Note on integrability of marginally deformed ABJ(M) theories

Abstract

We study the anomalous dimensions of operators in the scalar sector of [beta]-deformed ABJ(M) theories. We show that the anomalous dimension matrix at two-loop order gives an integrable Hamiltonian acting on an alternating SU(4) spin chain with the spins at odd lattice sides in the fundamental representation and the spins at even lattices in the anti-fundamental representation. We get a set of [beta]-deformed Bethe ansatz equations which give the eigenvalues of Hamiltonian of this deformed spin chain system. Based on our computations, we also extend our study to non-supersymmetric three-parameter γ-deformation of ABJ(M) theories and find that the corresponding Hamiltonian is the same as the one in [beta]-deformed case at two-loop level in the scalar sector.

Published for SISSA by Springer

Received: February 21, 2013 Accepted: March 6, 2013

Published: April 2, 2013

Song Hea,b and Jun-Bao Wuc

aState Key Laboratory of Theoretical Physics,

Institute of Theoretical Physics, Chinese Academy of Science, No. 55 Zhongguancun East Road, Beijing 100190, P.R. China

bKavli Institute for Theoretical Physics China, CAS,

No. 55 Zhongguancun East Road, Beijing 100190, P.R. China

cInstitute of High Energy Physics and

Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, 19B YuquanLu, Beijing 100049, P.R. China

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study the anomalous dimensions of operators in the scalar sector of -deformed ABJ(M) theories. We show that the anomalous dimension matrix at two-loop order gives an integrable Hamiltonian acting on an alternating SU(4) spin chain with the spins at odd lattice sides in the fundamental representation and the spins at even lattices in the anti-fundamental representation. We get a set of -deformed Bethe ansatz equations which give the eigenvalues of Hamiltonian of this deformed spin chain system. Based on our computations, we also extend our study to non-supersymmetric three-parameter -deformation of ABJ(M) theories and nd that the corresponding Hamiltonian is the same as the one in -deformed case at two-loop level in the scalar sector.

Keywords: Supersymmetric gauge theory, Gauge-gravity correspondence, Bethe Ansatz

ArXiv ePrint: 1302.2208

c

Note on integrability of marginally deformed ABJ(M) theories

JHEP04(2013)012

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP04(2013)012

Web End =10.1007/JHEP04(2013)012

Contents

1 Introduction 1

2 -deformation of superconformal Chern-Simons-matter theory 3

3 Two-loop anomalous dimensions in the scalar sector 6

4 The R matrices of deformed spin chain 9

5 Eigenvalues of deformed spin chain Hamiltonian and Bethe ansatz equations 11

6 The non-supersymmetric three-parameter -deformation of ABJ(M) theory 13

7 Conclusion and discussions 13

A Hamiltonian from deformed R-matrices 14

1 Introduction

There were great achievements on integrable structure in both sides of AdS/CFT correspondence in the last decade [1]. The best studied case is the integrability in the correspondence between four-dimensional N = 4 super-Yang-Mills theory and the type IIB superstring theory on AdS5 S5 [24]. In the eld theory side, it was found that, rst at lower loop level, the anomalous dimension matrix coincides with Hamiltonian of certain integrable spin chain [57]. The all-loop Bethe ansatz equations [8] were later proposed and they can be obtained from a S-matrix for the spin chain [9], though we do not know the corresponding Hamiltonian of the spin chain at all-loop level. In the string theory side, innity number of conserved charges on the worldsheet of Green-Schwarz superstring moving in AdS5S5 were constructed [10]. The integrable structure is a very powerful tool which enables us to compute, as an example, the cusp anomalous dimension for arbitrary value of the t Hooft coupling in the planar limit [11, 12].

Such integrable structure was also found for the recently proposed duality [13, 14] between three-dimensional N = 6 Chern-Simons-matter theory and type IIA theory on AdS4 CP 3 [15][42] (for reviews see [43, 44]). The dynamics in this case is more complicated and richer than the previous one, partly due to the fact that we now have less supersymmetries. As an example, there is still a to-be-determined function in the dispersion relation of the magnon [24, 25, 45]. In the N = 4 super Yang-Mills case, this function is trivial due to the fact that the theory is self-dual under the S-duality transformation [46].

1

JHEP04(2013)012

With the success of integrability in mind, it should be with great value to generalize the above studies to case with less supersymmetries. The theory in which the anomalous dimensions of gauge invariant operators are related to an integrable Hamiltonian seems to be quite rare. If we perform generic marginal deformations of the N = 4 super Yang-Mills theory [47] or ABJM theory, the obtained theory seems usually not to keep these integrable structure in the above sense, even when some supersymmetries and conformal symmetry are preserved. The - and -deformations of N = 4 super Yang-Mills theory are quite special since they preserve this remarkable integrable structure [48][51] and their gravity dual can be obtained through a certain solution generating technique [52]. Further studies of this integrability can be found in [53][73]. These marginal deformations are special also because they can be expressed elegantly using a star product which produces a certain phase factor for each interaction term in the Lagrangian. The solution generating transformation in the gravity side can be constructed by T-duality-shift-T-duality transformations in string theory [52]. The understand of the gauge-gravity correspondence in this case was improved in [74]. The - and - deformed ABJM theories and their gravity duals were also studied in [74]. Some classical string solutions in these deformed backgrounds of type IIA string theory have been studied in [75][78]. The aim of this paper is to explore the integrable structure in the eld theory side of these deformed AdS4/CF T3 correspondence.

We begin with the scalar sector of the -deformed ABJ(M) theory which has N = 2 supersymmetries. We compute the anomalous dimensions of these operators at the two-loop level in the planar limit and for all operators with length larger than 2. We express the result as a Hamiltonian of an alternating SU(4) spin chain. Comparing with the unde-formed case [1517], we nd that, in the Hamiltonian, only the terms from the interaction terms with six scalars are deformed. Though the interaction terms with two scalars and two fermions are also deformed, their contributions to the Hamiltonian coincide with the unde-formed case. We also nd that the Hamiltonian in non-supersymmetric three-parameter -deformed ABJ(M) theory is the same as the one of the -deformed theory, at two-loop level in the scalar sector. We expect the di erences will appear in other sectors and/or at higher loop order. As in [50], we deform the R-matrices constructed in [15, 16] by introducing suitable phase factors. We show that the obtained transfer matrices will produce essentially the Hamiltonian from the perturbative computations in the Chern-Simons-matter theories. This result shows that the Hamiltonian is integrable. By diagonalizing the transfer matrices, we obtain the Bethe ansatz equations and the eigenvalues of the Hamiltonian.

The organization of the remaining parts of this paper is as follows. In section 2 we briey review -deformation of ABJM theory. In section 3, we compute the two-loop corrections to the anomalous dimensions of operators in the scalar sector in both ABJM and ABJ theories. In section 4, we constructed the R matrices after deformation and show that the Hamiltonian obtained in section 2 is integrable. Based on these results, we derive the eigenvalues of the Hamiltonian of the deformed spin chain system in section 5. A brief discussion on non-supersymmetric three-parameter -deformation is put in section 6. Section 7 is devoted to conclusions and discussions. We put some details of the computations in section 4 in the appendix.

2

JHEP04(2013)012

2 -deformation of superconformal Chern-Simons-matter theory

The ABJM theory [13] is three dimensional N = 6 supersymmetric Chern-Simons-matter theory. The gauge group of this theory is U(N) U(N), and the Chern-Simons level of these two subgroups are k and k, respectively. The matter elds are four complex scalars Y I, I = 1, 2, 3, 4 and four fermions I, I = 1, 2, 3, 4 in the (N, N) representation. The action of this theory is1

S = Z

d3x(LCS + Lkin. VF VB), (2.1)

with

LCS = k

4 Tr

JHEP04(2013)012

AA + 2i

3 AAA

k

4 Tr

+ 2i 3

, (2.2)

Lkin. = 12Tr((DY )DY I + i ID I) +

1

2Tr(DY I(DY )I

+i ID I), (2.3)

VB = 1 3

2 k

2Tr

h

Y IY JY JY KY KY I + Y IY IY JY JY KY K

+4Y IY JY KY IY JY K 6Y IY IY JY KY KY J

i, (2.4)

VF = 2ik Tr

hY IY I J J 2Y IY J I J + IJKLY I JY K L

i

2i

k Tr

hY IY I J J 2Y IY J I J + IJKLY I JY K L

i. (2.5)

The covariant derivatives are:

DY I = Y I + iAY I iY I, DY I = Y I + iY I iY I, (2.6) D I = I + iA I i I, D I = I + i I i I. (2.7)

In the following part, we will discuss the deformation of the theory following the convention given by [74]. The deformed theory will preserve three dimensional N = 2 supersymmetries. The -deformation can be performed by replacing all of the ordinary product fg of two elds f and g in the Lagrangian by the following star product:

f g = ei(Q

f1 Qg2Qf2 Qg1)fg , (2.8)

where Qfi, i = 1, 2 are two global U(1) charges carrying by the eld f and is a real deformation parameter.2 For the -deformation of ABJM theory, we choose the charges for the scalars as in table 1. The fermionic super-partner I carries the same charge as Y I, and the gauge eld is neutral under these symmetries.

1We follow the convention of [16] closely.

2We denote the deformation parameter as to stress that it is real, However the supersymmetric one-parameter deformation is still called -deformation and the non-supersymmetric three-parameter deformation to be introduced in section 6 will be called -deformation. We hope this will not produce confusions for the readers.

3

Y 1 Y 2 Y 3 Y 4

U(1)1 +12 12 0 0

U(1)2 0 0 12 12

Table 1. U(1)2 charges of the scalars of the ABJM theory used for -deformation.

I J K phase factor 1 2 3 i1 2 4 i1 3 2 i1 3 4 i1 4 2 i1 4 3 i

Table 2. The non-vanishing phase factors for the third term of VB.

It is easy to see that this star product is associative and for several elds F1, . . . , Fn,

we haveF1 . . . Fn = ei [summationtext]i<j (QFi1Q

This deformation just adds phase factors according to the above equation to the interaction terms in the Lagrangian. One can see from the above rule for the star product that the deformation does not change the kinetic terms of the Lagrangian. In the supereld formulism, the deformation will only deform the superpotential W in the following manner given in [74]:

W ! W deformed = 4k Tr

2

JHEP04(2013)012

Fj2 QFi2Q

Fj1 )F1 . . . Fn. (2.9)

, (2.10)

where one can go back to the original quartic superpotential by setting deformation parameter to be zero.

For later use, we now give the interaction terms after the deformation. One can see that only VB and VF will be deformed. By a bit calculations, we found that the third term in scalar potential, eq. (2.4), will be deformed by multiplying the phase factor exp (2i(QI QJ + QI QK + QK QI)). Here I, J, K take value of the integral number 1, 2, 3, 4, and we dene:

QI QJ QI1QJ2 QJ1QI2. (2.11)

Where QI are the U(1) charges of the elds.So the third term now becomes:

V deformedB,3rd

= 13

2 k

ei/2Y 1Y 3Y 2Y 4 ei/2Y 1Y 4Y 2Y 3

4

Xtwo of I, J, K are the sameTr(Y +IY JY +kY IY +JY K)

4

I J phase factor 1 3 12i

1 4 12 i

2 3 12 i

2 4 12i

3 1 12 i

3 2 12i

4 1 12i

4 2 12 i

Table 3. The non-vanishing phase factors for the second term of VF .

I J K L phase factor 1 3 2 4 12 i

1 4 2 3 12i

2 3 1 4 12i

2 4 1 3 12 i

3 1 4 2 12i

3 2 4 1 12 i

4 1 3 2 12 i

4 2 3 1 12i

Table 4. The non-vanishing phase factors for the third term of VF .

+4

Xall of I, J, K belong to (12) or (34)

Tr(Y +IY JY +kY IY +JY K)

+4

X(IJK)=(123),(143),(142)eiTr(Y +IY JY +kY IY +JY K) + cyclic permutations

+ 4

JHEP04(2013)012

X(IJK)=(132),(134),(124)eiTr(Y +IY JY +kY IY +JY K)+cyclic permutations

Y +IY I +J J 2

X(I,J)=(1,4),(2,3),(3,1),(4,2)ei2 Y +IY J +I J

5

. (2.12)

The non-vanishing phase factors in the third term of the potential VB are listed in table 2. The other three terms in eq. (2.4) are untouched by the -deformation.

Now we turn to consider the interaction terms between the fermions and scalars in eq. (2.5). After the deformation, these terms become

V deformedF = 2i

k Tr

XI = J, or I, J {1, 2}, or I, J {3, 4}

Y +IY J +I J

2

Figure 1. The contribution to anomalous dimension of O from two loop contribution from scalar sextet interaction. In this context, the horizontal lines represent the operators and the ordered vertical lines denote the contraction between the two operators of the elds included in trace.

2

X(I,J)=(1,3),(2,4),(3,2),(4,1)ei2 Y +IY J +I J

+

X(IJKL)=(1324),(2413),(3241),(4132)ei2 iY +I JY +K L

+

X(IJKL)=(1423),(2314),(3142),(4231)ei2 iY +I JY +K L

+

Xother termsIJKL +I JY +K L

+ h.c (2.13)

We list the non-vanishing phase factors multiplying the second and the third terms of VF get changed in tables 3 and 4.

3 Two-loop anomalous dimensions in the scalar sector

Now we compute the two-loop planar contributions to the anomalous dimensions for the composite local operators in the scalar sector:

OI1ILJ1JL Tr

JHEP04(2013)012

Y I1Y J1Y I2Y J2 Y ILY JL , L 2. (3.1)

The anomalous dimension matrix can be expressed as Hamiltonian acting on an alternating SU(4) spin chain with the spins at odd lattice sides in the fundamental representation (4) and the spins at even lattices in the anti-fundamental representation (4). The length of the spin chain is 2L. The involved Feynman diagrams are the same as the ones in [15, 16].

Since only the third terms in VB is modied by the deformation, one can see by some computations based on the Feynman diagram in gure 1 that the Hamiltonian from VB,

HB = 2 2

2L

Xi=1[I 2Pi,i+2 Ki,i+1 + Pi,i+2Ki,i+1 + Ki,i+1Pi,i+1] , (3.2)

is now changed into

eHB = 2 2

2L

Xi=1[I 2

ePi,i+2 Ki,i+1 + Pi,i+2Ki,i+1 + Ki,i+1Pi,i+1] , (3.3)

6

Figure 2. The contribution to anomalous dimension of O from two loop contribution of gauge and fermion exchange interaction. The internal waved lines and dashed line stand gluon and scalar respectively.

Figure 3. The contribution to wave function renormalization of Y , Y + from two loop contribution of diamagnetic gauge interactions.

Figure 4. The contribution to wave function renormalization of Y , Y + is from two loop contribution of paramagnetic gauge interactions.

where N/k is the t Hooft coupling of ABJM theory, and denition of I, P, K are

IIJKL = IKJL, PIJKL = ILJK, KIJKL = IJKL. (3.4)

The denition of

ePi,i+2 is

e

The contributions to the anomalous dimension of operator (3.1) from gauge and fermion exchange interaction are relevant to Feynman diagrams in gure 2. The wave

7

JHEP04(2013)012

IiIi+1Ii+2

JiJi+1Ji+2 exp(i(QJi QJi+1 + QJi+1 QJi+2 + QJi+2 QJi))

(Pi,i+2)IiIi+1Ii+2JiJi+1Ji+2 . (3.5)

Pi,i+2

Figure 5. The contribution to wave function renormalization of Y , Y + is from two loop contribution of Chern-Simons interaction.

Figure 6. The contribution to wave function renormalization of Y , Y + is from two loop contribution of fermion pair interaction to wave function renormalization.

Figure 7. The contribution to wave function renormalization of Y , Y + is from two loop contribution of vacuum polarization.

function renormalization of Y, Y + will also make contributions to anomalous dimension of composite operators in eq. (3.1). There are three kinds of nonzero contribution arise from interactions involving gauge boson loops, vertices given in V deformedF and gauge-matter interactions. The relevant Feynman diagrams are given in gures 35, gure 6 and gure 7, respectively. We nd that these contributions

HF = 2

2L

Xi=1Ki,i+1, (3.6)

Hgauge = 2

JHEP04(2013)012

2L

Xi=1

14I 12 Ki,i+1

, (3.7)

HZ = 2

2L

Xi=134I, (3.8)

are the same as the undeformed case.

8

By summing over all of these contributions, we get3

eHtotal =

eHB + HF + Hgauge + HZ

= 2

2L

Xi=1

I

ePi,i+2 + 12Pi,i+2Ki,i+1 + 12 Pi,i+2Ki+1,i+2

. (3.10)

We notice that, as the undeformed case [17], the computations in the ABJ theory [14] with gauge group U(N)k U(M)k is almost the same besides replacing the factor 2 in the Hamiltonian by the factor ~

, where ~

is dened to be ~

M/k. So the computations and discussions in the following applied to the scalar section in ABJ theory at two-loop level as well.

4 The R matrices of deformed spin chain

In this section, we will show that Hamiltonian obtained in the previous section is integrable by constructing the R matrix which satises the Yang-Baxter equation (YBE) and gives this Hamiltonian through the transfer matrix by the standard procedure.

For the alternating spin chain, we need four R-matrices

gR44ij(u),

gR44ij(u), gR44ij(u), gR44ij(u) acting on the space Vi Vj. Here the upper indices

of

JHEP04(2013)012

gR44ij(u) denote SU(4) representations related to the two spaces and u denotes spectral parameter. We dened these R-matrices as

gR44(u)IJKL = exp

i

2(QJ QI QK QL)

R44(u)IJKL, (4.1)

gR44(u)IJKL = exp

i2(QJ QI QK QL)

R44(u)IJKL, (4.2)

gR44(u)IJKL = exp

i2(QJ QI QK QL)

R44(u)IJKL, (4.3)

gR44(u)IJKL = exp

i

2(QJ QI QK QL)

R44(u)IJKL, (4.4)

where

R44(u) = uI + P, (4.5)

R44(u) = (u + 2)I + K, (4.6)

R44(u) = (u + 2)I + K, (4.7)

R44(u) = uI + P, (4.8)

3Here our convention is

(Pi,i+2Ki+1,i+2)KiKi+1Ki+2JiJi+1Ji+2

= (Pi,i+2)KiKi+1Ki+2LiLi+1Li+2(Ki+1,i+2)LiLi+1Li+2JiJi+1Ji+2. (3.9)

9

are the R-matrices beform deformation [15, 16]. From above formulas, we can get:

gR44(u)IJKL = u exp(iQJ QI)IKJL + ILJK, (4.9)

gR44(u)IJKL = (u + 2) exp(iQJ QI)IKJL + IJKL, (4.10)

gR44(u)IJKL = (u + 2) exp(iQJ QI)IKJL + IJKL, (4.11)

gR44(u)IJKL = u exp(iQJ QI)IKJL + ILJK. (4.12)

The R matrices before deformation satisfy YBE [15, 16]:

R4412(u v)R4413(u)R4423(v) = R4423(v)R4413(u)R4412(u v), (4.13)

R4412(u v)R4413(u)R4423(v) = R4423(v)R4413(u)R4412(u v), (4.14)

R4412(u v)R4413(u)R4423(v) = R4423(v)R4413(u)R4412(u v), (4.15)

R4412(u v)R4413(u)R4423(v) = R4423(v)R4413(u)R4412(u v). (4.16)

As in [50], the choice of the phases in eqs. (4.1)(4.4) are such that the R matrices after deformation still satisfy the YBE:

gR4412(u v) gR4413(u) gR4423(v) = gR4423(v) gR4413(u) gR4412(u v), (4.17)

gR4412(u v)

gR4413(u)

gR4423(v) = gR4423(v) gR4413(u)

gR4412(u v), (4.18)

gR4412(u v) gR4413(u) gR4423(v) = gR4423(v) gR4413(u) gR4412(u v), (4.19)

gR4412(u v)

gR4413(u) gR4423(v) = gR4423(v)

gR4413(u) gR4412(u v). (4.20)

By introducing an auxiliary space V0, we can dene the following two transfer T-matrices:4

eT0(u, a) = 2L

gR4401(u)

gR4402(u + a)

gR4403(u)

gR4404(u + a)

gR440(2L1)(u)

gR440(2L)(u + a) , (4.21)

eT 0(u,) = 2L

gR4401(u +) gR4402(u) gR4403(u +)

gR4404(u)

gR440(2L1)(u +) gR440(2L)(u) . (4.22)

The YBE will lead to the following relations:

gR4400 ( )

eT0(, a)

JHEP04(2013)012

eT0 (, a) =

eT0 (, a)

eT0(, a)

gR4400 ( ), (4.23)

gR4400 ( )

eT 0(,)

eT 0 (,) =

eT 0 (,)

eT 0(,)

gR4400 ( ), (4.24)

gR4400 ( + a)

eT0(, a)

eT 0 (, a) =

eT 0 (, a)

eT0(, a)

gR4400 ( + a). (4.25)

Then the traces of the two T-matrices

e(, a) = Tr 0

eT0(, a) , (4.26)

e(, a) = Tr 0

eT 0(, a) , (4.27) 4Comparing with the T-matrices in [16], we include the constant factor 2L as in [15].

10

satisfy

e(, a), e(, a)] = 0 [

e(,), e(,)] = 0 , [

e(, a),

[

e(, a)] = 0 . (4.28)

From now on, we restrict a to be purely imaginary so that

e(, a) and

e(,) commute

with each other.

Now we compute the Hamiltonian from the R-matrices:

H = log

e(u, a) u

JHEP04(2013)012

u=0 +

log

e(u,) u

u=0. (4.29)

After some computations, we get

H =

2L

Xi=1Hi, (4.30)

with

H2l1 = 1 a2 4

(a 2)I + (a2 4)

eP2l1,2l+1

(a 2)P2l1,2l+1K2l1,2l + (a + 2)P2l1,2l+1K2l,2l+1

, (4.31)

H2l = 1 a2 4

(a + 2)I + (a2 4)

eP2l,2l+2

+(a + 2)P2l,2l+2K2l,2l+1 (a 2)P2l,2l+2K2l+1,2l+2

. (4.32)

Here we have already used the relation = a. The details of the computations are deferred to the appendix. After setting a = 0, multiplying Hi by 1, then shifting Hi by

32 I, the above Hamiltonian coincides with the one in the previous section obtained from the perturbative computations in eld theory side.

5 Eigenvalues of deformed spin chain Hamiltonian and Bethe ansatz equations

In previous section, we constructed transfer matrix and Hamiltonian of the deformed spin chain. We now derive the Bethe ansatz equation through diagonalizing the transfer matrices. By choosing the ground state or highest-weight state as |1414 > and introducing three sets of Bethe roots (la, mb, rc), 1 a Nl, 1 b Nm, 1 c Nr, we

can get the eigenvalues of ~

(, 0) as

e () = 2L( + 1)L( 2)L exp

i 4L +

i4Nm

i 2Nr

Ya=1 + ila 12 + ila + 12(5.1)

Nl

11

+2L( + 1)L()L exp

Nr

i 4L +

i4Nm

i 2Nl

Yc=1 + irc + 52 + irc + 32

Nl

+2L(2)LL exp

i4Li4Nm+i 2Nr

Ya=1 +ila+ 32 +ila+ 12

Nm

Yb=1 + imb +imb+1

Nc

Yc=1 +irc+ 12 +irc+ 32

Nm

+2L(2)LL exp

i4Li4Nm+i 2Nl

Yb=1 +imb+2 +imb+1.

As the undeformed case [15, 16], Nl, Nm, Nr should satisfy:

2Nl L + Nm, 2Nr L + Nm, 2Nm Nl + Nr. (5.2)

Similarly, we can get the eigenvalues of ~

(, 0) as

e () = 2L()L( + 1)L exp

i

4L

JHEP04(2013)012

i 4Nm +

i 2Nr

Ya=1 + ila + 52 + ila + 32(5.3)

Nl

+2L( + 1)L( 2)L exp

i

4L

i 4Nm +

i 2Nl

Yc=1 + irc 12 + irc + 12

Nr

Nl

+2L(2)LL exp

i 4L +

i 4Nm

i 2Nr

Ya=1+ila+ 12 +ila+ 32

Nm

Yb=1 +imb+2 +imb+1

+2L(2)LL exp

Nm

i 4L +

i 4Nm

i 2Nl

Yb=1+imb +imb+1

Nr

Yc=1+irc+ 32 +irc+ 12.

By demanding the residue vanishes at every pole of

e (), we get the following set of

Bethe Ansatz equations.

exp

i

2L

i 2Nm+iNl

la i2 la + i2

!L=

Ya6=ala la i la la + i

Nm

Yb=1la mb + i2 la mb i2,

exp

i 2L +

i2Nm iNl

rc i2 rc + i2

!L=

Nm

Yb=1rc mb + i2 rc mb i2

Yc6=crc rc i rc rc + i,

Nr

exp

i2Nl

i 2Nr

=

Nl

Ya=1mb la i2 mb la + i2

Yb6=bmb mb i mb mb + i

Yc=1mb rc + i2mb rc i2. (5.4)

We will get the same set of equations if we start with

e () instead. It is a consistent check that the same set of Bethe ansatz equations remove potential simple pole terms for

e ()

and

e (). One can see the equations just above go back to Bethe ansatz equations given in [15, 16] by setting = 0.

From the above eigenvalues, we can get the total momentum as

Ptotal = 1 i

hlog

e (0) + log

e (0)

i

= 1

i

" i2Nr +i2Nl +Nl

Xa=1log ila 12 ila + 12+

Xc=1log irc 12 irc + 12

#

Nr

. (5.5)

12

Y 1 Y 2 Y 3 Y 4 1 2 3 4

U(1)1 12 12 0 0

1

2 12 0 0

U(1)2 0 0 12 12 0 0

1

2 12

1

2 12 12 12 12

Table 5. U(1)3 charges of the scalars and fermions of the ABJM theory used for -deformation.

From the vanishing of the total momentum, we can obtain the following constraint:

1 = exp

i

2Nr +

U(1)3 12

1

2

1

2

JHEP04(2013)012

i 2Nl

Ya=1ila 12 ila + 12

Nr

Yc=1irc 12irc + 12. (5.6)

The total energy is

Etotal =

Nl

dd log

e () + dd log

=0

e ()

Xa=11l2a + 14+

Nr

Xc=11r2c + 14

Nl

=

!. (5.7)

From the total energy, we can get the eigenvalues of the anomalous dimension matrix of the operators in the scalar sector eq. (3.1) in the -deformed ABJM theory.

6 The non-supersymmetric three-parameter -deformation of ABJ(M) theory

The three-parameter deformation can be performed by replacing all of the ordinary product fg of two elds f and g in the Lagrangian by the following star product [74]:

f g = eiiQ

fj Qgkijk fg , (6.1)

where Qfi, i = 1, 2, 3 are three global U(1) charges carrying by the eld f and is are three real deformation parameters. We choose the U(1)i charges for the scalars and the fermions as in table 5. The gauge eld is neutral under these symmetries. One can see that this deformation degenerates to the one of -deformation by setting deformation parameters 1 = 2 = 0, 3 = .

After analogous calculation as in section 3, one can nd that in the scalar sector and at the two-loop level, the Hamiltonian is still given by eq. (3.10). The di erences between - and -deformations may appear at higher loop orders or in other sectors of the theories.

7 Conclusion and discussions

In this note, we start a study on the integrable structure of - and -deformed ABJ(M) theory, beginning with the scalar sector at the two-loop level in the planar limit. We rst perform perturbative computations of the anomalous dimension matrix and express the result

13

as a Hamiltonian acting on alternating SU(4) spin chain. We nd that only one term in the Hamiltonian is deformed and that the di erences between -deformation and -deformation are invisible in this sector at two loop level. As the undeformed case, the di erence between Hamiltonian for deformed ABJM theory and deformed ABJ theory only appears in the prefactor. So in this sector and at this order of the perturbation theory, the violation of parity invariance in the deformed ABJ theory does not a ect the integrability. Based on the structure of the deformations, we choose a suitable deformation of the R-matrices. Then the Bethe ansatz equations are obtained through diagonalizing the transfer matrices.

There are several directions worth pursuing. The study here in the eld theory side can be extended to full sector and/or higher loop order as in the undeformed case [79][85]. It is also interesting to reproduce the Bethe ansatz equation starting from the S-matrix of the spin chain based on the studies in [18, 64, 65, 72]. In the string theory side, one could try to construct the Lax pair and the innite number of conserved currents on the worldsheet. Even for the undeformed case, the story of IIA string on AdS4 CP 3 has

already been much richer than the one of IIB string on AdS5 S5, partly because that now the OSp(6|4)/U(3) SO(1, 3) coset action can only describe a subset of the complete Green-Schwarz action [2123].

Acknowledgments

The authors are grateful to Dongsu Bak, Bin Chen, Peng Gao, Miao Li, Xiao Liu, Jian-Xin Lu, Jian-Feng Wu, Zhi-Guang Xiao, Jie Yang and Hossein Yavartanoo for useful discussions. JW would like to thank SISSA, ICTS-USTC and School of Mathematical Sciences, USTC for warm hospitality. He is also very thankful to the organizers and participants of Symposium on Strings and Particle Physics 2012 held at CTP, SCU and International Workshop on Quantum Aspects of Black Holes held in CQUeST, Sogang University. This work was supported in part by the National Natural Science Foundation of China (No.10821504 (SH), No.10975168 (SH), No.11035008 (SH), No. 11105154 (JW), and No. 11222549 (JW)), and in part by the Ministry of Science and Technology of China under Grant No. 2010CB833004. SH also would like to appreciate the general nancial support from China Postdoctoral Science Foundation No. 2012M510562. JW gratefully acknowledges the support of K. C. Wong Education Foundation and Youth Innovation Promotion Association, CAS as well.

A Hamiltonian from deformed R-matrices

In this appendix, we compute the Hamiltonian of deformed spin chain in eq. (4.29). For this, we should compute

e(0, a), where the prime denotes the derivative with respect to spectrum parameter u.

e(0, a) = 2L Tr

Xi

gR4401(0) . . . d

JHEP04(2013)012

gR440(2i1)(u)du |u=0 . . .

gR440(2L)(a)

+2L Tr

Xi

gR4401(0) . . . d

gR440(2i)(u + a)du |u=0 . . .

gR440(2L)(a). (A.1)

14

Where the spectrum parameter u has been set as vanishing. The i-th term in rst part of eq. (A.1) can be written down as following

2L (P01)K1I1K0J1

gR4402(a)K2I2K1J2 . . .

gR440(2i2)(a)K2i2I2i2K2i3J2i2K2i1K2i2eiQK2i1 QI2i1

I2i1J2i1

gR440(2i)(a)K2iI2iK2i1J2i . . . P0(2L1)

K2L1I2L1 K2L2J2L1

gR440(2L)(a)K0I2LK2L1J2L

= 2L

gR4402(a)I3I2J1J2 . . .

gR440(2i2)(a)K2i1I2i2J2i3J2i2eiQK2i1 QI2i1 I2i1J2i1

gR440(2i)(a)I2i+1I2iK2i1J2i

. . .

gR440(2L)(a)I1I2LJ2L1J2L. (A.2)

The i-th term in the second part of eq. (A.1) is

2L (P01)K1I1K0J1

JHEP04(2013)012

gR4402(a)K2I2K1J2 . . .

gR440(2i2)(a)K2i2I2i2K2i3J2i2 P0(2i1)

K2i1I2i1K2i2J2i1 (I)K2iI2iK2i1J2i

eiQK2iQI2i P0(2i+1)

K2i+1I2i+1 K2iJ2i+1

gR440(2i+2)(a)K2i+2I2i+2K2i+1J2i+2 . . .

gR4402L(a)K0I2LK2L1J2L

= 2L

gR4402(a)I3I2J1J2 . . .

gR440(2i2)(a)I2i1I2i2J2i3J2i2I2iJ2iI2i+1J2i1eiQI2i+1 QI2i

gR440(2i+2)(a)I2i+3I2i+2J2i+1J2i+2 . . .

gR440(2L)(a)I1I2LJ2L1J2L. (A.3)

The deformed

e1(0, a) is

e1(0, a) = 2L

gR44

1

0(2L)(a)

J2L1J2L I1I2L. . .

gR44

1

0(2)(a)

J1J2I3I2, (A.4)

where

gR44

1

0(2i)(a)

J2i1J2i I2i+1I2i=

1a + 2

IJ2i1J2iI2i+1I2ieiQ

J2i1 QJ2i + 1 a2 4

KJ2i1J2iI2i+1I2i

. (A.5)

One can use above formula to obtain that

e1(u, a) e(u, a)|u=0 =

L

Xi=1IK1K2i4J1J2i4

gR44

1

0(2i)(a)

K2i1K2i I2i+1I2i

gR44

1

0(2i2)(a)

K2i3K2i2 I2i1I2i2

eiQ

[tildewide]

K2i1 QI2i1 I2i1J2i1

gR440(2i2)(a)

~K2i1I2i2 J2i3J2i2

gR440(2i)(a)I2i+1I2i~K2i1J2i IK2i+1K2LJ2i+1J2L

+

L

Xi=1IK1K2i2J1J2i2

gR44

1

0(2i)(a)

K2i1K2i I2i+1I2i

I2i+1J2i1I2iJ2i eiQI2i+1 QI2i IK2i+1K2LJ2i+1J2L

=

L

Xi=1IK1K2i4J1J2i4

gR44

1

0(2i)(a)

K2i1K2i I2i+1I2i

gR44

1

0(2i2)(a)

K2i3K2i2J2i1I2i2eiQ[tildewide]

K2i1 QJ2i1

gR440(2i2)(a)

~K2i1I2i2 J2i3J2i2

gR440(2i)(a)I2i+1I2i~K2i1J2i IK2i+1K2LJ2i+1J2L

L

Xi=1IK1K2i2J1J2i2

gR44

1

0(2i)(a)

K2i1K2iJ2i1J2ieiQJ2i1 QJ2i IK2i+1K2LJ2i+1J2L

15

Xi=11a2 4((a 2)

I + (a2 4)

eP2i1,2i+1 (a 2)P2i1,2i+1K2i1,2i+(a + 2)P2i1,2i+1K2i,2i+1) (A.6)

Similarly, one can get:

e1(u, a)

e(u, a)|u=0 =

L

Xi=11a2 4((a + 2)

I + (a2 4)

eP2i,2i+2 + (a + 2)P2i,2i+2K2i,2i+1(a 2)P2i,2i+2K2i+1,2i+2), (A.7)

where we have use the fact that a is purely imaginary. From these two equations, we can get the Hamiltonian given in the main text.

References

[1] N. Beisert et al., Review of AdS/CFT integrability: an overview,

http://dx.doi.org/10.1007/s11005-011-0529-2

Web End =Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3982

Web End =INSPIRE ].

[2] J.M. Maldacena, The large-N limit of superconformal eld theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [http://dx.doi.org/10.1023/A:1026654312961

Web End =Int. J. Theor. Phys. 38 (1999) 1113 ] [http://arxiv.org/abs/hep-th/9711200

Web End =hep-th/9711200 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9711200

Web End =INSPIRE ].

[3] S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, http://dx.doi.org/10.1016/S0370-2693(98)00377-3

Web End =Phys. Lett. B 428 (1998) 105 [http://arxiv.org/abs/hep-th/9802109

Web End =hep-th/9802109 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9802109

Web End =INSPIRE ].

[4] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [http://arxiv.org/abs/hep-th/9802150

Web End =hep-th/9802150 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9802150

Web End =INSPIRE ].

[5] J. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, http://dx.doi.org/10.1088/1126-6708/2003/03/013

Web End =JHEP 03 (2003) 013 [http://arxiv.org/abs/hep-th/0212208

Web End =hep-th/0212208 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0212208

Web End =INSPIRE ].

[6] N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of conformal N = 4 super Yang-Mills theory, http://dx.doi.org/10.1016/S0550-3213(03)00406-1

Web End =Nucl. Phys. B 664 (2003) 131 [http://arxiv.org/abs/hep-th/0303060

Web End =hep-th/0303060 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0303060

Web End =INSPIRE ].

[7] N. Beisert, The complete one loop dilatation operator of N = 4 super Yang-Mills theory, http://dx.doi.org/10.1016/j.nuclphysb.2003.10.019

Web End =Nucl. Phys. B 676 (2004) 3 [http://arxiv.org/abs/hep-th/0307015

Web End =hep-th/0307015 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0307015

Web End =INSPIRE ].

[8] N. Beisert, The SU(2|3) dynamic spin chain, http://dx.doi.org/10.1016/j.nuclphysb.2003.12.032

Web End =Nucl. Phys. B 682 (2004) 487 [http://arxiv.org/abs/hep-th/0310252

Web End =hep-th/0310252 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0310252

Web End =INSPIRE ].

[9] N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [http://arxiv.org/abs/hep-th/0511082

Web End =hep-th/0511082 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0511082

Web End =INSPIRE ].

[10] I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS5 S5 superstring, http://dx.doi.org/10.1103/PhysRevD.69.046002

Web End =Phys. Rev. D 69 (2004) 046002 [http://arxiv.org/abs/hep-th/0305116

Web End =hep-th/0305116 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0305116

Web End =INSPIRE ].

[11] N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing,http://dx.doi.org/10.1088/1742-5468/2007/01/P01021

Web End =J. Stat. Mech. 01 (2007) P01021 [http://arxiv.org/abs/hep-th/0610251

Web End =hep-th/0610251 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0610251

Web End =INSPIRE ].

[12] L. Freyhult, Review of AdS/CFT integrability, chapter III.4: twist states and the cusp anomalous dimension, http://dx.doi.org/10.1007/s11005-011-0483-z

Web End =Lett. Math. Phys. 99 (2012) 255 [arXiv:1012.3993] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3993

Web End =INSPIRE ].

16

=

L

JHEP04(2013)012

[13] O. Aharony, O. Bergman, D.L. Ja eris and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, http://dx.doi.org/10.1088/1126-6708/2008/10/091

Web End =JHEP 10 (2008) 091 [arXiv:0806.1218] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.1218

Web End =INSPIRE ].

[14] O. Aharony, O. Bergman and D.L. Ja eris, Fractional M2-branes, http://dx.doi.org/10.1088/1126-6708/2008/11/043

Web End =JHEP 11 (2008) 043 [arXiv:0807.4924] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.4924

Web End =INSPIRE ].

[15] J. Minahan and K. Zarembo, The Bethe ansatz for superconformal Chern-Simons, http://dx.doi.org/10.1088/1126-6708/2008/09/040

Web End =JHEP 09 (2008) 040 [arXiv:0806.3951] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.3951

Web End =INSPIRE ].

[16] D. Bak and S.-J. Rey, Integrable spin chain in superconformal Chern-Simons theory, http://dx.doi.org/10.1088/1126-6708/2008/10/053

Web End =JHEP 10 (2008) 053 [arXiv:0807.2063] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.2063

Web End =INSPIRE ].

[17] D. Bak, D. Gang and S.-J. Rey, Integrable spin chain of superconformal U(M) (N) Chern-Simons theory, http://dx.doi.org/10.1088/1126-6708/2008/10/038

Web End =JHEP 10 (2008) 038 [arXiv:0808.0170] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0808.0170

Web End =INSPIRE ].

[18] C. Ahn and R.I. Nepomechie, N = 6 super Chern-Simons theory S-matrix and all-loop Bethe ansatz equations, http://dx.doi.org/10.1088/1126-6708/2008/09/010

Web End =JHEP 09 (2008) 010 [arXiv:0807.1924] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1924

Web End =INSPIRE ].

[19] G. Arutyunov and S. Frolov, Superstrings on AdS4 CP 3 as a coset -model, http://dx.doi.org/10.1088/1126-6708/2008/09/129

Web End =JHEP 09 (2008) 129 [arXiv:0806.4940] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.4940

Web End =INSPIRE ].

[20] B. Stefaski Jr., Green-Schwarz action for type IIA strings on AdS4 CP 3,

http://dx.doi.org/10.1016/j.nuclphysb.2008.09.015

Web End =Nucl. Phys. B 808 (2009) 80 [arXiv:0806.4948] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.4948

Web End =INSPIRE ].

[21] J. Gomis, D. Sorokin and L. Wul , The complete AdS4 CP 3 superspace for the type IIA superstring and D-branes, http://dx.doi.org/10.1088/1126-6708/2009/03/015

Web End =JHEP 03 (2009) 015 [arXiv:0811.1566] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0811.1566

Web End =INSPIRE ].

[22] D. Sorokin and L. Wul , Evidence for the classical integrability of the complete AdS4 CP 3

superstring, http://dx.doi.org/10.1007/JHEP11(2010)143

Web End =JHEP 11 (2010) 143 [arXiv:1009.3498] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.3498

Web End =INSPIRE ].

[23] D. Sorokin and L. Wul , Peculiarities of string theory on AdS4 CP 3,

http://dx.doi.org/10.1002/prop.201100009

Web End =Fortsch. Phys. 59 (2011) 775 [arXiv:1101.3777] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1101.3777

Web End =INSPIRE ].

[24] D. Gaiotto, S. Giombi and X. Yin, Spin chains in N = 6 superconformal Chern-Simons-matter theory, http://dx.doi.org/10.1088/1126-6708/2009/04/066

Web End =JHEP 04 (2009) 066 [arXiv:0806.4589] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.4589

Web End =INSPIRE ].

[25] G. Grignani, T. Harmark and M. Orselli, The SU(2) SU(2) sector in the string dual of N = 6 superconformal Chern-Simons theory, http://dx.doi.org/10.1016/j.nuclphysb.2008.10.019

Web End =Nucl. Phys. B 810 (2009) 115 [arXiv:0806.4959] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.4959

Web End =INSPIRE ].

[26] C. Kristjansen, M. Orselli and K. Zoubos, Non-planar ABJM theory and integrability, http://dx.doi.org/10.1088/1126-6708/2009/03/037

Web End =JHEP 03 (2009) 037 [arXiv:0811.2150] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0811.2150

Web End =INSPIRE ].

[27] P. Caputa, C. Kristjansen and K. Zoubos, Non-planar ABJ theory and parity, http://dx.doi.org/10.1016/j.physletb.2009.05.021

Web End =Phys. Lett. B 677 (2009) 197 [arXiv:0903.3354] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0903.3354

Web End =INSPIRE ].

[28] A. Mauri, A. Santambrogio and S. Scoleri, The leading order dressing phase in ABJM theory, arXiv:1301.7732 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1301.7732

Web End =INSPIRE ].

[29] D. Astol, V.G.M. Puletti, G. Grignani, T. Harmark and M. Orselli, Finite-size corrections in the SU(2) SU(2) sector of type IIA string theory on AdS4 CP 3,

http://dx.doi.org/10.1016/j.nuclphysb.2008.10.020

Web End =Nucl. Phys. B 810 (2009) 150 [arXiv:0807.1527] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1527

Web End =INSPIRE ].

[30] T. McLoughlin and R. Roiban, Spinning strings at one-loop in AdS4 p3,

http://dx.doi.org/10.1088/1126-6708/2008/12/101

Web End =JHEP 12 (2008) 101 [arXiv:0807.3965] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.3965

Web End =INSPIRE ].

[31] L.F. Alday, G. Arutyunov and D. Bykov, Semiclassical quantization of spinning strings in AdS4 CP 3, http://dx.doi.org/10.1088/1126-6708/2008/11/089

Web End =JHEP 11 (2008) 089 [arXiv:0807.4400] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.4400

Web End =INSPIRE ].

17

JHEP04(2013)012

[32] C. Krishnan, AdS4/CF T3 at one loop, http://dx.doi.org/10.1088/1126-6708/2008/09/092

Web End =JHEP 09 (2008) 092 [arXiv:0807.4561] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.4561

Web End =INSPIRE ].

[33] T. McLoughlin, R. Roiban and A.A. Tseytlin, Quantum spinning strings in AdS4 CP 3:

testing the Bethe ansatz proposal, http://dx.doi.org/10.1088/1126-6708/2008/11/069

Web End =JHEP 11 (2008) 069 [arXiv:0809.4038] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0809.4038

Web End =INSPIRE ].

[34] M.A. Bandres and A.E. Lipstein, One-loop corrections to type IIA string theory in AdS4 CP 3, http://dx.doi.org/10.1007/JHEP04(2010)059

Web End =JHEP 04 (2010) 059 [arXiv:0911.4061] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.4061

Web End =INSPIRE ].

[35] D. Bombardelli and D. Fioravanti, Finite-size corrections of the CP 3 giant magnons: the Lscher terms, http://dx.doi.org/10.1088/1126-6708/2009/07/034

Web End =JHEP 07 (2009) 034 [arXiv:0810.0704] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0810.0704

Web End =INSPIRE ].

[36] D. Astol, V.G.M. Puletti, G. Grignani, T. Harmark and M. Orselli, Full Lagrangian and Hamiltonian for quantum strings on AdS4 CP 3 in a near plane wave limit,http://dx.doi.org/10.1007/JHEP04(2010)079

Web End =JHEP 04 (2010) 079 [arXiv:0912.2257] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0912.2257

Web End =INSPIRE ].

[37] M.C. Abbott, I. Aniceto and D. Bombardelli, Quantum strings and the AdS4/CF T3 interpolating function, http://dx.doi.org/10.1007/JHEP12(2010)040

Web End =JHEP 12 (2010) 040 [arXiv:1006.2174] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.2174

Web End =INSPIRE ].

[38] D. Astol, V.G.M. Puletti, G. Grignani, T. Harmark and M. Orselli, Finite-size corrections for quantum strings on AdS4 CP 3, http://dx.doi.org/10.1007/JHEP05(2011)128

Web End =JHEP 05 (2011) 128 [arXiv:1101.0004] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1101.0004

Web End =INSPIRE ].

[39] M.C. Abbott, I. Aniceto and D. Bombardelli, Real and virtual bound states in Lscher corrections for CP 3 magnons, http://dx.doi.org/10.1088/1751-8113/45/33/335401

Web End =J. Phys. A 45 (2012) 335401 [arXiv:1111.2839] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.2839

Web End =INSPIRE ].

[40] D. Astol, G. Grignani, E. Ser-Giacomi and A. Zayakin, Strings in AdS4 CP 3: nite size spectrum vs. Bethe ansatz, http://dx.doi.org/10.1007/JHEP04(2012)005

Web End =JHEP 04 (2012) 005 [arXiv:1111.6628] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.6628

Web End =INSPIRE ].

[41] M. Beccaria, G. Macorini, C. Ratti and S. Valatka, Semiclassical folded string in AdS4 CP 3, http://dx.doi.org/10.1007/JHEP05(2012)030

Web End =JHEP 05 (2012) 030 [Erratum ibid. 05 (2012) 137] [arXiv:1203.3852] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.3852

Web End =INSPIRE ].

[42] M. Beccaria, G. Macorini, C.A. Ratti and S. Valatka, Semiclassical energy of the AdS4 CP 3

folded string, http://dx.doi.org/10.1088/1742-6596/411/1/012006

Web End =J. Phys. Conf. Ser. 411 (2013) 012006 [arXiv:1209.3205] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.3205

Web End =INSPIRE ].

[43] T. Klose, Review of AdS/CFT integrability, chapter IV.3: N = 6 Chern-Simons and strings on AdS4 CP 3, http://dx.doi.org/10.1007/s11005-011-0520-y

Web End =Lett. Math. Phys. 99 (2012) 401 [arXiv:1012.3999] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3999

Web End =INSPIRE ].

[44] A.E. Lipstein, Integrability of N = 6 Chern-Simons theory, arXiv:1105.3231 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1105.3231

Web End =INSPIRE ].

[45] T. Nishioka and T. Takayanagi, On type IIA Penrose limit and N = 6 Chern-Simons theories, http://dx.doi.org/10.1088/1126-6708/2008/08/001

Web End =JHEP 08 (2008) 001 [arXiv:0806.3391] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.3391

Web End =INSPIRE ].

[46] D. Berenstein and D. Trancanelli, S-duality and the giant magnon dispersion relation, arXiv:0904.0444 [http://inspirehep.net/search?p=find+EPRINT+arXiv:0904.0444

Web End =INSPIRE ].

[47] R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, http://dx.doi.org/10.1016/0550-3213(95)00261-P

Web End =Nucl. Phys. B 447 (1995) 95 [http://arxiv.org/abs/hep-th/9503121

Web End =hep-th/9503121 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9503121

Web End =INSPIRE ].

[48] R. Roiban, On spin chains and eld theories, http://dx.doi.org/10.1088/1126-6708/2004/09/023

Web End =JHEP 09 (2004) 023 [http://arxiv.org/abs/hep-th/0312218

Web End =hep-th/0312218 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0312218

Web End =INSPIRE ].

[49] D. Berenstein and S.A. Cherkis, Deformations of N = 4 SYM and integrable spin chain models, http://dx.doi.org/10.1016/j.nuclphysb.2004.09.005

Web End =Nucl. Phys. B 702 (2004) 49 [http://arxiv.org/abs/hep-th/0405215

Web End =hep-th/0405215 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0405215

Web End =INSPIRE ].

[50] N. Beisert and R. Roiban, Beauty and the twist: the Bethe ansatz for twisted N = 4 SYM, http://dx.doi.org/10.1088/1126-6708/2005/08/039

Web End =JHEP 08 (2005) 039 [http://arxiv.org/abs/hep-th/0505187

Web End =hep-th/0505187 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0505187

Web End =INSPIRE ].

[51] S. Frolov, Lax pair for strings in Lunin-Maldacena background, http://dx.doi.org/10.1088/1126-6708/2005/05/069

Web End =JHEP 05 (2005) 069 [http://arxiv.org/abs/hep-th/0503201

Web End =hep-th/0503201 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0503201

Web End =INSPIRE ].

18

JHEP04(2013)012

[52] O. Lunin and J.M. Maldacena, Deforming eld theories with U(1) U(1) global symmetry and their gravity duals, http://dx.doi.org/10.1088/1126-6708/2005/05/033

Web End =JHEP 05 (2005) 033 [http://arxiv.org/abs/hep-th/0502086

Web End =hep-th/0502086 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0502086

Web End =INSPIRE ].

[53] S. Frolov, R. Roiban and A.A. Tseytlin, Gauge-string duality for superconformal deformations of N = 4 super Yang-Mills theory, http://dx.doi.org/10.1088/1126-6708/2005/07/045

Web End =JHEP 07 (2005) 045 [http://arxiv.org/abs/hep-th/0503192

Web End =hep-th/0503192 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0503192

Web End =INSPIRE ].

[54] H.-Y. Chen and S.P. Kumar, Precision test of AdS/CFT in Lunin-Maldacena background, http://dx.doi.org/10.1088/1126-6708/2006/03/051

Web End =JHEP 03 (2006) 051 [http://arxiv.org/abs/hep-th/0511164

Web End =hep-th/0511164 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0511164

Web End =INSPIRE ].

[55] H.-Y. Chen and K. Okamura, The anatomy of gauge/string duality in Lunin-Maldacena background, http://dx.doi.org/10.1088/1126-6708/2006/02/054

Web End =JHEP 02 (2006) 054 [http://arxiv.org/abs/hep-th/0601109

Web End =hep-th/0601109 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0601109

Web End =INSPIRE ].

[56] N. Bobev, H. Dimov and R. Rashkov, Semiclassical strings in Lunin-Maldacena background, Bulg. J. Phys. 35 (2008) 274 [http://arxiv.org/abs/hep-th/0506063

Web End =hep-th/0506063 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0506063

Web End =INSPIRE ].

[57] S. Ryang, Rotating strings with two unequal spins in Lunin-Maldacena background, http://dx.doi.org/10.1088/1126-6708/2005/11/006

Web End =JHEP 11 (2005) 006 [http://arxiv.org/abs/hep-th/0509195

Web End =hep-th/0509195 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0509195

Web End =INSPIRE ].

[58] C.-S. Chu, G. Georgiou and V.V. Khoze, Magnons, classical strings and -deformations, http://dx.doi.org/10.1088/1126-6708/2006/11/093

Web End =JHEP 11 (2006) 093 [http://arxiv.org/abs/hep-th/0606220

Web End =hep-th/0606220 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0606220

Web End =INSPIRE ].

[59] N. Bobev and R. Rashkov, Spiky strings, giant magnons and -deformations, http://dx.doi.org/10.1103/PhysRevD.76.046008

Web End =Phys. Rev. D 76 (2007) 046008 [arXiv:0706.0442] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0706.0442

Web End =INSPIRE ].

[60] D.V. Bykov and S. Frolov, Giant magnons in TsT-transformed AdS5 S5,

http://dx.doi.org/10.1088/1126-6708/2008/07/071

Web End =JHEP 07 (2008) 071 [arXiv:0805.1070] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0805.1070

Web End =INSPIRE ].

[61] F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Finite-size e ects in the superconformal -deformed N = 4 SYM, http://dx.doi.org/10.1088/1126-6708/2008/08/057

Web End =JHEP 08 (2008) 057 [arXiv:0806.2103] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.2103

Web End =INSPIRE ].

[62] C. Ahn and P. Bozhilov, Finite-size dyonic giant magnons in TsT-transformed AdS5 S5,

http://dx.doi.org/10.1007/JHEP07(2010)048

Web End =JHEP 07 (2010) 048 [arXiv:1005.2508] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1005.2508

Web End =INSPIRE ].

[63] N. Gromov and F. Levkovich-Maslyuk, Y-system and -deformed N = 4 super-Yang-Mills,http://dx.doi.org/10.1088/1751-8113/44/1/015402

Web End =J. Phys. A 44 (2011) 015402 [arXiv:1006.5438] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.5438

Web End =INSPIRE ].

[64] G. Arutyunov, M. de Leeuw and S.J. van Tongeren, Twisting the mirror TBA, http://dx.doi.org/10.1007/JHEP02(2011)025

Web End =JHEP 02 (2011) 025 [arXiv:1009.4118] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.4118

Web End =INSPIRE ].

[65] C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, Twisted Bethe equations from a twisted S-matrix, http://dx.doi.org/10.1007/JHEP02(2011)027

Web End =JHEP 02 (2011) 027 [arXiv:1010.3229] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1010.3229

Web End =INSPIRE ].

[66] M. de Leeuw and T. Lukowski, Twist operators in N = 4 beta-deformed theory, http://dx.doi.org/10.1007/JHEP04(2011)084

Web End =JHEP 04 (2011) 084 [arXiv:1012.3725] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3725

Web End =INSPIRE ].

[67] S. Giardino and V.O. Rivelles, Pulsating strings in Lunin-Maldacena backgrounds, http://dx.doi.org/10.1007/JHEP07(2011)057

Web End =JHEP 07 (2011) 057 [arXiv:1105.1353] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1105.1353

Web End =INSPIRE ].

[68] C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lscher correction and double wrapping in twisted AdS/CFT, http://dx.doi.org/10.1007/JHEP12(2011)059

Web End =JHEP 12 (2011) 059 [arXiv:1108.4914] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.4914

Web End =INSPIRE ].

[69] M. de Leeuw and S.J. van Tongeren, The spectral problem for strings on twisted AdS5 S5,

http://dx.doi.org/10.1016/j.nuclphysb.2012.03.004

Web End =Nucl. Phys. B 860 (2012) 339 [arXiv:1201.1451] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1201.1451

Web End =INSPIRE ].

[70] C. Ahn, D. Bombardelli and M. Kim, Finite-size e ects of -deformed AdS5/CF T4 at strong coupling, http://dx.doi.org/10.1016/j.physletb.2012.03.001

Web End =Phys. Lett. B 710 (2012) 467 [arXiv:1201.2635] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1201.2635

Web End =INSPIRE ].

[71] K.L. Panigrahi, P.M. Pradhan and P.K. Swain, Three spin spiky strings in -deformed background, http://dx.doi.org/10.1007/JHEP06(2012)057

Web End =JHEP 06 (2012) 057 [arXiv:1203.3057] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.3057

Web End =INSPIRE ].

19

JHEP04(2013)012

[72] C. Ahn, M. Kim and B.-H. Lee, Worldsheet S-matrix of -deformed SYM, http://dx.doi.org/10.1016/j.physletb.2013.01.047

Web End =Phys. Lett. B 719 (2013) 458 [arXiv:1211.4506] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.4506

Web End =INSPIRE ].

[73] K. Zoubos, Review of AdS/CFT integrability, chapter IV.2: deformations, orbifolds and open boundaries, http://dx.doi.org/10.1007/s11005-011-0515-8

Web End =Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3998

Web End =INSPIRE ].

[74] E. Imeroni, On deformed gauge theories and their string/M-theory duals, http://dx.doi.org/10.1088/1126-6708/2008/10/026

Web End =JHEP 10 (2008) 026 [arXiv:0808.1271] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0808.1271

Web End =INSPIRE ].

[75] M. Schimpf and R. Rashkov, A note on strings in deformed AdS4 CP 3: giant magnon and single spike solutions, http://dx.doi.org/10.1142/S0217732309032113

Web End =Mod. Phys. Lett. A 24 (2009) 3227 [arXiv:0908.2246] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0908.2246

Web End =INSPIRE ].

[76] C. Ahn and P. Bozhilov, Finite-size giant magnons on AdS4 CP 3 ,

http://dx.doi.org/10.1016/j.physletb.2011.07.065

Web End =Phys. Lett. B 703 (2011) 186 [arXiv:1106.3686] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.3686

Web End =INSPIRE ].

[77] J.-B. Wu, Multi-spin strings in AdS4 CP 3 and its -deformations, arXiv:1208.0389 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.0389

Web End =INSPIRE ].

[78] C. Ratti, Notes on multi-spin strings in AdS4 CP 3 and its marginal deformations, arXiv:1211.4694 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.4694

Web End =INSPIRE ].

[79] B.I. Zwiebel, Two-loop integrability of planar N = 6 superconformal Chern-Simons theory,http://dx.doi.org/10.1088/1751-8113/42/49/495402

Web End =J. Phys. A 42 (2009) 495402 [arXiv:0901.0411] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0901.0411

Web End =INSPIRE ].

[80] J. Minahan, W. Schulgin and K. Zarembo, Two loop integrability for Chern-Simons theories with N = 6 supersymmetry, http://dx.doi.org/10.1088/1126-6708/2009/03/057

Web End =JHEP 03 (2009) 057 [arXiv:0901.1142] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0901.1142

Web End =INSPIRE ].

[81] N. Gromov and P. Vieira, The all loop AdS4/CF T3 Bethe ansatz, http://dx.doi.org/10.1088/1126-6708/2009/01/016

Web End =JHEP 01 (2009) 016 [arXiv:0807.0777] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.0777

Web End =INSPIRE ].

[82] J. Minahan, O. Ohlsson Sax and C. Sieg, Magnon dispersion to four loops in the ABJM and ABJ models, http://dx.doi.org/10.1088/1751-8113/43/27/275402

Web End =J. Phys. A 43 (2010) 275402 [arXiv:0908.2463] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0908.2463

Web End =INSPIRE ].

[83] J. Minahan, O. Ohlsson Sax and C. Sieg, Anomalous dimensions at four loops in N = 6 superconformal Chern-Simons theories, http://dx.doi.org/10.1016/j.nuclphysb.2011.01.016

Web End =Nucl. Phys. B 846 (2011) 542 [arXiv:0912.3460] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0912.3460

Web End =INSPIRE ].

[84] M. Leoni et al., Superspace calculation of the four-loop spectrum in N = 6 supersymmetric Chern-Simons theories, http://dx.doi.org/10.1007/JHEP12(2010)074

Web End =JHEP 12 (2010) 074 [arXiv:1010.1756] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1010.1756

Web End =INSPIRE ].

[85] D. Bak, H. Min and S.-J. Rey, Integrability of N = 6 Chern-Simons theory at six loops and beyond, http://dx.doi.org/10.1103/PhysRevD.81.126004

Web End =Phys. Rev. D 81 (2010) 126004 [arXiv:0911.0689] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.0689

Web End =INSPIRE ].

20

JHEP04(2013)012