aa r X i v : . [ h e p - t h ] S e p KCL-MTH-10-08 E , generalised space-time and IIA string theory Peter WestDepartment of MathematicsKing’s College, London WC2R 2LS, UKAs advocated in hep-th/0307098 we construct the non-linear realisation of the semi-directproduct of E and its first fundamental representation at lowest level from the IIA view-point. We find a theory that is SO (10 , ⊗ GL (1) invariant and contains the fields ofgravity, a two form and a dilaton but which depend on coordinates which belong to thevector representation of SO(10,10). The resulting Lagrangian agrees that of recent workon the so called doubled field theory. However, the construction given in this paper isstraightforward and systematic. It also reveals the relevant underlying symmetries andopens the way to include the Ramond-Ramond, and higher level, fields together withadditional coordinates of the generalised space-time.1hen the E symmetry was first conjectured space-time was encoded into the non-linear realisation by introducing the space-time translation generators in an ad hoc manner[1,2]. It was subsequently proposed [3] that one should introduce generators transformingin the fundamental representation l of E ; more precisely one should take the non-linearrealisation of the semi-direct product of E and a set of generators in the fundamentalrepresentation associated with node one ( see figure 1) , i.e. E ⊗ s l [3]. At lowest levelsthe l multiplet in eleven dimensions begins with the space-time translation generators P a ,then a two form Z a a , a five form generator Z a ...a and a generator Z a ...a ,b togetherwith an infinite number of other generators. In this approach the fields in eleven dimen-sions would depend on all the coordinates introduced in this non-linear realisation that is x a , x a a , x a ...a , x a ...a ,b , . . . [3].There is convincing evidence that all the brane charges are contained in the l rep-resentation [3,4,5,6,7] and the introduction of the different coordinates corresponds tomeasuring using the different brane probes. The non-linear realisation based on E ⊗ s l is a theory that possess many new unfamiliar coordinates, indeed at all levels an infinitenumber of coordinates and it is not clear how to recover a field theory which has only thedependence on the usual space-time. Nonetheless the non-linear realisation of E ⊗ s l hasbeen used to construct the gauged supergravities with maximal supersymmetry, at leastthe field strengths of the four dimensional theories, although the techniques used are applyto all dimensions [8]. In this case not all the coordinates of the l representation were usedbut instead of taking just the usual space-time coordinates x a one took a space-time to bea slice which lies in l and E and contains many new coordinates.A new generalised geometry was proposed in references [8,9] and then subsequentto reference [3] used in a large number of papers to reformulate parts of supergravitytheories. However, reference [3] automatically introduces a generalise tangent space with ageneralised vierbein which is easily computed and whose role in the theory is very stronglyconstrained by the symmetries of the non-linear realisation. It is clear that many features ofthe work using generalised geometry are automatically encoded in the non-linear realisationof E ⊗ s l .In this paper we will compute the non-linear realisation of E ⊗ s l at the lowestlevel appropriate for the IIA theory. This has a SO (10 , ⊗ GL (1) symmetry and thesame fields as the NS-NS sector of the superstring, however, these fields depend on thecoordinates x a , y a that belong to the vector representation of SO(10,10). Following theideas put forward in [11,1,2,12] we consider the fields to only depend on x a and demandthat the theory is invariant under diffeomorphisms. This fixes the coefficients in the La-grangian and we arrive at the Lagrangian for the Neveu-Schwarz-Neveu-Schwarz sector ofIIA supergravity. A key role is played by the GL (1) part of the symmetry without whicha group theoretic derivation of the result would not have been possible. The coordinates x a , y a were first introduced in the context of string theory in reference [13,14]. In a recentpaper [15] the the non-linear realisation of E ⊗ s l at lowest level appropriate for the IIAtheory was used to deduce a formulation [13] of the bosonic sector of the superstring.To find a ten dimensional theory from the E non-linear realisation one must selectan A , or Sl(10), subalgebra, the so called gravity line, as this subalgebra leads to tendimensional gravity. Looking at the Dynkin diagram of E , see figure 1, one sees that2here are only two possibilities; the nodes from one to nine inclusive and the nodes one toeight inclusive and node eleven. The latter leads to the IIB theory [16] and the former theIIA theory [2] which is the subject of this paper. There are two nodes in the IIA theorywhich are not associated with gravity, that is nodes ten and eleven. Deleting node ten(see fig 1) leaves the algebra D , in particular its real form SO(10,10), and it is useful todecompose the E adjoint representation, which is the one that occurs in the non-linearrealisation, into representations of SO(10,10). However, to recognise objects that are morefamiliar it is helpful to further decompose these representations into those of SL(10) whichcorresponds to deleting node eleven in addition.The representations that occur in a decomposition of the adjoint representation of E to a subalgebra associated with a deleted node can be classified in terms of increasing levels,whose precise definition is given in appendix A. The decomposition into representationsof SO(10,10), associated with the deletion of node ten, at level zero is just the adjointrepresentation of SO(10,10) together with one other generator which is in the Cartansubalgebra of E . When written in terms of representations of SL(10) the generators ofthe adjoint representation of SO(10,10) are K ab , R ab , ˜ R ab , a, b = 1 , . . . ,
10, where the K ab are the generators of the adjoint representation of the SL(10) of the chosen gravityline. There other generator we will denote as ˜ R . Their algebra is derived from E inappendix A and is given by[ K ab , K cd ] = δ cb K ad − δ ad K cb , [ K ab , R cd ] = δ cb R ad − δ db R ac , [ K ab , ˜ R cd ] = − δ ac ˜ R bd + δ ad ˜ R bc , [ R ab , ˜ R cd ] = 4 δ [ a [ c K b ] d ] + 23 ˜ R, [ R ab , R cd ] = 0 = [ ˜ R ab , ˜ R cd ] , [ ˜ R, R cd ] = 0 , [ ˜ R, ˜ R cd ] = 0 (1)where ˜ R = − P a =1 K aa + 3 K . We will show in appendix A that this algebra is thatof SO (10 , ⊗ GL (1) where the last factor has the generator ˜ R .In the non-linear realisation of E the generators of the Borel subgroup lead to fieldsand so at level zero we find the fields h ab , A a a and a that is the fields of the NS-NSsector of the IIA string. At the next level, i.e. level one, one finds generators, and sofields, which are all anti-symmetric tensors of SL(10) which have odd ranks, i.e. those ofone to nine and these belong to the Majorana Weyl spinor representation of SO(10,10).These include the fields of the Ramond-Ramond sector and their duals as well as a ranknine form field that gives rise to the massive IIA theory. At higher levels one finds infinitenumber of fields many of which have a complicated index structure. In fact the six formfield which is the dual of the two form field at level zero occurs at level two.As explained in reference [3] to introduce a generalised space-time into the non-linearrealisation we must also consider the fundamental representation of E associated withnode one, denoted by l , see appendix A. In particular we will consider the non-linearrealisation of the semi-direct product of E and a set of generators that transform under E transformations, i.e E commutators, like the l representation; we denote this semidirect product as E ⊗ s l . The coordinates of the generalised space-time arise as theparameters of the l generators in the general group element. At level zero with respect tonode ten the generators of the l representation are l = { P a , Q a , a = 1 , . . . , } whichbelong to the vector representation of SO (10 , SO (10 , ⊗ GL (1) are derived in appendix A from the E ⊗ s l algebra and are givenby[ K cb , P a ] = − δ ca P b + 12 δ cb P a , [ R ab , P c ] = −
12 ( δ ac Q b − δ bc Q a ) = − δ [ ac Q b ] , [ ˜ R ab , P c ] = 0 , [ K ab , Q c ] = δ cb Q a + 12 δ cb Q a [ ˜ R ab , Q c ] = 4( δ ca P b − δ cb P a ) = 2 δ c [ a P b ] , [ R ab , Q c ] = 0 , [ ˜ R, P a ] = − P a , [ ˜ R, Q a ] = − Q a (2)At level one the generators of l , and so the associated coordinates, are all tensorsof SL(10) of even rank and they belong to the Majorana Weyl spinor representation ofSO(10,10). It is of opposite chirality to those of the level one fields mentioned above.We can now construct the non-linear realisation of E ⊗ s l with local sub-algebra K ( E ) at level zero where K ( G ) denotes the Cartan involution invariant subalgebra ofthe algebra G . This is just the non-linear realisation of ( E ⊗ s l ) (0) = ( SO (10 , ⊗ GL (1)) ⊗ s l , whose algebra is given in equations (1) and (2) above. The local subalgebraof the non-linear realisation is just the Cartan involution invariant subalgebra of SO(10,10)which is SO (10) ⊗ SO (10). The non-linear realisation is built from a general element of( E ⊗ s l ) (0) which is taken to transform as g → g gh where the rigid transformation g ∈ ( E ⊗ s l ) (0) and the local transformation h ∈ SO (10) ⊗ SO (10). Using the latterwe can bring the general group element of ( E ⊗ s l ) (0) to be of the form g = g l g E , where g l = e x a P a + y a Q a and g E = e h · K e A · R e aR (3)where R = (9 K − P a =1 K aa ) and the fields h ab , A a a and a depend on x a and y a .We note that we are using a different scalar generator R rather than the ˜ R that appearedin equation (1) and it has the commutators[ R, P a ] = 0 , [ R, Q a ] = 12 Q a , [ R, K ab ] = 0 , [ R, R ab ] = 12 R ab , [ R, R ab ] = − R ab (4)The Cartan form is given by V = g − dg = g − E ( dx a P a + dy a Q a ) g E + g − E dg E (5)The first term is given by g − E ( dx a P a + dy a Q a ) g E ≡ ( dx a , dy a ) E (cid:18) P b Q b (cid:19) = dz T E L (6)where the 2 D by 2 D matrix E is given by E = ( dete ) − (cid:18) e eAe − a e − T e − a (cid:19) (7)4here e = e µa = ( e h ) µa and A is the matrix A a a . We can think of E as a generalisedvielbein. In the last line we have use the definitions dz T = ( dx a , dy a ) and L = (cid:18) P b Q b (cid:19) . Wenote that E does not have determinant one, but this is consistent with the presence of theadditional GL(1) generator in the algebra whose non-linear realisation we are constructing.The Cartan form is inert under the above rigid transformations but transforms underthe local transformation h as V → h − V h + h − dh . However, carrying out a rigid trans-formation g → g g on the general group element of equation (3) we find that g l → g g l g − and g E → g g E as the l generators form a representation of E . As a result the co-ordinates form a representation of E , and in particular at level zero x a and y a form arepresentation of E (0)11 ≡ SO (10 , ⊗ GL (1) and the action of rigid E (0)11 transformationson the coordinates is given by x a P a + y a Q a → x a ′ P a + y ′ a Q a = g ( x a P a + y a Q a ) g − (8)As expected they transform according to the vector representation of SO(10,10).We can define the action of the vector representation of k ∈ E (0)11 = SO (10 , ⊗ GL (1)using the generators of l (0)1 by U ( k )( L N ) ≡ k − L N k = D ( k ) N M L M where L N = (cid:18) P b Q b (cid:19) and D ( k ) N M is the matrix representative. As a result of equation (8), we find that dz T → dz T ′ = dz T D ( g − ) or putting in the indices dz T N → dz T N ′ = dz T M D ( g − ) M N whereas before dz T N = ( dx a , dy a ). The derivative ∂ N = ∂∂z N in the generalised space-timetransforms as ∂ ′ N = D ( g ) N M ∂ M . Examining equation (5), we note that E is equal tothe matrix D ( g E ) N M . Almost identical considerations hold at all levels in the E ⊗ s l non-linear realisation.As the Cartan form is inert under rigid transformations, their action on the co-ordinates must be compensated by that on E which we give the indices E N A and so E N A ′ = D ( g ) N M E M A . The generalised vielbein E N A ′ transforms on its upper indexby a local SO (10) ⊗ SO (10) transformation and so we can think of the upper index as atangent index and the lower index as a world index.Rather than use the Cartan forms we will construct the action out of M ≡ g E I c ( g − E )where I c is the Cartan involution. It is easy to see that M is inert under local transforma-tions but transforms as M → M ′ = g M I c ( g − ) under rigid transformations. Fortunatelyat level zero, that is for the group E (0)11 , I c ( g − ) = g T . It follows from the above discussionjust above, and equation (7) that in the vector representation M takes the form D ( M ) = E E T = ( dete ) − (cid:18) ee T − eAAe T e − a eAe − e − a − e − T Ae T e − a e − T e − e − a (cid:19) (9)Writing out the indices explicitly D ( M ) NM = ( E E T ) NM it is clear that D ( M ) is inert underlocal SO (10) ⊗ SO (10) transformations as the tangent index is summed but transforms asa SO(10,10) vector on both of its lower world indices. In what follows we write D ( M ) justas M for simplicity.We can construct an invariant Lagrangian out of M and ∂ N . These are inert underlocal transformation and so the most general such object bilinear in generalised space-time5erivatives is given by L = c ∂ S M P Q ∂ T ( M − ) P Q ( M − ) ST + c ∂ S ( M − ) P Q ∂ P M T Q ( M − ) ST + c ∂ S ( M − ) SR ∂ P ( M − ) P Q M RQ + c ∂ S detM ∂ T ( detM ) − ( M − ) ST + c detM ∂ T ( detM ) − ∂ S ( M − ) ST (10)where c , . . . , c are constants.It is instructive to evaluate this Lagrangian in terms of the fields and this is easi-est achieved by going to string frame. That is carrying out the field redefinitions e → e − a e s , A → e a A s whereupon we find that E = e − τ ˜ E , where ˜ E = (cid:18) e s e s A s e − Ts (cid:19) (11)and e τ = e − a dete s . The Lagrangian of equation (10) then becomes L = e τ ( c ∂ S ˜ M P Q ∂ T ( ˜ M − ) P Q ( ˜ M − ) ST + c ∂ S ( ˜ M − ) P Q ∂ P ˜ M T Q ( ˜ M − ) ST + c ∂ S ( ˜ M − ) SR ∂ P ( ˜ M − ) P Q ˜ M RQ ) − (20 c + c − c + c − c ) e τ ∂ S τ ∂ T τ ˜( M − ) ST − e τ ( c − c − c ) ∂ S τ ∂ T ( ˜ M − ) ST (12)where ˜ M = ˜ E ˜ E T . We note that although E does not have determinant one, ˜ E does havedeterminant one and is an element of SO(10,10).The field redefinition could also have been achieved by using a different choice ofgenerators in the group element. Indeed, we can rewrite the group element of equation (3)in terms of the generators of SO(10,10), given in appendix A, and ˜ R to find g E = e h s · K e A s · R e a ˜ R = e h s · ˜ K e A s · R e ˜ a ˜ R (13)where e s = e h s , as a matrix, and e − a = e τ . Thus string frame is associated with the groupelement when written in terms of the SO(10,10) generators and e τ is associated with theGL(1) factor.So far we have constructed a theory which is invariant under a rigid symmetry namely SO (10 , ⊗ GL (1) but we would like to construct a theory that has local symmetrieswhich replace the rigid symmetries of SO (10 , GL ( D ) ⊗ s P where P are the translations and then demands that the theory admit diffeomorphism, orequivalently, admit a simultaneous realisation with the conformal group [11]. This fixesthe constants in the Lagrangian which has just the rigid symmetries of the non-linearrealisation and one finds Einstein’s theory of gravity. A similar procedure was followed atlow levels for the E ⊗ l non-linear realisation to find maximal supergravity theories butin these theories one also finds that the theory was gauge invariant corresponding to the6resence of the gauge fields [1,2]. A more subtle strategy was carried out in the contextof maximal supergravity in four dimensions and its E symmetry [12]. An E ⊗ s l non-linear realisation was constructed where in this case the l representations is the fifty sixdimensional representation of E . It was shown [12] that the corresponding action admitteda diffeomorphism symmetry if one neglected the coordinate dependence on forty nine ofthe fifty six coordinates, leaving a seven dimensional dependence, and fixed the constantsin the action to a particular set of values. In fact this diffeomorphism invariance requireda contribution from the four dimensional metric of the eleven dimensional supergravitytheory in which the E theory was embedded. Unlike the case of gravity, the E theorycan not admit a conformal symmetry that would imply the rigid symmetries of the non-linear realisation become local.We now follow this same strategy here. We restrict the derivatives in the generalisedspace-time to be only in the x a directions. Carrying out this step let us evaluate theLagrangian of equations (10) and (12) by substituting the expression for ˜ E of equation(11), we find that L = c e τ (2 ∂ µ g ρκ ∂ µ g ρκ − ∂ µ A ρκ ∂ µ A ρκ − ∂ µ A ρκ ∂ µ g ρκ )+ c e τ ( g ρµ ∂ µ g νκ ∂ ν g ρκ − ∂ ρ A ν κ ∂ ν A ρκ + 14 ∂ µ g νκ ∂ ν A ρκ A µρ + 14 ∂ ν g ρκ ∂ µ A ν κ A µρ − ∂ ρ g νκ ∂ ν A ρκ )+ c e τ ( ∂ µ g µρ ∂ ν g νκ g ρκ + 14 ∂ µ A µρ ∂ ν A ν κ g ρκ + 12 ∂ µ A µρ ∂ ν g νκ A ρκ − ∂ µ g µρ ∂ ν g νκ A ρκ ) − (20 c + c − c + c − c ) e τ ∂ µ τ ∂ ν τ g µν − e τ ( c − c − c ) ∂ µ τ ∂ ν g µν (14)where ∂ µ = g µν ∂ ν and A µν = A µρ A ρν .To fix the coefficients in the Lagrangian it is simplest to carry out an infinitesimaldiffeomorphism and gauge transformation on the bilinear terms and in particular the vari-ations δh µν = ∂ µ ξ ν + ∂ ν ξ µ , where g µν = e µa e νa = η µν + 2 h µν at lowest order, and δA µν = ∂ µ Λ ν − ∂ ν Λ µ we find the action is invariant provided c = − c , c = 6 c , c = 0.In fact the coefficient c and c occur in the combination c − c and so we can set c = 0.We can choose the overall scale of the Lagrangian so that c = 14 , c = − , c = − Z d x det ee − a { R − . . F µ µ µ F µ µ µ + 2 ∂ µ a∂ µ a } (16)In deriving this result we have used the identity Z d D x p − detge − a R Z d D x p − detge − a {− ∂ τ g νλ ∂ ν g τλ + 14 ∂ ν g ρκ ∂ ν g ρκ + 14 ∂ ν (ln det g ) ∂ ν (ln det g ) − ∂ ν (ln det g ) ∂ µ g µν − ∂ ν a (2Γ ν − g ντ ∂ ρ g τρ ) } = Z d D x p − detge − a {− ∂ τ g νλ ∂ ν g τλ + 14 ∂ ν g ρκ ∂ ν g ρκ + ∂ µ τ ∂ µ τ − ∂ µ τ ∂ λ g µλ − ∂ µ a∂ µ a } (17)valid in D dimensions and the definitions Γ λµν = g λτ ( ∂ ν g τµ + ∂ µ g τν − ∂ τ g µν ) and R µν ρλ = ∂ µ Γ ρνλ + Γ ρµκ Γ κνλ − ( µ → ν ).It is interesting to trace how the familiar density factor det e arises in the aboveLagrangians. Each factor of M carries with it a (det e ) − factor which it inherited fromthe product of two E ’s each with factor (det e ) − , this in turn had its origin in the lastterms, with factor of one half, in the commutation relations of K ab with P a and Q a givenin equation (2). This additional term arises as P a is the highest weight state in the E representation. The two space-time derivatives in the Lagrangian ensure that there is onemore factor of M − than of M in order to balance the indices and as a consequence wefind the desired det e factor. It is interesting to note that this factor has its origins in the E algebra and the definition of the l representation.We now comment on the relation of this paper to the work on the so called doublefield theory. This result was developed in a number of substantial papers [17-20] whichhad their origins in earlier work on string field theory [21] and reference [22]. The finalresult was a field theory with a metric, two form and dilaton, i.e. the fields of the masslessNS-NS sector of closed strings, but which depend on the coordinates x a and y a whichtransformed as a vector of O(D,D). As the author of this paper understands it, doubledfield theory was found by a circuitous route beginning [17] by extracting the quadraticand cubic terms, together with their gauge transformations, from gauge covariant closedstring field theory [23]. These terms were then shown to be invariant under a set ofO(10,10) transformations introduced by ansatz. By assuming a set of requirements, gaugetransformations were found to all orders in the fields and written in an O(10,10) covariantway [18]. By introducing covariant derivatives an all orders Lagrangian was found thatagreed with that of closed string field theory at quadratic and cubic order [19] and shownto be invariant under the gauge transformations if a constraint held. In the general casethis constraint was the same as suppressing all dependence on y a .Finally, this Lagrangian was expressed in terms of an O(10,10) generalised metric H (the ˜ M in this paper) and e − d (the e − τ in this paper). In fact this Lagrangian waspreviously contained in a seminar [24] of one of the authors (OH). However, it was alsoclaimed in the seminar that H − ∂H was the Cartan form of O(10,10). It was pointedout [25] by the author of this paper that the final result was almost certainly the lowestlevel non-linear realisation of E ⊗ s l , i.e. the result of this paper, and that the non-linear realisation would contain ˜ M = gg T where g ∈ O (10 ,
10) with the transformation g → g gh being understood. This relation subsequently appeared in paper [20] but in theform ˜ M = g T g . Unfortunately this latter expression is not inert under the local h part tothe transformation g → g gh of equation (5.1) of paper [20].8lthough our final result of equation (10) with the constants of equation (15) agreeswith that of [20] the derivation presented in this paper is very different. We begin withan entirely group theoretic construction, i.e the non-linear realisation of E ⊗ s l , asproposed in reference [3], at lowest level. It is important to realise that this is not thesame as the traditional sigma model associated with internal symmetries. The fields, theirrigid O ( D, D ) ⊗ GL (1) transformations and the most general Lagrangian are found by avery straightforward calculation. The constants in the Lagrangian are fixed, following theideas of [11,1,2,12], by demanding that the Lagrangian be diffeomorphsim invariant whenthe fields are restricted to depend only on x µ . Crucial to this construction is the GL(1)factor in the symmetry which seems to play no role in the papers on double field theory.Such an additional factor is required from the non-linear realisation perspective since thegenerators not in the local subalgebra lead to fields in the final theory and we require a D + 1 commuting subalgebra in order to find the diagonal components of the metric andthe dilaton. As explained above it is also essential to get the usual det e density factor inthe Lagrangian.The E ⊗ s l non-linear realisation also provides a clear method to include theRamond-Ramond fields and indeed higher level fields and also the corresponding additionalcoordinates. Furthermore it makes it clear that the symmetry underlying the doubled fieldtheory, which had its origins in string field theory, is just that of the non-linear realisationof E ⊗ s l .It is instructive to consider the eleven dimensional theory formed from the E ⊗ s l non-linear realisation with local subalgebra K ( E ) at lowest level, i.e. level zero withrespect to the deletion of node eleven. The resulting subalgebra is GL(11) which is as-sociated with eleven dimensional gravity. At level zero one has the GL(11) algebra, withgenerators K ab , a, b = 1 , . . . ,
11 and the local subalgebra is just SO(10), while in the l representation there are only the usual space-time translations P a . Their commutationrelations, as derived from E , are given by[ K ab , K cd ] = δ cb K ad − δ ad K cb , [ K cb , P a ] = − δ ca P b + 12 δ cb P a (18)We note the presence of the second term in the last commutator whose origin was discussedin appendix A. We can now follow exactly the path as above, we take the group element g = g l g E = e x · P e h · K and find from the Cartan form that E µa = ( dete ) − e where e µa =( e h ) µa . The most general invariant action is formed from ∂ µ = ∂∂x µ and M = E E T andhas the above form of equation (10) with the indices M, N, . . . now being replaced by µ, ν = 1 , . . . ,
11. We find a diffeomorphism invariant result, which is general relativity, in D dimensions provided c = , c = − , c = 0 = c and c = − D − . Of course D = 11for the case of interest to us. However, we note that the det e factor has the same origin asthat just described above, namely in the second term of the last commutator of equation(18).It is amusing to consider the next level in eleven dimensions. At level one we havethe three form generator R abc in the adjoint representation of E and the generators Z ab in l representation. Thus we have the field A abc which now depends on the coordinates x a and y ab . The group element is g = g l g E with g l = e x · P e y · Z and g E = e h · K e A · R . The9artan form takes the form V = g − dg = g − E ( dx a P a + dy ab Z ab ) g E + g − E dg E = ( dx a , dy ab ) E (cid:18) P b Z ab (cid:19) + g − E dg E (19)where the generalised vierbein E is given as a matrix by E = ( dete ) − (cid:18) e µa − e µc A cb b e − ) [ b µ ( e − ) b ] µ (cid:19) (20)One can then construct M = E E T which is invariant under local transformations. We hopeto report elsewhere on the dynamics of the non-linear realisation. The same calculationcan easily be carried out for the IIA theory at the next level. The additional fields andcoordinates were listed earlier in this paper.In a previous paper [15] it was shown that quantising the SO(10,10) symmetric stringleads to a field theory with coordinates x a , or y a or a field theory with the coordinates x a and y a but which obey non-trivial commutation relations, that is a non-commutativefield theory. The different theories corresponding to the different choices of representationsof the fundamental commutators of the theory. It is tempting to assume that all thesedifferent theories lead to the same physical result, but it can not be excluded that theymay differ in subtle ways. It would be interesting to investigate the connection with theresults found in this paper. It is striking that some non-linear realisations admit, with asuitable choice of constants, local symmetries. The deeper meaning of this result has yetto be understood. Acknowledgment
The author wishes to thank Christian Hillmann for extensive discussions that sub-stantially helped the development of this work and the Erwin Schr¨odinger InternationalInstitute for Mathematical Physics, Vienna for providing financial support and hospitalityduring the author’s stay in Vienna where much of this work was carried out. He alsothanks the Physics department of the Technical University of Vienna for their hospitalityand the STFC for support from the rolling grant awarded to King’s.
Appendix A
In this appendix we calculate the algebra E ⊗ s l at lowest level for the decompositionappropriate to the IIA theory. We will begin with the known algebra of the E algebra interms of A , or SL(11), representations [1,3]. This algebra is found by deleting node elevenin the Dynkin Diagram of E ( see figure 1). For a decomposition of E corresponding toa subalgebra associated with the deletion of a particular node, the resulting generators canbe classified in terms of increasing level. As for any Kac-Moody algebra, the generatorsof E are constructed as multiple commutators of the Chevalley generators and the levelof a given positive (negative) root generator with respect to a particular node is justplus (minus) the number of times that the positive (negative) root Chevalley generators10ssociated with this node occurs in this multiple commutator. For the decompositionto SL(11) we consider the level associated with node eleven. At level zero we have thealgebra GL(11) with the generators K ab , a, b = 1 , . . .
11 and at level one and minus onethe rank three generators R abc and R abc . These level zero and one generators containall the Chevalley generators of E (the positive root Chevalley generator associated withnode eleven is R ) and so their multiple commutators lead to all generators of E .As a result the level of a generator that occurs in the SL(11) decomposition is just thenumber of times the generator R abc occurs minus the number of times the generator R abc occurs. The generators at level two and minus two are R a ...a and R a ...a respectively,while those at levels three and minus three are R a ...a ,b and R a ...a ,b respectivelyThe E algebra at levels zero and up three is given by [1,3][ K ab , K cd ] = δ cb K ad − δ ad K cb , ( A. K ab , R c ...c ] = δ c b R ac ...c + . . . , [ K ab , R c ...c ] = δ c b R ac c + . . . , ( A. K ab , R c ...c ,d ] = ( δ c b R ac ...c ,d + · · · ) + δ db R c ...c ,a . ( A. R c ...c , R c ...c ] = 2 R c ...c , [ R a ...a , R b ...b ] = 3 R a ...a [ b b ,b ] , ( A. . . . means the appropriate anti-symmetrisation.The E level zero and negative level generators up to level minus three obey therelations[ K ab , R c ...c ] = − δ ac R bc c − . . . , [ K ab , R c ...c ] = − δ ac R bc ...c − . . . , ( A. K ab , R c ...c ,d ] = − ( δ ac R bc ...c ,d + · · · ) − δ ad R c ...c ,b . ( A. R c ...c , R c ...c ] = 2 R c ...c , [ R a ...a , R b ...b ] = 3 R a ...a [ b b ,b ] , ( A. R a ...a , R b ...b ] = 18 δ [ a a [ b b K a ] b ] − δ a a a b b b D, [ R b ...b , R a ...a ] = 5!2 δ [ a a a b b b R a a a ] [ R a ...a , R b ...b ] = − . . δ [ a ...a [ b ...b K a ] b ] + 5! δ a ...a b ...b D, [ R a ...a , R b ...b ,c ] = 8 . . δ [ b b b [ a a a R b ...b ] c − δ [ b b | c | [ a a a R b ...b ] )[ R a ...a , R b ...b ,c ] = 7! .
23 ( δ [ b ...b [ a ...a R b b ] c − δ c [ b ...b [ a ...a R b b b ] ) ( A. D = P b K bb , δ a a b b = ( δ a b δ a b − δ a b δ a b ) = δ [ a b δ a ] b with similar formulae whenmore indices are involved.We have taken the liberty of listing the algebra to a somewhat11igher than required in this paper. In fact these equations correct the coefficients of oneof the equations contained in reference [3].The IIA theory arises when we consider the deletion of node ten in the E Dynkindiagram to leave a SO(10,10) algebra leading to a decomposition of E into representationsof this algebra. However, it is illuminating to then delete node eleven and analyse therepresentations of SO(10,10) in terms of SL(10). Thus in this case we have two levelswhich we may write as ( l , l ) and are associated with nodes ten and eleven respectively.At level l = 0 we find generators with l = 0 which are the generators of GL(10), denotedby K ab , a, b = 1 , . . . ,
10, and the generator of ˜ R which is some combination of P a =1 K aa and K as well as the generators R ab ≡ R ab and R ab ≡ R ab at levels l = 1 and l = − K ab , K cd ] = δ cb K ad − δ ad K cb , [ K ab , R cd ] = δ cb R ad − δ db R ac , [ K ab , R cd ] = − δ ac ˜ R bd + δ ad ˜ R bc , [ R ab , R cd ] = 4 δ [ a [ c K b ] d ] + 23 ˜ R, [ R ab , R cd ] = 0 = [ ˜ R ab , ˜ R cd ] , [ ˜ R, R ab ] = 0 , [ ˜ R, R ab ] = 0( A. R = − P a =1 K aa + 3 K .At first sight this is not obviously the algebra of SO (10 , ⊗ GL (1), however, if weredefine the GL(10) generators by ˜ K ab = K ab + 16 δ ba ˜ R , ( A. K ab , ˜ K cd ] = δ cb ˜ K ad − δ ad ˜ K cb , [ ˜ K ab , R cd ] = δ cb R ad − δ db R ac , [ ˜ K ab , R cd ] = − δ ac ˜ R bd + δ ad ˜ R bc , [ R ab , R cd ] = 4 δ [ a [ c ˜ K b ] d ] , [ R ab , R cd ] = 0 = [ ˜ R ab , ˜ R cd ] ( A. R that com-mutes with all the SO(10,10) generators.We also need the fundamental representation of E associated with node one. By def-inition this is the representation with highest weight Λ which obeys (Λ , α a ) = δ a, , a =1 , . . . ,
11 where α a are the simple roots of E . In the decomposition to Sl(11), cor-responding to the deletion of node eleven, one finds that the l representation containsthe objects P a , a = 1 , . . . , Z ab and Z a ...a corresponding to levels zero, one and tworespectively. We have taken the first object, i.e. P a , to have level zero by choice. Takingthese to be generators in a semi-direct product group denoted E ⊗ s l their commutationrelation with the level one generators of E are given by[ R a a a , P b ] = 3 δ [ a b Z a a ] , [ R a a a , Z b b ] = Z a a a b b , [ R a a a , Z b ...b ] = Z b ...b [ a a ,a ] + Z b ...b a a a ( A. l representation.Thecommutator of K ab with P c can only take the form [ K ab , P c ] = − δ ac P b + eδ ab P c where e is a constant. This commutator was found [3] to have e = as a result of the factthat l is an E representation. We summarise the argument here as this relation playsa crucial role in this paper. Since P corresponds to the highest weight state in therepresentation the action of the Chevalley generator H is given by [ H , P ] = ( α , Λ ) P where H = ( K + K + K ) − ( K + . . . + K ) is the Chevalley generatorin the Cartan subalgebra associated with node eleven. Since (Λ , α ) = 0 we must have[ H , P ] = 0 and then we find e = as claimed. Using the Jacobi identity and the factthat e = we conclude that[ K ab , P c ] = − δ ac P b + 12 δ ab P c , [ K ab , Z c c ] = 2 δ [ c b Z | a | c ] + 12 δ ab Z c c , [ K ab , Z c ...c ] = 5 δ [ c b Z | a | c ...c ] + 12 δ ab Z c ...c ( A. R a ...a , P b ] = − δ [ a b Z ...a ] , [ R a ...a , Z b b ] = Z b b [ a ...a ,a ] , ( A. R a a a , P b ] = 0 , [ R a a a , Z b b ] = 6 δ b b [ a a P a ] , [ R a a a , Z b ...b ] = 5!2 δ [ b b b a a a Z b b ] ( A. l = 0 generators of the l multiplet are P a , a = 1 , . . . ,
10 and Q a ≡ − Z a whichhave levels l = 0 and l = 1 respectively. We find from the above algebra that they obeythe commutation relations[ K cb , P a ] = − δ ca P b + 12 δ cb P a , [ R ab , P c ] = −
12 ( δ ac Q b − δ bc Q a ) = − δ [ ac Q b ] , [ ˜ R ab , P c ] = 0 , [ K ab , Q c ] = δ cb Q a + 12 δ cb Q a , [ ˜ R ab , Q c ] = 2( δ ca P b − δ cb P a ) = 4 δ c [ a P b ] , [ R ab , Q c ] = 0 , [ ˜ R, P a ] = − P a , [ ˜ R, Q a ] = − Q a ( A. R = ( − P a =1 K aa +9 K ), used in the non-linear realisation, which leads to[ R, P a ] = 0 , [ R, Q a ] = 12 Q a , [ R, R ab ] = 12 R ab , [ R, R ab ] = − R ab ( A. K cb , P a ] = − δ ca P b , [ R ab , P c ] = −
12 ( δ ac Q b − δ bc Q a ) = − δ [ ac Q b ] , [ ˜ R ab , P c ] = 0 ,
13 ˜ K ab , Q c ] = δ cb Q a , [ ˜ R ab , Q c ] = 2( δ ca P b − δ cb P a ) = 4 δ c [ a P b ] , [ R ab , Q c ] = 0 , ( A. • |• − • − . . . − • − • − • − • E Dynkin diagram
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