A Note on D1-D5 Entropy and Geometric Quantization

We quantize the space of 2-charge fuzzballs in IIB supergravity on $K3$. The resulting entropy precisely matches the D1-D5 black hole entropy, including a specific numerical coefficient. A partial match (ie., a smaller coefficient) was found by Rychkov a decade ago using the Lunin-Mathur subclass of solutions - we use a simple observation to generalize his approach to the full moduli space of $K3$ fuzzballs, filling a small gap in the literature.


Introduction and Result
Ever since the pioneering work of [1,2], it has been known that string theory can explain the entropy of (at least some classes of) black holes in terms of microstates in an appropriate ensemble. The argument is most under control when the system is BPS, so that one can tune the string coupling to near-zero, while the degeneracy (more precisely an index) remains invariant: the black hole then gets mapped to a system of weakly coupled D-branes whose microstates are easily counted.
Impressive as it might be, this result is indirect and kinematical. In particular, the origin of the event horizon in the gravity regime is entirely obscure from the weakly coupled picture. As a corollary, we do not understand various dynamical issues like the information paradox. An outstanding question in this context is this: what happens to the individual microstates as one cranks up the coupling? The fuzzball proposal [3][4][5] is an attempt to answer this question.
The claim of the fuzzball proposal is that at strong coupling, the microstates turn into smooth non-singular solutions of string theory that differ from the black hole solution at the horizon scale, and that the ensemble of these fuzzball microstate "geometries" 1 is what the black hole is comprised of. For the two charge (D1-D5) black hole in Type IIB string theory on K3 or T 4 there are various arguments that the entropy of the system at leading order can be reproduced entirely via fuzzball solutions that are visible within the supergravity description as smooth horizonless solutions [11]. One argument is to quantize the phase space of fuzzball solutions and counting the number of states to see whether it reproduces the D1-D5 entropy. This was done for the original Lunin-Mathur subclass of 2-charge fuzzball solutions [12] by Rychkov [13]. But Rychkov's result, does not reproduce the full entropy, which is for the K3 black hole 2 . This is not surprising because the approach of Rychkov [13] did not incorporate the complete family of 2-charge fuzzballs, in particular they do not include the excitations in the compact directions. The result is nonetheless suggestive because it does capture the correct scaling of the entropy with the D1 and D5 charges.
In this paper, we will consider the complete phase space of 2-charge fuzzball solutions for the specific case of compactification on K3. These were constructed by Kanitscheider, Skenderis and Taylor (KST) [14]. We will use geometric quantization of the phase space of those solutions. We will be able to argue that a simple generalization of the Rychkov symplectic form is the correct choice on them, enabling us to extend his result to the full phase space. We find that the result (1.2) is precisely reproduced, giving closure to a gap in the literature.
do not have a Coulomb branch analogue (and therefore possibly cannot be seen in supergravity) but have a direct interpretation as single-centred black hole microstates -they are stable under wall-crossing, and have a so-called Lefschetz SU (2) symmetry which can be interpreted as capturing the spherical symmetry of the black hole horizon. In particular, it has been argued in [10] that all 4D microstates must have zero angular momentum (J = 0). In [6], solutions that capture a finite fraction of the entropy of the 5D black hole were constructed. The solutions have no non-trivial circles, so one does not have contradictions with the arguments of [10] via dimensional reduction. As an aside -it is not clear to us if the 4D pure Higgs microstates are forbidden (or not) from having a supergravity description in terms of some hitherto undiscovered J = 0 microstates. In any event, our goal here is not to get into the debate on what fraction of the 3-charge microstates can be seen in supergravity. Our goals are modest and limited to the 2-charge system. But we feel it is necessary to give the reader some context regarding the status of SUGRA fuzzball microstates in the interest of transparency. 2 We will not consider the T 4 compactifications in this paper for reasons explained in the final section.

General Two Charge Fuzzballs on K3
The general two-charge fuzzballs for the K3 case were constructed in [14], we will refer to them as the KST fuzzball solutions. We follow the conventions in [11]: The metric is in the string frame. The ω γ ≡ (ω α + , ω α − ) are a basis of self-dual and anti-self-dual 2-forms on K3 with γ = 1, · · · , 22 where 22 is the second Betti number of K3. The labels 3 take the values α + = 1, 2, 3 and α − = 1, · · · 19. The intersection numbers of the forms are The integration constant in C (2) tz ensures that the potential vanishes at infinity -the solutions depend on the harmonic functions (H, K, A i , A, A α − ) via Hereμ = (t, z) and * 4 is the Hodge dual over flat R 4 . The Hodge dual in the Calabi-Yau K3 metric is given by ǫ ρστ π . The solutions in [14] correspond to the choice of Harmonic functions given by Here, |x − F (v)| 2 is to be understood as i |x i − F i (v)| 2 , we will often suppress summation over the index i. The 5-brane charge Q 5 and the length of the defining curve L in the D1-D5 system (see [11]) are related through the radius of the z-circle R via by A relation that is useful and important for us is the expression for the D1 charge Q 1 : The integral charges are given by Here (2π) 4 V is the volume of K3. Henceforth, we will set α ′ to unity. The Lunin-Mathur solutions correspond to setting F (v) = 0 = F α − (v) in the KST fuzzballs. The detailed form of the KST solution will not be necessary to follow most of our discussions, but we present it here for two reasons: • In checking (3.4), which is the key observation of this paper, from first principles, we will need the details of the solution.
• We want to emphasize that at least superficially, the general fuzzballs are substantially more complicated than the Lunin-Mathur fuzzballs [12].

The Consistent Symplectic Form
The basic idea of geometric quantization is to quantize the phase space, count the states in the Hilbert space and use that as the definition of the micro-canonical entropy. The phase space and the space of solutions have a one-to-one map, so we can also work with the latter. The goal then is to compute the symplectic form on the space of solutions and then quantization can proceed as usual. In principle this is straightforward, but it is bound to be a complicated problem for the fuzzball solutions presented in the last section. Indeed, even for the Lunin-Mathur solutions the task was complicated, and Rychkov used two simplifying facts to make the problem tractable, and to compute the restriction of the full IIB supergravity symplectic form 4 onto the moduli space of solutions. The first was that the the Lunin-Mathur solutions are time-independent, which is a fact that is trivially true for our more general KST solutions as well. The second was that the Hamiltonian, when restricted to the moduli space took a specific simple form [13]: where the subscript LM on the left hand side denotes the fact that we are working with the Lunin-Mathur subclass of solutions. Using these facts, it was argued in pages 7-8 of [13] that the symplectic form should take the form where α can only depend on the various integrals of motion determined by the curve functions F i (v): At first sight, the generalization from Lunin-Mathur to KST fuzzballs seems formidable. The solution is substantially more complex, and for the K3 case, there are 20 (=19+1) new independent functions in the solution now. It is also clear, that the Hamiltonian of the KST fuzzballs must be different from (3.1) 6 . Despite these potential complications, we will show in this paper that the IIB supergravity Hamiltonian, when restricted to the KST solutions, retains enough of the simple features of the Lunin-Mathur solutions that we can adapt the Rychkov arguments to get the complete answer without getting bogged down in the details.
The basic observation is that the energy in the KST case can be directly computed, and it takes the simple form despite the added complexity of the KST solutions 7 . This can be obtained straightforwardly via the ADM approach (we sketch it in an appendix), but it is easy to 5 Here, µ 2 = g 2 s R 2 V . 6 Indeed this is necessary, if one has hopes of reproducing the full entropy by doing geometric quantization. 7 The subscript KST on the left hand side denotes that we are working with the full family of KST fuzzballs. convince oneself that this answer is as it should be, as follows -Using (2.6) and (3.4) we can show immediately that the total mass of the system is and since the system is BPS, this is something we would expect 8 . Now, the form (3.4) is very closely related to the original energy functional in Rychkov's computation (3.1), with the crucial fact that all the independent functions enter democratically and quadratically in it. In effect, therefore the arguments leading to the symplectic form (3.2) in [13] go through exactly as before, with the only new ingredient that it should also involve terms from the new functions: Now, since the subclass of curves considered in [13] to argue that α must be the numerical constant πµ 2 is also a subclass of the curves considered here, it immediately follows that the α = πµ 2 here as well. This fixes the symplectic form for the KST solutions completely.

Entropy Match from Quantized Phase Space
Once we have the symplectic form, we have everything we need to quantize and compute the entropy. Since all the curve functions enter democratically in the discussion, we will define Note that I takes 24 values because of this definition. Now, the standard approach is to expand the curve functions into Fourier oscillators and to count the modes, see section 2 of [13] for a clear discussion. The only difference between there and here is that there the indices i in F i (v) took only four values reproducing a result that would be equal to that of four chiral bosons (ie., central charge c = 4). Here we get the analogue of 24 chiral bosons (c = 24). The latter is what is indeed expected for the D1D5 black hole on K3, see eg. p.28 of [11]. The answer can therefore be obtained via the Cardy formula reproducing the classical Bekenstein-Hawking entropy of the K3 hole.

Comments
We have found that the geometric quantization of the general fuzzball moduli space on K3 reproduces the corresponding D1-D5 entropy on the nose. The possibility that a more complicated structure for the symplectic form could arise and complicate the computation has been raised in the literature (see discussion after eqn (4.76) in [11]), but by working with the energy of the general fuzzballs we have shown that the problem can be solved by a simple generalization of the Rychkov argument. The final symplectic form is indeed simple and democratic in all the curve functions.
With the malice of hindsight, perhaps one could have taken the existence of 24 unknown functions in the KST solutions as a hint of this, already at the time they were constructed [11,14]. It will be interesting to repeat a similar computation in the T 4 case. However, unlike in the K3 case, the solution (2.1) in the T 4 case (ie., now α − only in the range α − = 1, 2, 3 corresponding to the anti-self dual 2-forms on the 4-torus) does not describe the most general fuzzball solution [11]. It describes only the bosonic excitations, one needs to further add fermionic excitations. Related questions seem to have been addressed in [15], we hope to come back to the T 4 computation sometime in the future.
standard Kaluza-Klein ansatz when thought of as a 10 D solution. See for example Appendix E of Kiritsis [16], whose notations we follow. The reduction there is done starting with the string-frame metric, which is exactly what we want. From eqns (E.3-E.4) in [16] and the structure of KST solution (specifically, the dilaton and the metric components in the compact directions), one can see that the reduction of the 10D string-frame KST metric gives rise to a 5D string(-like)-frame metric and a 5D "dilaton" (this is the field φ defined in eqn. E.4 of [16] [17] is useful for fixing ADM conventions when comparing fall-offs. Note that |x − F (v)| 2 can be approximated |x| 2 ∼ r 2 (where r is the radial coordinate in 4+1 D) and taken outside the integral to the order that is relevant for calculating the ADM mass.