On global aspects of duality invariant theories: M2-brane vs DFT
G. Abellan, C. Las Heras, M.P. Garcia del Moral, J.M. Pena, A. Restuccia
aa r X i v : . [ h e p - t h ] M a y On global aspects of duality invariant theories:M2-brane vs DFT
G. Abell´an ,a , C. Las Heras ,b , M.P. Garc´ıa del Moral ,c , J.M. Pe˜na ,d ,A. Restuccia , ,e Departamento de F´ısica, Universidad de Antofagasta, Aptdo 02800, Chile. Departamento de F´ısica, Facultad de Ciencias,Universidad Central de Venezuela, A.P. 47270, Caracas 1041-A, Venezuela. Departamento de F´ısica, Universidad Sim´on Bol´ıvar,Apartado 89000, Caracas 1080-A, VenezuelaE-mail: a [email protected]; b [email protected]; c [email protected]; d [email protected]; e [email protected] Abstract.
Supermembrane compactified on a M × T target space is globally described bythe inequivalent classes of torus bundles over torus. These torus bundles have monodromyin SL (2 , Z ) when they correspond to the nontrivial central charge sector and they are trivialotherwise. The first ones contain eight inequivalent classes of M2-brane bundles which at lowenergies, are in correspondence with the eight type II gauged supergravities in 9 D . The relationamong them is completely determined by the global action of T-duality which interchangestopological invariants of the two tori. The M2-brane torus bundles are invariant under SL (2 , Z ) × SL (2 , Z ) × Z . From the effective point of view, there is another dual invarianttheory, called Double Field Theory which describe invariant actions under O ( D, D ). Globally itis formulated in terms of doubled 2 D torus fibrations over the spacetime with a monodromy givenby O ( D, D, Z ). In this note we discuss T-duality global aspects considered in both theories andwe emphasize certain similarities between both approaches which could give some hints towardsa deeper relationship between them.
1. Introduction
In the context of M-theory, String Theory or Effective Field Theories the search for T-dual/U-dual invariant theories is a relevant goal. In this note we want to discuss and briefly compareaspects of the bundle construction of two different duality invariant theories: Supermembranetheory, also called M2-brane theory and Double field theory (DFT). We consider of interestto understand whether or not there is any hint of a connection between them. In [1, 2] itwas argued that a fundamental formulation of string/M-theory should exist in which T- andU-duality symmetries are manifest from the start. In particular, it was argued that manymassive, gauged supergravities cannot be naturally embedded in string theory without such aframework [3–6].Supermembranes are elements of M-theory, for that reason, dualities should appear as discretesymmetries of the theory. In this note we will restrict our analysis to the case of a supermembranetheory formulated on M × T . It contains two different topological sectors one with irreduciblewrapping and a second one that is reducible. The theory corresponding to the irreduciblerapping sector called ’supermembrane with central charges’ [7]. It is consistently defined atquantum level [8], in distinction with the reducible case [9, 10]. At low energies it is describedby the type II gauged supergravity theories [11, 12]. The supermembrane without the centralcharge condition corresponds to the type II maximal supergravity in nine dimensions. Themass operator and the hamiltonian are U-dual invariant [11] and when supermembrane isdimensionally reduced to string theory T-duality for supermembranes agree with the standardone in string theory compactified on a circle [13]. Moreover, for the supermembrane withnontrivial central charge, it was found all the inequivalent classes of torus bundles with SL (2 , Z )monodromy that describes globally the theory, [12]. The M2-brane torus bundles are classifiedby their coinvariants of the base and the fiber and their associated monodromies. T-dualityaction determines the structure of the T-dual M2-brane bundle. M2-brane torus bundles withparabolic monodromy are the unique class of bundles invariant not only locally but also globally,i.e. preserving the coinvariants. This implies that at low energy the T-dual of a type IIBparabolic gauged supergravity is another type IIA parabolic gauged supergravity. The preciseT-duality action on the classification of the M2-brane bundles: elliptic, parabolic, hyperbolicor with trombone monodromy is done in [12]. The construction is always globally well defined,hence geometric.DFT is an effective field theory that describes sigma models on toroidal compactifications inwhich the number of spatial coordinates has been doubled to mimic the effect of winding modesfor string theory. These coordinates form a doubled torus T d with d -coordinates conjugate to themomenta and the other d -coordinates conjugate to the winding modes [2]. In order to preservethe physical number of propagating degrees of freedom of the theory a constrain is imposed. Theglobal formulation of the theory is done in terms of torus fibrations over a spacetime manifoldsuch that the transition functions are evaluated not only under diffeomorphims and gauge shiftsbut also under the T-duality group O ( d, d, Z ) ∼ SL ( d, Z ) × SL ( d, Z ) × Z × Z transformations[14]. The corresponding action is invariant under the duality transformations. String theorycan be consistently defined in those backgrounds called non geometric backgrounds [1, 2]. Suchbackgrounds can arise from compactifications with duality twists [15] or from acting on geometricbackgrounds with fluxes with T-duality [1–3]. In special cases, the compactifications with dualitytwists are equivalent to asymmetric orbifolds which can give consistent string backgrounds[16–20]. Some global aspects of T-duality were analyzed in [3, 14–19, 21] . These torus bundleshave an O ( d, d, Z ) monodromy of the doubled torus over the base spacetime manifold. Examplesof generalized T-folds [16,21] can be obtained by constructing torus fibrations over base manifoldswith singularities or with non-contractible cycles, like for example over a torus.The paper is structured in the following way: in section 2 we are going to review thesupermembrane theory description in terms of bundles: we indicate the local (Hamiltonianand mass operator) and the T-duality global action on the Supermembrane theory. In section 3we review very basic facts of DFT in particular focusing on its global description to introducenext section. In section 4 we compare both approaches indicating differences and similaritiesand we present our conclusions.
2. Supermembrane theory on a symplectic torus bundle
Let us consider a supermembrane theory with central charges formulated in the Light ConeGauge (LCG) on a target space M × T [7, 22, 23]. X n are the embedding maps Σ → M where n = 3 , . . . , X = X + iX from Σ → T . They are scalars parametrizing the transversecoordinates of the supermembrane in the target space. The moduli of the target space 2-torus T are the radius R , and the complex Teichm¨uller parameter τ . The supermembrane has anontrivial embedding over all the coordinates of the target. We define a matrix W = (cid:18) l l m m (cid:19) whose entires are the winding numbers of the embedded supermembrane. When the wrappingof the supermembrane is irreducible det W = n = 0 the theory has discrete spectrum [8]. Forhe case det W = 0 the spectrum is continuous [9]. The topological condition n = 0 defines aprincipal line bundle with first chern class c = n that algebraically implies the existence of anon-vanishing central charge in the supersymmetric algebra of the supermembrane. This sectorof the wrapped supermembrane theory we denote as supermembrane with central charges. TheHamiltonian describing it is found in [11, 23, 24]: H = Z Σ T / M2 √ ρ (cid:20)
12 ( P n √ ρ ) + 12 P P √ ρ + T M2 { X m , X n } + T M2 D X n )( D X n ) (cid:21) ++ Z Σ T / M2 √ ρ (cid:20) + T M2 FF ) (cid:21) + ( n Area T )++ Z Σ T / M2 √ ρ (cid:20) − ΨΓ − Γ n { X n , Ψ } −
12 ΨΓ − Γ D Ψ } −
12 ΨΓ − Γ D Ψ } (cid:21) + Z Σ √ ρ L (cid:20) D ( P √ ρ ) + 12 D ( P √ ρ ) + { X n , P n √ ρ } − { ΨΓ − , Ψ } (cid:21) , (1) where L is a Lagrange multiplier, T M2 is the 11D tension of the supermembrane, ρ is thedeterminant of the spatial part of the non-flat worldvolume two torus of supermembrane Σ.The symplectic bracket is defined as { A , B } = ω ab ∂ a A ∂ b B with ω ab = iǫ ab √ ρ and a, b = z, z thecomplex coordinates defined on Σ. The densities P n are the canonical momenta associated tothe X n and P those of X . Ψ are scalars on the worldvolume but an SO (7) spinor on the targetspace. Γ n are seven Gamma matrices and Γ = Γ + i Γ , denoting by Γ its complex conjugate. F = DA − DA + { A, A } . is a symplectic curvature defined on the base manifold with A aconnection under the infinitesimal symplectomorphism transformation δ ǫ A = D ǫ . See [22, 24]for a detailed analysis. The symplectic covariant derivative is defined as D• = D • + { A, •} with D • = e ar ∂ a • a rotated covariant derivative defined in terms of a zwei-bein e ar as, [11, 23] e ar := − π R ( l r + m r τ )Θ sr ω ba ∂ b c X s , with r, s = 1 , d b X are the harmonic one-formbasis defined on Σ. Symmetries of the Theory
The Hamiltonian is invariant the residual symmetry under AreaPreserving Diffeomorphisms (APD) connected and not connected to the identity. The theory isalso invariant under two different SL (2 , Z ) discrete symmetries: There is a SL (2 , Z ) Σ associatedto the invariance under the change of the basis of the harmonic one forms defined on Σ [25] andthe windings, d b X → Sd b X, W → S − W , (2)with S ∈ SL (2 , Z ). This dependence is encoded in the symplectic covariant derivative ofthe Hamiltonian through the matrix Θ ∈ SL (2 , Z ) [23]. There is a second SL (2 , Z ) T globalsymmetry, the S-duality transformation, associated to an invariance of the mass operator relatedto the target 2-torus T , τ → aτ + bcτ + d , R → R | cτ + d | , A → Ae iϕ τ , W → (cid:18) a − b − c d (cid:19) W , Q → Λ Q , (3)where Λ = (cid:18) a bc d (cid:19) ∈ SL (2 , Z ), e − iϕ τ = cτ + d | cτ + d | and Q = (cid:18) pq (cid:19) the Kaluza Klein (KK)charges of the supermembrane propagating on the target 2-torus T considered. Bundle description of the Supermembrane
The embedding description did in [7, 22, 23] can bealso understood in terms of a symplectic torus bundle over a torus with monodromy in SL (2 , Z ).This global formulation show topological invariants that carry physical information. The totalundle space E is defined in terms of a fiber F = M × T and Σ as the base manifold nontrivially patched. The structure group G is the symplectomorphisms group leaving invariantthe canonical symplectic structure in T . The action of G on F produces a π ( G )-action on thehomology and cohomology of F . In [12], the authors give a geometrical interpretation of theKK charges in terms of the H ( T ) charges. The monodromy of the bundle M is defined as M : π (Σ) → π ( G ) , with G = Symp ( T ) and π ( G ) = SL (2 , Z ) . (4)Consequently M = Λ γ ∈ SL (2 , Z ) and it acts on the homology basis of the T target torus with γ = γ + γ an integer defined in terms of the integer basis of the fundamental group π (Σ).The global symmetries of the theory become restricted by the monodromy. The symplecticconnection A defined on the base manifold transforms with the monodromy as dA → dAe iϕ M where ϕ M = cτ + d | cτ + d | is a discrete monodromy phase for a given modulus τ . In distinction theHamiltonian is invariant. The torus bundles with a given monodromy M are classified accordingto the elements of the twisted second cohomology group H (Σ , Z M ) of the base manifold Σ withcoefficients on the module generated by the monodromy representation acting on the homology ofthe target torus [26]. There is a bijective relation between the elements of the coinvariant group C F = { C a } , a = 1 , . . . , j associated with a particular monodromy group M G . A coinvariantclass in the KK sector is given by C F = { Q + ( M g − I ) b Q } , (5)for any element M g ∈ M , and b Q is any arbitrary element of the KK sector. There is also aninduced action on the cohomology of the base manifold, which corresponds to a ’ monodromy ’group of the winding sector M ∗ G . A coinvariant class in the winding sector is given by C B = { W + ( M ∗ g − I ) c W } , (6)with M ∗ g ∈ M ∗ = Ω M Ω − with Ω = (cid:18) − (cid:19) , and W = (cid:18) lm (cid:19) ∈ H (Σ). The M ∗ actson the fields which define the Hamiltonian, through the matrix Θ = ( V − M ∗ V ) T that appearsin the symplectic covariant derivatives D r [23]. M and M ∗ groups lie in the same equivalenceclass but their respective coinvariants classifying bundles are not equivalent. Then in order tospecify the physical content of the M2-brane on a symplectic torus bundles one also needs todetermine ( C F , C B , M G ). Each coinvariant class is invariant under the action of any element ofthe monodromy group M g ∈ M . Among its elements we have the orbits of any element of theclass. So the coinvariant class may be considered itself as a class of orbits under the action of M . Given a symplectic torus bundle the Hamiltonian of the theory is defined for any orbit ofthe coinvariant class. The T-duality transformation acts on the Hamiltonian H and the mass operator M and ithas also a action on the structure of the bundle consequently on their topological invariantsdescribing the M2-brane theory. Globally the T-duality transforms a bundle into a dual one, byinterchanging the cohomological charges of the torus base manifold into the homological chargesdefined on the torus fiber with dual moduli. T-duality also interchanges the coinvariant class ofthe base and the fiber in the dual T-bundle, ( C F , C B ) = ( e C B , e C F ) , where we denote by tildes thequantities in the dual bundle. In general this transformation becomes non linear. At low energiesthis fact will be reflected in the change of the gauging group associated to the correspondingdual supergravity. In order to analyze it with more detail, we will consider separately the casesof trivial and non trivial monodromy: Trivial monodromy : In this case M = I , and the coinvariant classes, which classify theinequivalent torus bundles, have only one element Q in the KK sector and one element W in the winding sector. The duality transformation on the symplectic torus bundle has anaction on the charges but also on the geometrical moduli. Following the notation of [12],we define dimensionless variables Z = ( T M2 A Y ) / where A = (2 π R ) Imτ is the area of thetarget torus and Y = R Imτ | qτ − p | is a variable proportional to the R radius of the complex torus.The T-duality transformation is given by:The moduli : Z e Z = 1 , e τ = ατ + βγτ + α . (7)The charges : e Q = T Q , f W = T − W , (8)with T = (cid:18) α βγ α (cid:19) ∈ SL (2 , Z ). The symplectic torus bundles are classified in this caseby two integers, the elements Z ⊗ Z and they are in one to one correspondence with the U (1) × U (1) principle bundle over the base manifold. Since the monodromy is trivial, thestructure group may reduce to the group of symplectomorphism homotopic to the identity.The dual transformation is then completed by the transformation of the moduli as given in(7). • Non trivial monodromy:
In this case, the monodromy group M is non trivial and it isan abelian subgroup of SL (2 , Z ). The T-dual transformation maps as before coinvariantclasses on the KK sector onto coinvariant class in the winding sector and each class denotedas [ ] contains several elements. Hence the map between the coinvariant classes is onlydetermined by T which is constructed from one element of each class Q and W respectively.The monodromy M can be parabolic, elliptic or hyperbolic, which can be linearly or non-linearly realized (trombone symmetries). The T-duality transformation is given by (7) andfor the charges:[ f W ] = T [ Q ] , [ e Q ] = T − [ W ] , M Ω → M ∗ , ( C F , C B ) → ( e C B , e C F ) . (9)The variables associated to the geometric moduli Z , Y and their duals are invariant on anorbit generated by M contained in the respective coinvariant class, provided that τ and Q transform as in (3). The symmetry of the Hamiltonian related to the basis of harmonic one-forms of Σ [25] allows to define the class of orbits associated to the winding matrices [ W ].T-duality defines a nonlinear transformation on the charges of the supermembrane since T is constructed from them, in distinction with the SL (2 , Z ) T action on the moduli which isa linear one. The condition Z e Z = 1 ensure that (T-duality) = I . This transformationbecomes a symmetry for Z = e Z = 1 which imposes a relation between the tension, themoduli and the KK charges of the wrapped supermembrane, T M2 = | qτ − p | R ( Imτ ) . Given thevalues of the moduli and the charges, the allowed tension is fixed T M2 . For Z = 1 theHamiltonian and the mass operator of the supermembrane with central charges are invariantunder T-duality: M = ( T M2 ) n A + k Y + ( T M2 ) / H = n e Y + ( T M2 ) k e A + ( T M2 ) / e H, (10)with H = e H . The mass operator of the supermembrane is U-dual invariant. Thought themass operator is always invariant, under T-duality only the parabolic M2-brane bundleslass is invariant since the action of T-duality is linear and the structure of the coinvariants ispreserved, so the bundle. For the rest of the monodromies contained in SL (2 , Z ) with linearor non linear realization the action of T-duality is non linear, the monodromy dual is alwaysin the same monodromy class but the bundle is classified by different coinvariants [12]. Scherk-Schwarz compactifications of supergravity may be expressed in terms of principal fiberbundles over circles with a twisting given by the monodromy [27, 28]. The background possessesa group of global isometries G associated to the compactification manifold over which it isfibered. In the type II gauged supergravities in 9D, the monodromies are associated to the GL (2 , R ) = SL (2 , R ) × R + global symmetry group. In the SL (2 , R ) sector, there are threeinequivalent classes of theories, corresponding to the hyperbolic, elliptic and parabolic SL (2 , R )conjugacy classes, see [27]. The gauging of the R + scaling symmetry called trombone givessupergravities without lagrangians [29, 30]. At quantum level this last symmetry correspondsto an SL (2 , Z ) non linearly realized and gives rise to a different symplectic torus bundle incomparison to the previous constructions in terms of linear representations [12,13]. Indeed, onlywhen the monodromy is parabolic, the T-duality action is linear meanwhile for the elliptic andhyperbolic case, T-duality does not commute with the monodromy but forms a unique class oftorus bundles in the type IIA side with an scaling symmetry A (1). The T-duality of M2-branetrombone bundles maps also in the ’type IIA side’ into two inequivalent classes of M2-branedual trombone bundles. These classes are in perfect agreement with the eight type II gaugedsupergravities, four in the type IIB (elliptic, parabolic, hyperbolic and trombone) and other fourin the type IIA (two trombone, one parabolic and one non abelian with group A (1) [31, 32].
3. Global Aspects of T-duality in Double Field Theory
We review here very general facts of the global description of DFT. It is a reformulation ofsupergravity constructed in such a way that it is T-dual invariant and it makes explicit theduality group O ( D, D ) of the theory [33]. So far, it has been studied extensively for the bosonicsector of the theory. T-duality is a transformation that changes moduli at the same timethat KK charges are interchanged with the winding modes. Winding modes are a propertyof extended objects, hence to introduce T-duality invariance into an effective field theory, thespatial coordinates are duplicated ( x D , e x D ) so that e x i , i = 1 . . . D are conjugated to the windingmodes ω i , i = 1 . . . D [34] and for consistency of the theory many algebraic structures becomeenlarged, see for a review, [35]. The worldsheet description of DFT on tori backgrounds hasbeen studied in [14, 36]. The string action on a toroidal general background is the following [37], S = − π Z π dσ Z dτ (cid:16) √ γγ αβ ∂ α X i ∂ β X j G ij + ǫ αβ ∂ α X i ∂ β X j B ij (cid:17) , (11)where γ αβ is the induced worldsheet metric, B ij is a constant target space 2-form and G ij isthe background metric. The associated Hamiltonian can be written in terms of a 2 D × D generalized metric H G ( E ) constructed in terms of the background metric G ij and the two-form B in a non-trivial way [33] H G ( E ) = (cid:20) G ij − B ik G kl B lj B ik G kj − G ik B kj G ij (cid:21) , (12)such that worldsheet hamiltonian density H can be expressed as H = 12 Z T H G ( E ) Z + N + ¯ N , (13)here N y ¯ N are the left and right number operators and Z = (cid:18) ω i p i (cid:19) is a generalizedmomentum that includes the KK momentum p i and the winding modes ω i , and a backgroundmatrix E ij as E ij ≡ G ij + B ij = (cid:20) E mn g µν (cid:21) , (14)with E mn = G mn + B mn , G mn is the flat metric for the torus, B mn the Kalb-Rammond 2-formcomponents over the n -torus and g µν is background metric on a target space M D − n − , . Thehamiltonian density (13) is O ( D, D ) invariant when it is induced a transformation law on thegeneralized metric H G ( E ′ ) = h H G ( E ) h T , where h is an element of O ( D, D, R ) group andverifies hηh T = η , with η = (cid:18) I D × D I D × D (cid:19) , the O ( D, D, R ) invariant metric. Due to thecharge quantization condition, ω m and p m are restricted to take discrete values. This impliesthat the symmetry group is restricted to be its arithmetic subgroup O ( n, n, Z ), the T-dualitygroup of string theory. An element of the T-duality group h ∈ O ( n, n, Z ) can be expressed interms of an O ( D, D ) representation for that reason one can consider O ( D, D ), as the globalT-duality group.The idea behind DFT is to find a T-duality invariant action effective field theory describingsupergravity. To this end authors in [38, 39], construct an O ( D, D ) tensor H MN , where M, N = 1 , . . . , D are O ( D, D ) curved indices, of the target space metric g ij , the Kalb-Rammond2-form b ij and the dilaton φ , combined as in (12). The dilaton can be expressed as an O ( D, D )singlet e − d = √ ge − φ . A new set of coordinates ˜ x i conjugated to the winding modes ω i are defined and with them some new generalized coordinates X M = (˜ x i , x i ) considering theoriginal and the dual ones are introduced. These coordinates induce the generalized derivatives˜ ∂ M = (cid:16) ∂∂ ˜ x i , ∂∂x i (cid:17) . Additionally the fields depends on this generalized coordinates H MN ( X ) , d ( X )and it is possible to construct the O ( D, D ) invariant action as a generalization of Einstein Hilbertterms [40], S = Z d D Xe − d R , (15)where the generalized curvature is given by R = 4 H MN ∂ M ∂ N d − ∂ M ∂ N H MN − H MN ∂ M d∂ N d + 4 ∂ M H MN δ N d, + 18 H MN ∂ M H KL ∂ N H KL − H MN ∂ M H KL ∂ K H NL . (16)In the same way one generalizes the rest of supergravity terms and to do it is also necessary tomodify several mathematical structures as the Lie derivative among others. Since the physicaldegrees of freedom have been doubled, a constraint is imposed to preserve the correct numberof degrees of freedom. The strong constraint closes a generalized Lie derivative gauge algebra η MN ∂ M ( A ) ∂ N ( B ) = ∂ M ( A ) ∂ M ( B ) = ∂ M ∂ M ( AB ) = 0 , (17)where η MN is the O ( D, D ) invariant metric, A and B represents any DFT field or generalizedLie Derivative Parameter ξ M = (˜ λ i , λ i ) defined by L ξ A M ≡ ξ P ∂ P A M + ( ∂ M ξ P − ∂ P ξ M ) A P , (18) L ξ B M ≡ ξ P ∂ P A M + ( ∂ M ξ P − ∂ P ξ M ) B P . (19)After imposing the constraint, the field configuration depends only of the coordinates of the D -dimensional subspace, which in general is a linear combination of the original x i and the dualcoordinates ˜ x i . lobal description String theory can be consistently defined in backgrounds in which thetransition functions between the patches considers not only diffeomorphisms and gaugetransformations but also T-duality transformations [2]. Those backgrounds that include T-duality transformations are called non geometrical. If one restricts to the case in which eachtransition function patches the charts with an element of GL ( n, Z ) = SL ( n, Z ) × Z ⊂ O ( n, n, Z ),the large diffeomorphisms group acting on the fiber, then the background is geometrical.However, this is not the most general case since in string theory T-duality maps the transitionsfunctions S ∈ GL ( n, Z ) into S ′ = gSg − with g ∈ O ( n, n, Z ) and in general S ′ ∈ O ( n, n, Z ) so thebackground becomes non-geometric. M-theory, as a theory of unification, contains the dualitiesas symmetries of the theory and DFT aims to be an effective description of M-theory, then it isnatural to consider bundles whose transition functions are defined in O ( D, D, Z ). In the DFTglobal formulation that we are considering, the target space M is locally a torus bundle with fiber T D over the base manifold N , is extended by considering another T D on the fiber, where both aresubspaces of a doubled torus T D containing the original one and its dual, forming an extendedtarget space ˜ M that can be interpreted as a torus bundle with fiber T D over the base manifold N . Each of the two torus has an SL ( D, Z ) invariance associated to its mapping class group.˜ M is a 2 D torus bundle with a well defined monodromy in O ( D, D, Z ) ⊂ GL (2 D, Z ), since it iscontained on the large diffeomorphims group of the T n . If M is a geometric background andwe have an O ( D, D, Z ) monodromy on the fiber, then we have a fibration of ˜ M over M [2].However in general only locally one can define the T D fibration with a G and B defined in theinternal torus consequently there is no well defined global geometry on the physical space-time:it is locally the product of a D -torus embedded in T D and a sector of the base manifold N .This kind of construction which comes from torus fibrations over a torus base where the torusfiber undergoes a monodromy lying in the perturbative duality group O ( D, D, Z ), were calledmonodrofolds [41] or T-folds [16, 21].Since the theory contains more degrees of freedom than those physical by choosing apolarization one selects the physical subspace forming a torus contained in the doubled T D foreach point of the base manifold N . T-duality acts on the bundle by changing the polarization,that is, by changing the physical subspace T D ⊂ T D and for each chart U of the manifold N ,there is a local space-time chart U × T D embedded in the doubled space U × T D , while for localT-folds these do not form a spacetime manifold even though the entire doubled space forms amanifold which is a T D bundle over the base manifold N .
4. Discussion: A comparison of the global T-duality action in both approaches
In this section we would like to compare the bundle construction of Supermembrane theorywith respect to the one associated to the Double Field Theory. For the DFT case the bundleconstruction has been mainly explored in the T-dual invariant string theory framework. Itcontains nongeometric vacua solutions [16, 20, 41]. Both theories are clearly different, in spiteof this, their bundle constructions have certain resemblances and differences that we would liketo emphasize. To start with, DFT is an effective field theory while Supermembrane theory ispart of M-theory. Supermembrane theory contains a topological sector (the one associated tothe non trivial central charge) that can be consistently quantized.The Supermembrane bundle that we analyze in this note has the M2-brane worldvolume torusas its base and the fiber is the compactified target space M × T . The transition functionsbetween the charts are given by the symplectomorphisms group preserving the symplectic 2-form on the base, and the bundle can be a trivial principal torus over a torus when thewrapped M2-brane has a vanishing central charge and a nontrivial symplectic torus bundlesover a torus with monodromy in SL (2 , Z ) linearly or non linearly realized when the wrapping isirreducible,i.e. has a nontrivial central charge. The M2-brane torus bundle is specified by theircoinvariants: ( C B , C F ) which are defined in terms of two different monodromies ( M B , M F ) [12].he hamiltonian is consistently defined on these M2-brane bundles. The global action of T-duality maps supermembrane class of bundles into dual ones, the homological charges of thetorus of the fiber associated to the charges by the cohomological charges associated to thewindings of the torus of the base. The tori geometry is not interchanged but the dual moduliof the fiber torus is obtained through the T-duality rules. The mass operator and hamiltonianare U-dual invariant. The monodromy of the bundle is interchanged with the monodromy ofthe base [12]. The monodromy dual remains in the same conjugate class as the original one butin general the coinvariants that define the inequivalent classes of bundles, as we have discussed,are not preserved. The patching is always geometrical and consequently the global picture welldefined. There is a SL (2 , Z ) Σ × SL (2 , Z ) T × Z invariance on the M2-brane bundle, where the Z action is associated to nontrivial discrete interchange of global structures and topologicalinvariants that we have already signaled.On the other hand, in DFT global formulation, the fields are defined over the spacetimecoordinates in distinction with the previous case. Its global description has been developedmainly inspired in the context of the corresponding duality invariant string worldsheet bundle.The bundle contains the spacetime manifold as a base and a doubled torus bundle as a fiber T D .The transition functions in general are not restricted to be just the diffeomorphisms and thegauge invariance but the charts are patched through elements of the T-duality group O ( D, D, Z ).In the case in which duality group is included the torus bundle has a non trivial monodromy.The fibration of the dual torus over the original one is only possible when the monodromy ofthe fiber belongs to the duality group O ( D, D, Z ). In order to make contact between the twoapproaches, let us particularize to the case in which the DFT describes a T = T × T bundlewith O (2 , , Z ) monodromy. The group of duality invariance of the theory O (2 , , Z ) can bedecomposed as SL (2 , Z ) τ × SL (2 , Z ) ρ × Z × Z where SL (2 , Z ) τ is associated τ the Teichm¨ullerparameter of the T fiber, SL (2 , Z ) ρ with ρ the complexified K¨ahler parameter defined in termsof the constant element of the 2-form b and the area A , that is ρ = b + iA parameterizing thedual ˜ T and the Z discrete symmetries acts on both parameters as follows [14], τ ⇔ ρ, ( τ, ρ ) → ( − τ , − ρ ) . (20)In DFT theories, the T-duality action can be understood as a different choice of polarizationwhich selects the physical subspace T D inside the doubled tori T D . It mixes the spatialcoordinates ( X ) with the doubled coordinates ( e X ). In those cases in which T D = T D × T D and the polarization selects as the physical tori the one associated with the spatial coordinates,T-duality intertwine the two tori, interchanging the geometry and not only the moduli underT-duality. Consequence of this the two monodromies defined one associated to the T τ andanother T ρ over N (on top of the O (2 , , Z ) monodromy associated to the T bundle over thebase manifold). These two monodromies become also interchanged under T-duality. It seemsclear that are resemblances between both theories: Both theories realize similar symmetries ina different way, the worldvolume torus in the supermembrane and the dual 2-torus in the DFTare related to the winding charges. In both cases there are two tori on the bundle constructionone of which is related to the torus target and the other to the winding modes, under T-duality in DFT these tori are interchanged but in the supermembrane case only the charges areinterchanged but not the geometry. In both cases the two monodromies associated to each of thetori under T-duality are interchanged. Both theories share the same discrete invariance O (2 , , Z )in the example considered but differently realized since in the case of the Hamiltonian of thesupermembrane, it involves the moduli in a nontrivial way. Finally, DFT is associated genericallyto non geometrical backgrounds since globally it is not well-defined, while the supermembraneformulation -as it also happens for string theory- is always geometrical and globally well defined.Since DFT has its roots in string theory and the five string theories have its origin in thesupermembrane theory, a sector of M-theory, it is quite natural to expect -even when there areifferences-, that DFT describing the factorized doubled torus with a its monodromy in O (2 , , Z )could be related to an effective description of the M2-brane torus bundle with monodromy in SL (2 , Z ). Obviously this hint will need to be confirmed by a more exhaustive study which isoutside of the scope of the present note.
5. Acknowledgements
MPGM is supported by Mecesup ANT1656, Universidad de Antofagasta, (Chile). A.R. andMPGM are partially supported by Projects Fondecyt 1161192 (Chile). J.M. Pe˜na is gratefulto Universidad de Antofagasta for its hospitality and its partial support during part of therealization of this work.
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