# Perturbation of pulsating strings

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## Abstract

We discuss semiclassical quantization of circular pulsating strings in \( \text {AdS}_3 \times \text {S}^3 \) background with and without the Neveu-Schwarz–Neveu-Schwarz (NS–NS) flux. We find the equations of motion corresponding to the quadratic action in bosonic sector in terms of scalar quantities and invariants of the geometry. The general equations for studying physical perturbations along the string in an arbitrary curved spacetime are written down using covariant formalism. We discuss the stability of these string configurations by studying the solutions of the linearized perturbed equations of motion.

## 1 Introduction

The study of string dynamics in curved space-time has always been an exciting area of research in string theory. In the framework of AdS/CFT correspondence, the quantization of semiclassical string [1] has been extremely useful in establishing integrability structure [2] and dualities between two sides. The linearized perturbation of semiclassical string is helpful in matching the duality beyond the leading order. The main motivations behind studying perturbative solutions are to investigate the stability properties of the string solutions and to find the quantum string corrections to the expectation value of the Wilson loops.

A non-covariant approach was introduced in Refs. [3, 4, 5] to study the worldsheet fluctuation of string in curved space-time, especially in black hole or de-Sitter backgrounds which led to finding the physical quantities like mass spectrum and scattering amplitudes. On the other hand, a covariant approach in conformal gauge was developed in Ref. [8] and was used to find the perturbations of stationary strings embedded in Rindler, Schwarzschild and Reisner–Nordstrø̈m spacetimes. It has been found that the frequencies of first order as well as second order fluctuations of the string in (2+1)-dimensional black hole and black string backgrounds are real, which shows string is stable in these backgrounds [9]. It was shown further that perturbations of circular strings in a power law expanding universe grow much slower than the radius, implying the perturbation is suppressed by the inflation of the universe [10]. Similarly perturbation of the planetoid string in Ellis geometry and in (2+1)-dimensional BTZ black hole, shows the presence of world-sheet singularity at the edges [11]. All the above works were based on the perturbation of bosonic string action.

The study of worldsheet fluctuation of the classical string in \(\text {AdS}_5\times \text {S}^5\) background in Green–Schwarz formalism was developed in Refs. [12, 13] which was an important step in extension of the AdS/CFT correspondence beyond the classical level. It was shown that in conformal and static gauge, the one loop correction to Green–Schwarz action can be expressed in terms of differential operators, where the determinants of these operators give rise to well defined and finite partition function. The quadratic quantum fluctuation of the rigidly rotating homogeneous string solution [14, 15, 16, 17] helps to find out the stability condition of the string dynamics. It has been shown that the pulsation enhances the stability of the multispin string solitons [18]. In case of non-homogeneous string solution [19, 20] the quadratic fluctuation is expressed in terms of single-gap-Lamé operator. This approach has also been applied in \(\text {AdS}\) backgrounds [21]. There are some instances where the two loop worldsheet fluctuations have been carried out to match the subleading correction to cusp anomalous dimension of the strongly coupled gauge theory [22, 23, 24]. The pohlmeyer reduction formalism has also been used to find the quantum fluctuation of the semiclassical strings in \(\text {AdS}_5\times \text {S}^5\) where it matches with the results of one loop computation [25]. Some recent work in this line has been done in Refs. [26, 27].

The other interesting example of holographic dual pair is that of string propagation in \(AdS_3 \times S^3 \times T^4\) background and the dual \(N = (4,4)\) superconformal field theory. This duality has also been well explored from both sides and various semiclassical solutions in this background have been studied in the context of integrability [28, 29, 30]. String theory in this background supported by NS–NS type flux can be described in terms of a *SL*(2, *R*) Wess–Zumino–Witten model. It has been suggested recently that in \(AdS_3\times S^3\) background supported by both NS–NS and RR fluxes \(( H_3 = dB_2 ~\mathrm{and} ~F_3 = dC_2 )\), the string theory is integrable as well [31, 32]. In this context a class of semiclassical string solutions have been studied in some detail in [33, 34, 35]. The dynamics of such solutions in pure NS–NS flux case both in the large and small charge limit have been discussed. The fluctuations around rigidly rotating string in presence of NS–NS flux have been considered in Refs. [36, 37, 38]. It was shown that the presence of flux couple the fluctuation modes in a non-trivial way that indicates the changes in quantum correction to the energy spectrum. It would be interesting to understand the one loop corrections to the energy of pulsating strings at one loop level as well.

Motivated by recent surge of interest in studying semiclassical strings and their perturbations, in this paper we study the worldsheet perturbation of pulsating strings in \(\text {AdS}_3 \times \text {S}^3\) in the presence of background NS–NS flux. The pulsating strings are one of simple string solutions whose gauge theory duals are known to exist and their anomalous dimensions has been found out by Ref. [39]. For very large quantum numbers, the circular pulsating string expanding and contracting on the S\(^5\). These solutions are time-dependent as opposed to the usual rigidly rotating string solutions. They are expected to be dual to highly excited states in terms of operators. For example the most general pulsating string in \(S^5\) charged under the isometry group *SO*(6) will have a dual operator of the form \(\mathrm{Tr}( X^{J1}Y^{J2}Z^{J3})\), where \(X, Y, \mathrm{and}~Z\), are the chiral scalars and \(J^i\)’s are the R-charges of the SYM theory. Hence it will be interesting to know their fate for small worldsheet fluctuations in the presence of NS–NS flux. The perturbation of spiky strings in flat and AdS space, in covariant formalism, has been discussed in Refs. [27, 40].

The rest of the paper is organized as follows. In Sect. 2, we review the worldsheet perturbation formalism in the presence of background flux. In Sect. 3 we construct the analytical solution of the perturbation equation for pulsating string in \( R \times S^2 \) background in short string limit and comment on the stability of such solutions. Sections 4 and 5 are devoted to study the physical perturbations of the pulsating strings in short string limit in \(AdS_3\) background with and without three form fluxes. Finally, in Sect. 6, we present our conclusion.

## 2 Worldsheet perturbation of bosonic strings with NS–NS flux

*i*is a number index and \(\psi ^\alpha \) are just reparametrizations.

In general, the above equations give a set of coupled second order partial differential wave equations for scalar quantities. These coupled wave equations are manifestly covariant under worldsheet diffeomorphisms.

## 3 Perturbation of pulsating string in \(R \times S^2\)

*m*is the winding number. The corresponding Polyakov action becomes,

*n*, the first solution is a oscillatory while the second one is periodically diverging. Observing the Fig. 1, one can comment that when

*n*increases amplitude of perturbation decreases for \(P_1\). We can also see that when the string becomes point-like, \(P_1\) vanishes while \(P_2\) blows up which gives rise to instability. The physical perturbations for short string solution can be written as

## 4 Perturbation of pulsating string in \(AdS_3\)

*AdS*space whereas it spreads over the \(\theta \) direction with winding number

*m*. We can find the equation of motion for \(\rho \),

*AdS*energy is given by,

*AdS*energy

*n*, out of which one is oscillatory and other one is divergent solution. The scalar function \(\varphi (\tau , \sigma )\) in the perturbation equation takes the form

## 5 Pulsating string in \(AdS_3\) with NS–NS B-field

*q*

*B*-field influences the solution through its functional form as well as the parameters present there. Therefore, the perturbation equations are different as the classical field equation is different. It is easy to check that the physical perturbations of the pulsating strings in AdS background with flux is more stable compared to the case when there is no flux.

## 6 Concluding remarks

In this paper, first we discuss the general formalism for the construction of perturbation equation using Polyakov action and in the presence of NS–NS flux. The geometric covariant quantities like normal fundamental form and extrinsic curvature tensor have been introduced to write the perturbation equations. By using pulsating string ansatz, we obtain the solution of the equations of motion and constraints in different subspaces of \(AdS_3 \times S^3\) background. In our analysis, we find the resulting perturbation equations in form of hypergeometric differential equations. The solutions to the perturbation equations have been found to be linear combination of oscillatory and diverging part. We find the physical perturbation by taking only the oscillatory part of the solution.

We wish to mention that the second order fluctuation of pulsating strings in \(AdS_5\times S^5\) was discussed in Ref. [20] by single gap Lamé form. In the present paper, we generalize the results in the presence of NS–NS flux. Further, we write down explicit solutions to the perturbation equations by reducing them to problems in the exactly solvable models. The covariant formalism essentially presents the quadratic action of the string in bosonic sector in terms of scalar quantities and covariants of the geometry and it is an elegant method of studying the perturbations. This work is believed to be helpful in finding the first order correction to the energy which correspond to the anomalous dimension of the gauge theory operators in strong coupling regime. Further extension of perturbation of fermionic part of the Green–schwarz action would be worth attempting. We wish to comeback to this issue in future.

## Notes

### Acknowledgements

We would like to thank S. P. Khastagir and Sayan Kar for some useful discussions.

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