Counting Black Hole Microstates in String Theory

Jorge Laraña Aragón1 1Theoretical Physics Division, Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden. [email protected] Advisor: Giuseppe Dibitetto Subject Reader: Ulf Danielsson

In this project, we explore the area of black holes in String Theory. String Theory has had several successes in describing properties of black holes. Recent progress in String Theory points towards the possibility that black holes should not be thought of as fundamental objects, but rather as statistical descriptions of a huge number of smooth horizonless microstate geometries. We study this deep connection between the microscopic description of black holes and String Theory. The main goal is to understand and learn how some of the modern techniques in String Theory can be applied to model black holes, in particular, calculating an expression for the entropy. The main idea is to construct black hole solutions from intersecting certain types of branes, in the context of Supergravity theories that emerge as low-energy limits of Superstring theories. With this microscopic approach, the aim is to count the number of microstates and then compare with the macroscopic Bekenstein-Hawking entropies. We plan to construct such solutions, for both supersymmetric and non-supersymmetric black holes. Also, this microscopic origin of the black hole entropy could provide new insights to the black hole information paradox.

Sammanfattning

I detta projekt behandlar vi svarta hål i strängteori. Svarta hål är några av naturens mest fascinerande objekt. De är indirekt observerade, som mörka objekt av stjärnstorlek, vilka ﬁnns i binära system, eller som supermassiva objekt i galaxers centra. Svarta hål är vanligare förekom- mande i universum än vad vi tidigare har trott. Uppskattningsvis ﬁnns 100 miljoner mörka objekt enbart i Vintergatan. Svarta hål dyker upp i allmän relativitetsteori som klassiska materielösningar som gravitationellt har kollapsat till en punkt, en singularitet. Från en teoretisk synvinkel förser svarta hål oss med laboratorier för tankeexperiment, där nya teoretiska ideér kan testas. 1976 upptäckte Stephen Hawking att svarta hål inte är så svarta - de sänder ut värmestrålning motsvarande en karaktäristisk temperatur, Hawking-temperaturen. Ter- miska egenskaper, såsom Bekenstein-Hawking-entropi, associerades nu med svarta hål. 1996 härled- des denna entropi med strängteoretiska metoder. Man bestämde de mikroskopiska frihetsgraderna för ett speciellt slags svarta hål, och sedan dess har strängteori skördat många framgångar inom forskningsområdet. Här studerar vi detta djupa samband mellan strängteori och den mikroskopiska beskrivningen av svarta hål. Huvudmålet är att förstå hur vissa moderna strängteoretiska tekniker kan appliceras på svarta hålmodeller, framför allt när entropin ska beräknas. 2

CONTENTS

I. Introduction 4

II. Black Holes are not that black 5 A. Black Hole Thermodynamics 5 1. Zeroth Law 5 2. First Law 5 3. Second Law 6 4. Third Law 6 B. Temperature as cyclic imaginary time 7 C. Macroscopic Black Hole Temperatures and Entropies 8 1. Schwarzschild black hole 8 2. D-dimensional Schwarzschild black hole 10 3. Reissner-Nordström black hole 12 4. D-dimensional Reissner-Nordström black hole 14

III. String Thermodynamics 16 A. The Hagedorn temperature 16 B. Counting partitions 17 1. Bosonic partitions 17 2. Fermionic partitions and generalisation 19 C. Entropy and temperature of a relativistic string at rest 20 1. Bosonic string 20 2. Fermionic string 22 3. Open superstring 22 4. Closed superstring 23 D. The string partition function 24 E. The Hagedorn temperature, the black hole entropy and the stretched horizon 26

IV. The Long String construction and the Random Walk model 28 A. Long strings are entropically favoured 28 B. The size of a string state and the Random Walk model 30 C. Estimation of the Schwarzschild black hole entropy 31 D. Recent developments 33

V. Black Holes from Brane intersections 33 A. D=5 3-charge Extremal Reissner-Nordström black holes 34 1. Construction in the Supergravity approximation 34 2. Microstate counting 37 B. D=4 4-charge Extremal Reissner-Nordström black holes 38 1. Construction in the Supergravity approximation 38 2. Microstate counting 41 C. Dual brane conﬁgurations 41 1. From D1-D5-W to D0-D4-F1 42 2. From D2-D6-NS5-W to D2-D2-D2-D6 43 D. Counting in the dual CFT 43 1. Generating functions 44 2. Saddle point approximation 45 3. Bosonic strings 46 4. Type II superstrings 47 E. Non-extremal Black Holes 49 1. D=5 3-charge Non-extremal Reissner-Nordström black holes 49 2. D=4 4-charge Non-extremal Reissner-Nordström black holes 53

VI. The Black Hole Information Paradox and the Microscopic Interpretation 56

VII. Conclusions 56 3

Acknowledgments 58

References 58 4

I. INTRODUCTION

Black holes are among the most fascinating objects in Nature. They are indirectly observed, either as stellar mass dark objects in binary systems or as supermassive ones at the centres of galaxies and some of the most energetic emissions in the universe are associated with matter accreting onto black holes or black hole mergers. Black holes are more common in the universe than we used to think. It has been estimated that the Milky Way contains around 100 million black holes. From a theoretical point of view, black holes provide laboratories for gedanken experiments where we can generate and test new theoretical ideas. They appear in General Relativity as classical solutions to matter that has gravitationally collapsed to a point, known as a singularity. They are one of the most interesting scenarios to test physical theories because they have extreme conditions and it is where both physical descriptions of General Relativity and Quantum Mechanics clash apparently leading to a contradiction. That is why the real puzzles regarding black holes arise at the quantum level.

In 1976, Hawking found out black holes are not that black, instead, they emit thermal radiation at a characteristic temperature called the Hawking temperature [1]. This was the beginning of Black Hole Thermodynamics. Thermal properties were associated with black holes, such as the so called Bekenstein-Hawking entropy. This later led to the famous black hole information paradox [2]. The thermal properties of black holes are diﬃcult to understand at the fundamental level of statistical mechanics. However, it is natural to consider the possibility of calculating the black hole entropy by using this standard statistical mechanics framework provided that we are able to count the number of black hole microstates that a black hole can be in. This is sometimes referred as the black hole entropy problem. Together with the black hole information paradox they point out the necessity of a quantum theory of gravity.

String Theory has recently turned out to be very interesting for black hole physics, in particular, in understanding the microscopic origin of the black hole entropy since in 1996, the Bekenstein-Hawking entropy expression was derived by counting microstates for extremal and charged black holes [3], conﬁrming the relation between the macroscopic entropy with the microscopic construction from string theory. String theory then allows us to calculate the temperature and entropy of black holes from the density of the associated microstates, at least for certain types of black holes. For those, the string statistical approach makes it possible to compute the entropy, at least, up to numerical factors. The results agree with the Bekenstein-Hawking entropy. Furthermore, for a few cases where the black holes are invariant under a certain amount of supersymmetry (BPS black holes) the calculations give the exact numerical coeﬃcients. To some extent, string theory is capable to solve the problems of entropy and information considering black holes as ordinary quantum mechanical systems (which respect unitarity). This could revolutionise our understanding of black holes and quantum gravity in general: the classical black hole solution would be the analogue of the thermodynamic description of a gas, while the horizonless microstates would be the analogue of the statistical description of this gas.

Interestingly, the study of black holes has provided key insights into the nature of gravity and its interplay with quantum mechanics. Black holes and their higher-dimensional generalisations have played a crucial role in the development of modern string theory, in advancing our understanding of non-perturbative string dynamics and contributing to paradigm shifts such as string duality and the holographic principle. Indeed, it was the scaling properties of black hole entropy that revealed the holographic nature of gravity, in other words, its fundamental degrees of freedom live in fewer dimensions than the classical limit indicates [4, 5]. The non-extensive character of the black hole entropy (scaling with the area) and the large number of degrees of freedom led to the development of a holographic representation of gravity, called holographic duality and also known as gauge/gravity correspondence, which relates a gravitational theory in d+1 space-time dimensions with a negative cosmological constant to a strongly coupled quantum ﬁeld theory description of a non-gravitational system in d dimensions.

This project has the following structure. In the first chapter, section II, we will review the theory of Black Hole Thermodynamics consisting of four laws analogous to Classical Thermodynamics and we will perform semiclassical calculations of the temperature and entropy for different types of black holes, not using the original Hawking’s approach (the Bogoliubov formalism in the context of QFT in curved spacetime) but instead, a more modern approach based on path integral arguments. We try to justify the method and then apply it to two specific types of black holes: the Schwarzschild and the Reissner-Nordström solutions. We do this for both the realistic case of four spacetime dimensions as well as for the general case of an arbitrary number of spacetime dimensions. Later, in section III, we address thermodynamical aspects of String Theory and the so called concept of the Hagedorn temperature. By 5 counting partitions we count string microstates from where we can derive expressions for the entropy in terms of the energy. We perform these calculations in a simple way going to the high-temperature limit for different types of strings: bosonic, fermionic and superstrings, considering the cases of open and closed strings. Then, we try to compute an expression for the partition function of the open bosonic string and see how the Hagedorn’s temperature arises naturally. In the next chapter, section IV, we briefly present a way of modelling the Schwarzschild black hole as a long string with the so called Random Walk model. In chapter V, we deal with the main topic of this project constructing black holes from intersecting different types of branes. We compute the entropies in both the macroscopic supergravity description and in the microscopic approach using the previously derived results and we compare results for two particular cases in 4 and 5 dimensions. After presenting a few examples of dual configurations with the same counting of states, we explain how the microstate counting can be performed from the perspective of the dual conformal field theory (CFT) which was the procedure originally followed. This approach not only gives the expected results but interesting stringy corrections to the entropy. Last, we also study the two particular cases in the near extremal limit obtaining the same results than in the extremal cases. Finally, we finish this project discussing some implications of this work in relation to the famous Black Hole Information Paradox, section VI, and then we make some final remarks in section VII.

II. BLACK HOLES ARE NOT THAT BLACK

One of the most surprising results of combining Quantum Mechanics and General Relativity is the fact that black holes are not as black as it was thought at the beginning. Quoting Stephen Hawking: “Black holes ain’t as black as they are painted”. Indeed, they emit thermal radiation. As a consequence, it is possible to characterise them with thermal properties such as temperature and entropy and even build a theory of black hole thermodynamics from the analogy between the classical laws of black hole mechanics and the laws of thermodynamics, as we will see in this section.

A. Black Hole Thermodynamics

The four laws described below were proposed by J. M. Bardeen, B. Carter, S. W. Hawking in 1973 [6]. The ﬁrst three laws are proved in the previous reference and in [7] a proof for the third law is given. The mathematical expressions for the laws are given in units where c = ~ = k = 1.

1. Zeroth Law

This law is analogous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It states that the surface gravity κ of a stationary black hole is constant over the event horizon. Therefore, it establishes the analogy between surface gravity and temperature T at the event horizon of the black hole giving by the relation: κ T = . (1) BH 2π

2. First Law

The ﬁrst law of thermodynamics is an energy conservation law that relates the total change of internal energy U with the change in entropy S, volume V and number of particles N:

dU = T dS − P dV + µidNi. (2)

Similar to this, this law relates the change in the black hole mass M with the change of its area A, charge Q and angular momentum J: κ dM = dA + ΦdQ + ΩdJ, (3) 8πGD 6 where GD is Newton’s constant in D-dimensions, Φ is the electric potential and Ω is the angular velocity.

3. Second Law

As the ﬁrst law was already pointing out, there is a relation between the area A of the black hole and the entropy S, this relation is the famous Hawking-Bekenstein formula: A SBH = . (4) 4GD Thus, the second law of black hole thermodynamics takes the form:

δSBH ∼ δA > 0, (5) this last expression is known as the Hawking’s area theorem. We can think that the previous arguments lead to a violation of the standard thermodynamical laws. In particular, a violation of the second law of thermodynamics, which states that the total entropy of an isolated system always increases or remains constant over time, but never decreases. As the black hole evaporates, its mass decreases and therefore, does its entropy as well. In addition, one could think about the following example. Let us suppose an observer is orbiting around a black hole. The observer has a box with a gas at non-zero temperature. If the box is thrown towards the black hole, once it disappears behind the horizon, the amount of entropy of the gas in the box would have been erased from the universe and thus, the total entropy of the universe would have decreased. In 1973, J. D. Bekenstein realised about this problem [8]. In order to solve this entropy decrease, he suggested generalising the second law of Thermodynamics including the contribution of black hole entropy to the total entropy of the universe. So then, when the box full of gas falls into the black hole, that decrease of entropy will be compensated with an increase in the entropy of the black hole. To sum up, he reformulated the second law as it follows:

δSgen = δ(Sconv + SBH ) > 0, (6) where Sgen represents the total generalised entropy of the universe, Sconv the conventional entropy and SBH the entropy of black holes. Nevertheless, we would like to have a better understanding of the entropy formula, in other words, we would like to directly derive it from ﬁrst principles. The best way to do this is by counting the quantum physical degrees of freedom associated with a black hole. The key point is then whether the entropy can be computed via a statistical mechanics derivation, as in ordinary physical systems. Consequently, the black hole entropy would be nothing but the density of microstates corresponding to the macrostate of the black hole characterised by its mass, charge and angular momentum (no-hair theorem). This new insight of the area law requires a complete quantum theory of gravity and, as we will see, considerable progress was made in the frame of superstring theory. Another interesting feature of the Bekenstein-Hawking formula is its universality, i.e., it applies to all possible types of black holes. This idea is very puzzling and it led to the idea of holography in an attempt to explain the origin of this formula for the entropy [4, 5].

4. Third Law

The third law of thermodynamics states that it is impossible for any physical process to reduce the entropy of a system to its absolute zero value in a finite number of operations. Analogously, the third law of black holes thermodynamics states that it is impossible by any procedure, no matter how idealised, to reduce the surface gravity κ of a black hole to zero by a finite sequence of operations. However, this law is a bit subtle. At this point, an apparent contradiction appears. We will see later that extremal black holes have a vanishing surface gravity while having a non-vanishing finite entropy. To try to reconcile both laws we can think of another form of the classic third law of thermodynamics which says that the entropy of a system at absolute zero is a well-defined constant (and maybe not necessarily zero). This is because a system at zero temperature exists in its ground state and this could have an intrinsic entropy. More precisely, this happens when the ground state is degenerate. In other words, it is exactly the degeneracy of the ground state that leads to a non-vanishing entropy at zero temperature. Thus, at least, this formulation of the third law is not problematic in the case of extremal black holes since they are systems with a degenerate ground state. 7

B. Temperature as cyclic imaginary time

An elegant and direct way of computing black Hole temperatures is based on Quantum Field Theory (QFT) at finite temperature, specifically, using the path integral formalism in the context of Euclidean field theory [9]. In this section, we will describe and justify the method that we will later apply to different types of black holes.

The key idea starts from the similitude between the unitary evolution operator in Quantum Mechanics e−iHt/~ and the Boltzmann weight in Statistical Mechanics e−βH , representing the probability of finding a state in a configuration at thermal equilibrium at a temperature β = 1/kT , where T and k are the temperature and the Boltzmann constant respectively. This suggests an identification t ∼ ~β. In fact, both objects are related by analytic continuation as we will see. From quantum statistical mechanics we can write the thermal partition function of a quantum system characterised by its Hamiltonian H as: X Z(β) = Tre−βH = hn| e−βH |ni . (7) n We look for a path integral representation of this quantum partition function. The path integral formulation allows us to express the time evolution of a system between an initial and a final state, respectively qi and qf as:

Z i R ∆t −iH∆t/~ dtL(q,q ˙ ) hqf | e |qii = Dq(t)e ~ 0 , (8)

1 2 where L(q, ˙ q) = 2 mq˙ − V (q) is the Lagrangian of the system. So we are integrating over all possible paths q(t) with the conditions qi = q(0) and qf = q(∆t). Since we want this integral to converge (cancellation of oscillatory phase factors from different paths), we perform a Wick rotation to Euclidean time t → iτ rotating the integration contour in the complex plane: Z −1 R ∆t dτL(q,q ˙ ) Dq(τ)e ~ 0 , (9) which is the so called Euclidean path integral. Now, in order to get the path integral representation of our partition function Z(β) we make the already mentioned replacement ∆t → ~β and we also note that the trace operation sets |qii = |qf i so the path integral must have a periodic boundary condition (PBC) q(0) = q(β): Z −1 R ~β dτL(q,q ˙ ) Z(β) = Dq(τ)e ~ 0 . (10) PBC Finally, we can extend the result to a field theory in d = D + 1 dimensions defining a Lagrangian density such that L = R dDxL(φ), where φ is a bosonic scalar field: Z −1 R ~β dτ R dD xL(φ) Z(β) = Dφ(x, τ)e ~ 0 , (11) PBC with the P BC φ(x, 0) = φ(x, ~β).

The result is that by Wick rotating and imposing PBC it is possible to relate an Euclidean QFT in D+1 dimensions with a periodicity condition in time 0 ≤ τ < β to a QFT at finite temperature in D-dimensional Minkowski (or quantum statistical mechanics description). The physical interpretation is that the Wick rotation provokes that the quantum field perceives time as periodic with period ~β and the quanta of the scalar field would think that they are living in a thermal bath with temperature T = 1/kβ. This argument is not evident, which is probably why the temperature was not first computed in this way.

From a fundamental point of view, what remains unclear in this sense is how a temperature is equivalent to a cyclic imaginary time. It is suggested that there might be something deeper to understand at this respect.

The relevance of all this is that we will calculate temperatures directly from imaginary periodicities in time as we will see in the following. 8

C. Macroscopic Black Hole Temperatures and Entropies

As we have already said, black holes actually emit radiation like a black body at a certain temperature. In this subsection, we show simple calculations to obtain expressions for the temperature and entropy for different types of black holes, making explicit the aforementioned procedure. Note the first calculation of the temperature was not performed with this technique of the imaginary periodicity of time that we will use here, but instead involving Quantum Field Theory in curved spacetime. This was done for the first time by S. Hawking [1] obtaining a thermal spectrum from which one can read directly the temperature for the Schwarzschild black hole. In the following, we work in units where c = ~ = k = 1.

1. Schwarzschild black hole

Let us consider ﬁrst, for simplicity, the 4-dimensional Schwarzschild solution for a totally general, static and spherically symmetric gravitational body of mass M:

R R −1 ds2 = − 1 − S dt2 + 1 − S dr2 + r2dΩ2 , 4 r r (2) where RS = 2G4M is the so called Schwarzschild radius, with G4 the usual 4-dimensional Newton’s constant and 2 2 2 2 2 dΩ(2) represents the 2-dimensional diﬀerential element of solid angle dΩ(2) = dθ + sin θdφ . In principle, this metric has two singularities in r = 0 and r = RS, but it turns out to be that only the ﬁrst one is a true singularity while the second one is just a coordinate singularity (i.e., it disappears for a given choice of coordinates, for example in Kruskal coordinates). However, the surface at r = RS is still physically of interest. The fact that for a certain solid angle the metric becomes timelike for r > RS and spacelike for r < RS indicates the presence of an event horizon exactly at this value of the radius, which protects the internal singularity at zero radius of being a naked singularity, according to Penrose’s cosmic censorship conjecture. Now, we are going to describe a series of steps based on this idea of considering the temperature as an imaginary periodicity in Euclidean time that will allow us to compute the Hawking temperature:

1. The ﬁrst key point consists of performing the Wick rotation, a complex rotation on the coordinate time t:

t → −iτ ⇒ dt2 = −dτ 2, (12)

so the metric takes the following form:

R R −1 ds2 = 1 − S dτ 2 + 1 − S dr2 + r2dΩ2 . (13) 4(E) r r (2)

Note that in Euclidean space with the new time coordinate τ the metric has the same sign for the time and spatial components.

2. For convenience, we want a change of coordinates such that in the new coordinate ρ(r) the horizon is located at the origin r(ρ = 0) = RS, so we perform a horizon translation ρ = r − RS. Including a factor to make the calculations a bit easier we can write:

r = RS(1 + ρ) ⇒ dr = RSdρ, (14)

and the metric takes the form: ρ 1 + ρ ds2 = dτ 2 + R2 dρ2 + R2 (1 + ρ)2dΩ2 . (15) 4(E) 1 + ρ S ρ S (2)

3. Next, we take the so called near horizon limit ρ → 0. We use the Taylor expansion of the geometric series to write the following fraction up to second order in ρ: ρ = ρ(1 − ρ + O(ρ2)), (16) 1 + ρ 9

so in the near horizon limit the metric reduces to: dρ2 ds2 = ρdτ 2 + R2 + R2 (1 + 2ρ)dΩ2 . (17) 4(E) S ρ S (2)

4. We again look for a new radial coordinate R such that when the metric is written in this new coordinate, its radial component is the unity, gRR = 1. In this way, if we only consider radial trajectories for a ﬁxed angular direction into the black hole, we have dΩ(2) = 0 and thus, the metric starts to be similar to the 2-dimensional Euclidean ﬂat space:

ds2 = R2dα2 + dR2, (18)

where R and α are the radius and the angle in polar coordinates respectively. dρ √ We look at gρρ and to reduce it to the unity in the new coordinates we impose dR = RS ρ , so by solving this simple diﬀerential equation and inverting we get the change of coordinates:

R 2 ρ(R) = , (19) 2RS so the metric becomes:

2 2 R 2 2 ds4(E) = 2 dτ + dR . (20) 4RS By identifying the Euclidean time with the polar angle coordinate we write: dτ dα = , (21) 2RS and the Euclidean time in terms of the polar angle is given by:

τ = 2RSα. (22)

We can check indeed that the metric ﬁnally reduces to the one in (18), which is a parametrisation in polar coordinates of a disc. This result is not surprising since it makes sense that the observer will see the horizon as a disc when falling radially into the spherical black hole. 5. Finally, as we saw before, α is the polar angle so that it must have a periodicity α → α + 2π, in order to avoid conical singularities. Imposing this condition using (22) we get the periodicity in τ:

α → α + 2π,

τ → τ + 4πRS. (23)

The inverse of the temperature β = 1/T can be directly taken as the periodicity in τ, thus the Hawking temperature TBH obtained for the 4-dimensional Schwarzschild black hole is: 1 1 TBH = = . (24) 4πRS 8πG4M

From this result, we can also calculate the entropy applying the already mentioned ﬁrst law of Black Hole Ther- modynamics. In this case, there is no thermodynamical work in the system dU = T dS, and all the internal energy is due to the mass of the black hole dU = dM, hence: 1 dS dM = . (25) 8πG4 M We impose the condition that when the black hole has totally evaporated the entropy also has to vanish, M = 0 ⇒ SBH = 0 and we get:

2 SBH = 4πG4M . (26) 10

We can also check that the Bekenstein-Hawking formula holds in this case if we consider the event horizon as the 2 2 surface S located at the Schwarzschild radius, which area is simply A = 4π(2G4M) , so then:

A 2 SBH = = 4πG4M , (27) 4G4 which coincides with the result from the thermodynamical approach (26).

2. D-dimensional Schwarzschild black hole

In total analogy to the 4-dimensional case, it is possible to generalise this result to a D-dimensional Schwarzschild black hole repeating the same procedure. In D dimensions the 4-dimensional Schwarzschild black hole invariant under SO(3) can be generalised to a D-dimensional Schwarzschild black hole invariant under SO(D-1) [10]:

" # " #−1 R D−3 R D−3 ds2 = − 1 − S(D) dt2 + 1 − S(D) dr2 + r2dΩ2 , (28) D r r (D−2) where RS(D) is the D-dimensional Schwarzschild radius

1 D−3 16πGDM RS(D) = , (29) (D − 2)A(D−2)

n with Newton’s constant GD in D dimensions and A(n) is the surface area of the unit n-sphere S

Z Z n−1 n+1 n Y i 2π 2 An = dΩ = dϕ sin θidθi = n+1 . (30) n n Γ( ) S S i=1 2

Now, we repeat exactly the same steps than in the former procedure:

1. Again we start doing the Wick rotation

" # " #−1 R D−3 R D−3 ds2 = 1 − S(D) dτ 2 + 1 − S(D) dr2 + r2dΩ2 , (31) D(E) r r (D−2)

2. We perform the same change of coordinates to translate the horizon to the origin (from now on we drop the angular part of the metric since it does not play any role in our calculation)

" # " #−1 1 D−3 1 D−3 ds2 = 1 − dτ 2 + R2 1 − dρ2, (32) D(E) 1 + ρ S(D) 1 + ρ

3. Taking the near horizon limit

" D−3# 1 D−3 1 − = 1 − 1 − ρ + O(ρ2) = (D − 3)ρ + O(ρ2), (33) 1 + ρ

R2 dρ2 ds2 = (D − 3)ρdτ 2 + S(D) . (34) D(E) (D − 3) ρ

4. Performing a new change of coordinates in the radial variable

R dρ (D − 3) dR = S(D) ⇒ ρ = R2, (35) 1/2 √ 2 (D − 3) ρ 4RS(D) 11

the metric results

2 2 (D − 3) 2 2 2 dsD(E) = 2 R dτ + dR . (36) 4RS(D)

From this last expression of the metric we obtain:

(D − 3) 2R dα = dτ ⇒ τ = S(D) . (37) 2RS(D) (D − 3)

5. From the periodicity condition we have: 4πR β = S(D) , (38) (D − 3)

and the Hawking temperature turns out to be:

1 D−3 (D − 3) (D − 3) (D − 2)A(D−2) TBH = = . (39) 4πRS(D) 4π 16πGDM

Finally, we can check that this last expression is consistent with the one from the 4-dimensional case (24). We just need to note that the expression for the D-dimensional Schwarzschild radius (29) gives the expected value of RS(4) = 2G4M. This works out since for D = 4 we have A(2) = 4π.

Again, from this last expression for the temperature (39) we can calculate the entropy applying the ﬁrst law of Black Hole Thermodynamics dM = T dS and we have:

1 (D − 3) (D − 2)A D−3 dM = (D−2) dS. (40) 4π 16πGDM We can now integrate the diﬀerential equation in M with the same condition than before and we get:

1 D−3 4π 16πGD D−2 SBH = 4 M D−3 , (41) (D − 2) (D − 2)A(D−2) which can be re-expressed in terms of the Schwarzschild radius,

D−2 A(D−2)RS(D) SBH = . (42) 4GD Not surprisingly, it is possible to get this same result for the entropy from the Bekenstein-Hawking formula, taking into account the D-dimensional generalisation of the area of the black hole:

D−2 AD = A(D−2)RS(D), (43) and therefore,

D−2 AD A(D−2)RS(D) SBH = = . (44) 4GD 4GD Note that both expressions for the entropy, the ﬁrst law of Black Holes Thermodynamics and the Bekenstein-Hawing formula (close related with the second law), have the same form independently of the dimensionality of the black hole. The dimension of the black hole is taken into account by the Hawking temperature. In this sense, these laws of Black Hole Thermodynamics are universal. 12

3. Reissner-Nordström black hole

First of all, we start considering the 4-dimensional case for simplicity like we did with the Schwarzschild black hole. The Reissner-Nordström black hole is the generalisation of a Schwarzschild black hole including an electric charge Q but no angular momentum. It is still therefore a stationary solution with spherical symmetry given by:

2G M G2Q2 2G M G2Q2 −1 ds2 = − 1 − 4 + 4 + 1 − 4 + 4 dr2 + r2dΩ2 . (45) r r2 r r2 (2)

The metric presents a true physical singularity for r = 0. But also, there are two values of the radius r±, often called inner and outer horizon, that makes gtt = 0, these are:

p 2 2 r± = G4M ± r0, r0 = G4 M − Q , (46) and thus we can rewrite the metric as:

(r − r )(r − r ) (r − r )(r − r )−1 ds2 = − + − dt2 + + − dr2 + r2dΩ2 . (47) r2 r2 (2)

However, neither of the two values for the radius is a curvature singularity (they could be removed by a change of coordinates). Instead, it turns out that r+ is an event horizon whereas r− is a Cauchy horizon. It is clear than in (46) we need to impose the physical condition |Q| ≤ M in order to have real roots. On the contrary, if |Q| > M the metric has a naked singularity at r = 0. The particular case |Q| = M and thus r± = G4M is physically of interest. This corresponds to the situation of maximal charge and it is called the extremal Reissner-Nordström solution. Its physical relevance concerns the relation between black holes and quantum gravity. In supersymmetric theories, extremal black holes can leave certain symmetries unbroken. The saturation of the bound for the charge is equivalent to the so-called BPS bound, which implies, precisely, that the extremal Reissner–Nordström black hole solution has some unbroken supersymmetry, which makes certain calculations a lot easier as we will see. We will ﬁrst focus on the general case (not necessarily extremal) to derive expressions for the temperature and entropy following the same procedure than before. 1. Wick rotation:

(r − r )(r − r ) (r − r )(r − r )−1 ds2 = + − dτ 2 + + − dr2 + r2dΩ2 . (48) r2 r2 (2)

2. Change of coordinates (dropping the angular part): r = r+(1 + ρ), dr = r+dρ. −1 2 r+ρ[r+(1 + ρ) − r−] 2 r+ρ[r+(1 + ρ) − r−] 2 2 ds = 2 2 dτ + 2 2 r+dρ . (49) r+(1 + ρ) r+(1 + ρ)

3. Taking the near horizon limit ρ → 0 up to O(ρ2):

r − r r3 dρ2 ds2 = + − ρdτ 2 + + . (50) r+ r+ − r− ρ

4. Change of coordinates to R such that gRR = 1,

3 1/2 2 r+ dρ r+ − r− R dR = √ ⇒ ρ = 3 , (51) r+ − r− ρ r+ 4

2 2 2 (r+ − r−) R 2 2 ds = 4 dτ + dR , (52) r+ 4

2 (r+ − r−) 2r+ dα = 2 dτ ⇒ τ = α. (53) 2r+ r+ − r− 13

5. Cyclic periodicity in Euclidean time:

4πr2 β = + , (54) r+ − r− and ﬁnally, the Hawking temperature takes the form:

p 2 2 r+ − r− r0 1 M − Q TBH = = = . (55) 2 2 p 2 2 2 4πr+ 2πr+ 2πG4 (M + M − Q )

In principle, we could calculate the entropy as we did before from the ﬁrst law of black hole thermodynamics:

dM = T dS + ΦhdQ, (56)

Q where Φh = Φ(r+) = is the electrostatic potential evaluated at the horizon. However, this procedure is rather r+ complicated since the diﬀerential equation not only depends on M but also on Q. Instead, in this case, we will use directly the Bekenstein-Hawking formula that we were checking in former cases, taking into account that the area of 2 the event horizon is now A = 4πr+, so the entropy gives:

πr2 2 A + p 2 2 SBH = = = πG4 M + M − Q . (57) 4G4 G4 Furthermore, we will examine the particular case of extremality. The metric of an extremal Reissner–Nordström black hole takes the form:

r − G M 2 r − G M −2 ds2 = − 4 dt2 + 4 dr2 + r2dΩ2 . (58) r r (2)

Taking the extremality condition |Q| = M in the results (55) and (57) we can calculate the temperature and entropy for the extremal case, obtaining:

2 TBH, extremal = 0,SBH = πG4M , (59) which clearly violates the third law of black hole thermodynamics getting a non-vanishing entropy when the temperature is zero. However, in this case, our procedure has a problem as one can easily see from the expression for the radius R (51), which diverges in the extremal case: r+ = r−. In order to solve this, we apply once again the Euclidean time procedure directly to the extremal metric (58) following the series of steps.

1. The metric (58) written in Euclidean time and ignoring the angular components is:

G M 2 G M −2 ds2 = 1 − 4 dτ 2 + 1 − 4 dr2. (60) r r

2. Change of coordinates: r = r+(1 + ρ) with r+ = G4M,

ρ 2 ρ −2 ds2 = dτ 2 + G2M dρ2. (61) 1 + ρ 4 1 + ρ

3. Let us take the near horizon limit ρ → 0 but note that this time the ﬁrst non-vanishing contribution is O(ρ3), so the metric reduces to:

2 2 2 2 2 −2 2 ds = ρ dτ + G4M ρ dρ . (62)

4. Change of coordinates to R such that gRR = 1, dρ dR = G M ⇒ ρ = eR/G4M , (63) 4 ρ 14

ds2 = dR2 + e2R/G4M dτ 2. (64)

5. Instead of getting the metric of a disc ds2 = dR2 + R2dα2 and imposing the periodicity condition to avoid a conical singularity, we ended up with the metric of an hyperbolic plane which has no singularities. Thus, one does not need to impose any periodic condition since the period is inﬁnite β → ∞ corresponding indeed to:

TBH, extremal = 0. (65)

Despite the problematic behaviour of our previous calculation in the extremal case, we got the same vanishing temperature results. This is a very important result because this is the only type of black hole for which a reliable microscopic calculation of the entropy from string theory has been done. This apparent contradiction of the third law of black hole thermodynamics is very interesting. We will see that we get the same result from string theory when constructing black hole solutions from branes and in fact, for extremal black holes we will recover the Bekenstein-Hawking formula for the entropy from counting microstates. However, it appears that the cases of extreme dyonic black holes in D = 4 and D = 5 are the only ones which lead to a non-vanishing Bekenstein-Hawking entropy, as it is argued in [11]. These are the two cases discussed in this project. Extremal black holes cannot be obtained as limits of non-extremal ones since it is impossible for non-extremal black holes to transform into extremal ones through a series of physical processes. This is studied in [12] for processes such as Hawking radiation and superradiance. Both cases are not connected and this works in both directions: neither it is possible to study non-extremal black holes from near-extremal limits nor to extrapolate results from extremal black holes to non-extremal ones. Despite that, we will see in section V.E. that we are able to reproduce the Bekenstein-Hawking entropy in the near-extremal limit, although the result of this calculation is not robust enough. All in all, in order to reconcile the extremal black holes behaviour with the third law of black hole thermodynamics, this law can be thought in terms of the impossibility to reduce the temperature (i.e., the surface gravity) of a black hole to zero by a ﬁnite sequence of operations, in other words, the extreme limit cannot be reached by a series of physical processes, as we have already said.

Finally, it is interesting to study the asymptotic behaviour in the near horizon limit. In order to do so, we deﬁne the constant R0 := G4M and we perform the change of coordinates rˆ = r − R0,

R −2 R 2 ds2 = − 1 + 0 dt2 + 1 + 0 drˆ2 + (ˆr + R )2dΩ2 , (66) rˆ rˆ 0 (2) and we now take the near horizon limit rˆ → 0,

R −2 R 2 ds2 = − 0 dt2 + 0 drˆ2 + R2dΩ2 , (67) rˆ rˆ 0 (2)

2 R0 and again, performing a new change of coordinates ρ = rˆ we get:

R 2 ds2 = 0 (−dt2 + dρ2) + R2dΩ2 . (68) ρ 0 (2)

2 which corresponds to a metric of the form AdS2 × S , where R0 is the 2-dimensional anti-de Sitter(AdS) radius and also the radius of the 2-sphere.

4. D-dimensional Reissner-Nordström black hole

The most general static higher dimensional charged black hole solution with spherical topology is given by the generalisation of the former 4-dimensional Reissner-Nordström black hole to D dimensions. The solution was originally ﬁrst given in [10] but it has been well-known since then [13, 14].

The higher dimensional metric takes the form: 15

2µ q2 2µ q2 −1 ds2 = − 1 − + dt2 + 1 − + dr2 + r2dΩ2 , (69) D rD−3 r2(D−3) rD−3 r2(D−3) (D−2) where the two parameters µ and q are directly related to the mass M and the electric charge Q of the black hole, 8πG M 2G Q µ = D , q = D , (70) (D − 2)A(D−2) (D − 3)(D − 2) and the inner and outer horizon can be written as:

1 D−3 p 2 2 r± = µ ± µ − q . (71)

One last time, we are going to apply the Euclidean time approach to calculate the temperature of the black hole, following the same list of steps than before:

1. Wick rotation:

2µ q2 2µ q2 −1 ds2 = 1 − + dτ 2 + 1 − + dr2 + r2dΩ2 . (72) D rD−3 r2(D−3) rD−3 r2(D−3) (D−2)

2. Change of coordinates (dropping the angular part): r = r+(1 + ρ), dr = r+dρ, we get:

2 2 −1 2 2 dsD = f(ρ)dτ + f(ρ) r+dρ , (73) with ! 2µ q2 f(ρ) = 1 − + . (74) D−3 D−3 2(D−3) 2(D−3) r+ (1 + ρ) r+ (1 + ρ)

3. Now we take the near horizon limit ρ → 0 up to O(ρ2). In order to do so, we will use the general Taylor expansion:

(a + bρ)n = an + nban−1ρ + O(ρ2), (75)

where a and b are constants and n is an integer number. In this way, we arrive at:

2 2 −1 2 2 dsD = (a + bρ) dτ + (a + bρ) r+dρ , (76) with 2µ q2 2(D − 3)µ 2(D − 3)q2 a = 1 − + , b = − . (77) D−3 2(D−3) D−3 2(D−3) r+ r+ r+ r+

4. Change of coordinates to R such that gRR = 1,

2 2 −1/2 1 b R dR = r+(a + bρ) dρ ⇒ ρ = 2 − a , (78) b 4r+

2 2 b 2 2 2 dsD = 2 R dτ + dR , (79) 4r+

b 2r dα = dτ ⇒ τ = + α. (80) 2r+ b 16

5. Cyclic periodicity in Euclidean time: 4πr β = + , (81) b and ﬁnally the Hawking temperature takes the form:

b D − 3 p 2 2 D − 3 D−3 D−3 TBH = = D−2 µ − q = D−2 (r+ − r− ). (82) 4πr+ 2πr+ 4πr+

Note we easily recover the expression from the 4-dimensional case (55) by just setting D = 4.

Lastly, we can compute the entropy using the Bekenstein-Hawking formula as we did before in (57):

D−2 D−2 A r A D−3 AD (D−2) + (D−2) p 2 2 SBH = = = µ + µ − q . (83) 4GD 4GD 4GD It is interesting to note that the extremal case occurs for µ = q and we will have a vanishing Hawking temperature TBH, extremal = 0, regardless of the dimension D with the only restriction D 6= 3 and a ﬁnite non-vanishing entropy D−2 A(D−2) SBH = µ D−3 , as expected. 4GD

III. STRING THERMODYNAMICS

In this section, we introduce the thermodynamics of strings which is, in the high-temperature limit, given by the exponential growth of the number of states as a function of the energy. The most amazing fact that we will find is that the linear dependence of the entropy with the energy produces that the temperature is bounded by a finite constant value, known as the Hagedorn temperature. We first motivate the historical origin of this limiting temperature and then we will derive it in our search for a string partition function. Out of this high-temperature limit, the theory is not fully understood and still presents many open questions. We review here the approach in [15] where this exponential growth is estimated by counting the number of partitions of large integers in the high-temperature limit as we said. At the end of this section, we will review some implications of the Hagedorn temperature for black hole physics.

A. The Hagedorn temperature

−1 Originally, the Hagedorn temperature TH = βH arose in the study of hadrons at high energy [16]. Hagedorn’s approach was to estimate the density of hadronic states ω(E) at large energy E as:

ω(E) = eβH E, (84) with a mass scale of the order of the pion mass TH ∼ mπ and thus, the partition function is: Z ∞ Z ∞ Z = dEω(E)e−βE = dEe−(β−βH ), (85) 0 0 which diverges if β < βH so this temperature can be seen as a limiting temperature in order to have a well-deﬁned partition function T < TH . The physical meaning of this bound is that as the energy is increased, rather than having an increase in the energy of the already existing states, new states with higher mass are produced and therefore, the temperature of the existing states is bounded. Although it seems quite restrictive to have a limiting temperature in these physical processes, the assumption made by Hagedorn is supported by the string model of hadrons and this model has in favour that it is able to reproduce linear Regge trajectories. As we already mentioned, we will see how this temperature bound appears when looking for the string partition function. It can also be thought as the critical temperature at which a phase transition takes place (the analogy with the string model of hadrons is evident by considering the QCD deconﬁnement) The interest here is that, although this Hagedorn temperature was originally introduced in the context of hadron physics, as we saw, it is also applicable to fundamental strings which are thought to be the key for a consistent quantum theory of black holes. 17

B. Counting partitions

1. Bosonic partitions

Let us consider a non-relativistic bosonic string with ﬁxed endpoints. We can think of it as a collection of inﬁnitely many simple harmonic oscillators with frequencies given by multiples of a basic frequency ω0. From basic Quantum Mechanics we can write the Hamiltonian of the system as: ˆ ˆ H = ~ω0N, (86) where Nˆ is the number operator (with eigenvalue the total number of states) given by the creation and annihilation operators

∞ ˆ X † N = lal al, (87) l=1

† that satisfy [am, an] = δmn and al |0i = 0 ∀l, where |0i is the vacuum state of the string. An arbitrary state |ψi is given by acting with creation operators on the vacuum

∞ Y † nl |ψi = (al ) |0i , (88) l=1 P∞ where nl are the occupation numbers satisfying N = l=1 lnl with N ∈ N, so from (86) and (87) the energy of the state is

E|ψi = ~ω0N. (89) We want to generalise a bit by considering B transverse directions along which the string can vibrate. These represent physically B bosonic excitations. Thus, for every harmonic oscillator we have a degeneracy accounting for the transverse directions and the occupation numbers need to take that into account, so we redeﬁne the total number of states as

∞ B X X (b) N = lnl . (90) l=1 b=1

Now, we want to count, for a given number of states, how many states |ψi there are such that Nˆ |ψi = N |ψi, equivalently, with energy given by (89). This number is known as the number of partitions of N, p(N). A partition of N is deﬁned as a set of positive integers that add up to N:    + X  pi(N) := kj ∈ Z | kj = N , (91)  j  and consequently, the number of partitions is X p(N) = pi(N). (92) i

We look for a formula for p(N). We are going to estimate ln p(N) in the high-temperature limit, i.e., at high energies, from a physical point of view: by applying Statistical Mechanics. In the microcanonical ensemble we can write the entropy as the logarithm of the number of microstates accessible to the system for a certain energy Ω(E):

S(E) = k ln Ω(E) = k ln p(N), (93) hence, by calculating the entropy of the system we can ﬁnd the number of partitions and since E is proportional to N (89), for large energy we will have a large number of states. We will use this fact to make approximations. To ﬁnd S we calculate the partition function Z for this system and then we will obtain the entropy as the derivative of the 18 free energy F with respect to the temperature at constant volume:

∂F ∂ ∂ ln Z S = − = − (−kT ln Z) = k ln Z + T . (94) ∂T V ∂T ∂T The partition function is given by

X − Eα Z = e kT , (95) α where the summing index α sums over the set of all possible states of a given energy. It is then straight forward to write the sum labelled by the set of occupation numbers and to sum over all the possible occupation numbers and the B vibrational transverse directions:

B ∞ ∞ (b) ω0 P P (b) ω l n X − ~ l n Y Y X − ~ 0 l Z = e KT l b l = e KT , (96)

nl b=1 l=1 nl=1

P∞ n 1 where the last sum can be performed as a geometric series n=0 x = 1−x , |x| < 1,

B ∞ " ∞ #B Y Y 1 Y 1 Z = = , (97) − ~ω0l − ~ω0l b=1 l=1 1 − e KT l=1 1 − e KT so taking the natural logarithm we can write

∞ X − ~ω0l ln Z = −B ln 1 − e KT . (98) l=1

~ω0 Now, we make use of the high-temperature limit, kT 1, so we can approximate the sum with an integral: Z ∞ Z ∞ − ~ω0l BkT −x ln Z ' −B dl ln 1 − e KT = − dx ln 1 − e , (99) 1 ~ω0 0

~ω0l where we have performed the change of variables x := KT and we have assumed that the lower limit of integration ~ω0 which is kT is suﬃciently small that we can start integrating from zero, in good approximation. Since this number P∞ xn is small, x is also small and we can use the Taylor expansion for the logarithm, ln(1 − x) = − n=1 n , getting:

∞ ∞ BkT X Z ∞ (e−x)n BkT X 1 ln Z ' dx = , (100) ω n ω n2 ~ 0 n=1 0 ~ 0 n=1 where we have commuted the sum and the integral and evaluated this last one. The last sum is convergent and easy to perform. In fact, it is the Riemann’s zeta function of 2,

∞ X 1 ζ(2) = = π2/6 (101) n2 n=1

sin(x) P∞ n x2n+1 This result is easy to prove using the Taylor expansion of the function x = n=0(−1) (2n+1)! , the factor expansion sin(x) Q∞ x x 2 in terms of the zeros of this function x = n=1 nπ − 1 nπ + 1 and setting equal the coeﬃcients of x from −1 P∞ 1 both expansions: 3! = − n=1 n2π2 . Hence, π2BkT ln Z ' , (102) 6~ω0 19 thus, from (94) we get the entropy:

π2Bk2T S ' . (103) 3~ω0 However, we are not done yet since we want to express the entropy as a function of the energy so we need to calculate it. We can do it from the partition function:

2 2 2 r ∂ ln Z π Bk T 6~ω0 E = − ' ⇒ T = 2 2 E, (104) ∂β 6~ω0 π Bk and plugging this last expression for the temperature in (103) we ﬁnally get

r2BE r2 S ' kπ = kπ NB, (105) 3~ω0 3 where we have used (89). Last, using (93) we get the number of partitions for large N:

r2 ln p(N) ' π NB. (106) 3

2. Fermionic partitions and generalisation

The former expression was valid for a bosonic string, but for a fermionic string we cannot create a state by acting with the same creation operator on the vacuum more than once. Therefore, we need to perform a new partition counting. However, it is easy to modify the previous result for the case of fermionic strings. The numbers that enter a partition into unequal parts are called fermionic numbers and we call the number of partitions of N into unequal parts q(N). As before, we use a non-relativistic string, but this time fermionic, to determine q(N) at large N. We consider the same system of simple harmonic oscillators. Analogously to the bosonic case, we have F fermionic excitations:

∞ F X X (f) N = lnl , (107) l=1 f=1 but however, in the case of fermionic oscillators the occupation number nl can only take the values 0 or 1 (Pauli’s exclusion principle). As before, we start computing the partition function from (95):

F F ∞ n(f) " ∞ # X − ~ω0 P∞ PF ln(f) Y Y X − ~ω0 l l Y − ~ω0 l Z = e kT l=1 f=1 l = e kT = 1 + e kT , (108)

nl=0,1 f=1 l=1 nl=0,1 l=1 taking the logarithm in the high energy limit and performing the same change of variables than before we have:

∞ Z ∞ X − ~ω0 l F kT −x ln Z = F ln 1 + e kT ' dx ln 1 + e . (109) ω0 l=1 ~ 0

n+1 P∞ (−1) n We now make use of the Taylor expansion of the logarithm ln(1 + x) = n=1 n x , so we get:

∞ ∞ F kT X (−1)n+1 Z ∞ F kT X (−1)n+1 ln Z ' dxe−nx = . (110) ω n ω n2 ~ 0 n=1 0 ~ 0 n=1

P (−1)n+1 P (−1)n+1 1 P 1 π2 It is easy to express this last sum in terms of the previous sum (101): n n2 − 2 n (2n)2 = 2 n n2 = 12 . Hence,

π2F kT ln Z ' . (111) 12~ω0 20

Again, we compute the entropy using (94):

π2F k2T S ' . (112) 6~ω0 and we calculate the energy to express the entropy in terms of the energy:

2 2 2 r ∂ ln Z π F k T 12~ω0 E = − ' ⇒ T = 2 2 E, ∂β 12~ω0 π F k r FE r1 S ' kπ = kπ NF, (113) 3~ω0 3 where we have made use of (89) in the last step. Therefore, from (93), the number of fermionic partitions is given by:

r1 ln q(N) ' π NF. (114) 3 At this point, it is easy to generalise the previous results to the number of partitions of a non-relativistic string with B-bosonic and F -fermionic modes P (N,B,F ) = p(N) × q(N): s 1 2 ln P (N,B,F ) = ln p(N) + ln q(N) ' π N F + B . (115) 3 3

C. Entropy and temperature of a relativistic string at rest

Now, we consider relativistic strings but before calculating the partition function for strings carrying spatial momentum we want to study the case of open and closed strings with zero spatial momentum, so the energy coincides with the mass at rest, and see how the Hagedorn temperature arises from the linear dependence of the entropy with the energy.

1. Bosonic string

First, for open strings the mass spectrum is: 1 M 2 = (N − 1), (116) α0 and in the high-temperature limit we have large N, so that, N E2 ' . (117) α0 We consider the number of dimensions D = 26 and thus we have 24 transverse direction with bosonic modes, B = 24. Then, as we saw before, the number of microstates is given by the number of partitions Ω(E) = p24(N) and using result (106) we can write the entropy as: √ S = k ln p24(N) ' 4kπ N. (118)

Now, using (117) we have the entropy as a function of the energy: √ S(E) = 4kπ α0E. (119)

From this entropy, it is easy to compute the associated Hagedorn temperature using the standard procedure from Statistical Mechanics:

∂S −1 1 TH = = √ . (120) ∂E 4kπ α0 21

This Hagedorn constant temperature emerges as a consequence of the linear dependence of the entropy with the energy. The physical interpretation is rather simple: strings are limited by a constant thermal energy kTH even if we keep increasing the energy and therefore, the entropy and the temperature will remain constant. However, this constant thermal energy is rather small compared to the energy associated with the mass of the string at massive levels. For example, we can compare the energy for the ﬁrst massive level that corresponds to the second level of the 2 1 string N = 2, M = α0 and using (120) we have: M = 4π, (121) kTH thus, the thermal energy is small compared to any massive state of the string.

For closed strings we will have instead: 2 4 M 2 = (N + N˜) = (N − 1), (122) α0 α0 where N and N˜ are the total number of states of left and right movers respectively and where we have used the level matching condition N = N˜. In the high-temperature limit we have large N, N˜ and thus, 4N E2 ' . (123) α0 The number of microstates is now given by the number of partitions of the left moving modes and the right moving modes so we get:

2 Ω(E) = p24(N)p24(N˜) = p24(N) , (124) where again, we have made use of the level matching condition. As before, we compute the entropy: √ S = 2k ln p24(N) = 8kπ N, (125) where we have used the result for the logarithm of the number of bosonic partitions (106). It looks like we got exactly double the entropy than in the open case. This picture makes sense since we can think of the closed string as two open strings: left and right movers. Expressing it in terms of the energy using (123): √ S(E) = 4kπ α0E, (126) and the Hagedorn temperature turns out to be: 1 TH = √ , (127) 4kπ α0 so when the entropy is expressed in terms of the energy we get the same result than in the open case, leading to the same expression for the Hagedorn temperature as in (120). In terms of the thermal energy, there is no distinction between open and closed strings. 22

2. Fermionic string

We can perform the same calculation for pure fermionic strings assuming the same mass levels than for the open and closed bosonic strings and making use of result (114). This leads to the following results using the former procedure.

For open strings: √ √ √ √ S = 2 2kπ N = 2 2kπ α0E, 1 TH = √ √ . (128) 2 2kπ α0 For closed strings: √ √ √ √ S = 4 2kπ N = 2 2kπ α0E, 1 TH = √ √ . (129) 2 2kπ α0

We see that, as before,√ both open and closed strings have the same entropy and the same Hagedorn temperature which is a factor of 2 bigger than the one in the bosonic case. However, fermionic strings do not have much physical sense, especially in this model where we have built the string as a collection of inﬁnitely many harmonic oscillators which correspond to scalar ﬁelds (bosonic). The only way to involve fermionic degrees of freedom is considering Supersymmetry (ensuring that the number of bosonic and fermionic degrees of freedom are the same), so we will consider next the superstring cases. Moreover, superstrings do not contain tachyons in the spectrum of states so in that sense they should be more physical than bosonic strings.

3. Open superstring

For open superstrings, the state space is divided into two sectors: the Neveu-Schwarz (NS) sector containing bosonic states (anti-periodic boundary condition) and the Ramond sector (R) containing fermionic ones (periodic boundary condition). Since supersymmetry reduces the dimension of the spacetime to D = 10 we will have 8 transverse modes, 4 bosonic and 4 fermionic. For all this, one could think that the number of states we need to compute is P (N,B=8,F =8) but however, we must account for the degeneracy of the ground states. It is easy to see that in the R sector the four creation operators allow us to construct 16 degenerate ground states [15]. Because of supersymmetry, we can extend this degeneracy factor to the NS sector as well. In this way, if the open superstring belongs to one sector or the other it will have the same number of degenerate ground states and thus, the total number of states is 16P (N, 8, 8).

For the NS sector we have: 1 1 M 2 = − + N , (130) α0 2 and for the R sector: 1 M 2 = N. (131) α0 However, both sectors acquire the same massive level structure in the high-temperature limit, i.e., for large number of states N, so then N ' α0E2. We follow the same procedure as before to compute the entropy and the Hagedorn 23 temperature: √ √ ln Ω(E) = ln 16P (N, 4, 4) ' 2 2π N + 4 ln 2, √ √ √ S = k(2π α0E + 4 ln 2) ' 2 2kπ α0E, 1 TH = √ √ . (132) 2 2kπ α0 √ Note the Hagedorn temperature is a factor of 2 larger in the case of an open superstring than for the bosonic one.

4. Closed superstring

For closed superstrings the situation is more complicated since we have both left and right movers which can belong to either the NS or the R sector. A priori, we will have four closed superstring sectors: two possible bosonic states {(NS, NS); (R, R)} and two fermionic {(NS, R); (R, NS)}. In order to get a closed superstring theory we need to truncate the sectors. This means, in terms of the number of states, that we divide each into equal parts. For example, for superstring theory type IIA we truncate the left sector {NS+;R-} and the right one {NS+;R+}. Therefore, we get two bosonic sectors {(NS+,NS+); (R-, R+)} and two fermionic {(NS+,R+); (R-, NS+)}. If R+ and R- are interchanged it is still the same type IIA. Type IIA arises then when the left and right truncated R sectors are of diﬀerent types. If instead of having R+ and R- we choose the same type R- for both left and right truncated R sectors, i.e, left sector {NS+;R-} and right sector {NS+;R-}, we will have superstring theory type IIB: two bosonic {(NS+, NS+); (R-, R-)} and two fermionic {(NS+, R-); (R-, NS+)}.

To count the total number of states we recall that we have split every sector so they have half of the number of states. For instance, for the (NS+,NS+) sector we will have

1 2 1 1 Ω(NS+,NS+) = Ω(NS) = Ω(NS)2 = (16P (N, 8, 8))2 = 64(P (N, 8, 8))2. (133) 2 4 4

In fact, this calculation looks right since the degeneracy factor is 64 in each sector and we have 4 sectors, giving 64 × 4 = 256 in total, so 128 bosonic and 128 fermionic which exactly corresponds to the total number of the degrees of freedom of type IIA.

As before, we still consider the high-temperature limit and the mass states are given in terms of the mass states of the left ML and right movers MR 1 α0M 2 = α0M 2 + α0M 2 , (134) 2 L R and using the former relation for the open superstring and imposing the level matching condition we get: 4 M 2 ' N, (135) α0

1 0 2 thus, in this case, we have N ' 4 α E , so the entropy and the Hagedorn temperature are: √ √ ln Ω(E) = ln 64(P (N, 8, 8))2 ' 4 2π N + 6 ln 2, √ √ √ √ S = k(2 2π α0E + 6 ln 2) ' 2 2kπ α0E, 1 TH = √ √ . (136) 2 2kπ α0 Note the Hagedorn temperature and the entropy for the closed superstring is the same than in the case of an open superstring, as it was happening for the bosonic string. 24

D. The string partition function

In this section, we compute the partition function for a bosonic open string in the low thermal energy approximation. As we saw before, when the temperature approaches the Hagedorn temperature, the Hagedorn thermal energy is smaller than the energy due to the mass. The aim is to see how the Hagedorn temperature arises from the requirement to have a well-deﬁned partition function.

Qd Let us consider a single open string in D = d + 1 dimensions inside a box of volume V = i=1 Li in thermal contact with a reservoir at temperature T . The quantum states of the string are labelled by the momentum p and the P occupation numbers λn,i, which determines the total number of states N = n,i nλn,i. This states can be constructed by acting with the light-cone creation operators on the momentum eigenstates:

∞ 25 Y Y i† λ + |λ, pi = (an )n,i |p , ~pT i , (137) n=1 i=2 where the mass of the string M arises from the on-shell condition:

2 2 + − i M = −p = 2p p p pi, (138) with energy, p E = M 2 + ~p2. (139)

Using this last relation and recalling the mass levels formula for the open string (116) and the former relation between the total number of states and the occupation numbers, we can write the string partition function to be computed as: √ 2 2 X −βEα X X −β M +~p X Zstr = e = e = Z(M), (140)

α λn,i ~p λn,i where Z(M) is the partition function of a relativistic particle of mass M.

In an aside, we compute the partition function of a relativistic particle to later use the result in our calculation. In the same conditions than before we have a relativistic particle with energy given by (139) and we need to sum over all quantum states of the particle in the box which are labelled by their respective quantised momenta. These momenta i ~p·~x depend on the size of the box. The wave functions ψ ∼ e ~ are periodic with a periodicity condition piLi = 2π~ni, ni ∈ Z, and thus,

Li ni = pi , (141) 2π~ so we can relate the sum over momenta with a sum over integers ni and if the dimensions of the box are suﬃciently large we can go to the continuum limit and replace the sum by an integral:

Z ∞ d d Z ∞ X X Y Li V Y = ' dpi = d dpi. (142) −∞ 2π (2π ) −∞ ~p ni i=1 ~ ~ i=1

Hence, the partition function is:

d Z ∞ √ X −βE(~p) V Y −β M 2+p2 E = e = dp e i . (143) (2π )d i ~p ~ i=1 −∞ 25

We perform the change of variables pi = Mui, so we have:

d Z ∞ √ V Y −βM 1+u2 Z = du e i . (144) (2π )d i ~ i=1 −∞

As we pointed out at the beginning of this subsection, we will consider the temperature limit T → TH so the thermal energy is small compared to the mass, βM 1. This means that the momenta of the states is small compared to the mass, i.e., the energy at rest is bigger than the thermal energy (associated with the energy of the collisions of the particle with the boundaries of the box). If |~p|/M 1, |~u| 1 and we can Taylor expand the squared root in the p 2 1 2 3 exponent 1 + ui = 1 + 2 ui + O(ui ), so then,

d d Z ∞ d Z ∞ d VM −βM Y − 1 βMu2 VM −βM − 1 βMu2 Z = e du e 2 i = e du e 2 i . (145) (2π )d i (2π )d i ~ i=1 −∞ ~ −∞

R ∞ −ax2 p The last integral is now just a Gaussian integral: −∞ dxe = π/a, so putting factors together we ﬁnally get the following result:

d M 2 Z(M) = V e−βM . (146) 2π~2β

We now go back to the calculation of Zstr. Recalling (116) we can write the sum over the occupation numbers (140) as a sum over the total number of states, taking into account that there are p24(N) states with the same N. Therefore, we have: X Zstr = p24(N)Z(M(N)). (147) N

We are going to assume that there is a suﬃciently large number of states N0 such that for N > N0 our estimation of the number of partitions (106) works well enough, √ 4π N p24(N) = e . (148)

We can write the partition function as:

∞ X Zstr = Z0 + p24(N)Z(M(N)), (149)

N=N0 where we have deﬁned Z0 as:

N −1 X0 Z0 = p24(N)Z(M(N)). (150) N=0

As the temperature approaches the Hagedorn temperature, Z0 becomes negligible since the second term will become large.

The density of states can be written as a function of the mass, p24(N)dN = ρ(M)dM. In the large N limit we have M 2 ' N/α0, so dN = 2α0MdM. Using this and the previous relation together with (148) we get: √ 0 0 4π α M 0 βH M p24(N)dN = ρ(M)dM = 2α Me dM = 2α Me dM, (151) where we have used the expression for the Hagedorn temperature and β = 1/kT . Considering this large N limit 26

(change the sum to an integral) and plugging this last result in the expression for the partition function it gives: Z ∞ 0 βH M Zstr = Z0 + 2α √ dMMe Z(M). (152) 0 M0= N0/α

At this point, we recall result (146) and we plug it in,

∞ 1 Z 27 0 2 −(β−βH )M Zstr ' Z0 + 2α V 2 2 dMM e . (153) 2π~ β M0 √ 0 TH Finally, recalling (127), we can use the relation β = 4π α T and we are left with:

Z ∞ √ 0 T 27 −4π α0( TH −1)M Zstr ' Z0 + 2α V √ dMM 2 e T , (154) 2 2 0 8π ~ α TH M0 thus, we can clearly see that the integral only converges if T < TH , so the thermal energy is bounded to kTH as we know. Also, note that in the high-temperature limit the factor multiplying the integral grows linear and for high enough temperatures we could neglect the contribution from Z0.

We have performed this computation for the open string for simplicity, but the computation for the closed string is analogous. One only needs to take into account the momentum states for the right and left movers and use the mass levels formula for the closed string. The procedure is straight forward and leads to the same limiting temperature which is precisely the Hagedorn temperature.

In our case, we did not end up with a closed result. In [15], by using a better approximation than our expression for the bosonic number of partitions (106), it is possible to cancel out the powers in M in the integrand of (154) and get a trivial integral and hence, a closed result. This better approximation is based on the Hardy-Ramanujan asymptotic expansion of the number of partitions which we will not describe now but it will appear in a later section. All in all, the important idea is that from√ mathematics we can get a more accurate approximation for the number of partitions of the form p(N) ∼ βN −γ eδ N , where β, γ, δ are positive constants, which allows us to write a closed result for the string partition function in the limit T → TH from below: ρ TH Zstr ∼ λ(TH ) , (155) TH − T with λ, ρ positive constants. As we can see, this quantity grows without bound as T → TH and does the energy as well: ∂lnZ 1 T E := − str ∼ ∼ H . (156) ∂β β − βH TH − T

E. The Hagedorn temperature, the black hole entropy and the stretched horizon

In this section, we will describe more precisely the black hole entropy problem. In field theories, it was seen that the local temperature closed to the horizon can grow without limit S = A/2 ( is a cut off at small distances) and this contradicts the finite character of the Bekenstein-Hawking entropy [4, 17]. String theory responds to the need for a modification of QFT at short distances.

As we have seen before, for a system with ﬁxed energy E it is possible to derive its entropy from the standard 27 statistical mechanics procedure. In the microcanonical ensemble we have:

S(E) = k ln ρ(E), (157) where ρ(E) is the density of states with energy E. If we consider a simple Schwarzschild black hole we can identify the energy with the mass E ∼ M and a UV complete theory of gravity should provide a method to calculate this density of states ρ(M) from which we should be able to recover the macroscopic expression of the Bekenstein-Hawking entropy for large M. As we have already seen, this entropy scales with the mass as S ∼ M 2, so then:

2 ρ(M) ∼ eM . (158)

The canonical partition function is: Z − E Z(T ) ∼ dEρ(E)e T , (159) which will diverge since the number of black hole states of a certain mass M grows extremely fast, as we can see from (158). The interest of string theory is that the density of states grows (roughly) exponentially with the mass, ρ(M) ∼ eM , for large mass, as we can see from (151), and the partition function diverges only above the Hagedorn temperature (154). If we consider p-branes instead (see [18] and [19] for further references):

2p ρ(M) ∼ eλM p+1 , (160) so for p > 1 it already diverges at T = 0. In conclusion, we already have a constraint for potential candidates to a satisfactory theory of quantum gravity and this is the need to explain a faster growing of the black hole density of states.

If we recall again√ (127), we see that the Hagedorn temperature scales with the inverse of the string length, which is 0 defined as ls := α , so TH ∼ 1/ls. As we have seen before, for all types of strings the entropy always scales linearly with the energy, or equivalently with the mass, and with the string length Sstr ∼ lsM. Hence, the density of states scales as eSstr ∼ e1/TH . In this way, the Hagedorn temperature can be understood by considering the fluctuations of multiple strings as a function of the temperature, i.e., as the percolation temperature at which multiple strings fluctuations fusion to fluctuations of a single string. The key idea behind modelling large black holes as fluctuations of a single string is the concept of stretched horizon, which was proposed in the context of looking for solutions to the Black Hole Information paradox [20, 21]. When constructing a consistent QFT in Rindler space (describing uniformly accelerated observers), a boundary condition is required since this space has a boundary at ρ = 0, where ρ represents the spatial dimension. This is accomplished by introducing at a very small distance ρ0 an effective membrane at a constant temperature given by the temperature of the Rindler horizon located at ρ0, so T = 1/(2πρ0). From the equivalence principle and in the presence of a black hole, an infalling observer sees neither this effective membrane, called stretched horizon, nor the real event horizon, while for an external observer at fixed coordinates, the stretched horizon is a thin layer containing the degrees of freedom that gives rise to the entropy encoded in the area of the horizon. The degrees of freedom of the infalling observer would be deposited in this layer and there would not be information loss caused by the black hole. The interest of the stretched horizon is that we can consider a single stringy mass on this horizon which is located at a very small distance from the event horizon given by the length of the string ρ = ls and at a proper temperature characteristic of a Rindler horizon T = 1/(2πρ), and thus we have T ∼ 1/ls, which is the Hagedorn temperature. Therefore, we see the relation between the temperature of the stretched horizon and the Hagedorn temperature in this long string model of a black hole. We will try to explain more this construction of the long string in a simple way following the review done in [21] in the next section. 28

IV. THE LONG STRING CONSTRUCTION AND THE RANDOM WALK MODEL

Despite of the problem pointed out in the previous section regarding the diﬀerent mass scaling of the string entropy 2 Sstr ∼ M and of the Bekenstein-Hawking entropy SBH ∼ M , it was suggested that near the horizon there exists a single long string covering the whole area of the black hole. This long string picture was proposed initially by L. Susskind [20, 22]. The way to circumvent the scaling problem and be able to relate the number of degrees of freedom of the black hole to the number of degrees of freedom of a long string, Sstr = SBH , is to assume that one needs to take into account the self-gravitation of the long string [23].

Microscopic calculations of the entropy in string theory consider adiabatic processes where a control parameter is varied. For a given value of this control parameter, the system allows an easy state counting. The control parameter, in this case, is the strength of the string coupling g and the adiabatic invariant is the total number of states, i.e., the entropy. In our case, we take the string coupling to zero g → 0 and consider a neutral black hole, which will evolve into a collection of free strings, since among all objects in string theory only free strings have ﬁnite energy as g → 0. If we consider a very massive black hole, as g → 0, either it will evolve into a large number of low mass strings or into a single very highly excited string. This last situation is more interesting since it can be thought of a mass of a tangled string that forms a random walk in space changing in time. Such a string random walk have a large entropy. In fact, if one considers the option of a number of low mass strings and consider the limit where the number of those is large, one sees that the single string construction is favoured. Hence, for a given total mass the statistically most likely state in free string theory is a single excited string and thus, when g → 0, most of the black hole states will evolve into a single excited string.

A. Long strings are entropically favoured

We can see with a simple calculation that the situation of having a single highly excited string is more entropically favoured than a collection of low mass strings. In order to do this, we consider the more accurate approximation to the result of the number of bosonic partitions (106) that we mentioned in section III.D., as in [15]: √ p(N) ∼ βN −γ eδ N , (161) where β, γ, δ > 0. We must note that this is still an approximate formula for large N. For simplicity, let us consider the case of an open bosonic string with large excitation number N = N0 and energy E0. We can relate both recalling (117), so α0E2 ' N, assuming relativistic strings at rest, i.e., with zero spatial momentum.

Let us assume that we have a single string that breaks into two strings with each carrying half of the energy. To calculate the change in entropy we start computing the ratio of the total number of available states of the two strings 0 2 and the number of states of the initial single string. This last one satisﬁes α E0 ' N0 and its number of states is just p(N0). The total number of states when the string breaks into two strings is given by the product of the number 0 2 of states of each. These two strings carry half of the energy E0/2 and thus satisfy α (E0/2) ' N0/4, so the total 2 number of states of both is given by p(N0/4) . Hence, the ratio is:

2 p(N0/4) 2γ −γ = 4 βN0 . (162) p(N0) The change in entropy of the process is then

2 ∆S p(N0/4) = ∆lnp(N) = ln = 2γln4 + lnβ − γlnN0, (163) k p(N0) 29 which, if we consider N0 large enough to neglect the contributions of the constants, is negative ∆S/k < 0. That means that the reverse process when two half-strings merge into a single string is entropically favoured. Note that the constant γ is the one responsible for this effect, so we would not have observed this effect working with the less refined result of (106).

In general, the combination of two open strings with energies E1 and E2, with large excitation numbers such that 0 2 0 2 0 2 α E1 ' N1 and α E2 ' N2, into a single open string with energy E0 = E1 + E2, satisfying α E0 ' N0, is a process that increases the entropy. To see this, we study the reverse process, the breaking of one string into two as before and we will get that the change of entropy is negative, so the direct process (two strings merging into one) has a positive change in entropy. In this case, we have:

∆S p(N1)p(N2) √ p p = ∆lnp(N) = ln = δ( N1 + N2 − N0) + lnβ + γ [lnN0 − (lnN1 + ln N2)] . (164) k p(N0)

√ √ √ We must note that we need to impose the constraint E0 = E1+E2 on this result, which is equivalent to N1+ N2 = N0, so that the ﬁrst term vanishes and for large excitation numbers, we are left with the last term: ∆S ' γ(ln N − ln(N N )), (165) k 0 1 2 √ √ √ which can be rewritten using N0 = N1 + N2 in the ﬁrst logarithm as:

∆S h p i ' γ ln(N + N + 2 N N ) − ln(N N ) . (166) k 1 2 1 2 1 2

For large excitation numbers N1,N2 we can see that the change in entropy will be negative since it behaves as ∆S O(N) ∆S k ∼ ln O(N 2) < 0. Hence, the process of merging two strings into one has a positive change in entropy k > 0, at least for the large excitation numbers regime.

We can investigate if all this holds not only for the large excitation numbers approximation but also for smaller ones. To do this, we can no longer trust expression (106) because it was derived for large N. Instead, we need to use the better approximation from before, the Hardy-Ramanujan asymptotic expansion [15]: √ 1 π 2 NB pH−R(N) ' √ e 3 . (167) 4 3N

We also need to take into account the exact relation between the energy and the excitation number α0E2 = N − 1.

0 2 As an example, we can consider an open string with excitation level N = 9 with an energy α E0 = 8 breaking into two strings with half of the energy and therefore, from the relation between the energy and the excitation level, these two strings will have an excitation level N = 3. In the case of open bosonic string, we have B = 24. We can now compute the change in entropy numerically:

∆S p(N = 3,B = 24)2 = ln = 2 ln p(N = 3,B = 24) − ln p(N = 8,B = 24) ' −15.83 < 0, (168) k p(N = 8,B = 24) so then, even if this is not a general proof but rather a concrete example, it looks like also for small excitation numbers the single long string scenario is entropically favoured. 30

B. The size of a string state and the Random Walk model

We are going to use the picture of a string of length L made out of string bits each of which has length ls to estimate the size of an open string state at zero string coupling. We use the so called Random Walk model which assumes that every string bit can point randomly in any of the d = D − 1 spatial orthogonal directions. The random walk model applies to strings with no angular momentum since the size of a rigidly rotating open string is proportional to the mass. In this model, it is assumed that the size of the string Rstr is much smaller than its length L. First of all, we can easily estimate the total length of the string: 1 L M ∼ TL ∼ 0 L ∼ 2 , (169) α ls

2 so then L ∼ lsM.

We start from the random walk formula for the average value of the square of the displacement in terms of a ﬁnite variance σ and the number of steps n:

< x2 >= σ2n. (170)

In this case, we can take the number of steps of the model as the number of bits of the string, i.e, the entropy of the string (in units of Boltzmann constant k = 1), n = S. This could be easily checked using (169):

L # string bits = ∼ lsM ∼ S. (171) ls

It is reasonable to assume in this picture a variance equal to the string length σ = ls. Then, the formula (170) allows us to estimate the size of a string state making use of the previous result on how the entropy scales with the mass and the string length: √ √ 2 3/2 1/2 1/4 Rstr = < x > = ls S ∼ ls M ∼ lsN , (172) where we have used in the last equality the mass spectrum for the bosonic open string for large N, M 2 ∼ N/α0. We 1/2 have seen that the size of the string grows like the square root of the mass Rstr ∼ M , while the total length of the string L grows like the mass ls ∼ M, so the assumption Rstr < L is satisﬁed as one can see from expressions (172) and (169).

Now, we want to see if suﬃciently massive strings could form a black hole and calculate the mass bound that a string must satisfy. First of all, we consider the case D = 4. We want to see if we can form a 4-dimensional Schwarzschild black hole of radius RS so we impose

Rstr < RS. (173)

Next, we use expression (172) on the left hand side and the dependence of the Schwarzschild radius RS ∼ G4M ∼ 2 2 g lsM on the right hand side and we get: 1 M > 4 , (174) g ls so we can deﬁne a mass bound M0 deﬁned as:

1 mP M0 := 4 ∼ 3 . (175) g ls g This means that for any ﬁxed string coupling there is a mass beyond which any excited string state is smaller than the Schwarzschild radius, so that suﬃciently massive strings such that M > M0 can form a black hole. 31

We can also see that in the general case with arbitrary dimension D, if the previous mass bound is satisﬁed then, the size of the string is smaller than the Schwarzschild radius. This is easy to check. On the one hand, we have that if M > M0 then, l R > s , (176) str g2 using (172). On the other hand, the D-dimensional Schwarzschild radius scales as

1/(D−3) 2 D−2 1/(D−3) RS(D) ∼ (GDM) ∼ (g ls M) , (177) and plugging this into the mass bound condition we get:

ls RS(D) > 2 . (178) g D−3

We can see that for D > 4 we have in fact that Rstr < RS as expected, so the mass bound that we found is valid for higher dimensional cases.

2 As we can see from expression (176), for M = M0 the bound is saturated and thus Rstr = ls/g , so if the coupling becomes very small Rstr becomes too big to trust the string model. Also, using the previous result (172) we have: l R = s ∼ l N 1/4, (179) str g2 s and from this last expression we can estimate the value of N for a string of mass M0 at zero string coupling: 1 N ∼ . (180) g8 As we are dialling down the string coupling, N grows considerably fast according to the picture of having a highly excited string.

C. Estimation of the Schwarzschild black hole entropy

This long string model allows for making an easy estimation of the entropy of a D-dimensional Schwarzschild black hole. It all relies on a series of assumptions. The idea, as we explained at the beginning of this chapter, is to adiabatically take the string coupling to zero g → 0, assuming that there will be a transition from the black hole to a single highly excited long string. This transition takes place when the Schwarzschild radius is of the order of the string length RS(D) ∼ ls. If we assume that this process of taking g → 0 does not change the entropy, then the entropy of the black hole can be directly computed by calculating the entropy of the long string in the ﬁnal state. Recalling from the last part of the former chapter, the entropy of such string scales linearly with the mass and the string length Sstr ∼ lsM.

We start from the relation between Newton’s constant in D dimensions, the string coupling constant and the string length (in natural units):

D−2 2 D−2 GD = lP (D) = g ls , (181) where the Planck length in D dimensions is deﬁned as r G l := ~ D . (182) P (D) c3 32

As we said before, we vary the string coupling g → 0 while keeping ﬁxed ls, so we will reach the limit when gravity decouples GD → 0. We start with a D-dimensional Schwarzschild black hole with large mass M0 and with an initial value of the string coupling g0 and of the Newton’s constant G0(D). We recall here expression (29) for the D-dimensional Schwarzschild radius in terms of M0 and we substitute G0(D), using former expression (181):

1 1 D−3 2 D−3 D−2 16πG0(D)M0 16πg0M0 D−3 RS(D) = = ls . (183) (D − 2)A(D−2) (D − 2)A(D−2)

From this last expression, we can see that for ﬁxed g0, if the mass M0 is large enough, the Schwarzschild radius will be bigger than the string length RS > ls, but as we take g → 0, there will be a moment when the Schwarzschild radius and the string length are comparable RS ∼ ls and there will be a transition from the black hole to the single long string, as we mentioned before.

Taking g → 0 will also produce, in general, that the mass varies in an adiabatic process. We will have a mass D−2 function M = M(g) such that M(g0) = M0. From (41) we have that SBH ∼ (GDM ) =constant, as we assumed from the beginning. As a consequence, we have:

D−2 2 D−2 D−2 GDM = g ls M = constant, (184) so from this last expression we have:

1 constant D−2 M(g) = . (185) 2 D−2 g ls

We can easily determine the constant from the condition M(g0) = M0 and after making some manipulations we get:

1 g2 D−2 M(g) = M 0 . (186) 0 g2

On the other hand, from expression (183) we have:

D−2 2 RS(D)(g) ∼ ls g M(g). (187) As we send g → 0, the Schwarzschild radius will get closer to the string length and at some point the transition will D−3 D−3 D−3 occur when RS(D)(g) ∼ ls, so dividing by a factor ls the former expression we have that RS(D)(g) /ls ∼ 2 lsg M(g) and thus, 1 M(g)l ∼ . (188) s g2

Now, we are going to combine expressions (186) and (188). From this last expression, we have g in terms of M(g) and ls and we plug it into (186). Expressing g0 in terms of G0(D) and ls using (181) and after some algebra we get:

D−2 1 −1 D−3 D−3 M(g) ∼ ls M0 G0 . (189) We ﬁnally got how the microscopic entropy of a single fundamental long string scales:

D−2 1 D−3 D−3 Sstr ∼ M(g)ls ∼ M0 G0 ∼ SBH , (190) which scales exactly in the same way than the macroscopic Bekenstein-Hawking entropy of the D-dimensional Schwarzschild black hole that we calculated in (41). Although this qualitative calculation is missing the correct factors, it points out that modelling black holes as a single highly excited string leads to the correct form of the macroscopic entropy and thus, this model is a good candidate to understand the microscopic structure of black holes. More precisely, this indicates that the Schwarzschild black hole is the strong coupling version of a highly excited string. 33

D. Recent developments

Recently, this idea has been more technically developed in a series of papers [24–26] by considering a mean field approximation. In this approximation, it is possible to match the entropy of large black holes with the stringy degrees of freedom in a quantitative way. This is done by considering a thin sell of matter of mass δM that falls into a black hole of mass M, so that the self-interaction within the shell is neglected. The shell ends up as a long string near the horizon. The framework used is type II and heterotic string theory in Rindler space. The formalism used is based on the construction of the so called thermal scalar in curved space. It is an alternative way to the original mechanism of the Hagedorn divergence in the partition function, explained in terms of new high mass states being created. Instead, one considers a Higgs phase transition when the mass squared of a complex field changes its sign. The equivalence between both formulations can be demonstrated in the random walk formulation of Euclidean field theory (see former references for more details). Furthermore, particularly in [24], it is claimed that only long strings having a Hagedorn density of states, i.e., S ∼ E, with THagedorn = THawking are in equilibrium with a present black hole and thus, one can think of the black hole, modelled in this way, as a set of long strings in equilibrium. It is interesting to think about that the equality between the Hagedorn temperature and the Hawking temperature somehow encodes how long strings behave in gravitational fields and more importantly, how the long string construction is capable of yielding the correct number of states to account for the microstructure of the black hole, in agreement with the macroscopic Bekenstein-Hawking entropy.

V. BLACK HOLES FROM BRANE INTERSECTIONS

In order to construct black holes in string theory, one needs to use a large number of elementary excitations. One way to do it, is by intersecting different types of branes, as we already said. Nevertheless, this is not enough because we also need to stabilise the object, so that it will not shrink. In other words, since we are interested in getting massive excitations we need to stretch these extended objects to a non zero size and this can be achieved by compactifying some directions of space, i.e., wrapping the branes around stable structures, like Calabi-Yau manifolds. The idea is then engineering string theory compactifications for vacuum solutions, making sure that the lower dimensional structure presents a horizon that prevents the unphysical situation of having a naked singularity, according to Penrose’s conjecture. In addition, given that extended objects can carry masses and charges, we need to make sure that the combined mass and charge of the total configuration coincides with the mass and charge of the black hole. A system of strongly interacting D-branes in string theory is indistinguishable from a black hole in the supergravity approximation. One could even talk about a brane/black hole correspondence in this sense. More concretely, there are some interesting results in the form of theorems claiming that at least 4 different extended objects (branes) are needed in order to produce a 4-dimensional black hole with a smooth horizon and therefore, leading to a non-vanishing entropy. It is curious that for the 5-dimensional case this number goes down to only three objects and decreases even more for higher dimensional objects, such as 6-dimensional black rings arising from intersecting only 2 different types of branes.

This possibility of obtaining black holes as supergravity solutions associated with configurations of extended objects in superstring theories, where this association can be understood as a strong-weak coupling limit, is a feature of great interest of these theories. However, this association is not always possible. In order to extrapolate some results computed in this higher dimensional picture to the black hole solution, we require that the results in question remain invariant when going from weak coupling (brane picture) to strong coupling (the black hole solution). The tool used to keep these results protected is supersymmetry and this makes total sense since the black hole solutions are given in the context of a supersymmetric theory of gravity. In general, every extended object that we introduce will break some supersymmetry. Nevertheless, the aim is that the final configuration will have some supersymmetry left, enough to keep the properties of the system intact while varying the string coupling. These type of objects protected by supersymmetry are called BPS states. Hence, we can define a BPS state in a given supersymmetric theory as a state which preserves some fraction of the total supersymmetry of the theory. Once more, these BPS states are certain representations of the supersymmetry algebra which are invariant under deformations of parameters. Thus, one can hope to use information derived at weak coupling to obtain nontrivial results at strong coupling. 34

As a consequence of demanding that some supersymmetry remains, we will have supersymmetric black holes in the lower dimensional theory. In particular, in the cases of study, they are extremal black holes. However, one must be careful when establishing a connection between extremal and BPS objects. The main diﬀerence between these resides in the fact that not all extremal black holes are BPS. Extremal black holes satisfy the condition M = |Q| (as we saw in section II.C.3.), while in order to have some supersymmetry preserved and get BPS states, only one of the possible signs is valid and we require M = +Q.

A. D=5 3-charge Extremal Reissner-Nordström black holes

The simplest black hole construction where we can perform the microstate counting to reproduce the entropy is the case of supersymmetric (extremal) Reissner-Nordström black holes in D = 5 dimensions with three charges. As we said before, supersymmetry makes the calculations easier since we can perform the counting of states in the non- interactions regime g = 0 and extrapolate the result when we turn on interactions in order to have a black hole g 6= 0. Thanks to supersymmetry, the counting is invariant under the value of string coupling. This is a simple example of a type of black hole with this property.

1. Construction in the Supergravity approximation

The way to construct them is to consider the N = 4, D = 5 supergravity approximation (the low energy limit of string theory) to type IIB closed superstring theory (D = 10) compactified on a 5-dimensional torus T 5. Superstring theory lives in D = 10 Minkowski spacetime and in order to get a lower dimensional supergravity approximation we need to compactify the extra dimensions in a way that some supersymmetry is preserved. This compactification can be done by curling up the last five dimensions. If we consider 10-dimensional coordinates x0...x9 we will compactify x5...x9 and thus, we will be left with a black hole living in 5 dimensions: x0...x4 (5-dimensional Minkowski). In order to end up with a supersymmetric black hole in the remaining non-compact dimensions, we need to consider three types of objects that correspond to three different black hole charges. Type IIB string theory has p-brane states for odd values of p (since the R-R sector has gauge fields C(q) for even q and the worldvolume has dimension p + 1 = q). 5 5 9 1 These objects are QD5 D5-branes wrapped on the torus T (x ...x ), QD1 D1-branes wrapped on a sphere S of radius R inside the 5-dimensional torus (x5 for example), since T 5 = T 4 × S1 and n units of Kaluza-Klein momentum ( momentum wave W ) also along S1 , i.e., in the coordinate x5 since along the direction of this coordinate the D-branes are transitionally invariant and therefore, they cannot carry momentum. The integers QD1, QD5 and n are precisely the three black hole charges. Each of the former objects breaks half of the initial supersymmetry, so we end up with 1/8 BPS states (7/8 of the supersymmetry is broken). From the 32 initial supercharges of the theory we are left with 4 conserved supercharges (since the Majorana-Weyl spinors have 16 real degrees of freedom in D = 10).

Let us now construct this solution by dimensional reduction of the 10-dimensional string metric to the low-energy 5-dimensional supergravity one. In order to do so, we first need to construct the 10-dimensional string metric for the configuration described. As we already mentioned, we have three kind of objects spatially located in the compact dimensions as the following table shows. The “ X ” describes the worldvolume dimension of each object and “ * ” describes transverse directions along which the objects have been smeared (this smearing allows to have a consistent compactification on T 5 in order to end up with a localised charge):

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D1 X X * * * * D5 X X X X X X W X X * * * * 35

To write the 10-dimensional metric we apply the so called Harmonic Superposition principle which allows us to simply combine the metrics of the D1 and the D5-branes by multiplying the components of each coordinate of the metric:

2 −1/2 −1/2 2 5 2 1/2 1/2 2 2 2 1/2 −1/2 6 2 7 2 8 2 9 2 ds10 = −HD1 HD5 [dt −(dx ) ]+HD1 HD5 [dr +r dΩ(3)]+HD1 HD5 [(dx ) +(dx ) +(dx ) +(dx ) ], (191) where we have parametrised the non-compact dimensions in polar coordinates. HDp is a any general harmonic function in (d − 2) Euclidean space which depends on the number of Dp-branes.

The procedure to add the Kaluza-Klein momentum wave is to perform a boost along the direction of x5 in the above metric. Given the metric of a momentum wave along a certain direction z in d dimensions:

2 2 2 2 2 2 ds = −dt + dz + (dr + r dΩ(d−3)), (192) if we perform a Lorentz boost in the positive or negative z direction: ! ! ! t cosh γ ± sinh γ t → , γ > 0, (193) z ± sinh γ cosh γ z we will get a new solution of Einstein’s equations:

2 −1 2 −1 2 2 2 ds = −HW dt + HW [dz + (HW − 1)dt] + (dr + dΩ(d−3)), (194) where HW is any general harmonic function in (d − 2) Euclidean space which depends on the number of units of Kaluza-Klein momentum. Applying this procedure to the metric in (191) we obtain:

2 −1/2 −1/2 −1 2 5 −1 2 ds10 = −HD1 HD5 HW dt + HW [dx + (HW − 1)dt] 1/2 1/2 2 2 2 1/2 −1/2 6 2 7 2 8 2 9 2 + HD1 HD5 [dr + r dΩ(3)] + HD1 HD5 [(dx ) + (dx ) + (dx ) + (dx ) ], r2 H = 1 + i , with i = D1, D5, W , (195) i r2 where we have parametrised the non-compact dimensions in polar coordinates, as before. The idea behind the Kaluza- 2 2 2 Klein dimensional reduction is that we want to express this metric as ds10 = ds(1,4) + ds(S1×T 4). To do so, we will 2 2 2 write the metric of the compact dimensions in terms of moduli fields ρ1 and ρ2, such that ds(S1×T 4) = ρ1dsS1 +ρ2dsT 4 . These moduli fields are scalars describing the size and shape of the compact dimensions. We dimensionally reduce first on S1(x5) and then on T 4(x6, ..., x9). These moduli fields turn out to be [18]:

H1/2 ρ = W , 1 1/4 1/4 HD1 HD5 HD1 ρ2 = . (196) HD5

2 By demanding these moduli ﬁelds to be constant we enforce that ds(1,4) only depends on the non-compact coordinates (x0, ..., x4) and we obtain the N = 4, D = 5 type IIB supergravity metric which has the following form:

2 −2/3 2 1/3 2 2 2 ds5 = −λ dt + λ (dr + r dΩ(3)),

3 2 Y ri λ = H H H = 1 + . (197) D1 D5 W r i=1

These black holes are extremal, as we anticipated. They have the minimal mass compatible with the three charges. Thus, as we saw in section II.C., extremal Reissner-Nordström black holes have a vanishing Hawking temperature and therefore, they do not radiate. This makes full sense since otherwise the emission of radiation would reduce the mass 36 in an evaporating process, whereas the charge would not change. Thanks to this, they are stable solutions. We also saw that they have a ﬁnite macroscopic entropy. We can easily calculate it from the Bekenstein-Hawking formula to then compare it to the microstate counting result.

The area of the horizon (at r=0) is

2 A5 = A(3)r1r2r3 = 2π r1r2r3, (198) where r1, r2, r3 are related with the three charges as we will see next. The key to do this is that the metric (197) reduces to the extremal Reissner-Nordström black hole given in (69) with D = 5, µ = q, if we set r1 = r2 = r3 = r±. This usual 5-dimensional extremal Reissner-Nordström black hole can be easily constructed from particularising expressions (69), (70) and (71):

2µ µ2 2µ µ2 −1 ds2 = − 1 − + dt2 + 1 − + dr2 + r2dΩ2 , r2 r4 r2 r4 (3) √ 4G M 1/2 r : = r = µ = 5 , (199) 0 ± 3π so then,

3πr2 µ = 0 . (200) 4G5

3 2 3 Since the area of the horizon is A = A(3)r0 = 2π r0 the relation between this black hole and the 3-charge one is 3 given by r0 = r1r2r3. The three-charge black hole satisﬁes that the total charge is q = QD1 + QD5 + n and since it is 3 3 extremal, the total mass of the black hole is µ = M1 + M2 + M3. Now, if we set r1 = r2 = r3, we have r0 = ri (where i = 1, 2, 3) and QD1 = QD5 = n, so q = 3qi = µ = 3Mi, and plugging this last expression into (200) we have:

2 πri Mi = . (201) 4G5

We can express the 5-dimensional Newton’s constant G5 in string units. To do this, we ﬁrst write it as the quotient 6 2 8 4 of the 10-dimensional constant G10 = 8π gs ls and the volume of the compactiﬁed dimensions. The volume of T is (2π)4V and the volume of S1, 2πR:

G πg2l8 G = 10 = s s . (202) 5 (2π)5RV 4RV

From (201) and using the previous result we relate the radii with the masses:

4G M g2l8 r2 = 5 i = s s M i. (203) i π RV

The masses Mi can be computed at weak string coupling from the mass of winding and momentum modes [27]. For the D-branes they come from multiplying the volume of the compactiﬁed dimensions times the tension of each brane (TD1 or TD5) times the respective charge. The tension of a Dp-brane is deﬁned as: 1 TDp = , (204) p p+1 (2π) gsls hence,

QD1R M1 = (2πR)TD1QD1 = 2 , gsls 5 QD5RV M2 = [(2π) RV ]TD5QD5 = 6 , (205) gsls 37 and for the n units of Kaluza-Klein momentum: n M = . (206) 3 R This is the point where supersymmetry becomes crucial for the validity of this calculation since it allows us to extrapolate these quantities from weak coupling to strong coupling, so that, results in both limits can be compared. Here, the extrapolated property is the number of quantum states. Then, using these last expressions and (203), we can express the three radii in terms of the respective three charges (Q1 := QD1, Q2 := QD5 and Q3 := n):

2 ri = ciQi, i = 1, 2, 3, (207) where ci are constants defined as: g l6 c = s s , 1 V 2 c2 = gsls, g2l8 c = s s . (208) 3 R2V We can finally compute the macroscopic entropy for this black hole configuration from the Bekenstein-Hawking formula using (198), (202) and the previous results for the radii:

A π2 p SBH = = r1r2r3 = 2π QD1QD5n, (209) 4G5 2G5 which only depends on the three charges and not on any of the string or the compactiﬁcation parameters.

2. Microstate counting

The objective is now to reproduce this last expression of the entropy (209) by counting states. In string theory we can only count states with no interactions gs → 0 but, as we said before, supersymmetry guarantees that the counting of states at gs → 0 will hold when interactions are turned on, g 6= 0.

The entropy comes from all possible ways to add momentum to the configuration of the D1 and D5-branes. As we explained before, this is done adding n units of Kaluza-Klein momentum along x5. Since along the direction of this coordinate the D-branes are transitionally invariant, the momentum must be carried by open strings attached to the D-branes. Supersymmetry demand that all these open strings must carry momentum in the same direction along x5. The different states come from the different strings stretching between the QD1 D1-branes and the QD5 D5-branes. Depending on which type of D-branes these strings have their endpoints attached to, we have: (1,1) strings going from D1-brane to D1-brane, analogously (5,5) strings connecting D5-branes and (1,5) or (5,1) ones connecting a D1-brane with a D5-brane. However, it is only this last type of strings we need to worry about to count states since the (1,1) and (5,5) strings become massive and thus, they have no correspondence with massless states in the dual CFT. They are not present in the 5-dimensional supergravity approximation considered, so we do not include them in the counting of states. Only the (1,5) and (5,1) are massless and contribute, so that, n must be split among all these open strings. We need to find a partition of n, taking into account that there are QD1QD5 ways of picking a D1-brane and a D5-brane.

Furthermore, let us recall that we have 4 supercharges in this configuration corresponding to 8 ground states in each string: 4 bosonic and 4 fermionic degrees of freedom. Then, we have that the total number of bosonic and fermionic degrees of freedom is B = F = 4QD1QD5. To find a partition of n with this conditions we can use what we derived 38 in section III, in particular, expression (115): p Sstr = ln P (N=n, B=4QD1QD5,F =4QD1QD5) ' 2π QD1QD5n, (210) which coincides with the expression for the macroscopic entropy of this black hole configuration (209). However, this result is not general enough. We need to remember that (115) is an approximate formula for large N. In our case, the result from this formula is only valid if N B,F so we require that one of the black hole charges is much bigger than the other two n QD1QD5 and this is a very concrete case.

Hopefully, there is a solution to the former problem of lack of generality. We can get the same result but with 1 a more general procedure. The key is to see the QD1 branes as a single D1-brane wrapped QD1 times around S 5 and the same for the QD5 D5-branes wrapped around T . With this new picture in mind, we need to see how the momentum of the string is quantised. Despite the fact that we said we do not consider (1,1) or (5,5) strings, they will help us to understand how (1,5) and (5,1) strings have their respective momentum quantised. Let us start considering 1 (1,1) strings stretching between D1-branes wrapped QD1 times around S . We have that a string needs to travel (2πR)QD1 to return to its starting point, so rather than having a quantised momentum in units of 1/R, it will be in units of 1/(QD1R). The same is true for (5,5) strings between D5-branes, having their momentum quantised in units of 1/(QD5R). Now, if we think of the (1,5) or (5,1) strings which are the ones we are interested in, it is easy to see that their momentum will be quantised in units of 1/(QD1QD5R) (one can consider the approximation where QD1 and QD5 are assumed to be relatively prime which is enough accurate if QD1 and QD5 are large enough). Nevertheless, the total momentum of the system of branes must be quantised in units of 1/R since the branes carry no momentum along x5 as we said before. This can also be seen as the system being described by a conﬁguration with one single long string of length QD1QD5R which is entropically favoured (as we saw in the previous chapter). One can interpret this as a fractionation of momentum along the string. Hence, we can write the momentum as: Q Q n p = D1 D5 , (211) QD1QD5R to keep it unchanged. We see then that with this approach we are not looking for partitions of n anymore but for partitions of N = QD1QD5n, instead. We still have 4 bosonic and 4 fermionic degrees of freedom in the (1,5) and (5,1) strings and hence, we ﬁnally have: p Sstr = ln(N=QD1QD5n, B=4,F =4) ' 2π QD1QD5n. (212)

Despite we got the same result than in (210), our current result is more general since the only assumptions are that the product of the three charges is large enough QD1QD5n 1, but we do not have any relations between the charges. Now, there is a total agreement with (209).

B. D=4 4-charge Extremal Reissner-Nordström black holes

The next simple example of a black hole construction from intersecting branes where we can count states to reproduce the macroscopic entropy is the 4-dimensional extremal Reissner-Nordström with 4 charges. The construction is very similar to the previous 5-dimensional case and the counting of states is analogous.

1. Construction in the Supergravity approximation

Again, we will use N = 4 supergravity, but this time it will be a D = 4 approximation to type IIA superstring theory. Surprisingly, we will still have a 1/8 BPS object preserving 4 out of the 32 supercharges. Even though we 39 have 4 objects which separately break 1/2 of the supersymmetry, the combination of them is still 1/8 BPS (instead of 1/16, since one of the SUSY projectors can be written in terms of the product of the other 3). The difference relies on the distinct type of compactification since in order to end up with only four non-compact dimensions, we will curl up the last six dimensions x4...x9 on a 6-dimensional torus T 6, as we will see. Another difference is that, as we said, we consider 4 different types of objects with their respective charges. In particular, the construction is based on: QD2 D2-branes, QD6 D6-branes, QNS5 NS5-branes and n units of Kaluza-Klein momentum (as before). The D2-branes are wrapped around T 2 = S1 × S1 (x4 and x9), the D6-branes around the full set of compact dimensions T 6 (x4...x9), the NS5 branes around T 5 = S1 × ... × S1 (x4...x8) and the n units of Kaluza-Klein momentum also around S1 (x4).

We follow the same path as before. First, we construct the black hole in the supergravity approximation from the 10-dimensional metric and compute the macroscopic entropy from the Hawking-Bekenstein formula. Then, we perform the counting of states and compare both results.

In this case, the conﬁguration is based on 4 diﬀerent types of objects, showed in the following table:

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D2 X X * * * * X NS5 X X X X X X * D6 X X X X X X X K-K X X * * * * *

The construction of the 10-dimensional metric is obtained using the same recipe that we have already discussed and we therefore omit the intermediate steps. This metric turns out to be:

2 −1/2 −1/2 −1 2 4 −1 2 −1/2 1/2 5 2 6 2 7 2 8 2 ds10 = −HD6 HD2 HW dt − HW [dx + (HW − 1)dt] + HD6 HD2 [(dx ) + (dx ) + (dx ) + (dx ) ] −1/2 −1/2 9 2 1/2 1/2 2 2 2 + HD6 HD6 HNS5(dx ) − HD6 HD2 HNS5[dr + r dΩ2], r H = 1 + i , with i = D2, D6, NS5, W. (213) i r2 We dimensionally reduce first on S1(x4), then on T 4(x5, ..., x8) and then on S1(x9). We will have three moduli fields 2 2 2 2 2 2 2 ρ1, ρ2, ρ3, such that ds10 = ds(1,3) + ds(S1×T 4×S1), with ds(S1×T 4×S1) = ρ1dsS1 + ρ2dsT 4 + ρ3dsS1 . These moduli fields are [18]:

H1/2 ρ = W , 1 1/4 1/4 HD6 HD2 H1/2 ρ = NS5 , 2 1/4 1/4 HD6 HD2 HD2 ρ3 = . (214) HD6

2 Again, by demanding that these moduli ﬁelds have to be constant we assure that ds(1,3) only depends on the non- compact coordinates (x0, ..., x3) and we get the N = 4,D = 4 type IIA supergravity metric. In analogy with (197), we have:

2 −1/2 2 1/2 2 2 2 ds4 = −λ dt + λ (dr + r dΩ(2)), 4 Y ri λ = H H H H = 1 + . (215) D6 D2 W NS5 r i=1

The area of the horizon (at r = 0) is in this case √ √ A4 = A(2) r1r2r3r4 = 4π r1r2r3r4. (216) 40

Again, we will relate the four radii r1, r2, r3, r4 with the four charges using the fact that the metric above (215) reduces to the 4-dimensional extremal Reissner-Nordström black hole (58) if we set r1 = r2 = r3 = r4 = r±. To construct this usual 4-dimensional extremal Reissner-Nordström black hole we particularise expressions (69), (70) and (71). Its metric is given by

2G µ G2µ2 2G µ G2µ2 −1 ds2 = − 1 − 4 + 4 + 1 − 4 + 4 dr2 + r2dΩ2 , r r2 r r2 (2)

r0 : = r± = G4µ, (217) and thus, r µ = 0 . (218) G4

2 2 Again, by comparing the area of the horizon of this metric which is A = A(2)r0 = 4πr0 we can establish the 2 √ relation with the four radii r0 = r1r2r3r4. The total charge is q = QD2 + QD6 + QNS5 + n and the total mass 2 2 µ = M1 + M2 + M3 + M4. If we set r1 = r2 = r3 = r4 we have r0 = ri (i = 1, 2, 3, 4) and QD2 = QD6 = QNS5 = n, so q = 4qi = µ = 4Mi and thus,

ri Mi = . (219) 4G4 Likewise, we need to express Newton’s constant in string units. We follow the same procedure than before but now the 6 6 Q6 volume of the compactiﬁed dimensions T is (2π) V , where V is the product of the six radii of each circle V = i=1 Ri and, in this case, we obtain:

G g2l8 G = 10 = s s . (220) 4 (2π)6V 8V

Hence, the relation between the radii and the mass is

g2l8 r = s s M . (221) i 2V i We perform the same computation for the masses in terms of the volume of the compactiﬁed dimensions and the tension of the respective brane:

R1R6QD2 M1 = (2πR1)(2πR6)TD2QD2 = 3 , gsls 6 VQD6 M2 = (2π) VTD6QD6 = 7 , gsls (R1...R5)QNS5 M3 = (2πR1)...(2πR5)TNS5QNS5 = 2 6 , (222) gs ls and for the Kaluza-Klein momentum: n M4 = . (223) R1

Using these and (221) we have the relation between each radius and the respective charge (Q1 := QD2, Q2 := QD6, Q3 := QNS5, Q4 := n):

ri = ciQi, i = 1, 2, 3, 4, (224) 41 where ci are constants deﬁned as: g l5 c = s s R R , 1 2V 1 6 g l c = s s , 2 2 2 ls c3 = , 2R6 2 8 gs ls 1 c4 = . (225) 2V R1

Lastly, we compute the macroscopic entropy applying the Hawking-Bekenstein formula to (216) and make use of (220) and (224):

A π √ p S = = r1r2r3r4 = 2π QD2QD6QNS5n. (226) 4G4 G4

2. Microstate counting

Now, we want to check if we can reproduce the last result for the entropy by counting states. The procedure is totally analogous to the former 5-dimensional case. As we explain when we were treating that case, we can get a more general result and not just a very specific approximation if we think of this type of black hole configuration as single branes of each type but wrapped around the compact geometries a number of times equal to the respective charges. 2 Thus, in the present case we have a single D2-brane wrapped QD2 times around T , a single D6-brane wrapped QD6 6 5 times around T and a single NS5-brane wrapped QNS5 times around T . Like in the previous example, the strings connecting the three branes will be the only ones effectively contributing to the counting of states, so that, the product of their three charges will determine how the Kaluza-Klein momentum is quantised. In particular, it will be quantised in units of 1/(QD2QD6QNS5R1), and since the total momentum of the system must satisfy the original Kaluza-Klein expression in units of 1/R1 we can write: Q Q Q n p = D2 D6 NS5 , (227) QD2QD6QNS5R1 and therefore, we just need to find partitions of N = QD2QD6QNS5n. In the same way as before, we have 4 bosonic and 4 fermionic degrees of freedom. Hence, using (115) we arrive at: p Sstr = ln(N=QD2QD6QNS5n, B=4,F =4) ' 2π QD2QD6QNS5n. (228)

We see that the agreement between the macroscopic entropy computed from the Bekenstein-Hawking formula and the entropy of the strings states computed counting partitions is complete.

C. Dual brane conﬁgurations

In addition to the D1-D5-W configuration described above, there are other types of brane intersections which lead to a 5-dimensional extremal Reissner-Nordström black hole in the supergravity approximation. These other black hole configurations can be obtained from the one presented here by applying the different S or T-dualities in type II string 42 theory along the compact directions. The same idea holds also for the case of obtaining a 4-dimensional extremal Reissner-Nordström black hole from the system D2-D6-NS5-W. The point is that all these dual configurations are totally equivalent since they give rise to the same expression for the entropy in terms of the dual charges. This is because string dualities (along the compact directions) relating brane configurations do not change the lower dimensional supergravity metric except by an overall constant function which, of course, does not affect the entropy result. In this section, we present a few concrete examples of dual configurations of these two cases and check that they, indeed, lead to the expected entropies when constructed in the supergravity regime.

1. From D1-D5-W to D0-D4-F1

A ﬁrst simple example is to perform an S-duality, so we still have a conﬁguration of type IIB, by replacing the D1-branes for F1-branes (fundamental strings), the D5-branes for NS5-branes and leaving the Kaluza-Klein momenta unchanged. It is easy to check that this case leads to the expected entropy: p S = 2π QF 1QNS5n. (229)

A more interesting example arises from performing a T-duality along the direction where the Kaluza-Klein momenta propagate, in our case, x5. With this duality we obtain a type IIA conﬁguration where the D1-branes are now D0- branes, the D5-branes D4-branes and the n units of Kaluza-Klein momenta are equivalent to an F1-brane wrapped n times on the dual circle S1(x5). The compactiﬁcation is of the same type: T 5 = T 4 × S1, as the following table shows:

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D0 X * * * * * D4 X * X X X X F1 X X * * * *

We can easily compute the masses in the same way as we did in section V.D.1. from the product of the volume of 2 the compact dimensions, the tension of the brane (for a Dp-brane given by (204) and for a fundamental string 1/ls), and the charge:

QD0 M1 = , gsls QD4V M2 = 5 , gsls QF 1R M3 = 2 . (230) ls

Now, we use the same procedure of writing the three radii ri in terms of the masses. Plugging the former results, we have the radii in terms of the charges and we can compute the entropy as in (209):

A π2 p S = = r1r2r3 = 2π QD0QD4QF 1, (231) 4G5 2G5 which has the same form than before but in terms of the dual charges. 43

2. From D2-D6-NS5-W to D2-D2-D2-D6

Like in the former case, it is possible to get dual configurations which give rise to the same form of the entropy. In this case, from the D2-D6-NS5-W configuration in type IIA we can perform a T-duality along x4, x5, x6, then an S-duality to obtain a configuration in type IIB and again a T-duality, this time along x9, giving back a type IIA system described by three types of D2-branes wrapping different directions (orthogonal T 2s) and D6-branes wrapping the whole compact space T 6, as we can see from the following table:

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9

D2 (xQ1) X X X * * * * D2 (xQ2) X * * X X * * D2 (xQ3) X * * * * X X D6 (xQ4) X X X X X X X

The masses are:

R4R5Q1 M1 = 3 , gsls R6R7Q2 M2 = 3 , gsls R8R9Q3 M3 = 3 , gsls (R4...R9)Q4 M4 = 7 , (232) gsls and using the same techniques than before, we can compute the entropy as in (226):

A π √ p S = = r1r2r3r4 = 2π Q1Q2Q3Q4. (233) 4G4 G4

D. Counting in the dual CFT

In this section, we will compute the microscopic entropy from counting states in the dual Conformal Field Theory (CFT). This method is more complicated than the counting of partitions but we will see that they both give the same leading contributions and moreover, string corrections to the entropy will arise. We will start from the case of the 3-charge supersymmetric black hole in 5 dimensions but this procedure can also be applied to the 4-dimensional 4-charge black hole. As we will see, we can use, in fact, the same derivation but substituting the diﬀerent product of charges, N = QD2QD6QNS5n.

In the case of the three charges black hole, this is the approach originally followed by Strominger and Vafa in [3]. However, their derivation is different that the configuration described in section V.C.1. They considered a different type of compactifications: N = 2 type IIB on K3×S1 (1/4 BPS). We follow their procedure of counting the degeneracy of BPS bound states but in the simpler case of toroidal compactifications described in the aforementioned section.

The low energy physics of the configuration D1-D5-W is described by an orbifold CFT defined on S1. The fields in the CFT correspond to zero modes of the open strings connecting the D1 and the D5-branes. Let us recall that it is 1 equivalent to consider (QD1QD5) strings than one single string wound around S (QD1QD5) times. In order to have 44 this string localised in the non-compact dimensions, the zero modes in the worldvolume theory must correspond to its position in the four transverse compact dimensions. We also remember that in order to have a localised object in the non-compact dimensions, we need that the D1-D5-W system leads to a bound state (BPS) and hence, the system is supersymmetric which implies that there are 4 bosonic and 4 fermionic zero modes on the string worldvolume. For simplicity, we will consider the 4 fermionic degrees of freedom in the R sector (the choice will be clear later).

As we have already said, this system can be described as a conformal ﬁeld theory. We will try to explain this a bit more without going into many details (for further details see [28]). In string theory, an orbifold is an object globally described by an orbit space given by the quotient M/G, where M is a manifold (in which a theory lives) and G is the group of its isometries. In our particular case, we consider a tensor product of (QD1QD5) theories describing singly wound strings, modding out the permutations (QD1QD5)!. This theory has many sectors but, interestingly, it is the sector corresponding to a single string wounded (QD1QD5) times the one giving the more low energy degrees of freedom than the others. These others involve multiple strings and are therefore entropically suppressed as we already know from section IV.A. Thus, we only need to count excitations of this long string wounded around S1 to get the entropy in good approximation. The question is now how many diﬀerent ways there are of constructing supersymmetric excitations of energy n/R (given by the n units of Kaluza-Klein momentum).

Before going directly into the D1-D5-W system, we will introduce two useful tools used in the counting of states. One is the generating function from which we can compute the number of states and the other is the mathematical technique of the saddle point approximation that will allow us to extract the number of states from the generating function. We will ﬁrst calculate the number of states in the simpler case of the bosonic open string and then we will use the same procedure in the aforementioned system where we will count states of a type II superstring.

1. Generating functions

First of all, we need to introduce here a useful tool called the generating function that allows us to calculate the N degeneracy or number of physical states with a certain energy dN . This is given by the coeﬃcient of w in the power series expansion of the generating function:

∞ X N G(w) = dN w . (234) N=0 In this way, generating functions encode the number of states that a string theory has at any given mass level.

† Consider a single oscillator given by the creation operator a1, for N = 0 we only have the ground state |0i, for † † k N = 1 we have one state |1i := a1 |0i and in general, for N = k we have one state |ki := (a1) |0i. The generating function of the system is then 1 G (w) = 1 + w + w2 + ... = , with |w| < 1. (235) 1 1 − w

† † 2 Suppose now another single oscillator given by a2 := (a1) . Then, for N = 0 we still have the ground state |0i, for † N = 1 we get one state |2i = a2 |0i and so on...Thus, the generating function, in this case, is 1 G (w) = 1 + w2 + w4 + ... = . (236) 2 1 − w2 45

† † The system formed by these two oscillators a1 and a2 has the following generating function: 1 1 G(w) = G (w)G (w) = . (237) 1 2 1 − w 1 − w2

The above multiplication rule is general. If we have certain oscillators of type A with generating function GA and oscillators of type B with GB, then the generating function for the space of states with oscillators A and B is GAB = GAGB. We will make use of this property of generating functions in the next section.

Finally, we can compute the total number of states from (234) using a contour integral on a small circle about the origin:

1 I G(w) d = dw, (238) N 2πi wN+1 which is easy to evaluate for large N using a saddle point approximation, as it is described in the following section.

2. Saddle point approximation

We brieﬂy describe here the general recipe to perform the saddle point approximation to complex integrals that we will apply later.

We want to compute an integral of the form: Z I = dwg(w)eNf(w) with N 1, (239) where f, g are analytic complex functions. We look for a saddle point w∗ of the function in the exponential of the 0 integrand such that f (w∗) = 0. Since we can always deform the contour of integration in such a way that it passes through the saddle point (Cauchy’s integral theorem), we expect the integral to be dominated by this saddle point of f since N is large. Hence, around this point we can Taylor expand the functions f, g as: 1 f(w) ' f(w ) + f 00(z )(z − z )2, ∗ 2 ∗ ∗ g(w) ' g(w∗). (240)

iφ 00 00 iθ Plugging this into (239) and deﬁning w − w∗ := re , f (z∗) := |f (z∗)e |, we have an integral depending on the two complex phases. However, we can eliminate one of them, without loss of generality, since we are free to choose φ. We have only one phase dependence if we choose θ + 2φ = π. Hence,

Z 1 00 2 Nf(w∗) iφ − N|f (z∗)|r I ' g(w∗)e e dre 2 . (241)

This last integral is gaussian giving:

2π 1/2 Nf(w∗) iφ I ' g(w∗)e e 00 . (242) N|f (z∗)| This is the approximation result that we will apply in the following. 46

3. Bosonic strings

From the previous simple example (237), it is easy to write the generating function for a bosonic string. We just † † need to think of the bosonic string as a chain of inﬁnitely many single oscillators a1, a2, ..., so then,

∞ Y 1 G(w) = . (243) 1 − wm m=1 However, we also need to take into account that the bosonic string has extra degrees of freedom coming from the fact that every oscillator of the chain can oscillate in 24 diﬀerent transverse directions. The generating function for the bosonic open string is

∞ Y 1 G(w) = . (244) (1 − wm)24 m=1

Next, we want to calculate the asymptotic density of states dN for large N. First of all, we must note that (244) becomes singular when w → 1−. We need to estimate the behaviour of G(w) near that point. This can be done making use of the Dedekind eta function η(τ) and its modular property:

∞ iπτ Y 2πimτ η(τ) = e 12 1 − e , m=1 1 η(τ) = √ η(−1/τ). (245) −iτ

By deﬁning w := e2πiτ , it is easy to see that

∞ Y ln w (1 − wm) = w−1/24η , (246) 2πi m=1 and making use of the modular property and the deﬁnition of the eta function we get:

∞ 1/2 1/2 ∞ 2 2 Y m −1/24 −2π −2πi −1/24 −2π π Y 4π m (1 − w ) = w η = w e 6 ln w 1 − e ln w , (247) ln w ln w ln w m=1 m=1 which is known as the Hardy-Ramanujan formula. From this and using (244), we can get an asymptotic expression for G(w) when w → 1−, since then ln w → 0− :

−12 2 2π −4π G(w → 1) ∼ e ln w . (248) ln w

As we anticipated before, for large N we can evaluate the integral (238) using a saddle point approximation. G(w) vanishes rapidly for w → 1− and if w < 1, wN+1 → 0 for large N. Thus, for large N there is a saddle point near w = 1. From (238) the integral to be computed is

−12 I 2 dw 1 2π −4π −N ln w d ' e ln w , (249) N 2πi w ln w so the condition to ﬁnd the saddle point is setting the derivative of the function in the exponential to zero:

1 d −4π2 f 0(w) = − N ln w = 0, (250) w d ln w ln w 47 which leads to −2π ln w∗ = √ , (251) N where we have discarded the positive sign for physical reasons. We have picked only the negative solution in order to 00 have a positive number of states. Evaluating g(w∗), f(w∗), f (w∗) and since this last one is real we can set θ = 0, φ = π/2. Now, we can apply the result from (242) to get

√ 1 −27/4 4π N dN ' √ N e , (252) 2 which coincides with the typical result one can ﬁnd in standard string theory books such as [29].

4. Type II superstrings

In order to be able to write the generating function for superstrings, we ﬁrst need to see how to count fermionic degrees of freedom for the NS and R sectors. From Pauli’s exclusion principle, it is clear that if we consider a single fermionic oscillator with creation operator fr, we can only form two states, the vacuum |0i and an excited state |1i := fr |0i, for a certain level r. So the generating function looks like:

r Gr(w) = 1 + w . (253)

For the NS sector α0M 2 = (N − 1/2) so then,

∞ Y (1 + w1/2)(1 + w3/2)(1 + w5/2)... = (1 + wN+1/2). (254) N=1 Taking into account the expression for the bosonic degrees of freedom from the previous section, the fact that there are 16 degenerate ground states in the open superstring, as we saw in section III.C.3., and that there are, in principle, 8 transverse oscillating modes that reduce to only 4 bosonic and 4 fermionic modes (due to the boundary conditions between the endpoints of the open superstring and the D1/D5-brane), we have:

∞ 4 Y 1 + wN+1/2 G (w) = 16 . (255) NS 1 − wN N=1

By contrast, for the R sector α0M 2 = N, so then,

∞ 4 Y 1 + wN G (w) = 16 . (256) R 1 − wN N=1

In our case, we are interested in the R sector to provide the fermionic degrees of freedom to the superstring. However, we have that for large N the choice of sector does not make a big diﬀerence since α0M 2 ' N. Therefore, we can choose the R sector for simplicity without lack of generality.

With all these tools and the previous example for the bosonic string, we can compute the number of states for the superstring connecting the D1-brane and the D5-brane with N = QD1QD5n. We use the generating function from (256) and as before, we need to study how it behaves when w → 1. In order to do that, in this case, we use the Jacobi 48 theta function and its modular property. Analogously, we deﬁne w := eiπτ :

∞ Y 1 − wm θ (τ) = , 4 1 + wm m=1 ∞ X (m−1/2)2 θ2(τ) = w , m=−∞ 1 θ4(τ) = √ θ2(−1/τ). (257) −iτ Hence,

2 " ∞ #−4 2 " ∞ #−4 2 2 2 −4 ln w X π (m−1/2)2 ln w −π X π (m2−m) G (w) = 16(θ (τ)) = 16 e ln w = 16 e ln w e ln w . (258) R 4 π π m=−∞ m=−∞

So as w → 1, we have:

2 2 ln w −π G (w → 1) ∼ e ln w . (259) R π

We have the same situation as before, meaning that there is a saddle point near w = 1. In this case, the integral to be computed is

2 I 2 dw 1 ln w −π −N ln w d ' e ln w , (260) N 2πi w π so we impose

1 d −π2 f 0(w) = − N ln w = 0, (261) w d ln w ln w to get the saddle point: −π ln w∗ = √ , (262) N where again, we only pick the negative solution for the same reasons as before. Evaluating the functions, the derivatives and fixing the complex phases exactly like in the former case, we obtain the final expression for the total number of states of the superstring: 1 √ d ' N −7/4e2π N . (263) N 2 Note that this is a quite general expression which does not depend on the type of brane configuration considered since it only depends on the mass level N of the superstring connecting the branes.

For the D1-D5-W system we just need to substitute N = QD1QD5n as we pointed out at the beginning of this section: √ −7/4 2π QD1QD5n d(QD1,QD5, n) ∼ (QD1QD5n) e , (264) and ﬁnally, the entropy is given by the logarithm of the above number of states: 7 S = ln d(Q ,Q , n) ∼ 2πpQ Q n − ln(Q Q n). (265) D1 D5 D1 D5 4 D1 D5 49

This result is very interesting. First of all, the leading term in (265) coincides with the result obtained in (212) and reproduces therefore the famous result of a quarter of the area of the horizon of the black hole computed in the supergravity approximation (209). Secondly, we have obtained the ﬁrst correction to the entropy which is proportional to the logarithm of the product of the three charges. This is a stringy correction to the supergravity entropy and it turns out that it goes like the logarithm of the area of the horizon since ln(QD1QD5n) ∼ ln(A/G5). This is thought to be a general rule. However, the factor of the logarithm is not the famous 1/4 of the leading term and its physical meaning, in case there is any, remains unclear.

Analogously, for the D2-D6-NS5-W conﬁguration we obtain: √ −7/4 2π QD2QD6QNS5n d(QD2,QD6,QNS5, n) ∼ (QD2QD6QNS5n) e , (266) and the entropy is 7 S = ln d(Q ,Q ,Q , n) ∼ 2πpQ Q Q n − ln(Q Q Q n), (267) D2 D6 NS5 D2 D6 NS5 4 D2 D6 NS5 and the same ﬁnal remarks than in the previous case can be done.

E. Non-extremal Black Holes

The main problem with the case of non-extremal black holes is that they are not supersymmetric, i.e., they do not lead to BPS states and therefore, the counting is not protected when going from weak to strong coupling. In addition, they have a non-zero Hawking temperature and thus, they will emit thermal radiation which makes them unstable and this is why the validity of the microstate counting is at risk. Nevertheless, it is possible to consider non-extremal conﬁgurations in the so called near extremal limit, that is, introducing a small perturbation in the extremal solution making it slightly non-extremal. We need to remark here again, as well as we did in section II.C.3., that it is not guaranteed that the counting of states will be preserved when we turn on interactions, when going from the brane intersections picture to the black hole picture.

1. D=5 3-charge Non-extremal Reissner-Nordström black holes

It is possible to construct a non-extremal generalisation to the case of the extremal D=5 3-charge Reissner- Nordström black hole. This model was suggested in [30]. We will follow the same procedure which consist of taking the metric from the extremal case (197) and add a non-extremality factor h, also called blackness since it measures how far the black hole is from extremality. In this way, we are considering the near extremality regime. This metric can be written as:

2 −2/3 2 1/3 −1 2 2 2 ds5 = −hλ dt + λ (h dr + r dΩ(3)), (268) where r2 h = 1 − 0 , r2 3 Y r2 λ = 1 + i , r2 i=1 2 2 2 ri = r0 sinh αi, i = 1, 2, 3. (269) 50

Note that the extremal case is recovered if we take the limit r0 → 0 while keeping ri ﬁxed. Also, if we take the limit α → 0 while keeping r0 ﬁxed, we get the 5-dimensional Schwarzschild metric.

We need to recall that in the non-extremal case we have two horizons. In this coordinate system, the outer horizon is located at r = r0 while the inner horizon is at r = 0. In order to calculate the area of the horizon we need to calculate the radius of the outer horizon which can be easily read oﬀ from the metric by comparing it with the metric of the sphere S3 (for a ﬁxed time and radius). So then, we identify

2 2 2 dsS3 = R dΩ(3), (270) with (268) for a constant time and radius,

2 1/3 2 2 ds (t = r = const.) = λ r dΩ(3), (271) and thus, R = λ1/6r and the radius of the horizon is

3 !1/6 1/6 Y 2 rH = R(r = r0) = λ(r0) r0 = r0 cosh αi . (272) i=1 The area of the horizon is therefore,

3 3 2 3 Y A = A(3)rH = 2π r0 cosh αi, (273) i=1 and the Bekenstein-Hawking entropy

3 3 A π2r3 Y 2πr3RV Y S = = 0 cosh α = 0 cosh α , (274) 4G 2G i g2l8 i 5 5 i=1 s s i=1 where we have expressed the Newton’s constant in string units using (202), as previously.

The black hole charges QD1 and QD5 (referred to as Q1, Q2 to simplify the notation) can be computed from the 10-dimensional string metric by integrating over the compact dimensions the different fields (see [30], [31]). The D1 charge is an electric charge and thus, it comes from integrating the Hodge star of the generalised gauge field C(2) which this brane is a source of. The D5 charge is a magnetic charge resulting from integrating the generalised gauge field C(6). To obtain n we must recall p = n/R and that the total momentum can be obtained from integrating the momentum density which is given by the energy momentum tensor T0i. Interestingly, these results are given in terms of the parameters r0, αi and the constants ci defined previously in the following form:

2 r0 Qi = sinh 2αi, (275) 2ci so recalling the deﬁnition of this constants (208) we have:

2 V r0 Q1 : = QD1 = 6 sinh 2α1, 2gsls 2 r0 Q2 : = QD5 = 2 sinh 2α2, 2gsls 2 2 R V r0 Q3 : = n = 2 8 sinh 2α3. (276) 2gs ls

On the other hand, if we set the tree charges equal, the mass of the black hole can be read oﬀ from the time 51

2 component of the metric gtt by identifying the terms with 1/r to 2µ in expression (69) and recalling (70). After some algebra involving the hyperbolic functions, it leads to:

3 πr2 X M = 0 cosh 2α . (277) 8G i 5 i=1

Both the form of the expression for the charges and the expression for the mass suggest an interesting interpretation. If we consider the deﬁnition of the sinh 2αi in the expression of the charges we can see it as a diﬀerence of two terms, 2αi −2αi the two exponential functions e and e . Similarly, the expression of the mass depends on cosh 2αi which can be expanded in terms of the sum of the two former exponentials. Therefore, we can see the system made of both branes and antibranes. These antibranes can be seen as branes with negative charge (in analogy with antiparticles). Therefore, the total charge is given by:

Qi := Qi,+ − Qi,−, (278) where Qi,+ represents the number of branes and Qi,− the number of antibranes. Analogously, the Kaluza-Klein momentum n can be decomposed in left-moving nL := Q3,+ and right-moving momentum nR := Q3,−. We can easily check that this is satisfied in the expressions of the charges in (275) if we define the charges of the branes/antibranes as: r2 0 ±2αi Qi,± = e . (279) 4ci Equipped with this definition, we can check if the expression for the mass can be derived from:

3 X M = (Mi,+ + Mi,−), (280) i=1 where Mi,± are deﬁned in terms of Qi,± using (205) and (206):

2 ±2αi 2 ±2αi RV r0 e πr0 e Mi,± = 2 8 = . (281) 2gs ls 2 8G5 2

Last, using the deﬁnition of cosh 2αi in terms of the exponentials and (280) we get:

3 πr2 X M = 0 cosh 2α , (282) 8G i 5 i=1 which is in total agreement with the result for the mass obtained by reading oﬀ from the metric.

Now, we want to re-express the result for the entropy (274) in terms of the partial charges of the branes and antibranes which can be done by using again the deﬁnition of cosh αi and (279):

3 Y p p p p p p √ √ S = 2π Qi,+ + Qi,− = 2π QD1,+ + QD1,− QD5,+ + QD5,− ( nL + nR) . (283) i=1 It is satisfying to note that this last expression is a generalisation of the previous result for the entropy in the extremal case (209), which is recovered setting all Qi,− = 0, as one would expect.

We can try to perform the counting of states and compare the result with the macroscopic entropy. We consider the same conﬁguration based on branes and antibranes, left-moving and right-moving momentum. From (283) we can see that every factor of the entropy is additive and is given by the sum of two contributions, one coming from branes Qi,+ and one coming from antibranes Qi,− so in principle the most logical thing to do will be to treat separately the 52 counting of states from the branes and the antibranes. We treat them separately applying the formula for counting partitions (115):

Sstr = ln P (N=QD1,+QD5,+nL,B=F =4) + ln P (N=QD1,−QD5,−nR,B=F =4) p p = 2π QD1,+QD5,+nL + QD1,−QD5,−nR . (284)

Clearly, this does not work since it does not match with the previous result (283). This happens because the counting that we performed is not reliable far from extremality, since all supersymmetry is broken and the states are not BPS-protected anymore, which means that it is not guaranteed to extrapolate from weak coupling to strong coupling. Due to this lack of control when we tried to derive the former general case using a controlled approximation we did not succeed.

Nevertheless, something else can be done in a well controlled way. We can consider the case of of near extremal black holes for which the non-extremality can be treated as a perturbation. We consider the near extremal approximation where the only antibranes are right-moving Kaluza-Klein excitations. In other words, we will leave extremality by just adding opposite Kaluza-Klein momentum to the system. This approximation where we can perform the counting of states correctly is called the dilute gas approximation [30, 32] which holds for

R ls, 4 V ∼ ls, QD1 ∼ QD5 ∼ n. (285)

In this approximation, very few antibranes are excited compared to the number of antimomenta. Hence, the dominant contribution to the entropy comes from momentum modes:

QD1 = Q1,+,

QD5 = Q5,+,

n = nL + nR. (286)

Then, previous formula (283) reduces to p √ √ S = 2π QD1QD5 ( nL + nR) , (287) where the extremal case can be recovered by just setting nR = 0. In this approximation, we can perform the counting. While for the extremal case we had NL := QD1QD5n, NR = 0 in this near extremal case we have separate contributions, as we explained before, coming from NL := QD1QD5nL and from NR := QD1QD5nR. So then, p p p √ √ Sstr = ln P (N=NL,B=F =4) + ln P (N=NR,B=F =4) = 2π NL + NR = 2π QD1QD5 ( nL + nR) , (288) which is in exact agreement with the macroscopic formula (287). The interpretation of this result is analogous to the former cases. The D1, D5-branes are wrapped QD1, QD5 times, respectively, so there is a total winding factor of QD1QD5 and thus nL as well as nR are quantised in units of 1/(QD1QD5R) in the same way than in (211), (212).

Finally, it is interesting to go back again to the general case with the entropy given in (283). If we expand this result by multiplying the sums of the square roots, we can see that we will get 8 terms which correspond to all the possible combinations of having a product of charges of a D1-brane/antibrane, a D5-brane/antibrane and a left/right-moving momentum QD1,aQD5,bnc, with a = +, −; b = +, −; c = L, R. So if there exists a branes construction that could reproduce this macroscopic formula, the microscopic formula will be computed from a sum of eight terms:

3 X X p Y p p Sstr = ln P (N=QD1,aQD5,bnc,B=F =4) = 2π QD1,aQD5,bnc = 2π Qi,+ + Qi,− , (289) a,b,c a,b,c i=1 53 where the sums are referred to all possible combinations of the three elements. It is not trivial to think of a brane configuration that will give this results. Rather than that, it looks like if it was necessary to consider 8 different brane configurations whose interpretation still remains obscure.

2. D=4 4-charge Non-extremal Reissner-Nordström black holes

In analogy with the construction of the D=5 3-charge non-extremal Reissner-Nordström black hole in the near extremal limit, we can consider the 4-dimensional case using the metric in (215) corrected by a non-extremality factor h. This metric can be written in the following way:

2 −1/2 2 1/2 −1 2 2 2 ds4 = −hλ dt + λ (h dr + r dΩ(2)), (290) where r h = 1 − 0 , r 4 Y ri λ = 1 + , r i=1 2 ri = r0 sinh αi, i = 1, 2, 3, 4. (291)

The extremal case is again recovered if we take the limit r0 → 0 while keeping ri ﬁxed and we recovered the 4- dimensional Schwarzschild metric taking α → 0 while keeping r0 ﬁxed.

Like before, the outer horizon is located at r = r0 and we can calculate the radius of this horizon using the same 2 procedure, that is, by comparing the metric with the metric of S and evaluating this radius of the sphere at r = r0. The radius of the horizon is

4 !1/4 1/4 Y 2 rH = R(r = r0) = λ r0 = r0 cosh αi . (292) i=1 The area of the horizon is, in this case,

4 2 2 Y A = A(2)rH = 4πr0 cosh αi, (293) i=1 and the macroscopic Bekenstein-Hawking entropy

4 4 A πr2 Y 8πr2V Y S = = 0 cosh α = 0 cosh α , (294) 4G G i g2l8 i 4 4 i=1 s s i=1 where we have used (220).

The four black hole charges QD2, QD6, QNS5 and n (referred to as Q1, Q2, Q3 and Q4 for convenience) can be derived from the 10-dimensional string metric like before [33]. The results are given in terms of the parameters r0, αi and the former constants ci as follows:

r0 Qi = sinh 2αi, (295) 2ci 54 plugging the value of the constants from (225) we get:

V r0 Q1 : = QD2 = 5 sinh 2α1, gslsR1R6 r0 Q2 : = QD6 = sinh 2α2, gsls r0R6 Q3 : = QNS5 = 2 sinh 2α3, ls r0VR1 Q4 : = n = 2 8 sinh 2α4. (296) gs ls

Again, we can compute the mass of the black hole reading oﬀ from the metric. This time the algebra leads to:

4 4 r0 X r0 r0 X M = sinh2 α + = cosh 2α , (297) 4G i 2G 8G i 4 i=1 4 4 i=1 where we have used the hyperbolic trigonometry property cosh 2α = 1 + 2 sinh2 α.

We make again the interpretation in terms of branes and antibranes. The total charge is still given by (278) with the same notation as in the previous case. The same can be done for the Kaluza-Klein momentum in terms of left-moving nL := Q4,+ and right-moving momentum nR := Q4,−. In order to satisfy (295) the charges of the branes/antibranes must be deﬁned as: r 0 ±2αi Qi,± = e . (298) 4ci We now check that we can derived the former expression for the mass (297) from summing the masses of the branes and antibranes:

4 X M = (Mi,+ + Mi, −) , (299) i=1 where, in this case, Mi,± are deﬁned in terms of Qi,± using (222) and (223):

±2αi ±2αi V r0 e r0 e Mi,± = 2 8 = . (300) gs ls 2 8G4 2

Finally, using the deﬁnition of cosh 2αi and (299) we obtain:

4 r0 X M = cosh 2α , (301) 8G i 4 i=1 which perfectly coincides with (297).

We re-express now the entropy from (294) in terms of the partial charges using (298) and the deﬁnition of cosh αi:

4 Y p p S = 2π Qi,+ + Qi,− i=1 p p p p p p √ √ = 2π QD2,+ + QD2,− QD6,+ + QD6,− QNS5,+ + QNS5,− ( nL + nR) . (302)

As expected, this is a generalisation of the previous extremal case (226), which is recovered setting all Qi,− = 0. 55

We want to investigate if the counting of the microscopic states matches the macroscopic result again. We tried to do it naively in the general case. As well as before, we treat separately the branes and antibranes since also in (302) every factor of the entropy is given by the sum of two contributions and we obtain, in this case:

Sstr = ln P (N=QD2,+QD6,+QNS5,+nL,B=F =4) + ln P (N=QD2,−QD6,−QNS5,−nR,B=F =4) p p = 2π QD2,+QD6,+QNS5,+nL + QD2,−QD6,−QNS5,−nR , (303) which, again, is a wrong result not matching with result (302) because of the same reasons we already explained in the previous case.

However, we can use the dilute gas approximation which now assumes that:

Ri ls, 6 V ∼ ls, QD2 ∼ QD6 ∼ QNS5 ∼ n. (304)

In this approximation, the dominant contribution to the entropy is coming from both the left-moving and the right- moving momentum modes and thus, the contribution of the antibranes can be neglected:

QD2 = Q2,+,

QD6 = Q6,+,

QNS5 = Q3,+,

n = nL + nR. (305)

Therefore, (302) simpliﬁes to: p √ √ S = 2π QD2QD6QNS5 ( nL + nR) . (306)

The extremal case is recovered by setting nR = 0. In this near extremal case, we have separate contributions: NL = QD2QD6QNS5nL and NR = QD2QD6QNS5nR. Hence, p p p √ √ Sstr = ln P (N=NL,B=F =4) + ln P (N=NR,B=F =4) = 2π NL + NR = 2π QD2QD6QNS5 ( nL + nR) . (307) This last result agrees with the macroscopic one in (306). The interpretation is analogous to the former case. The total winding factor is QD2QD6QNS5 and the two momentum modes are quantised in units of 1/(QD2QD6QNS5R).

Finally, we can speculate about the general case (302). Expanding this result, as before, we end up with 16 terms corresponding to all combinations of the four charges. Using a similar notation than before, we have that N = QD2,aQD6,bQNS5,cnd with a, b, c = +, − and d = L, R. Assuming that there is such a construction reproducing the macroscopic result, it will have a microscopic entropy formula as follows:

4 X X p Y p p Sstr = ln P (N=QD2,aQD6,bQNS5,cnd,B=F =4) = 2π QD2,aQD6,bQNS5,cnd = 2π Qi,+ + Qi,− . a,b,c,d a,b,c,d i=1 (308) Unfortunately, the situation is similar than before in the sense that it still remains unclear how to construct such a brane configuration or if it would be necessary to consider 16 different configurations. 56

VI. THE BLACK HOLE INFORMATION PARADOX AND THE MICROSCOPIC INTERPRETATION

The apparent absence of internal structure of black holes leads to the so called information puzzle. This problem arises from the fact that if some matter falls into a black hole and this black hole evaporates emitting thermal radiation which lacks in information content then, the information of the infalling matter seems to be erased from the universe. According to quantum mechanical postulates, the information of a system is always preserved, so that, this cannot be a valid physical process. This paradox is at the heart of the internal structure of black holes and it seems that a good way to try to understand more about this internal structure is to study the entropy of black holes.

String Theory is claimed to be a theory of quantum gravity, reconciling quantum theory with gravity and therefore, if this is true, it should provide an explanation to the black hole information paradox. The success of the microscopic computation of the entropy from intersected branes configurations points towards a possible solution to the paradox. That is, black holes are microscopically smooth horizonless geometries consisting of different types of wrapped branes. Therefore, from a microscopic point of view, there is neither a singularity nor a horizon. Those are only seen from a macroscopic effective description outside the black hole horizon in the lower dimensional theory. In this sense, although information appears to be lost in the lower dimensional effective description, there is no such an issue when one looks at the microscopic degrees of freedom of the black hole.

The former interpretation is a success of this microscopic approach. However, it has some important disadvantages. The problem is that this interpretation is straight forward in the case of extremal black holes but remains obscure in the non-extremal cases. Since in those cases the microstate counting is not reliable we do not know how to interpret these arguments regarding the information problem. One could think that this is just a small problem but it is rather a crucial issue because it is precisely non-extremal black holes the ones that have full physical meaning. They have a non-vanishing Hawking temperature and emit thermal radiation. It is well accepted that the Hawking radiation plays a vital role in the information loss paradox, so that, the microscopic interpretation is far from being a complete proposal to solve this problem. Despite the fact that these calculations did not solve the paradox, they provided an important hint on how information could be stored in black holes and it is possible that the supersymmetric cases can help to understand the physics near the singularity and, hopefully, the potential results would be independent of the supersymmetric character of the object. Furthermore, it is crystal clear that the microscopic approach is a very interesting starting point to tackle this problem. In fact, it has already been the starting point of some solution proposals to the paradox, such as fuzzballs [34], gravastars [35] and AdS bubbles [36], among others.

VII. CONCLUSIONS

The possibility of deriving the black hole entropy from a statistical mechanical approach and establishing its microscopic origin is one of the successes of string theory. The aim has been to extend this kind of calculation for diﬀerent types of black holes, not only extremal ones but for non-extremal as well. Although there has been progress made in this direction considering near extremal black holes, exact microscopic derivations of the entropy of non- extremal black holes are still a challenge nowadays. Particularly, Schwarzschild black holes remain a bit obscure, in this sense, which is unfortunate since it is one of the most simple but realistic models of a black hole and it might be very useful to solve the information puzzle. We have seen that string theory is capable of counting the zero-coupling degrees of freedom of a system of branes that turns into a black hole when the string coupling is turned on and the counting of states is still valid. However, we do not know how these degrees of freedom are distributed when the black hole is formed, i.e., at strong coupling.

The importance of this statistical mechanics derivation of the entropy lies in the fact that it is a ﬁrst step in the search for a microscopic theory of black holes which would allow us to derive the laws of black holes from some fundamental principles. Also, since the black hole microstates are manifestly quantised, the original calculation by Strominger and 57

Vafa was a real test of String Theory, since it involved the quantum nature of black hole states. Another important remark is that thanks to these computations, we know that there cannot be additional substructures in branes because they give the correct result of the total number of states whose logarithm leads to the right entropy.

As we already mentioned in the introduction, the black hole entropy suggested that the information of the black hole was contained in the area of the horizon (rather than the volume). This led to the gauge/gravity duality whose most well-known example is the AdS/CFT correspondence, originally proposed by Maldacena [37]. In our particular examples, we saw how in the weak coupling limit the conﬁguration of branes wrapping compact dimensions can be described in terms of a ﬁeld theory ignoring gravitational interactions. In fact, we performed this explicitly in section V. D. where we counted states in the dual CFT. Then, when we go to strong coupling, the system turns out to be a black hole in the lower dimensional theory which is described by classical supergravity. Thus, the validity of these calculations backs the AdS/CFT conjecture.

All in all, one of the mayor successes of string theory is this idea of counting black hole microstates that we tried to present here. It is also the ﬁrst time that string theory had solved (at least to some extent) a problem from another area of physics.

To sum up, in this project we have studied black holes in string theory. After introducing the basic ideas about black hole entropy we have developed the tool of counting partitions from string thermodynamics. Next, we have motivated the use of long strings in modelling black holes and the Random Walk model which provides a qualitative estimation of the entropy of a Schwarzschild black hole. The main part of the thesis is devoted to explain how to obtain extremal black holes from intersecting branes. We have presented two particular brane constructions that lead to two types of Reissner-Nordström black holes. In each case, we first compute the expected Hawking-Bekenstein (macroscopic) entropy in the supergravity approximation to then apply our techniques of counting partitions to count the number of microstates of these brane configurations. We compare both results and find a total agreement for both cases. Although we have picked two particular configurations, we briefly present how to relate them with other possible configurations by string dualities and give specific examples. For those, we check that they, indeed, lead to the same result for the entropy in terms of the dual charges. In addition, we explain the original procedure based on counting microstates in the dual CFT and we find with this approach stringy corrections to the entropy which do not appear to be universal. We have tried to study non-extremal black holes in the near extremal limit where we can still reproduce the macroscopic entropies by counting microstates. We have done this for our two concrete cases. Next, we have discussed the importance of these calculations and the relevance of the microscopic interpretation of black holes in the context of the information loss paradox, even though this approach is not capable to solve completely the problem. Last but not least, we finish this project making emphasis in the physical importance of these computations.

Despite the fact that this project seems to be purely theoretical, there are certainly many attractive observational prospects. Experiments using systems of radio telescopes, such as the EHT project, have become more important, but maybe the most promising projects are those based on the detection of gravitational waves. Special attention received the LIGO experiment when in 2015 gravitational waves were detected for the first time. At this respect, new possibilities with high-precision detectors might be around the corner. Currently, in addition to LIGO, there are more scientific collaborations working in gravitational waves detection, such that VIRGO or LISA, among others. A complete review of all these observational perspectives has recently been published [38]. All in all, fascinating perspectives are arising as gravitational waves could become a useful tool to study black hole horizon structures. Particularly, future observations of black hole event horizons could provide interesting insights into the information paradox and confirm or discard many proposals that try to solve this problem. In the future, we hope we could even get observational data that might take us one step closer to the formulation of a UV-complete Quantum Theory of Gravity that could describe our universe. In conclusion, we are currently living exciting times for gravitational physics since many theoretical models are about to be tested experimentally for the first time. 58

ACKNOWLEDGMENTS

First of all, I would especially like to thank my supervisor Giuseppe Dibitetto, for all the help and attention received during the realisation of this master thesis project, as well as, for giving me the opportunity to get in contact with this interesting ﬁeld of study, while transmitting me his passion and enthusiasm. Sincere thanks to Ulf Danielsson for reviewing my manuscript and useful advice. Also, many thanks to Suvendu Giri and Luigi Tizzano for helpful discussions.

I am very thankful to Anton, Antonio, Emmi, Hans, Jonatan, Lukas, Marcus, Melisa, Mirco, Roberto, Robin, Viktor for both physics and non-physics related conversations but most importantly, for making oﬃce hours more pleasant. Last but not least, special thanks to my family and friends (either named here or not they know who they are) for all their support.

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