On the Hagedorn Temperature in Holographic Confining Gauge Theories Francesco Bigazzia Tommaso Cannetiab Aldo L. Cotroneab

2025-05-02 1 0 584.46KB 38 页 10玖币

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On the Hagedorn Temperature

in Holographic Conﬁning Gauge Theories

Francesco Bigazzia, Tommaso Cannetia,b, Aldo L. Cotronea,b

aINFN, Sezione di Firenze; Via G. Sansone 1; I-50019 Sesto Fiorentino (Firenze), Italy.

bDipartimento di Fisica e Astronomia, Universit´a di Firenze; Via G. Sansone 1;

I-50019 Sesto Fiorentino (Firenze), Italy.

bigazzi@ﬁ.infn.it, canneti@ﬁ.infn.it, cotrone@ﬁ.infn.it

Abstract

The divergence of the string partition function due to the exponential growth of states is

a well-understood issue in ﬂat spacetime. It can be interpreted as the appearance of tachyon

modes above a certain temperature, known as the Hagedorn temperature TH. In the lit-

erature, one can ﬁnd some intuitions about its generalization to curved spacetimes, where

computations are extremely hard and explicit results cannot be provided in general. In this

paper, we present a genus-zero estimate of TH, at leading order in α′, for string theories on

curved backgrounds holographically dual to conﬁning gauge theories. This is a particularly

interesting case, since the holographic correspondence equates THwith the Hagedorn tem-

perature of the dual gauge theories. For concreteness we focus on Type IIA string theory

on a well known background dual to an SU(N) Yang-Mills theory. The resulting Hagedorn

temperature turns out to be proportional to the square root of the Yang-Mills conﬁning

string tension. The related coeﬃcient, which at leading order is analytically determined, is

the same as the one for Type II theories in ﬂat space. While the calculation is performed

in a speciﬁc model, the result applies in full generality to conﬁning gauge theories with a

top-down holographic dual.

arXiv:2210.09893v3 [hep-th] 6 Jun 2023

Contents

1 Introduction 1

2 The classical string conﬁguration 3

2.1 The reference classical conﬁguration . . . . . . . . . . . . . . . . . . . . . . . 5

3 Semi-classical quantization 9

3.1 Fermionicmodes ................................. 13

3.2 Summary ..................................... 18

4 Mass-shell condition and TH19

4.1 The Hagedorn temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Conclusions 28

A About the spin-connection and the Ramond-Ramond ﬁeld strength of the

Witten background 29

A.1 Thespin-connection ............................... 29

A.2 The Ramond-Ramond ﬁeld strength . . . . . . . . . . . . . . . . . . . . . . . 31

A.3 The pull-back on the world-sheet of Dpat the tip of the cigar . . . . . . . . 31

B The generalized level-matching condition 32

1 Introduction

Theories with a density of states which grows exponentially with the energy are well deﬁned

below a certain temperature, called the Hagedorn temperature TH. The latter is deﬁned

as the temperature above which the partition function Zdiverges [1]. Examples of theories

with this behavior include string theories but also ordinary gauge theories, as pure Yang-Mills

(YM), where the spectrum of hadrons (glueballs) indeed is believed to grow exponentially.

The computation of the Hagedorn temperature for generic conﬁning theories is not an easy

task, even on the lattice, since THis larger than the critical temperature for deconﬁnement

(but see e.g. [2] for an estimate in pure YM and [3] for a discussion). In this paper we consider

conﬁning theories having a dual holographic description with a reliable supergravity regime.

In this case the Hagedorn temperature of the gauge theory is given by the one of the dual

string theory on a curved background.

The problem of calculating THis well understood for string theories in ﬂat space. The

starting point is the asymptotic formula for the number of all partitions of a large-integer

Ngiven by Hardy and Ramanujan [4]. In fact, it can be interpreted as the degeneracy of

the N-th level in the spectrum of a one-dimensional bosonic string theory (e.g., see [5]).

This result has been ﬁrst generalized in whatever dimension by Huang and Weinberg [6]

and then extended to the superstring case in several works [7–10]. The exponential behavior

of the resulting density of states is such that Zconverges only below a particular limiting

temperature, that is TH.

Unfortunately, it is not known how to compute Zfor a string theory on a generic curved

background. What we can always do is to expand the string action around a classical

conﬁguration. The divergence in Zwe are looking for seems to originate from temporal

winding modes which become tachyonic above TH[11–13]. In this direction, Atick and

Witten proposed a genus-zero method for computing THin ﬂat space [14]. Their results

reproduce the known values of THexploiting just the mass-shell condition of the theory,

instead of well-known expressions of Zin ﬂat space.

In this paper, we provide an estimate for the Hagedorn temperature of conﬁning theories

with a gravity dual, by applying the Atick-Witten genus-zero method to curved backgrounds,

in the Green-Schwarz (GS) formalism.

The estimate of THis derived from the semi-classical quantization of the string around

the (classical) temporal winding conﬁguration. The string is placed in the region of the

geometry corresponding to the deep infra-red regime of the dual ﬁeld theory. The winding

mode is supposed to be the lightest state becoming tachyonic as the temperature is increased,

providing the value of TH. The latter is an estimate, rather than the precise value, of the

Hagedorn temperature for at least three reasons: because we extrapolate the validity of

the semi-classical quantization down to zero mass of the classical conﬁguration; because we

cannot completely exclude the (unlikely) possibility that a diﬀerent conﬁguration becomes

tachyonic at a smaller temperature; and because, not being able to fully solve the string

theory on curved backgrounds, the result is correct only at leading order in α′.

The mode we consider is the simplest classical conﬁguration (i.e. it solves the equations

of motion and the Virasoro constraints) winding the temporal direction. It is basically (the

Euclidean counter-part of) the conﬁguration providing the ﬁeld theory string tension Tsin

holography or on the lattice. This elementary observation explains the direct link of THto

Ts.

In this paper we work out the explicit example of the Type IIA string on Witten’s back-

ground dual to a YM theory coupled to adjoint Kaluza-Klein modes [15]. Nevertheless, it

will be clear from the computation that the result is completely general in Type II theories

dual to conﬁning theories. The semi-classical quantization of the string provides eight free

bosonic modes and their superpartners.1We calculate the masses of these modes consis-

1See [16] for a geometrical approach to this problem.

tently with the absence of conformal anomaly. Their values are directly connected to the

ones found in the Wilson loop calculation [17].

With this spectrum at hand, we derive in the standard way the mass-shell condition,

which provides the temperature dependence of the mass of the (quantized) conﬁguration.

The condition of zero mass, giving the Hagedorn temperature, will correspond to vanishing

values of the stringy mode masses. Thus, the calculation reduces to the one in ﬂat space, up

to the value of the string tension.

The resulting estimate for the Hagedorn temperature reads

TH=pTsr1

4π,(1.1)

where Tsis the string tension in the dual ﬁeld theory. We will also extract a subset of the

leading α′corrections to this result.

For Type II strings in ﬂat space the result is the same as (1.1) with Ts= 1/2πα′. In

general, for stringy models the expectation is that TH/√Ts∼1/√cef f , where ceff is some

eﬀective central charge (see e.g. [18, 19]). Clearly the point is to understand in each case

what are the correct values of Ts,ceff . Our explicit computation provides evidence to the

intuitive expectation that, in Type II theories dual to conﬁning models and at leading order

in α′,Tsis the dual ﬁeld theory string tension and ceff is the same as in ﬂat space.

The rest of the paper is organized as follows. In section 2 we describe the classical conﬁg-

uration winding the temporal direction, which will be the base of our calculation. Section

3 describes the semi-classical quantization of this conﬁguration and the mass spectra of

world-sheet bosonic and fermionic modes. These data are employed to derive the mass-

shell condition and, imposing vanishing mass of the quantized mode, the estimate for the

Hagedorn temperature in section 4. We conclude with a few comments in section 5. The

appendices contain further technical details of the computations.

Note added: after the completion of this work we have realized that, due to subtleties in

the light-cone gauge quantization, some statements in this paper are not accurate, although

the main results, i.e. the mass-shell condition (4.43) and the leading order expression (1.1)

for TH, turn out to be correct. We refer to [20] for a more rigorous treatment of the problem,

in which we explain how to deal with the non-physical (gauge) modes in a proper way and,

in particular, we consistently impose the light-cone gauge condition that removes them from

the spectrum.

2 The classical string conﬁguration

In this section, we aim to describe the string background dual to the so-called Witten-Yang-

Mills (WYM) theory and the classical conﬁguration that will be quantized in the following.

WYM is the theory on the world-volume of a stack of ND4-branes wrapped on a circle

with anti-periodic boundary conditions for fermions [15]. At low energies, it reduces to four-

dimensional SU(N) Yang-Mills coupled to massive adjoint Kaluza-Klein (KK) modes. In

the regime where the theory has a reliable supergravity dual, i.e. the strongly coupled planar

(large N) limit, the KK modes are at the same mass scale as the glueballs. The theory

displays linear conﬁnement and a mass gap and it is believed to be in the same universality

class of pure YM.

The Type IIA supergravity background dual to the WYM theory in the conﬁning phase

at ﬁnite temperature Tis given by

ds2=1

0u

R3/2δµν dxµdxν+4

9f(u)dθ2+R

u3/2du2

f(u)+R3/2u1/2dΩ2

0=u0

R3, f(u) = 1 −u3

u3, eϕ=gs

u3/4

R3/4, R = (πNgs)1/3α′1/2, F4= 3R3ω4,

(2.1)

where µ,ν= 0,1,2,3 and ω4is the volume form of the transverse S4. Notice that, with

this notation, the coordinates xµare dimensionless. The above ten-dimensional string frame

metric, the dilaton ϕand the constant Ramond-Ramond ﬁeld strength F4make up the

so-called Witten background, taken with Euclidean signature and a time direction which is

compactiﬁed on a circle of length 1/T . Here, u∈[u0,+∞) is the holographic coordinate

and θis an angular coordinate θ≃θ+ 2πwhich parametrizes a shrinking circle along the

u-direction. Its radius is asymptotically ﬁxed by the inverse of the glueball and KK mass

scale

MKK =3

2m0(2.2)

and vanishes at u=u0. The value u=u0corresponds to the position of the tip of the cigar

in the (u, θ)-plane. The region u∼u0is dual to the IR regime of the WYM theory.

In the following, we will ﬁnd a classical conﬁguration containing winding modes in the

temporal direction and around which we will expand the world-sheet action for type IIA

closed strings localized at the tip of the cigar. The latter requirement comes from the fact

that we search for the lighter winding mode which solves the equations of motion. In order

to compute a generalized mass-shell condition, we just need to know the ﬁeld content of the

action and so we have to expand it up to quadratic order in quantum ﬂuctuations. THwill

be deduced ﬁxing to zero the mass of the physical ground state.

The discussion of fermions in a Euclidean background is often problematic. So we will

adopt the strategy of Wick-rotating the x1-direction in (2.1) and work with a Lorentzian

background given by

ds2=1

0u

R3/2eηµν dxµdxν+4

9f(u)dθ2+R

u3/2du2

f(u)+R3/2u1/2dΩ2

0=u0

R3, f(u) = 1 −u3

u3, eϕ=gs

u3/4

R3/4, R = (πNgs)1/3α′1/2, F4= 3R3ω4,

(2.3)

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OntheHagedornTemperatureinHolographicConfiningGaugeTheoriesFrancescoBigazzia,TommasoCannetia,b,AldoL.Cotronea,baINFN,SezionediFirenze;ViaG.Sansone1;I-50019SestoFiorentino(Firenze),Italy.bDipartimentodiFisicaeAstronomia,Universit´adiFirenze;ViaG.Sansone1;I-50019SestoFiorentino(Firenze),Italy.bigazzi@...

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On the Hagedorn Temperature in Holographic Confining Gauge Theories Francesco Bigazzia Tommaso Cannetiab Aldo L. Cotroneab

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