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One-Loop Corrections to Type IIA String Theory in AdS4 x CP3

We study various methods for computing the one-loop correction to the energy of classical solutions to type IIA string theory in AdS4 x CP3. This involves computing the spectrum of fluctuations and then adding up the fluctuation frequencies. We focus on two classical solutions with support in CP3: a rotating point-particle and a circular spinning string with two angular momenta equal to J. For each of these solutions, we compute the spectrum of fluctuations using two techniques, known as the algebraic curve approach and the world-sheet approach. If we use the same prescription for adding fluctuation frequencies that was used for type IIB string theory in AdS5 x S5, then we find that the world-sheet spectrum gives convergent one-loop corrections but the algebraic curve spectrum gives divergent ones. On the other hand, we find a new summation prescription which gives finite results when applied to both the algebraic curve and world-sheet spectra. Naively, this gives three predictions for the one-loop correction to the spinning string energy (one from the algebraic curve and two from the world-sheet), however we find that in the large-j limit (where j=J/sqrt(2pi^2 lambda)), the j^(-2n) terms in all three cases agree. We therefore obtain a unique prediction for the one-loop correction to the spinning string energy.


Introduction
One of the most intriguing ideas in theoretical physics is the AdS/CF T correspondence, which relates string theory or M-theory on a background geometry consisting of AdS d+1 times some compact space to a conformal field theory living in d-dimensional Minkowski space. By far the best studied example of this correspondence is the duality between type IIB superstring theory on AdS 5 × S 5 and N = 4 SYM proposed by Maldacena in [1]. A little over a year ago, Aharony, Bergman, Jafferis, and Maldacena (ABJM) discovered a new example of this duality which relates type IIA string theory on AdS 4 × CP 3 to a three dimensional N = 6 Chern-Simons theory [2]. Some basic properties of this gauge theory are reviewed in Appendix A.
Since then, much of the analysis that was done to test the AdS 5 /CF T 4 duality has been repeated for the AdS 4 /CF T 3 duality. For example, various sectors of the planar Chern-Simons theory were shown to be integrable up to four loops in perturbation theory, i.e. it was shown that the dilatation operator in these sectors corresponds to a spin-chain Hamiltonian which can be diagonalized by solving Bethe equations [3,4,5,6]. Moreover, the classical string theory dual to the planar gauge theory was also shown to be integrable, i.e. the equations of motion for the string theory sigma model were recast as a flatness condition for a certain one-form known as the Lax connection [7,8,9,10]. It should be noted that classical integrability has only been demonstrated in the subsector of the AdS 4 × CP 3 superspace described by the OSp(6|4)/(U (3) × SO(3, 1)) supercoset, and that κ-symmetry in the coset sigma model breaks down for string solutions that move purely in AdS 4 [9]. Demonstrating integrability in the full superspace requires more general methods [11]. The pure spinor string theory on AdS 4 × CP 3 was studied in [12,13]. An important consequence of the Lax connection is that any classical solution to the sigma model equations of motion can be mapped into a multi-sheeted Riemann surface known as an algebraic curve [14,15,16]. The AdS 4 /CF T 3 algebraic curve was constructed in [17].
Following these developments, a set of all-loop Bethe equations, which interpolate between the gauge theory Bethe equations at weak coupling and the string theory algebraic curve at strong coupling, were proposed in [18]. The all-loop Bethe ansatz is a powerful tool for testing the AdS/CF T correspondence.
While the AdS 4 /CF T 3 duality shares certain features with the AdS 5 /CF T 4 duality, it also exhibits several new features. For example, when one looks at quantum excitations to the string theory sigma model in the Penrose limit of type IIA string theory on AdS 4 ×CP 3 , one finds that half of the excitations are twice as massive as the other half [19,20,21]. The latter are subsequently referred to as "light" and the former are referred to as "heavy". This is in contrast to what was found when looking at the Penrose limit of type IIB string theory on AdS 5 × S 5 , where all the excitations have the same mass [22]. Various properties of the heavy and light modes were studied in [23,24,25]. Furthermore, the AdS 4 /CF T 3 magnon dispersion relation was found to be = 1 2 1 + 8h(λ) sin 2 p 2 where h(λ) = λ for λ 1 and h(λ) = 2λ 2 for λ 1. This is in contrast to the magnon dispersion relation for for all values of λ. One possible reason why the AdS 4 /CF T 3 magnon dispersion receives corrections at strong coupling is that the theory only has 3/4 maximal supersymmetry. Another consequence of the less-than-maximal supersymmetry is that the radius of AdS 4 × CP 3 varies as a function of λ, although this only becomes relevant at two loops in the sigma model [26].
Perhaps the most puzzling new feature of the AdS 4 /CF T 3 correspondence arises when computing the one-loop correction to the energy of classical solutions to type IIA string theory in AdS 4 × CP 3 . Note that the one-loop corrections we are describing correspond to quantum corrections to the world-sheet theory and α corrections to the classical string theory. In particular, several groups found a disagreement with the all-loop Bethe ansatz after computing the one-loop correction to the energy of the folded spinning string in AdS 4 ×CP 3 . In computing the one-loop correction, these groups used the same prescription for adding up fluctuation frequencies that was used in AdS 5 ×S 5 [27,28,29]. The authors of [30] subsequently proposed an alternative summation prescription which achieves agreement with the all-loop Bethe ansatz by treating the frequencies of heavy and light modes on unequal footing. This prescription is not applicable to type IIB string theory on AdS 5 × S 5 because there is no distinction between heavy and light frequencies in this theory. Hence, the prescription proposed in [30] is special to the AdS 4 /CF T 3 correspondence. Reference [31] pointed out that the discrepancy can also be resolved if one takes h(λ) = √ λ + a 1 + O 1/ √ λ with a 1 = 0 when doing world-sheet calculations. Although the algebraic curve calculation in [32] found that this correction should be zero, the authors in [31] argue that different values of a 1 can be consistent because a 1 may be scheme-dependent.
In this paper we extend the study of one-loop corrections in AdS 4 × CP 3 by computing one-loop corrections for solutions with nontrivial support in CP 3 and trivial support in AdS 4 , notably a rotating point-particle and a circular string with two equal angular momenta in CP 3 , which we refer to as the spinning string. The latter solution is the AdS 4 /CF T 3 analogue of the SU (2) circular string which was discovered in [33] and studied extensively in the AdS 5 /CF T 4 correspondence [34,35,36]. The point-particle and spinning string solutions are especially interesting to study in the AdS 4 /CF T 3 context because they avoid the κ-symmetry issues described above (since they have trivial support in AdS 4 ).
Various string solutions with support in CP 3 were also constructed in [37,38], however one-loop corrections were not considered in those papers.
In order to compute the one-loop correction to the energy of a classical solution, we must first compute the spectrum of fluctuations about the solution. This can be computed by expanding the Green-Schwarz (GS) action to quadratic order in the fluctuations and finding the normal modes of the resulting equations of motion. We refer to this method as the world-sheet (WS) approach. Alternatively, the spectrum can be computed from the algebraic curve corresponding to this solution using semi-classical techniques. We refer to this as the algebraic curve (AC) approach. This approach was developed for type IIB string theory in AdS 5 × S 5 in [39] and then adapted to type IIA string theory in AdS 4 × CP 3 in [17]. In this paper, we compute the spectrum of fluctuations about the point-particle and spinning string using both approaches and find that the algebraic curve frequencies agree with the world-sheet frequencies up to constant shifts and shifts in mode number.
Although the algebraic curve and world-sheet spectra look very similar, they have very different properties. In particular, the algebraic curve spectrum gives a divergent oneloop correction if we use the same prescription for adding up the frequencies that was used in AdS 5 × S 5 . Since the point-particle is a BPS solution we expect that its oneloop correction should vanish. Furthermore, since the spinning string solution becomes near-BPS in a certain limit, we expect its one-loop correction to be nonzero but finite.
Hence the algebraic curve does not give one-loop corrections which are compatible with supersymmetry if one uses the standard summation prescription.
We propose a new summation prescription that gives a vanishing one-loop correction for the point-particle and a finite one-loop correction for the spinning string when used with both the algebraic curve spectrum and the world-sheet spectrum. This prescription has certain similarities to the one that was proposed by Gromov and Mikhaylov in [30], however our motivation for introducing it is somewhat different. Whereas they proposed a new summation prescription in order to get a one-loop correction to the energy of the folded spinning string which agrees with the all-loop Bethe ansatz, we find that a new summation prescription is required for a much more basic reason: consistency of the algebraic curve with supersymmetry. In principle, we obtain three predictions for the one-loop correction to the spinning-string energy; one coming from the algebraic curve and two coming from the world-sheet (since the world-sheet spectrum gives finite results using both the old and new summation prescriptions). However, if we expand in the large-J limit (where and J is the spin) and evaluate the sums at each order of J using ζ-function regularization, we find that all three predictions are the same (up to so-called non-analytic and exponentially suppressed terms which are sub-dominant). In this way we get a single prediction for the one-loop correction to the spinning string energy. Furthermore, we show that this result is consistent with the predictions of the Bethe ansatz.
The structure of this paper is as follows. In section 2, we review the world-sheet approach, the algebraic curve approach, and summation prescriptions. It should be noted that our versions of the world-sheet and algebraic curve formalisms have some new features. In particular, we recast the quadratic GS action in the AdS 4 × CP 3 supergravity background in a way that removes half of the fermionic degrees of freedom explicitly and we reformulate the algebraic curve approach using off-shell techniques which make calculations much more efficient. In section 3.1, we present the classical solution for a point-particle rotating in CP 3 and describe the gauge theory operator dual to this solution. In the rest of section 3 we summarize the fluctuation frequencies for the point-particle solution and compute the oneloop correction using the standard summation prescription used in AdS 5 × S 5 as well as our new summation prescription. * In section 4.1, we present the classical solution for a spinning string with two equal angular momenta in CP 3 and propose the gauge theory operator dual to this solution. In the rest of section 4 we summarize the fluctuation frequencies for the spinning-string solution, analyze various properties of the one-loop correction to its energy, and make a prediction for the anomalous dimension of its dual gauge theory operator. † Section 5 presents our conclusions. Appendix A reviews some basic properties of the dual gauge theory. Appendices B and C review the geometry of AdS 4 × CP 3 as well as our Dirac matrix conventions, and the remaining appendices provide detailed derivations of the fluctuation frequencies for the point-particle and spinning string using both the algebraic curve approach and the world-sheet approach. In Appendix H, we use the Bethe ansatz to compute the leading two contributions to the anomalous dimension of operator dual to the spinning and verify that they agree with the prediction we obtain using string theory. * Although the spectrum of the point-particle was already computed using the algebraic curve in [17], we present it again using more efficient techniques. † The authors in [37] made similar conjectures for the gauge theory operators dual to the point-particle and spinning string, however the classical solutions considered in that paper have different charges than the ones constructed in this paper.

World-Sheet Formalism
The world-sheet approach for computing the spectrum of fluctuations about a classical solution in AdS 5 × S 5 was developed in [40]. In this section we review how to compute the spectrum of fluctuations around a classical solution to type IIA string theory in a supergravity background which consists of the following string frame metric, dilaton, and Ramond-Ramond forms [2]: where R 2 is the radius of curvature in string units, J is the Kähler form on CP 3 , and k is an integer corresponding to the level of the dual Chern-Simons theory. Note that the AdS 4 space has radius R/2 while the CP 3 space has radius R. The metric for a unit AdS 4 space given by and the metric for a unit CP 3 space is given by where 0 ≤ ξ < π/2, 0 ≤ ψ < 2π, 0 ≤ θ i ≤ π, and 0 ≤ ϕ i < 2π. More details about the geometry of AdS 4 × CP 3 are given in Appendix B.
Using the metric in Eq. (1), the bosonic part of the string Lagrangian in conformal gauge is given by where a, b = τ, σ are world-sheet indices, η ab = diag [−1, 1], and we have set α = 1.
Because AdS 4 has two Killing vectors and CP 3 has three Killing vectors, any solution to the bosonic equations of motion has at least five conserved charges. In particular, the two AdS 4 charges are given by and the three CP 3 charges are given by where E is the energy and S, J ψ , J φ 1 , and J φ 2 are angular momenta.
A solution to the bosonic equations of motion is said to be a classical solution if it also satisfies the Virasoro constraints Note that these are the only constraints that relate motion in AdS 4 to motion in CP 3 .
The spectrum of bosonic fluctuations around a classical solution can be computed by expanding the bosonic Lagrangian in Eq. (4) to quadratic order in the fluctuations and finding the normal modes of the resulting equations of motion.
In the examples we consider, we find that two of the bosonic modes are massless and the other eight are massive. While the eight massive modes correspond to the physical transverse degrees of freedom, the two massless modes can be discarded. One way to see that the massless modes can be discarded is by expanding the Virasoro constraints to linear order in the fluctuations [40].
To compute the spectrum of fermionic fluctuations, we only need the quadratic part of the fermionic GS action for type IIA string theory. This action describes two 10-dimensional Majorana-Weyl spinors of opposite chirality which can be combined into a single non-chiral Majorana spinor Θ. The quadratic GS action for type IIA string theory in a general background can be found in [41]. For the supergravity background in Eq. (1), the quadratic Lagrangian for the fermions is given by Note that M is a base-space index while A, B = 0, ..., 9 are tangentspace indices. Explicit formulas for e A M , ω AB M , Γ · F 2 , and Γ · F 4 are provided in Appendix B. Explicit formulas for the Dirac matrices are provided in Appendix C.
We will now recast the fermionic Lagrangian in Eq. (9) in form that allows us to compute the fermionic fluctuation frequencies in a straightforward way. First we note that after rearranging terms, Eq. (9) can be written as where we define K = ∂ τ X M e 0 M , Θ + = P + Θ, and Note that P + = P † + and if the classical solution satisfies then P + is a projection operator, i.e. P 2 + = P + . In addition, if the classical solution satisfies then the fermionic Lagrangian simplifies to Finally, if we consider the Fourier mode Θ (σ, τ ) =Θ exp (−iωτ + inσ), whereΘ is a constant spinor, then the equations of motion for the fermionic fluctuations are given by One can choose a basis where P + has the form Only half of the fermionic components appear in the Lagrangian in Eq. (15). Hence, a natural choice for fixing kappa-symmetry is to set the other components to zero by imposing the gauge condition Θ = Θ + . This gives the desired number of fermionic degrees of freedom.
In particular, before imposing the Majorana condition, Θ has 32 complex degrees of freedom.

Algebraic Curve Formalism
The procedure for computing the spectrum of excitations about a classical string solution using the AdS 4 /CF T 3 algebraic curve was first presented in [17]. In this section, we reformulate this procedure in terms of an off-shell formalism similar to the one that was developed for the AdS 5 /CF T 4 algebraic curve in [42]. The off-shell formalism makes things much more efficient. First we describe how to construct the classical algebraic curve.
Then we describe how to semi-classically quantize the curve and obtain the spectrum of excitations.
This algebraic curve corresponds to the fundamental representation of OSp(6|4), which is ten-dimensional. Furthermore, the quasimomenta are not all independent. In particular where x is a complex number called the spectral parameter. To compute the quasimomenta, it is useful to parameterize AdS 4 and CP 3 using the following embedding coordinates where λ ∈ R. A classical solution in the global coordinates of Eqs. (2) and (3) can be converted to embedding coordinates using Eqs. (57) and (62) provided in Appendix B. One can then compute the following connection: j a (τ, σ) = 2 n i ∂ a n j − n j ∂ a n i 0 0 where a ∈ {τ, σ}, D a = ∂ a + iA a , and A a = i 4 I=1 z † I ∂ a z I [17]. This connection is a 9 × 9 matrix and transforms under the bosonic part of the supergroup OSp(6|4), notably (4). A key property is that it is flat, which allows us to construct the following monodromy matrix: where P is the path-ordering symbol and the integral is over a loop of constant world-sheet time τ . It can be shown that the eigenvalues of Λ(x) are independent of τ .
The quasimomenta are related to the eigenvalues of the monodromy matrix. In particular, if we diagonalize the monodromy matrix we will find that the eigenvalues of the AdS 4 part are in general given by wherep 1 (x) +p 4 (x) =p 2 (x) +p 3 (x) = 0, while the eigenvalues from the CP 3 part are given by where 4 i=1p i (x) = 0. The classical quasimomenta are then defined as where we have suppressed the x-dependence. From this formula, we see that q 1 (x) and q 2 (x) correspond to the AdS 4 part of the algebraic curve, while q 3 (x), q 4 (x), and q 5 (x) correspond to the CP 3 part of the algebraic curve.

Semi-Classical Quantization
The algebraic curve will generically have cuts connecting several pairs of sheets. These cuts encode the classical physics. To perform semiclassical quantization, we add poles to the algebraic curve which correspond to quantum fluctuations. Each pole connects two sheets.
In particular the bosonic fluctuations connect two AdS sheets or two CP 3 sheets and the fermionic fluctuations connect an AdS sheet to a CP 3 sheet. See  Table 1.
Notice that every fluctuation can be labeled by two equivalent polarizations because every pole connects two equivalent pairs of sheets as a consequence of Eq. (17). Fluctuations connecting sheet 5 or 6 to any other sheet are defined to be light. Notice that there are eight light excitations. All the others are defined to be heavy excitations. The physical significance of this terminology will become clear later on. When we compute the spectrum of fluctuations about the point particle in section 3 for example, we will find that the heavy excitations are twice as massive as the light excitations. When adding poles, we must take into account the level-matching condition where N ij n is the number of excitations with polarization ij and mode number n. Furthermore, the locations of the poles are not arbitrary; they are determined by the following equation: where x ij n is the location of a pole corresponding to a fluctuation with polarization ij and mode number n.
In addition to adding poles to the algebraic curve, we must also add fluctuations to the classical quasimomenta. These fluctuations will depend on the spectral parameter x as well as the locations of the poles, which we will denote by the collective coordinate y. The functional form of the fluctuations is determined by some general constraints: Table 1: Labels for the fluctuations (heavy,light) of the AdS 4 × CP 3 algebraic curve.
The coefficients of the other terms on the right hand side of equation Eq. (27) can be determined using arguments similar to those in [39].
• Finally, when the spectral parameter approaches the location of one of the poles, the fluctuations have the following form: where the proportionality constants can be read off from the coefficient of N ij in the k'th row of Eq. (27).
After computing the anomalous part of the energy shift, the fluctuation frequency is given by It is useful to consider the fluctuation frequency without fixing the value of y. In this case, the fluctuation frequency is said to be off-shell.
Using arguments similar to those in [42], we find all the relations among the off-shell frequencies. First, all the light off-shell frequencies are related by where i = 1, 2, 3, 4.
Second, all the heavy off-shell frequencies can be written as the sum of two light off-shell frequencies as summarized in table 2.
Finally, any off-shell frequency Ω ij is related to its mirror off-shell frequency Ω ij by where C = 1, 1/2, or 0 for AdS, Fermionic, or CP 3 polarizations respectively. The mirror polarization i, j of the polarization (i, j) can be readily found using Eq. (26), e.g. 1, 10 = Table 2: Relations between heavy and light off-shell frequencies.
Heavy Light AdS Fermions , etc. Using these relations, only two of the eight light off-shell frequencies are independent. For example, In conclusion, if we compute the off-shell frequencies Ω 15 and Ω 45 , then we can determine all the other off-shell frequencies automatically from the relations in Eqs. (30,31) and table   2.
The on-shell frequencies are then obtained by evaluating the off-shell frequencies at the location of the poles which are determined by solving Eq. (24), i.e. ω ij n = Ω ij x ij n . It will be convenient to organize them into the following linear combinations: where L stands for light and H stands for heavy. It should be noted that heavy and light frequencies are not as well-defined in the world-sheet approach. In general, the only way to identify heavy and light frequencies in the world-sheet approach is by comparing the world-sheet spectrum to the algebraic curve spectrum, i.e. a world-sheet frequency is said to be heavy/light if the corresponding algebraic curve frequency is heavy/light.

Summation Prescriptions
Given the spectrum of fluctuations about a classical string solution, we compute the oneloop correction to the string energy by adding up the spectrum. The standard formula is where κ is proportional to the classical energy (the exact formula is given in sections 3 and 4), B/F stands for bosonic/fermionic, n is the mode number, and i is some label.
For example, if we are dealing with frequencies computed from the algebraic curve, then where ω L n /ω H n are referred to as heavy/light frequencies and are defined in Eqs. (32) and (33). For later convenience, we note that Eq. (34) can be written in terms of heavy and light frequencies as follows: In the large-κ limit, Eq. (35) can be approximated as the following integral: In [30] it was shown that Eq. (37) gives a one-loop correction which agrees with the all-loop Bethe ansatz when applied to the spectrum of the folded spinning string.
In this paper we propose a new summation prescription: This sum can be motivated physically using the observation in [30] that heavy modes with mode number 2n can be thought of as bound states of two light modes with mode number n. This suggests that only heavy modes with even mode number should contribute to the one-loop correction. The formula for the one-loop correction should therefore have the form The coefficients A and B can then be fixed uniquely by requiring that the integral approximation to this formula reduces to Eq. (37) in the large-κ limit, ensuring that this summation prescription gives a one-loop correction to the folded spinning string energy which agrees with the all-loop Bethe ansatz. One then finds that A = 2 and B = 1. 3 Point-Particle

Classical Solution and Dual Operator
In terms of the coordinates of Eqs. (2) and (3), the solution for a point-particle rotating with angular momentum J in CP 3 is given by where J = J 4πg and g = λ/8. This version of the solution will be useful for doing calculations in the world-sheet formalism. Alternatively, we can write this solution in embedding coordinates by plugging Eq. (39) into Eqs. (57) and (62): This version of the solution will be useful for doing calculations in the algebraic curve formalism. The energy and angular momenta of the particle can be read off from Eqs. (5- Furthermore, the Virasoro constraints in Eq. (8) give κ = J , or equivalently E = J. Note that this is a BPS condition. We therefore expect that the dimension of the dual gauge theory operator should be protected by supersymmetry.
The gauge theory operator dual to the point-particle rotating in CP 3 should have the This can be understood heuristically by associating the scalars Z 1 ,Z 2 , Z 3 , Z 4 with the embedding coordinates z 1 , z 2 , z 3 , z 4 and noting that Since the transformation on the right hand side is an SU (4) transformation, the solution Furthermore, the engineering dimension of this operator is J, which matches the energy of the point-particle solution, and the two-loop dilatation operator in Eq. (54) vanishes when applied to this operator, which is consistent with our expectation that the anomalous dimension of the operator should vanish.

Excitation Spectrum
We summarize the spectrum of fluctuations obtained with the algebraic curve and the world sheet in Table 3. The algebraic curve frequencies have been re-scaled by a factor of κ in order to compare them to the world-sheet frequencies. The derivations of these frequencies are described in Appendices D and E. Note that the fluctuations in this table are labeled by polarizations. Although this notation was only defined for the algebraic curve formalism, we find that the world-sheet frequencies match the algebraic curve frequencies up to constant shifts, so it is convenient to label the world-sheet frequencies with polarizations as well.
Also note that both sets of frequencies agree with the spectrum of fluctuations that were found in the Penrose limit (up to constant shifts) [19,20,21].
While the constant shifts in the world-sheet spectrum occur with opposite signs and can be removed by gauge transformations, this is not the case for the algebraic curve Table 3: Spectrum of fluctuations about the point-particle solution computed using the world-sheet (WS) and algebraic curve (AC) formalisms (ω n = √ n 2 + κ 2 ). Polarizations (heavy/light) indicate which pairs of sheets are connected by a fluctuation in the AC formalism, and ± indicates that half of the frequencies have a + and the other half have a −.
In fact, the constant shifts in the algebraic curve frequencies have physical significance, which can be seen by taking the mode number n = 0. In this limit, the AdS frequencies reduce to κ, the CP 3 frequencies reduce to 0, and the fermi frequencies reduce to κ/2. In this sense, the n = 0 algebraic curve frequencies have "flat-space" behavior.
This property was also observed for algebraic curve frequencies computed about solutions in AdS 5 ×S 5 [39]. On the other hand, the world-sheet frequencies do not have this property.
In the next subsection, we will see that the constant shifts in the algebraic curve spectrum have important implications for the one-loop correction to the classical energy.

One-Loop Correction to Energy
Using Eqs. (32) and (33) we see that ω H and ω L are constants for both the world-sheet and algebraic curve spectra. In particular, for the world-sheet spectrum we find that ω H (n) = Thus we find that both summation prescriptions are consistent with supersymmetry if we use the spectrum computed from the world-sheet, but only the new summation is consistent with supersymmetry if we use the spectrum computed from the algebraic curve.

Classical Solution and Dual Operator
In the global coordinates of Eqs. (2) and (3), the solution for a circular spinning string with two equal nonzero spins in CP 3 is where J = J 4πg and m is the winding number. Using Eqs. (57) and (62), we can also write this solution in embedding coordinates (which are useful for doing algebraic curve calculations): Equations (5-7) imply that E = 4πgκ, S = 0, J ψ = 0, and J φ 1 = J φ 2 = J. Furthermore, the Virasoro constraints in Eq. (8) give κ = √ m 2 + 4J 2 , or equivalently E = 2J 1 + π 2 m 2 λ 2J 2 . In the limit J m, this reduces to the BPS condition E = 2J, so we expect that the dual operator should have engineering dimension 2J and a finite but non-zero anomalous dimension. Furthermore, the dispersion relation has a BMN expansion in the parameter λ/J 2 , which allows us to make a prediction for anomalous dimension of the dual operator.
Expanding the dispersion relation to first order in the BMN parameter gives To extrapolate this formula to the gauge theory, we must make the replacement λ → 2λ 2 .
One way to understand this replacement is by comparing the magnon dispersion relation at strong and weak t'Hooft coupling, as explained in the introduction. We therefore get the following prediction for the anomalous dimension of the dual gauge theory operator The higher order terms in the expansion of the classical string energy in Eq. (43) correspond to O λ 4 /J 3 corrections to the anomalous dimension, but the one-loop correction to the energy provides O λ 2 /J 2 corrections to the anomalous dimension (see Eq. 53).
The dual gauge theory operator should have the form where the dots stand for permutations of Z 1 Z † 2 and Z 3 Z † 4 . Note that the engineering dimension of the operator is 2J, as expected. When we apply the two loop dilatation operator in Eq. (54) to the operator in Eq. (45), it reduces to This is the Hamiltonian for two identical Heisenberg spin chains; one located on the even sites and the other on the odd sites. If one thinks of Z 1 and Z † 2 as being up spins and Z 3 and Z † 4 as being down spins, then each spin chain has J up spins and J down spins. In appendix H, we use the Bethe ansatz to show that the anomalous dimension is indeed given by Eq. (44).

Excitation Spectrum
We summarize the spectrum of fluctuations about the spinning string in Tables 4 and 5.
The algebraic curve frequencies have been re-scaled by a factor of κ in order to compare them to the world-sheet frequencies. The derivations are presented in Appendices F and G.
We find that the algebraic curve spectrum matches the world-sheet spectrum up to constant shifts and shifts in mode number. Furthermore, if we set the winding number m = 0 and take J → J /2, we find that all the frequencies in Table 4 reduce to the corresponding frequencies in Table 3, which is expected since setting the winding number to zero reduces the string to a point-particle. ‡ This is an important check of our results for the spinning string. On the other hand, if we set the mode number n = 0, we find that the algebraic curve frequencies once again have flat-space behavior, i.e., the AdS frequencies reduce to κ, the CP 3 frequencies reduce to 0, and the fermi frequencies reduce to κ/2. ‡ In showing that the (3,5), (3,6), (4,5), and (4,6) frequencies in Table 4 reduce to those in Table 3, the identity in Eq. (80) is useful. Table 4: Spectrum of fluctuations about the spinning string solution computed using the world-sheet (WS) and algebraic curve (AC) formalisms. The notation for the frequencies is given in Table 5. Polarizations (heavy/light) indicate which pairs of sheets are connected by a fluctuation in the AC formalism.
eigenmodes notation Finally, we would like to point out that both the algebraic curve and world-sheet spectra have instabilities when |m| ≥ 2. For example if we set m = 2, then the algebraic curve frequencies labeled by (3,5) and (3,6) become imaginary for n = −3 and n = −1 and the corresponding world-sheet frequencies become imaginary for n = ±1. § § We would like to thank Victor Mikhaylov for showing us his unpublished notes on the spinning string algebraic curve [43]. In these notes, he also derives the algebraic curve for the spinning string and uses it to compute the fluctuation frequencies, however the asymptotics that he imposes on the algebraic curve are different than the asymptotics we use in Eq. (27). The differences occur in the signs of several terms on the right hand side of Eq. (27). As a result, we obtain frequencies with different constant shifts.

One-Loop Correction to the Energy
For the spinning string, ω H (n) and ω L (n) defined in Eqs. (32) and (33) are nontrivial: where WS stands for world-sheet, AC stands for algebraic curve, and we used the notation in Table 5.
To compute the one-loop correction, we must evaluate an infinite sum of the form : where old/new refers to the summation prescription.
To gain further insight, let's look at the summands in Eq. (48) in two limits: the largen limit and the large-J limit. By looking at the large-n limit, we will learn about the convergence properties of the one-loop corrections, and by looking at the large-J limit and evaluating the sums over n using ζ-function regularization, we will be able to compute the J −2n contributions to the one-loop corrections. These are referred to as the analytic terms. In general there can also be terms proportional to J −2n+1 , which are referred to as the non-analytic terms, and exponentially suppressed terms, i.e. terms that scale like e −J .
These terms are sub-dominant compared to the analytic terms in the large-J limit. Table 6: Large-n limit of Ω (J , n, m) + Ω (J , n, −m) for the old summation (where Ω (J , n, m) = ω H (n) + ω L (n)) and the new summation (where Ω (J , n, m) = 2ω H (2n) + ω L (n)) applied to the world-sheet (WS) spectrum and algebraic curve (AC) spectrum.
The large-n limit of Ω (J , n, m) + Ω (J , n, −m) for the four choices of Ω (J , n, m) is summarized in Table 6.
From this table we see that all one-loop corrections are free of quadratic and logarithmic divergences because terms of order n and order 1/n cancel out in the large-n limit. At the same time, we find a linear divergence when we apply the old summation prescription to the algebraic curve spectrum since the summand has a constant term. In all other cases however, the summands are at most O(n −3 ), which suggests that the one-loop corrections are convergent. Hence we find that both summation prescriptions give finite one-loop corrections when applied to the world-sheet spectrum, but only the new summation prescription gives a finite result when applied to the algebraic curve spectrum. This is the same thing we found for the point-particle. The new feature of the spinning string is that the one-loop correction is nonzero and therefore provides a nontrivial prediction to be compared with the dual gauge theory.

Large-J limit
In the previous section we found that when Ω (J , n, m) = Ω old,AC (J , n, m), the one-loop correction is divergent but for the other three cases in Eq. (48), it is convergent. This means we have three possible predictions for the one-loop correction, however by expanding the summands in the large-J limit and evaluating the sums over n at each order of J using ζ-function regularization, we find that all three cases give the same result. The technique of ζ-function regularization is convenient for computing the analytic terms in the large-J expansion of one-loop corrections but does not capture non-analytic and exponentially suppressed terms [44]. We now describe this procedure in more detail.
If we expand in the summand in the large-J limit, only even powers of J appear: For each power of J , the sum over n can be written as follows (Ω k (n, m) + Ω k (n, −m)) .
If we expand Ω k (n, m) in the limit n → ∞, we find that it splits into two pieces: whereΩ k (n, m) is O n −2 . We will refer toΩ k (n, m) as the finite piece because it converges when summed over n, and 2k j=−1 c k,j (m)n j as the divergent piece because it diverges when summed over n. Furthermore, we find thatΩ k (n, m) =Ω k (n, −m) and c k,j (m) ∝ m 2k−j .
Hence, the odd powers of n cancel out of the divergent piece when we add Ω k (n, m) to Ω k (n, −m) and we get Noting that ζ(0) = −1/2 and ζ (2j) = 0 for j > 0, we see that only the constant term in the divergent piece contributes if we evaluate the sum over n using ζ-function regularization: (Ω k (n, m) + Ω k (n, −m)) → −c k,0 + 2 ∞ n=1Ω k (n, m).
Using the procedure described above, we obtain a single prediction for the one-loop correction to the energy of the spinning string: In showing that Eq. (47) where a = m 2 /4 + ∞ n=1 n n 2 − m 2 − n + m 2 /2 .
Note that the first term in Eq. (53) came from expanding the classical dispersion relation for the spinning string to first order in the BMN parameter λ/J 2 and then making the replacement λ → 2λ 2 . We verify Eq. (53) from the gauge theory side in Appendix H.

Conclusions
In this paper, we study various methods for computing one-loop corrections to the energies of classical solutions to type IIA string theory in AdS 4 × CP 3 . Previous studies which computed the one-loop correction to the folded spinning string in AdS 4 found that agreement with the all-loop AdS 4 /CF T 3 Bethe ansatz is not achieved using the standard summation prescription that was used for type IIB string theory in AdS 5 × S 5 . Rather, a new summation prescription seems to be required which distinguishes between so-called light modes and heavy modes. We extend this investigation by analyzing the one-loop correction to the energy of a point-particle and a circular spinning string, both of which are located at the spatial origin of AdS 4 and have nontrivial support in CP 3 . The spinning string considered in this paper has two equal nonzero spins in CP 3 and is the analogue of the SU For the point-particle, we find that the world-sheet and algebraic curve fluctuation frequencies match the spectrum of fluctuations obtained in the Penrose limit up to constant shifts.
Furthermore, for the spinning string we find that the algebraic curve spectrum matches the world-sheet spectrum up to constant shifts and shifts in mode number. In particular, the AC and WS frequencies for the spinning string both reduce to the corresponding pointparticle frequencies when the winding number is set to zero and become unstable when the winding number |m| ≥ 2. This is familiar from the SU (2) spinning string in AdS 5 × S 5 [33,45] which has instabilities for |m| ≥ 1.
Although the algebraic curve spectrum looks very similar to the world-sheet spectrum, it exhibits some important differences. For example, we find that the algebraic curve frequencies have flat-space behavior when the mode number n = 0. This was also found for algebraic curve frequencies in AdS 5 × S 5 . More importantly, if we compute one-loop corrections by adding up the algebraic curve frequencies using the standard summation prescription that was used in AdS 5 × S 5 , then we get a linear divergence. This is inconsistent with supersymmetry because we expect the one-loop correction to vanish for the point-particle and to be nonzero but finite for the spinning string. We propose a new summation prescription in Eq. (38) which gives precisely these results when applied both to the algebraic curve spectrum and the world-sheet spectrum. This summation prescription has certain similarities to the one that was proposed in [30]. In particular, it also gives a oneloop correction to the folded spinning string which agrees with the all-loop Bethe ansatz.
At the same time, it has some important differences which are described in section 2.3. For example, we find that our summation prescription generally gives more well-defined results for one-loop corrections.
In principle we can get three predictions for the one-loop correction to the spinning string (one coming from the algebraic curve and two coming from the world-sheet since the world-sheet gives finite results using both the old summation prescription in Eq. (36) and the new summation prescription in Eq. (38)), but by expanding the one-loop corrections in the large-J limit (where J = J √ 2λπ and J is the spin) and evaluating the sum at each order in J using ζ-function regularization, we find that all three cases actually give the same result. This is very nontrivial considering that our new summation prescription looks very different than the old one. Furthermore, we show that this result agrees with the predictions of the Bethe ansatz. Thus, while the old summation prescription only seems to work when applied to the world-sheet frequencies of solutions with trivial support in AdS 4 , our summation prescription works more generally. Fully understanding why the old summation prescription breaks down for solutions with nontrivial support in AdS 4 warrants further study. It would be useful to confirm our results using methods more rigorous than ζ-function regularization. This can be done using the contour integral techniques developed in [46], which can also be used to compute 1/J 2n+1 and exponentially suppressed terms in the large-J expansion of the one-loop corrections. It would also be interesting to evaluate the one-loop correction to the spinning string energy in a way that does not rely on summation prescriptions. The basic idea would be to identify the one-loop correction with a normal ordering constant which can be then determined by demanding that the quantum generators of certain symmetries preserved by the classical solution have the right algebra. Something along these lines was done for the type IIB superstring in plane-wave background in [47].
Ultimately, fully understanding how to compute one-loop corrections to type IIA string theory in AdS 4 × CP 3 may lead to important tests of the AdS 4 /CF T 3 correspondence. In particular, we would like to thank VM for sharing his unpublished notes on the spinning string algebraic curve and its semi-classical quantization, SSN for helping us with various calculations in this paper and for providing many useful explanations, and JHS for his guidance and many useful comments.

A Review of ABJM
The ABJM theory is a three-dimensional superconformal Chern-Simons gauge theory with N = 6 supersymmetry. The bosonic field content consists of four complex scalars Z 1 ,Z 2 , The kinetic and Chern-Simons terms for these fields are For k N k 5 , the theory is conjectured to be dual to type IIA string theory on

For operators of the form
the two-loop dilatation operator is given by where λ = N/k, P is the permutation operator, and T is the trace operator [3]. Note that the indices are periodic, i.e. 2J + 1 ∼ 1 and 2J + 2 ∼ 2.
The embedding or homogeneous coordinates (z I ∈ C) are defined by where λ ∈ R. The embedding coordinates are related to the global coordinates by Note that the metric in Eq. 59 can be written in terms of embedding coordinates as follows:

B.3 Fluxes
Using the vielbein, we can convert between base-space and tangent-space coordinates. In particular, by writing the four-form field strength in Eq. (1c) in tangent-space coordinates, one finds that where 0123 = 1 and all other non-zero components are related by antisymmetry. Furthermore, if one takes the exterior derivative of Eq. (60), plugs this into Eq. (1d), and converts to tangent-space coordinates, one finds that where 45 = 67 = 89 = 1 and all other non-zero components are related by anti-symmetry.

C Dirac Matrices
We use the following representation of the 10d Dirac matrices ( Γ A , Γ B = 2η AB ): where I is the 2 × 2 identity matrix, the γ s are 4d Dirac matrices given by and the Pauli matrices are given by Finally, we define the 10d chirality operator as D Point-Particle Spectrum from the World-Sheet

D.1 Bosonic Spectrum
To compute the spectrum of bosonic fluctuations about the point-particle, first add fluctuations to the classical solution in Eq. (39): where i = 1, 2, 3 and j = 1, 2. Expanding the bosonic Lagrangian in Eq. (4) to quadratic order gives We immediately see that the fluctuations δt and δψ are massless, while δη i and δξ have mass κ. If we consider Fourier modes of the form f (τ, σ) =f e i(ωτ +nσ) , then the equations of motion for the remaining fields reduce to The dispersion relations for the normal modes of this system are obtained by taking the determinant of the matrix on the left-hand side, setting it to zero, and solving for ω. The positive solutions are Each of these solutions has multiplicity two, giving a total of four positive solutions.
In summary, we find that there are eight massive modes and two massless modes. Three of the massive modes come from AdS 4 . Their dispersion relations are given by The remaining five massive modes come from CP 3 . One of them has the dispersion relation Noting that (∂δt) 2 − (∂δψ) 2 = ∂ (δt − δψ) ∂ (δt + δψ), we see that the equation above implies that all terms in the action involving δt and δψ vanish.

D.2 Fermionic Spectrum
In order to compute the spectrum of fermionic fluctuations about the point-particle solution given by Eq. (39), we only need to know the pullback of the vielbein and the spin connection in the background of this classical solution. These are given by and Plugging these expressions into Eq. (11) gives where we used K = ∂ τ X µ e 0 µ = (R/2)J . It is straightforward to check that Eqs. (13,14) are satisfied for the point-particle solution. Therefore, by plugging Eqs. (65,68,69) into Eq. (16) and using the Dirac matrices in appendix B, we obtain an explicit form of the equation of motion for the fermionic fluctuations. The frequencies are then determined using the procedure described at the end of section 2.1. In particular, the positive fermionic frequencies are given by where ω 2 and ω 3 have multiplicity two while ω 1 has multiplicity four, for a total of 8 fermionic frequencies.
E Point-Particle Algebraic Curve

E.1 Classical Quasimomenta
In this section, we compute the algebraic curve for the classical solution given in Eq. (40).
First we plug this solution into Eq. (18): Note that this connection is independent of σ, so it is trivial to compute the monodromy matrix in Eq. (19) since path ordering is not an issue. Diagonalizing the monodromy matrix and comparing the eigenvalues to Eqs. (20) and (21) then givesp 1 = −p 4 = 4πκx Recalling that κ = J and plugging these results into Eq. (22), we find that the classical quasimomenta are The algebraic curve corresponding to these quasimomenta is depicted in Fig. (2). Note that all sheets except those corresponding to q 5 and q 6 have poles at x = ±1.

E.2 Off-shell Frequencies
Recall from Eqs. (30,31) and table 2 that if we know the off-shell frequencies Ω 15 (y) and where α(x) is defined in Eq. (28), ± stands for the sum over the positive and negative mode number, and y is a collective coordinate for the positions of the two poles x 15 ±n . We have not made an ansatz for δq 3 and δq 4 because they are not needed to compute Ω 15 (y).
Notice that this ansatz satisfies the inversion symmetry in Eq. (26) and has pole structure in agreement with Eq. (28). In the large-x limit, the fluctuations reduce to where we neglect O x −2 terms. Comparing these expressions to Eq. (27) implies that the anomalous energy shift is given by The off-shell fluctuation frequency is then obtained by plugging this into Eq. (29) and recalling that the (1,5) fluctuation is fermionic: This implies that the off-shell frequency for a single fluctuation between q 1 and q 5 is given by Ω 15 (y) = 1 2 Now let's compute Ω 45 (y). Once again, let's suppose that we have two fluctuations between q 4 and q 5 which have opposite mode numbers ±n. We make the following ansatz for the fluctuations: where α ± (y) are some functions to be determined. Note that this ansatz satisfies the inversion symmetry in Eq. (26) and has pole structure in agreement with Eq. (28). Taking the large-x limit gives Comparing these limits with Eq. (27) implies that Furthermore, the residues of the poles at x = ±1 must be synchronized according to Eq. (25). For example, if we equate the residues of δq 1 and δq 4 near x = +1 we find that Combining this with the Eq. (71) implies that At this point it is useful to recall that x 45 n is a root of the following equation (which comes from plugging Eq. (70) into Eq. (24)): Note that this equation has two roots. The convention that we will follow is to assign the pole to the root with larger magnitude. Hence, if n < 0 then x 45 n = J n − 1 + J 2 n 2 and if n > 0 then x 45 n = J n + 1 + J 2 n 2 . The point to take away from this discussion is that Using this fact, Eq. (72) can be written as follows: The off-shell fluctuation frequency is then obtained by plugging this into Eq. (29) and recalling that the (4, 5) fluctuation is a CP 3 fluctuation: (1, 10); (2,9); (1,9) Fermions y 2 +3 2(y 2 −1) y 2 +1 2(y 2 −1) It follows that the off-shell frequency for a single fluctuation between q 4 and q 5 is given by The remaining off-shell frequencies are now easily computed from Eqs. (30,31) and table 2. We summarize the off-shell frequencies in Table 7.

E.3 On-shell Frequencies
To compute the on-shell frequencies, we must compute the locations of the poles by solving Eq. (24). Recall that fluctuations that connect q 5 or q 6 to any other sheets are referred to as light, and all the others are referred to as heavy. A little thought shows that for light fluctuations, Eq. (24) reduces to J x n x 2 n − 1 = n, and for heavy fluctuations it reduces to Each of these equations admits two solutions. We will assign the location of the pole to the solution with greater magnitude. Assuming n > 0, the location of the pole for light excitations is then given by and the location of the pole for heavy excitations is given by Plugging these solutions into the off-shell frequencies in Table 7 readily gives the on-shell algebraic curve frequencies in Table 3.
F Spinning String Spectrum from the World-Sheet where i = 1, 2, 3, j = 1, 2, and κ = √ 4J 2 + m 2 . Expanding the bosonic Lagrangian in Eq. (4) to quadratic order in the fluctuations gives Note that the AdS 4 fluctuations are the same as those of the pointparticle. In particular, we see that δt is massless and δη i have mass κ. If we consider Fourier modes of the form f (τ, σ) =f e i(ωτ +nσ) , then the equations of motion for the CP 3 fluctuations reduce to Because the matrix in Eq. (73) is block diagonal, the fluctuations δ ξ , δ φ − , δ θ + and δ ψ , δ φ + , δ θ − are decoupled. The frequencies are determined by taking the determinant of the matrix and finding its roots. The equation we must solve is This polynomial has 12 roots, which come in opposite signs. Of the six positive roots, three correspond to the fluctuations δ ξ , δ φ − , δ θ + : and three correspond to the fluctuations δ ψ , δ φ + , δ θ − : Note that the solution ω = |n| corresponds to a massless mode, which can be discarded along with the other massless mode δt. The remaining eight modes are massive and correspond to the transverse degrees of freedom.

F.2 Fermionic Spectrum
In this section we compute the spectrum of fermionic fluctuations about the spinning string solution in Eq. (41). The pullback of the vielbein and the spin connection in the background of this classical solution are given by and Plugging these expressions into Eq. (11) then gives where we used K = ∂ τ X µ e 0 µ = (R/2)κ. It is straightforward to check that Eqs. (13,14) are satisfied for the spinning string solution. Therefore, by plugging Eqs. (65,75,76) into Eq. (16) and using the Dirac matrices in appendix B, we obtain an explicit form of the equation of motion for the fermionic fluctuations. The frequencies are then determined using the procedure described at the end of section 2.1. In this way, the positive fermionic frequencies are given by where ω 1 and ω 2 have multiplicity two while ω 3 has multiplicity four, for a total of 8 fermionic frequencies. In obtaining these expressions, Eq. 80 is useful.

G.1 Classical Quasimomenta
Since the spinning string has the same motion in AdS 4 as the point particle, the AdS 4 quasimomenta have the same structure and are given by where κ = √ 4J 2 + m 2 for the spinning string. Therefore, we just have to find the CP 3 quasimomenta. For the classical solution in Eq. (42), one finds that the connection in Eq. (18) is given by Using Eq. (19), the CP 3 part of the monodromy matrix is given by and we set τ = 0 since the eigenvalues of Λ(x) are independent of τ . At this point, it is useful to observe that under a gauge transformation of the form J(σ, x) → g −1 (σ)J(σ, x)g(σ)− g −1 (σ)∂ σ g(σ), the monodromy matrix transforms as Λ(x) → g −1 (0)Λ (x) g(2π). If g(0) = ±g(2π), then the eigenvalues of Λ(x) are gauge invariant up to a sign. Furthermore, if we can choose g(σ) such that the σ-dependence of J(σ, x) is removed, then the monodromy matrix would be trivial to evaluate since path-ordering would not be an issue. This can be accomplished using the gauge transformation g (σ) = diag(e −imσ/2 , e imσ/2 , e −imσ/2 , e imσ/2 ) [43].
Under this transformation, J(σ, x) becomes When m is odd, g(0) = −g(2π) so we must supplement this gauge transformation with Diagonalizing Λ(x) and comparing to Eq. (21) gives where K(x) = J 2 + m 2 x 2 /4. In deriving Eq. (79), we made use of the following identity: From these quasimomenta, we see that the spinning string algebraic curve has a cut between q 3 and q 8 and between q 4 and q 7 (by inversion symmetry). The classical algebraic curve is depicted in Fig. (3).

G.2 Off-shell Frequencies
Since q 1 and q 5 have the same structure as they did for the point-particle solution, a little thought shows that Ω 15 (y) should be the same as we found for the point-particle. In particular, Ω 15 (y) = 1 2 From Eqs. (30,31) and table 2, it follows that the only off-shell frequency we need to compute is Ω 45 (x).
Let's suppose that we have two fluctuations between q 4 and q 5 ; one with mode number +n and the other with mode number −n. These fluctuations correspond to adding poles to the classical algebraic curve. The locations of the poles will be denoted x 45 ±n . Looking at Eq. (81), we see that q 4 is proportional to a square root coming from K(1/x). We therefore expect that δq 4 (x) should be proportional to ∂ x K(1/x) ∝ 1/K(1/x) and make the following ansatz for the fluctuations: where ± stands for the sum over positive and negative mode number, y is a collective coordinate for x 45 ±n , α(x) is defined in Eq. (28), and α ± (y) are some functions to be determined. Note that this ansatz is consistent with the inversion symmetry in Eq. (26). We also make the following ansatz for h(x, y): For this choice of h(x, y), the residue of δq 4 at x = x 45 ±n agrees with Eq. (28) and the residues of all the fluctuations are synchronized x = ±1 according to Eq. (25). To compute the anomalous energy shift, we must look at the large-x behavior of the fluctuations and compare them to Eq. (27). At large x, δq 2 and δq 4 are given by where we neglect terms of O(x −2 ). Comparing the asymptotic forms of δq 2 and δq 4 with Eq. (27) gives where κ = 2K (1). Recalling that the (4,5) fluctuation is a CP 3 fluctuation, Eq. (29) implies that This implies that for a single fluctuation Ω 45 (y) = 2 κ Now it is trivial to write down all the other off-shell frequencies using the relations in Eqs. (30,31) and table 2. The off-shell frequencies are summarized in Table 8.

G.3 On-shell Frequencies
The structure of this section is as follows: for each row of Table 8, we find the solutions of the equation in the second column, plug the solution with greatest magnitude into the off-shell frequency in the first column, and simplify the resulting expression to obtain the on-shell algebraic curve frequency in the corresponding row in Table 4. We use the following notation for on-shell frequencies: ω n = Ω (x n ) .
Solving for the pole location gives x n = 1 n κ ± κ 2 + n 2 .
Choosing solution with larger magnitude and plugging it into Eq. (82) then leads to ω n = 1 + n 2 κ 2 .
The solutions to the equation for the pole location are x n = (m + n) n(2m + n) κ ± 4J 2 + (m + n) 2 .
The solution with larger magnitude is the one with relative − sign in denominator.

H Comparison with Bethe Ansatz
In this Appendix we will compute the leading two contributions to the anomalous dimension of the operator dual to the spinning string in AdS 4 × CP 3 . First let's consider the operator dual to the SU (2) spinning string in AdS 5 × S 5 which has the form where Z and W are complex scalar fields in N = 4 SYM. In this sector, the one-loop planar dilatation operator corresponds to the Hamiltonian of a Heisenberg spin chain of length 2J [51]: The dilatation operator can be diagonalized by solving a set of Bethe ansatz equations [52]: where m is an integer which is introduced after taking the log of both sides of equation 92b. In the large-J limit, the Bethe equations simplify and can be solved using the methods described in [15,53,36]. In particular, [36] found that the anomalous dimension is given by a = m 2 + ∞ n=1 n n 2 − 4m 2 − n 2 + 2m 2 .
Now let's turn to the operator in Eq. (45). In this case, the two-loop planar dilatation operator is given by Eq. (46). As explained in section 4.1, this corresponds to the Hamiltonian for two identical Heisenberg spin chains of length 2J which are only coupled by a momentum constraint. With this in mind, the Bethe equations are