Supertubes as Black Hole Microstates

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University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

Supertubes as Black Hole Microstates

by

Nick Mabjaia

0513687

August 2014

54 ECTS

January 2013 - August 2014

Supervisor:

Prof. Jan de Boer

Examiner:

Dr. Ben Freivogel

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Abstract

In string theory D-branes have been studied in order to deepen the understanding of black holes from a theory of quantum gravity. The study of microstates of these D-branes sheds light on interesting properties of these systems. Here, we are interested in a gravitational description of the microstates of a black hole.

The supersymmetric bound state of D1-branes and D5-branes takes part in the so-called ‘supertube effect’ in which the system appears as a different dipole object of higher dimension. The D1D5 system has two dual descriptions, one as a field theory and the other in terms of gravity. In this study we discuss world-volume field theories and the geometric realizations of the microstates. We explore the relation between the two descriptions in context of the Higgs and Coulomb branches arising in the supersymmetric gauge theory of the D1D5 system. A map between the two branches is derived which can potentially be a valuable tool in the understanding of black hole physics.

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6 Conclusion and Outlook 61 7 Appendix 62 7.1 Supertube calculations . . . 62 7.1.1 Lagrangian . . . 62 7.1.2 Hamiltonian . . . 63 7.2 Supersymmetry . . . 64 7.2.1 Conventions . . . 64 7.2.2 Superspace . . . 65 7.2.3 N=2 Vectormultiplet Lagrangian . . . 66 7.2.4 N=2 Hypermultiplet Lagrangian . . . 69

7.2.5 Potentials: D- and F -terms . . . 70

7.2.6 Further potential terms . . . 72

7.2.7 Full 4d N=2 Lagrangian with vector and hypermultiplets 72 7.2.8 Rewriting the D-term . . . 73

7.2.9 R-symmetry . . . 74

7.2.10 Dimensional reducing 6d N = 1 to 4d N = 2 to (1+1)d N = (4, 4) . . . 75

7.3 More supersymmetry . . . 80

7.3.1 Supersymmetry condition for branes . . . 80

7.3.2 Spinor checks . . . 81

7.3.3 Supergravity . . . 81

7.4 Complex Geometry . . . 82

7.4.1 Characteristic Classes . . . 82

7.4.2 Covariantly constant J and the K¨ahler potential . . . 84

7.4.3 Volume of a K¨ahler manifold . . . 85

7.4.4 Calabi-Yau manifolds and holonomy . . . 86

7.4.5 Hyper-K¨ahler manifolds . . . 86

7.4.6 Orbifolds . . . 87

7.5 Miscellaneous . . . 87

7.5.1 Why 2 or 3 charges? . . . 87

7.5.2 Invertibility of the Higgs-Coulomb Map . . . 88

7.5.3 Bianchi Identity . . . 89

7.5.4 Curvature in the Chern Simons action . . . 89

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Figure 1: In (a) we scetched a representation of a black hole with a singularity in the middle and en event horizon as seen in a general space Rd_{. (b) Shows}

that the same black hole could be an extended object in a higher dimensional theory. The higher dimensional object which extends in the extra dimensions looks like a point in the lower dimensional subspace. If it is heavy enough (or the coupling is strong enough) the black p-brane describes a black hole on the hypersurface.

1 Introduction

For many years, black holes have been an interesting object of study. They are solutions to the Einstein equations of general relativity. In classical gravity various interesting aspects of black holes have been examined, for example the characterization by a singularity and an event horizon. As Hawking noticed that black holes should radiate due to quantum effects at the surface of the horizon, this opened the door for black hole thermodynamics to apply the notion of entropy to the object, which appeared to be proportional to the size of the horizon. These semi-classical observations do not describe all of the physics of black holes. One ideally would like to have a theory of quantum gravity to fully describe the object. This is where string theory comes in. String theory is a quantum theory of strings which automatically describes gravity. In string theory there are various ways of looking at back holes. In general, we make heavy black holes by putting stacks of branes or strings at the origin of a space. Black p-branes were originally solutions of supergravity which describe black holes in higher dimensional theories. A higher dimensional object which extends in not all the dimensions will look like a point in the spatial directions transverse to its world-volume directions. This is illustrated in figure 1. The black holes we study in this thesis are black holes in five dimensions, which means that there are four spatial directions transverse to the p-branes in which the branes look like a black hole.

As noted above, gravity allows one to assign some kind of entropy to the black hole as long as it has a nonzero event horizon. This entropy is given by the Bekenstein-Hawking area law

SBH =

A

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Statistical mechanics tells us that if a system has entropy, it must also have microstates. The number of these microstates, given by the exponential of the entropy

N = eSBH_, ₍₂₎

should be very big for a typical black hole. These microstates were lacking in the gravity description of black holes in which black holes produce identical geometries for a given set of conserved charges mass, charge and angular mo-mentum. This is the so-called ’no-hair’ theorem. This implies zero entropy as there is only one state given the three quantum numbers.

In string theory there are closed strings and open strings. If one incorporates open strings in the theory then, defined by the boundary conditions of the end-points, there must also be objects on which these strings end which are called D-branes.1 _{Suprisingly, these objects appeared to be the same objects as the}

black p-branes [1]. They are two descriptions of the same thing at different val-ues of the coupling constant. At low coupling we can describe the brane by the endpoints of open strings and by increasing the coupling, which is proportional to the gravitational constant G ∼ g2_{, we bring in gravity and get a black hole}

in the transverse space. A strong confirmation for this view is the counting of entropy for the D-brane. The open string description of D-branes allowed one to count states of the system so that one can assign an entropy to the system by counting the degeneracy of states with the same conserved charges. This proce-dure has exactly reproduced the entropy given by its horizon area calculated at strong coupling for a certain class of supersymmetric black holes [2]. The system must be supersymmetric for the entropy to match, underlining the importance of supersymmetry in string theory. This remarkable result solidifies the simi-larity of these descriptions and the two ways of looking at entropy. Somewhat later, Maldacena [3] specified this relation even further in his AdS/CFT conjec-ture. The correspondence states that the theory of near horizon gravity which is a result of the backreaction of the black hole on spacetime equals the theory on the D-brane which has no gravity an has one dimension less. Taking this lit-erally one expects degrees of freedom that are found in the D-brane theory also to appear in the gravity theory. The idea is that the microstates that provided the entropy of the system should also have gravitational geometric realizations. Going from small coupling to strong coupling, one expects the microstates to map into these microstate geometries [4]. This could suggest that black holes do have ’hair’. The object of this thesis is to study the relation between the two descriptions of black hole microstates.

In order to read this thesis a fairly good knowledge of quantum field theory and some introductory knowledge of string theory are required. In constraining the length of the report, I have chosen not to include the basics of string the-ory, such as the world-sheet analysis and mode expansions etcetera. Although it should be readable for non-string theorists, it is advised to have some back-ground knowledge in order to understand the theory presented in this thesis. Where necessary extra review material is mentioned in footnotes. Some intro-ductory references providing also background material on strings and D-branes are [5][6][7][8][9].

Firstly we will describe some of the basics of D-brane technology in order to 1_{The D in D-brane is due to the Dirichlet boundary conditions that the open string has in}

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have the right tools to get to a good description of the systems we are working with. Then we turn to the supertube effect, which essentially is the blowing up of the geometry of a D-brane system. The supertubes describe the microstate geometries that we are looking for. In order to examine the microstate geome-tries that account for the entropy in a geometric way, we study a black hole model constructed by a bound state of D1 and D5 branes. This is a subsystem of the D1D5P system which was used to get an entropy matching with the area law. Also, this system lies at the basis of the discovery of the AdS/CFT conjec-ture and is well studied in that context. The supersymmetric gauge theory that this system produces is extensively discussed in the chapter devoted to this. In the last chapter the Higgs-Coulomb map is introduced, which gives a mapping between the D-brane description and the geometrical description of the system. In this section I try to use this map on our system to learn more about the relation between the two branches of moduli space, and equivalently between the microstates and their geometric realizations.

2 D-branes

As pointed out in the introduction, the higher dimensional membranes in string theory that we call branes have originally appeared from two distinct viewpoints. On the one hand D-branes were objects that were necessary in the description of open strings with Dirichlet boundary conditions. Hence the D in D-branes. On the other side, by looking at the gravity of closed string theory, it was seen that the theory provides the ingredients to make higher dimensional generaliza-tions to black holes. The metrics produced by these black hole generalizageneraliza-tions, initially called ”black p-branes”, have been well studied. Only later it appeared that these were just two descriptions of the same thing [1]. Note that when we speak of Dp or p-branes, the p denotes the number of spatial world-volume

directions.

The D-branes from the open string theory are well studied at low string coupling where they become heavy and their centers of mass static.2 Since the radius of the back reaction on the surrounding spacetime is proportional to N gs this does not play a role in the gs → 0 limit and the theory on the

branes decouples from free supergravity. The supergravity description of closed strings on the other hand is well described at large N gs. So the two theories

describe the same thing at different values of the string coupling constant gs.

We will come back to these limits in more detail in context of the AdS/CFT correspondence in section 4.3.3.

2.1 Black holes in String theory

In this subsection we will quickly review some aspects of the study of back hole entropy in string theory. This discussion here is quite brief and introductory information is omitted as the focus of this thesis lies more on the D-brane 2_{Only perturbations around the center of mass in the form of waves are possible. For}

compactified branes one can totally fix the brane in space by only looking only at very low energies, since the waves are quantized in KK modes and one can go to energies below the first excitation. This in contrast to the infinite flat brane case.

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analysis. For introductions to the concepts used there are many reviews on this.3

The black hole model that was originally studied in order to find the entropy on a microscopic level matching the entropy given by its event horizon through the Bekenstein Hawking entropy formula was the the 3-charge D1D5P system [2]. This produces a stable black hole in five dimensions with a nonzero event horizon. For completeness we include here the solution for the metric for the extremal D1D5P back hole [13]:

ds2 = f₁−1/2f₅−1/2(−dudv + (fn− 1)du2) (3) +f1/2f₅1/2dxidxi+ f 1/2 1 f −1/2 5 dxadxa C₀₅(2) = −1 2(f −1 1 − 1) F_ijk(3) = ijkl∂lf5, F(3)= dC(2) e−2φ = f5f1−1 f1,5,n = 1 + ( r1,5,n r ) 2_. ₍₄₎

Here, v = t − y, u = t + y, where y = x1 parametrizes the circle S1 along

which the D1 is wrapped as well as the D5. The momentum P also along y. The xa, a = 2, ..5 parametrize the internal directions of the D5 transverse to the

D1 which are compactified on a four torus T4 _{whereas the x}

i, with i = 6, .., 9

parametrize the flat transverse directions. C(2) _{is the RR 2-form potential that}

couples to the D1 and the D5 as the magnetic dual. The harmonic functions corresponding to the D1, D5 and momentum respectively f1,5,n are given in

terms of the parameters r1,5,n as

r21= gsα0 ˜ v Q1, r 2 5= gsα0Q5, r2n= gs2α02 ˜ vR2 5 N, (5)

with Q1, Q5 and N the numbers of D1-, D5-branes and the quantized

momen-tum.

Coupling back to figure 1, in this picture we have a D-brane system wrapping the extra dimensions S1× T4 _{(of y and x}

a) which describes a black hole in the

transverse space of R1

× R4_{(t and x}

i). This is naively the supergravity solution

for the 3-charge system, describing a black hole in 5 spacetime dimensions. The 2-charge system which is the object of study in this thesis has a similar metric. The big difference is that the 2-charge system does not have a nonzero horizon area. This is because the length of the circle on which the D1 is wrapped will go to zero as one goes to r = 0, resulting in a zero area horizon, whereas in the 3-charge case it stabilizes to a constant radius as the r’s in the f₁−1/2f₅−1/2 and the fn cancel out.4 Since it has a nonzero horizon at r = 0 one can assign an

entropy to the black hole by the Bekenstein-Hawking entropy law. Rewriting the Q1, Q5 and N in terms of integer charges n1, n2 and np gives

SBek= 2π

√

n1n5np.

3_{Some other good reviews of the gravity side of the system are [10], [11], and also [12] and}

[3] clearly review aspects of the system. Here we mainly follow [13].

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This is shown to agree with the entropy calculated from the microstates on the D-brane.

In this thesis the focus is on the 2-charge subsystem. Looking at the entropy formula above, setting npto zero gives a vanishing entropy. From this and from

the absence of a nonzero horizon area one could conclude that the 2-charge system is not a good black hole. At the same time the microscopic counting does give an entropy. In making microstates of 3-charge black holes one follows very similar procedures as we will use in the 2-charge case [12]. So from this fact and give that the 2-charge case exhibits quite some richness by itself already, makes it certainly worth studying. As the above discussion has been very brief, many of the concepts will become more clear in the body of the thesis. Having briefly described a gravity solution for black holes in string theory we will turn our focus to the world volume theory on D-branes.

2.2 D-brane Actions

In this section we will discuss open string theory on D-branes and describe bosonic D-brane actions. As noted in the introduction, the branes in string theory have an interpretation of being the endpoints of open strings. The spec-trum of an open string describes a vector field in the NS sector and a fermion in the Ramond sector.5 With these fields one gets a gauge theory on the D-brane world-volume.

The general action, incorporating these fields and their couplings to closed string NSNS and RR fields will consist of a Dirac-Born-Infeld term and a Chern-Simons term,6

SDp = SDBI+ SCS.

The DBI action holds the coupling of the gauge fields of the brane Fµν with

NS-NS fields Gµν, Bµν and φ. The CS term incorporates the coupling of the

closed string R-R fields to the brane. In describing the general action above we will first focus on the bosonic D-brane action. Later, when we discuss the effective action obtained by taking the low energy limit for the brane we will use supersymmetry to arrive at a supersymmetric action with both fermions and bosons. We will work in the string frame.7

We will focus on D-branes as the main objects of study in this thesis. String theory has various other interesting solitonic objects. After D-branes other well studied solitons are NS5 branes and Kaluza-Klein monopoles, of which world volume actions are discussed for instance in [14] and [15]. The IIB NS5 brane being S-dual to the D5-brane and the KK T-dual to the NS5 [15]. These objects will come back later but are of lesser importance.

5_{As mentioned in the introduction, these statements will not be motivated in this thesis}

as they are well described in introductory string theory courses. We refer to the literature for derivations and explanations, for instance [5][7][9].

6_{For completeness: in a massive IIA supergravity background also an extra Chern-Simons}

term mICS should be added. This will not be discussed here.

7_{The string frame metric is related to the Einstein frame metric by g}E

µν= e2Φgµν, which,

using a field redefinition, can be convenient for describing gravity as it gives the canonical form to the dilaton kinetic term Einstein-Hilbert part in the low energy effective action of the bosonic string.

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2.2.1 DBI Action

The DBI action started off as an educated guess for a covariant action, but Lorentz covariance of T-dual descriptions turns out to require this form. It is given by8

SDBI= −Tp

Z

dp+1σe−φq−det(Gµν+ Bµν+ 2πα0Fµν). (6)

Here, µ,ν are worldvolume indices. The Gµν, Bµν and φ represent pullbacks

of the metric, anti-symmetric Kalb-Ramond 2-form field and the dilaton arising from the NS-NS sector of closed string states. Fµν is the usual fields strength

of the gauge field Aµ on the world volume on the brane. Pullbacks to the word

volume of the brane are done as follows:

P [GAB+ BAB]µν= (GAB+ BAB)∂µXA∂νXB,

where A, B goes over all of spacetime and µ, ν are the world volume indices. This pullback gives a kinetic term for the scalars Φirelated to the maps XA(σa) from the brane world volume to spacetime, where Xi = 2π`2Φi. With string coupling constant given by the dilaton gs= eφ the tension of a Dp brane is9

τp= Tp gs = 1 gs √ α0_(2π√_α0₎p. (7) 2.2.2 Nonabelian DBI

There is also a nonabelian form of this action which was derived by Myers [16]. A brief description of his derivation will be given below.

The Born-Infeld action given above (6) is compatible with T-duality. Dualiz-ing down from the action for a D9-brane shows that all Dp-brane world volume actions will be of this form. The scalar derivatives appearing in a reduction of the field strength conveniently match the pull backs of Gµνand Bµν. If one then

allows for non commutativity of the fields it appears that the above action does not satisfy the essential condition of being compatible with T-duality transfor-mations. Without going into details we will present the nonabelian action which is arrived at by dimensional reduction from a D9 action to Dp while keeping track of commutator terms resulting from the non commutativity. Some Matrix manipulations within the determinant gives the more obscure form

˜ SDBI = −Tp Z dp+1σTre−φ q −det(P [Eab+ Eai(Q−1− δ)ijEjb] + kFab)det(Qij) , (8) where k = 2πα0, Eµν = Gµν + Bµν and Qij+ iλ[Φi, Φk]Ekj. The Φ’s are the

adjoint scalars arising at dimensional reduction and related to the position of the brane.10 _{Here, the indices are in a different convention, a, b go over the}

8_{The DBI action is actually not an exact action as it requires the derivative of the field}

strength to be very small. This is even more so the case for the nonabelian case in which it is not so obvious how to satisfy this condition for reasonably big F .

9_{This is derived by T-duality from the D0 tension T}

D0 = (gs`s)−1 which follows from

M-theory where the D0 brane is seen as the first Kaluza-Klein excitation of the supergravity multiplet on the circle in the eleventh dimension. The circle has radius R11= gs`

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internal directions and i, j, k denote transverse directions of the brane. This notation will only be used in this subsection. For other purposes our starting notation is more convenient. The trace is over gauge indices that are suppressed here. What is interesting here is that the second determinant of Q appearing in this action gives the standard quartic scalar potential term. In a trivial NS background, thus taking the metric Minkowskian an B = 0, the expansion of this factor in string length becomes11

q detQi j= 1 − k 4[Φ i_{, Φ}j_][Φi_{, Φ}j_{] + ....,} ₍₉₎

where we see the SYM scalar potential [Φ, Φ]2 appearing. Eventually we will want to take a low energy action which incorporates this expansion in α0.

In section 4.1.1 we will follow a slightly different route by considering the (9+1) dimensional DBI action (6) which is expanded to first order in k, after which the consequences of non-commutativity and T-duality are evaluated. This is consistent with directly taking the limit from (8). In 4.1.1 more details of this procedure will be provided.

In string theory the non-commutativity in brane theory arises when one considers stacks of N branes. Both ends of the open strings can then end on different branes which leads one to write each open string state and eventually the fields arising from the string as N × N matrices, each index denoting one of the two endpoints of the strings. In the non abelian brane actions one encounters matrix multiplications and traces over these indices. This can be interpreted as reflecting the fact that a string stretching from the i’th to j’th brane can only interact with another j to k string. The open string gauge field Aµwill become

a matrix that is in the adjoint representation of a non abelian U (N ) gauge symmetry. The scalars Xi (and thus also Φi) also become adjoint matrices and the position of the branes will be given by their eigenvalues.

2.2.3 Chern-Simons gauge Couplings

In this subsection we mostly follow Johnsons notes on D-branes [9].

In type II string theory we have p+1 form fields Cp+1arising in the

Ramond-Ramond sector of the closed string spectrum, which have been found [1] to couple to Dp branes. As in type IIA we have only fields of uneven form, there

are only branes of even p: D0, D2, D4, ... Likewise in IIB we have uneven branes

D1, D3, D5, .. In order to incorporate such a coupling in an action of an RR

charged brane we introduce a Chern-Simons type of term:

µp Z C(p+1)= µp (p + 1)! Z Cµ1....µp+1 ∂Xµ1 ∂σ0 ...∂X µp+1 ∂σp dσ0...dσp,

where the integral is over the Dp brane world volume. µp is the charge of the

brane under the (p+1) form.

Now, if we require the CS action to be compatible with T-duality it needs to be modified. To illustrate how this is done we consider these R-R couplings for a D1 brane in the 1-2 plane

11_{Making use of the fact that the determinant is equal to the exponentiated trace of the}

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µ1 Z C(2) = µ1 2 Z Cµ1µ2 ∂Xµ1 ∂σ0 ∂Xµ2 ∂σ1 dσ0dσ1. If we then apply the static approximation, X1_{= σ}1_{, X}0_{= σ}0 _{we set}

∂X0 ∂σ0 = 1,∂X 1 ∂σ1 = 1,∂X 1,2 ∂σ0 = 0,∂X 0 ∂σ1 = 0, which will yield

µ1 Z C(2)= µ1 Z dσ0dσ1[C01+ C02 ∂X2 ∂σ1 ]. (10)

By analysis of the T-duality procedure one can find that performing a duality will transform the differential form fields and derivatives in the T-dualized direction.12 Dualizing in the X2direction gives

C01→ C012

C02→ C0

∂1X2→ kF12,

where k = 2πα0= πls2. This gives us the T-dual action:

µ1

Z

C(2)= µ2

Z

dσ0dσ1dσ2[C012+ C0kF12]. (11)

The first term in the integral is just the expected 3-form field that is to couple to the Dp+1 = D2-brane. The secon term, however, indicates that the gauge

field flux is coupled to the R-R one form field C0which is the field that naturally

couples to a D0-brane. This means that the new flux is a source of D0-brane

charge. At this stage it seems counterintuitive to view a gauge field flux as a source of charge, later we will see how this plays a part in interesting models in string theory. Furthermore the integral over the D2-brane worldvolume also

shows that the D0-brane is not localized in the D2 world volume, but rather

smeared out over it. It is on the 2-plane on the X1 _{and X}2 _{axes at a field}

strength F12. The above generalizes to other Dp brane situations as well. As

one T-dualizes a Dp-brane one can get a Dp+1brane with an extra Dp−1-brane

coupling. The effective Dp−1-brane will be dissolved in the Dp+1 worldvolume.

As an aside, this can be interpreted as a brane that is tilted with respect to the x1 axis, as sketched in figure 2. The value of the F12 is then given by

F = (1/2πα0) tan θ which is not so surprising since the ∂X2

∂σ1 in (10) gives the

change in X2 along the length of the brane, which essentially is the tilt that it has with an angle θ from the X1 axis. The situation is illustrated as follows. From ordinary geometry of a tilted line in a 2d surface we have

X2= dx

2

dx1X 1

= tan θX1.

Also, we can choose a gauge in which A2= X1F12so that after T dualizing

X2→ kA2= kF12X1.

12_{For more background on T-duality and its closed and open string origin, the reader is}

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Figure 2: A static D1 brane tilted in the x1− x2 plane under an angle θ with

the x1 axis.

Now the value of the gauge field strength after T-dualizing is actually given by the tilt of the D1 brane in the initial situation as given by F = (1/k) tan θ.

A more general Chern-Simons coupling term compatible with T-duality can now be constructed by repeating the T-dualisation procedure and applying the above results. The general expression for a Dp-brane becomes:

µp Z Mp+1 " X p C(p+1) # ∧ Tr ekF +B ₍₁₂₎

Here, we only integrate over the total p+1 terms from the sum and the expo-nential expansion that are proportional to the Dp-brane volume form, where the

sum is finite and over possible forms in the embedding spacetime. B is included next to F in order to assure spacetime gauge invariance as before. The pullback of the R-R fields and NS-NS B-field to the brane is implied and the trace is over gauge indices.

A first consequence of this form is that it signals a relationship to 4d instan-tons. Writing out the first nontrivial term in the expansion of the exponential (for B = 0) gives us µpk Z Cp−1F = µp−2 2π Z Cp−1F,

which corresponds to a D(p−2)-brane dissolved in the 2 dimensional sub-plane

of the Dp-brane world volume. This is the generalization of the term that we

had before in the D2-brane case. We used the expression for the charge of a

p-dimensional brane µp= 2π/g(2π

√ α0₎p+1_.

The second term in the expansion is µp k 2 Z C(p−3)∧ TrF ∧ F = µp−4 8π2 Z C(p−3)∧ TrF ∧ F. (13)

Here the Cp−3 form, which naturally couples to Dp−4-branes appears to couple

to this gauge configuration. In this term we recognise the pure gauge instanton configuration as arising on the 4 dimensional sub-space of the Dp worldvolume.

So an instanton on the subspace corresponds to a unit of Dp−4-brane charge

in the Dp world volume. So an instanton appearing on a Dp-brane carries

Dp−4-brane charge.13 In sections 4.3.4 and 4.3.5 we will come back to this

13_{If one considers this term for a D}

3 brane (which could be our spacetime for a 4d Yang

Mills theory in a brane-world scenario) one can write out the wedge product in terms of the anti-symmetric giving the θµνρσ_F

µνFρσ= θF ˜F term which counts the instanton number.

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term in relation to instantons. The correspondence between DpDp−4-brane

configurations and instantons will later become more convincing when we see that moduli spaces agree.

2.2.4 Nonabelian Chern-Simons terms

In this subsection, which mostly follows the notation of [16], we take the previ-ously found Chern-Simons action term involving couplings of the R-R fields and the NS-NS Kalb-Ramond and gauge fields (12) and extend it to incorporate for noncommuting world-volume fields and background fields with nonabelian scalar dependence. The nonabelian generalization of the flat space Chern-Simons ac-tion will be:

SCS= µp Z Tr eiλiΦiΦX i C(i)ekF +B ! .

We will not go through the details of how this action came about but stress that in the motivation for this action T-duality again plays a central part. Some im-portant consequences of this action will be briefly pointed out here. As Meyers showed [16], by including the nonabelian covariant derviatives and taking care of these in duality transformations, this action, where the background fields are functionals of the nonabelian scalars Φi_{, is compatible with T-duality. Since}

T-duality is a key property of string theory this is a strong nonabelian gener-alization of (12) for the coupling of the R-R fields with the branes and other fields.

To clarify the added elements of this action we see that there is a gauge trace wich is over the gauge indices just like in the nonabelian extension of the Born-Infeld action. The iΦ that appears twice in the exponential, which is

implicitly pulled back together with the C(i)_{and B fields, is the interior product}

in the direction of the vector Φi_{. When it acts on diferential forms it acts as an}

operator of form degree −1, which means that it produces a form of one rank lower. On a two form C(2)₌ 1

2CijdX i_dXj _{it works as follows:} iΦC(2)= ΦiCijdXj, so that iΦiΦC(2)= ΦiΦjCij= 1 2[Φ i_{, Φ}j_]C ij,

due to the antisymmetry of the 2-form. In this way it adds commutator terms of the matrix valued vectors Φi _{of the nonabelian theory which vanish for an}

abelian theory. An action without the above commutator terms would not be compatible with T-duality in the noncommutative case.

An interesting and important consequence of these terms is the fact that they make possible couplings with higher rank forms than one would expect to naturally couple to the brane. Since the double action on a p + 1-form creates a (p − 1)-form, also higher rank forms can couple to the brane worldvolume. After a quick glance at the Dp brane action without F and B fields one expects a coupling to a (p+3) form at first order in the interior derivative expansion, which

the kinetic term of the vectormultiplet superspace action (138) when one omits the Hermitian conjugate of WαWα|θ2. This term is a total derivative, not contributing to the local physics.

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indicates a coupling to charges naturally associated with a Dp+2 brane. We saw before a Dp brane couples to the R-R potential with form degree n = p + 1 naturally and to those with n = p−1, p−3, .. when one includes interactions with the Kalb-Ramond B two-form field and world volume gauge field strength F . Now we see, also looking at higher terms in the expansion, that the nonabelian situation allows couplings to R-R potentials with n = p + 3, p + 5, ... involving the commutators of the nonabelian scalars Φi_.

2.2.5 Myers Effect

In this section the Myers effect is reviewed, which is a blowing up of the geometry from a D0 brane situation to a D2 brane situation with D0 brane charge. This was witnessed by Myers while studying the nonabelian Chern-Simons action for D-branes in the presence of an external 4-form field strength [16]. We will review this here.

In the previous subsection we found the nonabelian Chern-Simons action. This allows us to look at the nonabelian case of N D0-branes (in IIA) Note the vanishing gauge field strength F on the one dimensional world volume. We have a = t, and take a vanishing Kalb-Ramond field B = 0. The Chern-Simons action becomes: SCS= µ0 Z Tr eikiΦiΦX i C(i) ! = µ0 Z Tr P C(1)+ ikiΦiΦC(3)− k2 2 (iΦiΦ) 2_C (5)− i k3 6 (iΦiΦ) 3_C (7)+ k4 24(iΦiΦ) 4_C (9) . Focusing on the C(1) and C(3), the interior derivatives and subsequently the

pullbacks give us = µ0 Z Tr P C(1)+ i k 2[Φ, Φ]C(3) = µ0 Z Tr Ct+ kDtΦiCi+ i k 2([Φ j_{, Φ}k_]C tjk+ kDtΦi[Φk, Φj]Cijk) . (14) Obviously the non-commutativity comes from the N D0-branes that offer endpoints for open strings so that the scalar fields here are N × N matrices. In the third term of the last line we see that there appears to be a source for electric D2 brane charge due to the commutator of the scalar fields! Clearly this only arises in the case of nonabelian fields.

We will now focus on the RR potential C(3) which we can take to be a

functional of Φi, ikµ0 Z TrP [iΦiΦC(3)] = ikµ0 Z Tr([Φj, Φk]Ctjk+ kDtΦi[Φk, Φj]Cijk). (15)

One can then do a nonabelian Taylor series expansion of C(3) and keep only the

non-derivative terms of C by considering a constant background of F(4)_{= dC}(3)_.

After partial integration of the relevant terms cubic in the scalars it can be written as i 3k 2_µ 0 Z dtTr(ΦiΦjΦk)F_ijkt(4)(t). (16)

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This extra therm modifies the quartic scalar potential term from the BI part of the action (9), so for the present nonabelian D0 brane problem we have the potential V (Φ) = −k 2_T 0 4 Tr([Φ i_{, Φ}j_]2_{) −} i 3k 2_µ 0 Z dtTr(ΦiΦjΦk)F_ijkt(4)(t), (17) with D0-brane tension T0. We can now take field strength F(4) to take the

simple form

F_tijk(4) =

−2f ijk for i, j, k ∈ {1, 2, 3}

0 otherwise.

So we have a background of electric flux and the other background fields are set to zero. Now if we vary the potential with respect to δΦ we get the equation

[[Φi, Φj], Φj] + if ijk[Φj, Φk] = 0.

For Φi _{= f α}i_{/2 there is a class of solutions where α}i _{is an N -dimensional}

representation of SU (2)

[αi, αj] = 2iijkαk.

The irreducible N-dimensional matrix representation of SU (2) satisfies Tr[(αi N)2] = N

3(N

2_{− 1), which, inserted back in (17), gives the potential}

VN = −

π2`3sf4

6g N (N

2_{− 1),}

which shows that the solution has lower energy for noncommuting matrices than for the abelian case. The N-dimensional representation gives lower energy than lower dimensional ones. Note that a diagonal matrix denotes separated branes since there are no strings between them. This solution shows that for the case of N D0-branes in a background of a 4-form field a separated brane system is not stable and will condense towards the nonabelian solution. In the absence of an electric field only the first term in (17) remains, so the minima of the potential then simply yield arbitrary abelian matrices for the scalar Φi_{, characterizing}

the classical moduli space where the matrices are diagonal. By turning on the electric fields, the Φi_{’s characterizing the position of the branes will not commute}

any more in a potential minimum. This indicates a different way of looking at the position of the branes in terms of noncommutative geometry. The result for the ground state is a nonabelian fuzzy two-sphere, of which the extent of non commutativity (or fuzziness) characterized by the average radius14

R2=k 2 N 3 X i=1 Tr[(Φi)2] = (πα0f )2(N2− 1). (18) One sees that the radius is proportional to the number of branes N and the strength of the background electric field f . The sphere becomes less fuzzy for large N and the radius will approximate R0= π`2sf N . There is an interpretation

of this as a fuzzy D2 brane. For large N the solution is actually a spherical (abelian) D2-brane with N dissolved D0-branes. Starting the analysis from the D2, the theory of a spherical D2-brane admits dissolved D0-charges. There is 14_{Recall for the transverse positions of the branes X}i_{= kΦ}i_{. One averages over N branes.}

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Figure 3: Myers effect: a collection of D0 branes blows up to a nonabelian sphere in the presence of a 4-form field strength, which couples to the dipole D2.

no total D2-brane charge, only D0-brane charge and local dipole charge which couples to F(4)_{. As opposite sides of the sphere have opposite orientation and}

thus opposite charge, there will be no net D2-charge, only locally.

The reason why one can really identify this fuzzy sphere that we find in our D0 theory as a D2 brane is the following. A D2 brane naturally couples to a 3-form RR potential via the Chern-Simons term of its 3-dimensional world-volume. Looking at this term for a spherical D2 brane one finds a leading order dipole coupling term that matches the coupling of the field strength to the D0 world-volume (16) in our irreducible ground state solution up to 1/N2 corrections [16]. This blow up effect due to the external field in the nonabelian theory has been dubbed the Myers effect. It is illustrated in figure 3.

This is a first result where we find a dielectrically charged object with non-trivial global charges in string theory, which can be obtained through initially unrelated viewpoints. In following chapters we will see more of this in the form of supertubes and Lunin-Mathur geometries. In the supertube system we will see that adding momentum to a bound state of fundamental strings and D0-branes will yield a dipole tube which is interpreted as a tubular D2-brane with world-volume magnetic and electric fields where these fields yield global D0 and F1 charge.

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3 The Supertube Effect

The main goal of this thesis is to learn more about black holes in string theory. In the previous section we described some of the D-brane technology and theory that is used to study D-branes which make the black holes in string theory. By analyzing the nonabelian theory for D0 branes in the presence of a 4-form field strength we found it to describe the myers effect which made the system ”blow up” in space to a D2 brane rather than a collection of point like objects. As we need branes to make black holes, it is natural to study this type of behaviour better and understand in what role it plays in an actual black hole. In this sec-tion we will discuss a particular type of blowing up that happens for two-charge systems in stead of the one-charge stack of D0 branes.15_{. This effect is dubbed}

the ”supertube effect” as it was first described by analyzing the world volume theory of supersymmetric cylindrical D2 branes in which fundamental string and D0 brane charges appear. In contrast to the blow up in the Myers effect which is a consequence of the addition of the extra field strength, the supertube effect describes a spontaneous blowing up of the 2-charge system itself. We will see this appearance of structure in the transverse space also happening on the geometry side when we consider a system of a fundamental string with momen-tum running along it which yields the Lunin-Mathur geometries. The idea is that these blow up geometries are actually the microstate geometries that are the geometrical analog of the gauge theory microstates.

3.1 Supertubes

In 2001 David Mateos and Paul Townsend studied the properties of a tubu-lar supersymetric D2 brane with angutubu-lar momentum generated by electric and magnetic fields on the world volume. Surprisingly this object turned out to be the ”blown up” version of a system of fundamental strings and D0 branes. The F1 and D0 charges appear dissolved on the world volume and source the magnetic and electric flux. In this section the supertube action and some of its interesting properties will be briefly described. Most of this analysis is based on the paper by Mateos and Townsend [17], next to which [18],[19] are also used.16

3.1.1 Properties of supertubes

To describe the original D2 supertube we take the DBI Lagrangian of a D2-brane with unit surface tension, which is given by

L = −∆ = −pdet(g + F ). (19)

Here g is the metric on the world volume and F is the gauge field strength on the worldvolume. We choose spacetime coordinates such that we have a 10 dimensional Minkowski metric

ds2= −dT2+ dX2dR2+ R2dΦ2_{+ ds2(E}6),

15_{As mentioned before, we turn to the two charge system in order to eventually learn more}

about more realistic 3-charge black holes. We also describe the mappings to the D1-D5 system as a sub-system of the D1-D5-P system.

16_{A nice introductory review of supertubes and the D3 supertube version can be found in}

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Figure 4: The supertube with world-volume electric and magnetic fields E and B, yielding an angular momentum J .

where Φ is a periodic coordinate. In order to describe the cylindrical D2 brane we choose the following world volume coordinates

T = t, X = x, Φ = φ.

We allow for time independent fields on the world volume in the form of an electric field E in de x-direction and a magnetic field B that form a field strength17

F = Edt ∧ dx + Bdx ∧ dφ. This yields (See appendix 7.1)

L = −pdet(g + F ) = −q(R2_{+ R}2

φ)(1 − E2) + R2R2x+ B,

where Rx= ∂xR and Rφ= ∂φR.

In what follows we will now focus on the D2 brane supertube with constant radius R. So this is the case of a cylindrical tube of which the radius is constant along all directions, Rx = Rφ = 0. For the perfect cylindrical supertube the

Lagrangian simplifies to

L = −pR2_{(1 − E}2_{) + B.} ₍₂₀₎

We can define the momentum conjugate to E, or the ”electric displacement” as

Π = ∂L ∂E.

With this we can express the electric field E in terms of the electric displacement, the B-field and the radius:

E =Π R

r

R2_{+ B}2

R2_{+ Π}2 (21)

Now we can write the Hamiltonian density corresponding to the Lagrangian density which is defined as

H = ΠE − L,

17_{Note that the electric field is a vector on the 3 dimensional world volume and the magnetic}

field, being the dual, is a scalar. Here the electric field lies only in the x-direction, which is upward in figure 3.1.1.

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in the following expression

H = R−1p(B2_{+ R}2_)(Π2_{+ R}2_). ₍₂₂₎

So we have a Hamiltonian density as a function of B, Π and radius R.

In order to find a stable low energy state of the system we minimize the hamiltonian density which gives us an expresso for the minimized radius

R =p|ΠB|. (23)

With this the minimal value of the Hamiltonian density (22)becomes

H = |B| + |Π|. (24)

Also, putting (23) into (21) gives us a constraint for the electric field E depend-ing on the sign of Π:

E = ±1. (25)

Later, in section 3.1.5, we will see that supersymmetry admits this choice of E. Putting this in (20) gives L = −B which means that this E-field would be the critical field in absence of B. Without the magnetic field there would be no stable supertube solution since the radius (23) then also goes to zero.

In the supertube configuration the electric potential is subject to the Gauss law constraint which becomes ∂xΠ = 0. Since the B-field now also is

indepen-dent of x we see that H is also x-indepenindepen-dent. We can now get a constant energy per unit length by integrating H over the circle that is parametrized by φ. In accordance with [17], we call this the tube tension

τ = I

dφH

The following integrals are (for an appropriate choice of units) the conserved IIA string charge and the conserved D0-brane charge per unit length carried by the tube: qs≡ 1 2π I dφΠ and q0≡ I dφB. (26)

With this the supersymmetric radius is R =p|qsq0|.

Using the hamiltonian density we got at (119) we get a corresponding tube tension which is minimized at

τ = |qs| + |q0|. (27)

Now we see that in the D2 supertube, charges appear which are associated with other objects, namely fundamental IIA strings and D0 branes. We need these charges to stabilize the supertube configuration at constant radius. The E and B fields generate a Poynting 2-vector density with nonzero φ- component Jφ= ΠB. Integrating this over φ gives an angular momentum per unit length

J = ΠB = |qsq0| (28)

which is the angular momentum that prevents the cylindrical supertube from collapse. It is clear now that the supertube is characterized by a fundamental string charge, a D0 brane charge and a corresponding angular momentum. It is this angular momentum that stabilizes the system against collapsing due to the tension in the brane.

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3.1.2 Dipole charges

Now we have found that there are D0 and F1 branes naturally appearing in a su-persymmetric tube that is stabilized by angular momentum created by magnetic and electric fields. The D2 brane has no net D2 charge, this charge will only be felt locally where extra supersymmetry is regained to 1₂BPS. But globally it does have F1 and D0 charge and preserves 1₄ of the supersymmetry.

The D2 preserves locally 16 supersymmetries (infinite and planar). The specific supersymmetries that are preserved depend on the orientation of the particular part of the worldvolume. Suprisingly there are 8 supersymmetries that remain preserved at all orientations of the brane which appear to be the same 8 supersymmetries that are preserved by the D0-F bound state. This enables the supertube to keep 1₄ of the supersymmetries at arbitrary shapes.

The appearance of D0 brane and F1 string charges associated with the mag-netic and electric fields on the D2 can be seen in the following light. Recall the Born infeld action (19) in which the gauge field strength Fµν _{and the}

anti-symmetric 2-form field Bµνappear together as a gauge invariant combination18.

The appearing of an electric field strength F0i _{is then analogous to turning on}

B0i which is the field that couples to the fundamental string [20]. For justifying the presence of the magnetic D0 we look at a previously derived phenomenon that showed a D(p−2)charge appearing in the chern simons term of a Dpbrane

by imposing the rules of T-duality on a D1 brane (13). In the case of a D(p=2)

brane we got (11)

µ2

Z

[C(3)+ C(1)∧ kF ]. (29)

These terms must be added to the DBI term to give the full action for the D2 supertube. So here we see that the magnetic field Fij appears as a source for the electric component of C(1) which says as much as that the magnetic field

acts as a D0 brane.

The interpretation of these findings is that if one looks at the angular momen-tum configurations of a D0/F1 system, this blows up to the D2 brane supertube configuration where the angular momentum is induced by the electric flux on the brane. So the supertube is the angular momentum version of the first system. The electric field lines on the supertube can be thought to be induced by the oppositely charged endpoints of strings on the brane, where the endpoints are placed at infinity, x = ∞ and x = −∞. This picture seems to be very different to the D0/F1 system in which the F1’s must end on the D0’s. The D0’s need to have the same amount of F1’s ending as starting on them since no flux can propagate on a D0. Thus in the static F1D0 case we are left with a necklace of D0’s connected by strings. The mechanism behind this ”puffing up” of 2-charge configurations as well as the geometrical interpretations remain open questions. These findings in the case of this simple D2-brane supertube configuration can be extended to more general situations involving supertubes of various types and of different dimensions. Some extensions to, or rather, implications of the model will be briefly discussed in the following subsections. 19

18_{Gauge invariance of the born infield term is achieved is the gauge transformation of B is}

accompanied by the A → A − Λ. The gauge invariant field strength becomes Fij= Fij− Bij.

So D-brane states with nonzero F carry B-charge, which thus corresponds to fundamental string charge.

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3.1.3 Mixed Configurations

If one incorporates ’mixed’ configurations which consist of the described system of a D2-brane supertube and its charges together with parallel IIA strings and D0-branes with charges q0_s and q0₀ so that Qs = qs+ qs0 and Q0 = q0+ q00.

The quantity |QsQ0| is then the upper bound on the angular momentum of

a supersymmetric state with these charges. The combined system still has a total angular momentum of J = qsq0 since the extra strings and D0-branes do

not have angular momentum. Now you can have a system with less than the maximal value of the upper bound. Transferring charge from the strings and D0’s to the supertube will increase the angular momentum at fixed total charges Qsand Q0. Supersymmetry allows this up to the situation where qs= Qs and

q0= Q0. The supersymmetric radius is thus

R2= |J | N , with the angular momentum bound

J2≤ R2_|Q

sQ0|. (30)

It is important to note that much of the results obtained for the cylindrically symmetric supertube are also applicable to arbitrary shaped supertubes. For a radius that depends arbitrarily on the angle φ the D2 brane is still 1/4 super-symmetric [19] and the blowup effect is still there. It appears that situations of arbitrary shape other that the perfect circle also correspond to lower angular momentum. A part of the energy of the system will be in the deformation away from the perfect circular profile of the tube. Although the mixed supertube + F1D0 are allowed, this arbitrary shaped supertube appears to be the right way of thinking about lower momentum configurations of these systems, which is in accordance with the microstate geometries described in following sections. 3.1.4 Overlapping Tubes

One can also consider the case of overlapping tubular D2-branes where the local field theory will get a U (N ) gauge symmetry as a result of N coincident branes, one brane wound N times around the φ coordinate or combinations of those, provided that the radius of the tube is much larger than the string scale. In this case Π and B become U (N ) matrices and in the F1 and D0 charges the traces of these appear. One can assume that these are diagonal matrices with diagonal entries Π and B. The charges become

qs= N Π and q0= N B,

while the tube tension stays the same as in (27). If we have the case of a multiply-wound D2-brane this simply incorporates the length which is now N 2πR. The Hamiltonian density changes as wel, but it is still minimized by R = p|ΠB| which now gives us

R = p|ΠB

N and J =

|qsq0|

N ,

which are related as

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R2= |J | N , while the general bound on J (30) still holds [18].

3.1.5 Supersymmetry preserving Tube

Now, let us check the extend to which the supertube is supersymmetric. The precise amount of supersymmetry that eventually is preserved by the config-urations is determined by the world volume gauge fields. We will see this by considering the supersymmetry equation (178)

Γε = ε

for a tubular D2 brane, with killing spinors ε. The matrix Γ is constructed out of Dirac matrices and the fields appearing in the Born-Infeld action. We introduce the induced polar world volume Dirac matrices (γt, γx, γϕ) and Γ11

the matrix that anti commutes with all spacetime Dirac matrices and Γ2₁₁= 1. For completeness we write the general expression for Γ in IIA as constructed in [24] and elaborated on in [25], which is given by

Γ = p|g|

p|g + F|se

1

2F_jkγjkΓ₁₁Γ0

(0),

where, F = B − F .20 We will not discuss further details of this function here, these can be found in [24] and [25].21 Carefully writing out the general expres-sion in the case of our supertube Lagrangian (20) will give us

Γ = ∆−1(γtxϕ+ EγϕΓ11+ BγtΓ11).

In this expression we see E and B appearing. After rewriting the Kiling spinor ε and writing the world volume γ-matrices in terms of spacetime Γ ma-trices22 _{the Killing supersymmetry equation reproduces the condition (120) on}

the sign of E and it gives us two separate constraints:

ΓT XΓ110 = −sign(E)0

ΓTΓ110 = sign(B)0. (31)

These conditions are compatible with each other (they commute) which im-plies that our minimal energy solution for a tubular D2-brane is a supersymmet-ric object which preserves 1/4 of the supersymmetries. Hence, it is a supertube. Also, it is particularly interesting to note that these constraints are associated with the supersymmetry constraints of the IIA superstring in the longitudinal X direction, and the D0-brane. Indicating that the amount of supersymmetry that it preserves derives from the fundamental string and D0-brane that are 20_{Note that this B is not the magnetic field but the Kalb-Ramond antisymmetric 2-tensor.} 21_{In this expression ”se” is the exponential function with antisymmetrized indices of the}

gamma matrices, Γ0₍₀₎is an antisymmetrized product structure of γ matrices and the metric.

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dissolved in the world volume of the tube.23 _{So the original D2-brane does not}

appear in these equations. This is reflected in the fact that the tubular D2-brane does not carry net D2-brane charge.

3.1.6 D1D5 Supertube

As we have seen the supertube effect shows how in some 2 charge system locally an extra charge can appear. Since we are discussing string theories in which du-alities are known to play an intrinsic part, we can dualize this 2 charge system to a range of other configurations that are identified by U-dualities. In performing these duality transformations the ”puffed up” dipole charge will also transform along the rules of the dualities. The main dual system of interest in this text is the D1-D5 system of a D1 brane and a D5 brane. The system of D5 charge along a four-torus in the x6, .., x9 direction and an S1 in x5on which also D1’s are wrapped are related by the following duality chain:

F 19D0 T678 −−→ F 19D3678 S − → D19D3678 T95 −→ D15D556789. (32) If we look at cases in which the blowing up only happens in the R4_transverse

to the compact directions24 _{the corresponding dipole D2 transforms as}

D2ϕ9 T678 −−→ D5ϕ6789 S − → N S5ϕ6789 T95 −→ KKMϕ56789. (33)

Thus, the D1D5 supertube is a Kaluza-Klein monopole that is wrapped along the compact directions and along a profile in R4 _{parametrized by ϕ.}

As M-theory was proposed to be the basic theory underlying all 10d string theories, the blowup models can be viewed clearly in M-theory using the M2 and M5 branes. The initial naive charges will be M2 branes and the dipole charges will be M5 branes. Via the following duality chain the whole system is mapped into M-theory:

D0F 19+ D2ϕ9 T78

−→ D278F 19+ D4ϕ789 M11

−−→ M 278M 29,10+ M 5ϕ789,10. (34) So we see 2 M2 branes blowing up to a M5 on a contractible cycle in transverse space. In this picture it is particularly nice to see how one would make 3-charge 23_{In type II the preserved supersymmetry charges for a Dp-brane stretched along the}

hy-perplane (x1, ..., xp) satisfy

L= Γ0Γ1....ΓpR.

There is a similar relation for the NS5 brane which differs between type IIA and IIB.

24_{Blow up profiles in the torus directions do seem to be necessary to be taken into account}

for describing the whole system and counting microstates as argued in [21]. In the D1D5 supertube these are extra dissolved dipole charges that appear as fluxes on the world volume.

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black holes. For a third charge one would add another M 256transverse to the

others. By adding this one one sees that more possibilities arise. As in (32) and (33) the first and second M2 blow up to an M5, also other combinations with the third M2 can now be made yielding different M5’s in the common directions and the transverse profile ϕ. One can now take the 5,..,10 directions to be compactified an a T6

, to make a black hole effectively in 5 dimensions of R1,4_.

In doing the M-theory reduction one arrives at the R1,4_{× T}4_{× S}1_{geometry in}

which we will work for the most part of the thesis, just with one charge left out.

3.2 Microstate Geometries

An insightful dual system in which the blow up effect also seems to appear is the F1-P system. By taking the D0-F1 system and performing a T-duality transfor-mation along the F1, followed by an S-duality transfortransfor-mation we obtain an F1 wrapped along an S1 _{with momentum P along it. As a fundamental string can}

not vibrate in its longitudinal direction it can vibrate transverse to it. Adding momentum in the direction of the string will create waves that travel along the string with the speed of light. After the above duality transformations the supertube, which was a D2 with a certain profile, will be a fundamental string F(ψ) that lies along some closed loop in transverse flat space ψ and has a mo-mentum P(ψ) along it. So it seems that in some way the supertube mechanism also shows itself in this appearance of structure in the space transverse to the initial directions of the system. In this section we will explore this system a bit more in the context of finding the microstate geometries of a black hole. The Lunin-Mathur geometries are reviewed by mapping the F1-P to the D1-D5 system. A clear overview of the black hole microstate proposal by Mathur and Lunin is given in [12], intuitively reviewing among others more detailed papers as [26], [4] and [27].

3.2.1 Black hole microstates

We begin the motivation for this with a particularly intuitive interpretation by Samir Mathur of Maldacena’s AdS/CFT results, which we briefly review in section 4.3.3. The Maldacena conjecture tells us that a field theory should be equivalent to a string theory containing gravity, which arose from two descrip-tions of a stack of D-branes that describe a black hole in the gravity side. Thus it states that the theory of an object (the black hole) is equal to the impact that this object has on spacetime. So we can study a theory of matter in flat spacetime or a curved spacetime without the matter. This is in contrast to the classical picture where one views mass as curving spacetime, thus describ-ing matter and gravity in the same image. Moreover, these theories should be describing the same thing.

One consequence of this is that whatever one finds in the field theory describ-ing the black hole should also appear in the gravity description. In the study of black hole entropy in string theory the D-brane picture has been shown by Strominger and Vafa to reproduce the Bekenstein hawking entropy. The field theory entropy arises due to the identification of different microstates that the D-brane theory can have and counting them. Essentially entropy can be de-scribed as the uncertainty of a system. So the possibility of a particular system being in different microstates gives rise to entropy. Now, since these microstates

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Figure 5: The multiply wound string with all momentum in its first harmonic which describes a wave that closes up to n1 times the length of the

compactifi-cation circle. Copies of the circle are sketched here by multiple intervals in the vertical direction.

were found on the field theory the Maldacena conjecture demands that there should be a description of these microstates in terms of string theory containing gravity. The idea is that every microstate should be a specific metric which differs from others at the level of microstates but at the macroscopic level they should be the same and thus describing a system with entropy.

Samir Mathur and Oleg Lunin arrived at a way to achieve these microstate geometries which, after coarse graining over the different microstates, describe a black hole as a so-called ”Fuzzball”. We will briefly outline some basic ingre-dients of this fuzzball proposal for black hole microstates and discuss how this fits in the rest of the thesis.

3.2.2 F1-P system

A key system in the microstate analysis is the bound state of n1 fundamental

strings F1 with npunits of momentum P. We consider the strings to lie along a

direction compactified along an S1_{, and 4 transverse directions are compactified}

on a T4_{. As one puts multiple strings on an S}1 _{this can make a ’long string’}

that is wound n1times around the circle. If momentum is bound to the string it

takes the form of traveling waves along the string. So we have a multiply wound string on an S1 with waves in one direction along the same circle. The wave on the string can be decomposed in Fourier modes or harmonics. The momentum of the k’th Fourier mode will be the usual KK momentum pk = 2πk/L where

L is the length of the circle. Now if the string on which the wave travels is wound multiple times along the same circle this means that the momentum wave only needs to close up to the total length of the string, which for the maximally wound long string is LT = n1L. This means that effectively the

total momentum is given by

p = np

R =

2πnpn1

LT

.

For this equation to be consistent with mi units of the above harmonic ki and

momentum p =P

imipki the following relation needs to be satisfied

X

i

miki= n1np. (35)

The picture of the F1-P bound state that we have is that of a momentum wave on a long string that is partitioned along the total length LT of the wrapped

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string. For each value of mi and ki that satisfies the relation (35) we have a

possible state of the geometry, which we now identify with a microstate. An important consequence of this, especially for entropy state counting, is that we now have a lot more units, n1np of momentum packet 2π/LT, than one would

have for a singly wound string giving np units of 2π/L. So, especially for big n

we now have a lot of possibilities for putting the momentum on the string, thus a lot of different microstate geometries yielding a large entropy for big n1 and

np.

As the momentum modes along the long string makes it vibrate in the trans-verse direction, there are no longitudinal vibrational modes of the string, so all the momentum is in the transverse vibrations of the string. This means that the strands will separate in the transverse spacetime and the total string will vibrate along a profile F (v) in the transverse space. So the string will not be confined to the point at r = 0 at which it naively would sit in transverse space, but bend away from this point. All different strands that make up the total string will have metrics that are characterized by the transverse displacement profiles, which obviously are related. All strands are given if we have the profile function ~F (t − y), where y parametrizes the S1_{. In the limit of equally big n}

1

and npand for states with wavelength much longer than the length of the circle

λ 2πR, the metric of the microstate solution is given by

ds2= H[−dudv + Kdv2+ 2Aidxidv] + 4 X i=1 dxidxi+ 4 X a=1 dzadza, (36) where H−1 = 1 + Q1 LT Z LT 0 dv |~x − ~F (v)|2 (37) K = Q1 LT Z LT 0 ( ˙F (v))2 |~x − ~F (v)|2dv (38) Ai = − Q1 LT Z LT 0 ˙ Fi(v) |~x − ~F (v)|2dv, (39)

where v = t − y, u = t + y and xi with i = 1, .., 4 parametrize the transverse

flat directions of which ~F is a vector and za, a = 1, ...4 parametrize the T4.

Note that in the case of separate strands with different vibration profiles the integrals over dv replace the sums over the different strands in the limit where neighboring strands are very similar.

These microstates appear to have no singularity and hence no horizon. This is no problem since they are only microstates and the macroscopic black hole, which is seen as an effective geometry obtained through coarse graining of these microstates, can still naively seem to have this singularity at the origin as long we do not see which exact microstate the system is in. The single microstate will also have no horizon area and thus no entropy, which is expected since entropy will only arise at the macroscopic level.

In order to do an entropy calculation for the microstate geometries one needs to assign a area of the horizon so that the Bekenstein relation can be checked. Mathur considered the distance at which the microscopic states seem

(28)

to differ as the radius of the effective horizon of the black hole. Before this distance the microstates differ and farther away the microstates are described by approximately the same metric which is the usual black hole metric. By using this definition of the horizon and calculating its area25_{the Bekenstein Hawking}

entropy for the fuzzball black hole agrees up to a constant to the known result for the 2-charge black hole

AE

4G10

∼√n1np∼ lnN .

3.2.3 D1-D5 microstates

Now we would like to see what implications these microstate have in the D1-D5 picture. Very convenient is the fact that the profile of the vibrating string in transverse space ~F (v) is invariant under the dualities that map the F1P to D1D5. We can now dualize the PF1 solution to a D1D5 system through the following duality chain:

P5F 15 ₋_→S P5D15 T6789 −−−→ P5D556789 S − → P5N S556789 T56 −→ F 15N S556789 S − → D15D556789. (40)

In undergoing these dualities some moduli such as the coupling g and radii of the compact directions will be changed. This yields the following solution

ds2= r H 1 + K[−(dt − Aidx i₎2_{+ (dy + B} idxi)2] (41) + r 1 + K H dxidxi+ p H(1 + K)dzadza, (42)

with the same harmonic functions just replacing Q1 by Q5/µ and F (v) by

µF (v), where µ2_{= V} T4R/g

2_R

6. Bi is given by dB = − ?4dA, where ?4 is the

hodge duality operator in the four dimensional transverse space. Comparing this metric to the ”naive” metric of the D1D5 system, one sees that the harmonic functions affect the metric at small distances and the metric asymptotes to the naive one. Also, note that there appears to be a mixing between the torus and the S1 coordinates through Ai and Bi. This mixing makes possible the

smoothness of the solution around ~x = µ ~F , as it appears that this is merely a coordinate singularity. The metric around this point becomes the same as the metric of a Kaluza-Klein monopole26 for the appropriate S1 radius [12]. Note that the supertube solution for the D1D5 is also a KK monopole! This strengthens the intuition that we can view the cylindrically symmetric supertube as a case of a Lunin-Mathur geometry.

25_{The area of the ”generic state” with harmonics of order k ∼} √_n

1n5 and wavelength

λ ∼pn1/npR is taken to arrive at the entropy agreement.

26_{The metric of a KK monopole is a smooth Taub-NUT metric. The Euclidean Taub-NUT}

can be viewed as a smooth S1 _{vibration over R}3_{. It is asymptotically R}3_{× S}1 _{and the S}1

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3.2.4 Microstates with angular momentum

One can also look at configurations of the F1-P system with angular momentum J , the value of which is bound by the product of the number of F strings and KK momentum modes respectively n1and np, J ≤ Jmax= n1np. In this system

a maximal angular momentum corresponds to having all energy in the lowest momentum Fourier mode corresponding to a one turn helix in the transverse space with a wavelength of LT. Lowering the angular momentum results in extra

energy allowing for higher modes to come in and perturbing the circular profile of the helix string. In the analysis of the dual 2 charge system of the supertube in section 3.1.1 we found the same upper bound for the momentum (30), where Q1= np, Q5= n1. In this case J ≤ Jmax corresponded to mixed configurations

of supertubes together with F1’s and D0’s, or preferably, to supertubes with no perfect circular profile, but arbitrary shapes. The appearing of blow up modes in 2-charge systems indicates that it is a quite natural phenomenon. The cylindrically symmetric supertube is just another way of looking at the Lunin Mathur microstate with highest angular momentum.

Lunin and Mathur interpreted this arbitraryness in shape of the supertube as the source for entropy of black holes. There the agreement of the bekenstein hawking entropy with microscopic counting of states comes from the identifica-tion of different states of the system to different ways of distributing the mo-mentum, or rather the different perturbations to the helical shape the system takes, in the F1-P case, which corresponds to the different shapes the supertube can take on [12][27].

In the F1-P picture, configurations with less angular momentum correspond to different vibrational profiles that have less angular momentum than the per-fect helix. So there will be vibrations around that helix. Since there is only one perfect helix the entropy of this system is approximately zero. Since any lower J ≤ Jmaxsituations can be obtained with different profiles, coarse graining over

microstates with a given J will give an entropy. 3.2.5 D1-D5 fractional strings

In previous sections we described the bound state of fundamental strings and momentum as an effective long string with momentum modes along the string. A consequence of this is that there were effectively more units of small momentum modes to count than for a bigger number of strings. The different microstates coming from different ways in which the momentum modes can be distributed along such a long string, so the degeneracy is given by counting partitions of the integer number n1np. In performing the dualization procedure (40) the charges

np, n1 effectively go to respectively n1, n5. Since now the n1 and n5 represent

the number of times the D1 and D5 are wrapped on the S1_{, the effective long}

string of length LT = 2πn1n5 here, can be seen as the distance one must travel

around the S1in order to arrive at the same point on the specific of D1 and D5 as one begun with.27

As before we now have n1n5 units of fractional charge coming from

”com-ponent strings”. The different ways of distributing the momentum modes cor-responds to the distribution of the total length between component strings of

27_{Assuming n}

Supertubes as Black Hole Microstates

University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

Supertubes as Black Hole Microstates

by

Nick Mabjaia

0513687

August 2014

54 ECTS

January 2013 - August 2014

Supervisor:

Prof. Jan de Boer

Examiner:

Dr. Ben Freivogel

Contents

1

Introduction

2

D-branes

2.1

Black holes in String theory

2.2

D-brane Actions

3

The Supertube Effect

3.1

Supertubes

3.2

Microstate Geometries