Integrability of Orbifold ABJM Theories

Integrable structure has played a very important role in the study of various non-perturbative aspects of planar ABJM theories. In this paper we showed that this remarkable structure survive after orbifold operation with discrete group $\Gamma(\simeq\mathbb{Z}_n)<SU(4)_R\times U(1)_b$. For general $\Gamma$, we prove the integrability in the scalar sector at the planar two-loop order and get the Bethe ansatz equations. The eigenvalues of the anomalous dimension matrix are also obtained. For $\Gamma<SU(4)$, two-loop all-sector and all-loop BAEs are proposed. Supersymmetric orbifolds are discussed in this framework.


Introduction
As a strong-weak duality, AdS/CFT correspondence [1]- [3] is very powerful in applications which use weakly coupled gravity to study strongly coupled field theory. However, this makes the non-trivial checks of this correspondence very hard since one needs to compute some quantities in the strong coupling limit of field theory to compare with results from the gravity side. Supersymmetric localization [4] and integrability [5] are two very important tools to perform such field theoretical computations. These two approaches are complemented by each other. Localization can be utilized beyond the planar limit but the quantities which it can compute usually should be invariant under the supercharges on which the localization based. When the integrable structure exists, we can compute some quantities which are even non-supersymmetric. However, such theories are quite rare and integrable structure usually only appears in the large N limit. These two tools also permit us to compute certain quantities at the intermediate values of the coupling constant where neither perturbative gauge theory nor weakly coupled gravity is applicable. Both four-dimensional N = 4 super Yang-Mills (SYM) theory and three-dimensional Aharony-Bergman-Jafferis-Maldacena (ABJM) theory [6] are integrable in the planar limit [7]- [9]. It is very interesting to see how far one can go by reducing the supersymmetries of the original theory while keeping integrable structure at the same time. For four dimensional case, people have explored a lot through at least three approaches including marginal deformations [10]- [13], orbifolding [14]- [19] and adding flavors [20]- [22]. Excellent reviews on these results include [23,24]. However in three-dimensional case, similar exploration is limited. In [25], integrability of planar β-and γ-deformed ABJM theories were established at two-loop order in the scalar sector. The anomalous dimension matrices can be expressed as a Hamiltonian acting on an alternative spin chain 1 . The obtained Hamiltonians have identical form for these theories in the scalar sector, though the former theory has only one deformation parameter, while the latter has three. Comparing with the two-loop scalar-sector Hamiltonian from planar ABJM theory, now in each summand of the Hamiltonian for β-deformed ABJM theory, the next-to-nearest permutation term attains a certain phase depending on the charges of the three involved sites under two global U (1)'s which are used to perform the β-deformations. To obtain the needed transfer matrices, we need to deform the four R-matrices by similar phase factors to satisfy Yang-Baxter equations and produce the wanted Hamiltonian at the same time. This deformation is of Drinfeld-Reshetikhin form. A double scaling limit of γ-deformed ABJM theory was considered in [28] which leads to an integrable theory of interacting fermions and scalars following four-dimensional consideration in [29] (some subtleties of this limit were also studied in [30]). This showed that integrable Chern-Simonsmatter theories with less supersymmetry can have new interesting feature. And as in four dimensional case [23], in β/γ-deformed and orbifold ABJM theories, states with single magnon can be physical and detailed study on them may be simpler in many aspects than on the excited states in ABJM theory where at least two magnons are needed.
In this paper, we will focus on integrability of planar orbifold ABJM theories. Orbifolding is a widely used technique to obtain gauge theories from a parent one [31][32][33]. It is carried out by starting with a discrete subgroup of the global symmetry group of the original theory. One can get various quiver gauge theories with less supersymmetry based on different discrete subgroups when the former one is supersymmetric. One of the advantages of the orbifolding operation is that the obtained theories inherit some good properties of the parent theory. In this paper, the parent theory is the ABJM theory [6] which is the low energy effective theory on the worldvolume of N coincident M2-branes at the C 4 /Z k orbifold singularity. The global (super)symmetry of ABJM theory is OSp(6|4) × U (1) b , direct product of a simple supergroup and a U (1) factor. This is distinct from the global symmetry of N = 4 SYM which is just a simple supergroup P SU (2, 2|4). In [34], two concrete quiver gauge theories with the residual N = 4 supersymmetries (nonchiral orbifold) and N = 2 supersymmetries (chiral orbifold) have been established through two different Z n orbifoldings in ABJM field theory. Other orbifold ABJM theories are discussed in [35][36][37]. In this paper, we only consider the case that Γ is isomorphic to Z n .
We start with planar two-loop order and focus on the scalar sector which is closed at this order. We consider the generic case with Γ < SU (4) R × U (1) b . The composite operators of the orbifold theory can be expressed compactly using the fields in the parent ABJM theory with twist matrix inserted in the trace and with the projection condition imposed on the fields in the parent theory . A straightforward computation shows that only two terms of the Hamiltonian were twisted by some phase factors whose precise forms depend on the charges of the involved sites under the action of Γ. To get transfer matrices which can produce this new Hamiltonian, we only need to insert certain constant matrices which act on the auxiliary spaces inside the traces. One can demonstrate that choosing the inserted matrices to be diagonal will make the RTT relations hold. By suitable choices of such matrices, we can produce the desired Hamiltonian. This completes the proof the integrability of general orbifold ABJM theories at planar two loop order in the scalar sector. Using algebraic Bethe ansatz, we find the Bethe ansatz equations (BAEs) in this sector at two-loop and give the constraints from the trace property and twist condition. The eigenvalues of the anomalous dimension matrix (ADM) are expressed using the Bethe roots.
Then we concentrate on the case with Γ < SU (4) R and generalize the above results to proposals for all-sector and all-loop order. The leading-order all-sector results can be employed based on the prescription of Beisert-Roiban [16] after obtaining the charges for each simple root of the superalgebra and the vacuum. The all-order asymptotic results are obtained similarly based on all-loop asymptotic BAEs for planar ABJM theory [41]. As non-trivial consistency checks, we show that the BAEs we obtained satisfy both the fermionic duality and dynamic duality conditions. Finally we analyse the condition on the charges for the orbifolding to preserve N = 2 and N = 4 supersymmetries. We also confirm these results by using the fact that the orbifold ABJM theory is the low energy effective theory of N membranes placed at the orbifold singularity C 4 /(Γ × Z |Γ|k ) [37].
The remaining part of this paper is organized as follows, in the next section and section 3 we study in detail the integrability of orbifold ABJM theories in the scalar sector at two loop level. In sections 4 and 5, we will obtain the two-loop all-sector and all-loop results. Finally, we will discuss the supersymmetric orbifold theories. Some technical details will be put in three appendices.

Two-loop Hamiltonian from orbifold ABJM theories
As mentioned in the introduction, in this section, we will consider orbifold based on group Γ( Z n ) < SU (4) R × U (1) b and focus on the scalar sector which is closed at two-loop order.

Basic Ingredients of Orbifolding in Gauge Theory
Now we will set up some necessary knowledge of orbifold gauge theory and our notation will follow that of [16,17] closely. We consider to perform orbifolding using discrete subgroup Γ Z n of SU (4) R × U (1) b which means to start with ABJM theory with gauge group U (nN ) × U (nN ) and impose the following projection condition on gauge fields and the scalar fields where R(g) is a matrix representation of Γ acting on the indices I, J = 1 · · · 4 of Y I and γ(g) is acting on the color space with the color indices suppressed. The projection condition on fermions is Notice that when g = (g 1 , g 2 ) ∈ SU (4) R × U (1) b , we have R(g) = R(g 1 )R(g 2 ) and R (g) = R (g 1 )R (g 2 ) = R(g 1 )R(g −1 2 ), sinceψ I and Y I have opposite U (1) b charges. The resulting theory is a quiver theory with gauge group U (N ) 2n . If the element g is the generator of Z n , the matrix representation γ(g) will have the form For the sake of simplicity, we also require that the field Y I has definite Γ(< SU (4) R × U (1) b ) charge, then R(g) will take the diagonal form R(g) I J = δ I J ω s I and the constraint on the field Y I becomes Here and the following by γ, we always mean γ(g). By orbifolding, the field Y I can be viewed as a n×n matrix with elements also being N ×N matrices and only some components will survive due to the condition (2.6). Then the orbifold theory can be formulated in terms of those decomposed fields however the action turns out to be quite complicated [14,34]. In our paper, we will use the field Y I in the parent theory and focus on the following single trace operators, Operators with the same m constitute the m-th twisted sector and m = 0 corresponds to the untwisted sector. If we move one γ to pass all the fields behind and use the cyclic property of the trace to move it back, we find an overall phase factor appear. The composite operators will have the possibility to be non-vanishing only when this phase factor is trivial, hence lead to the twist constraint Furthermore, this local operator can also be seen as a closed alternating spin chain state |O = |γ m ; I 1 ,J 1 , · · · , I L ,J L . (2.9)

Anomalous Dimension Matrix of Composite Operators in Twisted Sector
We now find the anomalous dimensions for these gauge invariant scalar operators. An important fact is that the operators belonging to different twisted sectors do not mix with each other. Thus in the following discussions we will stay in a fixed m-th twisted sector. Before any further computations, let us recall that for parent ABJM theory, in the planar limit and at 2-loop order, the anomalous dimension matrix Γ consists of local Hamiltonian of three adjacent sites [8,9], where λ = N/k and P , K are the permutation and trace operators acting on the tensor product of two vector spaces defined as For the orbifold ABJM theories, the anomalous dimension matrix is obtained by expressing local interaction terms H of ABJM theory in the operator basis (2.7). If γ m do not appear in the interaction region, we get the same local Hamiltonian as the parent ABJM theory, If γ m is present in the interaction region, we should move it away either to the left or to the right for the convenience of solving the problem. In our case, the non-trivial interactions only reside in the first and the last two sites of the spin chain and the modified local Hamiltonian are derived as follows. For the interactions among the (2L − 1)-th,the 2L-th and the 1st site, on one hand, we have, Similarly, when acting on the 2L-th ,the 1st and the 2nd sites, we shift the generator γ m to the left side and obtain Therefore, the orbifold ABJM Hamiltonian reads

Algebraic Bethe Ansatz of the Orbifold ABJM Model
The Hamiltonian derived above can be seen as a spin chain Hamiltonian with twisted boundary conditions. In this section we will give an explicit construction to show the integrability of this model and compute the eigenvalues of the Hamiltonian.

Integrability of Orbifold ABJM Hamiltonian
In order to demonstrate the integrability, the starting object is the R-matrix which satisfy the Yang-Baxter equation (YBE). For the orbifold ABJM theories, we use the same four R-matrices as those defined in the case of period spin chain [8,9], where V i and Vī denote the fundamental and anti-fundamental representation space of SU (4) respectively. The R-matrices satisfy the following six YBEs [8,9], By the standard procedure, the next step is to construct the monodromy matrices using these R-matrices, we have where 0 and0 refer to auxiliary spaces in the SU(4) fundamental and anti-fundamental representations respectively. Comparing with the T-matrices for the periodic spin chain, we modify them by inserting two additional matrices M andM in the auxiliary spaces V 0 and V0 so that they can generate the twisted boundary terms in equation (2.17) [38]. The precise form of these two matrices will be determined later by demanding that the obtained Hamiltonian is the same as the one from the orbifold ABJM theories (up to an overall constant factor and shifting by term proportional to identity operator). Here we first show that when M andM are diagonal, the obtained Hamiltonian is integrable. In this case it is easy to show that R ab (u), M aMb = 0, (3.14) Rāb(u),MāMb = 0. (3.15) where the indices of M andM denote on which site they act. Therefore we have the following important equations known as the RTT relations in the literature, By tracing over the auxiliary spaces of monodromy T-matrices, we obtain the transfer matrices Then the above RTT relations lead to for arbitrary u and v. Expanding τ (u) andτ (u) in terms of u, we find that the coefficients are mutually commuting and can be seen as the conserved charges. Of our interests is a certain combination of these conserved quantities given below because they will correspond to the Hamiltonians of our system, After some computations ,we find The details of the computations are put in the Appendix A. We would like to know the component forms of the boundary terms of the above Hamiltonian and for this purpose we first clarify our convention for the matrix indices as follows 2 , Hence we have where m i andm i i = 1, 2, 3, 4 are the diagonal elements of M andM . Comparing these results with the equations (2.15) and (2.16), one can fix the matrices M andM as So our conclusion is that by inserting the above two diagonal matrices into the monodromy matrices, we derived a Hamiltonian nearly the same as the one obtained in the field theory side only up to a shift of 3LI and an overall factor λ 2 which do not affect the integrability of the model. This completes the proof of the integrability of planar orbifold ABJM theories in the scalar sector at the two-loop order.

Eigenvalues of Spin Chain Hamiltonian and Bethe Ansatz Equations
In this section we consider the diagonalisation of the corresponding transfer matrices. In the seminal paper [39], the eigenstates of the Hamiltonian for a very general inhomogeneous spin chain with different spin on each site were constructed by means of the nested algebraic Bethe ansatz method. We find the related results can also apply to our alternating spin chain with twisted boundary conditions. However, here, we will use a much simpler method to obtain the Bethe ansatz equations. 3 First we select the ground state as which corresponds to the chiral primary operator Tr(γ m (Y 1 Y † 4 ) L ). Then we write the monodromy matrix as For this selected vacuum, we find the three super-diagonal elements serve as the creation operators while the other three sub-diagonal ones C 1 = T 2 1 ,C 2 = T 3 2 ,C 3 = T 4 3 as the annihilation operators. They also correspond to the simple roots of SU (4) Lie algebra.
The excited states can be constructed by acting three kinds of creation operators on the vacuum state, where u 1j = iu j + 1/2, u 2k = ir k + 1, u 3n = iv n + 3/2 with 1 ≤ j ≤ K u , 1 ≤ k ≤ K r , 1 ≤ n ≤ K v are three sets of Bethe roots. Then the eigenvalue of the transfer matrix τ (u) can be found by using the commutation relations between T i , i = 1, 2, 3, 4 and B i , i = 1, 2, 3 originated from the eq. (3.16) by throwing the unwanted terms, 3 Such treatment for SU (N ) spin chain can be found in the lecture notes by N. Beisert [40].
-8 -For the eigenvalue ofτ (u), it can be found from the conjugation conditionΛ(u) = Λ(2 − u * ) * [8] The Bethe ansatz equations (BAEs) can be readily obtained by demanding that the residue vanishes at each potential pole of Λ(u), The consistency of the theory guarantees that we could get the same sets of Bethe ansatz equations from Λ(u) instead, as one can easily check. Now let us investigate the twist constraint for the excited state which is largely due to an implicit charge conservation condition. Note that the component of monodromy matrix T is where a, b, c n , n = 1, · · · , 2L represent the indices of the auxiliary space and i n , j n , n = 1, · · · , 2L are the indices of quantum spaces. If allocating each index i of V i a phase s i and i ofV i a phases i with obvious relations i = −s i , we find the total phases are conserved under the action of three braiding operators I, P and K, Since the building blocks of the monodromy matrix are R-matrices which entirely consists of these three operators and M is diagonal, the whole process obey the phase conservation law, So the net phase of the quantum space is s a − s b under the action of T b a . Note that the phase of vacuum state is L(s 1 − s 4 ), then the phase of the excited state (3.34) become Therefore the twist constraint turn out to be The shift operator and the corresponding total momentum are defined as In the Appendix B we will show that the shift operator acts trivially on physical state. Now given the eigenvalues above, we find which is the zero momentum condition for the twisted spin chain. As mentioned above, by a shift of 3L and then multiplied by λ 2 , we find the energy of the spin chain which is dual to the anomalous dimension γ of the orbifold ABJM theories,

Orbifold Bethe Ansatz
Having obtained the orbifold Bethe equations for SU (4) sector, now we go toward all-sector results. From now on, we will restrict to the case with Γ < SU (4) R . The leading order 4 Bethe ansatz equations for ABJM theory read [8,9], where J = 5 is the rank of the algebra osp(6|4), M j,j is the symmetric Cartan matrix and V j are the Dynkin labels which specify the representation of spin sites. The distinguished simple root system is we label the simple roots as α 1 , α 2 , α 3 , α 4 , α4 in the given order above. For more details for the algebra osp(6|4), see Appendix C. As shown in Fig. 1, These equations can be written in a compact form,

Orbifolding the Bethe Ansatz
The leading order orbifold Bethe ansatz equations has the general form [16] (in the twist m sector for Z n orbifold), Where the q j , j = 1, 2, 3, 4,4 are the SU (4) charges of simple roots under orbifolding, and q 0 is the charge of vacuum. In the distinguished simple root system, the charges are, with t 1 , t 2 , t 3 integers. The q is related the charges s I by If we restrict to the scalar sector, one can recover the eqs. The energy is given by, It is useful to represent the twist field γ by a new type of quasi-excitation j = −1, with excited number Then the leading order Bethe equations for a Z n orbifold theory can also be written in a compact form, (4.13) We now consider a simple example in the SU (2) × SU (2) sector at two loops to verify our orbifold Bethe ansatz. The SU (2) × SU (2) sector is made of the elementary excitations (Y 2 |Y † 3 ) on the even and odd sites above the vacuum Tr((Y 1 Y † 4 ) L ), and it is closed at any order [43,44]. At leading order of ABJM theory, the Hamiltonian reduces to the sum of two decoupled Heisenberg XXX 1/2 Hamiltonians, one acting on the even sites and the other acting on the odd sites 5 (4.14) In orbifold case, the l-th term in the Hamiltonian is the same as above for 1 ≤ l ≤ 2L − 2, and the 2L − 1-th term and the 2L-th term are multiplied by the phases indicated in eqs. (2.15) and (2.16), respectively. We consider two excitations above the "twist vacuum" Tr(γ m (Y 1 Y † 4 ) L ), one on the even sites and another on the odd sites. The obtained operators are Tr(γ m (Y 2 Y † 3 )(Y 1 Y † 4 ) L−1 ) and the ones with permutations among even sites and odd sites independently. For the above operator to be non-vanishing, the twist constraint must be imposed. For concreteness, we take L = 3. In the basis, The Hamiltonian takes the form, To write the Hamiltonian in a compact form, we have used the eq. (4.10). With the aid of Mathematica, it is easy to find the eigenvalues Let's compute it using our orbifold Bethe ansatz equations. In the above simple case L = 3, we have all excitation numbers to be zero except K 4 = K4 = 1. The Bethe equations are simplified to be However, these three equations are not independent if we impose the twist constraint The energy is given by (4.23) The solutions of eqs. (4.19)-(4.21) are Substituting this in eq. (4.23) reproduces the result (4.18) obtained by diagonalising the Hamiltonian directly.

Higher Loops
We want to generalize our orbifold Bethe equations to higher loops. Firstly, we know the all loop AdS 4 /CF T 3 asymptotic Bethe equations read [41], where the x ± are Zhukowski variables, Note that the eqs. (5.1) still have the form of eq. (4.4) except that one uses the rapidities x j,k instead of u j,k and the scattering phases S j,j (x j,k , x j ,k ) between the various Bethe roots are modified to accommodate the higher-loop interactions.
Unlike the leading order Bethe equations, not all simple root systems are possible for writing down higher loops Bethe ansatz equations. One of the possible Dynkin diagrams is shown in Fig. 2. Another possible "Higher" Dynkin diagram is given by performing Weyl reflections with respect to the 1st and the 3rd simple roots in succession, and the result is the diagram on the right side in Fig. 3. See Appendix C for details. The two corresponding all loop Bethe equations are mapped to each other by "fermionic duality" which is consistent with odd Weyl reflection. The all loop Bethe equations has another "dynamic transformation" symmetry which transform the Bethe roots of type 1 into type 3 and change the spin chain length [42]. We will prove these two dualities after we give the all-order Bethe ansatz equations. The study of fermionic duality makes sure that these two simple root systems do give the equivalent BAEs for orbifold ABJM theories and helps us to identify simple root systems which can be used at all loop level. The valid of dynamic duality admits the dynamical nature of the higher loop BAEs which takes into the fact that some operators with different length can mix with each other at higher loop level. The Cartan matrix of two gradings η = ±1 can be summarized as Now we give a bit details of the derivation of q + , the one of q − is similar. To do this we need the Cartan matrix for the Dynkin diagram in Fig. 2, Then we begin with the distinguished simple root system (4.2). First we apply w α 2 and give then apply another w α 1 with α 1 being the first simple root δ 1 − 1 in new basis Now we get the "higher" simple root system with the Dynkin diagram shown in the Fig. 2. The original three SO(6) simple roots 1 − 2 , 2 − 3 , 2 + 3 can be found in this basis as Now, adding the first and two rows in eq. (5.6) to the third one, and multiplying the obtained matrix from the right to (0, 0, t 1 , t 2 , t 3 ), we get (5.10) and the non-vanishing Dynkin labels are the same with the distinguished simple root system because we have merely dualized the first and the second simple root. Also because of this, q 0 does not change and we get eq.(5.4). From eqs. (5.4) and (5.5), we observe that The all loop AdS 4 /CF T 3 orbifold Bethe equations read, and the twist condition (4.8) Here σ BES is the BES kernel for ABJM theory whose concrete expression can be found in [41]. The spectrum of energy is 19) and h(λ) is an interpolating function [43][44][45][46] and it also replaces √ λ/(4π) appearing in the BES kernel for 4d N = 4 Super Yang-Mills theory. We have the following relations, (5.20) They are easily confirmed using the definition (5.2).

Dynamic Duality
The equation for x 3 is, We now transform one type 3 root 1/x 3,k → x 1,k . For this transformation, u 3,k → u 1,k , and one of the x 3 equations transforms as, Using the momentum condition (5.17), we find where the relation(5.11) has been used. We recognize that this is the equation for x 1 with the same grading. Under this transformation, the scattering phases in x 4 and x4 equations also get changed. For example, 24) and the equation for x 4 transforms as, 25) thus with this transformation in addition to the following replacements which is called dynamic transformation, This dynamic duality is closely related to properties of the all-loop S-matrix. We only demonstrate this for grading η = 1. In fact, for the ABJM case, the needed property is for j = 1, 2, 3, 4,4.

The Fermionic Duality
We now prove that two choices of grading in (5.16) are equivalent based on some fermionic duality. In order to investigate this duality of the eqs.(5.16), we rewrite the equation of x 3 for η = +1 as, We further introduce the following polynomial P (x), .
Obviously there already exists K 3 + K 1 roots of P (x), The remaining solutions can also be grouped into two classes, type 3 roots and type 1 roots.
We now calculate P (x ± 4,k ) using two equivalent expressions of P (x), Using the relations (5.20) and (5.12), we find, Using the relation (5.34), we arrive at, Thus the equation for x 4 in the grading η = +1 is equivalent to one in the grading η = −1 and similar calculations can be done to show the equivalence of two gradings for x4 equation. It still remains to prove the equivalence for other equations. For this purpose, we calculate the combination P (x ± 2,k )P (1/x ± 2,k )in two ways, Using the relations (5.20) and(5.13), we can rewrite the above equation as, which proves the equivalence of two gradings of the type 2 equation. From eq.(5.33), we know that P (x 3,k ) = P (1/x 1,k ) = 0. While substituting back to (5.31), we findx 3 ,x 1 satisfy the same equation as x 3 , x 1 . Now flip the fractions in thex 3 andx 1 equations, while using the relations (5.14-5.15), we get equations for the alternative grading. In the end, we have proved that (5.1) are equivalent for the two choices of grading. Because q + 0 = q − 0 , the momentum conservation condition and the expression for the total energy are not changed under the fermionic duality as well. Notice that the relations among charges (5.11-5.15) play important roles in the verification of fermionic duality and dynamics duality. These relations are automatically satisfied by the charges calculated from the Cartan matrices, instead of imposing by hands in the twisted Bethe ansatz equations studied in [47]. In this sense, the check of these two duality for orbifold ABJM theories is a non-trivial check of these all loop BAEs, especially the computations of these charges.

Two applications
As an application, we now compare our results with the all-loop BAE equations for the β-deformed ABJM theory [27]. For η = 1 grading, the phase factors appearing in eqs. (5.17) and (5.16) (5.40) are replaced by It is easy to see that if these two groups of phases are the same. This means if these conditions are satisfied, the all-loop BAEs (for η = 1 grading) for orbifold ABJM theories and β-deformed ABJM theory coincide for states with these excitation numbers. Notice here β should be a rational number and t 1 should vanish. As will be discussed in the next section, t 1 = 0 is the condition for the orbifold theory to have at least N = 2 supersymmetry. This condition is not surprising since the β-deformed ABJM theory is N = 2 supersymmetric [26]. We now turn to relation between cusp anomalous dimension in orbifold ABJM theories and orbifold N = 4 SYM theories. As in [41], we start with grading η = −1 and focus on the solutions to all loop BAEs with only non-vanishing roots u 4,k = u4 ,k . Then the consistency of the BAEs leads to q − 4 = q − 4 . However from eq. (5.5), we have already q − 4 = t 2 − t 3 = −q − 4 . Then we are restricted to the case with q − 4 = q − 4 = 0. We also demand the phase in the zero momentum condition is trivial, The above conditions leads to t 2 = t 3 , exp(4πimt 2 /n) = 1. (5.46) As for the orbifold SYM side with sl(2) grading (corresponding to η 1 = η 2 = −1 in [48]), the phase for the momentum-carrying node is automatically zero (see eq. (3.15) of [49]). The triviality of the phase in the zero momentum condition gives exp(2πimt SY M as the one obtained in [41], under the assumption that wrapping contributions for twist operators are still subleading in the large spin limit with twist being finite.

Supersymmetric orbifold theories
Let us finally discuss the supersymmetric orbifold theories. Based on the results in previous sections, all we need is to determine the t i 's (or equivalently q's in the distinguished simple root system) which are compatible with certain number of supersymmetries. Here we follow the argument of [16]. We also check the result by determining the spinors of SO(8) preserved by the orbifolding.

N = 2 Orbifolds
To get an N = 2 theory, we need at least one fermionic (odd) generator commuting with the orbifold action. For example, when considering E α 2 corresponding to the only odd simple root α 2 = − 1 + δ 2 in the distinguished simple root system, this is equivalent to set q 2 = −t 1 = 0. We now demonstrate that this is enough. Γ can be naturally embedded into a U (1) subgroup of OSp(6|4). Denote the generator of this U (1) as P, we have that P † = P.
Note that E 1 −δ 2 locates in the first line of weight diagram in Fig. 4, and E − 1 +δ 2 locates in the last line. By using Sp(4) invariance 6 , we obtain that all generators in these two lines commute with P, then we get an N = 2 theory with the charges, We can also get the charges of 4 of SO(6) as (t 2 , −t 2 , t 3 , −t 3 ), where t 2 , t 3 are arbitrary integers except ones satisfying t 2 ± t 3 = 0, because such t 1 , t 2 will give N = 4 supersymmetries, as we will show below. This includes the chiral orbifold theory in [34] as a special case. We further demonstrate this method is indeed correct by counting the spinors of SO(8) preserved by the orbifold. Notice that the orbifold ABJM theory is the low energy effective theory of N coincident M2-branes at C 4 /(Γ × Z |Γ|k ) orbifold singularity where |Γ| is the order of Γ and Z |Γ|k acts as overall phase rotations of the four complex coordinates [37]. Under the action of the generator of finite group Γ as , where s 1,2,3,4 = ±1/2. The equation subject to s 1 + s 2 + s 3 + s 4 ∈ k|Γ|Z (6.3) has exact two solutions (s 1 , s 2 , s 3 , s 4 ) = ±(1/2, 1/2, −1/2, −1/2), (6.4) for generic t 2 , t 3 and n, and this demonstrates our conclusion above 7 .

Discussions
In this paper, we studied the integrability of planar orbifold ABJM theories. We first carried out perturbative computations of ADM in the scalar sector at two-loop order. We found that in the corresponding spin chain Hamiltonian, only two terms are deformed by certain phases. This deformation can be expressed in terms of twisted boundary condition. By inserting certain diagonal matrices inside the transfer matrices, we proved the integrability of this Hamiltonian. BAEs and eigenvalues of ADM were obtained through algebraic Bethe ansatz method. Restricting Γ to be inside SU (4) R , we obtained the all-loop all-sector BAEs which pass some non-trivial consistency checks. There are several interesting directions worth pursuing. One of them is that to explore all-loop BAEs for general Γ in SU (4) R × U (1) b . This study is beyond the framework of Beisert-Roiban [16] since U (1) b does not correspond to a node in the Dynkin diagram used for BAEs. To obtain some hints for the structure of the result, it may be helpful to first perturbatively compute the ADM of composite operators involving fermions as the computation in ABJM theory [50]. It is also interesting to find some solutions in the thermodynamical limit and study their holographic dual in term of semi-classical string/membrane solutions in the dual string/M theories.
Supersymmetric condition for the orbifold was studied in this framework of integrability. The obtained condition is consistent with the result that orbifold ABJM theory is the low energy effective theory of N membranes put at C 4 /(Γ × Z |Γ|k ) [37]. However the study in the integrability side seems only give condition for Z n orbifolds which is N = 2 or N = 4 simultaneously for all n. Let us consider the following examples taken from [35]. Take n to be even. The cases with (t 1 , t 2 , t 3 , t 4 ) = ±(n/2, n/2, (−1) l n, 0), l = 0, 1 is N = 2 supersymmetric and the case with (t 1 , t 2 , t 3 , t 4 ) = ±(n/2, −n/2, (−1) l n, 0), l = 0, 1 is N = 4 supersymmetric. The preserved supersymmetries can be easily obtained by counting the SO(8) spinors preserved by the orbifolds. Also notice that all these cases satisfy Γ < SU (4) R . We speculate that these cases do not appear in the analysis of supersymmetric orbifold here because they only appear for even n, not for all integer n. It is still interesting to see whether we can probe such cases through some refinements of the studies here. We leave this and directions mentioned previously as suggestions for further studies.

A Hamiltonian of the twisted spin chain
In this appendix, we give the detailed derivation of eq. (3.26). We employ a new set of indices i = 1, 2, · · · , 2L to relabel the quantum spaces of the alternating spin chain. Then the monodromy matrices in eqs. (3.11) and (3.12) are rewritten as At the special point u = 0, the transfer matrices become where we use Σ i ,Σ i ,i=1,· · · 4 to label each part in eq.(A.5) and (A.6) for the convenience of writing. Then let us first deal with τ (0) and give some intermediate results of the calculations Thus we find Similarly, forτ (0) we havē Therefore, So it turns out that the nearest neighbor interactions still cancels even for the twisted spin chain.

B Zero momentum condition
We change the transfer matrices into a form much easier for us to compute by means of permutation operators. τ which leads to The0,1 refer to the Z 2 grading, and the 6, 4 means that the odd part generators G1 are in the 6 and 4 representations of the even part G0, i.e. in the 6 of so(6) and 4 of sp(4). The total 24 odd generators are presented on the Fig. 4, where we denote E α as the generators of the algebra, for α ∈ ∆.