Spin chain for the deformed ABJM theory
aa r X i v : . [ h e p - t h ] S e p LPTENS-09/21
Spin chain for the deformed ABJM theory
Davide Forcella a Waldemar Schulgin a,b a Laboratoire de Physique Th´eorique de l’ ´Ecole Normale Sup´erieureand CNRS UMR 854924 Rue Lhomond, Paris 75005, France b Laboratoire de Physique Th´eorique et Hautes Energies,UPMC Univ Paris 06, Boite 126, 4 place Jussieu, F-75252 Paris Cedex 05 France
Abstract
In this short note we begin the analysis of deformed integrable Chern-Simons the-ories. We construct the two loop dilatation operator for the scalar sector of the ABJMtheory with k = − k and we compute the anomalous dimension of some operators. [email protected] [email protected] ontents SU (2) R × SU (2) invariant potential 5 In the context of the
AdS /CF T correspondence a very interesting development was theunderstanding of the existence of an integrable structure at the both sides of the correspon-dence [1, 2]. In the N = 4 SU ( N ) field theory the one loop dilatation operator in the scalarsector was identified with the Hamiltonian of an integrable spin chain [1]. Many and veryinteresting developments followed, see for example [3]-[18]. In this note we will be mainlyinterested in studying the integrability properties of the field theories with less supersym-metry. In four dimensions to remain in the perturbative regime, which allows a field theorycomputation, one is forced to take orbifold or marginal deformations of the original N = 4theory [19]-[22].Recently we gained a better understanding of the AdS /CF T correspondence [23]-[46].Indeed, it turns out that the three dimensional conformal field theories are Chern-Simonstheories with matter. In particular, the authors of [32] proposed a field theory dual to the1 / Z k singularities, the so called ABJM theory. This is an N = 6 Chern-Simons mattertheory with gauge group U ( N ) × U ( N ) and two Chern-Simons levels satisfying the constraint k + k = 0. This theory appears to be integrable at least at the leading order in perturbationtheory. Namely, the two loop dilatation operator can be identified with the Hamiltonian ofan integrable spin chain [47]. Many nice developments followed also in this context, see forexample [48]-[59].In particular we are interested to understand if integrability is present in the less super-symmetric theories. One could think that the possible generalizations of the basic examplein three dimensions, the ABJM theory, are very similar to the related generalizations of the N = 4 four dimensional case but they are slightly different.To compute the field theory dilatation operator, it is important that the theory has aweak coupling limit in which the elementary fields have canonical scaling dimensions. In fourdimensions this is possible if the superpotential is a cubic function of the chiral superfields,while in three dimensions it is possible if the superpotential is a quartic function. Thissimple observation points out that in three dimensions there could be more theories whichcan be analyzed perturbatively than in four dimensions. Indeed, it turns out that in threedimensions also the non-orbifold theories can have a perturbative limit [28, 38, 46].The second observation is due to the presence of Chern-Simons levels that do not exist inthe four dimensional case. There are Chern-Simons levels k i associated to every gauge group.They are integer numbers and we can vary their values without spoiling the superconformalsymmetry. It turns out that, for a class of N = 2 Chern-Simons matter theories, if P k i = 0the field theory moduli space has a four complex dimensional branch that is a Calabi-Yaucone and can be understood as the space transverse to the M2 brane [43, 44]. If instead P k i = 0 the four dimensional branch typically disappears and this effect can be interpretedas turning on a Roman’s mass F in the type IIA limit [38, 60]. Let us suppose that atheory has an integrable structure for some specific relations among the k i such that theysatisfy P k i = 0. It easy to see that there exist two possible interesting deformations of thisintegrable point. We can move in the space of possible integer values of the k i in such a waythat we preserve the constraint or in a way in which we break the constraint. It is importantto underline that these kind of deformations do not exist in four dimensions and offer a newlaboratory for studying integrability in the weak coupling regime.In this paper we start the analysis of these deformed theories. We take as basic examplethe ABJM theory and deform it in such a way that k + k = 0. We plan to return tothe other type of deformation in the near future. To be sure to remain in the perturbativeregime it is important to deform the theory in such a way that it preserves at least N = 3supersymmetry in three dimensions. Indeed, for N > k i . These theorieshave a quartic superpotential and could be dual to the non-orbifold M theory backgrounds.The organization of the paper is as follows. In Section 2 we introduce our main example.In Section 3 we rewrite the theory in the explicit invariant form under the global symmetries.In Section 4 we compute the two loop mixing operator for the scalar sector of the theory.In Section 5 we compute the anomalous dimension of some operators. We observe that thedegeneracy which due to integrability is present in the ABJM theory is lifted in the generic k = − k case. We finish with some conclusions and the appendix collects some useful2ormulae which we used in the main text. We are interested in studying the Chern-Simons theories described by the following action S = k π S CS ( V (1) ) + k π S CS ( V (2) ) + S kin ( Z i , Z † i , W j , W j † ) + Z d θW ( Z i , W j ) + c.c. , where S CS ( V ( l ) ) = Z d x Tr (cid:20) ǫ µνλ (cid:18) A ( l ) µ ∂ ν A ( l ) λ + 2 i A ( l ) µ A ( l ) ν A ( l ) λ + i ¯ χ ( l ) χ ( l ) − D ( l ) σ ( l ) (cid:19)(cid:21) ,S kin ( Z i , Z † i , W j , W j † ) = Z d θ Tr (cid:16) Z † i e − V (1) Z i e V (2) + W j † e − V (2) W j e V (1) (cid:17) ,W ( Z i , W j ) = 2 πk Tr (cid:0) Z i W i Z j W j (cid:1) + 2 πk Tr (cid:0) W i Z i W j Z j (cid:1) . (2.1)It is a three dimensional Chern-Simons theory with matter. The gauge group is U ( N ) × U ( N ) and the N = 2 bifundamental chiral superfields Z i and W j transform in the funda-mental of the first factor of the gauge group and antifundamental of the second one and viceversa for Z † i and W † j (see figure 1). k , k are integer numbers which we call from now on Z i W j U(N) k , U(N) k , Figure 1:
The quivers for the ABJM theory with generic Chern-Simons levels.
Chern-Simons levels. The three dimensional theory represented by the Lagrangian (2.1) is N = 3 superconformal. It admits a perturbative limit for the large values of the k i . TheLagrangian has SU (2) R × SU (2) global symmetry, where the first factor is the R symme-try associated to the N = 3 superconformal symmetry, while the second SU (2) is a globalsymmetry under which Z i and W j transform in the fundamental representation.In the particular case k = − k the supersymmetry of the Lagrangian is enhanced to N = 6 and the global symmetry group to SU (4) R . In this case the lower bosonic com-ponents of the chiral superfields can be organized in the fundamental representation : Y A = ( Z , Z , W † , W † ) and the upper ones in the antifundamental. Indeed, in this limitthe Lagrangian (2.1) reduces to the ABJM one [32] which is supposed to describe the threedimensional superconformal field theories living on N M2 branes at C / Z k singularities. Inthis particular case the theory is integrable in the planar limit at least at the two loop order. We use the same symbols Z i , W j for the superfields and for their lowest scalar components. We hopethis will not cause too much confusion. SU (4) spin chain with the sites transforming under and ¯4 [47].A natural question is if the generic theory in eq. (2.1) is still integrable. In the case k = − k the supersymmetry and the global symmetries are reduced, and the theory issupposed to be dual to some flux background. The four dimensional Calabi-Yau branch inthe field theory moduli space disappears and the theory is proposed to be dual to a typeIIA background with the Romans mass F turned on: k + k = F [38]. It is important tostress that this kind of deformation is not an orbifold deformation and this is a peculiarityof the Chern-Simons theories.In this paper we would like to do the first step towards understanding the questionconcerning integrability for this theory. We compute the dilatation operator in the scalarsector at the leading order which we use then to find anomalous dimensions of some operators.To make the computation more transparent we rewrite the eq. (2.1) in such a way that the SU (2) R × SU (2) symmetry becomes apparent. We group the scalar fields into the tensors O ai and O † ia , where the indices from the beginning of the alphabet correspond to the SU (2) R and from the middle to the SU (2) symmetry group O = (cid:18) Z † W Z † W (cid:19) , O † = (cid:18) Z Z W † W † (cid:19) . (2.2)They transform in the ( , ) of SU (2) R × SU (2) as U OV † , V O † U † , where U ∈ SU (2) and V ∈ SU (2) R .The class of the gauge invariant operators we are interested in has the form O = Tr (cid:16) O † i a O a i O † i a O a i .......O † i L − a L − O a L i L (cid:17) χ a i a i ...a L − ,i L i a i a ...i L − ,a L , (2.3)where χ is some tensor of SU (2) R × SU (2). These operators need to be renormalized O Mren = Z MN (Λ) O Nbare , (2.4)where M and N label all the possible operators, Λ is an UV cutoff, and Z subtracts all theUV divergences from the operator correlator functions. The object we are interested in isthe matrix of anomalous dimensions Γ. It is defined asΓ = d ln Zd ln Λ . (2.5)The eigenstates of Γ are conformal operators and the eigenvalues are the correspondinganomalous dimensions.It is convenient to represent the operators (2.3) as states in a quantum spin chain with2 L sites. Every site transforms in (2 ,
2) representation of SU (2) R × SU (2). The spin chain isalternating between the O † and O . In this language the mixing matrix (2.5) can be regardedas the Hamiltonian acting on the Hilbert space ( ¯ V ⊗ V ) ⊗ L .4 SU (2) R × SU (2) invariant potential In this section we would like to write the action (2.1) in terms of component fields andin particular we would like to have an explicit expression for the potential in terms of SU (2) R × SU (2) invariant objects. We start by integrating out all the auxiliary fields. Inparticular the spinorial fields χ ( l ) and the bosonic fields σ ( l ) , D ( l ) are all auxiliary fields andcan be eliminated using the equations of motion. From the chiral super fields Z I , W j we getthe complex scalars Z i , W j and the Dirac spinors ζ i , ω j . The potential V can be dividedinto a part V bos containing only bosonic operators and a part V ferm containing bosonic andfermionic operators. Let consider first the bosonic part. The bosonic potential V bos gets contributions from the superpotential and from the Chern-Simons interactions V bos = V bosW + V bosCS . The superpotential part is V bosW = Tr X i,j (cid:12)(cid:12)(cid:12) ∂ Z i W (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ W j W (cid:12)(cid:12)(cid:12) ! , (3.1)where W is the superpotential given in eq. (2.1). The Chern-Simons part is V bosCS = Tr (cid:16) Z † i Z i σ − Z i σ (1) Z † i σ (2) + Z i Z † i σ (cid:17) +Tr (cid:0) W † i W i σ − W i σ (2) W † i σ (1) + W i W † i σ (cid:1) , (3.2)where σ (1) = 2 πk (cid:16) Z † i Z i − W i W † i (cid:17) , σ (2) = 2 πk (cid:16) W † i W i − W i W † i (cid:17) . (3.3)If we write a general ansatz by use of operators in eq. (2.2) there exist 18 structurescompatible with the symmetries and the canonical dimension of the bosonic fields . V bosa n = a Tr O ai O † ia O bj O † jb O ck O † kc + a Tr O ai O † ia O bj O † kb O ck O † jc + a Tr O ai O † ja O bk O † ib O cj O † kc + a Tr O ai O † ja O bj O † kb O ck O † ic + a Tr O ai O † ib O bj O † jc O ck O † ka + a Tr O ai O † ib O bj O † kc O ck O † ja + a Tr O ai O † jb O bk O † ic O cj O † ka + a Tr O ai O † jb O bj O † kc O ck O † ia + a Tr O ai O † ia O bj O † jc O ck O † kb + a Tr O ai O † i a O bj O † kc O ck O † jb + a Tr O ai O † ja O bk O † ic O cj O † kb + a Tr O ai O † ja O bj O † kc O ck O † ib + a Tr O ai O † ic O bj O † ja O ck O † kb + a Tr O ai O † ic O bj O † ka O ck O † jb + a Tr O ai O † jc O bk O † ia O cj O † kb + a Tr O ai O † jc O bj O † ka O ck O † ib + a Tr O ai O † ja O bj O † ic O ck O † kb + a Tr O ai O † ja O bk O † kc O cj O † ib , (3.4) In principle we can write 36 structures which would correspond to the singlets of SU (2) R × SU (2). Fromthe group theory computation we get that there are only 25 singlets. It means that there 11 linear relationsamong the structures. 36 structures are equivalent to 18 different structures modulo cyclic permutation andwe find that invariance under cyclic permutation reduces the 11 relations to only 7. a n are 18 arbitrary real parameters, which we need to fix by use of the explicitexpressions for the bosonic potential in components V bosa n = V bos . If we apply † -operation onthe ansatz (3.4) we find that the first 16 terms are mapped into themselves, while the lasttwo are mapped into each other. It means that the reality of the potential forces a = a .On top of that it appears that some of the 18 structures are linear dependent. If we call O n the operators corresponding to the coefficients a n . We can find the seven linear relations3 O − O − O − O = 0 , O − O − O − O = 0 , O − O − O − O = 0 , O − O − O − O = 0 , O − O − O − O = 0 , O − O − O − O = 0 ,O − O − O − O + O + O = 0 . (3.5)In particular, if we try to solve the equation V bosa n = V bos as a function of the a n we finda family of solutions parameterized by seven parameters a n due to the relations (3.5). Tofind the coefficients for the potential we need first to reduce the ansatz by use of (3.5) to11 linearly independent structures and then solve V bosa n = V bos for the coefficients. This wayto proceed means that there are no unique form of the potential if we use the notion of the SU (2) R × SU (2)-fields. The concrete form of the mixing operator descends from the choiceof these 11 structures but the eigenvalues of the mixing operator are independent of thischoice. See additional comments in Appendix C. We found a choice of a n where 11 of the18 coefficients are zero. The remaining non-zero coefficients are a = − π k , a = − π k , a = − π k k a = 16 π k k ,a = 16 π ( k + k )3 k k , a = 16 π ( k + k )3 k k . (3.6)In the following we will use these coefficients. The bosonic potential written in the explicit SU (2) R × SU (2) invariant form is V bos = − π k Tr O ai O † ia O bj O † jb O ck O † kc − π k Tr O ai O † jb O bj O † kc O ck O † ia − π k k Tr O ai O † ia O bj O † kc O ck O † jb + 16 π k k Tr O ai O † jc O bk O † ia O cj O † kb + 16 π ( k + k )3 k k Tr O ai O † ic O bj O † ja O ck O † kb + 16 π ( k + k )3 k k Tr O ai O † jc O bj O † ka O ck O † ib . (3.7)With this choice of the coefficients the ABJM limit is apparent. Namely for k + k = 0the last two terms drop out and we obtain the ABJM potential written in SU (2) R × SU (2)invariant way. Indeed in this limit the R-symmetry and flavor indices of the O fields do notmix anymore due to the R symmetry enhancement to SU (4) . The remaining coefficientsare exactly the ones in [32] Le us now proceed with the fermionic potential V ferm . Our final goal is to compute thetwo loops mixing matrix in the planar limit. Part of the contribution to the renormalization6f the scalar operators O in eq. (2.3) comes from fermions running in the loops. Thisinteraction is due to the fermionic potential. The fermionic potential is a quartic functionin the fields, each term contains two bosons and two fermions. The contributions are of twotypes, the first one V fermffbb contains terms consisting of two fermions followed by two bosons,the second one V fermbfbf has the coupling fermions-boson-fermion-boson. It is easy to see thatthe terms of the second type do not contribute to the mixing matrix at the planar level forthe scalar operators. That’s why it is enough to consider only the terms of the first type.The fermionic potential has two contributions, one is coming from the superpotential V fermW and the other one coming from the Chern-Simons interactions V fermCS . After integratingout the auxiliary fields we get V ferm W = 4 πk (cid:16) ω i ζ i W j Z j + ζ i ω j Z j W i − ζ † i ω † i Z † j W † j − ω † i ζ † j W † j Z † i (cid:17) + 4 πk (cid:16) ω i ζ j W j Z i + ζ i ω i Z j W j − ζ † i ω † j Z † j W † i − ω † i ζ † i W † j Z † j (cid:17) + . . .V ferm CS = 2 πik (cid:16) ζ i ζ † i − ω † i ω i (cid:17) (cid:16) Z j Z † j − W † j W j (cid:17) + 2 πik (cid:16) ζ † i ζ i − ω i ω † i (cid:17) (cid:16) Z † j Z j − W j W † j (cid:17) + 4 πik (cid:16) ζ † i ζ j Z † j Z i + ω i ω † j W j W † i (cid:17) + 4 πik (cid:16) ζ i ζ † j Z j Z † i + ω † i ω j W † j W i (cid:17) + . . . (3.8)The ellipsis corresponds to couplings in V fermbfbf which are not relevant for our computation.We would like to rewrite the fermionic potential in the SU (2) R × SU (2) invariant way. In theABJM case the superpartners of the scalar field transform in the conjugated representationof the one of the scalars. This is the manifestations of the fact that the SU (4) correspondsto the R-symmetry group of the fields. It means that in the case of the fermionic objectstransforming under SU (2) R × SU (2) the R-symmetry index should transform in the conju-gated representation of the scalar superpartner. However, since we expect that the scalarsand spinors belong to the same flavor multiplet they should transform under the same rep-resentation of the SU (2) flavor symmetry group. This suggests the following ansatz ψ † i = − iζ i , ψ † i = ω † i ,ψ j = iζ † j , ψ j = ω j . (3.9)The index i , j transform under SU (2) flavor symmetry and the written out indices 1 , SU (2) R -symmetry. The SU (2) R × SU (2) invariant ansatz is then V ferm f n = f Tr O † ia O ai ψ † bj ψ bj + f Tr O † ia O aj ψ † bj ψ bi + f Tr O † ia O bi ψ † aj ψ bj + f Tr O † ia O bj ψ † aj ψ bi + f Tr O ai O † ia ψ bj ψ † bj + f Tr O ai O † ja ψ bj ψ † bi + f Tr O ai O † ib ψ aj ψ † bj + f Tr O ai O † jb ψ aj ψ † bi + . . . (3.10)The equation V ferm f n = V ferW + V ferCS gives the solution f = − πik , f = 0 , f = 4 πik , f = 4 πik ,f = − πik , f = 0 , f = 4 πik , f = 4 πik . (3.11)7nd the SU (2) R × SU (2) invariant fermionic potential is: V ferm = − πik Tr O † ia O ai ψ † bj ψ bj + 4 πik Tr O † ia O bi ψ † aj ψ bj + 4 πik Tr O † ia O bj ψ † aj ψ bi − πik Tr O ai O † ia ψ bj ψ † bj + 4 πik Tr O ai O † ib ψ aj ψ † bj + 4 πik Tr O ai O † jb ψ aj ψ † bi + . . . (3.12)The fermionic potential reduces to the ABJM one in the limit k + k = 0. Namely, by use ofthe relation δ il δ jk − δ ik δ jl = ǫ ij ǫ kl and appropriate redefinition of the fields O ai = Y A , ǫ ij ψ † aj = ψ † A where A is an SU (4) index. The last two terms in each line are combined into the termswhich mix the SU (4) flavor and the first terms in each line give then flavor non-mixingcontributions. Right now we have all the tools to compute the dilatation operator Γ. The contributions tothe dilatation operator come from the logarithmic divergences (ln Λ) of the renormalizationfunction Z (Λ). The lowest contributions come at two loops and the non vanishing logarithmicdivergences come from the graphs in figure 2. The renormalization of the composite operators (a) (b) (c) Figure 2:
The graphs that contribute to the mixing operator (a) only scalar bosons are runninginside the loops, (b) scalar bosons and fermions are running in the loops, (c) scalar boson andgauge bosons in the loops. O in equation (2.3) comes from three different kind of graphs where (a) only scalar fields, (b)scalar and fermionic fields and (c) scalar and gauge fields are running in the loops. We cananalyze them separately. Before doing it let us fix some notation. We are going to computethe Hamiltonian of an SU (2) R × SU (2) spin chain in representation ( , ), with alternatingsites corresponding to the fields O , O † in the operators O . At every site of the spin chainwe have two indices of SU (2) and the final Hamiltonian can be nicely expressed in terms oftwo basic operators acting on the group indices: the trace operator K : V ⊗ ¯ V → V ⊗ ¯ V or¯ K : ¯ V ⊗ V → ¯ V ⊗ V ; and the permutation operator P : V ⊗ V → V ⊗ V or P : ¯ V ⊗ ¯ V → ¯ V ⊗ ¯ V .We can distinguish between the operators acting on the R indices ( K , P ) and the operatorsacting on the flavor indices ( ˆ K, ˆ P ): K a ′ bb ′ a = δ a ′ b ′ δ ba , K i ′ jj ′ i = δ i ′ j ′ δ ji ,P a ′ b ′ ba = δ a ′ b δ b ′ a , P i ′ j ′ ji = δ i ′ j δ j ′ i . K acts on the nearest neighbor sites, while the permutation operator P acts on next to nearest neighbor sites. The ’t Hooft couplings λ i = N/k i are our perturbativeexpansion parameters. The final expression for the mixing operator Γ is a polynomial in K and P with coefficients that are functions of λ , λ . In this subsection we give the part of the Hamiltonian which comes from the diagram withonly scalar fields in the loops. The graph (a) in figure 2 gets the contribution from thevarious monomials in the sextic bosonic potential (3.7). The computation is done in twosteps. Firstly, one computes the logarithmic divergent part, and then carefully computes the SU (2) R × SU (2) combinatoric structure. To write down the final result in a most transparentway we distinguish between trace operators ¯ K l,l +1 and K l,l +1 . The first one acts as usual onthe sites ¯ V ⊗ V and gives zero on V ⊗ ¯ V , while the second one acts as usual on V ⊗ ¯ V andgives zero on ¯ V ⊗ V . The part of the mixing operator coming from this graph isΓ bos = 12 L X l =1 (cid:16) − λ ¯ K l,l +1 ˆ¯ K l,l +1 − λ K l,l +1 ˆ K l,l +1 + 2 λ λ P l,l +2 ˆ P l,l +2 − λ λ (cid:0) K l,l +1 P l,l +2 ˆ K l,l +1 ˆ P l,l +2 + ¯ K l,l +1 P l,l +2 ˆ¯ K l,l +1 ˆ P l,l +2 + P l,l +2 K l,l +1 ˆ P l,l +2 ˆ K l,l +1 + P l,l +2 ¯ K l,l +1 ˆ P l,l +2 ˆ¯ K l,l +1 (cid:1) +4( λ λ + λ ) P l,l +2 ˆ¯ K l,l +1 + 4( λ λ + λ ) P l,l +2 ˆ K l,l +1 (cid:17) . (4.1) The fermionic potential (3.12) gives two types of contributions to the graph (b) in figure2: a contribution proportional to the identity in the SU (2) R × SU (2) indices, namely avacuum energy contribution coming from the first two monomials in the two lines of (3.12)and an interacting contribution containing the K , ˆ K trace operators. The constant part ofthe full mixing matrix gets contribution also from other graphs than the ones in figure 2,for example, from the renormalization to the propagator < O † O > . We are not going tocompute these diagrams. Later, we fix this constant part using supersymmetry. For thisreason we concentrate here only on the contributions coming from the last two monomialsin each lines in (3.12). After computing the logarithmic divergent part of the graph (b) infigure 2 and computing the combinatorial SU (2) R × SU (2) structure, we obtain the fermioniccontribution to the mixing operatorΓ ferm = L X l =1 (cid:16) λ + λ λ ) ¯ K l,l +1 ˆ1 + λ ¯ K l,l +1 ˆ¯ K l,l +1 + 2( λ + λ λ ) K l,l +1 ˆ1 + λ K l,l +1 ˆ K l,l +1 (cid:17) . (4.2) The last contribution to the mixing operator comes from the graph (c) in figure 2. Thegauge bosons do not carry SU (2) R × SU (2) indices and we just need to compute the two9oop diagram with the correct coupling constants coming from the scalar-gauge interactionsin the Lagrangian. The final result isΓ gauge = − L X l =1 (cid:16) λ ¯ K l,l +1 ˆ¯ K l,l +1 + λ K l,l +1 ˆ K l,l +1 (cid:17) . (4.3) The complete two loop mixing operator is obtained summing up Γ bos , Γ ferm and Γ gauge . Beforewriting down the final expression we need to fix the constant contribution. Supersymmetryimplies that the anomalous dimension of the symmetric traceless operators is equal to zero.This fact fixes the constant contribution. The complete Hamiltonian can be written asΓ full = 12 L X l =1 (cid:16) ( λ − λ ) ¯ K l,l +1 ˆ¯ K l,l +1 + ( λ − λ ) K l,l +1 ˆ K l,l +1 +4( λ λ + λ )( P l,l +2 ˆ¯ K l,l +1 + K l,l +1 ˆ1) + 4( λ λ + λ )( P l,l +2 ˆ K l,l +1 + ¯ K l,l +1 ˆ1) − λ λ (cid:0) − P l,l +2 ˆ P l,l +2 + K l,l +1 P l,l +2 ˆ K l,l +1 ˆ P l,l +2 + ¯ K l,l +1 P l,l +2 ˆ¯ K l,l +1 ˆ P l,l +2 + P l,l +2 K l,l +1 ˆ P l,l +2 ˆ K l,l +1 + P l,l +2 ¯ K l,l +1 ˆ P l,l +2 ˆ¯ K l,l +1 (cid:1)(cid:17) . (4.4)The last two lines are the only contributions to the mixing operator in the ABJM case.Indeed in the limit k + k = 0 the Hamiltonian reduces toΓ ABJM full = λ L X l =1 (cid:16) − P l,l +2 ˆ P l,l +2 + K l,l +1 P l,l +2 ˆ K l,l +1 ˆ P l,l +2 + P l,l +2 K l,l +1 ˆ P l,l +2 ˆ K l,l +1 (cid:17) (4.5)that is exactly the mixing operator in [47] written in SU (2) R × SU (2) invariant form, wherewe didn’t distinguish between K , P and ¯ K , ¯ P .It is nice to observe that one can define a parity operator P acting on the spin chain. Itsaction reverses the orientation of the chain from clockwise to anticlockwise or vice versa. Inparticular it acts on the operators as P Tr (cid:16) O † i a O a i ...O † i L − a L − O a L i L (cid:17) = Tr (cid:16) O a L i L O † i L − a L − ...O a i O † i a (cid:17) . The parity operation on the Hamiltonian (4.4) exchanges λ and λ . The parity transformed We would like to stress here that since there are relations between the trace and permutation operatorsacting on two-dimensional indices the above form of the Hamiltonian is not unique. The action of theHamiltonian is of course independent of the concrete representation in terms of K s and P s. If we act with the parity operator on the Hamiltonian the transformed one should act on the paritytransformed states as the original Hamiltonian on the non transformed states. The new vertices of a suchtransformed Hamiltonian are obtained from the full potential by acting on all the terms with the parityoperator. This corresponds exactly to the exchange of λ and λ in eq. (4.4) or alternatively to the exchangeof K, ˆ K and ¯ K, ˆ¯ K . P Γ full P = 12 L X l =1 (cid:16) ( λ − λ ) ¯ K l,l +1 ˆ¯ K l,l +1 + ( λ − λ ) K l,l +1 ˆ K l,l +1 +4( λ λ + λ )( P l,l +2 ˆ¯ K l,l +1 + K l,l +1 ˆ1) + 4( λ λ + λ )( P l,l +2 ˆ K l,l +1 + ¯ K l,l +1 ˆ1) − λ λ (cid:0) − P l,l +2 ˆ P l,l +2 + K l,l +1 P l,l +2 ˆ K l,l +1 ˆ P l,l +2 + ¯ K l,l +1 P l,l +2 ˆ¯ K l,l +1 ˆ P l,l +2 + P l,l +2 K l,l +1 ˆ P l,l +2 ˆ K l,l +1 + P l,l +2 ¯ K l,l +1 ˆ P l,l +2 ˆ¯ K l,l +1 (cid:1)(cid:17) . (4.6)For λ = ± λ the parity symmetry of the Hamiltonian is broken by the terms in the firstand second line. The only values of λ and λ which correspond to the parity invariantHamiltonian are λ = ± λ . A typical sign of integrability of a system is the presence of different operators with the sameanomalous dimensions. [7, 37] In the ABJM case this happens for example for operators oflength four [47]. In that case the system is an SU (4) spin chain alternating between fun-damental representation and antifundamental ¯4 representation. The is associated withthe vector: Y A = ( Z , Z , W † , W † ) and the length four operators are Tr (cid:16) Y A Y † B Y A Y † B (cid:17) .If we decompose these operators in representations of SU (4) we find that they contain twosinglets , two adjoints , one and one representations. It happens that the twoadjoint operators have the same anomalous dimension 6 λ . The natural question is, whathappens to these operators in the case in which k = − k ? Are they still degenerate? Toanswer these questions we consider the following operatorsTr O † i a O a i O † i a O a i . (5.1)They decompose in representation of SU (2) R × SU (2). In particular the of SU (4) de-composes under SU (2) R × SU (2) as → ( , ) + ( , ) + ( , ) . For this reason in this section we will be interested to apply the Hamiltonian (4.4) to oper-ators in (5.1) in representations ( , ), ( , ) and ( , ). Operators with the same quantumnumbers typically mix among each other under renormalization. We need to consider all theoperators of the same length that transform in the same representation. The operators inthe representation ( , ) and ( , ) come only from the decomposition of the of SU (4),but there exist other three operators in the ( , ) representation coming respectively: onefrom the and two from the . As result we have two operators in the ( , ), two in the( , ), and five in the ( , ). In the following subsections we are going to analyze separatelytheir anomalous dimensions and to check if the degeneracy which is present in the integrableABJM case is still there or is lifted. 11 .1 Operators in (3,1) Let us start with the operators in representation ( , ). From the decomposition in the list(D.2) in the Appendix we know that there are six structures transforming in the represen-tation ( ), four come from and two from and of SU (4). Only the structuresdescending from the of SU (4) can form operators invariant under trace. Indeed cyclicityrelates four states and we get just two operators:Tr | − i ( , ) = Tr O † ia O ai O † mb O cm − trace , Tr | − i ( , ) = Tr O † mb O ai O † ia O cm − trace . (5.2)The first label enumerates the operators and the second one gives the corresponding SU (4)multiplet.Applying the mixing operator we obtain Γ Tr | − i ( , ) = 2( λ − λ λ + λ )Tr | − i ( , ) + (5 λ − λ )( λ + λ )Tr | − i ( , ) +6 λ ( λ + λ ) (cid:16) Tr O † ia O ci O † jb O aj + Tr O † ib O ai O † ja O cj (cid:17) = 2( λ + 5 λ λ + 7 λ )Tr | − i ( , ) + (5 λ − λ )( λ + λ )Tr | − i ( , ) Γ Tr | − i ( , ) = 2( λ − λ λ + λ )Tr | − i ( , ) + (5 λ − λ )( λ + λ )Tr | − i ( , ) +6 λ ( λ + λ ) (cid:16) Tr O † ia O aj O † jb O ci + Tr O † ib O cj O † ja O ai (cid:17) = 2( λ + 5 λ λ + 7 λ )Tr | − i ( , ) + (5 λ − λ )( λ + λ )Tr | − i ( , ) The application of the mixing operator on the states Tr | − i ( , ) and Tr | − i ( , ) produces structures which we cannot immediately match with the basis states. This comesfrom the fact that there are more structures than the linearly independent ones. There are 6ways to organize the R-symmetry indices in such a way that they transform in representation of SU (2) R and two ways to organize the flavor indices that transform in of SU (2). Usingthe relations from Appendix B these 12 structures can be related to the 6 basis structureswhich come from the decomposition of
15 , 45 and of SU (4). The eigenvalues are8 λ + 10 λ λ + 8 λ ± ( λ + λ ) q λ − λ λ + 31 λ . (5.3)For physical real values of λ , λ the eigenvalues are degenerate only for λ = − λ = λ . Inthis case our result reduces to the ABJM one [47] and the two operators in (5.2) have thesame anomalous dimension, 6 λ . In all the other cases the degeneracy is lifted. The operators in representation ( , ), similarly to the previous case, appear also in thedecomposition of the of SU (4). As in the ( , ), we get only two operatorsTr | − i ( , ) = Tr O † ia O ai O † jb O bk − trace , Tr | − i ( , ) = Tr O † jb O ai O † ia O bk − trace . (5.4)12gain using the relations from the Appendix B we obtainΓ Tr | − i ( , ) = 2(3 λ − λ λ + λ )Tr | − i ( , ) +( λ + λ )(5 λ + 7 λ )Tr | − i ( , ) , Γ Tr | − i ( , ) = 2(3 λ − λ λ + λ )Tr | − i ( , ) +( λ + λ )(5 λ + 7 λ )Tr | − i ( , ) . (5.5)The eigenvalues are:2(2 λ + λ λ + 2 λ ) ± ( λ + λ ) q λ + 22 λ λ + 13 λ ) . (5.6)As in the previous case the mixing and the anomalous dimensions reduce to the ABJM ones[47] in the limit λ = − λ , otherwise the degeneracy is lifted. The ( , ) case is a bit more involved. As we can see in the list (D.2) there are nine structurestransforming in ( , ) which come from the decomposition of the length four structuresof SU (4). Two of them coming from and , due to the antisymmetrization, do notcorrespond to any operators. From the remaining seven structures the four coming from of SU (4) correspond to two trace invariant operators. Altogether we have the followingbasis for the operators in ( , ). Tr | − i ( , ) = Tr O † ia O ai O † jb O ck − trace , Tr | − i ( , ) = Tr O † jb O ai O † ia O ck − trace , Tr | − i ( , ) = Tr (cid:16) O † [ i ( b O [ a ( k O † l ] e ) O d ] m ) − traces (cid:17) ǫ ad ǫ ce ǫ il ǫ jm = 4 Tr O † [ i ( b O [ a ( k O † j ] a ) O c ] i ) − Tr | − i ( , ) + Tr | − i ( , ) − traces , Tr | − i ( , ) = Tr (cid:16) O † ( j ( b O [ a [ i O † m ) e ) O d ] l ] − traces (cid:17) ǫ ad ǫ ce ǫ il ǫ km = 4 Tr O † ( j ( b O [ a [ i O † i ) a ) O c ] k ] −
13 Tr | − i ( , ) −
13 Tr | − i ( , ) − traces , Tr | − i ( , ) = Tr (cid:16) O † [ i [ a O ( c ( k O † l ] d ] O e ) m ) − traces (cid:17) ǫ ad ǫ be ǫ il ǫ jm = 4 Tr O † [ j [ b O ( a ( i O † i ] a ] O c ) k ) −
13 Tr | − i ( , ) −
13 Tr | − i ( , ) − traces . (5.7)The first number enumerates the operators and the second one gives the representation of SU (4) to which it corresponds. The states Tr | − i ( , ) and Tr | − i ( , ) in the definition In principle we can write two operators which would correspond to the decomposition of , the one withupper indices symmetrized and lower antisymmetrized and vice versa. By use of the relations in AppendixA one can show that one of these two structures can be written as a linear combination of the remainingone, Tr | − i ( , ) and Tr | − i ( , ) .
13f the last three operators come from the decomposition of the traces of the SU (4) operators, and .To obtain the mixing matrix of anomalous dimensions we apply the Hamiltonian (4.4)to the above basis states. In general the result will contain structures which do not matchwith the five basis operators in the list(5.7). We used the relations listed in Appendix A.The mixing matrix is (cid:0) λ + 3 λ λ + 5 λ (cid:1) ( λ + λ ) (7 λ + 5 λ ) 0 − λ ( λ + λ ) − λ ( λ + λ ) ( λ + λ ) (5 λ + 7 λ ) (cid:0) λ + 3 λ λ + 7 λ (cid:1) − λ ( λ + λ ) − λ ( λ + λ )0 0 2( λ − λ ) λ − λ ) − λ − λ ) − ( λ + λ )(2 λ + λ ) − ( λ + λ )( λ + 2 λ ) − λ + λ λ + λ ) ( λ + λ ) − ( λ + λ )(2 λ + λ ) − ( λ + λ )( λ + 2 λ ) − λ + λ ( λ + λ ) λ + λ ) In the ABJM-limit, λ = − λ = λ , the eigenstates and their corresponding eigenvaluesare [47] Tr | − i ( , ) : 6 λ , Tr | − i ( , ) : 6 λ , Tr | − i ( , ) : 8 λ , Tr | − i ( , ) : 0 , Tr | − i ( , ) : 0 . (5.8)There are other particular values of λ , λ . For λ = λ the theory is still parity invariant,but we don’t find any degeneracy pairs among the eigenstates which would map one intoeach other under the parity transformation. For the values of λ , λ outside the regime λ = − λ we can find degeneracy among the eigenvalues of the mixing matrix, but sincethe theory is not parity invariant the operators with the same anomalous dimensions do notform parity pairs. These results suggest that the ABJM integrability is broken for genericvalues of λ and λ . Let us try to get some conclusions related to the integrability of the system. As we claimedat the beginning of this section a generic feature of integrability is the presence of degeneracypairs [7, 37]. Namely, the existence of couples of operators which have the same anomalousdimension and which are mapped one into each other by the parity operator P . In theABJM spin chain the first example of degeneracy pairs is in the set of length four operators:they are the operators in the adjoint representation of SU (4). In this section we checkedthat all the SU (2) R × SU (2) operators which are contained in the decomposition of theABJM degeneracy pairs are no longer degeneracy pairs for generic k , k . This fact could beinterpreted as a weak evidence of the absence of integrability of the system for k = − k .Let us explain why this is just a weak evidence. First of all the parity symmetry is brokenby the Hamiltonian (4.4) for generic values of k , k . A nice observation is that parity isrestored for k = ± k . One of these two points is the ABJM limit where degeneracy pairsappear and the system is integrable. The other point is still parity invariant but there isno degeneracy in the anomalous dimensions. Even this observation is not conclusive: the14riginal eigenvectors of the ABJM mixing matrix are no longer eigenvectors of the newHamiltonian. The new eigenvectors do not form pairs under parity, they are actually parityeigenvectors and we cannot claim that integrability is broken because they do not have thesame anomalous dimension. To say something stronger about the integrability of the theoryone should compute for example the mixing of longer operators, or directly compute theintegrable Hamiltonian associated to the SU (2) R × SU (2) spin chain, but also in this casethe claim could be not definitive.Even with all these subtleties in mind we would like to take the lifting of the degeneracy,which is present in the ABJM limit, as a hint against the integrability of the system. Ofcourse, a more rigorous analysis is required. In this note we started the analysis of the deformed integrable Chern-Simons theories. As afirst example we considered the ABJM theory with arbitrary Chern-Simons levels k , k . Weconstructed the complete two loop mixing operator for the bosonic scalar sector of the theoryand we computed the anomalous dimension for some length four operators. We observed thatthe degeneracy of anomalous dimensions which is present in the integrable limit (the ABJMtheory) disappears for generic k and k . We interpreted this fact as a weak evidence of theabsence of integrability for these theories, namely, when k + k = 0 the ABJM integrabilityseems to be destroyed. A possible future direction could be to start a deeper investigationof the integrability of these theories, in field theory and maybe in the IIA string dual, tosupport or contradict our conclusions.Another nice application of the ideas presented in this note could be a more generalanalysis of the integrability of Chern-Simons quiver gauge theories. For example it would benice to see what happens to the integrable properties of Chern-Simons theories that comeby orbifolding ABJM, once we allow non orbifold values for the various k i . We hope to comeback to this problem in the near future.We hope to have convinced the reader that three dimensional Chern-Simons theories area nice laboratory to study integrability, and in a sense, due to the quartic interactions andthe presence of Chern-Simons levels, they allow a perturbative weak coupling analysis ofmore general deformations than the four dimensional examples. Acknowledgments
We are happy to thank first of all Konstantin Zarembo for many nice discussions, andClaudio Destri, Giuseppe Policastro, Alessandro Tomasiello, Jan Troost, Alberto Zaffaroni,Andrey Zayakin for valuable conversations. W.S. would like to thank Ludwig-Maximilians-University and University of Hamburg for kind hospitality where part of this work was done.D. F. is supported by CNRS and ENS Paris. The research of W. S. was supported in partby the ANR (CNRS-USAR) contract 05-BLAN-0079-01.15 ppendices
A Relations among the operator structures of (3,3)
If we hold one type of the coefficients fixed we can can write down six structures correspond-ing to representation of the other type of coefficients. | i = O † a O a O † b O c − trace | i = O † b O a O † a O c − trace | i = O † b O c O † a O a − trace | i = O † a O c O † b O a − trace | i = O † ( b O a O † e ) O d ǫ ad ǫ ce | i = O † a O ( c O † d O e ) ǫ ad ǫ be (A.1)From the group theory computation we know that there should be only three independentstructures transforming in the representation . ⊗ ⊗ ⊗ = ⊕ ⊕ (A.2)If we consider the following relations O † b O c O † a O a = ǫ be ǫ ad O † e O c O † a O d = (cid:0) δ db δ ae − δ ab δ de (cid:1) O † e O c O † a O d = O † a O c O † a O b − O † d O c O † b O d = ǫ ad ǫ be O † d O c O † a O e + O † a O c O † b O a O † b O c O † a O a = ǫ ae ǫ cd O † b O d O † e O a = (cid:0) δ ce δ da − δ ca δ de (cid:1) O † b O d O † e O a = O † b O a O † c O a − O † b O a O † a O c = ǫ ad ǫ ce O † b O d O † e O a + O † b O a O † a O c O † a O b O † c O a = ǫ ad ǫ be O † d O e O † c O a = (cid:0) δ ea δ bd − δ ed δ ba (cid:1) O † d O e O † c O a = O † b O a O † c O a − O † a O a O † c O b = ǫ bd ǫ ae O † d O e O † c O a + O † a O a O † c O b O † b O a O † a O c = ǫ ad ǫ be O † e O d O † a O c = (cid:0) δ ae δ db − δ ab δ de (cid:1) O † e O d O † a O c = O † a O b O † a O c − O † a O a O † b O c = ǫ ad ǫ be O † d O e O † a O c + O † a O a O † b O c (A.3)16e can find the following set of the relations among the six structures listed in (A.1) | i + | i − | i − | i − | i = 0 | i − | i − | i + | i − | i = 0 | i − | i + | i − | i = 0 (A.4)Let us write down the structures coming from , and of the SU (4) operators. | − i ( , ) = O † ia O ai O † jb O ck − traces | − i ( , ) = O † jb O ai O † ia O ck − traces | − i ( , ) = O † jb O ck O † ia O ai − traces | − i ( , ) = O † ia O ck O † jb O ai − traces | − i ( , ) = (cid:16) O † [ i ( b O [ a ( k O † l ] e ) O d ] m ) − traces (cid:17) ǫ ad ǫ ce ǫ il ǫ jm | − i ( , ) = (cid:16) O † ( j ( b O [ a [ i O † m ) e ) O d ] l ] − traces (cid:17) ǫ ad ǫ ce ǫ il ǫ km | − i ( , ) = (cid:16) O † [ i [ a O ( c ( k O † l ] d ] O e ) m ) − traces (cid:17) ǫ ad ǫ be ǫ il ǫ jm (A.5)The first number is just an enumerating label, the second one gives the multiplet of SU (4)to which it corresponds. In the case of the last three operators we let the redundant anti-symmetrizing brackets to make it more transparent.The flavor and R-symmetry indices can be labeled by use of the structures in represen-tation . Let us adapt the following notation: | − i ( , ) = (cid:16) | i , | i (cid:17) (A.6)where the first | i means that the R-symmetry indices correspond to the first structure inrepresentation of the list (A.1) and the second | i to the first structure of the correspondinglist for the flavor indices. | − i ( , ) = (cid:16) | i , | i (cid:17) , | − i ( , ) = (cid:16) | i , | i (cid:17) , | − i ( , ) = (cid:16) | i , | i (cid:17) , | − i ( , ) = (cid:16) | i , | i (cid:17) , | − i ( , ) = (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) + 12 (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) + 12 (cid:16) | i , | i (cid:17) , | − i ( , ) = (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) , | − i ( , ) = (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) . (A.7)We see that it is possible to write down 36 different structures, there are six different waysto put R-symmetry indices and 6 ways for the flavor indices. Since there are only threeindependent structures for one type of indices there are only 9 linear independent structuresif we consider both types of the indices at the same time. In (A.7) we wrote down only 717inear independent structures, 2 remaining ones correspond to and of SU (4) and don’tcorrespond to any trace operators, that’s why we are not considering them.In general if we act with the Hamiltonian on these structures we will have structureswhich not immediately match with the structures in (A.7). Let us go give here the list ofthe relations which we used to obtain the mixing matrix in the main text.We can immediately find the relationsTr (cid:16) | i + | i , | / i (cid:17) = 0 , Tr (cid:16) | i + | i , | / i (cid:17) = 0 , (A.8)where we usedTr (cid:16) | i , | / i (cid:17) = − Tr (cid:16) | i , | / i (cid:17) , Tr (cid:16) | i , | / i (cid:17) = − Tr (cid:16) | i , | / i (cid:17) . (A.9)Other relations which we used are Tr (cid:16) | i , | i (cid:17) = − Tr (cid:16) | i , | i (cid:17) + 2 Tr (cid:16) | i , | i (cid:17) − (cid:16) | i , | i (cid:17) = Tr | − i ( , ) − Tr | − i ( , ) − Tr | − i ( , ) , Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) = −
13 Tr | − i ( , ) + 23 Tr | − i ( , ) −
12 Tr | − i ( , ) −
12 Tr | − i ( , ) , Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) = 23 Tr | − i ( , ) −
13 Tr | − i ( , ) −
12 Tr | − i ( , ) −
12 Tr | − i ( , ) , Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) = 13 Tr | − i ( , ) + 13 Tr | − i ( , ) −
12 Tr | − i ( , ) −
12 Tr | − i ( , ) , Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) + 12 Tr (cid:16) | i , | i (cid:17) = 13 Tr | − i ( , ) + 13 Tr | − i ( , ) + 12 Tr | − i ( , ) −
12 Tr | − i ( , ) , Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) + 12 Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) = 13 Tr | − i ( , ) + 13 Tr | − i ( , ) + 12 Tr | − i ( , ) −
12 Tr | − i ( , ) , Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) = Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) −
12 Tr (cid:16) | i , | i (cid:17) = 13 Tr | − i ( , ) + 13 Tr | − i ( , ) −
12 Tr | − i ( , ) −
12 Tr | − i ( , ) . (A.10) Relations among the operator structures of (3,1)and (1,3)
We can write down the singlet structures for the SU (2) indices of the length four structuresin two ways and the decomposition of ⊗ ⊗ ⊗ tells us that there only two singlets.It means that there are no linear relation among the singlet structures and we can considerthem as the basis structures.The singlets are | i = O † a O a O † b O b , | i = O † a O b O † b O a . (B.1)Since there are 6 different ways to put the indices corresponding to the representation , there are 12 structures which correspond to the structures of the type ( , ) and 12 for( , ). Since the relations for ( , ) or ( , ) are similar we concentrate here only on the( , ) structures.The four of the total six ( )-structures come from of ABJM. Let us see how theylook like. Consider | i + | i and replace Y A by O † ia Y C Y † C Y B Y † A = O † ia O ai O † jb O ck = ǫ bd ǫ jl O † ia O ai O † dl O ck = ǫ bd ǫ jl O † ia O ai (cid:16) O † ( d ( l O c ) k ) + O † ( d [ l O c ) k ] + O † [ d ( l O c ] k ) + O † [ d [ l O c ] k ] (cid:17) (B.2)The second term corresponds to the structure which transforms in representation (3,1) . Letus consider it. 14 ǫ bd ǫ jl O † ia O ai (cid:16) O † dl O ck + O † cl O dk − O † dk O cl − O † ck O dl (cid:1) = 14 O † ia O ai (cid:0) O † jb O ck + ǫ bd ǫ ce O † je O dk − ǫ jl ǫ km O † mb O cl − ǫ bd ǫ jl ǫ ce ǫ km O † me O dl (cid:17) = 14 O † ia O ai (cid:0) O † jb O ck + ( δ eb δ cd − δ cb δ ed ) O † je O dk − (cid:0) δ jm δ lk − δ jk δ lm (cid:1) O † mb O cl − ( δ eb δ cd − δ cb δ ed ) (cid:0) δ jm δ lk − δ jk δ lm (cid:1) O † me O dl = 12 δ jk O † ia O ai O † mb O cm − δ jk δ cb O † ia O ai O † md O dm (B.3)Therefore, the four ( )-structures descending form the four -structures are | i ( , ) = O † ia O ai O † mb O cm − trace , | i ( , ) = O † mb O ai O † ia O cm − trace , | i ( , ) = O † mb O cm O † ia O ai − trace , | i ( , ) = O † ia O bm O † mc O ai − trace . (B.4)The remaining two structures come from the ABJM multiplet transforming under 45 and 45of ABJM and do not correspond to any trace invariant operators.To find the necesarry relations among the 12 different structures of ( ) we use the sametrick as in the previous section of the appendix. We write | i ( , ) = (cid:16) | i , | i (cid:17) , | i ( , ) = (cid:16) | i , | i (cid:17) , | i ( , ) = (cid:16) | i , | i (cid:17) , | i ( , ) = (cid:16) | i , | i (cid:17) . (B.5)19f we apply the Hamiltonian to these structures we find structures which do not matchwith the above structures. The structures which we need to identify are O † ia O ci O † jb O aj + O † ib O ai O † ja O cj = (cid:16) | i , | i (cid:17) ,O † ia O aj O † jb O ci + O † ib O cj O † ja O ai = (cid:16) | i , | i (cid:17) . (B.6)By use of (A.4) we find O † ia O ci O † jb O aj + O † ib O ai O † ja O cj = (cid:16) | i , | i (cid:17) = (cid:16) | i , | i (cid:17) = | i ( , ) + | i ( , ) ,O † ia O aj O † jb O ci + O † ib O cj O † ja O ai = (cid:16) | i , | i (cid:17) = (cid:16) | i , | i (cid:17) = | i ( , ) + | i ( , ) . (B.7) C The general form of the mixing operator from thesix-vertex diagram
The terms in the equation (3.4) are not linearly independent and in the main text of thisarticle we have chosen a specific choice of the a n which allowed to eliminate the lineardependencies and write the potential only with six terms. It is not necessary to make aconcrete choice. Actually, to obtain the mixing operator of the main text we first computedthe mixing operator starting from the full ansatz of the potential (3.4) and only then insertedthe coefficients a n from the solution of V bosa n = V bos . We put this formulae into the appendixsince it allows the reader to write down the mixing operator in a different form then in eq.(4.4). Which terms in the ansatz V bosa n are allowed to be set to zero can be decided lookingat the linear relations (3.5).The mixing operator from the six-vertex diagram derived from the ansatz in eq. (3.4) isΓ V ⊗ ¯ V ⊗ V = N π (cid:16) K ¯ V ⊗ V (cid:16) a ˆ K ¯ V ⊗ V + 3 a ˆ K V ⊗ ¯ V + a (cid:16) ˆ1 + ˆ K ˆ P + ˆ P ˆ K (cid:17) + 3 a ˆ P (cid:17) + K V ⊗ ¯ V (cid:16) a ˆ K ¯ V ⊗ V + 3 a ˆ K V ⊗ ¯ V + a (cid:16) ˆ1 + ˆ K ˆ P + ˆ P ˆ K (cid:17) + 3 a ˆ P (cid:17) + KP (cid:16) a ˆ K ¯ V ⊗ V + a ˆ K V ⊗ ¯ V + a ˆ P + a ˆ K ˆ P + a ˆ P ˆ K + a ˆ1 (cid:17) + P K (cid:16) a ˆ K ¯ V ⊗ V + a ˆ K V ⊗ ¯ V + a ˆ P + a ˆ P ˆ K + a ˆ1 + a ˆ K ˆ P (cid:17) + 1 (cid:16) a ˆ K ¯ V ⊗ V + a ˆ K V ⊗ ¯ V + a ˆ P + a ˆ1 + a ˆ K ˆ P + a ˆ P ˆ K (cid:17) + P (cid:16) a ˆ K ¯ V ⊗ V + 3 a ˆ K V ⊗ ¯ V + a (cid:16) ˆ1 + ˆ K ˆ P + ˆ P ˆ K (cid:17) + 3 a ˆ P (cid:17) (cid:17) . (C.1)To obtain the Γ V ⊗ ¯ V ⊗ V piece of the dilatation operator one needs to exchange K ¯ V ⊗ V by K V ⊗ ¯ V and additionally the following coefficients a ↔ a , a ↔ a , a ↔ a ,a ↔ a , a ↔ a , a ↔ a . (C.2)Formally, Γ V ⊗ ¯ V ⊗ V looks the same, but P K and KP act now on the ¯ V ⊗ V ⊗ ¯ V spaces.20 Representations of length four structures
A general length four operator transforming under ( , ) of SU (2) R × SU (2) will decomposeinto the irreducible representations as follows( , ) = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , )(D.1)From [47] we know that in the case of the length four operators in ABJM the integrabilitymanifests in the degeneracy of the the trace invariant operators transforming in the repre-sentation of SU (4). In the main text of this article we checked if the degeneracy stillholds among the operators ( , ), ( , ) and ( , ) which descend from those in the ofABJM in the notation of [47].From (D.1) we see that there are actually more than 4 structures in each of the repre-sentations ( , ), ( , ) and ( , ). This comes from the fact that some of them are alsopresent in other multiplets. The decomposition of all length four operators of ABJM under SU (2) R × SU (2) goes as follows → ( , ) → ( , ) + ( , ) + ( , ) + ( , ) → ( , ) + ( , ) + ( , ) → ( , ) + ( , ) + ( , ) + ( , ) + ( , ) → ( , ) + ( , ) + ( , ) + ( , ) + ( , ) → ( , ) + ( , ) + ( , ) + ( , ) + ( , ) + ( , ) + ( , ) (D.2)The structures coming from the and the ¯45 do not correspond to any trace invariantoperators because they get a minus under cyclic permutation. References [1] J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-Mills,” JHEP (2003) 013 [arXiv:hep-th/0212208].[2] I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the
AdS (5) × S super-string,” Phys. Rev. D (2004) 046002 [arXiv:hep-th/0305116].[3] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, “Strings in flat space and ppwaves from N = 4 super Yang Mills,” JHEP , 013 (2002) [arXiv:hep-th/0202021].[4] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A semi-classical limit of thegauge/string correspondence,” Nucl. Phys. B , 99 (2002) [arXiv:hep-th/0204051].[5] N. Beisert and M. Staudacher, “The N = 4 SYM Integrable Super Spin Chain,” Nucl.Phys. B , 439 (2003) [arXiv:hep-th/0307042].[6] M. Kruczenski, “Spin chains and string theory,” Phys. Rev. Lett. , 161602 (2004)[arXiv:hep-th/0311203]. 217] N. Beisert, C. Kristjansen and M. Staudacher, “The dilatation operator of N = 4 superYang-Mills theory,” Nucl. Phys. B (2003) 131 [arXiv:hep-th/0303060].[8] S. Frolov and A. A. Tseytlin, “Multi-spin string solutions in AdS × S ,” Nucl. Phys.B (2003) 77 [arXiv:hep-th/0304255].[9] N. Beisert, V. Dippel and M. Staudacher, “A novel long range spin chain and planar N = 4 super Yang-Mills,” JHEP , 075 (2004) [arXiv:hep-th/0405001].[10] G. Arutyunov, S. Frolov and M. Staudacher, “Bethe ansatz for quantum strings,” JHEP , 016 (2004) [arXiv:hep-th/0406256].[11] N. Beisert, “The dilatation operator of N = 4 super Yang-Mills theory and integrability,”Phys. Rept. , 1 (2005) [arXiv:hep-th/0407277].[12] N. Beisert, “The su (2 |
2) dynamic S-matrix,” Adv. Theor. Math. Phys. , 945 (2008)[arXiv:hep-th/0511082].[13] R. A. Janik, “The AdS × S superstring worldsheet S-matrix and crossing symmetry,”Phys. Rev. D , 086006 (2006) [arXiv:hep-th/0603038].[14] R. Hernandez and E. Lopez, “Quantum corrections to the string Bethe ansatz,” JHEP , 004 (2006) [arXiv:hep-th/0603204].[15] D. M. Hofman and J. M. Maldacena, “Giant magnons,” J. Phys. A , 13095 (2006)[arXiv:hep-th/0604135].[16] N. Beisert, R. Hernandez and E. Lopez, “A crossing-symmetric phase for AdS × S strings,” JHEP , 070 (2006) [arXiv:hep-th/0609044].[17] N. Beisert, B. Eden and M. Staudacher, “Transcendentality and crossing,” J. Stat.Mech. , P021 (2007) [arXiv:hep-th/0610251].[18] N. Gromov, V. Kazakov and P. Vieira, “Integrability for the Full Spectrum of PlanarAdS/CFT,” arXiv:0901.3753 [hep-th].[19] D. Berenstein and S. A. Cherkis, “Deformations of N = 4 SYM and integrable spinchain models,” Nucl. Phys. B (2004) 49 [arXiv:hep-th/0405215].[20] X. J. Wang and Y. S. Wu, “Integrable spin chain and operator mixing in N = 1,2supersymmetric theories,” Nucl. Phys. B (2004) 363 [arXiv:hep-th/0311073].[21] N. Beisert and R. Roiban, “The Bethe ansatz for Z S orbifolds of N = 4 super Yang-Millstheory,” JHEP , 037 (2005) [arXiv:hep-th/0510209].[22] A. Solovyov, “Bethe Ansatz Equations for General Orbifolds of N=4 SYM,” JHEP , 013 (2008) [arXiv:0711.1697 [hep-th]].[23] J. H. Schwarz, “Superconformal Chern-Simons theories,” JHEP , 078 (2004)[arXiv:hep-th/0411077]. 2224] J. Bagger and N. Lambert, “Modeling multiple M2’s,” Phys. Rev. D , 045020 (2007)[arXiv:hep-th/0611108].[25] J. Bagger and N. Lambert, “Gauge Symmetry and Supersymmetry of Multiple M2-Branes,” Phys. Rev. D , 065008 (2008) [arXiv:0711.0955 [hep-th]].[26] J. Bagger and N. Lambert, “Comments On Multiple M2-branes,” JHEP , 105(2008) [arXiv:0712.3738 [hep-th]].[27] A. Gustavsson, “Algebraic structures on parallel M2-branes,” Nucl. Phys. B , 66(2009) [arXiv:0709.1260 [hep-th]].[28] D. Gaiotto and X. Yin, “Notes on superconformal Chern-Simons-matter theories,”JHEP (2007) 056 [arXiv:0704.3740 [hep-th]].[29] A. Gustavsson, “Selfdual strings and loop space Nahm equations,” JHEP , 083(2008) [arXiv:0802.3456 [hep-th]].[30] M. Van Raamsdonk, “Comments on the Bagger-Lambert theory and multiple M2-branes,” JHEP , 105 (2008) [arXiv:0803.3803 [hep-th]].[31] S. Mukhi and C. Papageorgakis, “M2 to D2,” JHEP , 085 (2008) [arXiv:0803.3218[hep-th]].[32] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, “N = 6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals,” JHEP , 091(2008) [arXiv:0806.1218 [hep-th]].[33] M. Benna, I. Klebanov, T. Klose and M. Smedback, “Superconformal Chern-SimonsTheories and AdS /CF T Correspondence,” JHEP , 072 (2008) [arXiv:0806.1519[hep-th]].[34] T. McLoughlin and R. Roiban, “Spinning strings at one-loop in
AdS × P ,” JHEP , 101 (2008) [arXiv:0807.3965 [hep-th]].[35] L. F. Alday, G. Arutyunov and D. Bykov, “Semiclassical Quantization of SpinningStrings in AdS × CP ,” JHEP (2008) 089 [arXiv:0807.4400 [hep-th]].[36] C. Krishnan, “AdS4/CFT3 at One Loop,” JHEP (2008) 092 [arXiv:0807.4561[hep-th]].[37] C. Kristjansen, M. Orselli and K. Zoubos, “Non-planar ABJM Theory and Integrabil-ity,” JHEP , 037 (2009) [arXiv:0811.2150 [hep-th]].[38] D. Gaiotto and A. Tomasiello, “The gauge dual of Romans mass,” arXiv:0901.0969[hep-th].[39] N. Akerblom, C. Saemann and M. Wolf, “Marginal Deformations and 3-Algebra Struc-tures,” arXiv:0906.1705 [hep-th]. 2340] K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, “N = 4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets,” JHEP , 091 (2008)[arXiv:0805.3662 [hep-th]].[41] K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, “N = 5,6 Superconfor-mal Chern-Simons Theories and M2-branes on Orbifolds,” JHEP , 002 (2008)[arXiv:0806.4977 [hep-th]].[42] M. Schnabl and Y. Tachikawa, “Classification of N = 6 superconformal theories ofABJM type,” arXiv:0807.1102 [hep-th].[43] D. Martelli and J. Sparks, “Moduli spaces of Chern-Simons quiver gauge theories and AdS /CF T ,” Phys. Rev. D , 126005 (2008) [arXiv:0808.0912 [hep-th]].[44] A. Hanany and A. Zaffaroni, “Tilings, Chern-Simons Theories and M2 Branes,” JHEP , 111 (2008) [arXiv:0808.1244 [hep-th]].[45] A. Hanany, D. Vegh and A. Zaffaroni, “Brane Tilings and M2 Branes,” JHEP ,012 (2009) [arXiv:0809.1440 [hep-th]].[46] D. L. Jafferis and A. Tomasiello, “A simple class of N = 3 gauge/gravity duals,” JHEP , 101 (2008) [arXiv:0808.0864 [hep-th]].[47] J. A. Minahan and K. Zarembo, “The Bethe ansatz for superconformal Chern-Simons,”JHEP (2008) 040 [arXiv:0806.3951 [hep-th]].[48] D. Bak and S. J. Rey, “Integrable Spin Chain in Superconformal Chern-Simons Theory,”JHEP , 053 (2008) [arXiv:0807.2063 [hep-th]].[49] D. Gaiotto, S. Giombi and X. Yin, “Spin Chains in N = 6 Superconformal Chern-Simons-Matter Theory,” JHEP , 066 (2009) [arXiv:0806.4589 [hep-th]].[50] G. Arutyunov and S. Frolov, “Superstrings on AdS × CP as a Coset Sigma-model,”JHEP , 129 (2008) [arXiv:0806.4940 [hep-th]].[51] B. Stefanski, jr. “Green-Schwarz action for Type IIA strings on AdS × CP ,” Nucl.Phys. B , 80 (2009) [arXiv:0806.4948 [hep-th]].[52] G. Grignani, T. Harmark and M. Orselli, “The SU (2) × SU (2) sector in the stringdual of N = 6 superconformal Chern-Simons theory,” Nucl. Phys. B , 115 (2009)[arXiv:0806.4959 [hep-th]].[53] C. Ahn and R. I. Nepomechie, “N = 6 super Chern-Simons theory S-matrix and all-loopBethe ansatz equations,” JHEP (2008) 010 [arXiv:0807.1924 [hep-th]].[54] N. Gromov and P. Vieira, “The AdS /CF T algebraic curve,” JHEP , 040 (2009)[arXiv:0807.0437 [hep-th]].[55] N. Gromov and P. Vieira, “The all loop AdS /CF T Bethe ansatz,” JHEP (2009)016 [arXiv:0807.0777 [hep-th]]. 2456] D. Bak, D. Gang and S. J. Rey, “Integrable Spin Chain of Superconformal U ( M ) × U ( N )Chern-Simons Theory,” JHEP , 038 (2008) [arXiv:0808.0170 [hep-th]].[57] B. I. Zwiebel, “Two-loop Integrability of Planar N = 6 Superconformal Chern-SimonsTheory,” arXiv:0901.0411 [hep-th].[58] J. A. Minahan, W. Schulgin and K. Zarembo, “Two loop integrability for Chern-Simonstheories with N = 6 supersymmetry,” JHEP (2009) 057 [arXiv:0901.1142 [hep-th]].[59] A. Agarwal, N. Beisert and T. McLoughlin, “Scattering in Mass-Deformed N ≥≥