Asymptotic Expansion of the Dyon Degeneracy
Abstract:
We study various aspects of power suppressed as well as exponentially suppressed corrections in the asymptotic expansion of the degeneracy of quarter BPS dyons in supersymmetric string theories. In particular we explicitly calculate the power suppressed corrections up to second order and the first exponentially suppressed corrections. We also propose a macroscopic origin of the exponentially suppressed corrections using the quantum entropy function formalism. This suggests a universal pattern of exponentially suppressed corrections to all extremal black hole entropies in string theory.
1 Introduction and Summary
One of the major successes of string theory has been the matching of the BekensteinHawking entropy of a class of extremal black holes and the statistical entropy of a system of branes carrying the same quantum numbers as the black hole[1]. The initial comparison between the two was done in the limit of large charges. In this limit the analysis simplifies on both sides. On the gravity side we can restrict our analysis to two derivative terms in the action, while on the statistical side the analysis simplifies because we can use certain asymptotic formula to estimate the degeneracy of states for large charges. However given the successful matching between the statistical entropy and BekensteinHawking entropy in the large charge limit, it is natural to explore whether the agreement continues to hold beyond this approximation. On the gravity side this requires taking into account the effect of higher derivative corrections and quantum corrections in computing the entropy. The effect of higher derivative terms is captured by the Wald’s generalization of the BekensteinHawking formula[2]. For extremal black holes this leads to the entropy function formalism for computing the entropy[3]. Recently it has been suggested that the effect of quantum corrections to the entropy of extremal black holes is encoded in the quantum entropy function, defined as the partition function of string theory on the near horizon geometry of the black holes[4]. On the other hand computing higher derivative corrections to the statistical entropy requires us to compute microscopic degeneracies of the black hole to greater accuracy. Here significant progress has been made in a class of supersymmetric field theories, for which we now have exact formulæ for the microscopic degeneracies[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. (For a similar proposal in supersymmetric theories, see [30].)
Our eventual goal is to compare the statistical entropy computed from the exact degeneracy formula to the predicted result on the black hole side from the computation of the quantum entropy function (or whatever formula gives the exact result for the entropy of extremal black holes). However in practice we can compute the black hole side of the result only as an expansion in inverse powers of charges, by matching these to an expansion in powers of derivatives / string coupling constant. Thus we must carry out a similar expansion of the statistical entropy if we want to compare the results on the two sides. A systematic procedure for developing such an expansion of the statistical entropy has been discussed in [5, 6, 10, 13]. Our main goal in this paper is to explore this expansion in more detail, and. to whatever extent possible, relate it to the results of macroscopic computation.
The rest of the paper is organized as follows. In §2 we give a brief overview of the exact dyon degeneracy formula in a class of supersymmetric string theories, and discuss the systematic procedure of extracting the degeneracy for large but finite charges. We also organise the computation of the statistical entropy by representing the result as a sum of contributions from single centered and multicentered black holes, and then express the single centered black hole entropy as an asymptotic expansion in inverse powers of charges, together with exponentially suppressed corrections. In §3 we examine the leading exponential term in the expression for the statistical entropy and compute the statistical entropy to order . Previous computation of the statistical entropy was carried out to order . We compare these results with the exact result for the statistical entropy and find good agreement. We also find that the agreement is worse if we compare the result with the exact statistical entropy in a domain where besides single centered black holes, we also have contribution from two centered black holes. This confirms that the asymptotic expansion is best suited for computing the entropy of single centered black holes. From the gravity perspective these corrections should be captured by six derivative corrections to the effective action; however explicit analysis of such contributions has not been carried out so far.
In §4 we analyze the contribution from the exponentially subleading terms to the entropy of single centered black holes. While power suppressed corrections to the statistical entropy have been compared to the higher derivative corrections to the black hole entropy in various approximations, so far there has been no explanation of these exponentially suppressed terms from the black hole side.^{1}^{1}1Note that this expansion is quite different from the Rademacher expansion studied in [31, 32] since we scale all the charges uniformly. In §5 we suggest a macroscopic origin of the exponentially suppressed contributions to the entropy from quantum entropy function formalism. In this formalism the leading contribution to the macroscopic degeneracy comes from path integral over the near horizon geometry of the black hole with appropriate boundary condition. We show that for the same boundary conditions there are other saddle points which have different values of the euclidean action. These values have precisely the form needed to reproduce the exponentially suppressed contributions to the leading microscopic degeneracy.
2 An Overview of Statistical Entropy Function
In this section, we briefly review the systematic procedure for computing the asymptotic expansion of the statistical entropy of a dyon in a class of supersymmetric string theories. The approach mainly follows [5, 6, 10, 13, 22]. Our notation will be that of [23].
2.1 Dyon degeneracy
Let us consider an supersymmetric string theory with a rank gauge group. We shall work at a generic point in the moduli space where the unbroken gauge group is . The low energy supergravity describing this theory has a continuous symmetry which is broken to a discrete subgroup in the full string theory. We denote by and the dimensional electric and magnetic charges of the theory, by the invariant metric and by the combinations . Then for a fixed set of values of discrete Tduality invariants the degeneracy , – or more precisely the sixth helicity trace [33] – of a dyon carrying charges is given by a formula of the form:
(1) 
where , and are three complex variables, is a function of which we shall refer to as the inverse of the dyon partition function, and is a three real dimensional subspace of the three complex dimensional space labeled by , given by
(2) 
The periods , and of , and are determined by the the quantization laws of , and . , and are large but fixed numbers. The choice of the ’s depend on the domain of the asymptotic moduli space in which we want to compute . As we move from one domain to another crossing the walls of marginal stability, changes. However this change is captured completely by a deformation of the contour labelled by without any change in the partition function [17, 18]. A simple rule that expresses in terms of the asymptotic moduli is[21]:
(3) 
where is a large positive number,
(4) 
denotes the asymptotic value of the axiondilaton moduli which belong to the gravity multiplet and is the asymptotic value of the symmetric matrix valued moduli field of the matter multiplet satisfying .
A special point in the moduli space is the attractor point corresponding to the charges . If we choose the asymptotic values of the moduli fields to be at this special point then all multicentered black hole solutions are absent and the corresponding degeneracy formula captures the degeneracies of single centered black hole only[21]. This attractor point corresponds to the choice of for which
(5) 
Substituting this into (2.1) we get
(6) 
We can invert the Fourier integrals (1) by writing
(7) 
where are the coefficients of Fourier expansion of the function :
(8) 
Different choices of in (1) will correspond to different ways of expanding and will lead to different . Conversely, for associated with a given domain of the asymptotic moduli space, if we define via eq.(7), then the choice of is determined by requiring that the series (8) is convergent for .
A special case on which we shall focus much of our attention is the supersymmetric string theory obtained by compactifying type IIB string theory on or equivalently heterotic string theory compactified on . In this case the function is given by the well known Igusa cusp form of weight 10:
(9) 
where is defined via the equation
(10) 
2.2 Asymptotic expansion and statistical entropy function
In order to compare the statistical entropy with the black hole entropy we need to extract the behaviour of for large charges. We shall now briefly review the strategy and the results. For details the reader is referred to [22].

Beginning with the expression for given in (1), we first deform the contour to small values of (say of the order of 1/charge). In this case the contribution to from the deformed contour can be shown to be subleading, and hence the major contribution comes from the residue at the poles picked up by the contour during the deformation.

For any given pole, one of the three integrals in (1) can be done using residue theorem. The integration over the other two variables are carried out using the method of steepest descent. It turns out that in all known examples, the dominant contribution to computed using this procedure comes from the pole of the integrand ı.e. zero of at
(11) Furthermore near this pole behaves as
(12) where
(13) is related to the rank of the gauge group via the relation
(14) and is a known function which depends on the details of the theory. Typically it transforms as a modular function of weight under a certain subgroup of the group. In the variables the pole at (11) is at . The constant of proportionality in (12) depends on the specific string theory we are considering, but can be calculated in any given theory.

Using the residue theorem the contribution to the integral (1) from the pole at (11) can be brought to the form
(15) where and are two complex variables, related to and via
(16) and
(17) Even though and are complex, we have used the notation , , , and . Note that also depends on the charge vectors , but we have not explicitly displayed these in its argument. The in (15) denotes equality up to the (exponentially subleading) contributions from the other poles.

We can analyze the contribution to (13) using the saddle point method. To leading order the saddle point corresponds to the extremum of the first term in the right hand side of (3). This gives
(18) (19) We can regard the result for as the extremal value of the 1PI effective action in the zero dimensional quantum field theory, with fields (or equivalently , ) and action . A manifestly duality invariant procedure for evaluating was given in [13] using background field method and Riemann normal coordinates. The final result of this analysis is that is given by
(20) where is the sum of 1PI vacuum diagrams calculated with the action
(21) where
(22) Here is a fixed background value, , are zero dimensional quantum fields and
(23) for any arbitrary ordering of and in .
This finishes the required background for generating the asymptotic expansion of the statistical entropy to any given order in inverse powers of charges, – all we need is to compute to the desired order and then find its value at the extremum. The function is called the statistical entropy function.
2.3 Exponentially suppressed corrections
In our analysis we shall also be interested in studying the exponentially subleading contribution to the statistical entropy. These come from picking up the residues at the other zeroes of . The details of the analysis has been reviewed in [22]; here we summarize the results for the special case of heterotic string theory on [5]. In this case , is given by the Siegel modular form , and the periods are all equal to 1. has second order zeroes at
(24) 
Since eqs.(2.3) are invariant under , we can use this symmetry to set . For any given we can use the symmetry of under integer shifts in to bring , and in the range
(25) 
Using this symmetry we can fix in this range, but then we must extend the integration range over to be over the whole real axes. For given , , , , the last equation in (2.3) then determines in terms of the other variables. This equation also forces to be odd, and to be an integer multiple of . We can now evaluate the contribution from each of these poles using saddle point method. To leading order the location of the saddle point from the pole associated with a given set of values of , and is given by[5, 22]
(26) 
Besides these there are also contributions from the poles corresponding to . These are in fact the poles responsible for the jump in the degeneracy as we cross walls of marginal stability[17]. In particular for the wall associated with a decay of the form
(27) 
(28) 
(29) 
the jump in the index is given by the residue at the pole at
(30) 
Unlike the residues from the poles at (2.3), which grow as exponentials of quadratic powers of charges, the residues at the poles at (30) grow as exponentials of linear powers of charges. Thus one expects them to be suppressed compared to the contribution from all other poles of the form given in (2.3). Nevertheless we shall see that for small charges the residues at (30) give substantial subleading contribution to the statistical entropy.
2.4 Organising the Asymptotic Expansion
Consider the contour integral given in (1) with given as in (2.1). In order to find the asymptotic expansion of this expression we need to deform the contour so that it passes through the saddle point. Since the integral is done over the real parts of keeping their imaginary parts fixed, we shall deform the contour by varying the imaginary parts of . For this we first note that in the space, the point given in (6) corresponding to the choice of the contour for single centered black holes, and the values of given in (26) corresponding to various saddle points, lie along a straight line passing through the origin:
(31) 
Thus we can first deform the contour from its initial position to the position (6), keeping large all through, and then deform it along a straight line towards the origin. In the first step we shall only cross the poles of the type given in (30). This picks up the contribution to the entropy from the multicentered black holes which were present at the point in the moduli space where we are computing the entropy. In the second stage we pick up the contribution from all the saddle points with , but do not cross any pole of the type given in (30). These can then be regarded as the contribution to the entropy of a pure single centered black hole. Thus we see that the complete contribution to single centered black hole entropy comes from residues at the poles (2.3) with . This suggests that at least for finite values of charges where the jumps across the walls of marginal stability are not extremely small compared to the total index, the asymptotic expansion, based on the residues at the poles at (2.3) with , is better suited for reproducing the entropy of single centered black holes than that of single and multicentered black holes together. We shall see this explicitly in our numerical analysis.
3 Power Suppressed Corrections
In §2 we outlined a general procedure for computing the statistical entropy as an expansion in inverse powers of charges. In this section we shall use this method to compute the statistical entropy to order where stands for a generic charge. For comparison we note that the leading correction to the entropy is quadratic in the charges. Contribution to up to order has been computed in [6, 10, 13].
We begin with the expression for given in (3) and carry out the background field expansion as described in (21). For this we organise (21) as a sum of three terms
(1) 
where
(2) 
represent respectively the leading piece of order , the piece and all terms of the order . Since the loop expansion is an expansion in powers of , in order to carry out a systematic expansion in powers of we need to regard as the tree level contribution, as the 1loop contribution and as two and higher loop contributions. To compute up to a certain order, we need to compute 1PI vacuum diagrams in the zero dimensional field theory with action up to that order regarding as fundamental field. Thus for example in order to compute the contribution to to order we need to include all one and two loop diagrams involving vertices from , all one loop diagrams involving a single vertex of and the tree level contribution from , and .
To see more explicitly how the powers of appear, we expand in field variable around the background point . We then identify the quadratic term in in the leading action with the inverse propagator and all other terms (including quadratic terms in the expansion of and ) as vertices. Since is of order , this gives a propagator of order . All vertices coming from are of order , all vertices coming from are of order and the vertices coming from are of order with . Let us now consider a 1PI vacuum diagram with number of th order vertices coming from . Since there are no external legs, we have propagators. Thus the contribution from this diagram goes as
(3) 
Similar counting works for vertices coming from and , but every vertex coming from will carry an extra power of and every vertex coming from will carry two or more extra powers of . Thus an order contribution to the effective action can come from
(4) 
if all the vertices are from , and
(5) 
if this single two point vertex is from .^{2}^{2}2Note that does not give a two point vertex. The possible diagrams associated with (4) have been shown in Fig.1 whereas the diagram associated with (5) have been shown in Fig.2. Finally the order contribution from is obtained by just adding the term to .
The above analysis shows that in order to calculate the contribution to up to order , we need to expand to quartic order in , and to quadratic order in . This is done with the help of (21), (4). We get^{3}^{3}3Whenever a () appears without a vector sign, it should be interpreted as ().
(6)  
The quadratic term in the expansion of gives the propagator
(7) 
Using the vertices we can evaluate the order contribution to shown in the three diagrams in Figs. 1 and 2. The results are
(8) 
Combining this with the order and contribution to given in [13], the complete statistical entropy function goes as,
(9) 
The last term in vanishes at the extremum of where
(10) 
We can therefore get rid of this term by doing a field redefinition. Using this we can write
(11) 
We now note that is independent of the modular form . This fact has some important implications for our result; we will come back to it at the end of this section.
We can now extremize given in (3) with respect to to evaluate the blackhole entropy up to this order. For this it is enough to find the location of the extremum to order . Let be the extremum of given in (10). By extremizing we can find the extremum to order . We get
(12) 
where the derivative of is taken at fixed . Substituting this in the argument of the ’s we get
(13) 
where
(14)  
For type IIB string theory compactified on , , and . We have shown in table 1 the approximate statistical entropies calculated with the ‘tree level’ statistical entropy function, calculated with the ‘tree level’ plus ‘one loop’ statistical entropy function and calculated with the ‘tree level’ plus ‘one loop’ plus ‘two loop’ statistical entropy function and compared the results with the exact statistical entropy . The exact results for are computed using a choice of contour for which only single centered black holes contribute to the index for and both single and 2centered black hole solutions contribute for . We clearly see that the asymptotic expansion has better agreement with the exact results when only single centered black holes are present, in accordance with our general argument.
2  2  0  10.82  6.28  10.62  11.576  .2  0.756  
4  4  0  17.31  12.57  16.90  17.382  .41  0.072  
6  6  0  23.51  18.85  23.19  23.506  .32  .004  
8  8  0  29.71  25.13  29.47  29.71  .24  .000  
10  10  0  35.943  31.42  35.754  35.945  .189  0.002  
6  6  1  23.14  18.59  22.88  23.15  .26  0.01  
6  6  2  22.15  17.77  21.94  22.198  .21  0.05  
6  6  3  20.64  16.32  20.41  20.766  .23  0.13  
6  6  1  23.19  18.59  22.88  23.15  .31  .04  
6  6 