T-dualization of type II superstring theory in double space
aa r X i v : . [ h e p - t h ] A p r T-dualization of type II superstring theory in double space ∗ B. Nikoli´c and B. Sazdovi´c † Institute of Physics Belgrade,University of Belgrade,Pregrevica 118, Serbia
April 25, 2017
Abstract
In this article we offer the new interpretation of T-dualization procedure of type IIsuperstring theory in double space framework. We use the ghost free action of type IIsuperstring in pure spinor formulation in approximation of constant background fieldsup to the quadratic terms. T-dualization along any subset of the initial coordinates, x a , is equivalent to the permutation of this subset with subset of the correspondingT-dual coordinates, y a , in double space coordinate Z M = ( x µ , y µ ). Demanding thatthe T-dual transformation law after exchange x a ↔ y a has the same form as initialone, we obtain the T-dual NS-NS and NS-R background fields. The T-dual R-R fieldstrength is determined up to one arbitrary constant under some assumptions. Thecompatibility between supersymmetry and T-duality produces change of bar spinorsand R-R field strength. If we dualize odd number of dimensions x a , such change flipstype IIA/B to type II B/A. If we T-dualize time-like direction, one imaginary unit i maps type II superstring theories to type II ⋆ ones. T-duality is a fundamental feature of string theory [1, 2, 3, 4, 5]. As a consequence ofT-duality there is no physical difference between string theory compactified on a circle ofradius R and circle of radius 1 /R . This conclusion can be generalized to tori of variousdimensions.Mathematical realization of T-duality is given by Buscher T-dualization procedure [2].If the background fields have global isometries along some directions then we can localizethat symmetry introducing gauge fields. The next step is to add the new term in the ∗ Work supported in part by the Serbian Ministry of Education, Science and Technological Development,under contract No. 171031. † e-mail: bnikolic, [email protected] Z M = ( x µ , y µ ) ( µ = 0 , , , . . . , D − x µ and y µ are the coordinates of the D -dimensional initial and T-dual space-time, respectively.Interest for this subject emerged again with papers [21, 22, 23, 24, 25], where T-duality isrelated with O ( d, d ) transformations. The approach of Ref.[16] has been recently improvedwhen the T-dualization along some subset of the initial and corresponding subset of theT-dual coordinates has been interpreted as permutation of these subsets in the doublespace coordinates [26, 27].Let us motivate our interest for this subject. It is well known that T-duality is impor-tant feature in understanding the M-theory. In fact, five consistent superstring theoriesare connected by web of T and S dualities. In the beginning we are going to pay attentionto the T-duality. To obtain formulation of M-theory it is not enough to find all corre-sponding T-dual theories. We must construct one theory which contain the initial theoryand all corresponding T-dual ones.We have succeeded to realize such program in the bosonic case, for both constant andweakly curved background. In Refs.[26, 27] we doubled all bosonic coordinates and obtainthe theory which contains the initial and all corresponding T-dual theories. In such theoryT-dualization along arbitrary set of coordinates x a is equivalent to replacement of thesecoordinates with corresponding T-dual ones y a . So, T-duality in double space becomessymmetry transformation with respect to permutation group.Performing T-duality in supersymmetric case generates new problems. In the presentpaper we are going to extend such approach to the type II theories. In fact, doubling allbosonic coordinates we have unified types IIA, IIB as well as type II ⋆ [28] (obtained byT-dualization along time-like direction) theories. We expect that such a program couldbe a step toward better understanding M-theory.2n the present article we apply the approach of Refs.[26, 27] in the cases of com-plete (along all bosonic coordinates) and partial (subset of the bosonic coordinates) T-dualization of the type II superstring theory [1]. We use ghost free type II superstringtheory in pure spinor formulation [29, 30, 31] in the approximation of constant backgroundfields and up to the quadratic terms. This action is obtained from the general type II su-perstring action [32] which is given in the form of an expansion in powers of fermioniccoordinates θ α and ¯ θ α . In the first step of consideration we will limit our analysis on thebasic term of the action neglecting θ α and ¯ θ α dependent terms. Later, in the discussionof proper fermionic variables, using iterative procedure [32], we take into considerationhigher power terms and restore supersymmetric invariants Π µ ± , d α and ¯ d α as variables inthe theory.Rewriting the T-dual transformation laws in terms of the double space coordinates Z M we introduce the generalized metric H MN and the generalized current J ± M . Thepermutation matrix ( T a ) M N exchanges the places of x a and y a , where index a marks thedirections along which we make T-dualization. The basic demand is that T-dual doublespace coordinates, a Z M = ( T a ) M N Z N , satisfy the transformation law of the same formas initial coordinates, Z M . It produces the expressions for T-dual generalized metric, a H MN = ( T a HT a ) MN , and T-dual current, a J ± M = ( T a J ± ) M . This is equivalent tothe requirement that transformations of the coordinates and background fields, Z M → a Z M , H MN → a H MN and J ± M → a J ± M , are symmetry transformations of the doublespace action. From transformation of the generalized metric we obtain T-dual NS-NSbackground fields and from transformation of the current we obtain T-dual NS-R fields.The supersymmetry case includes the new features in both Buscher and double spaceT-duality approaches. In the bosonic case the left and right world-sheet chiralities havedifferent T-duality transformations. It implies that in T-dual theory two fermionic coor-dinates, θ α and ¯ θ α , and corresponding canonically conjugated momenta, π α and ¯ π α (withdifferent world-sheet chiralities), have different supersymmetry transformations. As it isshown in [33, 34] it is possible to make supersymmetry transformation in T-dual theoryunique if we change one world-sheet chirality sector. So, compatibility between super-symmetry and T-duality can be achieved by action on bar variables with operator a Ω, • ¯ π α = a Ω αβ a ¯ π β . As a consequence of the relation Γ a Ω = ( − d a ΩΓ it follows thatsuch transformations for odd d change space-time chiralities of the bar spinors. In such away operator a Ω for odd d maps type IIA/B to type IIB/A theory. Here d denotes thenumber of T-dualized directions.There is one difference comparing with bosonic string case [26, 27] where all resultsfrom Buscher procedure were reproduced. In the T-dual transformation laws of type IIsuperstring theory the R-R field strength F αβ does not appear. The reason is that R-Rfield strength couples only with the fermionic degrees of freedom which are not dualized.3his is analogy with the term ∂ + x i Π + ij ∂ − x j in the bosonic case, where background fieldΠ + ij couples only with coordinates x i which are undualized [21, 22]. To reproduce Buscherform of the T-dual R-R field strength we should make some additional assumptions.There is one appendix which contains block-wise expressions for tensors used in thisarticle and useful relations. In this section we will consider type II superstring action in pure spinor formulation[29, 30, 31] in the approximation of constant background fields and up to the quadraticterms. Then we will give the overview of the results obtained by Buscher T-dualizationprocedure [33, 34, 6, 7].
The sigma model action for type II superstring of Ref.[32] is of the form S = Z Σ d ξ ( X T ) M A MN ¯ X N + S λ + S ¯ λ , (2.1)where vectors X M and ¯ X N are left and right chiral supersymetric variables X M = ∂ + θ α Π µ + d α N µν + , ¯ X M = ∂ − ¯ θ α Π µ − ¯ d α ¯ N µν − , (2.2)which components are defined asΠ µ + = ∂ + x µ + 12 θ α (Γ µ ) αβ ∂ + θ β , Π µ − = ∂ − x µ + 12 ¯ θ α (Γ µ ) αβ ∂ − ¯ θ β , (2.3) d α = π α −
12 (Γ µ θ ) α (cid:20) ∂ + x µ + 14 ( θ Γ µ ∂ + θ ) (cid:21) , ¯ d α = ¯ π α −
12 (Γ µ ¯ θ ) α (cid:20) ∂ − x µ + 14 (¯ θ Γ µ ∂ − ¯ θ ) (cid:21) , (2.4) N µν + = 12 w α (Γ [ µν ] ) αβ λ β , ¯ N µν − = 12 ¯ w α (Γ [ µν ] ) αβ ¯ λ β . (2.5)Inserting the supermatrix A MN A MN = A αβ A αν E αβ Ω α,µν A µβ A µν ¯ E βµ Ω µ,νρ E αβ E αν P αβ C αµν Ω µν,β Ω µν,ρ ¯ C µν β S µν,ρσ , (2.6)4n (2.1), action gets the expanded form [32] S = Z d ξ h ∂ + θ α A αβ ∂ − ¯ θ β + ∂ + θ α A αµ Π µ − + Π µ + A µα ∂ − ¯ θ α + Π µ + A µν Π ν − + d α E αβ ∂ − ¯ θ β + d α E αµ Π µ − + ∂ + θ α E αβ ¯ d β + Π µ + E µβ ¯ d β + d α P αβ ¯ d β + 12 N µν + Ω µν,β ∂ − ¯ θ β + 12 N µν + Ω µν,ρ Π ρ − + 12 ∂ + θ α Ω α,µν ¯ N µν − + 12 Π µ + Ω µ,νρ ¯ N νρ − + 12 N µν + ¯ C µν β ¯ d β + 12 d α C αµν ¯ N µν − + 14 N µν + S µν,ρσ ¯ N ρσ − (cid:21) + S λ + S ¯ λ . (2.7)The world sheet Σ is parameterized by ξ m = ( ξ = τ , ξ = σ ) and ∂ ± = ∂ τ ± ∂ σ .Superspace is spanned by bosonic coordinates x µ ( µ = 0 , , , . . . ,
9) and fermionic ones θ α and ¯ θ α ( α = 1 , , . . . , π α and ¯ π α are canonically conjugated momentato θ α and ¯ θ α , respectively. The actions for pure spinors, S λ and S ¯ λ , are free field actions S λ = Z d ξw α ∂ − λ α , S ¯ λ = Z d ξ ¯ w α ∂ + ¯ λ α , (2.8)where λ α and ¯ λ α are pure spinors and w α and ¯ w α are their canonically conjugated mo-menta, respectively. The pure spinors satisfy so called pure spinor constraints λ α (Γ µ ) αβ λ β = ¯ λ α (Γ µ ) αβ ¯ λ β = 0 . (2.9)Matrix A MN containing type II superfields generally depends on x µ , θ α and ¯ θ α . Thesuperfields A µν , ¯ E µα , E αµ and P αβ are physical superfields, because their first compo-nents are supergravity fields. The fields in the first column and first row are auxiliarysuperfirlds because they can be expressed in terms of the physical ones [32]. The restones, Ω µ,νρ (Ω µν,ρ ), C αµν ( ¯ C µν α ) and S µν,ρσ , are curvatures (field strengths) for physicalsuperfields.The action from which we start (2.7) could be considered as an expansion in powersof θ α and ¯ θ α . In an iterative procedure presented in [32] it has been shown that eachcomponent in the expansion can be obtained from the previous one. So, for practicalreasons (computational simplicity), in the first step we limit our considerations on the basiccomponent i.e. we neglect all terms in the action containing θ α and ¯ θ α . As a consequence θ α and ¯ θ α terms disappear from Π µ ± , d α and ¯ d α and in the solutions for physical superfieldsjust x -dependent supergravity fields survive. Therefore we lose explicit supersymmetry insuch approximation. Later, when we discuss proper fermionic variables, we would gofurther in the expansion and take higher power terms, which means that supersymmetricinvariants, Π µ ± , d α and ¯ d α , would take the roles of ∂ ± x µ , π α and ¯ π α , respectively.We are going to perform T-dualization along some subset of bosonic coordinates x a .So, we will assume that these directions are Killing vectors. Since ∂ ± x a appears in Π µ ± , d α and ¯ d α , it essentially means that corresponding superfields ( A ab , ¯ E aα , E αa , P αβ ) should5ot depend on x a . This assumption regarding Killing spinors could be extended on allspace-time directions x µ which effectivelly means, in the first step, that physical superfieldsare constant. All auxiliary superfields can be expressed in terms of space-time derivativesof physical supergravity fields [32]. Then, in the first step, auxiliary superfields are zero,because all physical superfields are constant. On the other hand, constant physical super-fields means that their field strengths, Ω µ,νρ (Ω µν,ρ ), C αµν ( ¯ C µν α ) and S µν,ρσ , are zero. Inthis way, in the first step, we eliminated from the action terms containing variables N µν + and ¯ N µν − (2.5).This choice of background fields should be discussed from the viewpoint of space-timefield equations of type II superstring action [35]. Let us pay attention on the space-timefield equations for type II superstring given in Appendix B of [35]. The equation (B.7)from this set of equations represents the backreaction of P αβ on the metric G µν . If wetake constant dilaton Φ and constant antisymmetric NS-NS field B µν we obtain that R µν − G µν R ∼ (P αβ ) µν . (2.10)If we choose the background field P αβ to be constant, in general, we will have constant Riccitensor which means that metric tensor is quadratic function of space-time coordinates i.e.there is back-reaction of R-R field strength on metric tensor. If one wants to cancel non-quadratic terms originating from back-reaction, additional conditions must be imposedon R-R field strength - AdS × S coset geometry or self-duality condition (see the firstreference in [29]).Taking into account above analysis and arguments, our approximation can be realizedin the following way Π µ ± → ∂ ± x µ , d α → π α , ¯ d α → ¯ π α , (2.11)and physical superfields take the form A µν = κ ( 12 G µν + B µν ) , E αν = − Ψ αν , ¯ E αµ = ¯Ψ αµ , P αβ = 2 κ P αβ = 2 κ e Φ2 F αβ , (2.12)where G µν is metric tensor and B µν is antisymmetric NS-NS background field. Conse-quently, the full action S is S = κ Z Σ d ξ (cid:20) ∂ + x µ Π + µν ∂ − x ν + 14 πκ Φ R (2) (cid:21) (2.13)+ Z Σ d ξ (cid:20) − π α ∂ − ( θ α + Ψ αµ x µ ) + ∂ + (¯ θ α + ¯Ψ αµ x µ )¯ π α + 2 κ π α P αβ ¯ π β (cid:21) , where Π ± µν = B µν ± G µν . (2.14)Actions S λ and S ¯ λ are decoupled from the rest and can be neglected in the further analysis.The action, in its final form, is ghost independent.6S-NS sector of the theory described by (2.13) contains gravitation G µν , antisymmetricKalb-Ramond field B µν and dilaton field Φ. In NS-R sector there are two gravitino fieldsΨ αµ and ¯Ψ αµ which are Majorana-Weyl spinors of the opposite chirality in type IIA andsame chirality in type IIB theory. The field F αβ is R-R field strength and can be expressedin terms of the antisymmetric tensors F ( k ) [6, 36, 37, 38] F αβ = D X k =0 k ! F ( k ) Γ αβ ( k ) , h Γ αβ ( k ) = (Γ [ µ ...µ k ] ) αβ i (2.15)where Γ [ µ µ ...µ k ] ≡ Γ [ µ Γ µ . . . Γ µ k ] , (2.16)is completely antisymmetrized product of gamma matrices. The bispinor F αβ satisfieschirality condition, Γ F = ± F Γ , where Γ is a product of gamma matrices in D = 10dimensional space-time and sign + corresponds to type IIA while sign − to type IIBsuperstring theory. Consequently, type IIA theory contains only even rank tensors F ( k ) ,while type IIB odd rank tensors. Because of duality relation, the independent tensors are F (0) , F (2) and F (4) for type IIA while F (1) , F (3) and self-dual part of F (5) for type IIBsuperstring theory. Using mass-shell condition (massless Dirac equation for F αβ ) thesetensors can be solved in terms of potentials F ( k ) = dA ( k − . The factor e Φ2 is in accordancewith the conventions adopted from [39]. Let us start with the action (2.13) and apply standard T-dualization procedure [2, 13, 14].It means that we localize the shift symmetry for some coordinates x a . We substitute theordinary derivatives with covariant ones, introducing gauge fields v aα . Then we add theterm y a F a + − to the Lagrangian in order to force the field strength F a + − to vanish andpreserve equivalence between original and T-dual theories. Finally, we fix the gauge x a = 0and obtain S fix ( v a ± , x i , θ α , ¯ θ α , π α , ¯ π α ) = Z Σ d ξ (cid:20) κv a + Π + ab v b − + κv a + Π + aj ∂ − x j + κ∂ + x i Π + ib v b − + κ∂ + x i Π + ij ∂ − x j + 14 π Φ R (2) − π α Ψ αb v b − + v a + ¯Ψ αa ¯ π α − π α ∂ − ( θ α + Ψ αi x i ) + ∂ + (¯ θ α + ¯Ψ αi x i )¯ π α + 12 κ e Φ2 π α F αβ ¯ π β + κ v a + ∂ − y a − v a − ∂ + y a ) (cid:21) . (2.17)Varying the gauge fixed action with respect to the Lagrange multipliers y a we get thesolution for gauge fields in the form v a ± = ∂ ± x a , (2.18)7hile varying with respect to the gauge fields v a ± we have v a ± = − κ ˆ θ ab ± Π ∓ bi ∂ ± x i − κ ˆ θ ab ± ∂ ± y b ± θ ab ± Ψ α ± b π ± α . (2.19)Substituting v a ± in (2.17) we find S fix ( y a , x i , θ α , ¯ θ α , π α , ¯ π α ) = Z Σ d ξ " κ ∂ + y a ˆ θ ab − ∂ − y b + κ ∂ + y a ˆ θ ab − Π + bj ∂ − x j − κ ∂ + x i Π + ia ˆ θ ab − ∂ − y b + 14 π Φ R (2) + κ∂ + x i (Π + ij − κ Π + ia ˆ θ ab − Π + bj ) ∂ − x j − π α ∂ − ( θ α + Ψ αi x i − αa ˆ θ ab − Π + bj x j − Ψ αa ˆ θ ab − y b )+ ∂ + (¯ θ α + ¯Ψ αi x i + 2 ¯Ψ αa ˆ θ ab + Π − bj x j + ¯Ψ αa ˆ θ ab + y b )¯ π α + 2 π α Ψ αa ˆ θ ab − ¯Ψ βb ¯ π β + 12 κ e Φ2 π α F αβ ¯ π β (cid:21) . (2.20)Before we read the T-dual background fields, we must express this action in terms of theappropriate spinor coordinates, which we will discuss in the next subsections.Combining two solutions for gauge fields (2.18) and (2.19) we obtain transformationlaw between initial x a and T-dual coordinates y a ∂ ± x a ∼ = − κ ˆ θ ab ± Π ∓ bi ∂ ± x i − κ ˆ θ ab ± ( ∂ ± y b − J ± b ) . (2.21)Its inverse is solution of the last equation in terms of y a ∂ ± y a ∼ = − ∓ ab ∂ ± x b − ∓ ai ∂ ± x i + J ± a , (2.22)where we use ∼ = to emphasize that these are T-duality relations. Here we introduced thecurrent J ± µ in the form J ± µ = ± κ Ψ α ± µ π ± α , (2.23)where Ψ α + µ ≡ Ψ αµ , Ψ α − µ ≡ ¯Ψ αµ , π + α ≡ π α , π − α ≡ ¯ π α , (2.24)and the expression ˆ θ ab ± is defined in (A.9). One can see from (2.21) and (2.22) that left and right chiralities transform differently inT-dual theory. As a consequence, in T-dual theory we will have two types of vielbeins,two types of Γ-matrices, two types of spin connections and two types of supersymmetrytransformations. We want to have the single geometry in T-dual theory. So, we will showthat all these different representations of the same variables can be connected by Lorentztransformations [33, 34]. 8 .3.1 Two sets of vielbeins in T-dual theory
The T-dual transformations of the coordinates (2.22) we can put in the form ∂ ± y a ∂ ± x i ! = − ∓ ab − ∓ aj δ ij ! ∂ ± x b ∂ ± x j ! + J ± a ! , (2.25)which can be rewritten as ∂ + ( a X ) ˆ µ = ( ¯ Q − T ) ˆ µν ∂ + x ν + J +ˆ µ , ∂ − ( a X ) ˆ µ = ( Q − T ) ˆ µν ∂ − x ν + J − ˆ µ , (2.26)where we introduced the T-dual variables a X ˆ µ = { y a , x i } . Here and further on the leftsubscript a denotes the T-dualization along x a directions. For coordinates which containboth x i and y a we will use ”hat” indices ˆ µ, ˆ ν . The matrices Q ˆ µν = κ ˆ θ ab + − κ Π − ic ˆ θ cb + δ ij ! , ¯ Q ˆ µν = κ ˆ θ ab − − κ Π + ic ˆ θ cb − δ ij ! , (2.27)and theirs inverse Q − µ ˆ ν = − ab − ib δ ji ! , ¯ Q − µ ˆ ν = + ab + ib δ ji ! , (2.28)perform T-dualization for vector indices.Note that different chiralities transform with different matrices Q ˆ µν and ¯ Q ˆ µν . So, thereare two types of T-dual vielbeins a e a ˆ µ = e aν ( Q T ) ν ˆ µ , a ¯ e a ˆ µ = e aν ( ¯ Q T ) ν ˆ µ , (2.29)with the same T-dual metric a G ˆ µ ˆ ν ≡ ( a e T η a e ) ˆ µ ˆ ν = ( QGQ T ) ˆ µ ˆ ν = a ¯ G ˆ µ ˆ ν ≡ ( a ¯ e T η a ¯ e ) ˆ µ ˆ ν = ( ¯ QG ¯ Q T ) ¯ µ ˆ ν . (2.30)The Lorentz indices are underlined (denoted by a, b ).The two T-dual vielbeins are equivalent because they are related by particular localLorentz transformation a ¯ e a ˆ µ = Λ ab a e b ˆ µ , Λ ab = e aµ ( Q − ¯ Q ) T µν ( e − ) ν b . (2.31)From (2.27) and (2.28) we have( Q − ¯ Q ) T µν = δ ab + 2 κ ˆ θ ac + G cb κ ˆ θ ac + G cj δ ij ! , (2.32)which produces Λ ab = δ ab − ω ab , ω ab = − κe aa ˆ θ ab + ( e T ) bc η cb . (2.33)It satisfies definition of Lorentz transformationsΛ T η Λ = η = ⇒ det Λ ab = ± . (2.34)After careful calculations we have det Λ ab = ( − d , where d is the number of dimensionsalong which we perform T-duality. 9 .3.2 Two sets of Γ -matrices in T-dual theory Because in T-dual theory there are two vielbeins, it must also be two sets of Γ-matricesin curved space a Γ ˆ µ = ( a e − ) ˆ µa Γ a = ( a e − Γ) ˆ µ , a ¯Γ ˆ µ = ( a ¯ e − ) ˆ µa Γ a = ( a ¯ e − Γ) ˆ µ . (2.35)They are related by the expression a ¯Γ ˆ µ = a Ω − a Γ ˆ µ a Ω , (2.36)where a Ω is spinorial representation of the Lorentz transformation a Ω − Γ a a Ω = (Λ − ) ab Γ b . (2.37) The spin connection can be expressed in terms of vielbein as ω µab = 12 ( e νa c bµν − e νb c aµν ) − e ρa e σb c cρσ e cµ , (2.38)where c aµν = ∂ µ e aν − ∂ ν e aµ . (2.39)So, in T-dual theory there are two spin connections, defined in terms of two vielbeins. Asa consequence of (2.31) they are related as a ¯ ω ˆ µab = Λ ac a ω ˆ µcd (Λ − ) db + Λ ac ∂ ˆ µ (Λ − ) cb . (2.40)It is useful to introduce the spin connection in the form ω µ = ω µab Γ ab , (2.41)where Γ ab = Γ a Γ b − Γ b Γ a . (2.42)Then from (2.37) for a Ω = const we obtain a ¯ ω ˆ µ = a Ω − a ω ˆ µ a Ω . (2.43)10 .3.4 Single form of supersymmetry invariants in T-dual theory and newspinor coordinates So far we use the action from Ref.[32] which is an expansion in powers of θ α and ¯ θ α . Weperformed the procedure of bosonic T-dualization using first term in the expansion i.e. θ α and ¯ θ α independent part of the action. Consequently, supersymmetric invariants, Π µ ± , d α and ¯ d α , in that approximation became ∂ ± x µ , π α and ¯ π α . But if we would take higherpower terms into consideration, then these invariants would appear again in the theory.Consequently, we can use these invariants to find proper spinor variables.From compatibility between supersymmetry and T-duality we will find appropriatespinor variables changing the bar ones. We are not going to apply such procedure tobackground fields which transformation we will find from T-dualization. In subsection 2.5we will check that both T-dual gravitinos satisfy single supersymmetry transformationrule.Note that according to [29, 40] fermionic coordinates, θ α and ¯ θ α , and their canonicallyconjugated momenta, π α and ¯ π α , are parts of supersymmetry invariant variables d α = π α −
12 (Γ µ θ ) α ( ∂ + x µ + 14 θ Γ µ ∂ + θ ) ¯ d α = ¯ π α −
12 (Γ µ ¯ θ ) α ( ∂ − x µ + 14 ¯ θ Γ µ ∂ − ¯ θ ) . (2.44)In T-dual theory, as a consequence of two types of Γ matrices, there are two typessupersymmetry invariant variables a d α = a π α −
12 ( a Γ ˆ µ a θ ) α ( ∂ + a X ˆ µ + 14 a θ a Γ ˆ µ ∂ + a θ ) , (2.45) a ¯ d α = a ¯ π α −
12 ( a ¯Γ ˆ µ a ¯ θ ) α ( ∂ − a X ˆ µ + 14 a ¯ θ a ¯Γ ˆ µ ∂ − a ¯ θ ) . (2.46)We want to have both expressions with the same Γ matrices. Using relation (2.36) we canrewrite bar expressions as( a Ω a ¯ d ) α = ( a Ω a ¯ π ) α −
12 ( a Γ ˆ µa Ω a ¯ θ ) α ( ∂ − a X ˆ µ + 14 a ¯ θ a Ω − a Γ ˆ µ a Ω ∂ − a ¯ θ ) . (2.47)So, if we preserve expressions for a θ α = θ α and a π α = π α , change bar variables • ¯ θ α ≡ a Ω αβ a ¯ θ β , • ¯ π α ≡ a Ω αβ a ¯ π β , (2.48)and take Ω = 1 , (2.49)the transformation with bar variables will obtain the same form as those without barin a d α . Consequently, T-dual supersymmetric invariant variables a d α and a Ω αβ a ¯ d β areexpressed in unique form in terms of true T-dual spinor variables θ α , π α , • ¯ θ α and • ¯ π αa d α = d α , • ¯ d α = a Ω αβ ¯ d β = • ¯ π α −
12 ( a Γ ˆ µ • ¯ θ ) α ( ∂ − a X ˆ µ + 14 • ¯ θ a Γ ˆ µ ∂ − • ¯ θ ) , (2.50)if condition (2.49) is satisfied. 11 .3.5 Spinorial representation of the Lorentz transformation In order to find expressions for bar spinors in T-dual background we should first solveequation (2.37) and find expression for a Ω. We will do it for B µν →
0, so that ˆ θ ab + →− κ ( e G − ) ab , where e G ab is ab component of G µν . Then from (2.33) it follows a ω ab → e aa ( e G − ) ab ( e T ) bb ≡ a P ab , (2.51)where a P ab is some a dependent projector on the ab subspace a P ac a P cb = a P ab . If weintroduce Γ-matrices in curved spaceΓ µ = ( e − ) µa Γ a , (2.52)we can rewrite expression (2.37) in the form a Ω Γ µ = h Γ µ − e − ) µa a P ab Γ b i a Ω . (2.53)To simplify derivation from now on we will suppose that metric tensor is diagonal. Then( e − ) µa a P ab = δ µa ( e − ) aa and we have a Ω Γ µ = [Γ µ − δ µa Γ a ] a Ω . (2.54)For µ = a and µ = i we obtain a Ω Γ a = − Γ a a Ω , a Ω Γ i = Γ i a Ω . (2.55)The Γ-matrices in curved space for diagonal metric satisfy the algebra { Γ a , Γ b } = 2( G − ) ab , { Γ a , Γ i } = 0 , { Γ i , Γ j } = 2( G − ) ij . (2.56)We should find such a Ω that anticommutes with all matrices Γ a and commutes withall matrices Γ i . Let us first introduce Γ matrix asΓ = ( i ) D ( D − Q D − µ =0 G µµ ε µ µ ...µ D Γ µ Γ µ . . . Γ µ D , (2.57)where normalization constant is chosen so that Γ satisfies the condition (Γ ) = 1.Then we define analogy of Γ matrix in subspace spanned by T-dualized directions a Γ = ( i ) d ( d − d Y i =1 Γ a i = ( i ) d ( d − Γ a Γ a · · · Γ a d , (2.58)so that ( a Γ) = d Y i =1 G a i a i = 1 Q di =1 G a i a i . (2.59)12heir commutation (anticommutation) relations with one Γ matrix depend on numberof coordinates d , along which we perform T-dualizations. Therefore we have a Γ Γ a = ( − d +1 Γ a a Γ , a Γ Γ i = ( − d Γ i a Γ , (2.60)which means that the solution of eq.(2.55) is proportional to a Ω ∼ a Γ (Γ ) d . (2.61)Taking into account (2.49), a Ω = 1, we obtain a Ω = vuut d Y i =1 G a i a i a Γ ( i Γ ) d . (2.62)This is a general solution. Note that for a T a = 0 we have a Ω a Ω = ( − d d a Ω, where a = a S a .When the number of coordinates along which we perform T-duality is even ( d =2 k ), we have a Ω = ( − d qQ di =1 G a i a i a Γ. As a consequence of the relation Γ a Ω =( − d a Ω Γ we can conclude that in that case bar spinors preserve chirality. Whenthe number of coordinates along which we perform T-duality is odd ( d = 2 k + 1), wehave a Ω = ( − d − qQ di =1 G a i a i i a Γ Γ . As a consequence of the above relation suchtransformation changes chirality of the bar spinors.In the particular case, when we perform T-dualization along only one direction, x a ,then a Γ → Γ a , d → a Ω = i q G a a Γ a Γ . (2.63)This is the case of the transition between IIA and IIB theory, when T-duality changechirality of the bar spinors.When we perform T-dualization along all coordinates then d → D = 10, a Γ → Γ qQ D − µ =0 G µµ and from (2.62) we obtain ⋆ Ω = ( − D Γ = − Γ . (2.64) We have already learned that in order to have compatibility between supersymmetry andT-duality, we should choose the dual bar variables with ”bullet” in accordance with (2.48).So, before we read the T-dual background fields, we will reexpress the action (2.20) in terms13f the appropriate spinor coordinates (2.48) which, with the help of the relation a Ω = 1,produces a S ( y a , x i , θ α , • ¯ θ α , π α , • ¯ π α ) = Z Σ d ξ ( κ ∂ + y a ˆ θ ab − ∂ − y b + κ ∂ + y a ˆ θ ab − Π + bj ∂ − x j − κ ∂ + x i Π + ia ˆ θ ab − ∂ − y b + 14 π Φ R (2) + κ∂ + x i (Π + ij − κ Π + ia ˆ θ ab − Π + bj ) ∂ − x j − π α ∂ − ( θ α + Ψ αi x i − αa ˆ θ ab − Π + bj x j − Ψ αa ˆ θ ab − y b )+ ∂ + [ • ¯ θ γa Ω γα + ¯Ψ αi x i + 2 ¯Ψ αa ˆ θ ab + Π − bj x j + ¯Ψ αa ˆ θ ab + y b ] a Ω αβ • ¯ π β + 2 π α Ψ αa ˆ θ ab − ¯Ψ βb a Ω βγ • ¯ π γ + 12 κ e Φ2 π α F αβ a Ω βγ • ¯ π γ (cid:27) . (2.65)Consequently, applying the Buscher T-dualization procedure [2] along bosonic coordi-nates x a of the action (2.13) the T-dual action obtains the form a S = Z Σ d ξ (cid:20) κ∂ + ( a X ) ˆ µ a Π ˆ µ ˆ ν + ∂ − ( a X ) ˆ ν + 14 π a Φ R (2) (2.66) − π α ∂ − [ θ α + a Ψ α ˆ µ ( a X ) ˆ µ ] + ∂ + [ • ¯ θ α + a ¯Ψ α ˆ µ ( a X ) ˆ µ ] • ¯ π α + 12 κ e a Φ2 π α a F αβ • ¯ π β (cid:21) , where ( a X ) ˆ µ = ( y a , x i ), a Ψ α ˆ µ = ( a Ψ αa , a Ψ αi ) and a ¯Ψ α ˆ µ = ( a ¯Ψ αa , a ¯Ψ αi ).Now, we are ready to read the T-dual background fields a Π ab ± = κ θ ab ∓ , (2.67) a Π ± ia = − κ Π ± ib ˆ θ ba ∓ , a (Π ± ) ai = κ ˆ θ ab ∓ Π ± bi , (2.68) a Π ± ij = Π ± ij − κ Π ± ia ˆ θ ab ∓ Π ± bj , (2.69) a Ψ αa = κ ˆ θ ab + Ψ αb , a ¯Ψ αa = κ a Ω αβ ˆ θ ab − ¯Ψ βb , (2.70) a Ψ αi = Ψ αi − κ Π − ib ˆ θ ba + Ψ αa , a ¯Ψ αi = a Ω αβ ( ¯Ψ βi − κ Π + ib ˆ θ ba − ¯Ψ β ) , (2.71) e a Φ2 a F αβ = ( e Φ2 F αγ + 4 κ Ψ αa ˆ θ ab − ¯Ψ γb ) a Ω γβ , (2.72)when a Ω is defined in (2.62).The dilaton transformation in term Φ R (2) originates from quantum theory and will bediscussed in subsection 2.6. Note that in the expressions for T-dual fields a ¯Ψ αa , a ¯Ψ αi and a F αβ the matrix a Ω appearsas a consequence of T-dualization procedure and adoptions of ”bullet” spinor coordinates.In Refs.[33, 34] it appears as a consequence of compatibility between supersymmetry andT-duality. 14upersymmetry transformation of gravitino is expressed in terms of covariant deriva-tives δ ε Ψ αµ = D µ ε α + · · · , δ ¯ ε ¯Ψ αµ = D µ ¯ ε α + · · · , (2.73)with the same covariant derivative on both left and right spinors D µ = ∂ µ + ω µ . (2.74)In the T-dual theory, as a consequence of two kinds of spin connections, there are twokind of covariant derivatives a D ˆ µ = ∂ ˆ µ + a ω ˆ µ , a ¯ D ˆ µ = ∂ ˆ µ + a ¯ ω ˆ µ , (2.75)such that a δ ε a Ψ α ˆ µ = a D ˆ µ ε α , a ¯ δ ¯ ε a ¯Ψ α ˆ µ = a ¯ D ˆ µ ¯ ε α . (2.76)Let us show that improvement with a Ω in transformation of bar gravitionos just turns a ¯ D ˆ µ to a D ˆ µ . In fact, from a ¯ δ ¯ ε a ¯Ψ α ˆ µ = a Ω αβ (cid:16) ∂ ˆ µ ¯ ε β + a ¯ ω ˆ µβγ ¯ ε γ (cid:17) , (2.77)with the help of (2.43), for constant a Ω, we have a ¯ δ ¯ ε a ¯Ψ α ˆ µ = ∂ ˆ µ ( a Ω αβ ¯ ε β ) + a ω ˆ µαβ a Ω βγ ¯ ε γ = a D ˆ µ ( a Ω αβ ¯ ε β ) = a δ a Ω¯ ε a ¯Ψ α ˆ µ . (2.78)Therefore, it is clear that in order to preserve the same spin connection for both chiralitieswe should additionally change bar supersymmetry parameter • ¯ ε α ≡ ( a Ω) αβ a ¯ ε β . (2.79) In this subsection we will find transformation laws for pure spinors, λ α and ¯ λ α , which arethe main ingredient of pure spinor formalism.It is well known that pure spinors satisfy so called pure spinor constraints λ α (Γ µ ) αβ λ β = 0 , ¯ λ α (Γ µ ) αβ ¯ λ β = 0 . (2.80)After T-dualization they turn into a λ α ( a Γ ˆ µ ) αβ a λ β = 0 , a ¯ λ α ( a ¯Γ ˆ µ ) αβ a ¯ λ β = 0 . (2.81)The relation between matrices a Γ ˆ µ and a ¯Γ ˆ µ is given in (2.36). In order to have bothexpressions with same gamma matrices, as before, we preserve the expression for unbarvariables a λ α = λ α . (2.82)15nd change bar variables • ¯ λ α = a Ω αβ a ¯ λ β . (2.83)The varables w α and ¯ w α are canonically conjugated momenta to the pure spinors λ α and ¯ λ α , respectively. The transformation laws for pure spinor momenta can be found fromthe expressions for N µν + and ¯ N µν − (2.5) which would appear in the action if we would takehigher power terms in θ α and ¯ θ α . After T-dualization these expressions become a N +ˆ µ ˆ ν = 12 a w α ( a Γ [ˆ µ ˆ ν ] ) αβ a λ β , a ¯ N − ˆ µ ˆ ν = 12 a ¯ w α ( a ¯Γ [ˆ µ ˆ ν ] ) αβ a ¯ λ β . (2.84)Using Eq.(2.36) and definition of Γ [ µν ] (2.16) we see that relation between a Γ [ˆ µ ˆ ν ] and a ¯Γ [ˆ µ ˆ ν ] is the same as between gamma matrices (2.36). As in the previous case, in order to haveunique set of gamma matrices, we do not change unbar variables a w α = w α . (2.85)while choose bar variables in the form • ¯ w α = a Ω αβ a ¯ w β . (2.86)Let us note that free field actions S λ and S ¯ λ are invariant under T-dualization because a Ω = 1. To find the T-dual transformation laws for antisymmetric fields we will start with expres-sion (2.72). First, as it is explained in Refs.[2, 41] the quantization procedure producesthe well known shift in the dilaton transformation a Φ = Φ − ln det(2Π + ab ) = Φ − ln s det G ab det a G ab . (2.87)Together with (2.72) it gives relation between initial and T-dual background fields a F αβ = s det G ab det a G ab ( F αγ + 4 e − Φ2 κ Ψ αa ˆ θ ab − ¯Ψ γb ) a Ω γβ . (2.88)For B µν = 0 we have a G ab = ( G − E ) ab = ( G − ) ab , and consequently q det G ab det a G ab = p (det G ab ) = p | det G ab | . It is important to stress that unlike in expression (2.62) for a Ω here we haveabsolute value under square root. For diagonal metric G µν we have det G ab = Q di =1 G a i a i and taking into account expression (2.62) we find a F αβ = i d vuut sign ( d Y i =1 G a i a i ) d Y i =1 G a i a i (cid:16) F αγ + 4 e − Φ2 κ Ψ αa ˆ θ ab − ¯Ψ γb (cid:17) ( a Γ Γ d ) γ β . (2.89)16ote that we are going to T-dualize all D -directions. Then it is necessary to performT-dualization along time-like direction. Here the above square root has important conse-quences. For our signature (+ , − , − , . . . , − ), the square of the field strength ( a F αβ ) and,consequently the square of all antisymmetric fields will change the sign when we performT-dualization along time-like direction. This is just what we need to obtain Type II ⋆ theories in accordance with Ref.[28].In a simple case when gravitino fields and Kalb-Ramond field are zero and metric isdiagonal we will express transition from type IIB to type IIA theory. Taking d = 1 wehave a F αβ = i q sign ( G aa ) G aa F αγ (Γ Γ a ) γ β . (2.90)Let us choose type IIB as a starting theory. The matrix Γ turns F ( n ) to F (10 − n ) where ( F ( n ) ) αβ = 1 n ! F µ µ ··· µ n (Γ [ µ µ ··· µ n ]) αβ . (2.91)As a consequence of the chirality condition F Γ = − Γ F the independent tensors are F (1) , F (3) and self dual part of F (5) . So we can write F αγ (Γ ) γ β = (cid:18) F (1) + F (3) + 12 F (5) (cid:19) αβ , (2.92)Similarly, in T-dual theory (here it is IIA) we have a F αβ = (cid:16) a F (2) + a F (4) (cid:17) αβ , (2.93)where now ( a F ( n ) ) αβ = 1 n ! a F ˆ µ ˆ µ ··· ˆ µ n ( a Γ [ˆ µ ˆ µ ··· ˆ µ n ]) αβ . (2.94)The Γ-matrices on both sides are defined in curved space. For initial theory it is just(2.52) while for T-dual theory it is defined in the first relation (2.35) as a Γ ˆ µ = ( a e − ) ˆ µa Γ a .As a consequence of the first relation (2.29) between vielbeins a e a ˆ µ = e aν ( Q T ) ν ˆ µ we canfind the relation between Γ-matrices a Γ ˆ µ = ( Q − T ) ˆ µν Γ ν , (2.95)which produces ( a F ( n ) ) αβ = 1 n ! ( Qa F ) µ µ ··· µ n (Γ [ µ µ ··· µ n ] ) αβ , (2.96)where ( Qa F ) µ µ ··· µ n = a F ˆ µ ˆ µ ··· ˆ µ n ( Q − T ) ˆ µ µ ( Q − T ) ˆ µ µ · · · ( Q − T ) ˆ µ n µ n . (2.97)Using the standard relation between Γ-matricesΓ [ µ µ ··· µ n ] Γ a = Γ µµ µ ··· µ n a − n − G a [ µ n Γ µ µ ··· µ n − ] , (2.98)17e obtain F ( n ) Γ a = 1 n ! F µ µ ··· µ n Γ [ µ µ ··· µ n a ] − n − F µ µ ··· µ n − a Γ [ µ µ ··· µ n − ] . (2.99)So, from (2.90), (2.92), (2.93), (2.96), (2.97) and (2.99) we can find general relationconnected antisymmetric fields of Type IIA and type IIB theories a F ˆ µ ˆ µ ··· ˆ µ n = p signG aa G aa (cid:0) nF µ µ ··· µ n − δ aµ n − F µ µ ··· µ n a (cid:1) ( Q T ) µ ˆ µ ( Q T ) µ ˆ µ · · · ( Q T ) µ n ˆ µ n . (2.100)Under our assumptions we have( Q T ) µ ˆ µ = − G aa δ ij ! , (2.101)and consequently a F ij = − i p signG aa G aa F ij a , a F ia = − i p signG aa F i , (2.102) a F ijkq = − i p signG aa G aa F ijkqa , a F ijka = − i p signG aa F ijk . (2.103)For the space-like directions G aa < i √ signG aa is real. For time-like direction √ signG aa → √ signG = 1 and remaining imaginary unit causes that squares of theantisymmetric fields get additional minus sign and type II theories swap to type II ⋆ ones[28]. In this section we will introduce double space, doubling all bosonic coordinates x µ bycorresponding T-dual ones y µ . We will rewrite the transformation laws in double spaceand show that both the equations of motion and Bianchi identities can be written by thatsingle equation. Applying the Buscher T-dualization procedure [2] along all bosonic coordinates of theaction (2.13) the T-dual action has been obtained in Ref.[6]. This is particular case of ourrelations (2.67)-(2.72) where T-dual background fields are of the form ⋆ Π µν ± ≡ ⋆ B µν ± ⋆ G µν = κ µν ∓ , (3.1) ⋆ Ψ αµ = κ Θ µν + Ψ αν , ⋆ ¯Ψ αµ = κ ⋆ Ω αβ Θ µν − ¯Ψ βν , (3.2) e ⋆ Φ2 ⋆ F αβ = ( e Φ2 F αγ + 4 κ Ψ αµ Θ µν − ¯Ψ γν ) ⋆ Ω γβ . (3.3)18ere we use the notation G Eµν = G µν − BG − B ) µν , Θ µν = − κ ( G − E BG − ) µν , ⋆ Ω = − Γ , (3.4)and Θ µν ± = − κ ( G − E Π ± G − ) µν = Θ µν ∓ κ ( G − E ) µν , (3.5)so that (Π ± Θ ∓ ) µν = 12 κ δ µν . (3.6)From (3.1) and (3.5) it follows ⋆ G µν = ( G − E ) µν , ⋆ B µν = κ µν . (3.7)In this case the transformation laws (2.21) and (2.22) (the relations between initial x µ and T-dual coordinates y µ ) obtain the form ∂ ± x µ ∼ = − κ Θ µν ± ∂ ± y ν + κ Θ µν ± J ± ν , ∂ ± y µ ∼ = − ∓ µν ∂ ± x ν + J ± µ . (3.8) Rewriting equations (3.8) in the form where terms multiplied by ε ±± = ± ± ∂ ± y µ ∼ = G Eµν ∂ ± x ν − BG − ) µν ∂ ± y ν + 2(Π ± G − ) µν J ± ν , (3.9) ± ∂ ± x µ ∼ = ( G − ) µν ∂ ± y ν + 2( G − B ) µν ∂ ± x ν − ( G − ) µν J ± ν . (3.10)Let us introduce double space coordinates Z M = x µ y µ ! , (3.11)which contain all initial and T-dual coordinates. In terms of double coordinates therelations (3.9) and (3.10) are replaced by one ∂ ± Z M ∼ = ± Ω MN (cid:16) H NP ∂ ± Z P + J ± N (cid:17) , (3.12)where the matrix H MN is known in literatute as generalized metric and has the form H MN = G Eµν − B µρ ( G − ) ρν G − ) µρ B ρν ( G − ) µν ! . (3.13)The double current J ± M is defined as J ± M = ± G − ) µν J ± ν − ( G − ) µν J ± ν ! , (3.14)19nd Ω MN = D D ! , (3.15)is constant symmetric matrix. Here 1 D denotes the identity operator in D dimensions.Let us stress that matrix a Ω and Ω MN are different quantities.By straightforward calculation we can prove the relations H T Ω H = Ω , Ω = 1 , det H MN = 1 , (3.16)which means that H ∈ SO ( D, D ). In calculation of determinant we use the rule for blockmatrices det
A BC D ! = det D det( A − BD − C ) . (3.17)In Double Field Theory Ω MN is the SO ( D, D ) invariant metric and denoted by η MN . It is well known that equations of motion of initial theory are Bianchi identities in T-dualpicture and vice versa [9, 13, 16, 41]. As a consequence of the identity ∂ + ∂ − Z M − ∂ − ∂ + Z M = 0 , (3.18)known as Bianchi identity, and relation (3.12), we obtain the consistency condition ∂ + h H MN ∂ − Z N + J − M i + ∂ − h H MN ∂ + Z N + J + M i = 0 . (3.19)In components it takes a form ∂ + ∂ − x µ = − κ ( G − ) µν (cid:16) ¯Ψ αν ∂ + ¯ π α + Ψ αµ ∂ − π α (cid:17) ,∂ + ∂ − y ν = − κ G Eµν (cid:0) ⋆ ¯Ψ αµ ∂ + ¯ π α + ⋆ Ψ αµ ∂ − π α (cid:1) . (3.20)These equations are equations of motion of the initial and T-dual theory. Double spaceformalism enables us to write both equations of motion and Bianchi identities by singlerelation (3.12).The equation (3.19) is equation of motion of the following action S = κ Z d ξ h ∂ + Z M H MN ∂ − Z N + ∂ + Z M J − M + J + M ∂ − Z M + L ( π α , ¯ π α ) i , (3.21)where L ( π α , ¯ π α ) is arbitrary functional of fermionic momenta.20 T-dualization of type II superstring theory as permuta-tion of coordinates in double space
In this section we will derive the transformations of the generalized metric and current,which are consequence of the permutation of some subset of the bosonic coordinates withthe corresponding T-dual ones. First we will present the method in the case of the com-plete T-dualization (along all bosonic coordinates) and find the expressions for T-dualbackground fields. Then we will apply the receipt on the case of partial T-dualization.
In order to exchange all initial and T-dual coordinates let us introduce the permutationmatrix T M N = D D ! , (4.1)so that double T-dual coordinate ⋆ Z M is obtained as ⋆ Z M = T M N Z N = y µ x µ ! . (4.2)We demand that T-dual transformation law for double T-dual coordinate ⋆ Z M has thesame form as for initial coordinate Z M (3.12) ∂ ± ⋆ Z M ∼ = ± Ω MN (cid:16) ⋆ H NP ∂ ± ⋆ Z P + ⋆ J ± N (cid:17) . (4.3)Then the T-dual generalized metric ⋆ H MN and T-dual current ⋆ J ± M are ⋆ H MN = T M K H KL T LN , ⋆ J ± M = T M N J ± N . (4.4)Permutation of the coordinates (4.2) together with transformations of the backgroundfields (4.4) represents the symmetry transformations of the action (3.21).Using the corresponding expressions for T M N , H MN and J ± M , we obtain from thegeneralized metric transformation ⋆ G µν = ( G − E ) µν , ⋆ B µν = κ µν . (4.5)Taking into account that as a consequence of (2.48) the bar dual variable is • ⋆ ¯ π α =( ⋆ Ω T ) αβ ¯ π β , from current transformations we have ⋆ Ψ αµ = κ Θ µν + Ψ αν , ⋆ ¯Ψ αµ = κ ⋆ Ω αβ Θ µν − ¯Ψ βν , (4.6)where ⋆ Ω = − Γ .Consequently, using double space we can easily reproduce the results of T-dualization(3.7) and (3.2). The problem with T-dualization of the R-R field strength F αβ will bedisscussed in subsection 5.3. 21 .2 The case of partial T-dualization Applying the procedure presented in the previous subsection to the arbitrary subset ofbosonic coordinates we will, in fact, describe all possible bosonic T-dualizations. Let ussplit coordinate index µ into a and i ( a = 0 , · · · , d − i = d, · · · , D −
1) and denoteT-dualization along direction x a and y a as T a = T a ◦ T a , T a ≡ T ◦ T ◦ · · · ◦ T d − , T a ≡ T ◦ T ◦ · · · ◦ T d − , (4.7)where ◦ marks the operation of composition of T-dualizations. Permutation of the initialcoordinates x a with its T-dual y a we realize by multiplying double space coordinate bythe constant symmetric matrix ( T a ) M Na Z M ≡ y a x i x a y i = ( T a ) M N Z N ≡ a
00 1 i a i x a x i y a y i , (4.8)where 1 a and 1 i are identity operators in the subspaces spanned by x a and x i , respectively.It is easily to check the following relations( T a T a ) M N = δ M N , ( T a Ω T a ) M N = Ω
M N . (4.9)The first relation means that after two T-dualizations we get the initial theory, while thesecond relation means that T a ∈ O ( D, D ).Let us apply the same approach as in the case of the full T-dualization presented in theprevious subsection. We demand that double T-dual coordinate a Z M satisfy the T-dualitytransformations of the form as initial one Z M (3.12) ∂ ± a Z M ∼ = ± Ω MN (cid:16) a H NK ∂ ± a Z K + a J ± N (cid:17) . (4.10)Consequently, we find the T-dual generalized metric a H MN = ( T a ) M K H KL ( T a ) LN , (4.11)and T-dual current a J ± M = ( T a ) M N J ± N . (4.12)Note that equations (4.8), (4.11) and (4.12) are symmetry transformations of the action(3.21). The left subscript a means dualization along x a directions.22 T-dual background fields
In this section we will show that permutation of some bosonic coordinates leads to thesame T-dual background fields as standard Buscher procedure [6]. The transformationof the generalized metric (4.11) produces expressions for NS-NS T-dual background fields( G µν and B µν ). They are the same as in bosonic string case obtained in Ref.[26]. So, wewill just shortly repeat these results. From the transformation of the current J ± M (4.12)we will find T-dual background fields of the NS-R sector (Ψ αµ and ¯Ψ αµ ). Because R-R fieldstrength F αβ does not appear in T-dual transformations, we will find its T-dual undersome assumptions. G µν , B µν Demanding that the T-dual generalized metric a H MN has the same form as the initial one H MN (3.13) but in terms of the T-dual fields a H MN = a G µνE − a B a G − ) µν a G − a B ) µν ( a G − ) µν ! , (5.1)and using Eq.(4.11), one finds expressions for the NS-NS T-dual background fields a Π µν ± in terms of the initial ones a Π µν ± = ˜ g − β D − γ − A − ( ˜ β ∓ ) A − g T − g − β D − ( ¯ β T ∓ ) D − γ − γ − β T A − ( ˜ β ∓ ) ¯ γ − β T A − g T − D − ( ¯ β T ∓ ) ! , (5.2)where γ and ¯ γ are defined in (A.4), g and ˜ g in (A.5), while β , ˜ β and ¯ β are defined in(A.7). The quantities A and D are given in (A.11) and (A.13), respectively. In morecompact form we have a Π µν ± = κ ˆ θ ab ∓ κ ˆ θ ab ∓ Π ± bi − κ Π ± ib ˆ θ ba ∓ Π ± ij − κ Π ± ia ˆ θ ab ∓ Π ± bj ! , (5.3)where ˆ θ ab ± has been defined in (A.9). Details regarding derivation of the equations (5.2)and (5.3) are given in Ref.[26]. Reading the block components we obtained the NS-NS T-dual background fields in the flat background after dualization along directions x a , ( a = 0 , , · · · , d − a Π ab ± = κ ˆ θ ab ∓ , a Π a ± i = κ ˆ θ ab ∓ Π ± bi , (5.4) a Π ± ia = − κ Π ± ib ˆ θ ba ∓ , a Π ± ij = Π ± ij − κ Π ± ia ˆ θ ab ∓ Π ± bj . (5.5)These are just the equations (2.67)-(2.69). The symmetric and antisymmetric parts ofthese expressions are T-dual metric and T-dual Kalb-Ramond field, which are in fullagreement with the Refs.[6, 14]. 23 .2 T-dual NS-R background fields Ψ αµ , ¯Ψ αµ Let us find the form of T-dual NS-R background fields, a Ψ αa , a Ψ αi , a ¯Ψ αa and a ¯Ψ αi . TheT-dual current a J ± M (4.12) should have the same form as initial one (3.14) but in termsof the T-dual background fields a Π ± a G − ) ab ( a J ) b ± + 2( a Π ± a G − ) ai ( a J ) ± i a Π ± a G − ) ia ( a J ) a ± + 2( a Π ± a G − ) ij ( a J ) ± j − ( a G − ) ab ( a J ) b ± − ( a G − ) ai ( a J ) ± i − ( a G − ) ia ( a J ) a ± − ( a G − ) ij ( a J ) ± j = − ( G − ) aµ J ± µ ± G − ) iµ J ± µ ± G − ) aµ J ± µ − ( G − ) iµ J ± µ . (5.6)On the left-hand side of this equation we split the index µ in a and i components becausein T-dual picture index a has different position, it is now up. T-dual currents are writtenbetween the brackets to make distinction between left subscript a marking partial T-dualization and summation indices in the subspace spanned by x a .The information about T-dual NS-R background fields we can obtain from the lower D components of the above equation. In order to find the solution of these equations it ismore practical to rewrite them using block-wise form of matrices given in Appendix andRef.[26] − ˜ g ab ( a J ) b ± + 2( β ) ai ( a J ) ± i = 2( ˜ β ±
12 ) ab J ± b + 2( β ) ai J ± i , − β T ) ib ( a J ) b ± + ¯ γ ij ( a J ) ± j = γ ia J ± a + ¯ γ ij J ± j . (5.7)From the Eq.(3.18) of [26]( a G − ) µν = g ab − BG − ) aj G − B ) ib ( G − ) ij ! = ˜ g − β − β T ¯ γ ! , (5.8)(A.4) and (A.7), we find the components of a G − , G − and BG − , respectively. In thefirst equation on right-hand side for (Π ± G − ) ai stands just ( β ) ai because δ ai = 0.The difference ( a J ) ± i − J ± i = (¯ γ − ) ij h γ ja J ± a + 2( β T ) j b ( a J ) b ± i , (5.9)obtained from the second equation, we put in the first equation which produces2 (cid:20)(cid:18) ˜ β ± (cid:19) − β ¯ γ − γ (cid:21) a b J ± b = − (cid:16) ˜ g − β ¯ γ − β T (cid:17) ab a J b ± . (5.10)From the definition of quantity A ab (A.11) we get( a J ) b ± = 2 (cid:20) − A − ( ˜ β ±
12 ) + A − β ¯ γ − γ (cid:21) bc J ± c . (5.11)24sing the expression A ab = ˆ g ab (proved in [26]) and the relation (A.12), we recognize ab block-component of the relation (5.2). So, with the help of (5.3) it is easily to see that( a J ) b ± = 2 a Π bc ∓ J ± c = κ ˆ θ bc ± J ± c . (5.12)Note that now the T-dual current a J ˆ µ ± is of the form a J ˆ µ ± = ± κ a Ψ α ˆ µ ± a π ± α , (5.13)where a Ψ α ˆ µ + ≡ a Ψ α ˆ µ , a Ψ α ˆ µ − ≡ a ¯Ψ α ˆ µ , a π + α ≡ π α , a π − α ≡ • ¯ π α , (5.14)and as before J ± µ = ± κ Ψ α ± µ π ± α . (5.15)So, the a components of the T-dual NS-R fields are of the form a Ψ αa = κ ˆ θ ab + Ψ αb , a ¯Ψ αa = κ a Ω αβ ˆ θ ab − ¯Ψ βb . (5.16)Substituting (5.11) into (5.9) we obtain( a J ) ± i − J ± i = (cid:16) ¯ γ − + 4¯ γ − β T A − β ¯ γ − (cid:17) ij γ jb J ± b − (cid:20) ¯ γ − β T A − ( ˜ β ±
12 ) (cid:21) i a J ± a . (5.17)With the help of (A.13) the relation (5.17) transforms into( a J ) ± i − J ± i = 2 (cid:20) D − γ − γ − β T A − ( ˜ β ±
12 ) (cid:21) i a J ± a . (5.18)From ia component of (5.2) and (5.3) we finally have( a J ) ± i = J ± i − κ Π ∓ ib ˆ θ ba ± J ± a . (5.19)As in the previous case, using the expressions for currents (5.13) and (5.15), the final formof T-dual fields is a Ψ αi = Ψ αi − κ Π − ib ˆ θ ba + Ψ αa , a ¯Ψ αi = a Ω αβ ( ¯Ψ βi − κ Π + ib ˆ θ ba − ¯Ψ βa ) . (5.20)The relations (5.16) and (5.20) are in full agreement with the results from Ref.[6] givenby Eqs.(2.70) and (2.71).The upper D components of Eq.(5.6) produce the same result for T-dual backgroundfields. 25 .3 T-dual R-R field strength F αβ Using the relations a H = T a HT a and a J ± = T a J ± we obtained the form of NS-NS andNS-R T-dual background fields of type II superstring theory. But we know from Buscher T-dualization procedure that T-dual R-R field strength a F αβ has the form given in Eq.(2.72).In this subsection we will derive this relation within the double space framework.The R-R field strength F αβ appears in the action (2.13) coupled with fermionic mo-menta π α and ¯ π α along which we do not perform T-dualization. So, we did not doublethese variables. It is an analogue of ij -term in approach of Refs.[21, 22] where x i coor-dinates are not doubled. Consequently, as in [21, 22] we should make some assumptions.Let us suppose that fermionic term L ( π α , ¯ π α ) is symmetric under exchange of R-R fieldstrength F αβ with its T-dual a F αβ L = e Φ2 π α F αβ ¯ π β + e a Φ2 a π α a F αβ a ¯ π β ≡ L + a L , (5.21)for some F αβ and a F αβ . This term should be invariant under T-dual transformation a L = L + ∆ L . (5.22)Taking into account the fact that two successive T-dualization are identity transformation,we obtain from (5.22) L = a L + a ∆ L . (5.23)Combining last two relations we get a ∆ L = − ∆ L . (5.24)If ∆ L has a form ∆ L = π α ∆ αβ ¯ π β and consequently a ∆ L = a π α a ∆ αβa ¯ π β , then with thehelp of the first relation (2.48) we obtain the condition for ∆ αβa ∆ αβ = − ∆ αγ a Ω γβ . (5.25)So, we should find the combination of background fields with two upper spinor indiceswhich under T-dualization transforms as in (5.25). Using the expression for NS-R fields(2.70) and the equation ( a ˆ θ ± ) ab = κ Π ∓ ab = κ (ˆ θ − ± ) ab [see T-dual of (5.4) and (A.10)], itis easy to check that there are D different solutions∆ αβd = c Ψ αa ˆ θ ab − ¯Ψ βb , (5.26)where d = 1 , , . . . D and c is arbitrary constant. Consequently, when we T-dualize d dimensions x a ( a = 0 , , . . . d − e a Φ2 a F αβ = ( e Φ2 F αγ + c Ψ αa ˆ θ ab − ¯Ψ γb ) a Ω γβ . (5.27)For c = 4 κ we obtain the agreement with the expression (2.72). Note that the fermionicterm L d ( π α , ¯ π α ) depends on d , number of directions along which we perform T-duality aswell as in Ref.[21, 22]. 26 Conclusion
In this article we showed that the new interpretation of bosonic T-dualization procedurein double space formalism offered in [26, 27] is also valid in the case of type II superstringtheory. We used the ghost free action of type II superstring theory in pure spinor formu-lation in the approximation of quadratic terms and constant background fields. One canobtain this action from action (2.7), which could be considered as an expansion in powersof fermionic coordinates. In the first part of analysis we neglect all terms in the actioncontaining powers of θ α and ¯ θ α . This approximation is justified by the fact that action isa result of an interative procedure in which every step comes out from the previous one.Later, when we discussed proper fermionic variables, taking higher power terms we restoresupersymmetric invariants (Π µ ± , d α , ¯ d α ) as variables instead of ∂ ± x µ , π α and ¯ π α .We introduced the double space coordinate Z M = ( x µ , y µ ) adding to all bosonic initialcoordinates, x µ , the T-dual ones, y µ . Then we rewrote the T-dual transformation laws(3.8) in terms of double space variables (3.12) introducing the generalized metric H MN and the current J ± M . The generalized metric depends only on the NS-NS backgroundfields of the initial theory. The current J ± M contains fermionic momenta π α and ¯ π α alongwhich we do not make T-dualization and depends also on NS-R background fields. TheR-R background fields do not appear in T-dual transformation laws.The coordinate index µ is split in a = (0 , , . . . d −
1) and i = ( d, d + 1 , . . . D − a marks subsets of the initial and T-dual coordinates, x a and y a , alongwhich we make T-dualization. T-dualization is realized as permutation of the subsets x a and y a in the double space coordinate Z M . The main demand is that T-dual doublespace coordinates a Z M = ( T a ) M N Z N satisfy the transformation law of the same form asthe initial coordinates Z M . From this condition we found the T-dual generalized metric a H MN and the T-dual current a J ± M . Because the initial and T-dual theory are physicallyequivalent, a H MN and a J ± M should have the same form as initial ones, H and J ± M , butin terms of the T-dual background fields. It produces the form of NS-NS and NS-R T-dualbackground fields in terms of the initial ones which are in full accordance with the resultsobtained by Buscher T-dualization procedure [6, 7].The supersymmetry case is not a simple generalization of the bosonic one, but re-quires some new interesting steps. The origin of the problem is different T-duality trans-formation of world-sheet chirality sectors. It produces two possible sets of vielbeins inthe T-dual theory with the same T-dual metric. These vielbeins are related by particu-lar local Lorentz transformation which depends on T-duality transformation and whichdeterminant is ( − d , where d is the number of T-dualized coordinates. So, when weT-dualization along odd number of coordinates then such transformation contains paritytransformation. Consistency of T-duality with supersymmetry demands changing one oftwo spinor sectors. We redefine the bar spinor coordinates, a ¯ θ → • a ¯ θ α = a Ω αβ ¯ θ β , and27ariable a ¯ π α , a ¯ π α → • a ¯ π α = a Ω αβ ¯ π β . As a consequence bar NS-R and R-R backgroundfield include a Ω in their T-duality transformations. For odd number of coordinates d along which T-dualization is performed, a Ω changes the chirality of bar gravitino ¯Ψ αµ andchirality condition for F αβ . We need it to relate type IIA and type IIB theories.Transformation law (3.12) induces the consistency condition which can be considered asequation of motion of the double space action (3.21). It contains an arbitrary term depend-ing on undualized variables L ( π α , ¯ π α ). This is analogy with the term ∂ + x i Π + ij ∂ − x j in ap-proach presented in Ref.[21, 22]. So, to obtain T-dual transformation of R-R field strength F αβ we should make some additional assumptions. Supposing that term L ( π α , ¯ π α ) is T-dual invariant and taking into account that two successive T-dualizations act as identityoperator, we found the form of T-dual R-R field strength up to one arbitrary constant c .For c = 4 κ we get the T-dual R-R field strength a F αβ as in Buscher procedure [6].T-duality transformation of the R-R field strength F αβ has two contributions in theform of square roots. The contribution of dilaton produces the term q | Q di =1 G a i a i | . On theother hand contribution of spinorial representation of Lorentz transformation a Ω containsthe same expression without absolute value i d qQ di =1 G a i a i . Therefore, T-dual R-R fieldstrength a F αβ , besides rational expression, contains the expression i d q sign ( Q di =1 G a i a i )(2.89). If we T-dualize along time-like direction ( G > i and not canceled the one in front of the square root. So, T-dualization along time-like direction maps type II superstring theories to type II ⋆ ones[28].The successive T-dualizations make a group called T-duality group. In the case of typeII superstring T-duality transformations are performed by the same matrices T a as in thebosonic string case [26, 27]. Consequently, the corresponding T-duality group is the same.If we want to find T-dual transformation of F αβ without any assumptions, we shouldfollow approach of [26, 27] and, besides all bosonic coordinates x µ , double also all fermionicvariables π α and ¯ π α . In other words, besides bosonic T-duality we should also considerfermionic T-duality [40]. A Block-wise expressions for background fields
In order to simplify notation we will introduce notations for component fields followingRef.[26].For block-wise matrices there is a rule for inversion
A BC D ! − = ( A − BD − C ) − − A − B ( D − CA − B ) − − D − C ( A − BD − C ) − ( D − CA − B ) − ! . (A.1)28or the metric tensor and the Kalb-Ramond background fields we define G µν = ˜ G ab G aj G ib ¯ G ij ! ≡ ˜ G G T G ¯ G ! , (A.2)and B µν = ˜ b ab b aj b ib ¯ b ij ! ≡ ˜ b − b T b ¯ b ! . (A.3)We also define notation for inverse of the matric( G − ) µν = ˜ γ ab γ aj γ ib ¯ γ ij ! ≡ ˜ γ γ T γ ¯ γ ! , (A.4)and for the effective metric G Eµν = G µν − B µρ ( G − ) ρσ B σν = ˜ g ab g aj g ib ¯ g ij ! ≡ ˜ g g T g ¯ g ! . (A.5)Note that because G µν is inverse of G µν we have γ = − ¯ G − G ˜ γ = − ¯ γG ˜ G − , γ T = − ˜ G − G T ¯ γ = − ˜ γG T ¯ G − , ˜ γ = ( ˜ G − G T ¯ G − G ) − , ¯ γ = ( ¯ G − G ˜ G − G T ) − , ˜ G − = ˜ γ − γ T ¯ γ − γ , ¯ G − = ¯ γ − γ ˜ γ − γ T . (A.6)It is also useful to introduce new notation for expression( BG − ) µν = ˜ b ˜ γ − b T γ ˜ bγ T − b T ¯ γb ˜ γ + ¯ bγ bγ T + ¯ b ¯ γ ! ≡ ˜ β β β ¯ β ! . (A.7)We denote by hat ˆ expressions similar to the effective metric (A.5) and non-commutativityparameters but with all contributions from ab subspaceˆ g ab = ( ˜ G − b ˜ G − ˜ b ) ab , ˆ θ ab = − κ (ˆ g − ˜ b ˜ G − ) ab . (A.8)Note that ˆ g ab = ˜ g ab because ˜ g ab is projection of g µν on subspace ab . It is extremely usefulto introduce background field combinationsΠ ± ab = B ab ± G ab ˆ θ ab ± = − κ (ˆ g − ˜Π ± ˜ G − ) ab = ˆ θ ab ∓ κ (ˆ g − ) ab , (A.9)which are inverse to each other ˆ θ ac ± Π ∓ cb = 12 κ δ ab . (A.10)The quantity A ab is defined as A ab = (˜ g − β ¯ γ − β T ) ab . (A.11)One can prove the relation [26](˜ g − β D − ) ai = (ˆ g − β ¯ γ − ) ai , (A.12)where D ij is defined in Eq.(3.21) of [26] D ij = (¯ γ − β T ˜ g − β ) ij , ( D − ) ij = (cid:16) ¯ γ − + 4¯ γ − β T A − β ¯ γ − (cid:17) ij . (A.13)29 eferences [1] K. Becker, M. Becker and J. Schwarz String Theory and M-Theory: A Modern Intro-duction ; B. Zwiebach,
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