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JHEP05(2017)147

Published for SISSA by Springer

Received: April 13, 2017

Accepted: May 17, 2017

Published: May 29, 2017

More on microstate geometries of 4d black holes

M. Bianchi,

J.F. Morales,

L. Pieri

a,c

and N. Zinnato

Universit`a di Roma Tor Vergata and I.N.F.N, Dipartimento di Fisica,

Via della Ricerca Scientiﬁca, I-00133 Rome, Italy

I.N.F.N. — Sezione di Roma 2 and Universit`a di Roma Tor Vergata, Dipartimento di Fisica,

Via della Ricerca Scientiﬁca, I-00133 Roma, Italy

Center for Research in String Theory, School of Physics and Astronomy,

Queen Mary University of London, Mile End Road, London, E1 4NS, U.K.

E-mail: bianchi@roma2.infn.it, morales@roma2.infn.it,

lorenzo.pieri@roma2.infn.it, natale.zinnato@gmail.com

Abstract: We construct explicit examples of microstate geometries of four-dimensional

black holes that lift to smooth horizon-free geometries in ﬁve dimensions. Solutions consist

of half-BPS D-brane atoms distributed in R

. Charges and positions of the D-brane centers

are constrained by the bubble equations and boundary conditions ensuring the regularity

of the metric and the match with the black hole geometry. In the case of three centers, we

ﬁnd that the moduli spaces of solutions includes disjoint one-dimensional components of

(generically) ﬁnite volume.

Keywords: Black Holes in String Theory, D-branes

ArXiv ePrint: 1701.05520

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP05(2017)147

JHEP05(2017)147

a quantum theory however a black hole radiates as a black body with a ﬁnite temperature

and an entropy given by one quarter of the area of its event horizon. To explain the

microscopic origin of this entropy remains a primary task for any serious contender to a

quantum theory of gravity.

In string theory, black holes can be realised in terms of D-branes intersecting in the

internal space. The micro-states can be represented (and counted) in terms of excitations

of the open strings connecting the building brane bits. Alternatively, one may think of

the geometry generated by the excited brane state as the gravity representation of the

micro-state. Since the micro-state geometry describes a pure state with zero entropy, it

should have no horizon. This line of ideas motivates the “fuzzball” proposal that associates

to every black hole micro-state a regular and horizon-free solution of classical gravity. The

solutions, known as “fuzzballs” or “micro-state geometries”, share with the would-be black

hole the mass, charges and angular momentum but diﬀer from it in the interior [1–9].

The black hole horizon and its entropy arise from a coarse graining superposition of the

micro-state geometries.

In the last years, a large class of four and ﬁve dimensional black hole micro-state

geometries have been produced [10–20]. The micro-state geometries are typically coded

in smooth, horizon-less geometries with no closed time-like curves (CTC’s) in ﬁve or six

dimensions. This is the best one can achieve, since no-go theorems in four dimensions

exclude the existence of non-singular asymptotically ﬂat soliton solutions.

This is not the

case in ﬁve or higher dimensions where the existence of Chern-Simons interactions and

spatial sections with non-trivial topologies circumvent the no-go result [23]. From a four-

dimensional perspective, the ﬁniteness of the higher dimensional Riemann tensor (and its

derivatives) results into a ﬁnite eﬀective action with curvature divergences compensated

by the singular behaviour of the scalars and gauge ﬁelds.

A black hole with ﬁnite area in four dimensions can be realised in several diﬀerent

frames. Popular choices include bound states of D1-D5-KK-p, of D0-D2-D4-D6 branes [24–

30] or of intersecting D3-branes, wrapping three cycles in T

× T

[31–33]. After

reduction down to four dimensions, the solution can be viewed as a supersymmetric vac-

uum of an N = 2 truncation of N = 8 supergravity involving the gravity multiplet and

three vector multiplets characterizing the complex structures of the internal torus. In a

microscopic world-sheet description of the D3-brane system [32, 33], the harmonic func-

tions characterising the gravity solution are sourced by disk diagrams with a boundary

ending on a single, two or four diﬀerent branes [32, 34, 35]. Higher multipole modes are

generated by extra insertions of untwisted open string ﬁelds on the disk boundaries. The

micro-state multiplicities can be computed by counting open strings in the D1-D5-p-KK

system or vacua in the quantum mechanics associated to the D0-D2-D4-D6 realisation of

the black hole [36, 37].

The aim of this paper is to construct explicit examples of micro-state geometries of

four- dimensional BPS black holes. We follow the Bena-Warner ansatz [11, 23, 26] and

look for regular ﬁve-dimensional geometries generated by distributions of half-BPS D-

Regular solutions with AdS asymptotics have been recently found in [21, 22].

– 2 –

JHEP05(2017)147

brane atoms in R

. The regularity of the ﬁve dimensional geometry is coded in the so

called bubble equations that we generalise to account for the case of branes at angles.

Boundary conditions at inﬁnity further restrict the choices leading generically to a moduli

space that consists of disjoint components. There are two classes of solutions. Scaling

solutions are conﬁgurations that can be rigidly scaled (see [25, 38–40] for some results).

The other class includes solutions where the distances between the centers are bounded by

the charges.

We will consider micro-state geometries with both zero and non-zero angular momen-

tum. The existence of supersymmetric solutions with non-zero angular momentum may

look surprising, since single-center BPS black holes in four dimensions cannot carry angular

momentum, because rotations of a black hole horizon are not compatible with supersymme-

try. In our case, like for the multi center BPS black holes with non zero angular momentum

considered in [41], the angular momentum is generated by the cross product of electric and

magnetic ﬁelds of charges separated in R

. In the spirit of the fuzzball proposal, micro-

states with non-zero angular momentum can be viewed as members of a canonical ensemble

description of the black hole. The statistical average exposes zero angular momentum, even

though each micro-state can carry some.

The plan of the paper is as follows. In section 2 we present the BPS solutions de-

scribing systems of intersecting D3-branes on T

from the four dimensional perspective.

We consider both cases of orthogonal and of intersecting D3-branes at angles. The Bena-

Warner ansatz is introduced in section 3. The ansatz is generalised to accomodate for

non-orthogonally intersecting D3-branes. In section 4 and 5 solutions to the bubble equa-

tions and to the boundary conditions are found for the 3-center case with orthogonal or

D3-branes intersecting at angles. A preliminary discussion of the counting of the number

of micro-states with ﬁxed charges is presented in the Conclusions. In appendix A we derive

the four dimensional solution from dimensional reduction of the intersecting D3-branes

solution in ten dimensional type IIB supergravity.

2 Black holes from intersecting D3-branes

In this paper we consider a family of BPS solutions describing systems of intersecting D3-

branes on T

from the four dimensional perspective. We refer the reader to the appendix

for details on the ten dimensional solution and its reduction down to four dimensions. The

four dimensional solution has been ﬁrst explicitly derived in [26] in the D0-D2-D4-D6 type

IIA frame and lifted to a system of M5 branes in M-theory. Here, for convenience, we

consider the very symmetric formulation in terms of D3-branes, but the results can be

easily translated into the type IIA and M-theory frame.

The four-dimensional geometries can be viewed as solutions of an N = 2 truncation

of N = 8 supergravity involving the gravity multiplet and three vector multiplets. The

scalars U

in the vector multiplets, usually referred as STU, parametrise the complex

structures of the three internal T

’s and span the moduli space M

ST U

= [SL(2, R)/U(1)]

⊂

We thank Iosif Bena for clariﬁcations on this point.

– 3 –

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