Boosts superalgebras based on centrally-extended su(1|1)^2
Juan Miguel Nieto García, Alessandro Torrielli, Leander Wyss
aa r X i v : . [ m a t h - ph ] S e p DMUS-MP-20/08
Boosts superalgebras based on centrally-extended su (1 | Juan Miguel Nieto Garc´ıa , Alessandro Torrielli and Leander Wyss Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK
Abstract
In this paper, we studied the boost operator in the setting of su (1 | . We find a family of differentalgebras where such an operator can consistently appear, which we classify according to how thetwo copies of the su (1 |
1) interact with each other. Finally, we construct coproduct maps for each ofthese algebras and discuss the algebraic relationships among them. [email protected] [email protected] [email protected] ontents AdS /CF T d LR = d RL = ± H L d LR = ζ H R algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.1 Differential Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 d AB = ± case 12 d AB = 0 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.2 d AB = +1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.3 d AB = − The infinite-dimensional quantum supergroup underlying the integrable systems beneath the
AdS /CF T correspondence [1] is a constant source of new developments which extend our understanding of exact S -matrices and their (super)symmetries.Such an exotic Hopf superalgebra [2] is extraordinarily close to a Yangian [3] (see also [4]), but it isdistinct from it in ways that allow the possibility of in fact a much larger and more complicated structure.One can certainly recognise in it a filtration in levels, with the level-0 charges represented by Beisert’s psu (2 |
2) Lie superalgebra with central extension [5]. Almost all level-1 charges have a correspondentlevel-0 one, except for the secret or bonus symmetry [6] (see [7] for a review with further references).The absence of a level-0 hypercharge symmetry of the S -matrix prevents the straightforward enlarge-ment to a gl (2 |
2) Yangian. A more sophisticated construction is necessary [8], which utilises the
RT T relations. What [8] has shown (see also [9]) is that Beisert’s S -matrix naturally produces upon the RT T recipe the most part of the
AdS /CF T Hopf algebra, including the secret symmetry.The exploration of lower-dimensional instances of the
AdS/CF T integrable system has revealed amongother things how much more complex the situation can be, casting new light on the original
AdS /CF T problem as well. A new host of boost-like symmetries have made their appearance primarily in AdS ,nd they have been discovered in AdS as well.Integrability in AdS /CF T , both for the AdS × S × S × S and the AdS × S × T background [10](see also [11]), has permitted the adaptation of the largest part of the tools constructed for AdS /CF T ,at least in infinite volume, with the Quantum Spectral Curve and full TBA still to be determined. Thederivation of the finite-gap equations [12] and of the S -matrix [13], see also [14], follows the path of thefive-dimensional analysis. However, massless representations make their powerful appearance and forceus to dramatically rethink the entire algebraic establishment [15]. The relationship between masslessintegrable systems and conformal field theory makes its unavoidable entrance in the conversation [16, 17],after having realised the inadequacy of the standard massive techniques to deal with the massless sector[18] (see also [19, 20, 21]).The clearest manifestation of the new type of symmetries is obtained by thinking of their existence asa deformation, in the quantum group sense, of the natural Poincar´e supersymmetry of the original stringsigma model, which is lost upon gauge-fixing. In AdS × S this idea started with [22] (see also [23])and brought to the recent findings of [24]. In case of massless AdS /CF T excitations, a series of worksestablished a large number of surprising results [25, 26]. Ultimately, in [27, 28] a change of variableswas discovered which recasts the non-relativistic S -matrix for massless left-left and right-right movingmodes, with the inclusion of the dressing factor, in manifestly difference-form. Furthermore, the exactsame functional form applies to the non-relativistic as well as to the relativistic S -matrix as obtained in[17]. This has allowed to write the massless TBA in perfect analogy to [17] also in the non-relativisticlimit [28].Here we will revisit the su (1 | algebra that describes the massless sector of the AdS × S × M scattering problem and the modified Poincar´e structure which we can introduce to describe the remnantof Poincar´e symmetry after gauge-fixing the reparameterisation freedom of the string sigma model. In[32] we were able to construct a consistent coproduct for the boost operator associated to the modifiedPoincar´e algebra that was valid for short representations. In order to understand how to write a coproductmap for the boost operator, we will study the interplay between the two su (1 |
1) subalgebras by makingthem depend on different momenta. We will study the consistency conditions that this new Poincar´estructure has to satisfy, which amounts to checking if the generators satisfy the Jacobi identity. Thiswill provide us with six different possible algebras, which we proceed to study in some detail. Withthat information we will give the form of the coproduct for the boost operator for each of the differentalgebras.The article is structured as follows. In Section 2 we review the su (1 | algebra and its outer auto-morphisms group. We will also study the modified Poincar´e algebra built upon it, assuming in this casethat each copy have a different momenta, and check which restrictions we have to impose in order toget a consistent algebra. Section 3 is devoted to describing each of these consistent algebras separately.In Section 4 we will construct the coproduct of the boost operator associated to the modified Poincar´ealgebra for each of the different cases we described in Section 3. In Section 5 we summarise our resultsand present some concluding comments. We shall always understand that our algebras are superalgebras. q-Poincar´e algebra in AdS /C F T In this article we will be concerned with the massless sector of the
AdS × S × M superstring theory, sowe will study (one copy of) the su (1 | ≡ su (1 | L ⊕ su (1 | R scattering problem. However, in contrastwith what is usually done, we are going to consider the case where the momentum associated with eachof the su (1 |
1) algebras involved is in principle different: p L not necessarily equal to p R . The generatorsof the resulting algebra, together with the two independent associated boost generators, satisfy thefollowing (anti-)commutation relations { Q A , S A } = H A , [ J A , p A ] = i H A , [ J A , H A ] = i H A Φ A , [ J A , Q A ] = φ QA Q A , [ J A , S A ] = φ SA S A , (2.1)where A = L, R , µ = h and we have only reported explicitly the non vanishing (anti-)commutators.The generators Q A and S A are of fermionic (odd) nature ( supercharges ) and form the odd part of su (1 | L ⊕ su (1 | R , while the generators H A ( energies ), J A ( boosts ) and p A ( momenta ) are of bosonic(even) nature. Such generators come from extending su (1 | L ⊕ su (1 | R in order to incorporate adeformed Poincar´e subalgebra, with the added novelty that we shall now introduce with an L and a R copy of every single generator appearing. By φ QA , φ SA and Φ A we have denoted six non-identically-vanishing functions of the momentum generators p A . Their form, as we will amply discuss, is restrictedby the Jacobi identities, although not completely fixed: for instance, one has i H A Φ A = [ J A , H A ] = [ J A , { Q A , S A } ] = { [ J A , Q A ] , S A } + { [ J A , S A ] , Q A } = ( φ QA + φ SA ) H A . (2.2)Apart from the generators we have introduced, we are interested in centrally-extending the algebrain the following way { Q L , Q R } = P , { S L , S R } = K . (2.3)The algebra so defined is traditionally denoted by su (1 | c.e. , where c.e. indicates the central extension. The centrally-extended algebra admits a large outer automorphism group. One particularly prominentgenerator among the outer automorphisms is the so-called hypercharge , denoted by B , which acts on thefermionic generators as[ B , Q L ] = 2 i Q L , [ B , S L ] = − i S L , [ B , Q R ] = − i Q R , [ B , S R ] = 2 i S R . As described in [20], the full set of outer automorphisms form a GL (2) group that rotates thefermionic generators with the same quantum number under the hypercharge B . This group is definedby the action Q L S R ! λ Q L S R ! , S L Q R ! ρ S L Q R ! , (2.4) The odd vs. even grading of the generators can be evinced by the type of anti-commutation vs. commutation relationsthey satisfy. λ and ρ are GL (2) matrices. The action of the outer automorphism group on the centralelements is derived from the action on the supercharges. The algebra action associated to this group isthen given by[ t λ , Q L ] = [ t λ , Q L ] = Q L , [ t λ , S R ] = − [ t λ , S R ] = S R , [ t λ + , S R ] = Q L , [ t λ − , Q L ] = S R , [ t ρ , Q R ] = [ t ρ , Q R ] = Q R , [ t ρ , S L ] = − [ t ρ , S L ] = S L , [ t ρ + , Q R ] = S L , [ t ρ − , S L ] = Q R , (2.5)while the remaining commutation relations vanish. Notice that the hypercharge can be written in termsof these outer automorphisms as 2 i ( t λ − t ρ ). The careful reader might have noticed that we have only enumerated the action of the boost on generatorswith the same handedness among the non-zero (anti-)commutation relations. An important step we needto take is to generalise this construction to incorporate a non-zero action for the boost of one handednesson generators of the opposite handedness.Guided by physical input, we will first restrict the energy generators H A to being non-identically-vanishing positive even functions of the respective momenta, i.e. H A = H A ( p A ). Then, we postulate theaction of a boost from one handedness onto a momentum generator from the opposite handedness as[ J L , p R ] = i H L d LR , [ J R , p L ] = i H R d RL , (2.6)where d AB are functions of the generators p A . Determining such functions will be one of the mainpurposes of our work. We can further generalise this action and write a similar expression for anygenerator in our algebra: H B [ J A , X B ] = H A d AB [ J B , X B ] , (2.7)where X represents any generator with well-defined handedness and A = B . The action on the twogenerators that have mixed handedness, i.e. P and K , is inferred from the Jacobi identities involvingone boost and two supercharges, e.g. [ J L , P ] = ( φ QL H R + φ QR H L d LR ) P .In order to analyse the consistency of our two-handed algebra, first we have to elaborate on thevalue of [ J L , J R ]. There are essentially two arguments why only [ J L , J R ] = 0 is a sensible choice inthis setting: Firstly, we want to maintain the underlying Z -symmetry given by the L ↔ R exchangethat the algebra has up to this point. This choice will make sure that such symmetry manifests itselfthroughout our analysis (e.g. in (2.12)), as a Z -symmetry would require [ J L , J R ] = [ J R , J L ] for the(bosonic) J A . Secondly, we want (2.7) to make sense for all algebra elements X B , including the boostoperators. Considering (2.7) with two differently-handed boosts lets us see that [ J L , J R ] = 0 shouldvanish indeed. We will later revisit this point for the case of a particular representation of the boost.Now we can study the possible values of d AB . These two new functions are not arbitrary, but arefixed by the Jacobi identities involving two boost generators of different handedness and a momentum[ J L , [ J R , p R ]] − [ J R , [ J L , p R ]] + [ p R , [ J L , J R ]] = [ J L , i H R ] − [ J R , i H L d LR ] = 0 , (2.8)which can be simplified to H L d LR Φ R = − i H L [ J R , d LR ] + H R d RL Φ L d LR . (2.9)The other Jacobi identity gives the same equation with the labels L and R exchanged. In addition, wehave to consider the Jacobi identities involving the two boost operators and any of the energies, but they5ive us no extra information. Indeed, we have[ J L , [ J R , H R ]] = [ J R , [ J L , H R ]][ J L , H R Φ R ] = [ J R , H L d LR Φ R ][ J L , H R ]Φ R + H R [ J L , Φ R ] = [ J R , H L d LR ]Φ R + H L d LR [ J R , Φ R ] H L d LR Φ R = − i H L [ J R , d LR ]Φ R + H R d RL Φ L Φ R d LR which do not add extra information on top of (2.9). Jacobi identities involving two boosts and asupercharge are also congruent with (2.9).By direct examination of (2.9) we can already find five different solutions. The first solution is thetrivial case d RL = d LR = 0. The second solution is d RL = 0 and d LR = ζ H R , and the third solution isobtained from the second one by swapping handedness. ζ is in principle a function of the momentumgenerators, however it has to be central with respect to the entire algebra, including the boost generators,so the only option is for it to be a constant function independent of p A . The remaining two solutionsare given by d RL = d LR = ± We now prove that we can only have two categories of algebras, namely either i) d LR d RL = 1 or ii)one (or both) d AB = 0. We can use the above equations for the boost on d AB and equation (2.7) tocompute its action on d LR d RL , from which we get − i H L [ J R , d LR d RL ] = ( H L Φ R − H R Φ L d RL ) d LR d RL (1 − d LR d RL ) , − i H R [ J L , d LR d RL ] = ( H R Φ L − H L Φ R d LR ) d LR d RL (1 − d LR d RL ) . (2.10)Imposing the further consistency H L [ J R , d LR d RL ] = d RL H R [ J L , d LR d RL ] we get that[ H L Φ R (1 − d LR d RL ) − H R Φ L ( d RL − d LR )] d LR d RL (1 − d LR d RL ) = 0 , (2.11)and a similar condition for H R [ J L , d LR d RL ], which can be obtained by swapping the L and R labels in(2.11). These two consistency conditions can be solved simultaneously by either d LR = 0, d RL = 0 or d LR d RL = 1. One can show that no further solutions come from requiring H L Φ R (1 − d LR d RL ) − H R Φ L ( d RL − d LR ) = 0and H R Φ L (1 − d LR d RL ) − H L Φ R ( d LR − d RL ) = 0 (2.12)simultaneously, which would descend from assuming that d LR d RL (1 − d LR d RL ) is non-zero. In fact, aftersome manipulations, we can see that the only values that solve both equations (2.12) simultaneously are d LR = d RL = ± d LR to − i H L [ J R , d LR ] = H L d LR Φ R − H R Φ L . (2.13)A first step to solve this equation is to take out a factor of H R from d LR . If we define d LR = d LR H R ,with d LR a function of the momenta, we can see that − i H L [ J R , d LR ] = − i H L H R [ J R , d LR ] + H L H R d LR Φ R = H L H R d LR Φ R − H R Φ L . As we will more explicitly see later, d AB = ± H L ∝ H R . i H L [ J R , d LR ] = Φ L , which notably simplifies the constraint. Finally, let us prove that d LR H L should commute with J L , fixing it to be a central element with respect to all the elements of the algebra,including the boost: H R [ J L , d LR H L ] = H R [ J L , d LR ] H L + H R d LR [ J L , H L ] = d LR H L [ J R , d LR ] H L + i H R d LR H L Φ L = − i d LR H R Φ L H L + i H R d LR H L Φ L = 0 . We can prove in a similar way that d LR H L commutes with J R . Thus, we finally obtain the relation H L d LR = ζ H R , (2.14)where ζ is a constant. This establishes the sixth type of algebras we shall study. As we will explain inSection 3, there is a nomenclature that lends itself to our situation nicely, namely calling the algebraseither separable or differential : separable algebras = d AB = 0 d LR = 0 and d RL = ζ H L d RL = 0 and d LR = ζ H R differential algebras = d AB = +1 d AB = − H L d LR = ζ H R . The most natural representation from a physical standpoint is the one where we realise the momenta asreal variables, and the boost operator as a derivative operator: J A = i H A ddp A . (2.15)The derivative is understood in a convective fashion, for example ddp R = ∂∂p R + dp L dp R ∂∂p L . (2.16)This is especially relevant for d AB , P and K , as they depend explicitly on both momenta. In thisrepresentation we can understand Φ A = H ′ A = d H A dp A , and d LR and d RL are nothing but the derivative ofone momentum with respect to the other, where we understand that eventually one can choose eitherof the p A to be the only independent variable. In this respect, d AB assume the rˆole of Jacobians. Thisallows us to rewrite (2.16) as ddp R = ∂∂p R + d RL ∂∂p L . (2.17)In this representation we can also write equation (2.9) and its L ↔ R symmetric as H L H R d d LR dp R = ( H L Φ R − H R Φ L d RL ) d LR , H L H R d d RL dp L = ( H R Φ L − H L Φ R d LR ) d RL . (2.18)The six solutions naturally acquire an induced representation. Given our definition of d AB , we can seethat the first solution presented in the previous section, namely d LR = 0 = d RL , is the case where p R and p L are independent of each other. The second and third solutions imply instead p R = ± p L and H R ( p R = ± p L ) = const. H L . During the latter part of our discussion, we will indeed find that the case of H L d LR = ζ H R has connections to boththe differential and separable algebras.
7n addition, this differential representation can be used to construct the action of the outer automor-phisms on the boost operator, as it reduces to computing the action of the outer automorphisms on H A ,as e.g. [ t ρ − , J L ] = [ t ρ − , i H L ddp L ] = i [ t ρ − , H L ] ddp L , where we have [ t ρ − , H L ] = P .As a sanity check of the statement made in the previous section about the vanishing of [ J L , J R ], wequickly want to demonstrate that this is fulfilled within the differential representation framework. Thedifferential representation for the boost feature a convective derivative as in (2.16), and as d AB candepend on the momenta p A , p B , the expression for [ J L , J R ] also features such derivatives, as for example[ J L , J R ] (cid:12)(cid:12)(cid:12) ∂ L coeff. = H L H R ( ∂ p L d RL ) + H L d LR ( ∂ p R H R ) d RL + H L d LR H R ( ∂ p R d RL ) − H R d RL ( ∂ p L H L ) , which is exactly (the right) equation (2.18) after expressing the convective derivative in terms of partialderivatives - and thus vanishes. The vanishing of the ∂ p R coefficient can be proven in a similar fashion,as it has the same structure but with the L ↔ R handedness swapped.Above, we were able to see that a decisive difference of the convective differential with respect to theordinary (holonomic) partial derivatives is that two convective derivatives might not commute due to thepossible momentum-dependence of d AB . However, this in turn is crucial for the vanishing of [ J L , J R ] = 0. In this section we will substitute the acceptable values of the operators d AB we found above and studythe different algebras we obtain in the process, which we have divided into two categories. On the onehand, we denote the cases where at least one d AB = 0 as separable algebras, as in those cases we cancompletely decompose them into two independent subalgebras. On the other hand, we denote the othertwo cases as differential algebras because the relation d AB d BA = 1 can be understood as the inversefunction theorem and d AB can be interpreted as the Jacobian of a change of variables from p B to p A ,building upon the intuition provided by the differential representation presented in the previous section. Let us first examine the case where we set d LR = d RL = 0. The commutation relations involving theboost operators take the form[ J A , p A ] = i H A , [ J A , H A ] = i H A Φ A , [ J L , p L ] = H L , [ J R , p R ] = H R , [ J A , Q A ] = φ QA Q A , [ J A , S A ] = φ SA S A , [ J L , X R ] = 0 , [ J R , X L ] = 0 , [ J A , P ] = φ QA P , [ J A , K ] = φ SA K , (3.1)where X A represents any generator with well-defined handedness A . We can understand these algebras,by looking at their differential representation, as the ones in which all generators with left handednessonly depend on p L for all representations, and similarly for the right handedness, as the two momentahave to be independent. 8e want to better illustrate this separation of the algebra into two by looking at the particular caseof relativistic dispersion relation H A = p A + m . If we focus into the subalgebra formed by the boosts,the momenta and the energies, we can see that it reduces to a known finite-dimensional Lie algebrabecause of the constraint H A Φ A = p A . In that case we can compare it with the classification of solvable6-dimensional Lie algebras presented in [29]. For that we pick the basis x , = − i J L,R n , = H R,L − p R,L n , = H R,L + p R,L , where the n i span the nilradical (which in this case forms an abelian subalgebra) and the x i form itscomplement. With this, it is evident that our algebra corresponds to N αβγδ , with α = δ = 0 and β = γ = − N αβγδ , isindecomposable provided γ + δ = 0 and αβ = 0, the latter of which is not the case for us. Thus, ouralgebra actually corresponds to a simple direct sum of the 3-dimensional left-handed and right-handedsides. This is also the situation for the choice H A Φ A ∝ [ p A ] q , which corresponds to the magnonicdispersion relation for a particular value of q , i.e., H A = h A sin p A + m , with h A a constant.We also want to address the question of fixing the functions φ QA and φ SA in this setting. On the onehand, the Jacobi identity involving one boost operator and two supercharges with different handednessimposes the relations in the fifth line of (3.1) thanks to the decoupling of the left and right sectors. Thisrestriction can be combined with equation (2.2) to further constraint the form of the central elements.In particular i KP Φ A = K (cid:16) φ QA P (cid:17) + P (cid:16) φ SA K (cid:17) = [ J A , P ] + [ J A , K ] , (3.2)as Φ A only depends on p A while P and K depend on both momenta. On the other hand, the Jacobiidentity involving two different boost and a supercharge imposes the restriction[ J A , φ QB ] = [ J A , φ SB ] = 0 , (3.3)for A = B . In the differential representation this can be interpreted as restricting the functions φ SA and φ QA to depend only on the momentum p A . This is truly a very heavy restriction, as it forces the centralelements to be separable, i.e., P = P R P L and K = K R K L , for this algebra to be consistent. In thedifferential realisation of the boost, this is immediate, as i H A ddp A P = [ J A , P ] = φ QA P = ⇒ i ddp A log P = H − A φ QA , where the latter term only depends on p A . Even if J A is a priori not of differential form, the sameargument can be made as (3.3) still holds since the adjoint action of J A only vanishes on p A -independentelements because of d AB = 0. Thus, since J A vanishes on P − [ J B , P ] for A = B , the argument follows.Let us point out that the d LR = 0 = d RL case has no applications to AdS physics. If we wanted tostudy massless excitations in AdS , we would necessarily have to take the moves from central elementsof the form K ∝ P ∝ sin( p L + p R ), which do not fulfil the condition of separability.Let us move our attention to the case d LR = 0 and d RL = ζ H L . This case can be transformed backto the d LR = d RL = 0 case - after a redefinition of one of the boosts in terms of the d AB = 0 boosts J A ˆ J R = J R − ζ H R J L , (3.4)so all of the above arguments apply here with minimal changes. Similarly can be done for the oneobtained by swapping the handedness, d RL = 0 and d LR = ζ H R , with the redefinitionˆ J L = J L − ζ H L J R . (3.5)9or the d AB = 0, it also follows that [ J L , J R ] = 0, be virtue of the relationship amongst L -handed and R -handed generators in this case. (3.4) and (3.5) imply the same conclusion for d LR = 0 and d RL = ζ H L as well as d RL = 0 and d LR = ζ H R , respectively. d LR = d RL = ± algebra As we have pointed out earlier by referring to the differential representation, this choice of d AB imposesthat either p L = p R or p L = − p R , which means that either the two copies built on each su (1 |
1) areexactly the same or one is the parity-transformed of the other. We should also stress that this solutiononly exists provided H R [ p R ( p L )] = d LR h R h L H L ( p L ) , (3.6)with h A constants such that sign( d AB h B h A ) = +1. Although this case is very restrictive, the algebra we willpresent in the next section is a generalisation of this algebra without such constraints. Representationsof this particular algebra have been profusely studied for the case p L = p R , for example in [20].After performing the identifications (3.6), the algebra can be written as { Q A , S A } = H A , [ J A , p A ] = i H A , [ J A , H A ] = i H A Φ A , [ J A , Q A ] = φ QA Q A , [ J A , S A ] = φ SA S A , { Q L , Q R } = P , { S L , S R } = K ,h A [ J B , P ] = (cid:16) h L φ QR + h R φ QL (cid:17) P , h A [ J B , K ] = (cid:0) h L φ SR + h R φ SL (cid:1) K , [ J A , X B ] = h A h B [ J B , X B ] , (3.7)where A, B = L, R , A = B and X is any generator with well-defined handedness. Notice that we havemade explicit use of the condition h A H B = d AB h B H A . The above relations also imply [ J L , J R ] = 0, as J L and J R essentially coincide for d AB = ± J A , H A and p A with relativistic dispersion relation H A = p A + m ,which is reduced to a 5-dimensional one by identifying H R = H L = H , we find an algebra which has a2-dimensional centre spanned by the generators p L ∓ p R and J L − J R . In the case d LR = d RL = 1, theremaining 3-dimensional algebra obtained after we mod-out the centre has the following non-vanishingcommutation relations (cid:20) J L + J R , p L + p R ± H (cid:21) = ± (cid:18) p L + p R ± H (cid:19) , which correspond to one of the four irreducible 3-dimensional solvable algebras [34]. This constructioncan be extended to different dispersion relations, thanks to the restriction H R = H L which keeps p L ∓ p R and J L − J R as central elements.We shall now return to study general features of the d LR = d RL = ± Let us consider the differential representation for the d LR = d RL = ± AdS , we are going to choose the following particular10ependence on the momenta for the central generators H L = h L (cid:12)(cid:12)(cid:12)(cid:12) sin p L (cid:12)(cid:12)(cid:12)(cid:12) , H R = d LR h R (cid:12)(cid:12)(cid:12)(cid:12) sin p R (cid:12)(cid:12)(cid:12)(cid:12) , K ∝ P ∝ (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) p L + d LR p R (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) , (3.8)where is the identity matrix in (1boson , h L and h R are positive realnumbers. The quantity h L + h R is usually identifies as a coupling constant appearing in the dispersionrelation of the fundamental excitations in the context of the AdS /CF T duality. Moreover, we setthe two boost to be equal up to a multiplicative constant h R J L ≡ h L J R , which is consistent with theconstraints imposed by the Jacobi identities specialised to the algebra in object. These two identificationstogether with the constraint p L = ± p R also imply d LR h R H L = h L H R in this representation.We now restrict ourselves to p L ≡ p R ≡ p > φ Q = φ S , so that in this representation we have[ J A , Q B ] = i Φ A Q B , [ J A , S B ] = i Φ A S B , (3.10)for any combination of A and B . Using such commutation relations, we can strip the supercharges ofthe dependence on the momentum and write (keeping in mind equation (3.9)) Q L = r αh L sin p Q L , Q R = r βh R sin p Q R , S L = r h L α sin p S L , S R = s h R β sin p S R , (3.11)where α and β are constants and the hatted quantities are matrices satisfying { ˆ Q L , ˆ S L } = { ˆ Q R , ˆ S R } = , (3.12)in order for this algebra to agree with our definition of H A .Furthermore, we can repeat the same rewriting with the other two central elements P and K . It iseasy to see that P = (cid:12)(cid:12)(cid:12)(cid:12) h L h R αβ (cid:12)(cid:12)(cid:12)(cid:12) sin p P , K = (cid:12)(cid:12)(cid:12)(cid:12) h L h R αβ (cid:12)(cid:12)(cid:12)(cid:12) sin p K . (3.13)Furthermore, centrality imposes that ˆ P and ˆ K should be proportional to the identity. Combiningthat with their definition in terms of supercharges we can write { ˆ Q L , ˆ Q R } = ˆ P = γ , { ˆ S L , ˆ S R } = ˆ K = η (3.14)with γ and η being constants. The first two equalities come from our definition of H A . Although we canperform the redefinitions ˆ Q A → √ γ ˆ Q A and ˆ S A → √ γ ˆ S A , which are akin to setting γ = 1, there is nomethod to set η to 1 at the same time. In fact, this parameter η is actually related to the shorteningcondition, as short representations can only exist if η = 1. For general values of η we can see that thehatted quantities fulfil { (1 + xη ) ˆ Q L − (1 + x ) ˆ S R , x ˆ S L + ˆ Q R } = 0 , (3.15) { y ˆ Q L + ˆ S R , ( y + 1) ˆ S L − ( y + η ) ˆ Q R } = 0 , (3.16)11or any values of x and y .Thus, we can write a set of relations which are peculiar to this representation: (cid:20) J L + J R , p L + p R (cid:21) = H L + H R , [ J L + J R , H L + H R ] = i ( H L + H R )(Φ L + Φ R ) , { ˆ Q L , ˆ S L } = { ˆ Q R , ˆ S R } = { ˆ Q L , ˆ Q R } = η − { ˆ S L , ˆ S R } = , (3.17)which represents a complete separation of the su (1 |
1) algebra from the modified Poincar´e algebra. H L d LR = ζ H R algebra Substituting the value of d AB into equation (2.7) give us that the relation between the two handednessof this algebra is [ J A , X B ] = (cid:16) ζ + (1 − ζ ) d AB (cid:17) [ J B , X B ] , (3.18)for any generator X and for any combination of A, B = L, R . In contrast with the previous case, no linearcombination of p L and p R is central. Nevertheless, one can readily check that we can indeed constructthe boost operators J L and J R of this algebra from J L and J R - again the ones of the d LR = d RL = 0algebra - by making the following identifications: J L = J L + ζ J R , J R = J R + ζ J L . (3.19)Notice that (3.19) also implies [ J L , J R ] = 0, as we can just reexpress [ J L , J R ] in terms of [ J L , J R ], whichwe know to vanish. As we are interested in representations defined by H L ( p L ) = h L sin p L and H R ( p R ) = h R sin p R ,substituting these expressions into the expression for d LR we get d LR = ∂ p L p R = h R h L ζ csc p L p R ⇒ p R ( p L ) = 4 arccot (cid:16) κ cot γ p L (cid:17) , (3.20)where κ is an integration constant and γ = h R h L ζ . Depending on whether | κ | is greater or smaller than 1,we have to reduce the range of p R or the range of p L . This representation amounts to having H R ( p R ( p L )) = κh R h γL H γL κ cos γ p L + sin γ p L . (3.21)when H R is written in terms of p L .Notice that the representation from section 3.2.1 is included in this family of representations as thecase κ = γ = 1. d AB = ± case In this section we will continue the line of research we started in [32], where we picked a particularrepresentation and coproduct of the su (1 | c.e. algebra and constructed a boost operator whose coproductquasi-cocommutes with the associated R -matrix. However, in said article we were not able to write such We could also state the algebra without reference to handedness due to the above identification.
12 coproduct beyond the short representation of this algebra. Here we will construct a coproduct mapfor the boost operator .We will argue that it is impossible to construct such a map using only the elements of the algebraand that we need to make use of elements of the GL (2) outer automorphisms of su (1 | . Interestingly,something similar was observed in the study of the universal R -matrix of the centrally extended psu (2 | AdS scattering problem [31]. There it also was found that it is unavoidable to utilisethe generators associated to the outer automorphisms, which are dual in the sense of the Killing form tothe generators of the central extensions. The issue of the universal R -matrix encompassing the centralextension has not been fully studied in the context of the su (1 | algebra (for a partial analysis, seeAppendix C [33]). Let us take the case of the left sector and the coproduct∆ S L = S L ⊗ e ip L / + e − ip L / ⊗ S L , ∆ Q L = Q L ⊗ e ip L / + e − ip L / ⊗ Q L . (4.1)with similar coproduct for the right fermionic generators, with p L substituted by ± p R . It is important tostress that this choice of coproducts forces us to set the central elements to have the following dependenceon the momentum H L ∝ (cid:12)(cid:12)(cid:12)(cid:12) e ip L / − e − ip L / (cid:12)(cid:12)(cid:12)(cid:12) R ∝ (cid:12)(cid:12)(cid:12)(cid:12) e ip R / − e − ip R / (cid:12)(cid:12)(cid:12)(cid:12) , P ∝ K ∝ (cid:0) e i ( p L ± p R ) / − e − i ( p L ± p R ) / (cid:1) , (4.2)where the ± in (4.2) corresponds to the sign of the R-handed coproduct choice, not the choice of d AB .These constraints are due to the fact that central generators of the algebra have to be co-commutative.We should first comment on the issues this coproduct presents in the different algebras and how to dealwith them. First of all, the algebra with d AB = 0 needs the central elements P and K to be separable tobe consistent while the coproduct imposes them to be a trigonometric function of the sum (or difference)of the two momenta. The simplest form to deal with this issue is to set them to zero and consider thealgebra d AB = 0 as two completely independent su (1 |
1) algebras. In addition, there is an apparent issuewith the cases where p L ± p R = 0, but this is just a consequence of ∆ P and ∆ K becoming trivial in suchcases, therefore eliminating the restriction that allowed to fix them. We will address this point in depthlater.In order to construct the coproduct map, we should start by looking at the form it takes when it isevaluated in a tensor product of short representations in the case d AB = 1 (i.e., when S R ≡ Q L and S L ≡ Q R ). This was constructed in [32] and is given by ∆ J = ∆ J + e − i p ⊗ e i p S ⊗ Q + Q ⊗ S ] , (4.3)where ∆ J = J A ⊗ cos p + cos p ⊗ J A .Although such map was constructed specifically to be an algebra homomorphism when consideringthe tensor product of short representations, it is easy to see that it is not an algebra homomorphism for By definition, such a map will be independent of the choice of a representation. Equations (3.8) and (3.9) in [32] possess typos, here we present the corrected expressions. A different expression for this map was proposed in [25]. Although it has different quasi-triangularity properties, itsexpression is similar enough that the computation we are going to perform can be applied to it with minimal changes. Q L − S R and Q R − S L that vanish for the short representation. One way to rephrase this problem is that we would like to finda way for the commutation relation[( S L ⊗ Q L + Q L ⊗ S L ) , ∆ Q R ] = e − i p R S L ⊗ P + P ⊗ e i p R S L , (4.4)to become vanishing.These unwanted terms pose a problem because all of them involve only the fermionic generator S L while we started with Q R . There does not exist any generator in the algebra su (1 | that can transformeither of these two fermionic generators into the other, so we need to make use of the outer automorphismsto get rid of the unwanted terms. In particular, the combination of generators we are looking for is FT L = S L ⊗ Q L + Q L ⊗ S L − α R (cid:0) P ⊗ t ρ + + K ⊗ t λ + (cid:1) − β R (cid:0) t ρ + ⊗ P + t λ + ⊗ K (cid:1) , (4.5)where α A = − e i p A ⊗ e i p A and β A = e − i p A ⊗ e − i p A . We can check that now the commutators withall the labelled R fermionic generators vanish while all the new contributions vanish when applied to alabelled L fermionic generator. A similar combination can be written for the right copy of the algebra FT R = S R ⊗ Q R + Q R ⊗ S R − α L (cid:0) P ⊗ t λ − + K ⊗ t ρ − (cid:1) − β L (cid:0) t A − ⊗ P + t ρ − ⊗ K (cid:1) , (4.6)which commutes with the two labelled L fermionic generators. These two linear combinations of gener-ations are the cornerstone to tackle the problem of the coproduct for the d AB = ± d AB = 0 case We will start by focusing on the d LR = d RL = 0 case. This is the simplest to treat, as the restriction P = K = 0 completely separates the two algebras, so we can just consider equation (4.3) and write theappropriate subindices. However, it is interesting for later arguments to upgrade the fermionic tail withthe expressions we wrote above even though the new terms give no contribution∆ J L ( d AB = 0) = ∆ J L ( d AB = 0) + e − ip/ ⊗ e ip/ FT L , ∆ J R ( d AB = 0) = ∆ J R ( d AB = 0) + e − ip/ ⊗ e ip/ FT R . (4.7) d AB = +1 case Let us now focus on the case d LR = d RL = 1. If we choose to set J L = J R = J with d LR = d RL = 1, thenthe action of J is perfectly analogous to what the action of J L + J R is in the case where d LR = d RL = 0.At the level of the algebra, making this connection is unproblematic. At the level of the Hopf algebra, itis not. As we explained above, the d LR = d RL = 0 case exhibits problems with coproducts and centralelements, so we set the latter to zero. However, if we want to draw some parallels between how theboost operators act for the different values of d AB , the upgraded tails we used in equation (4.7) arenecessary. In this way, they have the properties they need for us to write a similar relation for thecoproduct of the boosts. Though, one needs to be be aware that by simple adding two d LR = d RL = 0coproducts, the derivative term would appear twofold, hence it is necessary to subtract the so-obtainedresult by one ∆ J . This happens because at the end we have to identify p L = p R , so we are adding thederivative factor twice instead of once. Thus, the combination of generators that makes the coproductmap a homomorphism and that reduces to the correct expression for the short representations can be14xpressed as ∆ J ( d AB = 1) = ∆ J L ( d AB = 0) + ∆ J R ( d AB = 0) − ∆ J , (4.8)∆ J = ∆ J + FT L + FT R = ∆ J + e − i p ⊗ e i p { S L ⊗ Q L + Q L ⊗ S L + S R ⊗ Q R + Q R ⊗ S R − α (cid:2) P ⊗ ( t ρ + + t λ − ) + K ⊗ ( t λ + + t ρ − ) (cid:3) − β (cid:2) ( t ρ + + t λ − ) ⊗ P + ( t λ + + t ρ − ) ⊗ K (cid:3)(cid:9) . (4.9) • Short Representation
A consistency check of our coproduct is to study the representation where S R = Q L = Q and Q R = S L = S , where it should reduce to the su (1 |
1) boost-symmetry computed in [32], which wereproduce in equation (4.3). One obstruction to evaluate ∆ J in this representation is the fact that thegenerators of the outer automorphism are not well-defined in this representation. However, we will seethat they appear in a particular linear combination that is well-defined. In this representation we alsohave H L = H R = P = K = H and the coproduct of the boost becomes∆ J = ∆ J + e − i p ⊗ e i p S ⊗ Q + 2 Q ⊗ S − α H ⊗ T − β T ⊗ H ] , (4.10)where T = t λ + + t λ − + t ρ + + t ρ − . For this representation we can see that [ T , Q ] = Q and [ T , S ] = S , whichimplies that [2 S ⊗ Q + 2 Q ⊗ S − α H ⊗ T − β T ⊗ H , ∆ X ] = [ S ⊗ Q + Q ⊗ S , ∆ X ] , (4.11)for X = Q or S . Thus our coproduct map for the boost for the d LR = d RL = 1 reduces to the appropriateone in this representation. d AB = − case Let us now move to the case d LR = d RL = −
1. First of all, we find that equation (4.2) seems to fix twoof the central elements to be trivial in this case, due to the relation between the momenta. This is notthe case because, although the boost behaves in this case very similarly to the case d LR = d RL = 1, thecoproduct structure allows us more freedom as the central elements P and K fulfil now∆ P = P ⊗ ⊗ P , ∆ K = K ⊗ ⊗ K . Thus any dependence on the momentum now satisfies the cocommutativity property and the behaviourfor them we deduced in equation (4.2) is not valid any more. Nevertheless, as there is nothing specialabout this point and from the commutation relations between P , K and J , we can fix the four centralelements to be H L ∝ H R ∝ P ∝ K ∝ ( e ip/ − e − ip/ ) , (4.12)and a similar construction to the case d LR = d RL = 1 follows.Finally, we want to comment that choosing the opposite sign for p R in (4.2) just exchanges thebehaviour between the cases d LR = d RL = ±
1, so the same arguments apply there.
In the above subsection, we constructed the coproduct of the boost generator for the case we denotedas “bosonically braided coproduct” in [32]. In this subsection, we want to complement it by computingthe coproduct of the boost in the case we denoted as “bosonically unbraided” in said article.15his bosonically unbraided coproduct is defined by the following choice of coproducts for the fermionicgenerators ∆ Q A = Q A ⊗ e i p + e − i p ⊗ Q A ∆ S A = S A ⊗ e − i p + e i p ⊗ S A , In contrast with the bosonically braided case, here the coproduct of the boost also involves the hyper-charge operator. In particular, for the short representation we found that∆ J L = A ( p , p ) J L ⊗ B ( p , p )1 ⊗ J L + F + ( p , p ) S ⊗ Q + F − ( p , p ) Q ⊗ S + G ( p , p ) [ B ⊗ − ⊗ B ] . (4.13)There are multiple subtleties and ambiguities in the process of fixing the functions A , B , F + , F − and G , which are discussed at length in [32]. One of the possible ways of fixing them is A ( p , p ) = B ( p , p ) = cot (cid:16) p (cid:17) cot (cid:18) p − p (cid:19) F ± ( p , p ) = e ± i p p cot (cid:18) p − p (cid:19) (cid:20) i csc (cid:16) p (cid:17) csc (cid:16) p (cid:17) − i ± cot (cid:18) p + p (cid:19)(cid:21) ,G ( p , p ) = − ih cos (cid:18) p − p (cid:19) csc (cid:18) p + p (cid:19) . In order to construct an analogous FT A for this case, we also need to substitute any B operatorpresent in ∆ J A by another operator that acts as the usual hypercharge on A -handed operators, whereasthey would have to vanish on operators with the opposite handedness. We call these operators B A andthey can be written in terms of the outer automorphisms as − i B R = t λ − t ρ − t λ − t ρ − i B L = t λ − t ρ + t λ + t ρ With those operators we can write the following upgraded version of the tail FT L = G [ B L ⊗ − ⊗ B L ] + F + (cid:2) S L ⊗ Q L − β R t ρ + ⊗ P + β R K ⊗ t λ + (cid:3) + F − (cid:2) Q L ⊗ S L − α R P ⊗ t ρ + + α R t λ + ⊗ K (cid:3) , (4.14) FT R = G [ B R ⊗ − ⊗ B R ] + F + (cid:2) Q R ⊗ S R + α L t ρ − ⊗ K − α L P ⊗ t λ − (cid:3) + F − (cid:2) S R ⊗ Q R + β L K ⊗ t ρ − − β L t λ − ⊗ P (cid:3) . (4.15)Here, we have again α A = − e i p A ⊗ e i p A and β A = e − i p A ⊗ e − i p A .Having constructed the FT A for this choice of fermionic coproducts, we can follow the prescriptionwe used above, that is∆ J L ( d AB = 0) = ∆ J L ( d AB = 0) + FT L , ∆ J R ( d AB = 0) = ∆ J R ( d AB = 0) + FT R , ∆ J ( d AB = 1) = ∆ J L ( d AB = 0) + ∆ J R ( d AB = 0) − ∆ J = ∆ J + G ( B ⊗ − ⊗ B )+ F + (cid:0) S L ⊗ Q L + Q R ⊗ S R − β (cid:0) t ρ + ⊗ P − K ⊗ t λ + (cid:1) − α (cid:0) P ⊗ t λ − − t ρ − ⊗ K (cid:1)(cid:1) + F − (cid:0) Q L ⊗ S L + S R ⊗ Q R − α (cid:0) P ⊗ t ρ + − t λ + ⊗ K (cid:1) − β (cid:0) t λ − ⊗ P − K ⊗ t ρ − (cid:1)(cid:1) keeping in mind that we identify p L = p R ≡ p and that B R + B L = B .16 L d LR = ζ H R d AB = +1 d AB = − d AB = 0 d LR =0 d RL = ζ H L d RL =0 d LR = ζ H R differential algebras separable algebras p L = ± p R ( . ) ( . ) (3.19), (3.18) ( . ) ( . ) Figure 1: On the left, we can see how the differentiable algebras are related to each other, whereas onthe right we can see the genealogy of separable algebras. Notice the curious relation of H L d LR = ζ H R with both algebra types. In this article, we have studied different consistent extensions of the centrally extended su (1 | algebrato the case where each copy depends on a different momenta. This shed some light on how to constructthe coproduct map for the boost operator we can add to such an algebra, generalising the results obtainedin [32] for short representations to any representation.First, we investigated possible ways the boost operator is able to adjointly act on generators ofopposite handedness (and by extension, also the central elements). By imposing consistency of theJacobi identities, we arrive at six different algebras, some of which are the handedness-transformed ofthe other. Essentially, these can be classified in terms of two categories, as schematically depicted inFigure 1.These names are explained by the most characteristic property of each of the two sets of algebras.The three separable algebras are characterised by the fact that they can be split into two well separatedsubalgebras that do not talk to each other. The differential algebras are characterised by the equation d LR d RL = 1, which can be understood as a consequence of the inverse function theorem if we understandthe boost as a differential operator. The H L d LR = ζ H R is interesting, as it can both be understood asdifferential and separable at the same time.After this close study of different algebras, we were able to construct coproduct maps for the boostoperator J A for each of the cases. The crucial point in our construction is that such coproducts cannotbe built with only elements of our algebra and we need to make use of the generators associated to theouter automorphisms group of su (1 | [8, 31]. They appear in the tail of ∆ J A as key ingredients tocancel out unwanted fermionic contributions in commutators. We have showed that it is enough four usto compute ∆ J A for the case d AB = 0, as the remaining cases can be constructed by appropriate linear17ombinations of the fermionic tails associated to them.The most immediate extensions of this work is to consider the massive case, as we have restrictedourselves to the massless one so far. Another task would be to construct the coproduct map associatedto the symmetry which in AdS × S was computed for short representations by [24]. As in the casestudied here, we expect the outer automorphisms to play a central rˆole also in those situations, as theyare known to be dual (in the sense of Hopf algebra) to the generators associated to the central extension[31]. In addition, it would be interesting to repeat the construction of the dual of the algebra performedin those two articles in our setting, as this will confirm that central generators and outer automorphismsare dual to each other. Furthermore, we would be able to check if the R -matrix obtained through thatprocedure still quasi-cocommutes with the coproduct of the boost we computed here.An interesting point we have not studied in this paper is the antipode map and its action on the newlyconstructed coproduct map. An antipode would extend the bi-algebraic structure we have described inthis article to a Hopf algebra. However, this computation would require a more profound knowledge ofthe action of the coproduct and the counit on the outer automorphisms that is beyond the scope of thisarticle. Nevertheless, we plan to address this point in future publications. Acknowledgements
We are grateful to Vidas Regelskis for reading the manuscript and providing very useful comments. LWis funded by a University of Surrey Doctoral College Studentship Award. This work is supported by theEPSRC-SFI grant EP/S020888/1
Solving Spins and Strings .No data beyond those presented and cited in this work are needed to validate this study.
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