation in equation ( deduce that: acting on the delta function and then integrate over the smeared constraint ( C variable numbers and requiring a generalized vector function with same properties. The sme JHEP10(2015)158 ), . ] has ) 4.12 (4.21) (4.19) (4.16) (4.17) (4.18) (4.15) (4.20) (4.22) ρ 1 [ X B − 2 X ( ation: ] and . δ λ [ B K 2 d 2 Π ξ , P ints 1 32 c and the corresponding 2 ) ∂ X X + ( ]. After using the definition the dependence of fields and rm: ed Lie derivative of the gen- − d λ kets in which we have to deal ymmetric tensors outside the , we see that the term on the e fact that in equation ( [ C P MN 2 ] . 1 ξ B ] Π , ,ξ . K C 2 2 H, ] ξ ) is just given by: ) [ ∂ 2 MN MN b + L X ∂X , ξ Π ( H 1 4.12 G = C d ξ . ] ) [[ 2 d C ) + ] N L 2 i ,ξ 2 δ e 1 C ,ξ F 2) X ξ P [ 1 ( M ( ξ b = [ δ L = + 2 ↔ ) and comparing the resulting expression with b L } } ] – 17 – d . ] R 2 ) and do similar kind of computations as for the (1 2 ≡ O MN ρ d ξ Π [ [ 2 ) + 2 i 4.16 Π X − 1 − O B C X ) e 4.12 , X , d X ] 2 ] X d be a function of coordinates, we define: and − λ 1 2 ( d [ ξ d 2 1 d [ O 1 Z B X ξ X λ X Z C { ( d b } L { 2 ) takes the following nice form: δ ) P d 2 P 2 ∂ ), one obtains: X ∂ P Z 2 4.12 ( ξ ∂X − 4.6 P 2 KL ξ ) H ] = N L 2 Now we turn to the smeared constraint ξ δ λ [ b L K M , ( . ) we obtain the result for the Poisson bracket under consider B δ ) } 1 ] ρ X 4.11 − ( B[ ) in equation ( = , X MN ] ( λ Π It is easy to see that the Poisson bracket between the constra Now, let us focus on the term involving the generalized metri { B B[ This notation will prove usefulwith when two computing integration Poisson variables brac three non trivial terms and can be written in the following fo We use this expression in equation ( { canonical momentum. Using theeralized expression metric and for the the symmetry generaliz properties arising from th combining this with the dilaton term ( definiton ( dilaton term. After asecond long line but in straightforward equation computation ( Poisson bracket, we find that: the indices inside the Poisson bracket are contracted with s of Hence, we conclude that the first line in equation ( To proceed, we introduce aparameters useful on piece coordinates. of notation Let to denote Now, a short computation shows that JHEP10(2015)158 . ) 2 , isson ) and X (4.29) (4.27) (4.28) (4.26) (4.24) (4.23) (4.25) ) 2 . After . λ − X H ) and the 1 2) ↔ − 1 X N . 1 ( 4.1 ρ ∂ i ( ↔ 1 δ X d ( 2) − (1 δ 1 M 1 1 M , ∂ ↔ ρ ∂ K − 1 K ∂ (1 d 1 2) , ) ∂ n for the generalized MN 1 2 1 N H ∂ X ↔ NK NK 8 } − 2) H − H (1 − + 4 1 ↔ 1 R MN 2 R cancel each other. 1 ). After some algebra which 1 N racket relations ( ∂ ng of the dummy integration X 2 d Π ∂ ∂ (1 ), one finds that: ( } − his result to evaluate 2 1 G 1 N 2 X λ. 1 δ M MR 2 d ∂ 4.1 ∂ N 1 − MR Π } − MN H ∂ d 1 N 2 , 1 H Π d, 2 d 1 ∂ + 1 M MN 1 , ρ X χ Π 1 ∂ 1 ( MN + L ρ b L , H L 4 {R ∂ δ ∂ 1 d MN 1 4 ) K H 1 K 1 d 2 {R ∂ ρ is simply proportional to − 2 Π 2 M ∂ ) d H 2 1 2 ∂ − MN − d 1 d − N 1 M G MN e 1 NL X − ∂ − d Π ∂ H { ρ d NL ) H 1 H 2 2 2( 2 1 d K M N d H 1 d K d – 18 – − 2 +4 ∂ ∂ M ∂ 2( ∂ ) = − e e 1 ∂ M − 1 2 Z − ∂ d e 1 1 KL 16 16 KL 1 MN X 1 2 KL 2( = H h d KL 1 H − H − H λ 2 2 2 1 N e λ involved in F H ρ ρ ρ 1 h ∂ 1 1 1 + 1 2 + 2 } 2 ≡ X + 1 λ λ λ M 2 ρ ( ρ 1 λ d ∂ 1 1 2 2 2 is easy to compute and it vanishes. The key observation is δ N M 4 − λ Π ∂ X X X χ 1 , M NL MN G 2 d d d 1 ∂ M 2 2 2 d H H X 2 d d d MN 2 d 1 1 1 1 1 N K λ∂ − 2 , is non trivial. To evaluate it, we will need the following Po 4 ∂ ∂ e d X X X { 1 d d d H 2 2 2 KL KL 1 1 X = 2Π d d d d MN H H 2 } Z Z Z Π d 1 1 M M ∂ ∂ X = = = MN 2 Z d 2) anti-symmetry the two terms appearing in 2 1 1 8 2 Π 8 1 d F G 2 H is defined as follows: + − ↔ = Z M MN χ F = Π , The third term, The second term, i.e., 1 H {R bracket. due to (1 that the commutator some manipulations, which are by now familiar, we obtain where involves repeated use ofvariables integration we see by that: parts and a relabeli Now we use the fact that This relation is obtained by using the fundamental Poisson b explicit expression for generalized curvature. Now we use t and the three terms are: Let us compute thesecurvature scalar terms and one the fundamental by Poisson one brackets ( now. Using the expressio JHEP10(2015)158 ). , a . 3.4 M ρ , it is ) (4.31) (4.32) (4.37) (4.30) (4.33) (4.38) (4.35) (4.36) (4.34) NK χ N ρ ∂ H i , NK M ↔ ) ∂ H ρ ρ ρ λ L ( M L + ∂ ∂ ↔ ∂ ρ, − K L λ NK KL ∂ ( ∂ ML identically zero if ρ H K L − H ML ∂ ∂ −H λ M ) ρ H K ρ∂ = given in equation ( L ∂ K L ∂ NK ∂ NK ∂ K NK H KL H − MN ∂ lling the definition of ρ, ). To see this, we write the KL H H L . + ML ρ f Π ∂ M NK 3.4 ρ L ML H ∂ −H K L ∂ MP H H ∂ ∂ η + ) λ NK KL + ML = , ML H H ρ H NK ) NK MN λ H η N JP + H Π ∂ K would also reduce to a similar expression. , we have: H ρ, λ and ρ ∂ ρ M ML M J X ML η ∂ M H N ∂ d N ( ∆ ( ∂ L 2 ↔ we conclude that: NK H H d η we obtain λ M λ − H + 2 G – 19 – = ∂ MN + 2 ρ ) = Z λ ) and comparing the resulting expression with that . H λ t JP = ↔ NK 2 λ, ρ and MN MN η ) D NK H λ MN ρ 4.30 λ Π H ρ F H d + ∆ ( NK Π ρ ML L NJ 2 ↔ L η η X H ∂ − KL ( ∂ X d e λ 2 as determined in equation ( d MN 1 ( H ρ 2 , we expect that d ML MN PL − L H d MN − F ∂ H χ N H H Z Π b MN λ Z 2 1 L λ λ h KL 2 − = ] + K H χ MN ∂ ) in equation ( [ λ ≡ = MN H MN ) C Π χ NK M Π b 4.32 L ) one can see that: = ∂ H X λ, ρ d X } 2 ] d MN d ρ 2 MN 4.29 MN ∆ ( [ d Π Π Π ) and ( Z B Z , X X X ] = d d d 2 λ 4.31 = 2 2 [ d d H d Following the result for The second term on the right hand side in the above equation is B Z { Z Z using ( Similarly, up to symmetric terms in where the last equality follows by using the on-shell value o Now, by using the identities, easy to see that up to symmetric terms in Now, consider the following: integrand in the second term as: in equation ( Combining this with the result for short computation yields the following. where we use the value of the Π In the following we will see that this is indeed the case. Reca JHEP10(2015)158 . ). 2 C X ( (4.50) (4.48) (4.49) (4.47) (4.43) (4.45) (4.44) (4.46) (4.41) (4.40) (4.39) (4.42) , ξ 2 and d 2 B ξ b L } 2 d Π , 1 , d } 2 2 e , { KL . MN 2 1 H , , 2 d 2 } ction of the canonical field ξ H ative of that function with ) , 2 Π , 2 the dependence of fields and b MN 2 L d ξ Y rms appearing in this Poisson d , 2 is to be understood as 1 nerated by the generalized Lie b H . 32 L ξ 2 , the most convenient approach ξ ξ − 2 } b L b ξ b 2 L U + ) = 0 L , MN 1 = 0 X 1 } d, ( Π 2 d 2 . ξ R 1 d MN as the gauge transformations of the ] NK δ b Π MN 1 L JP and ] ) vanishes identically and the Poisson χ Π Π { H [ d Π , , 2 H [ MN T MN 1 t d 1 1 δ R f C MN δd λ, ρ d Π ML Π D H R 2 { 1 δ H δ 1 J {R = MN d H − ξ d 1 M 2 e 2 b d } Π = d L KL 2 e ] 2 + − and a slight relabeling of indices one finds that: 2 H 1 δ ρ } e Π − δd [ − λ , in particular, ) 1 1 – 20 – { e + 2 Z, 1 1 λ λ 2 NK B PK Y and λ λ ( , η X 2 2 X + H ] d d 2 2 d MN X λe X λ ξ λ X X 2 [ Π d d ML b d d X X W d L H 2 2 , ML 2 2 η d d and t 1 d d B η ( ) 2 2 d d + 1 { D 1 1 d d X ) we see that ∆( X 1 1 d Z Z P X X X V 2 MN d d N X X d − − ]) ( 2 2 d d + H H 2 2 d d d 4.37 t [ Z d d = = U f D on both sides we get the following: Z Z ) becomes: ( 1 T + Z Z U t 16 { 2 2 D − − − − − T 4.34 ======Let us now turn our attention to the Poisson bracket between } T Z ] U V W ξ [ . ], we can understand } C ] 6 , ξ ] λ [ C[ , B ] { λ Let us compute these terms now. To compute B[ Since the gauge transformations inderivatives double [ field theory are ge using this fact, it is easy to obtain the following: parameters on the coordinates where we have used the notation introduced earlier to denote bracket in equation ( By applying operator { and using this in equation ( After a short calculation, webracket can as arrange follows: the five different te After multiplying this equation by is to use thewith fact the that corresponding the canonical Poissonrespect momentum bracket to is of the just canonical an field, the arbitrary i.e., fun deriv JHEP10(2015)158 ) is (4.51) (4.55) (4.52) (4.54) (4.58) (4.53) (4.56) (4.57) obtain 4.45 , is a gauge constraints d R , Π . d λ. Π 2 d P P ξ Π . Now we combine 1 ∂ 32 P ξ . 1 P . Since ∂ 32 ξ ξ + ll. In particular, only ]: P + R 6 MN ξ d he following: 2 P erms add up nicely to give: MN Π meter − ∂ Π MN 2 d , an compute the term in equa- Π MN served under time evolution. . X e MN elta function. The term ( d , i.e., use of fundamental Poisson R 2 Π MN } + 2Π λ d ] d 2 Π + 2Π ρ P [ 2 d − d ∂ Z e 2 Π P B e , ξ = P ξ ] MN P − ξ λ + 2 Π ∂ [ P R P ∂ B R d B ξ d X λδ 2 d { MN 2 2 d = − e 2 d Π − e – 21 – 2 d } e ] − P ξ P ξ d , we resort to the same kind of techniques which [ Z ∂ 2 λ , e V P P C − X λe P P ξ , ∂ d requires the use of the on-shell value of the canonical ] λ 2 ∂ = P d d λ P } P [ 2 and ∂ ξ U B X λ∂ Z P B , X ξ d U + ξ { d 2 1 B 2 16 d − T X λe { d d X λ − 2 Z d Z B d 2 = d − = = Z V = Z ), is the easiest one to compute. After a short computation we 2 Z 2 − U . As discussed earlier, the closure of the algebra of smeared + − transforms as a scalar density, we conclude that [ 4.47 = + W = d MN T 2 + W Z − e V ). denotes a gauge transformation generated by parameter + ξ U δ 4.46 + For the next two terms So we conclude that the algebra of constraints closes on-she T where the two terms and write them as follows: scalar and Now we can combine all five terms. After some algebra, all the t were used in computing the Poisson bracket the following expression: somewhat easier to compute and after some algebra we obtain t The last term, ( After using similar methods buttion rather ( tedious algebra, we c bracket relations, integration by parts and properties of d dilaton and the generalized metric generated by a gauge para hence we find that: the closure of the bracket under Poisson brackets ensures that the constraints are pre momentum Π JHEP10(2015)158 ] 57 (for (5.2) (5.1) (5.3) s [ M ∂ , i ) B (∆ ]. ace be characterized. i plicitly show that the . and low energy action eld theory. ∇ 52 ory Hamiltonian. To do g = 0, then the second term ains the vacuum Einstein s of general relativity and er limits. Afterwards we ian by including the total kes’ theorem which would − y a hyper-surface is the Ricci scalar, and latin ]: nergy and momentum which sence of the boundary terms he notions of ADM energy, pply our results to compute o construct these charges we . M √ il 52 R ∂ x econd term can be written as a 4 δg sion see [ d is given by: j space-time dimensions is given by: i and ∇ Z d ) gR, ij B + − g il g √ ij kj x g δg d d − Here we briefly review the key aspects of the R Z jl g ij – 22 – g ki ] = 1 2 g ij g − [ = ij k EH R ) ) which were neglected earlier. From the boundary terms S 1. To obtain equations of motions we look at the behavior B g − does not have a boundary, i.e., 3.18 (∆ − , d √ M x ··· 4 , d ) and ( 1 , Z 2.38 = while we were finishing this paper, we became aware of the work EH δS is the determinant of the space-time metric ] which also discuss conserved charges in double field theory g 58 We start by discussing boundary terms in general relativity The Einstein-Hilbert action for general relativity in indices take values 0 where the first term giveswhile vacuum Einstein the equations second in term the involves ab a total derivative. (∆ where, write the boundary termsderivative for terms ( the double field theory Hamilton of NS-NS string.enable Then us we discuss to a writeboundary generalized terms boundary version of terms of double inlow field Sto theory double energy reduce field effective to theory. action boundary term of We ex string theory upon taking prop of this action under a variation of the metric. One obtains [ If the space-time manifold boundary terms in general relativity, for a detailed discus Note added: and [ 5.1 Boundary terms Boundary terms in general relativity. of the Hamiltonian, we define theare notions of conserved generalized quantities ADM e inADM double energies field and theory. momenta for Finally some we known a solutions of double fi is zero and theequations. variational problem However, is if well the defined space-time and manifold one is obt bounded b In this sectionmomentum we etc construct in conserved general relativity)need charges in to (similar double add to field appropriate theory.this t boundary T we terms need to toThis the understand will double be how field the can the subject the of boundary this of section. a doubled sp 5 Conserved charges and applications simplicity we only consider time-likeboundary boundary), integral then using the the s Stokes’ theorem. JHEP10(2015)158 jk δg (5.9) (5.4) (5.5) (5.6) (5.7) (5.8) (5.11) (5.10) = 1. If j n , given by the i . jk n is described by jk ij δg M ) as follows: g i ∂ δg ∂ i M i with indices taking 2, then the Stokes’ 5.2 ∂ ∂ n i ij − n jk n ( . h , i.e., jk i , d ) hh h M B − and . − ∂ , ··· et boundary conditions, the whose variation cancels the i √ , mal vector to the boundary he standard Dirichlet bound- dary integral as follows: (∆ ij jk y J 1 i i h 1 = 0 . , EH δg − i i d hn ro boundary term: S hn S n ∂ d j k − = 0 ij − on the boundary ∂ n ) can be written as: h i ¯ √ Z √ ij j ¯ . ij i ¯ y g . If the boundary y h 1 − h 5.9 ij 1 . g ! − − M d = ij d j j ¯ ∂ S d S d j j jk δg ∂ , where ∂y ∂x ∂ Z i ¯ Z i i δg ij ¯ y i such that g S = R ∂ 1 – 23 – = i ∂y ∂x i i ∂ ij i , ∝ takes the following form: n ) J g j = 0, does not lead to vacuum Einstein equations. ij i i = n 1 2 jk g ij B n i j ¯ ∇ g i h ij ¯ EH n g p − , i.e., h (∆ δg − δS − i ij −

ij g jk √ ∇ R ij = g ) = constant, then the normal vector is just related to the g x δg i ) as follows: x i d − g ( n x = ∂ d ( and its inverse to deﬁne objects √ k − ij S = 0. The second term depends on the normal derivative of n Z x ij √ h S ij d g Σ g

d x d ij d = Z 1. In particular, i δg ) Z − is the unit normal vector to the boundary B = i , d (∆ n i EH n ··· acts as a projector to the boundary , δS 1 is the determinant of the induced metric ij , h h Suppose Using Stokes’ theorem we can write the second term in equatio push-forward of the bulk metric A short computation shows that the integrand in ( values 0 where One can use the metric Hence, a constraint function i.e, and we see that the requirement gradient of the function The way to resolve this is to add a boundary term to and is not zerovariation in of general. the So, Einstein-Hilbert we action conclude includes that, a non-ze with Dirichl which is given by: It is straightforward to obtain the properly normalized nor In the above equation, theary first condition, term vanishes i.e., if one imposes t coordinates on the boundary are given by theorem relates the bulk integral of a divergence to the boun JHEP10(2015)158 (5.15) (5.17) (5.18) (5.13) (5.19) (5.12) (5.14) (5.16) . ki ry conditions, g j ∂ k , . j n kj n ij ij g i i g ∂ δg ∂ i j k − h n ij ally transparent boundary R ], given by: ij g + 2 ij h k to both, bulk and boundary 60 g oundary term exactly cancels rivatives (derivative along the on under Dirichlet boundary ∂ ry conditions. One finds that: , jk ary conditions. This allows for − 1 2 ). , k i g i n the two boundary terms with re- m is not unique as it is defined 59 kj − B ∂ ij i g k i 5.17 , g ij n ∂ hK, i hn R ij h n k ij − − h Γ − kj g √ √ j ij h √ h ) and ( − n g y y i 1 1 − = y √ − 1 − ∇ − √ d d 5.12 x − kj ij (GHY) term [ j ji d d d d g y g Γ i d d 1 – 24 – are fixed at the boundary and hence we deduce ∂ Z Z ik − ≡ k i d Z g Z n d n = K + ij = 2 g Z = ) = k gR − = ). It turns out that the following boundary term achieves B GHY − kj EH-bdy S g √ S i x EH-bdy 5.11 ∂ d i S d n EH-bdy + S kj Z g − EH = S Gibbons-Hawking-York ( i δ GHY B i S EH-bdy= n and the normal vector S ) of the normal vector and the metric. Under Dirichlet bounda ij ]. + M g ∂ 52 is the trace of the second fundamental form and is given by EH S K the boundary term inconditions and the one variation obtains: of Einstein-Hilbert acti It is now an easy exercise to verify that the variation of the b the equivalence of the two boundary terms ( It is easy to verify thatspect the to difference the between variations metric of vanishes upon imposing Dirichlet bounda where and a short computation shows that: so that the total action can be written as: where so the difference of the twoboundary terms depends on the tangential de this goal [ boundary term in equation ( the metric This boundary term, however,coordinate is transformations. non-covariant with Evidently, respect thisonly up boundary to ter terms whose variationsan vanish improvement for and Dirichlet one bound canterm, indeed known introduce as a the more geometric JHEP10(2015)158 = M (5.24) (5.21) (5.22) (5.23) (5.26) (5.25) (5.20) ). Thus the 1)-dimensional , where 5.13 ki M − g j d X ∂ k ond field involve only n , 2 is (2 ij g H 2 M − 1 eneralized Stokes’ theorem. 12 H riginating from the strong of a doubled space, we refer ∂ . The correct boundary term ij , , 1 g i − 12 f Stokes’ theorem for a doubled blem. In light of our discussion k . For details on the nature of 2 B ∂ = 0 ) i − ar term needs to be compensated A k The low energy effective action of n To discuss boundary terms in the 2 n ) M φ ∂φ 2 ij 1. The boundary is characterized by is given in equation ( . N g with coordinates − ∂φ i − , M ] , he + 4 ( d φ B 2 2 N M jk − R , S b − + 4 ( i √ [ M R he ∂ φ ∂ y ··· 2 1 − , , − 2 − ≡ φ – 25 – √ = 3 d , 2 ) = constant ge d − y M 1 X − ijk ) = 0 = 1 ( ge N Z − √ H d − ¯ x ··· d = S is the field strength associated with the Kalb-Ramond M ( d √ d Z M x ijk d ∂ + Z d H M Eff-bdy = S Z , where N ¯ M = Y defined as: Eff Eff-bdy S M S -dimensional doubled space d N + Eff S . The boundary of the doubled space, denoted by d is the unit normal to the boundary and is the Ricci scalar, 2 and is defined as: i , R is the dilaton. The ‘kinetic’ terms for dilaton and Kalb-Ram n ij b φ ··· Consider a 2 , 2 , where the boundary is specified by for the boundary terms inwith general a relativity, the boundary Ricci term to scal haveis well the defined variational following: problem first derivative terms and pose a well defined variational pro a gradient vector and where field Boundary terms in doublecontext of field the double theory field action. theory,space. we need a generalization We o provide such a generalization in appendix This gradient vectorconstraint: has to satisfy following constraints o and it has coordinates boundary and gradient vectors associatewthe with reader the to boundary the appendix. Here we will briefly describe the g total action reads: where NS-NS sector of conventional string theory is written as: Boundary terms in low energy effective action. 1 JHEP10(2015)158 (5.30) (5.27) (5.28) (5.32) (5.29) (5.31) omor- . d N ∂ . So, the bulk ow focus on the R MN and get rid of the ld theory action. , which is a density H = , . These constraints 4 M d 2 i DFT J − R − d L e the Kalb-Ramond field, M N r N ∂ the ordinary direction, the MN d is the absolute value of the become: ction for double field theory 2 are the coordinates adapted m. However, in terms of the H ented as follows:

g. − r, i.e., this boundary term explicitly ′ MN N uble field theory is superfluous. e − M erms of ∂ lativity and low energy effective ′

H ∂X ∂X ′ √ 4

X ) involves only ﬁrst derivatives of M = − ∂X ∂X . 2.5 N d

. 2 d 2 − MN = 0 Y − e 1 e H S , where

− , , φ N ′ d ¯ M N 2 ∂ ′ M d ∂X Y = 0 ∂X ∂X d ∂X and

2 Z – 26 – ij is the dilaton. − e ≡ = Y , b ! d h 1 0 g M − ′ M d = 0 M ∂ 2 1 X ˜ d ∂ J 0 − + d g 2 Z R

), which specify the boundary is also subject to the strong − d e + 2 X = ( − R e d M H 2 S = − transforms as a density with respect to the generalized diﬀe X ∂ ) and dependence on dual coordinate vanish, i.e., d d φ 2 2 X e DFT d d − 2 L e d

Z ′ Z ) reduces precisely to the Einstein-Hilbert action. Let us n ∂X ∂X

is a generalized vector and = . In the following, we show that this is indeed the case. ’ contains any fields, parameters or their arbitrary product 5.30 M ··· 5.23 DFT J S Action for general relativity is obtained by demanding that Now we turn our attention towards boundary terms in double fie term in ( where imply that the function One can always express the generalized curvature scalar in t We stress here that this construction of boundary term for do The factor to to the boundary and defined as: explicit boundary term. However,and investigate it if is it reduces instructive toaction to boundary write terms of general re constraint. The generalized Stokes’ theorem can now be pres The generalized curvature scalar reduces to the Ricci scala space time dilaton ( determinant of the transformation matrix The Lagrangian for double field theory, as given in ( boundary term. Since the boundary has to be completely along fields and hence does notgeneralized require curvature this any additional Lagrangian boundary can be ter written as: including an explicit boundary term as follows: using the generalized Stokes’ theorem, we can now write the a This would imply that the generalized metric and the dilaton phisms on the boundary and it is the push-forward of the facto where ‘ with respect to the diffeomorphisms of the bulk space. JHEP10(2015)158 can (5.33) (5.35) (5.41) (5.38) (5.36) (5.34) (5.39) (5.40) (5.37) M N ! ). S , 0 i ki 5.5 ∂ g

j ∂ , = 2 ij .

g ′ kl h. M g − ∂x ∂x j N l ′

S l = g ∂x ∂ ∂x | c. First of all note that for , − i kl k ′ vity, we would like to express and the corresponding minor dd ements of the inverse matrix .    g i = i C ¯ be the transformed space-time ∂x | ∂x ˜ y ij x | S M and the gradient vector g g ows that: k =    M − M | ∂ M ij √ = , the space time metric transforms ′ ij , ′ ∂ g −

x . ) ! gg | i i

′ so that ij = and → ′ − ( ˜ x be an arbitrary non-singular matrix, then . x M ). It is easy to see then: → | ij C √ ′ ij ij d ∂x

∂x

, g , g

M j ¯ x j i ¯ = = ∂ = S h = . Now, let = S j l – 27 –

ij M S ∂ ij ′ = ∂ , g M i ′ d, j 1 kl kl M S kl g∂ i g g − = g ∂X ∂X ∂ j l l − i ′ j ¯ ,X

M ij √ − g ∂x ! ∂x ∂y ∂x and the boundary i i ¯ i i k ¯ ˜ y k x ′ = = = ) for (

M d ∂y ∂x ∂x ∂x dd ′ N = g MN ∂ = = 5.38 ¯ H j ¯ M = M i ′ ¯ ij N g N ) ∂ dd ) minor of matrix dd M → 1 is equal to the induced metric as deﬁned in equation ( ( ,Y ij MN − ) C N ij ! d H g i i dd 2 M d ( ˜ x x 2 − C e −

e ]: is the ( 4 = 52 ) M ij ( X C Using these expressions, a straightforward calculation sh be written as: as follows: To make connection with the boundarythe terms determinant of of general the relati the Jacobian case in at terms hand: of the induced metri where Now, under the coordinate transformation metric, i.e., an elementary result fromcan linear be algebra written in implies terms that of the the el determinant of the matrix coordinates on the bulk space as follows [ Let us apply the formula ( and the minor To proceed, we need the following result. Let JHEP10(2015)158 , . ki (5.47) (5.44) (5.48) (5.45) (5.46) (5.42) (5.43) g ki j g ∂ j )=0. The . ij ∂ ). However g k ··· n 2 − ( . ij H ˜ 5.33 ∂ ij g g φ 2 1 12 k − k ∂ H ∂ uble ﬁeld theory. A ij − ij 1 g 12 g k + 4 2 ∂ ) − ds the bulk term of low , tes, i.e., k ij ! ndary terms in the action 2 g n ki e following form: ) ∂φ nd the gradient vector are k boundary term for general g ij S ( j ∂ n. j g φ, ]: ∂ ∂φ i ij ∂ overall multiplicative constant − n in equation ( ∂ ij S g 48 i h φ k g . S −

n i ∂ − and after some algebra, the above ) φ ∂ + 4 ( kl 2 ), it is easy to see that the integrand √ φ ij g S − S R l g j y S ∂ 1 ∂ l he + 4 p 5.35 φ kl − ∂ − 2 d g S R h

kl i d − √ g ∂ h ). To do this, we solve the strong constraint y φ ge φ ij 1 Z 2 − is just equal to the unit normal vector to the 2 g φ ) and ( − − − ( 2 − √ + d 5.23 ) we can write the desired determinant in the − √ ). g – 28 – ge ge

= x ˜ xd S 5.34 ge − d j gR d − ) reduces to the low energy eﬀective action of string ∂ s 5.38 d − d √ S 5.15 − √ ˜ xd k S N i = d √ − x ∂ √ Z ∂ ∂ 5.30 d d

). Putting this all together we see that the double ﬁeld x kl ′ ij g d = = g ˜ ) in ( xd d Z +4 5.8 d MN d √ ∂X ∂X d N = Z

MN ∂ 5.41 Z − H H ˜ R x M d d N = 2 d N ∂ − MN R M Z d H ) and ( MN 2 N X e N = d H − d ∂ 2 d 2 2 5.40 − d =0 e − X e ,φ M e d Z 4 2 N =0 d d ,b 2 − Z =0 e ˜ ∂

′ Let us now compute diﬀerent terms in the boundary action of do Let us now see how the action ( DFT ∂X ∂X

S bulk term of the double field theory action then becomes [ boundary as given in equation ( The integration over the dual coordinatesand just amounts we to an obtain therelativity exactly standard matching Einstein-Hilbert equation action ( plus the By doing integration by parts for the term involving and we see thatenergy the action plus bulk a term boundaryof term. of double Let double us field field now theory. focus theorythe on same action Note the as yiel that bou in the the transformation previous matrix case a for general relativity, give expression becomes: short computation shows that: there is no restriction on the Kalb-Ramond field or the dilato by requiring that the fields do not depend on the dual coordina theory including the boundary terms as in ( theory action, upon proper truncation reduces to: following way: of the boundary term of the double field theory action takes th Using this expression and the equations ( Using equation ( We recognize that the factor JHEP10(2015)158 y , ), ) = d 5.8 X Π (5.50) (5.51) (5.49) (5.52) ( L , , S N ). For the 4 1 φ k − ki ∂ 3.18 g 4 j is the boundary ∂ RP − k H n ki bdy ) and ( g ij LR j H g M energy and momen- ∂ are defined as in equa- Π ij − nian is zero due to sec- 2 2.38 P g M or the case of double field g including the boundary ij nderstand those boundary H contributes to the on-shell N se some care in identifying g − ) and -dimensional doubled space N k ian of the double field theory 1 n for double field theory we 12 ∂ ij D + 2 k g erms ( − 3.22 k n and ∂ 2 ij ) LP ij g M ′ g H ∂φ , φ X P 2 ! ∂ bdy − − + 4 ( S H he on the full 2 j d ∂ R − + ˆ P S N ∂ k √ S ˆ i H M φ ∂ 2 y ∂ – 29 – b LP kl H 1 = − ij with the unit normal vector to the boundary ( g − H g d ge 4 d d p − N and characterized by the constraint function DFT S ∂ N j

Z √ ¯ d H ∂ . M x φ S 2 + 2 d k S − Y MN i ∂ d bdy − e ∂ kl ij H g he Z g L − H − = √ N ˜ x √

d ), the integrand of the boundary term in the double ﬁeld theor ′ d MN ). = ) respectively. H ∂X Z ∂X on-shell 5.42 N

5.23 ∂ = H 5.24 Y 1 M =0 with coordinates ˜ − ∂ d

N 2 d are the boundary adapted coordinates. M 2 d ) and ( ∂ is the bulk Hamiltonian as deﬁned in equation ( − DFT M e ′ Z S

H ′ X 5.28 = First, let us make the notion of asymptotic ﬂatness precise f Now we can deﬁne notions of conserved charges analogous to AD As discussed earlier, the on-shell value of the bulk Hamilto ∂X ∂X bdy

H theory. Consider the generalized metric which matches the lowterms energy as given effective in action ( for NS-NS strin tum. However, due tothe the correct strong expressions constraint, for we conserved need charges. to exerci tions ( where, boundary can be written as: 5.2 Boundary termsWe in recall the that Hamiltonian in andneglected our conserved boundary discussion charges terms. on Nowterms we the and are canonical include in them formulatio a in position the Hamiltonian. to better The full u Hamilton we finally obtain: takes the following form: By identifying the factor ondary constraints, we concludeHamiltonian, that i.e., only the boundary term term which can be written by including the total derivative t Using the relation ( constant, this boundary term takes the following form: where JHEP10(2015)158 . - 1 a d and ) = M , z 1 icted a oduct. ˜ (5.57) (5.55) (5.54) (5.56) (5.53) z ··· 3 , M ( : i 2 a y of the ∂ M -dimensional n. does not have d + . and it is given by: m DFT ˆ N LP = H ˆ M d H = 0 in the boundary b δ P for some asymptotically er-surface into ‘allowed’ ∂ M and xpression for energy given − se, suppose the coordinates N d DFT aint may not completely spec- is described by the constraint P cal quantity. The reason is the onserved energy by eliminating H arameters for a particular solu- ∂ , this definition is still not free = 0. Since , n, e would expect that the notion ds depend on coordinates on holds trivially. However we can and are independent of ˜ M LP . M i E ··· H y N ,    . 4 , 2 . 2 L = 1 MN δ N × M d direction is trivially satisfied, i.e., 1 0 of coordinates in such a way that it reduces 2 with fields being independent of × M -dimensional subspace of the 2 1 a − − d 1 2 z e 1 0 0 P M

0 0 0 – 30 – , m, a ′ M ∂ M − ) is only allowed to depend on the sub-space    = = ∂X ··· ∂X X ,

. We denote the flat metric by ( = M M and dual ˜ ˆ N ∂ S Y = 1 a ˆ 1 M z . We do not worry about global issues regarding this direct pr − → ∞ b δ 2 d 2 , i M d P a M that the boundary of the doubled hyper-surface is also restr ∂ , z i Z A , y a ˜ →∞ z lim , i P ) is proportional to an undesirable factor, i.e., the volume ˜ y ? ≡ ) = constant, then and ‘not-allowed’ subspace = 5.54 E X 1 ( M M X S -dimensional hyper-surface are arranged as follows: d In this paper we use the direct product just to denote that fiel 3 Then the strong constraint along function It is shown in appendix tion of double field theory only depend on a explicit time dependence, thenot identify conservation this of definition energy ofstrong conserved constraint. energy Due with to a the physi strong constraint, fields and p to be along the ‘allowed’ subspace. If the boundary of Now assume that fields depend only on coordinates stationary observer in doubled space-time i.e., N = 1 the integration over thefrom ‘not-allowed’ ambiguity. sub-space. The reason However forify this the is that ‘allowed’ the sub-space. strongof To constr the make 2 the argument more preci dimensional ‘not-allowed’ sub-space. So one can define the c are independent of coordinates on This definition of energy is just the numerical value of the to the flat metric in the limit doubled hyper-surface. So,subspace we can decompose the doubled hyp and suppose that it depends on a function so that we can write With these considerations inin mind, equation it is ( easy to see that the e Following the analogy withof general conserved relativity, on energy na¨ıvely can be defined by taking N = 1 and Hamiltonian: JHEP10(2015)158 , i ˜ x s of → (5.59) (5.61) (5.60) (5.58) i ce-time . x j conserved b-Ramond dx d i -dimensional ˜ x d d , , kj ) corresponds to ing different half- b ). d have 2 d ik -dimensional space Π 2 LP g n 5.58 , or a solution of dou- 2 M H e field theory given by P − N ··· ow choose the sub-space ∂ j ces are allowed and from 1 4 , le space-time spanned by deed reduces to the ADM ˜ x obtain different values for n agreement with 2 − d , − i different solutions related by d ˜ imensional subspace in equa- ory, we have a single solution geometry of the double space. x L ubled space-time. A complete P equation ( d = 1 ∂ N ij , ized metric on the 2 g ndary of the doubled space. These erved charges from the double field LP M KL -dimensional sub-space amounts to + lj H Π d b j 4 kl = 1 ( dx g L MK i ik N H M b d dx 2 2 N − −

e lj ′ ij b

-dimensional slice of the 2 g . On the other hand, since ﬁelds are independent – 31 – ′ kl . So, we see that the double ﬁeld theory solution n ∂X ij ∂X g b ij

∂X ∂X ik b = ˜

b Y ij 1 − Y 1 − 1 ij d − − g d ˜ g d and identify it with the usual space-time. Then, from the 1 ) corresponds to a solution with space-time metric given by d x 1 M = ∂ M 5.58 Z N ∂ Z dX →∞ lim M →∞ P lim ). Which one should we pick? Note that all the different choice and identify it with the usual space-time. Then, from the spa P ]. Similarly this solution will correspond to a different Kal a ≡ dX 4 x ≡ , z M a ) can lead to different space-time interpretation upon choos MN E z P H 5.58 ) as follows: = 2 ) = 0. Hence we can pick any 5.54 ds -dimensional slice are related by T-duality. So all the choi With these subtleties in mind we define the conserved energy f The notion of conserved momentum can be defined similarly. We To make this point more precise, consider a solution of doubl n ··· and the Kalb-Ramond field given by ( a ij ˜ space-time perspective, the double field theory solution of given by ( where the role of upper and lower indices has been exchanged i The conservation of momentum is also easy to establish. conserved charges by choosing differentunderstanding sub-spaces of the of physical the interpretation of do thesetheory cons perspective must await a better understanding of the field from the space-time perspective, dimensional sub-spaces of the double space-time. Hence, we spanned by the coordinates ˜ a solution with the space-time metric given by: g of all the coordinates, we can make a different choice. Let us n ∂ quantities associated with spatial translations onquantities the are bou obtained by setting N = 0 and as explained in [ It is now anenergy easy of exercise general to relativity. show that the above expression in spanned by (˜ the perspective of theT-duality. normal space-time However, from they the correspond perspectivewhich to of satisfies double strong field constraint. the The choice of the the choosing a duality frame. tion ( ble field theory by eliminating the integration over a half-d constant generalized metric and dilaton, where the general hyper-surface is given by the following line element: Let us now choosethe the coordinates half-dimensional sub-space of the doub perspective, the solution ( JHEP10(2015)158 = D → ∞ (5.63) (5.62) (5.65) (5.66) (5.64) (5.67) ]. KK- r ] and is 62 56 j , dy 61 i dy as follows: j ˆ 2 A M i d by the following line f X A , to some specific solutions. + zed in a three dimensional ]. ral relativity [ ates ij s: sarily compact) directions as δ 56 ˜ er three directions. As we will z, z and shift vector, it is useful to , , . me dimensions with one spatial i ained previously, i.e., the formu- j f 55 y n as ‘generalized KK-brane’. , y i i + y 2 ˜ y b ˜ z , ij ˜ x d δ a d 1 f a , x = ˜ x f. a + 2 d ˜ x ~ ∇ f. 2 ab δ = = dz ˜ t, t, b d ~ + , r A 2 – 32 – , b 2 2 r h − = f e A dx dz ~ a ∇ × coordinates only given by: j M ˜ y i 1 + dx d y ab ij ˜ t, X 3 so that the doubled space is 20 dimensional, i.e., ( ) = 1 + δ δ , f , r t, ˆ ( 2 N + − , + f ˜ z j = 2 dX ˜ d y ˜ t i ˆ = 1 d ˆ d M i M i ˜ dy y + is not important here but it is easy to see that it vanishes in X d dX i 2 i ij DFT monopole solution was discussed in great detail in [ ˆ N δ A A dt ˆ 2 1 f f M 6 and b , H − + + = = ··· , 2 ds is a harmonic function of = 1 a f = 9). The generalized metric for the DFT monopole is describe To describe this solution, we split the generalized coordin is a three dimensional divergence less vector which satisfie , d i where element: 10 limit. To recast this solution in terms of the lapse function well. Therefore it is appropriate to interpret this solutio inspired by the Kaluza-Kleinmonopole (KK) is monopole a solution solution indimension of compactified gene general and relativity a in divergencesee five localized space-ti below, in the the oth DFT-monopolesubspace has of a the full divergence doubled which space is which locali has other (not neces DFT monopole. A And the solution for dilaton is given by: where This section is devoted to thelae application for of conserved the energy results andIn obt momentum particular in we double will field consider theory, the solutions discussed in [ 5.3 Applications The precise form of JHEP10(2015)158 ) to i y (5.70) (5.76) (5.69) (5.75) (5.68) (5.72) (5.71) (5.73) (5.74) , ab δ vanish and we . The gradient = ) at the bound- 3 d ab Euclidean ( b , y H 2 → ∞ finition of conserved and r , y . ompute the conserved 1 y f. . 0 MN ring with the form of the 2 int. An obvious choice is , = , H 0 h e following way: , fr d ij i 2 δ r y i − lidean ( = , A dz, 0 ) is given by: 1 f , , e LP Z 0 − , 5 ˜ H zdθdφ . 0 5.25 P , . MN ) = ∂ ˆ θ b ij , N j H πhV L z , δ = ˆ ˜ ab xdzd r. M b 1 1 f f = 5 H δ N b δ sin ( d z = 4 2 and then take the limit 2 xd = = = = ) = ) for the DFT monopole solution vanishes. − r 5 , n ˜ r z z MN →∞ e z ij j −→ ˜ ab z X r – 33 – ˜ xdz ˜ = yd n b ( H b b b H 5 − H 3 H , H ˆ

5.61 N i d d ′ d , ˆ S M ) of equation ( P , n = Z i b ∂ ∂X ) it is easy to see that: H ∂X X ˜ n = 0

( the generalized metric reduces to the ﬂat form ,i.e., Y , πh LP 1 a M − 2.31 S H , d , n N L 2 = 4 a d → ∞ ˜ n N j E , d , r 2 2 A , i f − = i , ), after a short calculation, one obtains: 2 2 e A 1 A 4 f A M − 1 f + N = 1 5.60 N ij = = δ 1 + i ˜ z 00 b b f H H f ) coordinates, i.e., = = = = ), i.e., the function ˜ z then provides the correct Jacobian in transforming from the θ, φ ij i 00 zz

b b b b 5.60 H ′ H H H ∂X ) = 0. Using this solution for the strong constraint and the de ∂X

Note that in the limit ··· ( ˜ energy in equation ( ∂ vector takes the following form: deduce that the conserved momentum ( This limit definesenergy the ( boundary at which we do integration to c It is easy to see that the canonical momenta associated with Let us now choose a particular solution of the strong constra where we expect thegeneralized notation metric to in be equation self ( explanatory. By compa A short calculation shows that: record the non zero component of the generalized metric in th We will do the integration at a fixed radius spherical ( It is natural to use spherical coordinates instead of the Euc ary. So the surface element at the boundary can be written as: and JHEP10(2015)158 er ]. (5.79) (5.80) (5.78) (5.81) (5.77) 65 direction. is explains z . The gradient direction. The ). z . πG 1 ) → ∞ 16 , z R 0 , i hich can in principle be direction is compactified e: ventional (5-dimensional) ution which is dual to the omentum associated with , y z as the ‘winding’ coordinate localized NS5-brane [ e that this solution actually red along 0 nterpretation of NS-5 brane , , z to the flat form in the limit rgy and momentum is similar 0 i , ]. A solution which is localized ) = 0. Then a straightforward , y (0 ) reproduces the ADM energy of . From the expression for energy ]. To describe the solution, let us 56 i z ˜ 2 ··· y . ( z 55 R. 2 ˜ 5.76 ∂ 1 z + = ˜ z, z, 2 h 2 . + r 5 z 2 ˜ t, t, r r + direction [ hV 2 2 and then take the limit = z as normal coordinate then the DFT monopole r π = z R – 34 – z p ) describing the boundary is thus given by: ] (up to an overall factor of = 4 M , n ) = 1 + X 64 z as follows: E ( ) = = 1 and realize that the ˜ n Here we discuss the generalized pp-wave solution and , r, z 5 ˆ X i M ( S ( V ˜ t, t, X f X , n In the previous solution, the harmonic function is inde- i S direction as hinted in [ ˜ = n as the ‘normal’ coordinate and , z and so its divergence is not localized along the ˆ a ]. Then the expression ( M z ], if one treats z appearing in the expression for conserved energy. On the oth , n 63 X [ a 56 ˜ n dz h R = . The function M can then be understood as the energy of a single NS5-brane. Th N 5 → ∞ 2 πhV z + 2 r √ direction is obtained if the harmonic function is chosen to b z ≡ We choose the solution of the strong constraint as When reduced to thesolution usual in space-time, string this theory solution smeared has along the the i calculation yields that: This is exactly the result one would expect for the ‘mass’ of a vector in this case is given by: We perform integration at a fixed value of Generalized pp-wave solution. obtain associated conserved energy andcarries momentum. momentum We along will se thesplit ˜ the generalized coordinates in standard KK-monopole calculated in [ Localized DFT monopole. pendent of the coordinate solution corresponds to an infinite array of NS5-branes smea the physical origin of infinite. As discussed in [ The conserved energy is proportional to a volume element of w this solution isR still zero. The generalized metric reduces With this harmonic function,to the what analysis for was conserved done ene in previous subsection. In particular, the m hand if one chooses to treat ˜ coefficient 4 obtained above, one canKK-monopole. in To fact do recover this the we energy set for the con then the DFT monopole solutionNS5-brane corresponds with to KK-circle a along KK-brane the sol winding direction into a circle of radius 2 JHEP10(2015)158 i y 2) ) to pro- − i . → ∞ y

d (5.83) (5.82) (5.90) (5.84) (5.89) (5.88) (5.85) (5.86) (5.87) 1 ′ r . − ∂X ∂X ij

f δ ) with equa- = = ˜ tdz z d 0 ij b 5.85 H + H ˜ z = . , d 0 z ij dtd − rm in the limit δ b 2 . The factor H + 1) dimensional double r ) and ( = 1) d = 3) , icitly as follows: ij . − 0 e spanned by coordinates 2 5.83 z f ) − Euclidean coordinates ( H i f − b d H √ d = 2( g line element in doubled space: ( just gives surface area of ( , y . j 0 = h , f, D , y ˜ + 2 ( ydS i z 0 1 0 + y 2 = , , − b ˜ z 2 ij H d 1 f d (0 δ P fd r 1 − f, ∂ 1 f = + ˜ zdzd 2 = = − LP 2 d = constant = 1 2 1 i H zz dz − d z L b , r d 2 ) 2-dimensional unit sphere. Since the integrand , n H – 35 – r i 3 − N f N f ˜ e − d n − h 2 f, = d , − = = d . z r − j ˜ z − e Y ˜ zz y = N , n 4 1 N coordinates and is given by: . We will use the same procedure as above to obtain d z b d i H − r + (2 i 2 ˜ d ˜ n y = 2 y 2 2 , − d ˜ t , , = 1 + d e ij zz )

N = ) = f δ ′ fd b f = 0 , H X M + f − ( dX − ∂X 1 ∂X j LP 2 f, N √

M S H − dt dy i P + (2 1 and the solution is defined on 2 ) we obtain the following: dX = 2) ∂ 2 dy ˆ N = N L − ij − 00 ˆ M N δ 2.32 b 1 f f H b , d H + depends only on , − 2 f = = ( is the surface element of ··· 1 , f, − 2 2 1 dimensional. The gradient vector is given by: ) and ( − f f p ds 3) so we have d − = 1 = = = d i dS 2.31 d 00 zz 2 d > To integrate over the boundary we use spherical coordinates Again we see that the generalized metric reduces to the flat fo b − H H e The function The components of the generalized metric can be written expl spherical coordinates. So, we have: is independent of the angular coordinates, the integration where The solution for dilaton is given by: vides the appropriate Jacobian for the transformation from space-time. The generalized metric is given by the followin tions ( It is easy to see that, we choose the constant to be unity here. By comparing ( is now where the conserved energy, the only difference being that the spac (for JHEP10(2015)158 d → ∞ (5.97) (5.94) (5.96) (5.95) (5.91) (5.93) (5.92) r are: , i r M y . V i a conjugate to # r actually vanishes. y f d i − z Π z M + 2 2 H sion for the energy should ing the integration as usual rmation and hence it should + dz 1 f ed pp-wave smeared along the e field theory carry equal mo- grand in the boundary integral zi Z − Π . 1 ! d dz ) = 0. The final expression for the . i − Mz r 2 y 1 = 3, the generalized metric becomes Z 1 f ··· H r 2 2 − d − ( d " ) d d − ˜ 1 ∂ d 1 d 2 4 π 2 = 2 − − r − Γ d d − 2 i ) . r r π y f d

(3 ) 4 Γ ). After a short computation one finds that f d − = 0 h 3) Ki 4 cancels and one obtains a finite result in direction given as: 3) 3.4 d – 36 – − − (3 Π − r z 2 Π direction just as expected. − h d (3 = ( d z MK − h z ( h V H h = = are the following: = i = = z z ˜ z i ˜ = 2 z L P has some non-vanishing components which can be computed V MN E N = Π = = Π and the dual ˜ zi KL z z i MN z Π P and the dual ˜ H z = Π = Π = 3 the energy vanishes. For MK z zi i d ≡ H = Π = Π M iz iz V Π Π limit, it is easy to see that the only non zero components of of . A short calculation shows that the momentum conjugate to MN → ∞ H Notice that for To compute the conserved momentum, we need to compute moment r direction. This implies that thementum generalized density pp-wave along solution of doubl the non zero components of Π for the conserved momentum. Let’s now compute the following vector which appears as inte Using this in the formula forwe conserved get momentum momentum and along perform In dimensional unit sphere. Dependence on constant and can be putcorrespond into flat to form zero via energy. abe coordinate understood The transfo as coefficient the in energyz the density carried above by expres the generaliz limit. We choose to solve the strong constraint by and energy is: Momentum conjugate to straightforwardly by using the equation ( JHEP10(2015)158 ric dimensional and d which characterizes is 2 M uble field theory as in N canonical momenta and , we discussed the nature M ed geometry in complete oisson bracket algebra of ies. One can then try to ouble field theory and it is nserved charges associated version of Stokes’ theorem. ry. Starting from a double l value of Hamiltonian. For t way of computing physical plit of the coordinates, the -dimensional doubled space- hich are asymptotically flat. ry and secondary constraints f double field theory. tivity is written in terms of served energy of double field or to ian for double field theory can ion and to move forward one led space-times which are not D e trace of extrinsic curvature. e field theory. However these nstraints do not change under btain a better understanding of ved energy and momentum are t vector and the generalized lapse n. In addition we assumed that the and we want to write the volume integral of a – 37 – M ∂ -dimensional manifold decomposes into the generalized met D -dimensional doubled space, we split the 2 D ). In particular, we assume that our doubled space 1 dimensional boundary − 5.27 d The expression for conserved energy obtained here involves Our construction of conserved charges provides a convenien To deal with the boundary terms arising in double field theory divergence as a boundary integral in terms of a gradient vect Our goal hereequation is ( to write the Stokes’ theorem for the case of do generality. Progress in any ofthe these geometry directions of would double help field o theory. A Generalized Stokes’ theorem fields appearing explicitly.a ADM purely energy geometric of object,Is generalized i.e., same the rela sort boundary oftheory? geometric integral understanding of This possible th is forneeds a the to promising con develop direction a of theory further of investigat surfaces and embeddings in doubl An interesting direction forasymptotically future work flat is but to admitgeneralize consider some the doub notions other of asymptoticwith conserved symmetr these energy asymptotic and symmetries. momenta to co asymptotically flat doubled space-times,constructed notions and of result conser are applied to some known solutionsobservables o associated with aconserved charges particular are solution only of defined for doubl doubled space-times w of the gradient vector toAppropriate the boundary boundary terms are and added gavethen to a shown the generalized that Hamiltonian the of these d boundary terms provide the on-shel We have given thefield canonical theory formulation on for 2 double field theo 6 Conclusions and outlook it has a 2 time evolution. on the doubled spatial hyper-surface,function. the This generalized split shif also requiredfields a are re-definition of independent dilato of thethen dual be time computed coordinate. by following Hamilton thein standard procedure. the canonical Prima formalismsecondary are constraints derived. closes on-shell It implying is that shown the that co the P time into temporal andgeneralized metric spatial on the parts 2 explicitly. With this s JHEP10(2015)158 ) A.1 (A.1) (A.3) (A.4) (A.2) (A.6) (A.5) boundary of ) the bulk term , 1 are coordinates M A.5 − and we deduce that are the coordinates J d 2 M M M , ′ N y the following. ft side of equation ( N is carried by a derivative, X of a doubled space, it is d ) and ( ··· M 2 , M − ch a suitable gradient vector N e J A.4 aracterizing the boundary. In non trivial restrictions on the

= 1 ′ ing an inner product on the doubled ¯ M ∂X ∂X , re is a natural candidate for deﬁning the ,

¯ . . M . Y Y 1 and this will be the focus of our discus- . ) = 0 , = 0 − = 0 S d S M , 2 ··· M M ¯ M ) = constant and d ( M N M N ∂ N X Y M Z ( M ∂ M = – 38 – = ∂ N S M ≡ M N ], we avoid furnishing the doubled space with extra structure N M ′ 66 M ’, i.e., the vector index of X J ··· d 2 , then due to the conditions ( M − M ∂ e are coordinates on N ··· d M is defined as: ’, we can do a partial integration in the boundary integral to 2 = , M M ) is the gradient vector X ∂ ··· N J d , 2 A.1 d ’ contains any number of fields or their products. Since ‘ = 1 We will show that: Z ··· 4 is of the type ‘ ,M M M J . The boundary is specified by X then due to the strong constraint, the bulk integral on the le If the gradient vector should satisfy: vanishes and the boundary integrand is proportional to vanishes. So, we deduce that the gradient vector must satisf where, ‘ a boundary isdeduce zero that: Now, consider should satisfy. M One can try to formulate the boundary integration by introduc • • ∂ 4 M on where, To develop the generalimportant to theory understand of the nature integrationparticular, of on we the will the gradient vector see boundary ch thatgradient vector. the Let strong us constraint now putsN summarize some a set of conditions whi sion below. A.1 Conditions on gradient vector A key element in ( and work with the gradient vector. The gradient vector space and defining theinner notion product, of i.e., a the generalized normal metric vector. [ Although the adapted to the boundary, i.e., the boundary. JHEP10(2015)158 ng , in , we 1 (A.9) (A.8) (A.7) d (A.10) (A.13) (A.12) (A.11) 2 M − then just e . P ξ ′ P X ∂ subspace . We deduce that , 1 = . iest way to see this M ′ d ) M 2 d J − ) X . ′ M ( e , ke: ∂ ξ X ( δ , actually transforms as a M ′ , say d M d d 2 2 J M y, i.e., M J ) − M − ∂ 2 J e X e ( d M 2 d − 2 − J ′ , M − e M 2 e ∂ ′ J − X. X M M d ). We start by showing that the inte- d M ∂ M ∂ 2 2 ∂ P d d

J ξ

′ A.1 P ′ M Z ∂ ∂ ). This fact plays important role in deﬁning P P ∂X ∂X ∂X = ∂X ∂ ξ ) = constant

– 39 – 2 X = = 5.56 ) = ( − = ], under a generalized coordinate transformation, e ] regarding the gradient vector. The boundary of a ′ X d ( S 21 ) is invariant under generalized coordinate transfor- ′ = X 25 M d M M X ∂ transforms with the opposite factor, i.e., 2 ) as follows: ). The strong constraint restricts the fields to depend A.1 ) and ( J d J M M d ) X X 2 of [ 2 ( M A.1 . ′ X d J J − ( 2 d 2 5.55 d e J 2 d S 2 transforms as a scalar density, i.e., ) ′ − − − d e X e 2 M ( M ′ ∂ − d e 2 J M M − ξ ∂ M . The above conditions are then just a consequence of the stro e δ ∂ S X ∂ ) is some function of the coordinates. The gradient vector is ξ d ′ M M δ 2 X ∂ ∂ ( d Z S = ) does not change under generalized diffeomorphisms. The eas I , the integrand transforms as follows: ′ where given by constraint satisfied by only a half-dimensional subspace of the doubled space physical quantities for solutions of double field theory. doubled space, in general, can be specified by a constraint li This confirms the proposal of [ the sense of equations ( the boundary of the doubled space is completely along the the X A.1 → Then as discussed in section 2 deduce that the integrand as a whole transforms like a densit scalar, i.e., By using the fact that So, we deduce that the integral ( mations and we can write X gral ( and the integration measure A.2 General result Let us now turn to the proof of the relation ( Now a short calculation shows that the factor multiplying is to write the integrand of in ( JHEP10(2015)158 5 di- ) = X ( (A.18) (A.19) (A.16) (A.17) (A.15) (A.20) (A.14) M ommons N . We see , Y 1 − d S ), i.e., 2 , d 1 ,X − ′ d = 2 X ′ ( F ,Y X ) can be written as: 1 − Number DE-SC00012567. ··· d dinates and it is given by: d for numerous discussions 2 , oundary integral. ork. I would also like to ac- A.15 2 d redited. . . n Zwiebach. This work is sup- )) is just given by:

′ ,Y M ′ , , 1 S X that the transformed coordinates ( . ∂ M Y M ′ ∂X ) ∂X ∂X ) can be written as:

′ M J X J ( ] via the matrix = d = M N M d . M 2 2 21 ′ A.18 − N J d δ M ′ d ′ 2 e X d 2 d M δ ′

2 2 , the integral ( ′ − − N ′ P = M e e d ,X 2 ∂X ′ ′ ∂X N J ′ −

– 40 – P X X ··· M ′ M X 1 1 , = 2 − N − Y e ′ ) d d ′ 1 ∂X ∂X ′ M 2 2 = − X d d ( ,X N d ′ 1 2 d ′ M ′ ) = Z d 2 Z X − ), which permits any use, distribution and reproduction in J X = = e Z ( ′ M I I = M → N I N = constant, it is easy to do the integration using primed coor d 2 S ,X CC-BY 4.0 This article is distributed under the terms of the Creative C ··· ). ′ , 2 X ( ,X ′ N 1 N ) X ′ X,X ( This is equivalent to the transformation postulated in [ 5 N M Now, we use the fact that Putting this all together we see that the integral ( From this we can find the gradient vector in the un-primed coor F any medium, provided the original author(s) and source are c Now by using the fact that where the integration measurethat on the gradient the vector boundary in is the given primed coordinates by ( since at the boundary nates and one obtains that: Open Access. I would like to thankand Olaf very Hohm insightful for comments throughout suggestingknowledge several the this useful course project comments of and an suggestions thisported by w by Barto the U.S. Department of Energy under grant Contract This is how volume integral of a divergence isAcknowledgments related to the b are adapted to the boundary, i.e., Let us consider the following coordinate transformation so Attribution License ( JHEP10(2015)158 , , , ] ]. , , (1990) 394 (2010) 047 (2012) 080 SPIRE , 10 IN , , [ 03 , (2011) 116 (2011) 086 B 347 cs (1993) 2826 , ral scalars 11 JHEP 188 , JHEP , D 48 arXiv:0904.4664 , , [ (1990) 610 ]. JHEP , Nucl. Phys. , ]. ]. SPIRE B 335 ]. Phys. Rev. IN (2009) 099 , SPIRE ][ 09 IN SPIRE SPIRE ][ IN ]. ]. ]. ]. ]. ]. IN ][ ][ JHEP Nucl. Phys. Prog. Theor. Phys. 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