Semiclassical folded string in AdS4 X CP3

We consider type IIA superstring theory on the background AdS4 x CP3, and the classical solution describing a folded string spinning in AdS4 with angular momentum in CP3. In the 't Hooft limit, it is the gravity dual of twist operators in the ABJM superconformal theory. We quantize the classical solution by algebraic curve methods and determine the first semiclassical correction to the energy. We provide an integral representation for this quantity valid for all values of the charges. We analyze its properties in the special regimes associated with a short or long string providing various accurate analytical expansions. Finally, we investigate various properties of the so-called slope, the leading term of the energy for short strings, collecting information that could be useful in attempts to generalize the exact results recently proposed for the folded string in AdS5 x S5.


Introduction
The AdS/CFT correspondence [1][2][3] is an extremely deep and fruitful theoretical idea that played a central role in the past decade. In its simplest instance, it links the integrability of the world-sheet σ-model for type IIB superstring to the strong coupling behaviour of four dimensional gauge theories. In the specific case of planar N = 4 Super Yang-Mills theory, integrability methods determine the spectrum of anomalous dimensions as a function of the 't Hooft coupling λ, from weak to strong coupling.
After the foundational papers applying integrability methods in AdS/CFT [4][5][6][7][8][9][10] (see also the recent review [11]), a solution to the spectral problem was obtained for asymptotically long single trace operators by means of the Bethe Ansatz approach [12]. Later, the full spectrum was shown to be captured by suitable Y-system equations [13][14][15][16][17]. These equations are expected to be general and to be valid for any operator, including short ones. The Y-system equations are definitely involved, but in [16] they had been formulated for the single trace sl(2) twist operators in a form suitable for numerical studies. This opened the way to explorations of the flow of the anomalous dimension of the most famous among them, the Konishi operator. This investigation culminated in the analysis of [18] where the λ dependance has been studied interpolating from the weak coupling expansion, known explicitly from perturbative calculations in gauge theory up to four loops, to the strong coupling string theory prediction [19] known only to tree level.
The next logical step was the determination of the analytical strong coupling expansion of the Konishi anomalous dimension, i.e. the energy of its gravity dual state. The sub-leading coefficient in the string theory expansion was found independently by three groups [20][21][22] (see also [23] for related work on a different superconformal representative of Konishi). In the end, the numerical predictions of [18] have been confirmed. More recently a highly nontrivial observation was made in [24], which allows one to reproduce the one-loop result almost without any effort. Finally, the second nontrivial strong coupling expansion coefficient was derived analytically in [25] by using the one-loop expression for a general (S, J) folded string [26] found in [20] and the conjecture proposed in [24]. The comparison with the available numerical data from the thermodynamical Bethe Ansatz approach shows a rather promising agreement.
The folded string is indeed a very useful theoretical laboratory. At the classical level, it is convenient to describe it in terms of the scaled charges S = S/ √ λ, J = J/ √ λ. They have a simple and clear dynamical meaning. The spin S is associated with rotation around the center of AdS 5 while J is an angular momentum on the sphere. At small S, the string is short and admits a near-flat space description. At large S, the string stretches and reaches the boundary of AdS with the characteristic scaling E ∼ log S of its energy. Quantum corrections in the semiclassical approximation are fully under control and can be studied by algebraic curve tools. Also, the folded string is a clear example where integrability techniques are quite competitive with traditional field-theoretical methods [27,28] (see also [29] for a similar approach to the more complicated case of spiky strings).
Remarkably, very similar integrability structures as well as folded string classical solutions are known to be available in the ABJM theory [30]. This is a three dimensional U (N )×U (N ) gauge theory with four complex scalars in the (N, N ) representation, their fermionic partners, and a Chern-Simons action with levels +k, −k. This theory has N = 6 superconformal symmetry osp(2, 2|6). ABJM can be considered as the low energy theory of N parallel M2-branes at a C 4 /Z k singularity. In the large N limit this is M theory on AdS 4 ×S 7 /Z k . For fixed λ = N/k we can describe it by type IIA string on AdS 4 × CP 3 which is classically integrable [31][32][33]. The manifest (non abelian) part of the R symmetry is SU (2) × SU (2). The complex scalars can be written as two doublets transforming as (2,1) and (1,2). Under the gauge group they transform as (N, N ) and (N , N ). At leading order (two loops, λ 2 in 't Hooft coupling λ), the dilatation operator for single trace operators built with these scalars is a SU (4) integrable spin chain [34,35]. In [36], Gromov and Vieira have proposed a set of all-loop Bethe-Ansatz equations for the full osp(2, 2|6) theory consistent with the two loops analysis and with the superstring algebraic curve at strong coupling [37]. The equations depend on a dressed coupling h(λ) which takes into account the fact that the one-magnon dispersion relation is not protected [38][39][40][41]. Finally, an exact SU (2|2) A ⊕ SU (2|2) B symmetric S-matrix consistent with [36] has been presented in [42]. A comprehensive review of integrability methods in ABJM discussing also their available checks can be found in [43].
The important point for our investigation is that there is a sl(2)-like sector in ABJM as discussed in [36,44]. At strong coupling and large spin these operators behave quite similarly to the corresponding ones in AdS 5 × S 5 . In particular their dual string state is a folded string rotating in AdS 3 and with angular momentum J in CP 3 [45][46][47]. At weak coupling, they are composite operators in totally different theories. Nevertheless, both N = 6 Super Chern-Simons and N = 4 Super Yang-Mills are integrable and the all-loop Bethe equations in the sl(2) sectors are essentially the same 1 .
The current treatment of quantized folded string in AdS 4 × CP 3 [49] is still rather limited. The large spin limit has been studied at leading order in various regimes: (a) J/S fixed in [41], (b) J ∼ log S in [50]. The short string limit has never been addressed. On the numerical side, the analysis of [51] explored the TBA equations of AdS 4 × CP 3 , although in a small window of the coupling reaching at most λ ∼ O(1). In this paper, we present a complete analysis of the one-loop semiclassical energy of folded string in AdS 4 × CP 3 . In more details, (a) we derive a simple and compact integral representation for the one-loop energy correction valid for generic values of the charges.
(b) At small (semiclassical) spin, we discuss the explicit evaluation of the sum over frequencies and obtain various short string expansions like that in [25] or the resummed one discussed in [52].
(c) At large spin, we extract from the integral representation the energy expansion at next-to-leading order.
(d) Finally, we discuss in some detail the structure of the slope in ABJM, i.e. the linear term in the small spin expansion of the one-loop energy as a first step toward a generalized Basso's conjecture [24] for this theory.
2 Algebraic curve quantization in AdS 4 × CP 3 We give in this section a compact self-contained summary of the results of [37] using the language of off-shell fluctuation energies [53]. We shall work in the algebraic curve regularization and write all equations in terms of the σ-model coupling g. For large g, it is related to the 't Hooft coupling by but, contrary to the AdS 5 × S 5 case, this relation will get corrections at finite g. The classical algebraic curve for AdS 4 × CP 3 is a 10-sheeted Riemann surface. The spectral parameter moves on it and we shall consider 10 symmetric quasi momenta q i (x) (q 1 , q 2 , q 3 , q 4 , q 5 ) = (−q 10 , −q 9 , −q 8 , −q 7 , −q 6 ). (2.2) They can have branch cuts connecting the sheets with In the terminology of [37], the physical polarizations (ij) can be split into heavy and light ones and are summarized in the following Virasoro constraints require that the poles of the quasi-momenta q i (x) at x = ±1 are synchronized according to (q 1 , q 2 , q 3 , q 4 , q 5 ) = α + x − 1 (1, 1, 1, 1, 0) + · · · = α − x + 1 (1, 1, 1, 1, 0) + · · · .
(2.4) Inversion symmetry reads where m ∈ Z is a winding number. The asymptotic values of the quasi-momenta for a length L state with energy and spin E, S are where M r,u,v are related to the SU (4) representation of the state (2.7)

Semiclassical quantization
Semiclassical quantization is achieved by perturbing quasi-momenta introducing extra poles that shift the quasi-momenta q i → q i + δq i . Virasoro constraints and inversion properties of the variations δq i follow from those of the q i 's. In order to find the asymptotic expression of δq i in terms of the number N ij of extra fluctuations we can look at the details of polarized states and obtain where the sum is over all pairs (ij) ≡ (ji) of physical polarizations and integer values of n with Also, the residues at the extra poles are and k ij = 0, ±1, ±2 are the coefficients of N ij in (2.8). By linear combination of frequencies and inversion (as in the AdS 5 /CFT 4 case), we can derive all the off-shell frequencies in terms of two fundamental ones Their explicit expressions turns out to be , , , , , , , , , , , The folded string in AdS 4 × CP 3 We present the algebraic curve for the folded string in AdS 4 × CP 3 closely following the notation of [50]. In terms of the semiclassical variables the energy of the folded string can be expanded according to where the small S expansion of the classical contribution E 0 reads

Quasi-momenta
The quasi momenta are closely related to those of the AdS 5 × S 5 folded string since motion is still in AdS 3 × S 1 and the CP 3 part of the background plays almost no role. The only non trivial case is where the branch points obey 1 < a < b and The other quasi-momenta are The above expressions are valid for a folded string with minimal winding. Adding winding is trivial at the classical level, but requires non trivial changes at the one-loop level (see for instance [25] for a detailed analysis of the AdS 5 × S 5 case). The independent off-shell frequencies can be determined by the methods of [53]. The result is rather simple and reads 2 Integral representation for the one-loop correction to the energy The one-loop shift of the energy is given in full generality by the following sum of zero point energies where the sum over ij is over the 8 B + 8 F physical polarizations and x ij n is the unique solution to the equation (2.10) under the condition |x ij n | > 1 3 . In the same spirit as [20,25], the infinite sum over on-shell frequencies can be evaluated by contour integration in the complex plane. The result is quite similar to the AdS 5 × S 5 one and reads As in AdS 5 ×S 5 , the labeling of the various contributions reminds their physical origin. In particular, dressing and wrapping contributions have been separated in order to split the asymptotic contribution from finite size effects. As in AdS 5 ×S 5 , the anomaly terms are special contributions arising from the deformation of contours and ultimately due to the presence of the algebraic curve cuts. The representation (4.1) is a compact formula for E 1 and can be evaluated numerically with minor effort. In order to understand it better, we shall now analyze the short and long string limit. In the former case, we shall evaluate the explicit sum over frequencies clarifying the relation with the contour integrals. In the latter, we shall extract the analytical expansion at large spin directly from (4.1).

Short string limit
The short string limit is generically S → 0. Regarding J , we shall consider two cases.
The first amounts to keeping J fixed, expanding in the end each coefficient of powers of S at small J . This is precisely the procedure worked out in [25] in AdS 5 × S 5 . In the second case, we shall keep the ratio ρ = J / √ S fixed as in [52]. The two expansions are related, but not equivalent and provide useful different information.

Fixed J expansion
After a straightforward computation, our main result is This expansion is rather similar to the one derived in [25] for AdS 5 × S 5 , but there are two remarkable differences: 1. The leading terms at small J are O(S n /J 2n ). Instead, they were O(S n /J 2n−1 ) in AdS 5 × S 5 . Also, there are terms with all parities in J while in AdS 5 × S 5 , there appear only terms odd under J → −J . The additional terms are important and we shall discuss them in more details later. Remarkably, they imply that if one scales J ∼ √ S they give a constant contribution in the short string limit. This is different from AdS 5 × S 5 where the energy correction vanishes like √ S in this regime.
2. There are terms proportional to log(2). As we discuss in App. (A), these terms can be removed by expressing the energy correction in terms of the coupling in the world-sheet scheme. The scheme dependence is universal and agrees with that found in [41] for a circular string solution and in [54] for the giant magnon.

Fixed
The result in this limit is and C is related to the branch cut endpoints by the formula Its expansion at small S with fixed ρ = J / √ S is C = 1 − ρ ρ 2 + 2 − 2 ρ 3 + 5 ρ 4 (ρ 2 + 2) 3/2 S + ρ (12 ρ 6 + 68 ρ 4 + 126 ρ 2 + 73) 32 (ρ 2 + 2) 5/2 S 2 + · · · (5.6) Expanding E 1 at large ρ we partially resum the calculation at fixed J . Just to give an example, from the expansion we read the coefficients of all terms ∼ S n /J 2n . The first ones are of course in agreement with (5.1). As another non trivial example, the large ρ expansion of a 11 (ρ) is and allows to read the coefficients of all terms ∼ S n /J 2n−3 .

Summation issues
The explicit sum over the infinite number of on-shell frequencies requires some care and a definite prescription since the sums are not absolutely convergent due to physically sensible cancellations between bosonic and fermionic contributions. As discussed in [50], the following summation prescription is natural from the point of view of the algebraic curve (see [55] for a different prescription) 5 where K n is a particular grouping of heavy and light modes The short string expansion of K n takes the form where C, given in (5.5), is independent on p and the sum of K p (which start at O(S)) is convergent. The alternating constant C poses some problems because we have to give a meaning to − C + C − C + C + · · · . (5.14) An analysis of the integral representation shows that it automatically selects the choice Later, we shall provide various consistency checks of this prescription. In particular, we shall see that it is necessary in order to match the asymptotic Bethe Ansatz equations when wrapping effects are subtracted. Notice also that the expansion of C at fixed J is so, upon expanding at small J , it provides precisely the terms with even/odd J exponents in the coefficients of the odd/even powers of S in (5.1).
Apart from the C term, the integral representation implements the Gromov-Mikhailov (GM) prescription. The reason is that the singularities at |x| = 1 are avoided by implicitly encircling them by a small circumference. This cut-off on |x − 1| translates in a bound on the highest mode n that correlates heavy/light polarizations according to GM. In other words the highest mode for light polarizations is asymptotically half the highest mode for heavy polarizations.
As a numerical check of the agreement between the integral representation and the series expansion, we fix ρ = 1 in table (1) and show the value of E 1 from our analytical resummation and result from the integral. The agreement is very good already at moderately small S.  Table 1. Comparison between resummation at fixed ratio ρ = 1 and integral representation. The asymptotic value for S → 0 is ( √ 3 − 3)/6 −0.211, but already at S = 1/500 we have 6 digits agreement.
A similar check at fixed J is shown in Fig. (1) where we plot the asymptotic expansion (5.1) and the exact numerical E 1 as functions of S at J = 1/5.

The slope function
The one-loop correction E 1 tends to zero linearly with S when S → 0 at fixed J . The slope ratio contributions start at order S 2 [24]. It also does not receive wrapping corrections in AdS 5 . Instead, in the case of AdS 4 the slope has a non vanishing wrapping contribution. For instance, a rough evaluation at J = 1 gives a definitely non zero value around −0.042.
Indeed, an analytical calculation shows that the wrapping contribution to the slope in AdS 4 × CP 3 is exactly This formula is in perfect agreement with numerics since for instance The large J limit of this expression is exponentially suppressed as it should To analyze the small J limit it is convenient to split this contribution into the n = 0 term plus the rest. The result is very intriguing. For the n = 0 term, we find This is precisely the set of terms even under J → −J in the full slope which is the first term of (5.1). Similarly, we can consider the rest of σ wrap and expand at small J . We find Comparing again with (5.1), we see that we are reproducing all the irrational terms of the slope, involving zeta functions or log (2). The remaining terms are the same as in Thus, we are led to the following expression for the one-loop full slope The above analysis of the slope is a confirmation that the various terms in (5.1) are organized in the expected way. The asymptotic contribution is precisely the same as in AdS 5 × S 5 , while wrapping is different and is exponentially suppressed for large operators. This is a property of the integral representation and a confirm that the prescription (5.15) is correct.

Weak coupling
It is interesting to evaluate the slope at weak coupling. In principle, this requires the knowledge of the anomalous dimensions of short sl(2) operators in closed form as a function of the spin at a certain length (i.e. twist, in the gauge theory dictionary). This information is available for the asymptotic contribution, but not for the wrapping, which is only known as a series expansion at large spin and low twist [48,57]. Nevertheless, if we are interested in the correction to the slope only (so, just the first term at small spin), then the Lüscher form of the wrapping correction presented in [48] is enough 8 . At twist-1, and following the notation of [48], the wrapping correction enters at four loops and is expressed by the following function of the integer spin N of the gauge theory operator Here, S a (N ) are generalized harmonic sums while W(N ) is a complicated expression depending on the Baxter polynomial Q N (u) associated with the Bethe roots. The first factor γ 2 (N ) is nothing but the two-loop anomalous dimension of the twist-1 operators.
In the small N limit, it starts at O(N ). Thus, the factor W(N ) can be evaluated at N = 0 where the Baxter polynomial trivializes Q 0 (u) = 1. After a straightforward calculation, one finds that (on the even N branch), So, even at weak coupling, we find a correction to the slope coming from the wrapping terms 9 .

Long string limit
The large S behaviour of the one-loop energy E 1 can be computed starting from the integral representation. Let us first summarize the result valid for AdS 5 × S 5 from [20]. We scale J with S for S 1 according to where we assume > 0 10 . Then, the one-loop energy correction can be written 2) with f AdS 5 10 ( ) = The expansion in AdS 4 × CP 3 can be derived in the same way as in [20] and the result is simply .
This formula can be easily checked numerically from the explicit evaluation of the integral representation. Notice that the simple 1 2 rule for the leading two terms is in agreement with the result of [58]. The correction ∼ 1/ log S comes from the anomaly terms. It is twice bigger than in SYM.
The explanation of this fact is as follows 11 . The low energy effective theory of the Gubser-Klebanov-Polyakov (GKP) string in AdS 4 × CP 3 has two massless modes at finite chemical potential . Namely, one massless Dirac Fermion and one massless boson that gives a central charge 2 (in AdS 5 × S 5 one has only one massless boson giving central charge 1). Also, concerning the → 0 limit, the other low-energy modes acquire a mass proportional to at small and their contribution is exponentially suppressed with the effective length log S at fixed . When → 0 they become massless and contribute at leading order to 5 units of central charge (there are actually 4 bosons with mass and one with mass /2 while there were only four with mass in AdS 5 × S 5 ). In other words it should be true that in the small limit the 1/ log S gets corrected by 5 extra units of central charge giving a total −(2+5) π 12 log S for the energy of the vacuum state (i.e. the twist 1 state of the theory). Indeed, 2 + 5 = 7 is the correct central charge of the low-energy effective theory on the GKP background [46,59]. Instead, in AdS 5 × S 5 the final result for → 0 (i.e. for twist 2) was coming with 1 + 4 = 5 units of central charge, which is the correct central charge of the O(6) model.

Relation with marginality condition
Let us define Λ ≡ λ in AdS 5 × S 5 , and Λ = 16 π 2 g 2 in AdS 4 × CP 3 . The role of Λ is to emphasize the close analogy between the expressions in the two cases. For the folded string in AdS 5 × S 5 , the energy admits the following expansion where the following exact formula for the constants A i has been conjectured in [24]: Expanding at large λ, we find the first values The expansion (8.1) is very convenient since all powers of S have a coefficient with an expansion at large Λ starting with a more and more suppressed term. The simplicity of (8.1) is a special feature of the folded string with two cusps. If winding is allowed, it is known that such structure is lost as discussed in [25] (see also the results of [52]).
For the folded string in AdS 4 × CP 3 , the expansion with fixed J 12 has the general form (see the Appendices of [25]) where the classical energy is 13 and the semiclassical computation provides v (0) pq according to the results in (5.1). 12 Actually when we speak about fixed J we mean small S followed by small J . 13 Note that there is a typo in the S 3 term in the introduction to [52] Expanding E 2 , we find that (8.1) takes the following form This structure is different from (8.1) since higher powers of S are not associated with terms that are more and more suppressed at large Λ. This is possible since the new terms not present in (8.1) are associated with suitable inverse powers of J. The same phenomenon is discussed in [25] for the folded string in AdS 5 × S 5 with non-trivial winding. As we discussed in Sec. (6), wrapping corrections are responsible for these terms.

Conclusions
In this paper we analyzed in a systematic way the one-loop correction to the energy of a semiclassical folded string spinning in AdS 4 × CP 3 . We derived an integral representation for the energy correction and analyzed its properties for short and long strings. Also, for short strings, we studied the properties of the slope as a first step toward an eventual exact formula in the spirit of [24]. In this respect, the main difficulty appears to be the inclusion of wrapping effects in Basso's conjecture since we have shown that these finite size corrections affect the ABJM slope in contrast to the simpler case of N = 4 SYM. and expand at large g. The part linear in c takes precisely the same form E 0 = 4 π g E 0 = c 1 J − J 32π 2 g 2 + 3J 3 2048π 4 g 4 + · · · S + (A.4) + − 4πg J 3 + J 128π 3 g 3 − J 3 1024π 5 g 5 + · · · S 2 + + 24π 2 g 2 J 5 + 1 128π 2 g 2 J − 33J 2 13 π 4 g 4 + 775J 3 2 20 π 6 g 6 + · · · S 3 + · · · + · · · This means that choosing c = − log(2) (A.5) removes all log 2 terms from E 1 . The above redefinition of g connects the coupling in the algebraic curve regularization with the coupling in the world-sheet regularization and agrees with the calculation in [41] as well as [54].