Generalized geometry and nonlinear realization of generalized diffeomorphism on D-brane effective action
aa r X i v : . [ h e p - t h ] M a y Preprint TU-968
Generalized geometry and nonlinearrealization of generalized diffeomorphism onD-brane effective action
T. Asakawa ♯ , H. Muraki ♭ , S. Sasa ♭ , and S. Watamura ♭ ♯ Department of Integrated Design Engineering,Faculty of Engineering,Maebashi Institute of TechnologyMaebashi, 371-0816, Japan ♭ Particle Theory and Cosmology GroupDepartment of PhysicsGraduate School of ScienceTohoku UniversityAoba-ku, Sendai 980-8578, Japan
Abstract
The characterization of the DBI action of a D p -brane using the generalizedgeometry is discussed. It is shown that the DBI action is invariant under the dif-feomorphism and B -transformation of the generalized tangent bundle of the targetspace. The symmetry is realized non-linearly on the fluctuation of the D -brane. Based on talk given by S.W. at the Workshop on Noncommutative Field Theory and Gravity,Corfu, Greece, September 8-15, 2013. e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] Introduction
It is known that when a symmetry in the high energy theory is spontaneously broken toa smaller symmetry, the low energy effective theory(LEET) is strongly controled by thesymmetry breaking pattern. If the broken symmetry is a continuous symmetry, LEET isdescribed by the Nambu-Goldstone boson [1, 2] and the original symmetry in the highenergy is realized nonlinearly which characterizes the LEET action. This type of argumentcan also be applied to analyse the low energy D-brane action[3]. It is known that the D-brane effective action is given by DBI action. It was argued that the DBI action can becharacterized also as a system with the spontaneous symmetry breaking without refferingto the detail of the string theory [4, 5]. There the Poincare symmetry of the target spaceplays the role of the high energy symmetry and thus they are realized nonlinearly. Intheir formulation, the scalar boson which describe the transvers fluctuation of the braneare the Nambu-Goldstone modes, and on the contrary the gauge boson on the brane is acovariant field of the nonlinear realization of the Poincar´e group of the target space. Thevector field is thus not considered as a Nambu-Goldstone mode [4, 5].In this talk, I will show that the DBI action has an even bigger symmetry in the targetspace and there the vector field also appears as a Nambu-Goldstone mode. This becomespossible if we identify the D-brane as a Dirac structure in the generalized tangent bundleof the target space. We also discuss other consequences on the effective theory of theD-brane obtained from our formulation based on the generalized geometry. This talk isbased on the papers [6, 7].
First let us introduce some concepts appearing in Hitchin’s generalized geometry[8, 9],which we need to discuss our formulation. In generalized geometry, instead of the tangentbundle
T M of M , we consider the generalized tangent bundle T M ⊕ T ∗ M . Its section is ageneralized tangent vector which is denoted by u + ξ where u ∈ Γ( T M ) and ξ ∈ Γ( T ∗ M ).The Lie bracket is generalized to the Dorfman bracket which is not antisymmetric, andthus the Lie algebroid of the tangent vectors is generalized to a Courant algebroid. Asthe symmetry of the Lie bracket is the diffeomorphism, the symmetry of the Dorfmanbracket is called the generalized diffeomorphism which is the diffeo. × B-transformationgenerated by a vector u and a closed 2-form ω ∈ Ω closed . The generator of the generalizeddiffeomorphism is a generalized Lie derivative L ( u,ω ) . A Dirac structure is a subbundle L ⊂ T M ⊕ T ∗ M of rank D such that 1 Isotropic: for all a, b ∈ Γ( L ) , h a, b i = 0, • Involutive: L is closed under the Dorfman bracket,where h a, b i is the canonical inner product. The Dirac structure defines a Lie algebroid. The reason why we identify the D-brane with the Dirac structure is the following. AD-brane is a hyper surface accompanied by a line bundle. Thus, it is described by theembedding map ϕ from the p + 1 dimensional world volume Σ to the D -dimension targetspace M (in the following we consider M = R D for simplicity), ϕ : Σ ∋ σ a → x µ ( σ ) ∈ M , where σ a ( a = 0 , · · · , p ) are brane coordinates and x µ ( µ = 0 , , · · · , D −
1) are coordinatesof the target space. We take the static gauge and denote the brane coordinates by x a = σ a ( a = 0 , · · · , p ) and x i ( σ ) = 0. The fluctuation of the D -brane is then given by scalar fieldsΦ i ( x )( i = p + 1 · · · D −
1) corresponding to the transverse displacements and the gaugefield A a corresponding to the connection of the line bundle.First, a Dirac structure defines a singular foliation on the target space M . For example,the Dirac strucutre given by L p = span { ∂ , ∂ , · · · , ∂ p , dx p +1 , · · · , dx D − } , (2.1)defines a foliation, a leaf of which is a p + 1 dimensional hyperplane. We can identify aleaf as a D-brane then the translational invariance of the foliation is broken, generatingthe Nambu-Goldstone modes.The second reason is that the fluctuation can also be incorporated in the same man-ner. When we take the Dirac structure L p to describe the Dp-brane in static gauge, thefluctuation of the Dp-brane, given by a vector field A a and scalar fields Φ i , can be unifiedinto a generalized connection A ∈ L ∗ p , A = A a dx a + Φ i ∂ i . (2.2)where L ∗ p is the dual Dirac structure of L p . Then, the generalized field strength F of A is given by F = 12 F ab dx a ∧ dx b + ∂ a Φ i dx a ∧ ∂ i ∈ Γ( ∧ L ∗ ) , (2.3)and it defines the deformed Dirac structure as L F = e F L p = span { ∂ a + ∂ a Φ i ∂ i + F ab dx b , dx i − ∂ a Φ i dx a } , (2.4)together with the condition that d F = 0. 2 .3 Symmetry of the Dirac structure Since the generalized diffeomorphism is given by diffeo. × B -field gauge transformation, itsgenerator is a generalized Lie derivative given by L ǫ + λ . It transforms the deformed Diracstructure L F , but the effect can be absorbed into the transformation of the fluctuation δ F only. This corresponds to keep the static gauge, and causes the non-linear transformationlaw for F as δ F ab = ( ∂ [ a + F j [ a ∂ j )( λ b ] − ǫ c F cb ] − λ k F kb ] ) − ( ǫ k − ǫ c F kc ) ∂ k F ab , (2.5a) δ F ja = ( ∂ a + F ia ∂ i )( ǫ j − ǫ c F jc ) − ( ǫ k − ǫ c F kc ) ∂ k F ja . (2.5b)When we evaluate them on the leaf of x i = Φ i ( x a ), they correspond to the transformationlaw given by δA a = λ a − ǫ c F ca + λ k ∂ a Φ k ,δ Φ i = ǫ i − ǫ c ∂ c Φ i . (2.6)where we have imposed the static gauge condition on the fluctuation and the coordinate x i in the parameter is replaced by the field Φ i . As seen from this result, the transformationscaused by ǫ c and λ k are linear in fields. Since they preserve the leaf, the unbrokensymmetry is the worldvolume diffeomorphism and the U (1)-gauge transformation. Onthe other hand, the inhomogeneous terms ǫ i and λ a corresponds to the broken symmetryby specifying a particular leaf, and thus we can interpret that they are Nambu-Goldstonemode. Note that these transformation laws are extension of the non-linearly realizedPoincar´e symmetry in [4, 5]. In particular, the term ǫ i includes a translation along thetransverse direction as well as a Lorentz rotation in a a − i plane. We have derived the non-linearly realized transformation rule of the fluctuation of theD-brane under the generalized diffeomorphism of the target space M . The DBI actionis manifestly invariant under the worldvolume diffeomorphism on the D-brane and the U (1)-gauge transformation, but it actually invariant under the full symmetry of the targetspace. To see this invariance we first write the DBI action with the integration over thetarget space as S DBI = Z µ L DBI δ ( D − p − ( x i − Φ i ( x a )) dx ∧ · · · ∧ dx D − , (2.7)where δ ( D − p − ( x i − Φ i ( x a )) is a Dirac’s delta function seen as a distribution along x i -directions. The Lagrangian is given by L DBI = q det( ϕ ∗ Φ ( g + B ) − F ) ab , (2.8)3here g is the metric and B the 2-form B field,both defined in the target space evaluatedon the leaf defined by the embedding map ϕ Φ (Σ) of L F at x i = Φ i ( x a ).By using the rule (2.6), the transformation of the Lagrangian L DBI under the gener-alized diffeomorphism on the target space M is given by δ L DBI = − ǫ µ ∂ µ L DBI − ( ∂ c ǫ c + ∂ c Φ k ∂ k ǫ c ) L DBI . (2.9)On the other hand, the delta function transforms as δ [ δ ( D − p − ( x i − Φ i )]= − ǫ µ ∂ µ [ δ ( D − p − ( x i − Φ i )] − ( ∂ k ǫ k − ∂ k ǫ c ∂ c Φ k ) δ ( D − p − ( x i − Φ i ) . (2.10)By combining them, we obtain δ (cid:2) L DBI δ ( D − p − ( x i − Φ i ) (cid:3) = − ∂ µ (cid:2) ǫ µ L DBI δ ( D − p − ( x i − Φ i ) (cid:3) . (2.11)The transformation of the integrand in the DBI action (2.7) is a total derivative and thus,the DBI action is invariant under nonlinearly realized full target space diffeomorphismsand B-field gauge transformations. In the discussion, I want to talk about some other consequences following the identificationof the D-brane with a Dirac structure. In the following, we consider the simplest Diracstructure, i.e. L = T M which corresponds to the spacetime filling D-brane. It is knownthat adding a U (1) gauge flux ω as a fluctuation describes a bound state of such a D-braneand lower dimensional branes. As discussed in the previous sections, it defines a deformedDirac structure L ω provided dω = 0, whose section has the form V = v + ω ( v ) , v ∈ Γ( T M ) , (3.1)where ω ( v ) = i v ω . In the generalized geometry, the same Dirac structure can also bedescribed as the deformation of the dual Dirac structure T ∗ M , as V ′ = ξ + θ ( ξ ) , ξ ∈ Γ( T ∗ M ) , (3.2)where the θ is a Poisson 2-vector. We denote the corresponding Dirac structure by L ω .The equivalence of the two Dirac structures requires that there are always two descriptionsof the same vector, i.e. V = V ′ and gives the relation θ = ω − . (3.3)4e can also describe the generalized metric from these two different Dirac structures.The standard description is a description based on T M and given by a tensor E = g + B of T ∗ M ⊗ T ∗ M . On the other hand, we can also describe the same metric structure fromthe Dirac structure L θ . It is characterized by the tensor t of T M ⊗ T M . The relation ofthe two tensors can be derived by the equivalence of the two representations of the metricstructure by v + ( g + B )( v ) = ξ + θ ( ξ ) + t ( ξ ) , (3.4)and this gives the relation t + θ = ( g + B ) − . (3.5)Identifying t with G + Φ ∈ Γ( T M ⊗ T M ), we get the relation appearing in the Seiberg-Witten map [10].The fluctuation on the D-brane bound state has also an alternative description. In thestandard description based on L ω , the fluctuation is a 2-form F . It can be considered asa deformation of the Dirac structure L ω + F , when dF = 0. Now from the Poisson side L θ ,we can consider the fluctuation by the deformation of the Poisson tensor θ by a 2-vectorˆ F . Then the generalized vector in the deformed Dirac structure L θ + ˆ F is given by V = ξ + ( θ + ˆ F )( ξ ) . (3.6)The condition that L θ + ˆ F is again a Dirac structure is now a Maurer-Cartan equation[ θ, ˆ F ] S + 12 [ ˆ F , ˆ F ] S = 0 , (3.7)where [ · , · ] S is the Schouten bracket. This is a Bianchi identity of a new type of therepresentation of U (1) gauge theory. In fact, we can also consider the 1-vector potentialand gauge transformation corresponding to this field strength. Moreover, the explicitrelation between two gauge potentials is obtained, and the gauge-equivalence is shown in[7]. This type of gauge field may be interesting when we consider the non-geometric flux.See for example[11]. Acknowledgments