Microstates of a 2dBlack Hole in string theory Panos BetziosaOlga Papadoulakib aDepartment of Physics and Astronomy University of British Columbia

2025-04-29 0 0 1.39MB 79 页 10玖币

侵权投诉

Microstates of a 2dBlack Hole in string theory

Panos Betzios,aOlga Papadoulakib

aDepartment of Physics and Astronomy , University of British Columbia,

6224 Agricultural Road, Vancouver, B.C. V6T 1Z1, Canada

bPerimeter Institute for Theoretical Physics , Waterloo,

Ontario N2L 2Y5, Canada

E-mail: pbetzios@phas.ubc.ca,opapadoulaki@perimeterinstitute.ca

Abstract: We analyse models of Matrix Quantum Mechanics in the double scaling limit

that contain non-singlet states. The ﬁnite temperature partition function of such systems

contains non-trivial winding modes (vortices) and is expressed in terms of a group theo-

retic sum over representations. We then focus in the case when the ﬁrst winding mode is

dominant (model of Kazakov-Kostov-Kutasov). In the limit of large representations (con-

tinuous Young diagrams), and depending on the values of the parameters of the model such

as the compactiﬁcation radius and the string coupling, the dual geometric background cor-

responds to that of a long string (winding mode) condensate or a 2d(non-supersymmetric)

Black Hole. In the matrix model we can tune these parameters and explore various phases

and regimes. Our construction allows us to identify the origin of the microstates of these

backgrounds, arising from non trivial representations, and paves the way for computing

various observables on them.

Keywords:

Matrix Quantum Mechanics, Black Holes, Long Strings, Phase transitions, Group Repre-

sentations

arXiv:2210.11484v2 [hep-th] 28 Oct 2022

Contents

1 Introduction 1

2 The 2dEuclidean black hole and the FZZ duality 6

3 The matrix model with winding (vortex) perturbations and its dual 10

4 Partition function and representations 12

4.1 Expanding the τ-function in terms of representations 12

4.2 Measures in the space of representations 13

4.3 Microstates and the origin of entropy (thermodynamic limit) 16

5 The limit of continuous representations 17

5.1 Eﬀective action and saddle point equations 17

5.2 Determining the resolvent 21

5.3 The stable regime (R < RKT )22

5.3.1 A cut with no saturation of the density 22

5.3.2 Saturated cut 27

5.4 The unstable regime (R > RKT )29

6 Thermodynamics from continuous representations 32

7 A completely gauged ZZ/FZZT model 36

7.1 Canonical partition function 38

7.2 Grand Canonical partition function 38

8 Discussion 39

A Partitions - Representations 43

A.1 Partitions 43

A.2 Composite representations 44

B Free fermions and Integrable Hierarchies 45

B.1 Fermionic Algebra 46

B.2 Free fermions, currents and Young diagrams 48

B.3 The τ-function as a sum of partitions 49

B.3.1 Schur polynomials 50

B.4 τ-function from the reﬂection amplitude 51

B.5 τ-function from the gauge ﬁeld zero modes 54

– i –

C MQM in a ﬁxed representation 55

C.1 Free energy in a ﬁxed representation 55

C.1.1 From the reﬂection amplitude 55

C.1.2 From the gauge ﬁeld zero modes 59

C.2 Limit of continuous representations 60

C.3 Determining the density of boxes from a coherent state 63

D Saddle point equations for the model with dynamical bifundamentals 65

E A method to determine the general resolvent 66

F Determinantal formulae for the τfunction 67

F.1 A determinantal formula in the space of shifted weights 67

F.2 Determinantal formulae in Frobenius coordinates 67

F.3 The τfunction in terms of two coupled normal matrix models 69

G Ungauged Matrix Quantum Mechanics 70

G.1 The canonical partition function in the holonomy basis 72

1 Introduction

One of the most interesting physical systems, whose complete description requires a deep

understanding of the merging of quantum mechanics and gravity are Black Holes. The per-

tinent questions one would like to address are related to the spacetime physics of horizons

and singularities as well as to the possibility of acquiring a microscopic description of their

macroscopic thermodynamic properties such as their entropy. Traditional (worldsheet)

string theory can be used to describe only a few aspects of them, since it is mainly based

on perturbation theory (and is well developed around very symmetric backgrounds). Non

perturbative objects, such as D-branes greatly expanded the tools and prospects that string

theory has to attack this problem from a microscopic perspective. This eventually led to

some remarkable results in the study of supersymmetric (extremal) black holes, for which

non-renormalization theorems guarantee an explicit counting of their microstates using the

dual D-brane system [1]. Even though after the advent of the AdS/CF T correspondence,

we are in a rare position to have an in principle non-perturbative description of quantum

gravity in asymptotically AdS spaces, it is still fair to say that there are various aspects of

black holes1, even in AdS, that still defy a complete understanding.

The situation of course is even worse in the case of non-supersymmetric black holes

for which even a method to count their microstates is not available (unless one can invoke

the Cardy formula in special examples), let alone when one wishes to describe realistic

examples in asymptotically ﬂat spacetimes, such as the common Schwarzschild black hole.

1Most of them have to do with the structure and properties of their interior and singularities.

– 1 –

The main motivation behind our work is to ﬁnd a (matrix) model that is solvable (or at

least analysable via saddle point methods at large N), that can describe microscopically a

(non-supersymmetric) black hole in string theory2. Of course this is a tall order and the

price we have to pay is that such a model can be explicitly constructed in non-critical lower

dimensional versions of string theory, the most prominent example based on the duality

between c= 1 Liouville theory and Matrix Quantum Mechanics (MQM) in a double scaling

limit. With such a model at hand one would hope to address the deepest questions related

to black holes, such as the spacetime physics behind the horizon and the nature of the

black hole singularity, if the lessons to be learned exhibit any form of universality across

dimensions (see the discussion section 8for more details).

Most of the past works on MQM (some excellent reviews are [2–5]) have focused in its

singlet sector using an SU(N) gauged version of the model, that reduces to the dynamics of

(non-relativistic) free fermions in an inverted oscillator potential. The culmination of these

works resulted in matching numerous physical observables with the dual c= 1 Liouville

string theory on a linear dilaton background as well as making (exact) predictions for

observables that are not yet computable from the string theory side. In contrast, all the

eﬀorts to uncover some aspect of black hole physics in high energy scattering using the

singlet sector of MQM [6–8] proved fruitless, demonstrating thus that states resembling

black holes can only (potentially) exist in the non-singlet sector of the theory3. This

result is consistent with the inﬁnite W1+∞symmetry of the quantum inverted oscillator

that is in conﬂict with the expected thermalising and “chaotic” properties of systems that

are holographic duals of black holes4. At this point we should emphasize that it is not

entirely certain which properties of higher dimensional black holes should still persist in

the two-dimensional case. Some expected properties characteristic of black holes, are the

appropriate scaling of the entropy and the mass of the black hole with the string coupling,

and the quasi-normal mode behaviour for the retarded two point functions of localised

probes. Other expectations include the presence of an eﬀective non-zero absorption, and

an enhanced particle production when scattering highly energetic particles (strings) that

can form a black hole5.

Turning on to the non-singlet sectors, there have been two main proposals for models

that are dual to a two dimensional black hole in the literature [11,13]. They are closely

related, since they both involve the liberation of vortices on the string worldsheet [14–16]

and the presence of non-trivial winding modes around the target space thermal circle [17].

In the model of [11], it was demonstrated that the entropy scales in a consistent way with

the expectations coming from the Euclidean black hole background and its dual Sine-

Liouville theory [37–39]. On the other hand in their model, [11] did not clearly identify

2The matrix model is there to provide an in principle non-perturbative description of the black hole.

3An exponential degeneracy of states can be found more generally in models for which the aforementioned

fermions carry additional indices, such as those arising from dimensional reduction on extra compact space

dimensions, see for example [9].

4Recent works have argued that black holes should exhibit a fast scrambling and chaotic behaviour

(which can ascertained by the analysis of four point OTOC’s) [10].

5For example the 2 →Namplitude should peak at high values of Nwith the production of many “soft

particles” that would resemble Hawking radiation.

– 2 –

the origin of the microstates accounting for this entropy and neither provided a real time

description of the physics. Some subsequent developments related the analytic continuation

of Euclidean winding modes, with long-strings that stretch and scatter on the linear dilaton

background [18,19], but they focused in the regime when they do not backreact on the

geometry.

In relation to these works, in [13] it was found that even though the model of [11] does

not have an a priori obvious analytic continuation into Lorentzian signature, it nevertheless

appears in a particular scaling limit of a more general class of MQM models which do have

a real time description. These models contain in addition to the original N×NMQM

matrix Mij (and non dynamical gauge ﬁeld Aij), bi-fundamental ﬁelds χαi transforming

under an SU(Nf)×SU(N) symmetry6(that source the MQM SU(N) non-singlets that

were originally projected out). Going to the matrix eigenvalue basis, they can be equiva-

lently described in terms of dynamical SU(Nf) spin-Calogero models [13,59] in an inverted

oscillator potential. The models of [13] are well deﬁned both in Euclidean and Lorentzian

signature, and their Euclidean partition function constitutes a generalisation of that ap-

pearing in [11], obeying a discrete (Hirota-Miwa) soliton equation instead of the simpler

Toda diﬀerential equation. They are also related to the ungauged version of MQM, but

contain additional parameters (fugacities for vortices/winding modes) as the model of [11]

does. In addition due to the relation of the models of [13] with FZZT branes, these parame-

ters have a natural interpretation from the Liouville side: The masses of the bifundamentals

are related to the boundary cosmological constant via: 2m=σ , µB=√µcosh πσ,Nfto

the number of FZZT branes and so forth, see [13] for more details.

At this point we should mention a known diﬃculty relating the model of [11] with an

object that behaves like an actual target space black hole. As we review in section 2, this

matrix model is most directly related to Sine-Liouville theory, which in turn is related to

the SL(2, R)k/U(1) WZW coset description of the 2dblack hole [32,33,35], by a form

of strong/weak duality (FZZ duality [42]). The issue is that the coset is an actual CFT

only for a certain compactiﬁcation radius R2=k= 9/4 that is of string scale (in units

where α0= 1). This brings us to another important topic, that of the black hole - string

transition [20–23]. In short the basic idea is that once black holes become small and reach

the string scale (the so called correspondence point), the gravitational semi-classical black

hole geometry ceases to be a good description and is replaced by a condensate of (long)

strings. For the particular case of the two-dimensional black hole, it can be shown that

the winding mode becomes non-normalisable for radii R2=k < 3. This means that it is

explicitly sourced and below k= 3 the bosonic black hole ceases to be a normalisable state

in the theory [44,45], leaving only a long string condensate to survive, for more details

see the end of section 2. Of course this signals a possible trouble for the interpretation

of the WZW coset as a black hole for such a string scale radius, when it is actually a

CFT. Nevertheless, even though we do not currently have access to an exact worldsheet

description, generally one does anticipate the existence of black hole solutions for a wide

6The SU (Nf) symmetry in these models is a global symmetry. This comes from the open strings ending

on NfFZZT branes. We construct in section 7a new model for which both symmetries are gauged.

– 3 –

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Microstatesofa2dBlackHoleinstringtheoryPanosBetzios,aOlgaPapadoulakibaDepartmentofPhysicsandAstronomy,UniversityofBritishColumbia,6224AgriculturalRoad,Vancouver,B.C.V6T1Z1,CanadabPerimeterInstituteforTheoreticalPhysics,Waterloo,OntarioN2L2Y5,CanadaE-mail:pbetzios@phas.ubc.ca,opapadoulaki@perimeterin...

展开>> 收起<<

Microstates of a 2dBlack Hole in string theory Panos BetziosaOlga Papadoulakib aDepartment of Physics and Astronomy University of British Columbia.pdf

共79页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Microstates of a 2dBlack Hole in string theory Panos BetziosaOlga Papadoulakib aDepartment of Physics and Astronomy University of British Columbia

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: