Unification of Type II Strings and T-duality
aa r X i v : . [ h e p - t h ] O c t MIT-CTP-4277
Unification of Type II Strings and T-duality
Olaf Hohm, Seung Ki Kwak, and Barton Zwiebach
Center for Theoretical Physics,Massachusetts Institute of Technology,Cambridge, MA 02139, USA.
We present a unified description of the low-energy limits of type II string theories. This is achievedby a formulation that doubles the space-time coordinates in order to realize the T-duality group O (10 ,
10) geometrically. The Ramond-Ramond fields are described by a spinor of O (10 , ,
10) representative of the so-called generalizedmetric. This theory, which is supplemented by a T-duality covariant self-duality constraint, unifiesthe type II theories in that each of them is obtained for a particular subspace of the doubled space.
PACS numbers: 11.25.-w
Superstring theory in ten dimensions is arguably themost promising candidate for a unified quantum mechan-ical description of gravity and other interactions. Thistheory, however, takes different guises. For instance,there are two different string theories with maximal su-persymmetry, the type IIA and the type IIB theory. Theten-dimensional superstring theories, together with 11-dimensional supergravity, are different limits of a singleunderlying theory and are related through a web of du-alities (see, e.g., [1]). The simplest of these dualities isT-duality that, for instance, relates type IIA string the-ory on the circular background R , × S of radius R totype IIB string theory on the same background, but withradius 1 /R .In its low-energy limit string theory is described byEinstein’s theory of general relativity, coupled to partic-ular matter fields. In this description, T-duality resultsin the appearance of the hidden symmetry group O ( d, d )upon dimensional reduction on a torus T d . Moreover, thelow-energy limits of type IIA and type IIB give rise tothe same theory, consistent with their equivalence underT-duality [2].The general coordinate invariance of Einstein gravitynaturally explains the presence of the GL ( d ) subgroup,but the emergence of the full O ( d, d ) upon dimensionalreduction requires the precise matter couplings predictedby string theory, hinting at a novel geometrical struc-ture. Recently, a ‘double field theory’ (DFT) has beenfound which realizes a T-duality group prior to dimen-sional reduction [3, 4] (see also [5, 6]). By doubling thespace-time coordinates, the low-energy effective actionof bosonic string theory or, equivalently, of the Neveu-Schwarz–Neveu-Schwarz (NS-NS) sector of superstringtheory, can be extended to an action that has O ( D, D ) asa global symmetry, where D is the space-time dimension.In this Letter we introduce the extension to theRamond-Ramond (RR) sector of type II strings, whichwill lead to a theory that contains all type II theoriessimultaneously in different T-duality ‘frames’. Here wewill not present explicit derivations, but a more detailedexposition will appear elsewhere [7]. Related work hasappeared in [8, 9].We start by reviewing the NS-NS subsector. It consists of the metric g ij , the Kalb-Ramond 2-form b ij and thescalar dilaton φ , where i, j, . . . = 1 , . . . , D are space-timeindices. The DFT is formulated in terms of a dilatondensity d , which is related to φ via the field redefinition e − d = √ ge − φ , g = | det g | , and the ‘generalized metric’ H MN = (cid:18) g ij − g ik b kj b ik g kj g ij − b ik g kl b lj (cid:19) , (1)which combines g and b into an O ( D, D ) covariant tensorwith indices
M, N, . . . = 1 , . . . , D . All fields depend onthe doubled coordinates X M = (˜ x i , x i ). We can regard H as the fundamental field, taking values in SO ( D, D )and satisfying H T = H , and view (1) as just a particularparametrization. The action can be written as S = Z dx d ˜ x e − d R ( H , d ) , (2)where R ( H , d ) is an O ( D, D ) invariant scalar, cf. (4.24)in the second reference of [4], and we use the short-handnotation dx = d D x . The action is invariant under thegauge transformations δ ξ H MN = ξ P ∂ P H MN + 2 (cid:0) ∂ ( M ξ P − ∂ P ξ ( M (cid:1) H N ) P ,δ ξ d = ξ M ∂ M d − ∂ M ξ M , (3)with the derivatives ∂ M = ( ˜ ∂ i , ∂ i ). Here, O ( D, D ) indices
M, N are raised and lowered with the invariant metric η MN = (cid:18) (cid:19) , (4)and (anti-)symmetrizations are accompanied by the com-binatorial factor . The consistency of the above theoryrequires the constraint ∂ M ∂ M A = η MN ∂ M ∂ N A = 0 , ∂ M A ∂ M B = 0 , (5)for all fields and parameters A and B . This constraintimplies that locally the fields depend only on half of thecoordinates, and one can always find an O ( D, D ) trans-formation into a frame in which the fields depend onlyon the x i . If one drops the dependence on the ‘dual coor-dinates’ ˜ x i in (2) or, equivalently, sets ˜ ∂ i = 0, the actionreduces to the conventional low-energy effective action S = Z d D x √ ge − φ (cid:20) R + 4( ∂φ ) − H ijk H ijk (cid:21) , (6)where H ijk = 3 ∂ [ i b jk ] is the field strength of the 2-form.Moreover, for ˜ ∂ i = 0 the gauge transformations (3) withparameter ξ M = ( ˜ ξ i , ξ i ) reduce to the conventional gen-eral coordinate transformations x i → x i − ξ i ( x ) and tothe gauge transformations of the 2-form, δb ij = 2 ∂ [ i ˜ ξ j ] .Let us now turn to the extension by the RR sector.In this we make significant use of the work of Fukuma,Oota, and Tanaka [10]. (See also [11, 12].) The RRsector consists of forms of degrees 1 and 3 for type IIAand of degree 2 and 4 for type IIB, where the 5-formfield strength of the 4-form is subject to a self-dualityconstraint. Here, we will use a democratic formulationthat simultaneously uses dual forms, such that type IIAcontains all odd forms, and type IIB contains all evenforms, both being supplemented by duality relations [10].The set of all forms naturally combines into a Majoranaspinor of O (10 , D = 10 from now on. More precisely,these are representations of the double covering groupsPin(10 ,
10) of O (10 , ,
10) of SO (10 , { Γ M , Γ N } = 2 η MN . (7)A convenient representation can be constructed usingfermionic oscillators ψ i and ψ i , satisfying { ψ i , ψ j } = δ ij , { ψ i , ψ j } = 0 , { ψ i , ψ j } = 0 , (8)where ( ψ i ) † = ψ i . With (4) we infer that they realize thealgebra (7) via Γ i = √ ψ i , Γ i = √ ψ i . (9)Introducing a ‘Clifford vacuum’ | i with ψ i | i = 0 forall i , and the normalization h | i = 1, we can constructthe representation by successive application of the raisingoperators ψ i . A general spinor state then reads χ = D X p =0 p ! C i ...i p ψ i . . . ψ i p | i , (10)whose coefficients C i ...i p can be identified with p -forms C ( p ) . Any element S of the Pin group projects, via agroup homomorphism ρ : Pin(10 , → O (10 , h ∈ O (10 , S Γ M S − = Γ N h N M , h = ρ ( S ) , (11) where hηh T = η . Conversely, for any h ∈ O (10 , S ∈ Pin(10 ,
10) such that both ± S project to h . Aspinor can be projected to a spinor of fixed chirality, i.e.,to eigenstates χ ± of ( − N F with eigenvalues ±
1, where N F = P k ψ k ψ k is the ‘fermion number operator’. Thespinor χ + of positive chirality then contains only evenforms, and the spinor χ − of negative chirality containsonly odd forms. Imposing a chirality constraint reducesthe symmetry from Pin(10 ,
10) to Spin(10 ,
10) since onlythe latter leaves this constraint invariant. Finally, weneed the charge conjugation matrix satisfying C Γ M C − = (Γ M ) † . (12)A particular realization is given by C = ( ψ − ψ )( ψ − ψ ) · · · ( ψ − ψ ) , (13)which satisfies Cψ i C − = ψ i and thereby (12).Given a spinor (10) we can act with the Dirac operator /∂ ≡ √ M ∂ M = ψ i ∂ i + ψ i ˜ ∂ i , (14)which can be viewed as the O (10 ,
10) invariant extensionof the exterior derivative d . In fact, for ˜ ∂ = 0, it differ-entiates with respect to x i and increases the form degreeby one, thus acting like d . Moreover, it squares to zero, /∂ = 12 Γ M Γ N ∂ M ∂ N = 12 η MN ∂ M ∂ N = 0 , (15)using (7) and the constraint (5).In order to write an action that couples the NS-NSfields represented by the generalized metric H in (1) tothe RR fields represented by a spinor χ , we note that thematrix H is an SO (10 ,
10) group element and thus hasa representative in Spin(10 , SO (10 ,
10) has two connected components, SO + (10 , SO − (10 , g , H is actually an elementof SO − (10 , S H ∈ Spin(10 ,
10) of H cannot be constructed consis-tently over the space of all H . For instance, one mayfind a closed loop H ( t ), t ∈ [0 , H (0) = H (1), in SO − (10 , time-like T-duality, for which a continuously definedspin representative yields S H (1) = − S H (0) . As a result,time-like T-dualities cannot be realized as transforma-tions of the conventional fields g and b . Nevertheless, afully T-duality invariant action can be written if we treatthe spin representative itself as the dynamical field. Wethus introduce a field S , satisfying S = S † , S ∈ Spin − (10 , . (16)The generalized metric is then defined by the group ho-momorphism, ρ ( S ) = H . By (16) and the general prop-erties of the group homomorphism [7], H T = ρ ( S † ) = H and so, as required, H is symmetric.We are now ready to define the DFT formulation oftype II theories, whose independent fields are S , d and χ .The action reads S = Z dxd ˜ x (cid:16) e − d R ( H , d ) + 14 ( /∂χ ) † S /∂χ (cid:17) , (17)and is supplemented by the self-duality constraint /∂χ = −K /∂χ , K ≡ C − S . (18)For the special case of type IIA, a similar duality relationhas also been proposed in [8].The field equation of χ reads /∂ (cid:0) K /∂χ (cid:1) = 0 , (19)which also follows as an integrability condition from theduality relation (18), upon acting with /∂ and using (15).The field equation of S reads R MN + E MN = 0 , (20)where R MN is the DFT extension of the Ricci tensor [4],and the ‘energy-momentum’ tensor reads, using (18), E MN = − H ( M P /∂χ Γ N ) P /∂χ . (21)Let us now discuss the symmetries of this theory. First,it is invariant under a global action by S ∈ Spin + (10 , χ → Sχ , S → S ′ = ( S − ) † S S − , (22)implying /∂χ → S /∂χ . Specifically, χ is assumed to have afixed chirality, which breaks the invariance group of theaction from Pin(10 ,
10) to Spin(10 , + (10 , δ λ χ = /∂λ , (23)with spinorial parameter λ , leaving (17) and (18) man-ifestly invariant by (15), and the gauge symmetry (3)parametrized by ξ M . On the new fields S and χ it reads δ ξ χ = ξ M ∂ M χ + 12 ∂ M ξ N Γ M Γ N χ ,δ ξ K = ξ M ∂ M K + 12 (cid:2) Γ P Q , K (cid:3) ∂ P ξ Q , (24)where we have written the gauge variation of S in termsof K defined in (18). It can be checked that this gaugetransformation gives rise to the required variation (3) of H upon application of ρ .We will now evaluate the DFT defined by (17) and (18)in particular T-duality ‘frames’, starting with ˜ ∂ i = 0. Tothis end, we have to choose a particular parametrizationof S . Writing H = (cid:18) b (cid:19) (cid:18) g − g (cid:19) (cid:18) − b (cid:19) ≡ h Tb h − g h b , (25) we have to find spin representatives of the group elements h b and h g . The subtlety here is that, with g Lorentzian, h g takes values in SO − (10 ,
10) and thus is not in the com-ponent connected to the identity. It is then convenientto write g in terms of vielbeins, g = e k e T , h g = h e h k h Te , (26)where e has positive determinant, i.e., e ∈ GL + (10), and k is the flat Minkowski metric diag( − , , . . . , h e and h b are in the component connectedto the identity and so their spin representatives can bewritten as simple exponentials, S b = e − b ij ψ i ψ j , S e = 1 √ det e e ψ i E ij ψ j , (27)with e = exp( E ), as can be verified with (11). A spinrepresentative for the matrix k can be chosen to be [13] S k = ψ ψ − ψ ψ , (28)where 1 labels the time-like coordinate. This can also beverified with (11). A spin representative S H of H canthen locally be defined as S H ≡ S † b S − g S b , S g = S e S k S † e . (29)We now set S = S H , but we stress that this is just aparticular parameterization in much the same way that(1) is just a particular parametrization of H .It is now straightforward to evaluate the action (17) for˜ ∂ = 0. First, as noted above, /∂χ reduces to the exteriorderivatives of the C ( p ) , F ( p +1) ≡ dC ( p ) . The action of S b in S H then modifies this, using (27), to b F = e − b (2) ∧ F = e − b (2) ∧ dC . (30)Second, (29) implies for the action of S − g S − g ψ i ... ψ i p | i = −√ gg i j ... g i p j p ψ j ... ψ j p | i . (31)The Lagrangian corresponding to the RR part of (17)then reduces to kinetic terms for all forms, L RR = − √ g D X p =1 p ! g i j · · · g i p j p b F i ...i p b F j ...j p , (32)where we recall that the sum extends over all even or allodd forms, depending on the chirality of χ . Similarly,using (13), the self-duality constraint (18) reduces to theconventional duality relations (with the Hodge star ∗ ), b F ( p ) = ( − ( D − p )( D − p − ∗ b F ( D − p ) . (33)We have thus obtained the democratic formulation oftype II theories, whose field equations are equivalent tothe conventional field equations of type IIA for odd formsand of type IIB for even forms [10].Let us briefly comment on the gauge symmetries for˜ ∂ = 0. The transformations (24) for χ , parameterized by ξ M = ( ˜ ξ i , ξ i ), reduce to the conventional general coordi-nate transformations x i → x i − ξ i ( x ) of the p -forms C ( p ) ,but also to non-trivial transformations under the b -fieldgauge parameter ˜ ξ i , δ ˜ ξ C = d ˜ ξ ∧ C .We turn now to the discussion of other T-dualityframes, starting with ∂ i = 0, ˜ ∂ i = 0. For the analysisof this case it is convenient to perform a field redefini-tion according to the T-duality transformation J thatexchanges x i and ˜ x i and which, as a matrix, coincideswith η defined in (4), H ′ ≡ J H J = H − . (34)It has been shown in [4] that the NS-NS part of the DFTreduces for ∂ i = 0 to the same action (6), but written interms of the primed (T-dual) variables. Next, we definea corresponding field redefinition for the RR fields, usinga spin representative S J of J , χ ′ = S J χ , /∂ ′ = ψ i ˜ ∂ i + ψ i ∂ i , /∂ ′ χ ′ = S J /∂χ . (35)For the RR action we then find L RR = ( /∂χ ) † S H /∂χ = ( /∂ ′ χ ′ ) † ( S − J ) † S H S − J /∂ ′ χ ′ = − ( /∂ ′ χ ′ ) † S H ′ /∂ ′ χ ′ , (36)where we used that J contains a time-like T-duality suchthat, as mentioned above, this leads to a sign factor inthe transformation of S H . Thus, in the new variables theaction takes the same form as in the original variables, upto a sign. The transformed Dirac operator in (35) impliesthat setting ∂ i = 0 in the first form in (36) is equivalent to setting /∂ ′ = ψ i ˜ ∂ i in the final form in (36). This way toevaluate the action is, however, equivalent to our compu-tation above of setting ˜ ∂ = 0 in the original action, justwith fields and derivatives replaced by primed fields andderivatives. Thus, we conclude that the DFT action re-duces for ∂ i = 0 to a type II theory with the overall signof the RR action reversed. These are known as type II ⋆ theories and have been introduced by Hull in the contextof time-like T-duality [14]. They are defined such thatthe time-like circle reductions of type IIA (IIB) and typeIIB ⋆ (IIA ⋆ ) are equivalent. This result also implies thatthe overall sign of S has no physical significance in thatit merely determines for which coordinates ( x or ˜ x ) weobtain the type II or type II ⋆ theory.More generally, one finds that evaluating the DFT in aT-duality frame that is obtained by an odd (even) num-ber of T-duality inversions from a frame in which thetheory reduces, say, to type IIA, it reduces to the T-dualtheory, i.e., type IIB (IIA) for space-like transformationsand IIB ⋆ (IIA ⋆ ) for time-like transformations. Summa-rizing, the DFT defined by (17) and (18) combines alltype II theories in a single universal formulation. Wehope that this theory may provide insights into the stillelusive formulation of string theory as, e.g., for a yet tobe constructed type II string field theory. Acknowledgments
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