A Killing spinors in the attractor geometry

# Quantum entropy of supersymmetric black holes1

## Abstract:

We review recent progress concerning the quantum entropy of a large class of supersymmetric black holes in string theory both from the microscopic and macroscopic sides. On the microscopic field theory side, we present new results concerning the counting of black hole microstates for charge vectors with nontrivial arithmetic duality invariants. On the macroscopic gravitational side, we present a novel application of localization techniques to a supergravity functional integral to compute the quantum entropy of these black holes. Localization leads to an enormous simplification of a path integral of string theory in by reducing it to a finite dimensional integral. The localizing solutions are labeled by parameters, with the number of vector multiplets in the theory of supergravity. As an example we show, for four dimensional large black holes which preserve four supersymmetries in toroidally compactified IIB string theory, that the macroscopic degeneracy precisely agrees with all the terms in an exact Rademacher expansion of the microscopic answer except for Kloosterman sums which in principle can be computed. Generalizing previous work, these finite charge contributions to the leading Bekenstein-Hawking entropy can also be viewed as an instance of “exact holography” in the context of correspondence.

black holes, superstrings, dyons, holography
2

## 1 Introduction

Einstein’s general theory of relativity predicts that a sufficiently massive object can deform spacetime in such a way that it creates a region from where not even light can escape. This solution is called a black hole. The boundary of such a region is a null hypersurface called event horizon. This a surface of infinite redshift which then motivates the name black hole.

As pointed out first by Bekenstein [1] and then by Hawking [2] a black hole must carry entropy so that the second law of thermodynamics is not violated. The classic thought experiment is to throw a bucket of warm water inside the horizon. Since the entropy of the Universe cannot decrease the black hole must have entropy. It is well known that, in general theory of relativity, the black hole entropy is proportional to the area of the horizon in contrast with ordinary matter systems where it is proportional to the accessible volume,

 SBH=A4GNℏ. (1)

Here is the area of the horizon and is the Newton’s constant. Consistency with statistical mechanics naturally lead us to the following question: can we describe a black hole as an ensemble of quantum states in such way that we can relate the entropy to the logarithm of the number of accessible states?

 SBH=lnΩmicro (2)

To answer this question we need a theory of quantum gravity. String theory is the leading candidate for such a theory. Although we are still far from a description of the real world in terms of strings, this theory is able to incorporate gravity in a consistent way with other forces and it leads to the discovery of branes from where the holographic correspondence [3] was born. String theory gives us a systematic procedure to compute corrections to Einstein’s theory of gravity which can be important to understand finite size effects in quantum gravity.

The salient results covered by this article are:

• Finite charge corrections to Bekenstein-Hawking entropy

The main focus is the computation of finite charge corrections to the leading Bekenstein-Hawking entropy. Formula (1) is valid for an action with only the Einstein-Hilbert term. Since in string theory both the and string-loop corrections depend on the phase 3 of the theory, finite size corrections to the area law can give us information about the microscopic details of the phase.

To implement the effect of the higher derivative corrections we need to use the Wald formalism [4, 5]. The entropy is then given by a surface integral over the horizon geometry. To compute the Wald entropy we need first to find the black hole solution by solving the gravity equations and then perform the surface integral which is not an easy task. However, for extremal black holes, the near horizon geometry has enhanced symmetries which can be used to simplify the computation of the Wald entropy. The near horizon geometry has symmetries and is separated from infinity by an infinite throat. The moduli of the theory get attracted at the horizon and their value only depends on the charges. This is called attractor mechanism [6]. Combining both the symmetries of the near horizon geometry and the attracor mechanism Sen gives a simple prescription to compute the Wald entropy [7]. This method, called entropy function, resumes all the computation of Wald to a minimization problem and does not require solving the Einstein’s equations. The entropy function is proportional to the Lagrangian computed on the attractor background and the minimization parameters are the attractor values of the different fields. The black hole entropy is then given by the minimum of that function.

Unfortunately this formalism is not completely adequate in the full quantum theory. There can be non-local and/or non-analytic terms in the action coming from the integration of massless fields. In this case the Wald formalism can not be applied since it requires a local and gauge invariant action. Moreover it is in our interest to compute not just perturbative but also non-perturbative corrections to the Wald entropy as suggested by the microscopic answers. Thus even defining the proper notion of quantum entropy presents important conceptual problems. In an attempt to solve these issues Sen proposes a different formalism called quantum entropy [8, 9]. The idea is based on correspondence and gives a quantum version of the entropy. In summary, we are instructed to compute a path integral of string theory on with a Wilson line insertion at the boundary. The holographic correspondence then relates this observable to the degeneracy of the black hole. In contrast with higher dimensional cases, in the electric fields are non-normalizable modes and therefore they have to be fixed while performing the path integral. This is equivalent to fixing the charges instead of the chemical potentials which means that we are working in the microcanonical ensemble. For large charges the path integral is peaked at the classical attractor saddle point and the computation reduces to that of the entropy function. Since the equations of motion are no longer implied we can compute both perturbative and non-perturbative contributions in a systematic way.

• Supersymmetric Localization

For supersymmetric theories we can hope to use supersymmetric localization to compute exactly a path integral [10, 11, 12]. In a few words, localization means deforming the original physical action by a Q-exact term of the form , where is the action of some supersymmetry. If both the deformation and the observable, we are interested in computing, are supersymmetric by themselves, then it can be shown that the path integral does not depend on . This is very practical because we can choose a parameter where the computation is more convenient. In the limit the deformation dominates over the original physical action and the semiclassical approximation becomes exact since, in this case, plays the role of . Application of this technique in a path integral requires the supersymmetry to be realized off-shell. Fortunately for us, there is an off-shell formulation of supergravity even though only eight SUSYs are realized [13, 14, 15, 16]. When this formalism is applied to supergravity on the path integral localizes to a subspace where the scalar fields can be excited above their attractor values at the cost of exciting the auxiliary fields [17]. The solution is labeled by constants where is the number of vector multiplets in the theory. Using this technique we were able to reduce a very complicated path integral to a finite dimensional integral which resembles very much the formula proposed by Ooguri, Strominger and Vafa [18] but with some important differences. These differences include for example loop determinants, instantons or subleading orbifold saddle points. Once they are taken into account it should be possible to reproduce exactly the microscopic answers.

For large black holes in string theory the microscopic answer has a simple expression given in terms of a Jacobi form. The degeneracy can be written as a sum of Bessel functions in an exact expansion called Rademacher expansion. Using localization techniques we were able to reproduce all these terms for arbitrary values of the charges except for Kloosterman sums that can in principle be computed [19]. In this analysis we do a careful treatment of the measure on the localization locus which reveals crucial for the exact matching.

The big goal is to establish an exact equality between a degeneracy computed from the microscopic degrees of freedom and the quantum entropy computed from gravity. This obviously implies two big tasks,

• The first is to compute the expectation value of a Wilson line in by performing a path integral over the horizon string fields. The localization technique is extremely useful in this case.

• The second is to compute precisely the microscopic degeneracy using some weak coupling description in the same spirit of Strominger and Vafa [20].

For large charges both tasks simplify. In this regime we can use a Cardy formula to compute the microscopic degeneracy. On the gravity side large charge means large horizon radius and therefore we can neglect higher derivative corrections. The entropy area law suffices in this case. Performing both tasks exactly is equivalent as establishing an exact holography.

• Microscopic counting

The success of Strominger and Vafa black hole inspired many other works. Results in microscopic BPS counting flourished. For quarter BPS black holes in string theory the results are particularly interesting. The microscopic partition function is given in terms of the fourier coefficients of a Siegel modular form, which is a very rich object from the mathematical point of view. Part of this review is devoted to the analysis of the quarter-BPS dyon spectrum in these theories and to the construction of the corresponding partition functions. Previous works [21, 22, 23, 24, 25, 26, 27, 28, 29, 30] concern the spectrum of dyons which obey a particular primitivity constrain on the charges. As first noted in [31], is the only discrete invariant relevant in this problem, where denotes the dyonic electric and magnetic charges vectors respectively. Consider the charge lattice where both the electric and magnetic charge vectors live. These charge vectors generate a two dimensional lattice inside . The invariant basically counts the number of unit cells of inside a cell bounded by and . A primitive dyon corresponds to a unit cell. When the primitivity condition is relaxed additional difficulties arise in the microscopic counting mainly due to the analysis of multi-particle bound states at threshold [32]. Without loss of generality we consider the case when the electric charge vector is a multiple of a primitive vector while is primitive. In type IIB frame this implies studying a system of D-branes weakly interacting with a KK-monopoles. We study the low energy theory and propose a two dimensional supersymmetric sigma model [30]. A modified elliptic genus then gives an index which is consistent with previous constructions [33, 29] and passes many physical tests. In brief, the index found is given in terms of the fourier coefficients of the primitive answer and carries a non-trivial dependence on the divisors of . In [34] we propose a non-trivial check of the counting formula. We map a particular set of states to perturbative momentum-winding states of IIA string theory where the counting can be easily done and agreement is found for any value of .

• Cardy limit and correspondence

In the last section we focus on a different approach based mostly on rather than . Instead of computing the entropy valid for any charge we consider the simpler case when only one of the charges is very large keeping the other charges arbitrarily finite. The result is exact in the limit considered and is able to probe details of the phase we are working on [35]. The main result is: for black holes which preserve at least four supercharges the asymptotic growth of the index has a Cardy like formula with an effective central charge that is given by a linear combination of the coefficients of the Chern-Simons terms computed at asymptotic infinity. Whenever a black hole has a factor in the near horizon geometry we can view it as an extremal BTZ black hole living in space in the limit when the circle has a very large radius. The momentum along the circle corresponds to the angular momentum of the BTZ black hole. Then the extremal condition implies that we are counting states of large mass and therefore we can use a Cardy formula. In this case holography is extremely powerful since it relates the central charges, which are anomaly coefficients in the CFT, to the coefficients of the Chern-Simons terms living in the bulk of [36, 37, 38]. Note that the entropy formulas obtained are exact in the limit considered, that is, when only one of the charges is taken to be very large while keeping the remaining finite. During the analysis we found convenient to consider a macroscopic index that captures all the degrees of freedom from the horizon till asymptotic infinity. In the process we need to take into account additional contributions from external modes to the bulk of .

The review is organized as follows. In section §2 we give exact results on the microscopic counting of both primitive and non-primitive dyons. In section §3 we explain the quantum entropy formalism based on the correspondence and its relation with the microscopic index. In section §4 we use localization of supergravity on to reduce a very complicated path integral to a finite dimensional integral. We end discussing its relation to the OSV proposal. In section §5 we apply our results from localization in the problem of large black holes in string theory. Since the microscopic answer is known exactly we conclude by comparing both the macroscopic and microscopic answers which agree exactly for any finite charge. In the last section we study the index in the particular charge limit where only one of the charges is taken to be very large. In this regime of charges the point of view becomes more useful.

## 2 Microscopic counting

In string theory the Newton’s constant is proportional to the square of string coupling . As a consequence the gravitational attraction, proportional to , with the mass of the object, can be made arbitrarily small by decreasing . In particular, for fundamental strings and D-branes goes as and respectively while for the KK monopole or NS5-brane they are of order one. In this regime of very weak string coupling we can turn off gravity and “dissolve“ the black hole. The space becomes flat and these objects weakly interact. In this regime we can count the microscopic BPS states by quantizing the low energy theory of the system.

The first successful example in matching the microscopic degeneracy with the Bekenstein-Hawking entropy is the Vafa and Strominger five dimensional black hole [20]. They consider a system of D1 and D5-branes wrapping cycles of in type IIB string theory along with momenta through the circle. Effectively we see a five dimensional black hole carrying electric and magnetic charges. The low energy theory of the branes is a two dimensional supersymmetric conformal field theory. In the limit of large charges we can use a Cardy formula to compute the entropy of BPS states while on the black hole regime the entropy has the area law. Both answers perfectly agree.

For a large class of supersymmetric black holes it is known that the number of BPS states is constant over regions of the moduli space separated by codimension one walls where the states are marginally stable against decay [39, 40, 41, 42, 43, 44]. The constancy of the degeneracy follows from the non-renormalization of the mass of a state that saturates the Bogomol’nyi\unichar8211Prasad\unichar8211Sommerfeld bound, that is, of a BPS state. In other words the mass equals the central charge which is perturbatively not renormalized and therefore these BPS states sit in multiplets of shorter dimension. Due to this property, we can work in a region of the moduli space where string theory is weakly coupled, count the number of BPS states and then extrapolate this result to strong coupling, in the black hole regime. In the limit of large charges, or thermodynamic limit, the curvature of the horizon becomes small and the entropy is given by the Beckenstein-Hawking area law.

To count BPS states we use an index. This has the property of being invariant under continuous deformations of the theory. This is exactly what we mean by the constancy of the number of BPS states over the moduli space. In particular we use a helicity trace index or spacetime index. As a matter of fact, the index counts the number of bosonic minus fermionic states and therefore it can be zero or even negative. This is puzzling because ultimately we want to compare it with the exponential of the Beckenstein-Hawking entropy which is a strictly positive quantity. The usual understanding is that the number of states that get paired up is subleading in the large charge limit. Later we will see that the correct thing to do is to compare this microscopic index with an index constructed from the black hole solution. This issue will be analysed in section §6 where we make a clear distinction between index and degeneracy.

Since the index is invariant under U-duality it becomes important to classify duality orbits and corresponding charge invariants. For dyons in string theory with electric and magnetic charge vectors and we can construct many duality invariants out of the charges. Apart from the continuous T-duality invariants , and there is one discrete U-duality invariant which is particularly important in this problem. Very basically it encodes a primitivity condition in the dyon charge vector. A primitive dyon is one for which . Previous works in string theory concern the spectrum of primitive dyons [45, 46, 23, 22, 24, 26, 47, 48]. The main focus of this section is the counting of quarter-BPS states when the primitivity condition is relaxed. We propose a two dimensional supersymmetric sigma model whose index captures the spectrum of non-primitive dyons [30]. The resulting index is consistent with many physical tests including a perturbative test [34] and is in agreement with the answer proposed in [33, 29].

This section is organized as follows. In section §2.1 we consider the low energy theory of Heterotic string on and give general properties of quarter BPS dyons. In section §2.2 we focus on U-duality and classification of orbits via charge invariants. In particular we identify an important U-duality invariant on which the counting depends non trivially. Further in section §2.3 we analyse the role of invariant in the microscopic counting, explaining the construction of the partition functions in the cases and .

### 2.1 Heterotic string on T6: generalities

We consider heterotic string theory compactified on a six-dimensional torus . This is a four-dimensional string theory with supersymmetry or sixteen supercharges. It can have a dual description as IIA or IIB string theory compactified on .

The four-dimensional low energy theory contains the metric , the axion-dilaton and six gauge fields together with their susy partners sitting in the gravity multiplet. It contains in addition 22 vector multiplets. Each of these contains a gauge field and six real scalars plus susy partners. The axion-dilaton together with the 132=22x6 scalars from the vectors parametrize the moduli space of the theory

 SL(2,R)SL(2,Z)×SO(2)×SO(22,6;R)SO(22,6;Z)×SO(22,R)×SO(6,R). (3)

This theory has U-duality group

 G(Z)=SL(2,Z)×O(22,6;Z) (4)

where the first factor corresponds to electric-magnetic duality and the second factor corresponds to T-duality.

The 28 gauge fields can carry electric and magnetic charges which can be arranjed in the dyon charge vector

 Γi=[QiPi]

The index ’i’ stands for the vector representation of and the electric-magnetic duality acts on the pair by an transformation. This is also the S-duality symmetry of the four dimensional theory that acts on the axion-dilaton. Both the dyon and the axion-dilaton transform as

 λ→aλ+bcλ+d,[QP]→(abcd\par)[QP]with(abcd)∈SL(2,Z)

The superalgebra has central charges . A dyon with mass that saturates the BPS bound will preserve 1/4 of the supersymmetries. Note the dependence on the moduli measured at infinity. For certain values of the state can become marginally stable against decay into 1/2-BPS states. These regions are codimension one and are called walls of marginal stability [40] . As a consequence the index will jump.

A 1/4-BPS dyon breaks 12 supercharges out of 16. The 12 fermion zero modes associated with the broken susys make the Witten index vanish. To correctly account for the additional fermion zero modes we need to use a modified index [27]. Also known as helicity trace index or spacetime index, it is defined as

 B6(Γ,ϕ∞)=−16!Tr(−1)F(2h)6 (5)

where is the helicity quantum number and is the fermion number. The insertion of in the usual Witten index has the effect of rendering the trace over the fermion zero modes non-zero.

Lets work in more detail the contribution of the fermion zero modes. Each pair carries . To simplify the counting we compute first and the index becomes the sixth derivative of at . Tracing over the six complex fermion zero modes we obtain which, after differentiation, gives the net result of . In most of the cases we use the Witten index where the ’ denotes that the trace over the fermion zero modes has been carried out. Moreover long supermultiplets carry additional fermion zero modes so they won’t be captured by .

The index should be U-duality invariant. This translates to 4

 B6(Γ,ϕ∞)=B6(Γ′,ϕ′∞) (6)

where both and and and are related by a transformation. If two dyons belong to the same duality orbit, immediately we know that they have the same index. In the problem of microstate counting it is important to identify duality orbits through charge invariants.

Both the electric and magnetic charge vectors live in a Narain lattice from which we can construct the continuous T-duality invariants

 Q2=QTLQ,P2=PTLP,Q.P=QTLP (7)

with the invariant metric.

One important continuous U-duality invariant is the quartic Cremmer-Julia invariant

 Δ(Γ)=det(ΓΓT)=Q2P2−(Q.P)2. (8)

Later we will see that the entropy is proportional to . Because the U-duality group is discrete we can have more interesting invariants. One of major importance in the characterization of duality orbits is the arithmetic invariant [31]

 I=gcd(Q∧P)=gcd(QiPj−QjPj). (9)

This invariant will play an important role in the counting of 1/4-BPS dyons.

### 2.2 Duality orbits and invariants

As mentioned before, string theory has U-duality symmetry

 G(Z)=SL(2,Z)×SO(22,6;Z) (10)

composed of S and T-duality symmetries. As a consequence the index should be invariant under U-duality transformations of the charge vectors.

Under a rotation , the charge vectors transform as

 Q→ΩQ,P→ΩP, (11)

while the Lorentzian metric and the Narain lattice are left invariant

 ΩTLΩ=L,ΩΛ=Λ. (12)

As mentioned before we can construct the T-duality invariants , and which are left invariant under the continuous U-duality group. Additional discrete invariants can be constructed. These are necessary to completely characterize a T-duality orbit.

Consider a dyon with primitive charge vectors, that is, a dyon that cannot fragment into ”smaller” dyons. This means that the charge vector cannot be written as multiple of a vector but it doesn’t imply that the electric and magnetic charge vectors have to be individually primitive. We can represent these charge vectors in a sublattice generated by as

 Q=r1e1,P=r2(u1e1+r3e2),r1,r2,r3,u1∈Z+ (13)

such that and . Recent work on the classification of T-duality invariants [49] allows the identification of the set of integers

 Q2,P2,Q.P,r1,r2,r3andu1 (14)

as the complete set of T-duality invariants.

In these variables the discrete U-duality invariant becomes . This means that for a primitive dyon, that is, for a dyon with , and therefore the orbit becomes labelled by , and only. As a matter of fact the partition function for a primitive dyon depends only on the continuous invariants. For non-primitive dyons it is expected the index to have non trivial dependence on and the remaining integers.

We can also explore the consequence of S-duality on these integers. It was shown in [50] that the set can be brought to the form by an transformation. The charge vector acquires a much simpler representation

 Q=Ie′1,P=e′1+e′2. (15)

In this new ”frame” the derivation of the dyon partition function becomes easier since most of the invariants are trivial. Moreover the set is left invariant under the action of a subgroup of and therefore we expect the index to exhibit this symmetry explicitly. The subgroup is defined by matrices

 (amod(I)cd)∈SL(2,Z)

### 2.3 The dyon partition function

The Siegel modular form is for 1/4-BPS dyons as the ramanujan function is for 1/2-BPS states. The first is a modular form of , the modular group of genus two riemann surfaces, while the second is the lower dimensional version, that is, for genus one surfaces. Using this analogy and consistency with electric and magnetic duality, lead Dijkgraaf, Verlinde and Verlinde [45] long time ago to propose as the dyon partition function. This clue was remarkable and many other works followed in its derivation [51, 46, 22, 24].

In the work [26], which we review next, the authors gave a detailed derivation of the dyon partition function from first principles. Nevertheless only primitive dyons were concerned. Later it was shown in [31] that the discrete invariant plays a non-trivial role in the counting. In [29, 33] the authors consider the case and propose a degeneracy formula based on duality symmetries and consistency checks much like Dijkgraaf, Verlinde and Verlinde did. Following this proposal, in [30] we attempt to give a physical sigma model interpretation of that result.

#### Primitive dyons: I=1

Also known as Igusa cusp form, is the unique weight 10 form of . It depends on three complex numbers which encode the modular parameters of a genus two riemann surface. They can be packaged in a symmetric two dimensional matrix

 τ=(ρvvσ)

taking values in the Siegel upper half plane, defined as

 Im(ρ)>0,Im(σ)>0,Im(ρ)Im(σ)−Im(v)2>0. (16)

Under a transformation

 g=(ABCD)∈Sp(2,Z)

with matrices, the matrix transforms as

 τ→τ′=(Aτ+B)(Cτ+D)−1 (17)

in analogy with modular transformations in a torus. Correspondingly shows the modular property

 Φ10(τ′)=det(Cτ+D)10Φ10(τ). (18)

The subgroup can be realized in via matrices of the form

 g=(AT00A−1)withA∈SL(2,Z)

As can be easily checked this transformation leaves invariant. As explained before, invariance of the index concerns the set , that is, of primitive dyons.

The index is extracted performing an inverse fourier transform of

 B6(Γ,ϕ∞)=(−1)Q.P+1∫C(ϕ∞)d3τe−πiΓTτΓΦ10(τ). (19)

where the integration goes over a three dimensional torus

 0≤Re(ρ)≤1,0≤Re(σ)≤1,0≤% Re(v)≤1 (20)

at fixed large values of the imaginary part of

 Im(ρ)≫1,Im(σ)≫1,Im(v)≫1. (21)

This defines the integration countour . Note the dependence of the integration contour on the moduli space measured at infinity . Later we show that this dependence can lead to wall crossing. As expected from the analysis of duality orbits of the index shows dependence on only , and via .

#### Derivation from physical grounds

This section is based on [26] which we review in the following.

Without loosing generality we can restrict to a charge sub-lattice corresponding to the reduction on a particular two-torus . In this sector we have four electric and four magnetic charges. The charge configuration is taken be of the form

 Γ=[QP]=[~nn~wwW~W~KK]H.

where the indice denotes the heterotic frame. The charges denote momentum on the circles and respectively while stand for winding charges on the respective circles. The magnetic charges correspond to NS5-branes wrapped on and respectively. Additionally we can have Kaluza-Klein monopoles associated with the circles and respectively. We endow the lattice with a metric invariant under ,

 L=(02×212×212×202×2).

With this metric we construct the T-duality invariants

 Q2=2(~n~w+nw),P2=2(W~K+~WK),Q.P=~n~K+nK+W~w+~Ww (22)

In the presence of NS5-branes the microscopic theory is strongly coupled and there’s not much information we can extract. We avoid this problem by going to the IIB frame and consider a system of D-branes coupled to KK monopoles where a weakly coupled description is available. We perform the following chain of dualities. A string-string duality maps Heterotic string on to IIA on which is further T-dualized to give IIB on the dual circle and finally we do a ten dimensional S-duality. Lets see more carefully what is happening to the charges under this chain of transformations.

1. Six dimensional string-string duality, Het to IIA: the momentum and kaluza klein charges don’t transform while the Poincar\unichar233 electric-magnetic duality of the six dimensional NS-NS B field takes winding charge to NS5-brane charge and vice-versa.

 Γ=[QP]=[~nn~WWw~w~KK]IIA.
2. T-duality along , IIA to IIB: this duality maps IIA on the circle to IIB on the dual circle . The momentum and winding charges associated with this circle are exchanged. The same happens for NS 5-branes and KK monopoles.

 Γ=[QP]=[~wn~W~Kw~nWK]IIB.
3. Ten dimensional S-duality, IIB to IIB: this transformation maps winding charges to D1-branes and NS 5-branes to D5-branes. Other charges remain untouched.

 Γ=[QP]=[~Q1n~Q5~KQ1~nQ5K]IIB.

Here and represent charges associated with D1-branes wrapping a circle and respectively. Analogously and represent D5-branes wrapping and .

For simplicity we take a charge configuration of the form

 Γ=[QP]=[0n0~KQ1JQ50]H.

which corresponds to a system of D1-branes and D5-branes wrapping and respectively in the background of KK-monopoles, with momentum and along the circles and . This configuration is also known as D1-D5-KK system. If we impose primitivity on the charge vectors we get the following condition

 I=gcd(Q∧P)=gcd(Q1n,Q1~K,nQ5,J~K,Q5~K)=1 (23)

which can be satisfied imposing and . The general case with arbitrary number of KK monopoles will be studied later for non-primitive dyons. The condition is known to be a physical requirement for the existence of D1-D5 bound states at threshold [32, 27, 28].

In weak coupling limit both the D-branes and the KK-monopole are weakly interacting. We can see the D1-D5 brane system moving as a particle in the transverse four dimensional Taub-Nut (TN) geometry which is the solution of Einstein’s equations in the presence of a Kaluza-Klein monopole. The ten dimensional geometry is

 ds2=−dt2+ds2Taub−Nut+ds2K3×S1 (24)

with the Taub-Nut metric given by

 ds2Taub−Nut=(1+Rr)(dr2+r2dθ2+r2sin(θ)2dϕ2)+R2(1+Rr)−1(2dψ+cos(θ)dϕ)2

The TN space has the particularity that near the origin it looks like while for large it asymptotes to . From the point of view of the observer at infinity he sees a theory in four-dimensions. The TN geometry possesses in addition a normalizable 2-form .

The microscopic theory can be described by three weakly interacting parts. Each of these can be realized as a two dimensional supersymmetric sigma model on [26]. We denote the weakly interacting parts as

1. Higgs branch of D1-D5: describes the moduli space of vacua of the low energy theory of the D1-D5 brane system on . In the Higgs branch [52] the (1,5) strings acquire vevs forcing the D1 and D5 branes to sit on top of each other. In the IR, the low energy theory is described by a two dimensional SCFT with sigma model the Hilbert scheme of points on which is isomorphic to the symmetric product of at the orbifold point

 MD1−D5=SymQ1Q5+1(K3) (26)

This is a (4,4) SCFT with R-symmetry . The R-symmetry corresponds to rotations in the transverse space.

2. Center of mass motion of the D1-D5 system: it describes the vector multiplet degrees of freedom of (1,1) and (5,5) strings. We can see the D1-D5 as a particle moving in the TN space. From the motion on TN we have 4 scalars transforming under the vector representation of and 4 left-moving and 4 right-moving fermions transforming in the fundamental of and respectively. The Taub-Nut background breaks half the susy’s. This gives rise to a SCFT on . The sigma model has target space

 MCM=R4 (27)

This target space contrasts with the curved TN. We note that the index is a quantity that doens’t depend on the parameters of the theory. We use this property to compute the index for very large radius which is equivalent as putting the D1-D5 at , where TN looks like . This comment fails for the case of zero modes where we need to be more careful.

3. KK monopole closed string excitations: this describes the low energies excitations of closed strings in the Taub-Nut background. We have 3 massless scalars coming from the breaking of translation. Additionally the reduction of the Ramond-Ramond C 4-form on gives 19 left-moving scalars and 3 right-moving scalars. Additionally the NS-NS B field and the Ramond-Ramond 2-form give together 2 extra scalars. In total we have 24 left-moving and 8 right-moving scalars. The Taub-Nut preserves half susy’s of IIB on . This gives in addition 8 right-moving fermions. We denote the resulting sigma model by .

The analysis of the zero modes of the supersymmetric field theory on Taub-Nut requires special care. The dynamics is of a superparticle with 4 bosonic and 4 fermionic coordinates moving in the Taub-Nut space. So far we have been working in a point in the moduli space where and are orthogonal. A mixing between the circles can be achieved by a translation. As a consequence the tension of D1-D5 brane system generates a potential

 V(r)=a2R2(1+Rr)−1 (28)

which under supersymmetrization originates other fermionic terms. Under this potential supersymmetric bound states can form and contribute to the total index.

Concerning the fermionic zero modes resulting from the broken susy’s, the analysis goes as follows. IIB string theory on preserves 8 left-moving susy’s on the world volume theory. The breaking of 8 supersymmetries gives rise to 4 complex fermion zero modes. Additionally the D1-D5 breaks 4 of the remaining 8 susy’s contributing with 2 complex fermion zero modes. This gives a total of 6 complex fermion zero modes as expected for a 1/4-BPS dyon.

We now proceed to the construction of index.

We use the index where tracing over fermion zero modes has been carried out. We find convenient to compute the generating function, also known as elliptic genus,

 χ(q,¯¯¯q,y,~y;M)=TrR-R(−1)2JL−2JRqL0¯¯¯q¯¯¯L0y2JL~y2JR (29)

with

 q=e2πiρ,y=e2πiv,~y=e2πi~v, (30)

which corresponds to the partition function of the sigma model with Ramond-Ramond boundary conditions. The generators and are the usual left and right Virasoro dilatation generators while and correspond to the Cartan generators of , the little group in five dimensions. We contrast this with the little group in four dimensions which is . Due to the particular fibration structure of the TN space (2.3.2) there is an interesting connection between five and four dimensional black holes known as 4d-5d lift [46]. While at the tip of TN space the geometry looks like , at asymptotic infinity it looks like . If now we put the D1-D5 system at the tip of the TN, the transverse space looks five dimensional. Therefore we can relate the degrees of freedom of the five dimensional BMPV black hole [53] to the D1-D5-KK four dimensional black hole. The rotation generator measured at is further identified with translations on the circle at asymptotic infinity.

Due to supersymmetry, the right movers are forced to stay in the ground state and therefore the dependence of the function on drops meaning that only BPS states are being counted. We now show the different contributions to the generating function:

1. Higgs branch of D1-D5:

 ∞∑N=0pN−1χ(q,y;SymN(K3))=1p∏n≥1,m≥0,l∈Z1(1−pnqmyl)c(4nm−l2) (31)

with defined via the equation

 χ(q,y;K3)=8[v2(τ,z)2v2(τ,0)2+v3(τ,z)2v3(τ,0)2+v4(τ,z)2v4(τ,0)2]=∑n,j∈Zc(4n−j2)e2πin+2πijz (32)
2. CM contribution:

 χ(q,y;R4)=∏n≥1(1−qn)4∏n≥1(1−qny)2(1−qny−1)2 (33)
3. KK closed string excitations:

 χ(q,σKK)=Tr(−1)FqL0¯¯¯q¯¯¯L0=1q1∏n≥1(1−qn)24 (34)

This is a four dimensional index. States don’t carry charge and dependence on drops. This index is the same as the 1/2-BPS index that counts electric states in the heterotic string. In fact the system KK-P can be mapped to Heterotic momentum-winding states using duality symmetry.

4. superparticle in Taub-Nut:

 Tr(−1)Fy~J=∑j≥1je2πijv=e2πiv(1−e2πiv)2 (35)

where is the momentum charge on the circle . Note that the last expression can be expanded either in powers of or . This generates ambiguity when trying to extract the fourier coefficient. We show later that this is related to wall-crossing.

Putting all factors together we get

 −1pqy∏n,m≥0,l∈Zl<0fork=l=01(1−pnqmyl)c(4nm−l2) (36)

with

 p=e2πiσ,q=e2πiρ,y=e2πiv (37)

which is equal to .

The chemical potentials and couple to , and respectively. To extract the index we perform an inverse fourier transform

 B6(Q2,P2,Q.P)=(−1)Q.P+1∫Cdρdσdve−iπρQ2−iπσP2−2πivQ.PΦ10(ρ,σ,v) (38)

where the contour is as given in (20,21). The additional factor is reminiscent of going from five to four dimensions [46, 23].

#### Consistency checks

In the limit of large charges , and we can make an asymptotic expansion of (38). The leading term can then be compared with the black hole entropy valid in the same limit. The idea is to deform the contour such that it passes near a pole whose residue contribution is much leading than the left over integral [45, 26, 54]. The Siegel form has second order zeros at

 n2(ρσ−v2)+bv+n1σ−m1ρ+m2=0, (39)

with and obeying the condition . The residue at , modulo transformations, gives the leading term in the asymptotic expansion which is the correct result for the entropy

 B6(Q2,P2,Q.P≫1)≈eπ√Δ(Q,P)+O(eπ√Δ2) (40)

Subleading perturbative corrections to the microscopic answer can be computed. In fact for sufficiently large charges we can approximate by

 B6≈K0(−1)Q.P∫d2ττ22(26+πτ2|Q+τP|2)eπ2τ2|Q+τP|2−24lnη(τ)−24lnη(−¯¯τ)−12ln(τ2) (41)

which in the saddle point approximation reduces to (40). Subleading non-perturbative corrections are suggestive of multi-center black hole contribution [54].

The residues for are even more subleading. They encode phenomena associated with wall-crossing. Although the integrand in (38) is manifestly invariant the contour is not. After such transformation we may cross a pole in deforming the contour to its original form. It happens that only a pole can be crossed in the deformation. Take for example the residue at which corresponds to the pole . Near this pole the partition function behaves like

 1Φ10≈1v2η24(ρ)η24(σ) (42)

In this case the index jumps by the amount

 ΔB6=Resv=0=(−1)Q.P(Q.P)∫e−πiρQ2η24(ρ)∫e−πiσP2η24(σ) (43)

In a different context, this can be easily recognized as the Denef’s split attractor formula for 1/2-BPS black holes in supergravity [39, 43, 44]

 ΔΩ=(−1)⟨Γ1,Γ2⟩⟨Γ1,Γ2⟩Ω(Γ1)Ω(Γ2) (44)

with and is the index that counts 1/2-BPS states. In string theory the index that counts 1/2 BPS states is [55]

 d1/2(Q2)=∫e−πiρQ2η24(ρ) (45)

From a microscopic point of view we can understand the jump in the index as the decay of a 1/4-BPS dyon into its 1/2-BPS constituents, that is,

 (Q,P)→(Q,0)+(0,P). (46)

From the gravity point of view it corresponds to the appearance or disappearance of a two center black hole [56]. In fact for large charges the jump in the index can be interpret as coming from the contribution of two centers which are very far from each other

 ln(ΔB6)≈2π√Q2+2π√P2. (47)

Other poles with correspond to more complex decays which are basically related by a transformation to the case. For more details we refer the reader to [40].

Physically the picture is the following. The transformation acts not just on the charges but also on the axion-dilaton . Since the mass has a non-trivial dependence on it will change as we move on the moduli space. When the mass of the quarter-BPS dyons equals the sum of the masses of the half-BPS dyons for , that is,

 m1/4−BPS(Q,P)|λ∗=m1/2−BPS(Q)+m1/2−BPS(P) (48)

it becomes marginally stable and can decay into its 1/2-BPS constituents. The regions in the moduli space where the dyon becomes marginally stable are codimension one walls. Schematically we have the moduli space divided into chambers separated by codimension one walls. The index is piecewise constant in these chambers.

Consider the example of a dyon with and . We can easily extract the index from (35). It gives . Under a S-duality transformation

 [QP]→(01−10)[QP]

the T-duality invariants are mapped to and . This is equivalent to the change of contour . In deforming the countour to its original value we pick a residue at . In this case the jump is easy to compute and formula 43 gives . At the same time the axion-dilaton gets transformed to and the dyon jumps from one chamber to another separated by a wall at . In this new chamber the index (35) only contains positive powers of which gives a zero index consistent with the predicted jump.

In [57] the authors propose a contour which captures only the contribution from single center black holes. In this case the index becomes moduli independent and therefore the dyon is free from decaying. The prescription is the following

 Im(ρ)=Λ(|λ|2λ2+Q2R√ΔR) (49) Im(σ)=Λ(1λ2+P2R√ΔR) (50) Im(v)=−Λ(λ1λ2+QR.PR√ΔR) (51)

with , , and . The matrix is a symmetric matrix which encodes the 132 moduli of the theory and obeys the constraint , with the metric on . The scalar is the axion-dilaton. The parameter is taken to be very large to ensure the dyon doesn’t leave this chamber.

### 2.4 Non-primitive dyons: I>1

Derivation of the spectrum of non-primitive dyons from physical grounds is more complex. As a matter of fact, it was noted long time ago that the counting of non-primitive charge vectors, in the context of toroidally compactified IIB string theory, was a difficult problem [27]. The case of the D1-D5 system with and not coprime is a good example. Since the system can split at no cost of energy, this signals the presence of singularities in the moduli space of the low energy theory [32].

In the case of 1/4-BPS dyons with non-primitive charge vectors, similar difficulties are encountered. Consider a charge configuration of the form

 Γ=[QP]=[0nI0kIQ1JQ50]H. (52)

with coprime. We choose charges such that . In this case we have to consider a configuration multi KK-monopoles. If we were to repeat the analysis done in the case, we would face the following difficulties

1. Multi KK monopoles have collective coordinates which parametrize a non-trivial moduli space. The study of bound states in this background is a very difficult problem.

2. The multi KK geometry admits non-trivial 2-cycles. For each pair of KK monopoles there is a 2-cycle that touchs both of them [58]. The area of this 2-cycle is proportional to the distance between the two centers and approches zero when the monopoles touch each other. A D3-brane wrapping such cycle will give rise to tension less strings [59]. In the counting we should consider a possible contribution from these strings.

Aware of these problems, the authors in [29, 33] proposed an index formula much as Dijkgraaf, Verlinde, Verlinde have made for the case of primitive dyons. This formula is consistent with many properties known for non-primitive dyons and corresponding black holes. The proposed index has the form

 d(Q,P)=(−1)Q.P+1∑s|Is4∫C(s)d3τe−πiΓTτΓΦ10(ρ,s2σ,sv) (53)

with contour

 C(s):0≤Reρ≤1,0≤Reσ≤1s2,0≤Rev≤1s. (54)

After a simple manipulation we can write it in a more convenient form

 d(Q,P)=∑s|Isd1(Q2s2,P2,Q.Ps) (55)

where denotes the fourier coefficient extracted from the primitive answer (38). The main driving principle for the such construction is based on wall crossing for non-primitive decay. In the case of a primitive decay5, there is a one to one correspondence between the decay and the pole in . That is, take the most general primitive decay

 (Q,P)→(αQ+βP,γQ+δP)+(δQ−βP,−γQ+αP) (56)

with and . The set of integers gives the location of the pole at .

Once a marginally stable dyon can decay into products which are non-primitive. This allows for a larger set of integers . In [33] they postulate that such correspondence should remain even in the non-primitive case. This was helpful in suggesting part of the pole structure of the partition function which is indeed that of (53).

Take the example of a non-primitive decay

 (IQ0,P)→(IQ0,0)+(0,P). (57)

associated with the pole at . The wall crossing formula extracted from the residue of (53) gives a jump of the form

 Δd(Q,P)=(−1)Q.P(Q.P)∑s|Id1/2(Q2/s2)d1/2(P2) (58)

Again for large charges, the term in (53) gives the leading contribution reproducing correctly the black hole entropy

 d(Q,P)≈∑s|IeSBH/s. (59)

with .

One additional requirement is the invariance of the index under . S-duality invariance demands that with and . By embedding this subgroup in via

 g=((hT)−100h)withh∈Γ0(I)

we can show that the integrand (53) is left invariant due to

 Φ10(ρ′,s2σ′,sv′)=Φ10(ρ,s2σ,sv). (60)

There is yet another important check to this formula. At special points in the moduli space of the heterotic string on