Quantum fluctuation on the worldsheet of probe string in BTZ black hole

Black holes have become a natural platform for us to understand spacetime and gravity at a semi-classical level after Stephen Hawking proposed the exciting theory of black hole radiation. Then, drawing inspiration from black hole thermodynamics, physicists proposed the holographic nature of gravity Susskind:1994vu ; tHooft:1999rgb , which then gives rise to the gauge/gravity duality Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj . The extensive applications of this duality provide novel perspectives for exploring gravity and strongly coupled system.

Recently, quantum information theory has begun to transcend its traditional framework, providing more and more insights in the field of quantum gravity such as computational complexity Osborne_2012 ; TCS-066 ; Dvali000 ; PhysRevA.94.040302 ; PhysRevD.96.126001 ; Watrous:2008 ; Bao:2018ira . The complexity essentially measures the difficulty of changing a quantum state into another state, however, it is still not clear to define the initial states and target states when one applies complexity into quantum field theory (QFT). Though considerable attempts have been made in this field Vanchurin:2016met ; Chapman:2017rqy ; Molina-Vilaplana:2018sfn ; Bhattacharyya:2018wym ; Nielsen:2005mkt ; doi:10.1126/science.1121541 ; Jefferson:2017sdb ; Yang:2018nda ; Bhattacharyya:2018bbv ; Bhattacharyya:2019kvj ; Camargo:2022wkd ; Adhikari:2022whf ; Adhikari:2021pvv ; Adhikari:2022oxr , a widely acceptable understanding of the complexity still poses an unresolved query. Thanks to holography, two elegant descriptions of complexity have been proposed from gravity side. One is the “complexity=volume (CV)” conjecture, where V represents the volume of the Einstein-Rosen (ER) bridge linking the two sides of the AdS black hole’s boundary. The other is the “complexity=action (CA)” conjecture, where A denotes the classical action of a space-time region enclosed by the bulk Cauchy slice anchored at the boundaries, also known as the “Wheeler-Dewitt (WdW)” patch Chapman:2016hwi ; Brown:2015bva ; Brown:2015lvg .

Based on the CA conjecture, there are many studies on the stationary systems, see for examples Pan:2016ecg ; Guo:2017rul ; Momeni:2016ekm ; Tao:2017fsy ; Alishahiha:2017hwg ; Reynolds:2017lwq ; Qaemmaqami:2017lzs ; Sebastiani:2017rxr ; Couch:2017yil ; Swingle:2017zcd ; Cano:2018aqi ; Chapman:2018dem ; Chapman:2018lsv ; Auzzi:2018pbc ; Yaraie:2018hwz ; Alishahiha:2018tep ; An:2018xhv ; Cai:2016xho ; Ghodsi:2020qqb ; Frassino:2019fgr and references therein. Similar efforts are evident in dynamic systems, such as the investigation of complexity growth with probe branes Abad:2017cgl ; Santos:2022lxj and the exploration of non-local operator effects in the BTZ black hole Ageev:2014nva ; Zhou:2021vsm ; Zhou:2023nza ; Nagasaki:2018csh ; Nagasaki:2019icm ; Bravo-Gaete:2020lzs ; Santos:2020xox ; Nagasaki:2021ldz ; Nagasaki:2022lll . Besides in Andi-de Sitter (AdS) spacetime, complexity in de Sitter (dS) spacetime was recently studied, and it was found that in this case the holographic complexity exhibits ‘hyperfast’ growth Susskind:2021esx ; Jorstad:2022mls . Later, the authors of Santos:2023eqp investigated the phase transition between the dS and AdS spacetime regimes based on the holographic entanglement entropy and the renormalization group flow, which could provide insight to understand more holographic properties, such as complexity, in different energy scales. Moreover, holographic complexity was found to have potential to describe the information emission from the rotating BTZ black hole encoded by quantum complexity Brown:2017jil , even at zero temperature Santos:2024zoh .Though considerable remarkable progress has been made in holographic complexity, a well-defined reference state in holography is also open. Therefore, the study of quantum fluctuation or perturbation of complexity could circumvent the state puzzle and help to step forward to investigate the complexity.

Thus, the aim of this paper is to study the second-order normal fluctuation on a probe string whose two points end in the dual boundary of the BTZ black hole. As a first attempt, our goal is to build a possible connection between the complexity growth and correlation function. The purpose we consider the normal fluctuations on the probe string mainly consists of two aspects. On one hand, the probe string in BTZ black hole can be treated as a 2-dimensional quantum field theory in the curved space-time background, similar as the handling with holographic Brownian motion proposed in deBoer:2008gu ; Atmaja:2010uu . Then in this scenario, various fluctuation modes could be excited on the string’s worldsheet due to Hawking radiation in the black hole environment Lawrence:1993sg . Here for convenience, we shall fix the end points of the string onto the boundary and focus on the normal direction fluctuation of the string. This operation allows us to obtain the normal operator of those excited normal modes along the string, as we will show soon. On the other hand, this consideration allows us to work with the string field theory/boundary conformal field theory (SFT/BCFT) correspondence, stating that every classical field in worldsheet of the probe string can be described by a BCFT of an open string attached to the boundary Fuchs:2008cc ; Taylor:2003gn ; Rastelli:2005mz ; Taylor:2006ye ; Kiermaier:2008qu ; Kudrna:2012re . In particular, using the AdS/BCFT scenario in Horndeski gravity Santos:2021orr , the authors of Santos:2024zoh found that the corrections provided in the probe string worldsheet in a rotating BTZ black hole indeed contribute to the action growth. In this framework, we can treat the excited fields as a single object and construct the corresponding vertex operator on the boundary. This then further helps us to calculate the correlation function of the bulk fields in the Schr$\ddot{o}$dinger functional representation and extract the curved-dependent worldsheet excited field.

The remaining of this paper is organized as follows. In Sec. II, we briefly review the BTZ black hole and the action growth of probe string model. In Sec. III, we consider second order normal fluctuation on the Nambu-Goto action of the probe string in the black hole and then we get the normal fluctuation operator. In Sec.IV, we explore the two-point correlation function of these excited modes and extract the field function on the worldsheet. We summarize our work in the last section.

We start with the 3-dimentional BTZ black hole with a negative cosmological constant of which the metric is PhysRevLett.69.1849 ; PhysRevD.48.1506

where $-\infty\leq t\leq\infty$ and $0\leq\phi\leq 2\pi$, $M$ is the black hole mass and $l$ is AdS radius. Then for convenience, we shall set $l=1$ and rewrite metric (1) under Poincare coordinate as,

where we have $z=\frac{1}{r}$, the horizon location $z_{h}=\frac{1}{\sqrt{M}}$ and the Hawking temperature $T=\frac{1}{2\pi\sqrt{M}}$.

We proceed to consider a probe string in this background spacetime and its two endpoints are attached in the boundary subspace. The configuration is shown in FIG. 1 where we have omitted the time direction.

Subsequently, we can work in the worldsheet coordinates $\tau=t$ and $\sigma=\phi$, and further perform $X^{\mu}(t,\phi)$ as

Then the Nambu-Goto action of this string is

where $T_{s}$ is the tension of string. $h_{ab}$ is the induced metric of worldsheet, which can be written in the form

with $g_{\mu\nu}$ the metric defined in (2). Then directly from (4), we can define the action growth of the probe string as

Further calculations on the above action growth has been addressed in Nagasaki:2018csh ; Zhou:2023nza ; Zhou:2021vsm , in which the authors discussed the significant effects of the graviton mass and black hole mass, similar to that from CV approach Zhou:2019jlh . In the following studies, we shall concern the quadratic fluctuation on this action growth.

We shall then adopt the normal coordinate gauge, under which the fluctuation is normal to the probe string everywhere, to expand the action growth into the quadratic order. The reason exists in that the normal coordinate gauge could be the safest choice comparing to other coordinate gauges, as addressed in Kinar:1999xu . The explicit geometric description of this fluctuation is shown in FIG.2, in which the dashed curve describes a fluctuated string on the original probe string.

To proceed, we expand the action growth (6) up to the quadratic term as

where the first term in the right side denotes the initial classical action growth. It is noticed that in general we can have infinite higher order terms but in our following study we consider till quadratic term. Then we shall assume that the fluctuation field is a scalar field $\xi_{n}$ and account for the contributions from the quantum fluctuations. To this end, we first expand the string coordinates around (3) as

Here we introduce the unit normal vector $n_{\mu}=(n_{z},n_{\phi})$ and consider the normal fluctuation on the coordinate. Subsequently, (8) can be rewritten as

Noted that there is no fluctuation on the time direction, which means that we keep the time coordinate fixed. The normal vector should satisfy the condition Kinar:1999xu

where $\lambda_{\phi}$ and $\lambda_{z}$ are the parameters satisfying the tangent vector equation of string

Now we have to fix the normal vector. The Hamiltonian of the string can be extracted from (4) as

which does not explicitly depend on $\phi$, so it is conserved. Then $z^{{}^{\prime}}$ could be written as

where $z_{*}$ is the turning point at the $\phi_{0}$ (see FIG.1). Then combining (11)-(13) and (15), we can solve out $n_{z}$ and $n_{\phi}$ as

Under the fluctuated string coordinates

the induced metric (5) can be expanded as

After straightforward calculation, we can expand the $\det[\tilde{h}_{ab}]$ up to the second order and obtain the quadratic term

where we have used the relation of the normal vector in (13). By defining a normal fluctuation operator as

we can rewrite the quadratic terms (19) as

which was proposed in Kinar:1999xu .

In principle, we can solve out $\xi_{n}$ from (21) by the equation of motion

however, it is a very difficult mission for us to go ahead. The difficulties come from two aspects : the first is that $\xi_{n}$ here are not a single field but a collection of all possible fields, so we cannot solve it by the separation method proposed for single filed Kinar:1999xu of which the authors treated it as an eigenvalue problem and separated $\phi(v,t)$ into $e^{i\omega t}\phi(v)$ to solve it. The second is that the equation $O_{n}\xi_{n}$ is highly complex, so analytical solution is a big challenge and numerical method is called for. Therefore, we shall shift our strategy and adopt an alternative approach to find $\xi_{n}$, which will be elaborated in the next section.

In this section, we shall endow the quantum fluctuation on the probe string with a potential physical process and discuss what we can construct. The probe string living in BTZ black hole environment should be inevitably affected by the Hawking radiation deBoer:2008gu ; Atmaja:2010uu ; Lawrence:1993sg , besides, combining (8) and (21) further inspires us to interpret $\xi_{n}$ as the fluctuation fields caused by the Hawking radiation. It is noted that random fluctuations on the worldsheet caused by Hawking radiation can also lead to a random motion of the endpoint of the string on the boundary, which is dubbed holographic Brownian motion deBoer:2008gu . But here our case is different because we fix the two endpoints of the string on the boundary and only consider the fluctuation on the worldsheet. Then one interesting question we shall ask is which kind of physical quantity in the fluctuations can correspond to complexity/action growth caused by this process. We shall do some analysis and expect to give insight in this issue.

As we aforementioned that the quantum fluctuations could be excited by any kind of field so that the sum in (21) should collect all possible fields. Fortunately, inspired by the strategy in string field theory (SFT) Taylor:2003gn ; Fuchs:2008cc ; Erler:2019vhl , we can treat this set of fluctuations as a single object, called excited string fields (ESF) $\Xi[\phi_{i},A_{\mu},...]$, where $\phi_{i}$ and $A_{\mu}$ are the scalar fields and $U(1)$ gauge field, while the ellipsis denote other possible excited fields. Moreover, in our setup the worldline of the string is swept out by the endpoints attached to the boundary, so all of these components collectively form boundary conformal field theory. Thus, we shall apply SFT/BCFT correspondence Fuchs:2008cc ; Taylor:2003gn ; Rastelli:2005mz ; Taylor:2006ye ; Kiermaier:2008qu ; Kudrna:2012re to proceed.

We define a worldsheet’s excited space , $\mathscr{H}_{BCFT}$ , which can be viewed as a tensor productor of “matter ” and “ghost” sectors Polchinski

where $m$ and $gh$ denote matter and ghost sectors, respectively. The ghost sector is a $bc$ system that is characterized by anticommuting, holomorphic, and anti-holomorphic worldsheet fields $b,\bar{b},c,\bar{c}$. Since the ghost sector comes from the gauge fixing process and its physics is usually not well understood, so here we do not consider the ghost sector for the fluctuations. For a state $\ket{\xi_{n}}\in\mathscr{H}_{BCFT}$ , there exits a corresponding boundary operator $V_{\xi_{n}}$, called vertex operator, so that we have

Here $\ket{0}$ is the $SL$(2, $\mathbb{R}$) vacuum in the boundary without vertex operator. Then we extend the vertex operator at the point $\phi_{0}$ as $\mathscr{V}(\phi_{0})=:e^{i\beta_{n}X^{\mu}(\phi_{0})}:$ of which the integral form is

Here, $::$ means the normal ordering, and $\beta_{n}$ is the charge of $\xi_{n}$ which plays the role of space-time momentum along the $X^{\mu}$. $\gamma$ is the induced metric in the boundary. For a physical state $\ket{\xi_{n}}$, it must be a BRST invariant state in BCFT at ghost number 1, satisfying Erler:2019vhl ; Polchinski

where $Q$ is the BRST operator and $\ket{\Lambda}$ is ghost number 0 state. Recalling the equation of motion (22), we find that it is natural to treat the normal fluctuation operator $O_{n}$ as the BRST operator $Q$.

Next, we use the $Schr\ddot{o}dinger$ representation for the excited worldsheet fields and consider Dirichlet boundary conditions of the probe string on both endpoints

Subsequently, since the background metric is conformally flat near the boundary, so we can expand $X^{\mu}(t,\phi)$ in (3) as Maccaferri:2023wrg

or more explicitly in the form

and the vertex operator (24) could be

Then we define the overlap

where $\Xi_{n}$ denotes the collection of excited fields on the worldsheet, depending on the curve $X^{\mu}$ in the background. It is noticed that $\Xi_{n}$ could also include the $b(\phi)$ and $c(\phi)$ fields, but here we do not consider this ghost sector. For the chosen point $\phi_{0}$ in the boundary, we calculate the correlation of the vertex operator in the worldsheet as

where $I$ denotes the Belavin-Polyakov-Zamolodchikov (BPZ) conjugation, $I(a)=-\frac{1}{a}$. That is to say, the $I\circ V_{\xi_{n}}(\phi_{0})$ corresponds to another point in the boundary. Thus, (32) contains two vertex operators, which are at $\phi_{0}$ and $-\frac{1}{\phi_{0}}$, respectively, and the formula of the path integral over the worldsheet is

Then we factorize this integration into three parts, saying outside $(z>\left|z(\phi)\right|)$, on $(z=\left|z(\phi)\right|)$, and below $(z<\left|z(\phi)\right|)$ the probe string,

by using the BPZ inverse, which takes the form

From (35), we can extract the induced string field on the worldsheet or the projection of excited fields of Hawking radiation on the worldsheet as

In the second equality, we change the integration variable in term of the facts that $dX^{\mu}=dX^{0}+d(\delta X)$ and $dX^{0}$ is fixed. In the third equality, we recall (8) and take $n_{\mu}$ as a constant vector along the worldsheet. Then considering that $\int d\xi e^{-S_{NG}^{(2)}}$ is a Gaussian integral, we shall further reduce the above formula into

which could be defined as the induced fields excited by Hawking radiation under the Schr$\ddot{o}$dinger functional representation. As expected, $n_{\mu}$ appears here indicates that these fields are polarized and normal to the worldsheet. One interesting property we can read off from (37) is when the string tension $T_{s}$ approaches to zero, the fields will blow up. This could be understood as that the structure of string worldhseet will be changed by the high Hawking temperature (Hagedorn temperature) when it goes near the horizon Giddings:1989xe ; Bowick:1989us ; Bagchi:2015nca .

Before closing this section, we shall present some discussions on our results. Firstly, we argue that $\Xi_{n}[X^{\mu}]$ can be defined as the induced field excited by Hawking radiation based on the fact that the fluctuation of worldsheet can be caused by Hawking radiation as the probe string is in black hole environment, even though the physical quantities related to Hawking radiation are not reflected in (37). It is noticed that besides Hawking radiation, the fluctuations may also be triggered by other mechanisms, such as a scalar field on the worldsheet as a defect Garriga:1991tb , but here we take more care of the worldsheet fluctuation itself , which means that fields $\Xi_{n}[X^{\mu}]$ are only curve-dependent. Secondly, in our analysis, we see that under the second-order fluctuation on the action/complexity growth, the two-point correlation emerge from the worldsheet’s perturbation. Moreover, (35) shows that the second order fluctuation of action growth of probe string indeed contributes to the correlation of vertex operators in the boundary conformal field theory in the form of functional path integrals. We argue that this may indicate a profound connection between complexity and the correlation function, which deserves further clarification.

There are plenty of schemes to define complexity in quantum field theory and conformal field theory. Relating with gravity, one also has two holographic versions of complexity. So further studies on complexity in a suitable framework will help us to further understand complexity in physics. In this work, we handle with complexity in string field theory and expect to shed light on connecting complexity with correlation function in quantum field theory and holography.

We investigated the second-order quantum fluctuation of a probe string ending on the boundary of the BTZ black hole background. Since the string is in black hole environment, we assumed that the unique normal fluctuation on the string is introduced by the Hawking radiation. Considering that the fluctuation can also affect the complexity or action growth of the string and will bring in a high-order correction on the Nambu-Goto action. So we calculate the second-order correction on the Nambu-Goto action and obtain the normal fluctuation operator.

Moreover, in the SFT/BCFT framework, by treating the excited fields on the worldsheet as a single object, and further defining them as excited string field, we found that the normal fluctuation operator also can be viewed as a BRST operator. Then we calculated the correlation function of vertex operators which is given in the form of functional path integrals. This corresponds to the correlation function of the excited fields on the worldsheet. We then extracted the induced field excited by Hawking radiation under the Schr$\ddot{o}$dinger functional representation. Our study shows that under the second-order fluctuation on the action/complexity growth, the two-point correlation can emerge from the worldsheet’s perturbation. This indicates that somehow we may explain the complexity as scattering amplitude, which could be an interesting direct, for example, one should clarify the definition of the minimal path in the scattering amplitude, and so on. We hope to further study the related topics in the future.