Some semiclassical structure constants for AdS_4 x CP^3

We compute structure constants in three-point functions of three string states in AdS_4 x CP^3 in the framework of the semiclassical approach. We consider HHL correlation functions where two of the states are"heavy"string states of finite-size giant magnons carrying one or two angular momenta and the other one corresponds to such"light"states as dilaton operators with non-zero momentum, primary scalar operators, and singlet scalar operators with higher string levels.


Introduction
The AdS/CFT duality between string/M theories on Anti-de Sitter (AdS) background and conformal field theories (CFTs) on its boundary has been a most productive research direction ever since it was proposed [1][2][3]. As a strong-weak coupling duality, integrability has played crucial roles in non-perturbative computations [4]. It has been applied to compute conformal dimensions of the CFTs and energies of corresponding string states. A natural next challenge is to utilize integrability to compute structure constants which determine three-point functions.
A promising recent progress is so-called hexagon amplitude approach [5]. The structure constants are given by sums of hexagon amplitudes, which can be determined exactly in all orders of 't Hooft coupling constant λ. This approach has proved effective in the weak coupling limit [6][7][8]. However, technical difficulties such as summing up all intermediate states, finite-size effects, etc. become substantial in the strong coupling limit [9].
For two-heavy and one-light operators in the semiclassical limit λ ≫ 1, the "HHL" three-point functions can be obtained from explicit evaluation of light vertex operator with the heavy string configurations [10][11][12]. In spite of limited applicability, this method JHEP02(2018)110 is useful to obtain structure constants when the heavy operators have large but finite J ≫ √ λ values. The resulting HHL functions show exponential corrections e −J/ √ λ , which can be related to exact S-matrix, hence the integrability [13]. Type IIA string theory on AdS 4 × CP 3 background is dual to N = 6 super Chern-Simons theory in three space-time dimensions, known as ABJM theory [14]. Classical integrability [15,16] and giant magnon solutions have been studied in [17]- [20]. The HHL 3point functions have been computed for various string states in AdS 4 ×CP 3 [21]- [25]. In this paper, we will focus on finite-size effects of some normalized structure constants in AdS 4 × CP 3 in semiclassical limit where the heavy string states are finite-size giant magnons. We also consider various different light string states, such as dilaton operators with non-zero momentum, primary scalar operators, and singlet scalar operators on higher string levels.
The paper is organized as follows. In section 2, we introduce preliminary contents along with various giant magnons on CP 3 . We present the HHL functions of two giant magnons with dilaton operator with non-zero momentum in section 3, with primary scalar operators in section 4, and with singlet scalar operators on higher string levels in section 5. We conclude the paper in section 6.

Structure constants
It is known that correlation functions of CFTs can be determined in principle in terms of the basic conformal data {∆ i , C ijk }, where ∆ i are the conformal dimensions defined by two-point normalized correlation functions and C ijk are structure constants of three-point correlation functions The HHL three-point functions of two heavy operators and a light operator can be approximated by a supergravity vertex operator evaluated at the heavy classical string configuration [12]: For |x 1 | = |x 2 | = 1, x 3 = 0, the correlation function reduces to Then, the structure constants can be given by where c ∆ is the normalization constant of the corresponding light vertex operator.

The string Lagrangian and Virasoro constraints
String theory moving on certain background can be described by the Polyakov action where T is the string tension. We choose to work in conformal gauge γ mn = η mn = diag(−1, 1), in which the Lagrangian and the Virasoro constraints take the form The background metric g M N for AdS 4 × CP 3 is given by where R is related to the string tension T and the 't Hooft coupling constant (α ′ = 1) by There are also dilaton and RR-forms, which do not influence the motion of the classical strings. Further on, we set R = 1. The metric of CP 3 space can be written as where θ ∈ [0, π], ϑ 1 , ϑ 2 ∈ [0, π], ϕ 1 , ϕ 2 ∈ [0, 2π], ϕ 3 ∈ [0, 4π]. The angular coordinates in (2.6) can be expressed also by the following complex coordinates

Giant magnons
The giant magnon solutions in CP 3 can be found by the Neumann-Rosochatius (NR) integrable system with the following ansatz for the string embedding [18] t(τ, σ) = κτ,

CP 1 giant magnon
Let us start with the giant magnon living in the R t × CP 1 subspace. Such subspace can be obtained by setting θ = ϑ 2 = ϕ 2 = ϕ 3 = 0. What remains is 1 Then (2.7) becomes For this case, the induced metric on the string wordsheet is By using (2.8) and (2.9) in (2.3), one can write down the string Lagrangian in the following form (prime is used for d/dξ) from which the first integral for f 1 becomes with an integration constant C 1 . The first Virasoro constraint (2.4) along with (2.12) becomes while the second constraint (2.5) determines the constant C 1 = vκ 2 ω 1 . We will further restrict ω 1 = 1 since we can choose an appropriate unit of τ . In terms of χ ≡ cos 2 ϑ 1 , (2.13) can be written as (2.14) JHEP02(2018)110

RP 3 giant magnon
The RP 3 giant magnon lives in the R t × RP 3 subspace, which can be obtained from (2.6) by setting ϑ 1 = ϑ 2 = π 2 , ϕ 3 = 0. The resulting metric is Correspondingly, the coordinates (2.7) reduce to For the case at hand, the metric induced on the string worldsheet is given by The string Lagrangian in this case becomes (2.16) from which the first integrals for f 1 and f 2 become with integration constants C 1 , C 2 . Since the RP 3 giant magnon should be well-defined at θ = π, we impose an extra condition C 1 = 0. The two Virasoro constraints (2.4) and (2.5) are combined along with (2.17) to give a parametric relation 18) and the first integral for θ

JHEP02(2018)110
This time we fix ω 2 ≡ 1 and define to rewrite this equation as where χ p and χ m satisfy the equalities:

HHL of dilaton operator with non-zero momentum
The vertex for the dilaton operator with non-zero momentum j, originally defined for AdS 5 × S 5 in [12], is modified in the AdS 4 × CP 3 case to where we denote the scaling dimension ∆ d = 4 + j and (x m , z) as the Poincare coordinates on AdS 4 . The coordinates on CP 3 are represented by angular coordinates X k or equivalently by the complex coordinates z a defined in (2.7). The choice of the indices (a, b) determines the direction of the momentum in the CP 3 space. The AdS part of the giant magnon solution is given by (after Euclidean rotation, iτ = τ e , where τ is the worldsheet time) x 0e = tanh(κτ e ), x i = 0, z = 1 cosh(κτ e ) .
one finds

JHEP02(2018)110
The normalized structure constant can be obtained by integrating the vertex over the string worldsheet.
where we replaced the integration over σ with integration over χ in the following way using eq. (2.14) for χ ′ . The parameter L is introduced here in order to take into account the giant magnons in the finite-size worldsheet volume. The final expression of the integral in (3.6) becomes where F 1 (a; b 1 , b 2 ; c; z 1 , z 2 ) is a hypergeometric function of two variables (Appell F 1 ) and We used an integral representation for the F 1 in (3.7) [32] The structure constants are given by the parameters κ, v which are eventually related to the conserved angular momentum J 1 and the worldsheet momentum p (see (2.14), (3.8)) by Here K, E and Π are the complete elliptic integrals of the first, second, and third kinds, respectively.

Leading finite-size corrections
Since J 1 and p define the heavy operators of the dual gauge theory, it is important to express the semiclassical structure constants C CP 1 ,d

13
(j) in terms of them. For given J 1 and

JHEP02(2018)110
p, one can solve numerically (3.9) and (3.10) to find corresponding values of κ, v, which can be used to evaluate C CP 1 ,d

13
(j) from (3.6). Explicit computations are possible for the case where J 1 is large but finite J 1 ≫ T , equivalently, ǫ ≪ 1. We start by rewriting (2.14) in the following form Next, we use the small ǫ-expansions

Replacing (3.12) into (3.11), one finds relations
The expressions for v 0 , v 1 , v 2 in terms of the worldsheet momentum p can be found from (3.9) and (3.10) along with the expansion parameter in terms of J 1 and p The case of j = 0. Let us begin with the simplest case j = 0, i.e. dilaton with zero momentum, which is just the Lagrangian. The C CP 1 ,d

JHEP02(2018)110
The case of j = even integer. In this case, the third arguments −j/2 of the Appell F 1 functions in (3.7) are negative integers. With the help of Mathematica, we have found that these functions can be expressed in terms of the elliptic integrals, K and E. (The j = 0 result analysed above belongs to this case too.) We list explicit functional relations for a few simple cases in the appendix. Using small ǫ-expansion of the elliptic integrals, one can find the leading finite-size corrections of the HHL for any even j in principle. Here, we present explicit results for j = 2 (∆ d = 6) as an example: The case of j = odd integer. Since the Appell functions can not be written in terms of the elliptic integrals in this case, we express the F 1 in (3.7) using an infinite sum [31] (3.19) We can use small ǫ-expansions for the hypergeometric 2 F 1 functions and resum afterward. The resulting structure constants (3.7) in the leading-order of ǫ are Here γ is the Euler's constant and 2 F 1 (a; b; c; z) the Gauss hypergeometric function,

JHEP02(2018)110
3.2 With the R t × RP 3 giant magnons From (2.15), all combinations (a, b) in the dilaton vertex (3.1) are non-vanishing. However, we will focus on only V d 13 (j) and V d 24 (j) for which explicit expressions can be obtained. Working in the same way as the CP 1 case (see (3.4)), one derives which can be used to V d 13 (j) and V d 24 (j): By integrating V d 13 (j) and V d 24 (j) over the string worldsheet coordinates, we derive the corresponding structure constants C RP 3 ,d 13 again in terms of the F 1 and 2 F 1 : where χ p is given by (2.22). The parameters κ, u, v in (3.24), (3.25) are related to the conserved angular momenta J 1 , J 2 and the worldsheet momentum p along with (2.22) by For given J 1 , J 2 , p, one can find corresponding κ, u, v, with which the structure constants can be evaluated. infinite size case, one can take a limit of κ = 1 and ǫ = 0. The above results reduce to the zero momentum dilaton with j = 0. Small ǫ-expansions for RP 3 are straightforward for these cases since they are either in terms of the hypergeometric 2 F 1 functions or the Appell F 1 functions with special arguments which can be expressed in terms of the elliptic integrals. We will not present detailed expressions here.

HHL of primary scalar operators
The vertex for primary scalar operators is given by [12] V pr ab (j) = The scaling dimension is ∆ pr = j. This reduces for giant magnons (4.1) to In the case of the CP 1 giant magnons, we consider V pr 13 with 3) By using (4.2) and (4.3), we can integrate the vertex over the string worldsheet to obtain the corresponding structure constants as follows: The Appell F 1 functions in C CP 1 ,pr

13
(j) have the same arguments as in C CP 1 ,d

13
(j) in (3.7). Therefore, a similar analysis can be done for the leading finite-size corrections for C CP 1 ,pr 13 (j). We present here j = 2 which is the simplest case of even integer (j = 0 is trivial) after converting F 1 functions into the elliptic functions:

JHEP02(2018)110
For the HHLs with two RP 3 giant magnons, we consider again two primary scalar operators V pr 13 and V pr 24 and the results are given by These results are quite similar to those of the dilaton vertex in (3.24) and (3.25). As before, the case of RP 2 giant magnons can be obtained by taking u = 0 limit. One can make small ǫ-expansion by either using series expansions of 2 F 1 for V pr 13 or the Appell functions which can be expressed by elliptic integrals for V pr 24 .

HHL of singlet scalar operators with string levels
The vertex for singlet scalar operators is given by [12] V q = [cosh (κτ e )] −∆q ∂ + X k ∂ − X k q , (5.1) where q is related to the string level n by q = n + 1 and with the 't Hooft coupling λ.
Since the evaluation of ∂ + X k ∂ − X k for the giant magnons have been given in previous sections, we just present the results here.
For the CP 1 case,

JHEP02(2018)110
This approach is particularly useful to understand the finite-size corrections in the HHL in terms of the underlying integrability structure like the world-sheet S-matrix. In this sense, various HHL functions and their finite-size corrections which we have analysed in this paper can be used to understand new aspects of integrability in the planar limit of AdS 4 /CF T 3 .