On Small Black Holes in String Theory Bruno Balthazar Jinwei Chu David Kutasov Kadano Center for Theoretical Physics and Enrico Fermi Institute

2025-05-02 0 0 613.59KB 28 页 10玖币

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On Small Black Holes in String Theory

Bruno Balthazar, Jinwei Chu, David Kutasov

Kadanoﬀ Center for Theoretical Physics and Enrico Fermi Institute

University of Chicago, Chicago IL 60637

brunobalthazar@uchicago.edu, jinweichu@uchicago.edu, dkutasov@uchicago.edu

Abstract

We discuss the worldsheet sigma-model whose target space is the d+ 1 dimensional

Euclidean Schwarzschild black hole. We argue that in the limit where the Hawking

temperature of the black hole, T, approaches the Hagedorn temperature, TH, it can

be described in terms of a generalized version of the Horowitz-Polchinski eﬀective

theory. For d≥6, where the Horowitz-Polchinski EFT [1, 2] does not have suitable

solutions, the modiﬁed eﬀective Lagrangian allows one to study the black hole CFT

in an expansion in powers of d−6 and TH−T. At T=TH, the sigma model is

non-trivial for all d > 6. It exhibits an enhanced SU(2) symmetry, and is described

by a non-abelian Thirring model with a radially dependent coupling. The resulting

picture connects naturally to the results of [3–5], that relate Schwarzschild black holes

in ﬂat spacetime at large dto the two dimensional black hole. We also discuss an

analogous open string system, in which the black hole is replaced by a system of two

separated D-branes connected by a throat. In this system, the asymptotic separation

of the branes plays the role of the inverse temperature. At the critical separation, the

system is described by a Kondo-type model, which again exhibits an enhanced SU(2)

symmetry. At large d, the brane system gives rise to the hairpin brane [6].

arXiv:2210.12033v1 [hep-th] 21 Oct 2022

Contents

1 Introduction and summary 1

2 Eﬀective ﬁeld theory description of a small black hole 5

2.1 Beyond HP I: d= 6 −.............................. 7

2.2 Beyond HP II: d= 6 + ............................. 10

2.3 Relation to large Euclidean black holes . . . . . . . . . . . . . . . . . . . . . 13

3 Open string analog 14

4 Discussion 18

A Scaling analysis of the eﬀective action 20

B Derivation of the open string eﬀective action 21

1 Introduction and summary

In this note we continue our study [7] of the Horowitz-Polchinski (HP) string/black hole

transition in ﬂat spacetime [8].1This transition is often discussed in Lorentzian signature, but

we will focus on the Euclidean case, which is simpler, since one does not need to understand

the physics beyond the horizon of the black hole, or the singularity. The Euclidean and

Lorentzian problems are related, as discussed e.g. in [1].

The problem we will address can be posed as follows. A Euclidean Schwarzschild black

hole is a solution of Einstein gravity in an asymptotically ﬂat spacetime Rd×S1. It is

described by the metric

ds2=f(r)dτ2+dr2

f(r)+r2dΩ2

d−1.(1.1)

where (r, Ωd−1) are spherical coordinates on Rd,

f(r) = 1 −r0

rd−2,(1.2)

1We use many of the technical results of [7], but the overall picture we arrive at is diﬀerent.

r0is the Schwarzschild radius, which is related to the mass of the black hole via the relation

M=(d−1)ωd−1

16πGN

rd−2

0,(1.3)

ωd−1is the area of the unit (d−1)-sphere, GNis the d+1 dimensional Newton constant, and

τis Euclidean time, that lives on a circle of circumference β, equal to the inverse Hawking

temperature, β= 1/T . It is related to the Schwarzschild radius via the relation

β=4πr0

d−2.(1.4)

Since the background (1.1), (1.2) is obtained by solving the classical Einstein equations, it is

only valid for r0lp(the Planck scale). In weakly coupled string theory, there is a stronger

constraint, since classical string theory reduces to Einstein gravity only at distances much

larger than ls, the string scale. Thus, the regime of validity of (1.1), (1.2) is r0lslp,

where the second inequality is due to small string coupling, gs1.

For r0of order ls, the background (1.1), (1.2) is replaced by a worldsheet conformal ﬁeld

theory (CFT) which asymptotes to free ﬁeld theory on Rd×S1at large r, but is non-trival

at ﬁnite r. From the point of view of this CFT, r0(or β, (1.4)) parametrizes a conformal

manifold. The question is how does the CFT change when r0decreases from the classical

GR regime r0lsto r0∼ls. Of particular interest for the discussion of [1,8] is the nature

of this CFT in the limit where βapproaches the inverse Hagedorn temperature of string

theory in ﬂat spacetime, βH.

In this limit, the string mode that winds once around the Euclidean time circle becomes

massless [9–12]. Thus, if there is an eﬀective ﬁeld theory (EFT) description of the contin-

uation of the solution (1.1), (1.2) to this regime, the winding tachyon must be included in

it. Moreover, the winding tachyon is known to be non-zero in the solution. This is the case

already for large black holes [5, 13], and is expected for small ones as well.

A natural approach to the study of small Euclidean black holes is to write an eﬀective

action for the winding tachyon χ, the radion ϕ, that describes the variation of the radius of

the Euclidean time circle with the radial direction in Rd, and other light ﬁelds, like the dilaton

and the metric on Rd, and look for solutions of this action that have the same symmetries and

other properties as the Euclidean Black Hole (EBH). Horowitz and Polchinski (HP) wrote

the leading terms in this action in [1], and showed that for d < 6 it has suitable solutions.

It is natural to interpret the HP solution as the continuation of the EBH (1.1), (1.2) to

β∼βH(but, see [2] for a recent discussion of possible obstructions to this). Indeed, the

two solutions have some features in common. In particular, both involve a condensate of the

winding tachyon and break the U(1) winding symmetry. Furthermore, both solutions have

a ﬁnite classical entropy [1, 2, 7].

One problem with this interpretation is that the HP eﬀective action does not seem to

have solutions with the right properties for d≥6, while the black hole problem described

above appears to make sense there.2In section 2 of this note, we resolve this diﬃculty. To

do that, we treat das a continuous parameter, and focus (following our previous paper [7])

on the region near d= 6. We show that as d→6, the subleading terms to the ones that

were kept by HP need to be retained, and when one does that, a sensible picture emerges.

For d= 6 −with 0 ≤1, we show that the range of temperatures in which the

original HP solution is valid shrinks as →0. The behavior of the solution beyond this

region is sensitive to some subleading terms that were not included in the analysis of [1, 2].

Keeping these terms allows one to analyze the solution in this regime using the EFT, in a

double expansion in and β−βH. As β→βH, the solution of the modiﬁed equations for

all d≤6 goes to zero for all r, as in [1].

For d= 6 + , we ﬁnd that the eﬀective ﬁeld theory has solutions with the required

properties, whose existence is again due to the presence of the subleading terms in the

eﬀective action. The modiﬁed action that we study gives the leading behavior of the solution

in and β−βH. To compute higher order corrections, one needs to include higher order

terms in the eﬀective action. For of order one, the appropriate language to describe the

solution is the full classical string theory, i.e. the worldsheet CFT.

For d > 6, the solution does not go to zero as β→βH. We argue that at β=βH,

the corresponding CFT has an enhanced SU(2) symmetry. It is described by a certain

non-abelian Thirring model with an rdependent coupling, that was introduced in [7]. We

comment on the relation of the resulting picture to that of Euclidean black holes at large

d[3–5].

The resulting picture is reminiscent of the one found for two dimensional black holes (see

e.g. [14, 15]). The worldsheet CFT describing these black holes is exactly solvable, since it

corresponds to a coset CFT, SL(2,R)/U(1). Semiclassically, it describes a semi-inﬁnite cigar

geometry whose overall size is governed by k, the level of the underlying SL(2,R) current

algebra. The asymptotic radius of Euclidean time is given by R=√kls.

For large kone can view the CFT as a solution of two dimensional dilaton gravity, and

the stringy corrections are small. One of these corrections is a non-zero expectation value

of the tachyon winding around the Euclidean time circle. Since the radius of the circle far

from the tip of the cigar is large, this tachyon is very heavy in this regime, and its proﬁle

decays rapidly at inﬁnity. One can think of this tachyon as providing a non-perturbative (in

α0) correction to the worldsheet sigma-model.

2For large d, the authors of [5] provide strong evidence that the black hole CFT exists for all β≥βH. It

is natural to assume that this is the case for all d.

On the other hand, as kdecreases, the tachyon becomes lighter, and at3k= 2, its rate

of radial decay matches that of the geometric perturbation that deforms the asymptotic

cylinder R×S1to a cigar. At that point, the SL(2,R)/U(1) CFT develops an enhanced

SU(2) symmetry [16]. For k < 2, the eﬀect of the tachyon dominates over that of the

geometric deformation.

The above picture is known as the FZZ correspondence [15,17–20]. It plays an important

role in a number of applications of the two dimensional black hole in string theory, e.g.

[5, 15, 18]. The picture proposed in this note can be thought of as a generalization of the

FZZ correspondence to Euclidean black holes in asymptotically ﬂat spacetime.4

The analogy between the two cases is not perfect; for example, in the two dimensional

black hole, the parameter kthat controls the size of the black hole also controls the central

charge of the CFT, c= 3 + 6

k, while in ﬂat spacetime the corresponding parameter is the

mass of the black hole, and the central charge is independent of it. Also, the ﬂat spacetime

analysis gives rise to the analog of the region k≥2 in the two dimensional problem; β→βH

in ﬂat spacetime corresponds to k→2 in the SL(2,R)/U(1) EBH. At the same time, the two

systems are related via the large danalysis of [3–5]. The diﬀerences between them mentioned

above have a natural interpretation in that context.

In section 3 we discuss an open string analog of the EBH that describes two separated

parallel D-branes, as one varies the distance between them [2]. For large separation, there

is a solution of the DBI equations of motion where the branes are connected by a wide

tube [21], and one can ask what happens to this solution as the separation between the

branes decreases.

In particular, when the distance approaches a critical one, at which a string stretched

between the branes goes to zero mass, one can study the low energy eﬀective action that

includes this string. We show that this eﬀective action has a similar structure to the closed

string one, and thus the same kind of solutions. At the critical separation of the branes,

there is again an enhanced SU(2) symmetry (for d > 6), this time realized in terms of a

Kondo type Lagrangian (see [22] and references therein), with a coupling that depends on

the radial direction.

The resulting structure is again analogous to a known solution in a two dimensional

model – the hairpin brane in a linear dilaton space [6, 23–26]. We discuss this analogy, and

the role the hairpin brane plays in the D-brane system at large d.

3In the superstring; in the bosonic string the corresponding value is k= 4.

4Such a generalization was anticipated in [13].

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OnSmallBlackHolesinStringTheoryBrunoBalthazar,JinweiChu,DavidKutasovKadanoCenterforTheoreticalPhysicsandEnricoFermiInstituteUniversityofChicago,ChicagoIL60637brunobalthazar@uchicago.edu,jinweichu@uchicago.edu,dkutasov@uchicago.eduAbstractWediscusstheworldsheetsigma-modelwhosetargetspaceisthed+1dime...

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On Small Black Holes in String Theory Bruno Balthazar Jinwei Chu David Kutasov Kadano Center for Theoretical Physics and Enrico Fermi Institute

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