Supersymmetric Index for Small Black Holes

Supersymmetric elementary string states in the compactified heterotic string theory are described by small black holes that have zero area event horizon. In this paper we compute the supersymmetric index of such elementary string states using gravitational path integral. The dominant contribution to the path integral comes from an Euclidean rotating black hole solution of the supergravity theory with a finite area event horizon, but the logarithm of the index, computed from the saddle point, vanishes. Nevertheless we show that the solution is singular on certain subspaces of the horizon where higher derivative corrections can be important, and once the higher derivative corrections are taken into account the solution could yield a finite result for the logarithm of the index whose form agrees with the microscopic results up to an overall numerical constant. While the numerical constant is not determined in our analysis, we show that it is independent of the details of the compactification and even the number of non-compact dimensions, in agreement with the microscopic results.


Introduction
The excitations of a string lead to an infinite tower of massive states with large degeneracy.It has long been suspected that sufficiently heavy states of the string can be identified as black holes and that the logarithm of the degeneracy of such string states may provide an explanation of the black hole entropy [1][2][3][4][5][6].However since the counting of states of the elementary string can be carried out reliably at weak string coupling, while the description of the system as a black hole holds at large string coupling, quantitative comparison between the microscopic and macroscopic calculations is difficult.This problem can be overcome by considering a special class of elementary string states that preserve some of the supersymmetries of the theory and hence saturate the BPS bound [7].For such states there are general arguments that the log of the degeneracy computed at weak coupling remains the same at strong coupling and hence can be compared with the entropy of the black hole carrying the same charges.One finds however that the corresponding black hole has vanishing area of the event horizon and hence the entropy vanishes.It was argued in [8][9][10] that once we take into account the higher derivative corrections to the effective action, the black hole may acquire a finite entropy.Using the symmetries of the theory and a scaling argument, one could determine the dependence of the entropy on the charges up to an overall numerical factor whose value depends on the details of the higher derivative corrections [8].The dependence of the entropy on the charges, obtained this way, precisely matches with the known result from the microscopic counting [8][9][10].The numerical constant was determined later in [11] under the assumption that only certain four derivative terms contribute to the entropy of the black hole.
There is however one caveat in this analysis.The analysis described above relies on the degeneracy of BPS states being protected under quantum corrections so that it takes the same value for weak and strong coupling.However it is not the degeneracy, but a closely related quantity called the supersymmetric index, that is protected this way.So we need to make an extra assumption, namely that the degeneracy and the index should be equal both at strong and weak coupling.In the weak coupling limit, where the states can be described as excitations of the fundamental string, this is known to be the case for compactification of heterotic string theory.However for type II string compactification this fails -the index vanishes while the degeneracy continues to be large.At strong coupling where the black hole description is valid, there is no general argument showing the equality of degeneracy and the index either in the heterotic string theory or in type II string theory.See e.g.section 5 of [12] for some discussion on these issues.The goal of this paper is to circumvent this problem by directly analyzing the index of fundamental string states in heterotic string theory compactified on (10 − D) dimensional torus and other manifolds.On the microscopic side the computation is straightforward and the result has been known for many years.On the macroscopic side we use a procedure for computing index developed recently in [13], following earlier work on AdS spaces [14].This procedure has been successful in reproducing correctly the leading result [13,15] as well as logarithmic correction to the logarithm of the index of black holes with regular finite area event horizon [16,17].We use this method to calculate the leading contribution to the logarithm of the index of small black holes.
It turns out that the saddle point that contributes to the log of the index of small black holes is not a small black hole with zero area event horizon but a rotating Euclidean black hole with finite area event horizon.Nevetheless, when one follows the algorithm for computing the index using this black hole saddle point, one finds that in the two derivative theory the index vanishes, just as the entropy of the small black holes vanished in the two derivative theory.One finds however that despite having an event horizon of finite area, this black hole solution is not completely smooth, -it has singularity on a subspace of the horizon where higher derivative corrections may become important.We carefully analyze the geometry near the singularity, and by using the symmetries of the theory and a scaling argument very similar to the ones used in [8], we can completely determine the corrected logarithm of the index up to an overall numerical factor.This dependence of the index on the charges found this way turns out to be in perfect agreement with the microscopic results for the index.We furthermore show that the undetermined numerical factor is independent of the dimension D of the non-compact part of the space-time even though the full black hole solution depends non-trivially on D. This is also in agreement with the microscopic results.
The rest of the paper is organized as follows.In section 2 we review the microscopic results for the index of fundamental string states.In section 3 we describe the general strategy that we shall use for the macroscopic computation of the log of the index.In section 4 we describe the rotating black hole solution that contributes to the index and show that the log of the index computed from this solution vanishes in the two derivative theory.We also show that the solution is singular on a subspace of the horizon.In section 5 we zoom in on the region near the singularity and express the solution in a rescaled coordinate system that is useful for studying the solution near the singularities.In section 6 we present the argument based on scaling and the symmetries that determines the higher derivative correction to the log of the index up to an overall numerical constant.We show that the dependence of this correction on the charges carried by the state is precisely what is expected from the microscopic analysis.In section 7 we generalize these results to other compactifications of heterotic string theory.
Since the paper is technical, for the benefit of the reader we would like to point to the final results of our paper that are contained in section 6. Eqs.(6.1)-(6.6)describe the solution near the singularity.These configurations are universal, i.e. independent of the parameters of the solution like the charges or the asymptotic values of the moduli, except for two factors.First the string metric (6.1) contains a direct sum of a universal four dimensional metric and a nearly flat (D − 4) dimensional metric b 2 dΩ D−4 , whose only role in our analysis is to provide a factor of the volume of the (D − 4) dimensional space, proportional to b D−4 , in the evaluation of the action.The second source of non-universality is an additive constant − ln(g −2 s m 0 b D−4 ) in the expression for the dilaton field Φ as given in (6.6), which also provides an overall multiplicative factor in the action but does not affect our analysis in any other way.Expressing these nonuniversal factors in terms of the charges carried by the black hole, we arrive at the result (6.12) which is our final result for the logarithm of the index.Here K is an unknown numerical factor that can in principle be determined with a better understanding of the higher derivative terms in the action, or, equivalently, a better understanding of the world-sheet σ model with target space determined by the solutions (6.1)-(6.6).
Ref. [6] gives an argument that the black hole solution describing elementary string states and the string star solution of [5] belong to the same moduli space parametrized by the mass of the black hole.If a similar relation can be established for charged rotating black holes, then we may be able to use the correspondence to map the problem of computing K to the problem of computing the entropy of a rotating string star solution.It will be interesting to explore this avenue.

Microscopic results for small black hole entropy
We consider heterotic string theory compactified on a (10 − D) dimensional torus so that we have a theory in D non-compact space-time directions.The vacuum of this theory is characterized by the string coupling g s that can be identified as the vacuum expectation value of e Φ/2 , Φ being the dilaton field and the vacuum expectation value of a (36 − 2D) × (36 − 2D) matrix valued scalar field M , satisfying where I n denotes n × n identity matrix.We shall denote by Φ ∞ and M ∞ the asymptotic values of the fields Φ and M .
The elementary string states in this theory carry 36 − 2D different charges, which we can divide into a (10 − D) dimensional charge vector Q R and a (26 − D) dimensional charge vector If we denote by Q the full (36 − D) dimensional charge vector, then we can write We also define, Among the elementary string states there are a special class of states, known as half BPS states, that are invariant under eight of the sixteen unbroken supersymmetries of the theory.
The mass of such a state, measured in the string metric, is given by At this point we need to describe our convention.We shall work in the α ′ = 1 unit so that if we have a fundamental string carrying n units of momentum and w units of winding along an internal circle of radius R, it has mass wR + n/R.In our convention √ 2 for this state so that (2.4) holds and we have . The microscopic entropy of such BPS states, defined as the log of the degeneracy of the states, can be computed and yields the result (see e.g.[4,8]) in the limit of large charges.However this is not protected under quantum corrections.The more relevant quantity for our analysis is an appropriate index, known as the fourth helicity trace index, defined as where T r Q denotes trace over all states carrying charge Q and zero momentum, (−1) F takes value 1 and −1 for bosonic and fermionic states respectively, H is the Hamiltonian and J is the component of the angular momentum carried by the state in some fixed two dimensional plane.
β is an arbitrary parameter labelling the inverse temperature of the system but the index is supposed to be independent of β since only those states whose H eigenvalue is equal to m BP S contribute to the trace.For heterotic string theory on T 10−D this quantity can be shown to be equal to the degeneracy in the weak coupling limit, and so the log of the index is given by (2.5).However this index can also be shown to be protected under quantum corrections and hence takes the same value at strong coupling.
Our goal is to try to reproduce (2.5) from macroscopic analysis, i.e. from the analysis of the classical geometry describing a solution carrying these charges.

Strategy for macroscopic computation of small black hole entropy
In any two derivative theory in D space-time dimensions containing the metric g µν , 2-form field µ } and scalar fields {ϕ α }, there is a scaling relation under which and the action scales as λ D−2 .For a classical black hole solution carrying mass m, charges Q and angular momenta J, if we scale the fields as in (3.1), the various quantities parametrizing the solution scale as Under this scaling the Bekenstein-Hawking entropy S BH of a regular black hole scales in the same way as the action: These results can be derived as follows.First note that in the usual coordinate system where the asymptotic metric is taken to be η µν , the metric does not scale as given in (3.1).For this we introduce new coordinate system xµ = x µ /λ so that the the asymptotic metric has the form λ 2 η µν dx µ dx ν in accordance with the scaling laws in (3.1).In this coordinate system the usual fall-off properties take the form: We now see that the λ dependence of these equations can be compensated by scaling the fields as in (3.1) and m, Q, J as in (3.2).
We can take the macroscopic limit by taking λ to be large.Note that under the same scaling (2.5) scales by a factor of λ D−3 , leading to an apparent contradiction.This issue was resolved in [8] by noting that the classical BPS black hole in the two derivative supergravity theory has zero area event horizon and hence vanishing entropy and we need to take into account higher derivative corrections to get the correct scaling of the entropy.We shall carry out a similar analysis in this paper, but focussing from the beginning on the computation of the index on the macroscopic side as well.
We now outline the main steps in our analysis.
1. Since we are interested in computing the index, we follow the strategy described in [13], where we begin with an electrically charged black hole carrying a charge vector Q and angular momentum J in a two dimensional plane.The angular momentum J is adjusted so that the chemical potential Ω dual to this angular momentum, also called the angular velocity, is given by where β is the inverse temperature of the black hole.The effect of this chemical potential is to insert a factor of e 2πiJ into the path integral, which in turn plays the role of (−1) F that appears in (2.6).Therefore this rotating black hole solution provides the dominant saddle point in the computation of the index.Then the leading semiclassical result S macro for the log of the index, computed from the gravitational path integral, is given by where S wald is the Wald entropy of the black hole, which can also be computed following the approach of Gibbons and Hawking where we first compute the partition function by computing the path integral over all the fields and then use the appropriate thermodynamic relations to extract the entropy [18].We shall denote by S wald the expression for the black hole entropy including the effect of higher derivative corrections, irrespective of which way it is calculated.For two derivative theories this agrees with the Bekenstein-Hawking entropy S BH of the black hole given by A/(4G N ) where A is the horizon area and G N is the Newton's gravitational constant.The 2πiJ term in (3.6) is the effect of the chemical potential that inserts the explicit factor of e 2πiJ in the path integral.Note that according to (2.6) we also need to insert a factor of (2J) 4 into the path integral, but as discussed in [16,17], this factor just soaks up the fermion zero modes associated with broken supersymmetries in the path integral and does not have any other effect.
2. We then show that for this solution If this had been non-zero, then according to (3.2), (3.3) this would scale as λ D−2 .On the other hand (2.5) and (3.2) shows that S micro scales as λ D−3 .This would lead to an obvious contradiction which is avoided by (3.7).
3. We can now ask whether higher derivative corrections to (3.6) could resolve the disagreement.Since J is measured from the fields at infinity where the higher derivative corrections can be ignored, we focus on S wald .If the horizon had been smooth, then the four derivative corrections to S wald would be of order λ D−4 according to (3.1) since two extra derivatives need to be contracted with a g µν .This would be inconsistent with (2.5) which scales as λ D−3 .We show that the solution is singular near a subspace, and that in the neighbourhood of this subspace we can choose a coordinate system in which the string frame metric becomes independent of the charges and the dilaton takes the form where F is a function of the coordinates that does not depend on the charges.Since e Φ/2 expectation value gives the string coupling, (3.8) shows that for large charges, the string coupling at the horizon is small.Hence we can restrict our analysis to tree level string theory.On the other hand since the metric and ∂ µ Φ are of order unity (having no dependence on the charges) we see that the α ′ corrections are of order unity, and we need to take into account all order corrections in α ′ to compute the entropy.However these corrections are completely universal, independent of the charges.
4. This argument shows that if the higher derivative corrections generate a finite entropy, then the dependence of the entropy on the charges comes solely from the factor in (3.8).This will appear as an overall multiplicative factor in the expression for the on-shell action and hence the entropy.So we can write in agreement with (2.5).Here C is an unknown numerical constant that depends on the details of how the α ′ corrections modify the near horizon geometry.However C is independent of Φ ∞ and M ∞ .We also show that C is independent of D, in agreement with (2.5).
Finally we would like to remark that since the macroscopic computation of the index will be done using the full space-time geometry, it implicitly uses the grand canonical ensemble, i.e.
instead of computing the trace over fixed charge states as in (2.6) we sum over all charged states weighted by e −βµ.Q and then perform the Legendre transform of the logarithm of the result with respect to the chemical potential µ.Of course we shall not see this being done explicitly, but the gravitational formula (3.6) for the index that we shall use does this automatically.If the index is a smooth function of the charges then the result of this computation differs from the actual index by subleading logarithmic terms that we do not keep track of.However if the index is not a smooth function of the charges, e.g. if it switches sign as we move from one element of the charge lattice to its neighbouring element, then the procedure described above can give results that differ significantly from the actual index, -see e.g.[14] for similar issues for AdS back holes.For heterotic string compactification we know that the microscopic index is a smooth function of the charges for large charge.We shall assume that this holds on the macroscopic side as well.

The black hole solution
We begin by writing down the two derivative action describing the massless bosonic fields in heterotic string theory compactified on T 10−D [19]: where, Furthermore this symmetry is known to survive α ′ correction to the effective action, i.e. this is an exact symmetry of tree level string theory [20,21].G µν appearing in (4.1) is the string frame metric.The canonical Einstein frame metric g µν is related to this via the equation: where g 2 s is the asymptotic value of e Φ .The g s dependent factor is normally not included in the definition of the Einstein frame metric, but we have included it here so that if G µν approaches η µν asymptotically, so does g µν .We shall always work with the string frame metric unless mentioned otherwise.

The solution
Let α and γ be two boost parameters.Let ⃗ n be a (26 − D) dimensional unit vector and ⃗ p be a (10 − D) dimensional unit vector.The general non-extremal solution in D-dimensions with single rotation takes the form [22]: where, (4.7) {t, ρ, θ, ϕ} are the four spacetime coordinates and dΩ D−4 is the round metric on a (D − 4)dimensional unit sphere.The dilaton supporting the solution is: The U(1) vector fields A (j) for 1 ≤ j ≤ (26 − D) are determined by the unit vector ⃗ n: and the remaining vector fields A (j) for j ≥ (27 − D) are determined by the unit vector ⃗ p: (4.10) The two-form field takes the form, and the matrix of scalars takes the form, where, Since P and U fall off for large ρ, we see from (4.12) that M ∞ = I 36−2D .However using the symmetry transformation (4.4) we can generate any other M ∞ from this solution, and hence setting M ∞ to I 36−2D does not restrict the solution in any way.

BPS limit and Euclidean continuation
The BPS limit is obtained by taking m → 0, γ → ∞ keeping m 0 ≡ m cosh γ finite.The BPS solution is also written in [22].Let us in addition consider the analytic continuation to Euclidean space, together with a redefinition of the rotation parameter a and the fields: The last analytic continuation is required for consistency with (4.3).This gives the following Euclidean solution, where now The gauge fields are given by and The two form field takes the form, and the scalar fields are given by, where Note that with the redefinitions given in (4.14) the solution is real.The horizon is at ρ = b and the relevant region of space-time is ρ ≥ b.Also the solution is singular at ρ = b, cos 2 θ = 1.This is easiest to see by examining the dilaton given in (4.17), which blows up at these values.
We now compute the electric charges carried by the solution.For this we note that for large ρ, we have We define the electric charges Q (j) as the integral of1 2 δS δF (j) (E)ρτ over the (D − 2) dimensional sphere at a large but fixed value of ρ. 1 Using the fact that e −Φ∞ = 1/g2 s and M ∞ = I 36−2D , where ω k represents the volume of a unit sphere in k dimensions.Since n and p are unit vectors, we get We can also calculate the ADM mass of the black hole.For this we use (4.15), (4.17) and (4.5) to write down the expression for the canonical Einstein metric: Using this and eq.( 14) of [23], we get where the Newton's constant G N for the action (4.1) is given by This gives in agreement with (2.4).
The overall normalization of the charges depends on the normalization of the U(1) gauge fields appearing in (4.1).We have chosen the normalization so that we get (4.29).We have also carefully kept track of all dependence on the asymptotic moduli and the dimension D. In particular, if we use the transformation (4.4) to generate a solution with some other M ∞ , the dependence of m ADM on M ∞ and the charge Q will be captured by the dependence of The area of the ρ = b surface in the Einstein frame is (4.30) The Bekenstein-Hawking entropy of the black hole is given by The Euclidean angular momentum J E of the black hole is, Clearly, This reproduces (3.7) with the identification given in (4.14).

Periodicity of the τ circle
To understand the periodicity of the τ circle, let us introduce With these coordinate changes, the relevant part of the metric near ρ = 0 is In writing this metric, we have dropped the terms of order ρ 2 dτ d ϕ and various other higher order terms in the expansion in ρ.Away from the singular subspace on which sin θ vanishes, the ( ρ, τ ) part of the metric is smooth at ρ = 0 provided τ has periodicity in agreement with (3.5).

(D − 1)-dimensional base and the singular subspace
Remarkably, the string frame metric (4.15) takes the following form in general dimensions2 where and The base metric ds 2 base does not depend on the charges and is in fact flat.To see this, let us introduce, where ψ 1 , . . .ψ D−4 are the angles labelling points on the (D − 4) sphere.In this coordinate system the base metric takes the form: Despite having a flat base, the metric is singular is at ρ = b, cos 2 θ = 1 since f vanishes and ω ϕ blows up there.To determine the geometry of the singular surface, note from (4.44) that y 1 and y 2 vanish on the singular subspace and y 3 , • • • , y D−1 lie on the surface of a (D − 4) dimensional sphere of radius b.The range of the coordinate θ is (0, π) for D = 4 whereas for D > 4 it is (0, π 2 ).Therefore for D > 4, we have a connected singular surface at ρ = b, θ = 0 which is a S D−4 sphere.In D = 4 the singular subspace is a pair of points corresponding to ρ = b, θ = 0, π.Since S 0 consists of a pair of points, we can describe the singular surface as S D−4 for all D. This will be important later.
f computed from (4.42) can be written as where In D = 4, R ± have the interpretation of the distance of a point from the locations of the singularities at ρ = b, θ = 0, π.In this case f −1 is simply a product of two harmonic functions and the solution can be written in a standard IWP form [15,24].
5 The geometry near the singularity Eq. (4.33) shows that in the two derivative approximation to the effective action, the macroscopic index vanishes.According to the scaling argument given between (3.7) and (3.8), higher derivative corrections to S wald from the non-singular regions cannot produce the answer known from the microscopic counting.We shall now focus on the singular regions near the horizon and explore whether higher derivative correction to S wald from the singular region can produce the desired result.

Coordinates centered at the singular surface
In order to analyse the solution near the singular surface cos θ = 1, we introduce the following coordinates near the singular subspace.Define, The inverse coordinate transformations are, (5.2) θ has range (0, π) for all D. These coordinates have the important property that when R = 0, we have ρ = b and cos θ = 1, i.e., we are at the singular surface.For D = 4 the second copy of the singular subspace is at R = 2b, θ = π, but near that region it is more appropriate to introduce new coordinate system by replacing cos θ by − cos θ in (5.1).

Large charge limit
We now make a final coordinate transformation, and take the limit m 0 → ∞ and b → ∞ by taking keeping fixed R, θ, τ , ϕ, Ω D−4 .The scaling of b follows from (3.2) after noting that J E ∝ m 0 b.The limit gives (5.10) (5.11) We have kept terms in the expansion to the order that will be needed in our computation of the solution up to terms that vanish for large λ.Combining all this together and using (4.40), we arrive at a universal form for the metric: (5.12) Θ has range 0, π 2 and ϕ and τ have periodicities 2π.The metric is singular at R = 0. Since b is large, we can replace b 2 dΩ D−4 with D−4 i=1 dx 2 i locally.From this we see that near the singularity the metric has a universal form independent of D. If on the other hand we take a spherically symmetric solution by taking the rotation parameter b to zero, the size of the (D − 4) dimensional sphere goes to zero and the nature of the singularity becomes dependent on D [9].
We now turn to the matter fields in the large charge limit at fixed R, τ , ϕ, Θ.In this limit (5.13) (4.21) now implies, For the gauge fields, we get, for 1 ≤ j ≤ 26 − D, A and for j ≥ 27 − D, A (5.16) 2. In the large charge limit, since b ∼ λ and m 0 ∼ λ D−3 , e −Φ given in (6.6) is of order λ.Since e −Φ can be identified as the inverse square of the string coupling locally, we see that near the singularity the string coupling is small.Hence we can ignore string loop corrections and focus only on the α ′ correction to the solution.Taking into account the overall factor of C D in (4.1) we see that the net α ′ correction to S wald , and hence to S macro according to (3.6), (3.7), takes the form, where K is a universal numerical constant that can in principle be computed from α ′ correction to the solution (6.1)-(6.6).In particular, K is independent of D.
We now recall from (4.25): Using this we can express (6.7) as Substituting this into (6.9)we get This would agree with (2.5) if K = 8π 2 .While we do not have a way to calculate K at present, the D-independence of (6.12) is consistent with the D-independence of (2.5).Note that the dependence on the parameter b, that controls the temperature of the black hole via (4.36), has disappeared.This is consistent with the fact that the index should be independent of the temperature.Note also that the result is independent of the combination This can be traced to the α-independence of the result, -the O(26 − D, 10 − D) transformation that removes the α dependence of M near the singularity also removes the α dependence of the gauge fields.Such a dependence would have been inconsistent with the microscopic results which do not depend on M ∞ .This was also a feature in the analysis in [8].

Generalization to other compactifications
The above analysis can be easily generalized to other N = 4 or N = 2 supersymmetric compactifications of heterotic string theory.For this we use the observation of [10] that the black hole solution describing BPS elementary string states can be embedded in a universal consistent truncation of the supergravity theory used above.Furthermore the universality is maintained to all orders in α ′ .Therefore the analysis given above can be carried out without any further modification and we shall arrive at the expression (6.12) for the macroscopic entropy with the same K.The microscopic counting also remains unchanged, since the asymptotic growth of BPS states is controlled by the central charge of the left-moving matter sector of the world-sheet theory and this is always equal to 26.Note however that in order to get BPS fundamental string in the first place, we must have at least one internal circle on which the string can wrap, i.e. we cannot have such states in ten dimensional heterotic string theory or the six dimensional theory obtained by compactifying heterotic string theory on K3.
Finally we shall make a few remarks on the type II string compactification.For definiteness let us consider the case of torus compactification.The microscopic index for the fundamental string states vanishes in this case due to the cancellation between the contributions from the bosonic and fermionic states.On the macroscopic side the relevant part of the two derivative action is very similar to the one we have analyzed here and identical arguments lead to a form (6.12) for the logarithm of the index.Therefore for consistency the coefficient K must vanish for type II theory.This may be related to the fact that higher derivative corrections are relatively rare in type II string theory.For example the four derivative term, that was used in [11] to get a finite result for K in the heterotic string theory, is absent in type II string theory.However to get a complete picture we need to learn how to compute K by taking into account the effect of all the α ′ corrections to the effective action.
) and R G is the Ricci scalar associated with the metric G µν .M and L have been introduced in (2.1).C D is an arbitrary D-dependent constant, which can be absorbed into the dilaton field Φ.This action is invariant under an O(26 − D, 10 − D) symmetry under which (τ, ϕ) ≡ (τ + β, ϕ) where β = 2π m 0 b D−4 .(4.36) Using (4.34), this translates to (τ, ϕ) ≡ τ + β, ϕ − b course ϕ itself has a periodic identification under shift by 2π in order that the metric (4.41) is non-singular for ρ > b.From this we can determine the angular velocity of Euclidean black

3 .
Since α ′ correction to the effective action does not destroy the O(26−D, 10−D) symmetry, the O(26 − D, 10 − D) transformation that was used to bring the solution to the form (6.1)-(6.6) is a valid symmetry and leaves the α ′ corrected formula for S wald invariant.In fact we can use the O(10−D) subgroup of O(26−D, 10−D) to bring the vector p to some canonical form, e.g.only with the first component non-zero.In this case the solution is universal except for the b 2 dΩ D−4 term in the metric (6.1) and the overall factor of g −2 s b 4−D m 0 in the expression for e −Φ in (6.6).In the limit of large b the b 2 dΩ D−4 part of the metric describes almost locally flat (D − 4) dimensional space and the α ′ corrections are suppressed by inverse power of λ.At the leading order this part of the metric will give an overall multiplicative factor of b D−4 ω D−4 to the action and hence to S wald buthas no effect on the α ′ correction to the rest of the field configuration.On the other hand from (4.1) we see that the effect of the g −2 s b 4−D m 0 in the expression for e −Φ is to multiply the action by a factor of g −2 s b 4−D m 0 but it also does not affect the α ′ corrections to the solution since these corrections involve Φ only through its derivatives.Therefore α ′ corrections to the solution take a universal form, even independent of D, and modify only the R and Θ dependence of various fields in (6.1)-(6.6).