On finite-size spiky strings in AdS3× S3× T4 with mixed fluxes

We discuss finite-size corrections to the spiky strings in AdS space which is dual to the long 𝒩 = 4 SYM operators of the form Tr(Δ+J1ϕ1Δ+J2ϕ2…Δ+Jnϕn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varDelta}_{+}^{J_1}{\phi}_1{\varDelta}_{+}^{J_2}{\phi}_2\dots {\varDelta}_{+}^{J_n}{\phi}_n $$\end{document}). We express the finite-size dispersion relation in terms of Lambert W-function. We further establish the finite-size scaling relation between energy and angular momentum of the spiky string in presence of mixed fluxes in terms of W-function. We comment on the solution in pure NS-NS background as well.


Introduction
AdS/CFT correspondence relates type IIB superstring theory on AdS 5 × S 5 background and the N = 4 supersymmetric Yang-Mills theory in flat space R 1,3 with gauge group SU(N ) [1][2][3]. This holographic conjecture asserts one-to-one correspondence between various aspects of two models (e.g. global symmetries, spectra, correlation functions, Wilson loops, scattering amplitudes etc.). However, the precise matching of spectra on both sides of duality is really challenging and it is only recently, the problem has become tractable in infinite or large charge limit. The most remarkable development in this direction was made by Gubser, Klebanov and Polyakov (GKP) in 2002 [4], where they studied the closed and folded string spinning in AdS 3 ⊂ AdS 5 . The folded spinning string dispersion relation for large angular momentum was found out to be, The important observation they made was that the difference between the energy and the angular momentum of the long string (E − J), scales as the logarithm of the angular momentum (ln J), which was very similar to the anomalous dimension of twist two Wilsonian operators in perturbative QCD. It has been shown that this string configuration is dual to the twist two operators of N = 4 SYM theory which has the form Tr(φ∆ J + φ). 1 This observation made the single trace operators in large N gauge theory relevant in the AdS/CFT JHEP02(2020)071 correspondence. Thereafter the twist two operator has been generalised to higher twist operator with arbitrary number of fields of the form (1.2) In large spin limit, the anomalous dimension of this operator is completely dominated by the contribution from derivatives. The string solution dual to such a operator has been discussed in [5] for J 1 = J 2 = . . . = J n . The dispersion relation of dual string which has n spikes approaching the boundary is given by A lot of activities have followed this work, e.g. [6][7][8][9][10][11][12] and further, the spin chain connection have been studied in detail, e.g. [13][14][15][16].
Apart from most studied AdS 5 /CF T 4 duality, the other holographic set up which are amenable to integrability is AdS 3 /CF T 2 . In this case, the type IIB string theory on AdS 3 × S 3 × T 4 is shown to be dual to the N = (4, 4) superconformal field theory in two dimensions. Several semiclassical string solutions have been investigated in AdS 3 × S 3 × T 4 geometry which arises from the near horizon limit of the intersecting D1-D5 branes in supergravity. The superstring theory on AdS 3 × S 3 geometry supported by both NS-NS and RR fluxes has been proven to be integrable and the S-matrix has been constructed for the same [17][18][19][20][21][22]. 2 Recently, mirror thermodynamic Bethe ansatz (TBA) has been proposed in pure NS-NS AdS 3 × S 3 × S 3 × S 1 and AdS 3 × S 3 × T 4 backgrounds [24,25]. And also, the TBA has been contructed for low-energy string excitations as well as massless modes in RR background [26,27]. In [28,29], the complete Yangian symmetry underlying the integrability of type IIB superstrings on AdS 3 ×S 3 ×T 4 and AdS 3 ×S 3 ×S 3 ×S 1 supported by mixed R-R and NS-NS fluxes have been derived. The giant magnon solution in mixed fluxes background was studied in [30] followed by many other semiclassical string solutions [31][32][33][34].
In the context of AdS/CFT correspondence, it has long been known that the gauge theory as well as string theory are integrable in the limit of very large or infinite global charges. The dilatation operators of the N = 4 SYM theory acting on the gauge invariant single trace operators can be mapped to the Hamiltonian of an integrable spin chain model which can be diagonalized by Bethe ansatz [35][36][37][38]. The asymptotic Bethe ansatz predicts the correct form of the anomalous dimension up to the order of λ L only when the length of the spin chain (L) is infinite or larger than the loop order. However, when the range of spin chain interaction exceeds the length of spin chain, the virtual particles start circulating around the spin chain giving rise to wrapping effect. In other words, the wrapping corrections have to be taken into account at and above the L th loop order for the spin chain of length L. There has been many proposals in this direction such as thermodynamic Bethe ansatz, Y-system, and quantum spectral curve can be found in literature to correctly account for the wrapping corrections [39][40][41][42][43].
Not only the gauge theory but the string theory also witnessed the inefficiency of asymptotic Bethe ansatz in the limit of finite volume [44]. The finite-size corrections to the JHEP02(2020)071 one-loop calculations in the sigma-model on AdS 5 × S 5 has been observed as exponentially suppressed in the units of string length [45,46]. The finite-size corrections to giant magnon and dyonic giant magnon dispersion relation have been extensively studied at classical [47][48][49][50][51][52][53][54][55][56][57][58][59] as well as one-loop level [60,61]. There have been several instances of finite-size study for certain deformations [62][63][64][65][66][67]. Recently, the anomalous dimension of GKP strings for finite angular momentum has been shown in terms of Lambert W −functions in [68]. Motivated by all these developments, in this paper, we compute finite-size effects for n-spiky string using the known information of the infinite volume case.
The rest of the paper is organized as follows. In section 2, We study the finite-size n-spike string in terms of Lambert W-function. We show that for the limiting case of n = 2, it does reduce to the case of the GKP string. Section 3 is devoted to the study of finite-size n-spike string in AdS 3 × S 3 in the presence of both NS-NS and R-R fluxes. In section 4 we study the finite-size n spike strings in pure NS-NS flux. In section 5 we conclude with some discussions.

Spiky string in AdS
In this section, we study the finite-size effect on the Kruczenski spiky string in AdS 3 background. We start with the AdS 3 metric in global coordinates (2.1) To study the relevant string dynamics, we follow closely the analysis presented in [5]. We choose the following ansatz for the rigidly rotating closed string t = τ, ρ = ρ(σ), φ = ωτ + σ. (2. 2) The Nambu-Goto action for the F-string in AdS 3 geometry can be written as where λ is the 't Hooft coupling constant, X's are the background coordinates and the scalar products can be computed using metric (2.1). TheẊ and X refer to derivative of X with respect to τ and σ respectively. By solving the equation of motion for ρ, we get where ρ 0 is the integration constant. We can see from equation (2.4) that, at ρ = ρ 0 , ρ vanishes and at ρ = coth −1 ω, ρ blows up, which indicates the string profile has spikes at maximum value of ρ (i.e ρ = ρ 1 = coth −1 ω) and valleys at the zero of ρ (i.e ρ = ρ 0 ). The angle difference between the spikes and the valleys is given by,

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Now the energy and angular momentum of the string can be calculated as (2.7) In energy and angular momentum expression, the integration limit is taken from ρ 0 to ρ 1 which gives result for one segment arc of closed spiky string. To get the total energy and angular momentum it is multiplied by 2n. Now we make the following change of variables u = cosh 2ρ, u 0 = cosh 2ρ 0 , and u 1 = cosh 2ρ 1 . (2.8) Substituting equation (2.8) in equation (2.4) and integrating, we get the exact string solution, which has the following form where the modulus of complete elliptic integral of third kind Π is defined as The string profile produced by solution (2.9) gives a arc segment (half spike) where ρ goes from ρ 0 to ρ 1 . Hence the closed string profile can be obtained by gluing 2n arc segments.
Here, we need to use the closedness condition of the spiky string, which is ∆φ = 2n 2π . The plot of closed string profile for n = 5 spikes in figure 1 has been included for completeness. The change of variables (2.8) reduces the angular separation, the string energy and the string angular momentum in terms of elliptic integrals as Now, we proceed with the following substitution,  With the above substitution and using the formula appendix (A.5) for the complete elliptic integral of third kind, the angle difference between the spike and the valley can be written as,

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Correspondingly, the (E − J ) can be expressed as fuction of x and u 0 as Here the energy and the angular momentum of the string have been scaled as E → π n √ λ E and J → π n √ λ J. In figure 2, we plot the string energy and angular momentum against the parameter x. It is clear from the expression (2.16) that to get the string dispersion relation E(θ, J ), we have to first express the parameters u 0 and x in terms of angle differnce ∆φ = θ and string angular momentum J . For x → 0, the expressions (2.15) and (2.16) contain logarithmic singularity due to the presence of following elliptic functions Here the coefficients that appear in the above series are given by , (2.19) where H n = n k=1 1 k . The logarithms from the above mentioned expressions can be eliminated using the following expression for ln x, which is obtained using the substitution (2.14) and the elliptic integral Π formula (A.5) to the angular momentum expression Setting u = csc a, the equation (2.15) can be expanded in a double series around both x = 0 and a = θ, then the series can be inverted for a by using Mathematica. Now plugging the JHEP02(2020)071 (2.20) and exponentiating, we end up with the following equation where the coefficients a n = a n (θ, J ) can be obtained from the equation (2.20) and x 0 is given by (2.22) The expression x(θ, J ) can be obtained from equation (2.21). But we can see, it contains 1/x term in the exponent and also x 0 is exponentially increasing with J . So the general Lagrange-Bürmann formula cannot be applied to get x, rather a different approach has to be adopted. Following the procedure discussed in [68], let us define x * as the leading solution to (2.21) x which implies the functional form of x * as .

(2.24)
For J → ∞, x * approaches to zero (x * → 0). Hence, we have chosen W −1 , the lower branch of Lambert function over W 0 , the principal branch 3 with argument a 0 /x 0 = − 1 4 csc θ exp(−4J − A). Considering x * as the leading solution to (2.21), we can set the full solution x = x * e z , with z → 0. (2.25) Substituting the above equation into (2.21) and using (2.24), we get the following expression for z which leads to the x as a series in x *

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To see the explicit finite-size corrected structure, we would like to calculate the leading order correction terms (leading in J ) to the infinite-size dispersion relation by expanding the Lambert W −1 function for large value of angular momentum J . For this purpose we have taken the following few terms, which are expected to contribute to the leading terms of the scaling relation, Using the expansion of the W −1 function, we write the explicit form of 1/x * as The above expression have been written by considering J large and we discuss the Stirling numbers of first kind n + m n + 1 in appendix A. Plugging (2.30) to (2.29), we obtain the following expression for the leading order correction terms One may notice, the finite-size dispersion relation for spiky string reproduces the result of infinite string, which has been discussed in [5] with the correction terms. For n = 2 (closedness condition implies ∆φ = θ = π/2), our results (2.28) and (2.31) are in complete agreement with the results of finite-size correction to the closed folded GKP string dissused in [68].

Spiky string with mixed fluxes
In this section, we discuss the finite-size effect on the spiky string in AdS 3 background supported by mixed NS-NS and R-R fields where the parameter b varies from 0 to 1 (0 ≤ b ≤ 1). Since we are dealing with fundamental string solution, the R-R field is not relevant to us and can be ignored. The Nambu-Goto action for the F-string in presence of the mixed fluxes is written as where λ is the 't Hooft coupling as mentioned in the previous section and ab is the antisymmetric tensor with τ σ = − στ = 1. We use the same ansatz as equation (2.2). The equations of motion for t and θ are satisfied if which is consistent with the equation of motion for ρ. Here K in the above expression is an integration constant. It can be noticed that the presence of B-field does not change the maximum value of ρ (ρ 1 = coth −1 ω), the cusp position but the minimum value has been modified by the parameter b. Now we have two roots (ρ ± 0 ) corresponding to the zero of ρ , the valley positon but we will consider the one which reduces to ρ 0 for b = 0. Accordingly the conserved charges and angle difference between the spike and the valley have also been changed due to inclusion of mixed flux. The angle difference between the spike and the valley of one arc segment of closed spiky string is given by The energy and the angular momentum of the string are obtained as follows Here n is the number of spikes and using the string closedness condition, we can express ∆φ in terms of n as ∆φ = 2π 2n . Let's proceed by making the following change of variable like the previous section u = cosh 2ρ, u ± 0 = cosh 2ρ ± 0 , and u 1 = cosh 2ρ 1 .
The two roots corresponding to the valley of the string configuration can be calculated as, Clearly u + 0 is the correct one, we are looking for as it return to the original root u 0 for b = 0. Using the above substitutions to the equation (3.3) and performing the integration, JHEP02(2020)071 the exact string solution has been obtained in the following form where the arguments of elliptic function are written as (3.10) Again for the purpose of completeness and to make a clear understanding of the impact of flux on string profile, we have plotted closed string profiles in figure 3 and figure 4, for n = 2 spikes and n = 7 spikes respectively considering different values of flux parameter b. It can be observed that the inclusion of B-field makes the string profile more fatter and the string becomes almost circular when b → 1. Using the substitution of variables (3.7), the anglular separation and the conserved charges can be expressed in terms of complete elliptic integrals as follows Now we proceed to get the finite-size correction to the spinning string solution with B-field by making the following substitution as before for the argument of elliptic function . Now, we have paramere u 0 instead of u + 0 in our expressions. Henceforth applying the same procedure we have followed in the last section, we first express u 0 as u 0 (J , θ, x).
Substituting u 0 (J , θ, x) to the angular momentum expression, the following simplified expression has been computed where the coefficients a n = a n (θ, J ) can be evaluated from J expression and the x 0 has the form x 0 = 16 exp 4J − 1 + (π − 2θ) cot θ + cot 2 θ 2 + csc θ 2 + csc θ 2 + B , = 16 exp(4J + A + B). Now we can write the leading solution to (3.15) as

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using which the full solution for x(θ, J ) can be constructed (already discussed in the last section). It should be mentioned that, here the coefficient a 0 (θ, J ) has been modified to a 0 = −4 csc θ + 2 (3 + cos 2θ) cot θ csc θ b − 1 2 40 cot 2 θ csc θ + 7 sin θ − sin 3θ b 2 . (3.19) Finally plugging the parameters u 0 (J , θ, x) and x (J , θ) to the E − J expression, we write the finite-size dispersion relation for closed folded spiky string in presence of mixed fluxes as  The above scaling relation agrees well with the infinite-size result of spiky string with flux discussed in [34]. One may notice that turning B-field off, dispersion relation (3.20) gives us back the finite-size (E − J ) relation (2.28). For n = 2, we get the following dispersion relation in presence of mixed fluxes where the argument of W-function is − 1

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where the R-R flux vanishes. The substitution of b = 1 reduces one constant parameter from the theory which makes the problem more tractable. In this case, the equation of motion for ρ comes out to be ρ = sinh 2ρ 4 K + sinh 2 ρ which is simply the b = 1 limit of the equation (3.3). Like the previous case of mixed flux, here also the spike position remains same as ρ 1 = coth −1 ω, where as the position of valley ρ 0 has been modified. Using the following change of variables u = cosh 2ρ, u 0 = cosh 2ρ 0 , and u 1 = cosh 2ρ 1 , we obtain the exact string profile in terms of trigonometric functions as The explicit form of u 0 in terms of integration constant K can be written as The angular separation ∆φ between the valley (ρ = ρ 0 ) and the spike (ρ = ρ 1 ) in the limit ρ 1 ρ 0 1 takes the following form The string angular momentum and energy expressions in the above mentioned limit can be evaluated as follows

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Finally, the dispersion of closed rotated string with pure NS-NS flux has the following form Note that for J → ∞, the scaling relation reduces to E − J = π/2, which is the analogous relation in the infinite charge limit. 4

Conclusions
In this paper, we have studied finite-size spiky string in AdS 3 × S 3 background with and without flux. Firstly, we have discussed the finite-size dispersion relation for the spiky string in AdS 3 background in terms of Lambert W-function. When the background is supported by mixed NS-NS and R-R fluxes, we have followed the perturbation technique to analyse the associated string dynamics. The scaling relation between the energy and the angular momentum has been computed for general n in the presence of mixed fluxes. We have further discussed the limiting n = 2 case in detail, which resembles to the long folded string. For pure NS-NS case, the result has been presented by naive geometric series. The analysis presented here can be extended in various ways. First, it will be interesting to look for the spiky string solution from the D-string in this background and look for the finite-size effect. It is worth investigating the finite-size effect of an oscillating (m, n)-string in the mixed flux background.

A Useful elliptic integrals and Jacobi elliptic functions
This appendix is about the definitions and some basic properties of elliptic integrals and also we write some relevant elliptic integrals which have been used in our paper. The incomplete elliptic integral of first kind is defined as where the range of modulus m and amplitude ϕ are 0 ≤ m ≤ 1 and 0 ≤ ϕ ≤ π 2 respectively. We write complete elliptic integral when amplitude ϕ = Here are some formulas, we need for our calculation

B Lambert W-function
In this appendix, we briefly discuss Lambert W-function which has been used throughout our paper to present the result. The multi-valued Lambert W-function is defined by the following relation W(z)e W(z) = z ⇔ W(ze z ) = z. (B.1) The Lambert W-function has two branches on real line i.e W 0 (x) and W − (x), the complex variable z replaced by real x. For x ≥ 0, there is only one real branch but for −1/e ≤ x < 0, W-function is double-valued. In this interval, the branch satisfying W(x) ≥ −1 denoted by W 0 (x) and W(x) ≤ −1 by W −1 (x). The principal branch W 0 (x) > 0 for x > 0 and

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.