Comments on Charges and Near-Horizon Data of Black Rings
arXiv:0704.1819v3  [hep-th]  17 Dec 2007
Preprint typeset in JHEP style - HYPER VERSION TIT/HEP-570
arXiv:0704.1819
Comments on Charges and Near-Horizon Data
of Black Rings
Kentaro Hanaki1, Keisuke Ohashi2 and Yuji Tachikawa3
1 Department of Physics, University of Michigan,
Ann Arbor, MI 48109-1120, USA
E-mail: hanaki@umich.edu
2 DAMTP, Centre for Mathematical Sciences, Cambridge University,
Wilberforce Road, Cambridge CB3OWA, UK
E-mail: keisuke@th.phys.titech.ac.jp
3 School of Natural Sciences, Institute for Advanced Study,
Princeton, New Jersey 08540, USA
E-mail: yujitach@ias.edu
Abstract: We study how the charges of the black rings measured at the asymptotic
infinity are encoded in the near-horizon metric and gauge potentials, independent of
the detailed structure of the connecting region. Our analysis clarifies how different sets
of four-dimensional charges can be assigned to a single five-dimensional object under
the Kaluza-Klein reduction. Possible choices are related by the Witten effect on dyons
and by a large gauge transformation in four and five dimensions, respectively.
Keywords: Black Rings, Page Charges.
http://arxiv.org/abs/0704.1819v3
mailto:hanaki@umich.edu
mailto:keisuke@th.phys.titech.ac.jp
mailto:yujitach@ias.edu
http://jhep.sissa.it/stdsearch
Contents
1. Introduction 1
2. Near-Horizon Data and Conserved Charges 3
2.1 Electric charges 4
2.2 Angular momenta 6
2.3 Example 1: the black ring 9
2.3.1 Geometry 9
2.3.2 Electric charge 11
2.3.3 Angular momenta 11
2.4 Example 2: concentric black rings 12
2.5 Generalization 14
3. Relation to Four-Dimensional Charges 16
3.1 Mapping of the fields 17
3.2 Mapping of the charges 18
3.3 Reduction and the attractor 19
3.4 Gauge dependence and monodromy 20
3.5 Monodromy and Taub-NUT 21
4. Summary 24
A. Geometry of Concentric Black Rings 24
1. Introduction
One of the achievements of string/M theory is the microscopic explanation for the
Bekenstein-Hawking entropy for a class of four-dimensional supersymmetric black holes
[1, 2]. The microscopic counting predicts subleading corrections to the entropy, which
can also be calculated from the macroscopic point of view, i.e. from stringy modi-
fications to the Einstein-Hilbert Lagrangian [3]. Comparison of the two approaches
has proven to be very fruitful, e.g. it has led to the relation to the partition function
– 1 –
of topological strings [4]. Beginning in Ref. [5], it has been also generalized to non-
supersymmetric extremal black holes using the fact that the near-horizon geometry has
enhanced symmetry. The analysis has also been extended to rotating black holes [6].
There is a richer set of supersymmetric black objects in five dimensions, including
black rings [7], on which we focus. The entropy is still given by the area law macro-
scopically to leading order, and it can be understood microscopically using a D-brane
construction [8, 9]. The understanding of higher-derivative corrections remains more
elusive [10, 11, 12]. One reason for this is that the supersymmetric higher-derivative
terms were not known until quite recently [13]. Even with this supersymmetric higher-
derivative action, it has been quite difficult to construct the black ring solution embed-
ded in the asymptotically flat spacetime, and it is preferable if we can only study the
near horizon geometry. Then the problem is to find the charges carried by the black
ring from its data at the near-horizon region.
The usual approach taken in the literature so far is to consider the dimensional
reduction along a circle down to four dimensions, and to study the charges there [12,
14, 15, 16]. Then, the attractor mechanism fixes the scalar vacuum expectation values
(vevs) and the metric at the horizon by the electric and magnetic charges [17, 18].
Conversely, the magnetic charge can be measured by the flux, and the electric charge
can be found by taking the variation of the Lagrangian by the gauge potential. In
this way, the entropy as a function of charges can be obtained from the analysis of
the near-horizon region alone [5, 6]. Nevertheless, it has not been clarified how to
reconcile the competing proposals [8, 9, 19, 20, 21] of the mapping between the four-
and five-dimensional charges of the black rings embedded in the asymptotically flat
spacetime.
Thus we believe it worthwhile to revisit the identification of the charges directly
in five dimensions, with local five-dimensional Lorentz symmetry intact. It poses two
related problems because of the presence of the Chern-Simons interaction in the La-
grangian. One is that, in the presence of the Chern-Simons interaction, the equation
of motion of the gauge field is given by
d ⋆ F = F ∧ F, (1.1)
which means that the topological density of the gauge field itself becomes the source of
electric charge. To put it differently, the attractor mechanism for the black rings [22]
determines the scalar vevs at the near-horizon region via the magnetic dipole charges
only, and the information about the electric charges seems to be lost. Then the electric
charge of a black ring seems to be diffusely distributed throughout the spacetime.
Eq.(1.1) can be rewritten in the form
d(⋆F −A ∧ F ) = 0, (1.2)
– 2 –
(∗F − A ∧ F ) is independent of Σ. This integral is called the Page charge.
Similar analysis can be done for angular momenta, and Suryanarayana and Wapler [23]
obtained a nice formula for them using the Noether charge of Wald.
There is a second problem remaining for black rings, which stems from the fact
that A is not a well-defined one-form there because of the presence of the magnetic
dipole. It makes
(⋆F − A ∧ F ) ill-defined, because in the integral all the forms are
to be well-defined. The same can be said for the angular momenta. The aim of this
paper is then to show how this second problem can be overcome, and to see how the
near-horizon region of a black ring encodes its charges measured at the asymptotic
infinity.
In Section 2, we use elementary methods to convert the integral at the asymptotic
infinity to the one at the horizon. We apply our formalism to the supersymmetric black
ring and check that it correctly reproduces known values for the conserved charges. We
will show how the gauge non-invariance of
A∧F can be solved by using two coordinate
patches and a compensating term along the boundary of the patches. Then in Section
3 we will see that our viewpoint helps in identifying the relation of the charges under
the Kaluza-Klein reduction along S1. We will see that the change in the charges under
a large gauge transformation in five dimensions maps to the Witten effect on dyons [24]
in four dimensions. Proposals in the literature [8, 9, 19, 20, 21] will be found equivalent
under the transformation. We conclude with a summary in Section 4. In Appendix A
the geometry of the concentric rings is briefly reviewed.
2. Near-Horizon Data and Conserved Charges
To emphasize essential physical ideas, we discuss the problem first for the minimal
supergravity in five dimensions. Later in this section we will apply the technique to
the case with vector multiplets. The bosonic part of the Lagrangian of the minimal
supergravity theory is
⋆ R − F ∧ ⋆F − 4
A ∧ F ∧ F
. (2.1)
Our metric is mostly plus, and Rµν is defined to be positive for spheres. We define the
Hodge star operator for an n-form as
⋆ (dxµ0 ∧ · · · ∧ dxµn−1) =
(5− n)!ǫ
µ0···µn−1
µn···µ4dx
µn ∧ · · · ∧ dxµ4 . (2.2)
– 3 –
with the Levi-Civita symbol ǫ01234 = +1 and ǫ
01234 = −1 defined in local Lorentz
coordinates. The equations of motion are
Rµν = −
gµνFρσF
ρσ + 2FµρFν
ρ, (2.3)
d ⋆ F = − 2√
F ∧ F. (2.4)
2.1 Electric charges
From the equation of motion of the gauge field (2.4), we see that F ∧ F is the electric
current for the charge
⋆F . Thus, the charge is distributed diffusely in the spacetime
as was emphasized e.g. in [25]. However, the equation (2.4) can also be cast in the form
A ∧ F
= 0. (2.5)
At the asymptotic infinity, A ∧ F decays sufficiently rapidly, so that we have
A ∧ F
A ∧ F
, (2.6)
where the subscript ∞ indicates that the integral is taken at S3 at the asymptotic
infinity, and Σ is an arbitrary three-cycle surrounding the black object. Thus we can
think of the electric charge as the integral of the quantity inside the bracket, which is
called the Page charge.
One problem about the Page charge is that, even in the case where A is a glob-
ally defined one-form, it changes its value under a large gauge transformation. It is
completely analogous to the fact that
A for an uncontractible circle C is only de-
fined up to an integral multiple of 2π under a large gauge transformation. Indeed,
let us parametrize C by 0 ≤ θ ≤ 2π and we perform a gauge transformation by
g(θ) ∈ U(1), i.e. we change A to A + i−1g−1dg. Such a continuous g(θ) can be writ-
ten as g(θ) = exp(iφ(θ)). Then
A changes by
dφ(θ) = φ(2π) − φ(0), which can
jump by a multiple of 2π. Thus,
A is invariant under a small gauge transformation
φ(0) = φ(2π) but is not under a large gauge transformation φ(0) 6= φ(2π). Exactly the
same analysis can be done for
A ∧ F , and it changes under a large gauge transfor-
mation along C if Σ contains intersecting one-cycle C and two-cycle S and
F 6= 0.
However, this non-invariance under a large gauge transformation poses no problem if
Σ is at the asymptotic infinity of the flat space, because we usually demand that A
should decay sufficiently rapidly there, which removes the freedom to do a large gauge
transformation.
– 4 –
These facts are well-known, and have been utilized previously e.g. in [14]. It is
the manifestation of the fact that there are several notions of electric charges in the
presence of Chern-Simons interactions, as clearly discussed by Marolf in Ref. [26]. One
is the Maxwell charge which is gauge-invariant but not conserved, and another is the
charge which is conserved but not gauge-invariant. In our case
⋆F is the Maxwell
charge and
(⋆F + (2/
3)A∧F ) is the Page charge. Yet another notion of the charge
is the quantity which generates the symmetry in the Hamiltonian framework, which
can be constructed using Noether’s theorem and its generalization by the work of Wald
and collaborators [27, 28, 29, 30]. The charge thus constructed is called the Noether
charge, and in our case it agrees with the Page charge.
Unfortunately, the manipulation above cannot be directly applied to the black
rings with dipole charges. It is because A is not a globally well-defined one-form, and
the integrals are not even well-defined. The way out is to generalize the definition of
A ∧ F to the case A is a U(1) gauge field defined using two coordinate patches, so
F ∧ F = “
A ∧ F ” (2.7)
holds. Then the manipulation (2.6) makes sense. The essential idea is to introduce a
term localized in the boundary of the patches which compensates the gauge variation.
Copsey and Horowitz [31] used similar subtlety associated to the gauge transformation
between patches to study how the magnetic dipole enters in the first law of the black
rings.
Figure 1: Coordinate patches used to define
A ∧ F consistently without ambiguity.
Let us assume the whole spacetime is covered by two coordinate patches, S and
T , see Figure 1. We denote the boundary of two regions by D = ∂S = −∂T . The
gauge field A is represented as well-defined one-forms AS and AT on the patches S and
T , respectively. These two are related by a gauge transformation, AS = AT + β with
dβ = 0 on the boundary D. Suppose the region B has the boundary C = ∂B. Then
– 5 –
we have
F ∧ F =
F ∧ F +
F ∧ F (2.8)
C∩S+D∩B
AS ∧ F +
C∩T−D∩B
AT ∧ F (2.9)
AS ∧ F +
AT ∧ F ) +
(AS ∧ F − AT ∧ F ) (2.10)
AS ∧ F +
AT ∧ F ) +
AS ∧ β. (2.11)
Now we define the symbol
A ∧ F for a three-cycle M to mean
A ∧ F ” ≡
AS ∧ F +
AT ∧ F +
AS ∧ β, (2.12)
then the relation (2.7) holds as is. The important point here is that we need a term
D∩M AS ∧ β which compensates the gauge variation localized at the boundary of the
coordinate patches.
One immediate concern might be the gauge invariance of the definition (2.12),
but it is guaranteed for C = ∂B from the very fact the relation (2.7) holds. It is
because its left hand side is obviously gauge invariant. For illustration, consider the
case ∂B = C1 − C2. The Page charges measured at C1, C2 themselves are affected by
a large gauge transformation, but their difference is not. When one takes C1 as the
asymptotic infinity, it is conventional to set the gauge potential to be zero there, thus
fixing the gauge freedom. Then the Page charge at the cycle C2 is defined without
ambiguity.
In the following, we drop the quotation marks around the generalized integral
A ∧ F ”. We believe it does not cause any confusion.
2.2 Angular momenta
The technique similar to the one we used for electric charges can be applied to the
angular momenta, and we can obtain a formula which expresses them by the inte-
gral at the horizon. There is a general formalism, developed by Lee and Wald [27],
which constructs the appropriate integrand from a given arbitrary generally-covariant
Lagrangian, and the expression for the angular momenta was obtained in [23, 32]. In-
stead, here we will construct a suitable quantity in a more down-to-earth and direct
method. We will see that the integrand contains the gauge field A without the exterior
derivative, and that it is ill-defined in the presence of magnetic dipole. We will use the
technique developed in the last section to make it well-defined.
– 6 –
Firstly, the angular momentum corresponding to the axial Killing vector ξ can be
measured at the asymptotic infinity by Komar’s formula
Jξ = −
⋆∇ξ, (2.13)
where ∇ξ is an abbreviation for the two-form ∇µξνdxµ ∧ dxν = dξ. Using the Killing
identity, the divergence of the integrand is given by
d ⋆∇ξ = 2 ⋆ Rµνξµdxν , (2.14)
which vanishes in the pure gravity. Thus, the angular momentum of a black object of
the pure gravity theory can be measured by
⋆∇ξ for any surface S which surrounds
the object.
Let us analyze our case, where the equations of motion are given by (2.3) and (2.4).
We need to introduce some notations: £ξ denotes the Lie derivative along the vector
field ξ, ιξω denotes the interior product of a vector ξ to a differential form ω, i.e. the
contraction of the index of ξ to the first index of ω. Then £ξ = dιξ + ιξd when it acts
on the forms. For a vector ξ and a one-form A, we abbreviate ιξA as (ξ · A).
We will take the gauge where gauge potentials are invariant under the axial isometry
£ξA = 0. It can be achieved by averaging over the orbit of the isometry ξ. We
furthermore assume that every chain or cycle we use is invariant under the isometry ξ,
then any term of the form ιξ(· · · ) vanishes upon integration on such a chain or cycle.
Under these assumptions, the difference of the integral of ⋆∇ξ at the asymptotic
infinity and at C is evaluated with the help of the Einstein equation (2.3) to be
⋆∇ξ −
⋆∇ξ = 2
⋆Rµνξ
µdxν = 4
(ιξF ) ∧ ⋆F (2.15)
where B is a hypersurface connecting the asymptotic infinity and C. We dropped the
ιξ(⋆F
2) because it vanishes upon integration.
The right hand side can be partially-integrated using the following relations: one
d [⋆(ξ · A)F ] = −(ιξF ) ∧ ⋆F − (ξ · A)
F ∧ F (2.16)
and another is
d [(ξ · A)A ∧ F ] = (ξ · A)F ∧ F − (ιξF ) ∧ A ∧ F (2.17)
(ξ · A)F ∧ F − 1
ιξ(A ∧ F ∧ F ) (2.18)
– 7 –
of which the last term vanishes upon integration. Thus we have
dXξ[A] = −(ιξF ) ∧ ⋆F (2.19)
modulo the term of the form ιξ(· · · ), where
Xξ[A] ≡ ⋆(ξ · A)F +
(ξ ·A)A ∧ F. (2.20)
Xξ[A] is not a globally well-defined form. Thus, to perform the partial integration of
the right hand side of (2.19), compensating terms along the boundary of the coordinate
patches need to be introduced, just as we did in the previous section in the analysis of
the Page charge.
Let S and T be two coordinate patches, D = ∂S = −∂T be their common bound-
ary, and AS = AT + β as before. Let us call the correction term Yξ[β,AS] and we
define
Xξ[A] ≡
Xξ[AS] +
Xξ[AT ] +
Yξ[β,AT ]. (2.21)
We demand that it satisfies
Xξ[A] =
(ιξF ) ∧ ⋆F. (2.22)
Then Y [β,A] should solve
dYξ[β,AT ] = Xξ[AS]−Xξ[AT ]. (2.23)
The right hand side is automatically closed since dXξ[A] is gauge invariant. Thus the
equation above should have a solution if there is no cohomological obstruction. Indeed,
substituting (2.20) in the above equation, we get
Yξ[β,AT ] = (ξ · β)Z −
2(ξ · β)β ∧AT + (ξ · AT )β ∧ AT
(2.24)
modulo ιξ(· · · ), where dZ should satisfy
dZ = ⋆F +
AT ∧ F, (2.25)
the right hand side of which is closed using the equation of motion (2.4). Unfortunately
there seems to be no general way to write Z as a functional of A and β. We need to
choose Z by hand for each on-shell configuration. With these preparation, we can
finally integrate the right hand side of (2.15) partially and conclude that
(⋆∇ξ + 4Xξ[A]) . (2.26)
– 8 –
is independent under continuous deformation of C.
Taking C to be the 3-sphere at the asymptotic infinity, the terms X [A] vanish
too fast to contribute to the integral. Then, the integral above is proportional to the
Komar integral at the asymptotic infinity. Thus we arrive at the formula
Jξ = −
⋆∇ξ + 4 ⋆ (ξ ·A)F + 16
(ξ · A)A ∧ F
, (2.27)
where Σ is any surface enclosing the black object. The right hand side is precisely the
Noether charge of Wald as constructed in [23, 32].
The contribution
⋆∇ξ to the angular momentum is gauge invariant but is not
conserved. It is expected, since the matter energy-momentum tensor carries the angular
momentum. The rest of the terms in (2.27) was obtained by the partial integral of the
contribution from the matter energy-momentum tensor, and can also be obtained by
constructing the Noether charge. The price we paid is that it is now not invariant
under a gauge transformation.
2.3 Example 1: the black ring
Let us check our formulae against known examples. First we consider the celebrated
supersymmetric black ring in five dimensions [7].
2.3.1 Geometry
It has been known [33] that any supersymmetric solution of the minimal supergravity
in the asymptotically flat R1,4 can be written in the form
ds2 = −f 2(dt+ ω)2 + f−1ds2(R4) (2.28)
where f and ω is a function and a one-form on R4, respectively. For the supersymmetric
black ring [7], we use a coordinate system adopted for a ring of radius R in the R4 given
ds2(R4) =
(x− y)2
1− x2 + (1− x
2)dφ21 +
y2 − 1 + (y
2 − 1)dφ22
(2.29)
with the ranges −1 ≤ x ≤ 1, −∞ < y ≤ −1 and 0 ≤ φ1,2 < 2π.1 φ1, φ2 were denoted
by φ, ψ in Ref. [7].
1We fix the orientations so that
dx∧ dφ1 ∧ dφ2 > 0 and
dx∧ dφ1 < 0 for S2 surrounding the
ring.
– 9 –
The solution for the single black ring is parametrized by the radius R in the base
4 above, and two extra parameter q and Q. More details can be found in Appendix A.
q controls the magnetic dipole through S2 surrounding the ring,
q. (2.30)
Conserved charges measured at the asymptotic infinity are as follows:
Q, (2.31)
J1 = −
⋆∇ξ1 =
q(3Q− q2), (2.32)
J2 = −
⋆∇ξ2 =
q(6R2 + 3Q− q2) (2.33)
where ξ1, ξ2 are the vector fields ∂φ1 , ∂φ2 respectively.
There is a magnetic flux through S2 surrounding the ring, so we need to introduce
two patches S, T . We choose S to cover the region x < 1−ǫ and T to cover 1−ǫ < x < 1,
with infinitesimal ǫ. The boundary D is at x = ǫ and parametrized by 0 ≤ φ1, φ2 < 2π.
We choose the gauge transformation between the two patches to be
AT = AS +
qdφ1 (2.34)
which is chosen to make AT smooth at the origin of R
The horizon is located at y → −∞ and has the topology S1 × S2. The gauge
potential near the horizon is
AS = −
q(x+ 1)dχ, (2.35)
while the geometry near the horizon is given as
ds2 = 2dvdr +
rdvdψ + ℓ2dψ2 +
(dθ2 + sin2 θdχ2) (2.36)
where r = r(y) is chosen so that r → 0 corresponds to y → −∞, v is a combination of
t and y, x = cos θ, ψ = φ2 + C1/r + C0 for suitably chosen C0,1, χ = φ1 − φ2, and
ℓ2 = 3
(Q− q2)2
. (2.37)
It is a direct product of an extremal Bañados-Teitelboim-Zanelli (BTZ) black hole with
horizon length 2πℓ and curvature radius q and of a round two-sphere with radius q/2.
– 10 –
ℓ is a more physical quantity characterizing the ring than R is, so it is preferable
to express J2, (2.33), using ℓ in the form
−2ℓ2 + 3Q
. (2.38)
Our objective is to reproduce the conserved charges, (2.31), (2.32) and (2.38), purely
from the near-horizon data, (2.35) and (2.36).
2.3.2 Electric charge
We use the formula (2.6) to get the electric charge. Using the form of the gauge field
near the horizon (2.34) and (2.35), we obtain
A ∧ F
AS ∧ F +
AS ∧ β
Q+ q2
Q− q2
Q, (2.39)
which correctly reproduces the charge measured at the asymptotic infinity. Vanishing
⋆F at the horizon means that all the Maxwell charge of the system is carried
outside of the horizon in the form of
F ∧F , while all of the Page charge is still inside
the horizon.
One important fact behind the gauge invariance of the calculation above is that the
integral
AS along the ψ
′ direction is not just defined mod integer, but is well-defined
as a real number. It is because the circle along ψ, which is not contractible in the
near-horizon region, becomes contractible in the full geometry.
2.3.3 Angular momenta
The integral of the right hand side of (2.25) can be made arbitrarily small by choosing
very small ǫ, so that we can forget the complication coming from the choice of Z. Then
for ξ1 = ∂φ1 = ∂χ, we have
−1<x<1−ǫ
(ξ · AS)AS ∧ F +
x=1−ǫ
(ξ · β)β ∧ AT
(2π)2
(q3 + qQ) +
(−q3 + qQ)
q(3Q− q2), (2.40)
– 11 –
reproducing (2.32).
For ξψ = ∂ψ = ∂φ1 + ∂φ2 , we have a contribution from
⋆∇ξψ = 4π2qℓ2. Adding
contribution from X [A], we obtain
−2qℓ2 − q
+ 3qQ+
(2.41)
which matches with J1 + J2, see (2.32) and (2.38).
The second and the third terms in the expression above are obtained by the partial
integration of the contribution from the angular part of the energy-momentum tensor
of the gauge field. In this sense, a part of the angular momentum is carried outside of
the horizon and the part proportional to ℓ2 is carried inside the horizon. However, the
Noether charge of the black ring resides purely inside of the horizon.
2.4 Example 2: concentric black rings
The concentric black-ring solution constructed in Ref. [34] is a superposition of the
single black ring we discussed in the last subsection. We focus on the case where all the
rings lie on a plane in the base R4. For the superposition of N rings, the full geometry
is parametrized by 3N parameters qi, Qi and Ri, (i = 1, . . . , N). qi is the dipole charge
and Ri is the radius in the base R
4 of the i-th ring. For more details, see Appendix A.
We order the rings so that R1 < R2 < · · · < RN . The conserved charges measured at
infinity are known to be
Qi − q2i
, (2.42)
2s3 + 3s
(Qj − q2j )
, (2.43)
2s3 + 3s
(Qj − q2j ) + 6
(2.44)
where s is an abbreviation for the sum of the magnetic charges, i.e. s =
i=1 qi. Our
aim is to reproduce these results from the near-horizon data.
The near-horizon metric of i-th ring has the form (2.36) with q, Q, R replaced with
qi, Qi and Ri, respectively. The horizon radius ℓi is given by
ℓ2i = 3
(Qi − q2i )2
− R2i
. (2.45)
Since each ring has a magnetic dipole charge, we introduce coordinate patches S
and Ti so that the gauge field is non-singular in each patch. Let Ti be the patch covering
– 12 –
the region between (i− 1)-th and i-th ring and S be a patch covering the outer region.
More precisely, we introduce the ring coordinate (2.29) for each of the ring, and choose
S to cover −1 + ǫ < xi < 1 − ǫ for each ring while Ti to cover 1 − ǫ < xi < 1 for the
i-th ring and −1 < xi−1 < −1 + ǫ for the (i − 1)-th ring. Then, near the i-th horizon
the gauge field on S is given by
AS = −
− qi + 2s
qi(1 + x) + 2
j=i+1
. (2.46)
Its ψ component is determined in Appendix A, while the coefficient for dχ is determined
so that the field strength is reproduced, the gauge field is non-singular except for x = ±1
for the 1st to (N − 1)-th rings and non-singular except for x = −1 for the N -th ring.
The gauge field on Ti is given by
ATi = AS +
qjdφ1. (2.47)
The electric charge is given by using (2.6) and βi = AS −ATi = −
j=i qjdφ1 as
AS ∧ F +
Σi∩∂S
AS ∧ βi +
Σi−1∩∂S
AS ∧ βi
Qi − q2i
+ 2sqi
Qi − q2i
Qi − q2i
. (2.48)
This correctly reproduces the known result (2.42).
Let us move onto the evaluation of the angular momenta. Note that for certain
configurations of charges, the concentric black rings develop singularities on the rotation
axes. While the condition for the absence of singularities has not been known fully, it
was pointed out in Ref. [34] that there is no singularity on the rotation axes if all
Qi − q2i
(2.49)
are equal. We will show that we can obtain the correct angular momenta if this condi-
tion is satisfied.
– 13 –
The angular momentum associated with ξ1 = ∂φ1 = ∂χ is given by
J1 = −
(ξ1 · AS)AS ∧ F
Σi∩∂Ti
Σi−1∩∂Ti
(ξ1 · βi)βi ∧ATi . (2.50)
After summing up terms, we have
2s3 + 6
(Qi − q2i )
j=i+1
qj + 3
(qi(Qi − q2i ))
. (2.51)
If the condition (2.49) is satisfied, J1 computed above matches (2.43) and we have
2s3 + 3Λis
. (2.52)
Finally, let us consider the angular momentum associated with ξψ = ∂ψ = ∂φ1+∂φ2 .
In addition to (2.50) with ξ1 being replaced by ξψ, here we have to consider two more
contributions. Namely,
⋆∇ξψ −
Σi∩∂Ti
Σi−1∩∂Ti
(ξψ ·ATi)βi ∧ATi . (2.53)
It is easy to check that the sum of each term is given by
i + 4s
3 + 6s
(Qi − q2i )
. (2.54)
When evaluated under the condition (2.49), this gives
i + 4s
3 + 6Λis
(2.55)
and agrees with Jψ given as the sum of (2.43) and (2.44).
2.5 Generalization
It is straightforward to generalize the techniques we developed so far to the supergravity
theory with n of U(1) vector fields AI , (I = 1, . . . , n). There are (n−1) vector multiplets
because the gravity multiplet also contains the graviphoton field which is a vector field.
– 14 –
The scalars in the vector multiplet are denoted by M I , which are constrained by the
condition
N ≡ cIJKM IMJMK = 1. (2.56)
cIJK is a set of constants. The action for the boson fields is given by
⋆R− aIJdM I ∧ ⋆dMJ − aIJF I ∧ ⋆F J − cIJKAI ∧ F J ∧ FK
(2.57)
where R is the Ricci scalar, and
aIJ = −
(NIJ −NINJ) . (2.58)
In the last expression, NI = ∂N /∂M I and NIJ = ∂2N /∂M I∂MJ . This is the low-
energy action of M-theory compactified on a Calabi-Yau manifoldM with n = h1,1(M),
6cIJK =
ωI ∧ ωJ ∧ ωK (2.59)
is the triple intersection of integrally-quantized two-forms ωI on M . The action for the
minimal supergavity (2.1) is obtained by setting n = 1, c111 = (2/
3)3, and a11 = 2.
As for the calculation of the electric charges, one only needs to put the indices
I, J,K to the vector fields and the result is
⋆aIJF
cIJKA
J ∧ FK
. (2.60)
As for the angular momenta, there is extra terms coming from the energy-momentum
tensor of the scalar fields in the right hand side of (2.15). Its contribution to the angular
momenta vanishes upon integration, so that the result is
Jξ = −
⋆∇ξ + 2 ⋆ aIJ(ξ · AI)F J + 2cIJK(ξ ·AI)AJ ∧ FK
. (2.61)
For a more complicated Lagrangian, e.g. with charged hypermultiplets and/or with
higher-derivative corrections, it is easier to utilize the general framework set up by
Wald, than to find the partial integral in (2.15) by inspection. The charge constructed
by this technique has an important property [27] that it acts as the Hamiltonian for
the corresponding local symmetry in the Hamiltonian formulation of the theory, and it
reproduces the Page charge and the angular momenta (2.61). Consequently, the charge
as the generator of the symmetry is not the gauge-invariant Maxwell charge, but the
Page charge which depends on a large gauge transformation.
The integrands in the expressions above are not well-defined as differential forms
when there are magnetic fluxes, thus it needs to be defined appropriately as we did
– 15 –
in the previous sections. Generically, we would like to rewrite the integral of a gauge
invariant form ω on a region B to the integral of ω(1) satisfying
dω(1) = ω (2.62)
on its boundary ∂B. The problem is that ω(1) may depend on the gauge. On two
patches S and T , it is represented by differential forms ωS(1) and ω
(1) respectively. Since
ω is gauge-invariant, we have dωS
= dωT
. Thus, if we take a sufficiently small
coordinate patch, we can choose ω
(S,T )
such that
(S,T )
= ωS(1) − ωT(1). (2.63)
Then one defines the integral of ω(1) on C = ∂B via
ω(1) ≡
ωS(1) +
ωT(1) +
, (2.64)
where D = ∂S = −∂T . The equations (2.62), (2.63) are the so-called descent relation
which is important in the understanding of the anomaly. It will be interesting to
generalize our analysis to the case where there are more than two patches and multiple
overlaps among them. Presumably we need to include higher descendants ω
(S1,...,Sn)
the correction term at the boundary of n patches S1, . . . , Sn in the definition of the
integral (2.64).
3. Relation to Four-Dimensional Charges
We have seen how the near-horizon data of the black rings encode the charges measured
at the asymptotic infinity. We can also consider rings in the Taub-NUT space [19, 20,
21] instead in the five-dimensional Minkowski space. Then the theory can also be
thought of as a theory in four dimensions, via the Kaluza-Klein reduction along S1
of the Taub-NUT space. It has been established [35] that supersymmetric solutions
for five dimensional supergravity nicely reduces to supersymmetric solutions for the
corresponding four dimensional theory.
In four dimensions, there are no problems in defining the charges, because the
equations of motion and Bianchi identities yield the relations
dF I = 0, dGI = d(⋆(g
IJ )F
J + θIJF
J) = 0 (3.1)
where (g−2)IJ are the inverse coupling constants and θIJ are the theta angles. The
electric and magnetic charges can be readily obtained by integrating GI and F
I over
– 16 –
the horizon. Then it is natural to expect that our formulae for the charges will yield
the four-dimensional ones after the Kaluza-Klein reduction. One apparent problem is
that the Page charges changes under a large gauge transformation, whereas the four-
dimensional charges are seemingly well-defined as is. We will see that a large gauge
transformation corresponds to the Witten effect on dyons in four-dimensions.
3.1 Mapping of the fields
First let us recall the well-known mapping of the fields in four and five dimensions.
The details can be found e.g. in [11, 12, 15, 16]. When we reduce a five-dimensional
N = 2 supergravity with n vector fields along S1, it results in a four-dimensional N = 2
supergravity with (n+1) vector fields. The metrics in respective dimensions are related
ds25d = e
2ρ(dψ − A0)2 + e−ρds24d, (3.2)
where we take the periodicity of ψ to be 2π so that eρ is the five-dimensional radius of
the Kaluza-Klein circle. The factor in front of the four-dimensional metric is so chosen
that the four-dimensional Einstein-Hilbert term is canonical.
The gauge fields in four and five dimensions are related by
AI5d = a
I(dψ − A0) + AI4d (3.3)
where I = 1, . . . , n. It is chosen so that a gauge transformation of A0 do not affect AI4d.
We need to introduce coordinate patches when there is a flux for AI5d. We demand that
gauge transformations used between patches should not depend on ψ so that aI are
globally well-defined scalar fields.
Then, by the reduction of the five-dimensional action (2.57), the action of four-
dimensional gauge fields is determined to be 2
L = −
e3ρ + eρaIJa
F 0 ∧ ⋆F 0 − cIJKaIaJaKF 0 ∧ F 0
+ 2eρaIJa
IF 0 ∧ ⋆F J + 3cIJKaIaJF 0 ∧ FK
− eρaIJF I ∧ ⋆F J − 3cIJKaIF J ∧ FK . (3.4)
Partial integrations are necessary to bring the naive Kaluza-Klein reduction to the form
above. The resulting Lagrangian above follows from the prepotential
F (X) =
cIJKX
IXJXK
, (3.5)
2We take the following conventions in four dimensions: The orientations in four and five dimensions
are related such that
dx0 ∧ dx1 ∧ dx2 ∧ dx3 ∧ dψ = 2π
dx0 ∧ dx1 ∧ dx2 ∧ dx3. The Levi-Civita
symbol in four dimensions is defined by ǫ0123 = +1 and ǫ
0123 = −1 in local Lorentz coordinates.
– 17 –
if one defines special coordinates zI = XI/X0 by
zI = aI + ieρM I . (3.6)
This relation can be checked without the detailed Kaluza-Klein reduction. Indeed,
the ratio of aI and M I in (3.6) can be fixed by inspecting the mass squared of a
hypermultiplet, and the fact aI should enter in zI linearly with unit coefficient is fixed
by the monodromy.
3.2 Mapping of the charges
In many references including Ref. [12, 16, 23], the charge of the black object in five di-
mensions is defined to be the charges in four dimensions after the dimensional reduction
determined from the Lagrangian (3.4). It was motivated partly because the analysis
of the charge in five dimensions was subtle due to the presence of the Chern-Simons
interaction, whereas we studied how we can obtain the formula for the charges which
has five-dimensional general covariance in Section 2. Now let us compare the charges
thus defined in four- and five- dimensions.
Firstly, the magnetic charge
F 0 (3.7)
in four dimensions counts the number of the Kaluza-Klein monopole inside C. It is
also called the nut charge. The other magnetic charges in four dimensions
F I (3.8)
come directly from the dipole charges in five dimensions, as long as the surface C does
not enclose the nut. When C does contain a nut, the Kaluza-Klein circle is non-trivially
fibered over C. Thus, the surface C cannot be lifted to five dimensions. We will come
back to this problem in Section 3.5.
The formulae for the electric charges follow from the Lagrangian :
⋆2eρaIJ(F
J − aJF 0) + 6cIJKaJFK − 3cIJKaJaKF 0
, (3.9)
⋆e3ρF 0 − ⋆2eρaIJaI(F J − aJF 0) + 2cIJKaIaJaKF 0 − 3cIJKaIaJFK
(3.10)
It is easy to verify that the five-dimensional Page charges (2.60) and the Noether
charge Jψ (2.61) for the isometry ∂ψ along the Kaluza-Klein circle are related to the
– 18 –
four-dimensional electric charges via
QI = −
QI , Q0 = −
Jψ. (3.11)
An important point in the calculation is that the compensating term on the boundary
of the coordinate patches vanishes, since aI and F J4d are globally well-defined.
Thus we see that the four-dimensional charges are not the reduction of the gauge-
invariant Maxwell charges
⋆F or that of the gauge-invariant “Maxwell-like” part of
the angular momentum,
⋆∇ξ. They are rather the reduction of the Page or the
Noether charges, which change under a large gauge transformation.
3.3 Reduction and the attractor
In the literature, the attractor equation is often analyzed after the reduction to four
dimensions [12, 15, 16], while the five-dimensional attractor mechanism for the black
rings in [22] only determines the scalar vacuum expectation values via the magnetic
dipoles. As we saw in the previous sections, the electric charges at the asymptotic
infinity are encoded by the Wilson lines along the horizon. We show that how these
five-dimensional consideration reproduces the known attractor solution [36, 37] in four-
dimensions.
The five-dimensional metric is characterized by the magnetic charges qI through
the horizon, and the physical radius of the horizon ℓ = eρ there. From the attractor
mechanism for the black rings [22], the near-horizon geometry is of the form AdS3×S2,
and the curvature radii are q and q/2 in each factor, where q3 = cIJKq
IqJqK . The scalar
vevs are fixed to be proportional to the magnetic dipoles, i.e. M I = qI/q.
For the calculation of electric charges the Wilson lines aI along the horizon are
also important. Then we can evaluate the Page charges and angular momenta on the
horizon to obtain
QI = 6cIJKa
JqK , Q0 = qℓ
2 − 3cIJKaIaJqK . (3.12)
We can solve the equations above for ℓ and aI so that we have the formula for the
four-dimensional special coordinates zI in terms of the charges. The result is
zI = aI + ieρM I =
DIJQI + i
qI (3.13)
where
DIJ = cIJKq
K , DIJDJK = δ
K (3.14)
– 19 –
D = q3 = cIJKq
IqJqK , Q̂0 = qℓ
2 = Q0 +
DIJQIQJ . (3.15)
It is the well-known solution of the attractor equation in four-dimensions with q0 = 0
[36, 37].
Thus, the combination of the attractor mechanism in five dimensions and the tech-
nique of Page charges yield the attractor mechanism in four dimensions. The point is
that the Wilson lines aI along the horizon of the black string carry the information
of its electric charges. Conversely, the Wilson line at the horizon is determined by
the electric charge. The horizon length is also determined by the angular momentum.
In this sense, the attractor mechanism for the black rings also fixes all the relevant
near-horizon data by means of the charges, angular momenta and dipoles.
3.4 Gauge dependence and monodromy
Let us now come back to the question of the variation of the Page charges under large
gauge transformations. The problem is that the integral
A∧F depends on the shift
A→ A+β for dβ = 0 if C has a non-contractible loop ℓ and
β 6= 0. In the spacetime
which asymptotes to R4,1, the large gauge transformation can be fixed by demanding
that the gauge potential vanishes at the asymptotic infinity.
In the present case of reduction to four dimensions, however, the gauge potential
along the Kaluza-Klein circle is one of the moduli and is not a thing to be fixed. More
precisely, if the ψ direction is non-contractible, a large gauge transformation associated
to the Kaluza-Klein circle corresponds to a shift aI → aI + tI where tI are integers. In
four-dimensional language it is the shift
zI → zI + tI , (3.16)
and the gauge variation of the Page charge translates to the variation of the electric
charge under the transformation (3.16). It is precisely the Witten effect on dyons [24] if
one recalls the fact that the dynamical theta angles of the theory depends on zI . In the
terminology of N = 2 supergravity and special geometry, it is called the monodromy
transformation associated to the shift (3.16), which acts symplectically on the charges
(qI , QI) and on the projective special coordinates (X
I , FI)
For the M-theory compactification on the product of S1 and a Calabi-Yau, electric
charges QI and q
I correspond to the number of M2-branes and M5-branes wrapping
two-cycles ΠI and four-cycles ΣI , respectively. The relation (2.59) translates to 6cIJK =
#(ΣI ∩ ΣJ ∩ ΣK) in this language. The gauge fields AI arise from the Kaluza-Klein
reduction of the M-theory three-form C on ΠI . Thus, the results above imply that
– 20 –
the M2-brane charges transform non-trivially in the presence of M5-branes under large
gauge transformations of the C-field.
It might sound novel, but it can be clearly seen from the point of view of Type IIA
string theory on the Calabi-Yau. Consider a soliton without D6-brane charge. There,
the D2-brane charge QI of the soliton is induced by the world-volume gauge field F on
the D4 brane wrapped on a four-cycle Σ = qIΣI through the Chern-Simons coupling
(F +B) ∧ C (3.17)
where B is the NSNS two-form and C is the RR three-form. In this description, aI is
given by
B. The induced brane charge in the presence of the non-zero B-field is
an intricate problem in itself, but the end result is that the large gauge transformation
B → B + ω with
ω = tI changes the D2-brane charge of the system by 6cIJKq
ItJ .
It will be interesting to derive the same effect from the worldvolume Lagrangian [38]
of the M5 brane, which is subtle because the worldvolume tensor field is self-dual.
The change in the M2-brane charge induce a change in the Kaluza-Klein momentum
carried by the zero-mode on the black strings wrapped on S1, so that Q0 also changes
[2]. The point is that the momentum carried by non-zero modes, Q̂0 defined in (3.15),
is a monodromy-invariant quantity.
Before leaving this section, it is worth noticing that if an M2-brane has the world-
volume V , it enters in the equation of motion for G = dC in the following way:
d ⋆ G+G ∧G = δV (3.18)
where δV is the delta function supported on V . Thus, the quantized M2-brane charge
is not the source of the Maxwell charge. It is rather the source of the Page charge.
Essentially the same argument in five dimensions, using the specific decomposition
(2.28), was made in Ref. [39].
3.5 Monodromy and Taub-NUT
If we use the Taub-NUT space in the dimensional reduction, in other words if there
is a Kaluza-Klein monopole in the system, the Kaluza-Klein circle shrinks at the nut
of the monopole. As the circle is now contractible, one might think that one can no
longer do a large gauge transformation and that it is natural to choose aI = 0 at the
nut. Nevertheless, from a four-dimensional standpoint the monodromy transformation
should be always possible. How can these two points of view be reconciled?
Firstly, the fact that the five-dimensional spacetime is smooth at the nut only
requires that the gauge field strength is zero there and that the integral of the gauge
– 21 –
potential is an integer. There should be a patch around the nut in the five-dimensional
spacetime in which AI should be smooth, but it is not the patch connected to the
asymptotic region of the Taub-NUT space where aI is defined.
A similar problem was studied in Ref. [40]. There, it was shown how the winding
number can still be conserved in the background with the nut, where the circle on
which strings are wound degenerates. A crucial role is played by the normalizable self-
dual two-form ω localized at the nut, which gives the worldvolume gauge field A of the
D6-brane realized as the M-theory Kaluza-Klein monopole via C = A ∧ ω. It should
enter in the worldvolume Lagrangian in the combination dA+B, and the large gauge
transformation affects the contribution from B.
Indeed, the Kaluza-Klein ansatz of the gauge fields (3.3), one can make the com-
bined shift
aI → aI + tI , AI4d → AI4d + tIA0 (3.19)
without changing the five-dimensional gauge field strengths. Therefore, the magnetic
charge also transforms as
qI → qI + tIq0. (3.20)
The action of the transformation (3.16) on the electric charges then becomes
QI → QI + 6cIJKtJqK + 3cIJKtJ tKq0, (3.21)
Q0 → Q0 −QItI − 3cIJKtItJqK − cIJKtItJtKq0, (3.22)
which is exactly how the projective coordinates
X0, XI , FI = 3cIJKX
JXK/X0, F0 = −cIJKXIXJXK/(X0)2. (3.23)
get transformed by the monodromy aI → aI + tI . It was already noted in Ref. [21]
that the same symmetry acts on the functions which characterize the supersymmetric
solution on the Taub-NUT, (V,KI , LI ,M) in their notation. The point is that it
modifies the five-dimensional Page charges, and hence the four-dimensional charges.
If we neglect quantum corrections coming from instantons wrapping the Kaluza-
Klein circle, it is allowed to do the monodromy transformation zI → zI + tI even with
continuous parameters tI . It maps a solution of the equations of motion to another,
and the electric charges in four-dimensions depends continuously on the vevs for the
moduli aI at the asymptotic infinity. The issue concerning the stability of the solitons
can be safely ignored. In the analyses in Refs. [19, 20, 21], their proposals for the
identification of four-dimensional electric charges QI and of five-dimensional ones QI
were different from one another. The source of the discrepancy in the identification
is now clear after our long discussion. It can be readily checked that the differing
– 22 –
proposals for the identification can be connected by the monodromy transformation
with tI = 1
qI . Namely, the charges in the five-dimensional language are transformed
QI − 3cIJKqJqK , Jψ → Jψ − Jφ (3.24)
for Q0 ≫ q3 limit.3 Thus they are equivalent under a large gauge transformation.
The analysis above also answers the question raised in Section 3.2 how the dipole
charges in five dimensions are related in the magnetic charges in four dimensions in the
presence of the nut. It is instructive to consider the case of a black ring in the Taub-
NUT space. From a five-dimensional viewpoint, the dipole charge is not a conserved
quantity measurable at the asymptotic infinity. Correspondingly, the surface of the
Dirac string necessary to define the gauge potential can be chosen to fill the disc inside
the black ring only, and not to extend to the asymptotic infinity. It was what we did
in Section 2.3.1 in defining the coordinate patches. However, the gauge transformation
required to achieve it necessarily depends on the ψ coordinate, which is the direction
along the Kaluza-Klein circle. Hence it is not allowed if one carries out the reduction to
four dimensions. In this case, the Dirac string emanating from the black ring necessarily
extends all the way to the spatial infinity, thus making the magnetic charge measurable
at the asymptotic infinity. A related point is that dipole charges enter in the first law of
black objects because of the existence of two patches [31]4. It is easier to understand it
after the reduction because now it is a conserved quantity measurable at the asymptotic
infinity.
As a final example to illustrate the subtlety in the identification of the four- and
five-dimensional charges, let us consider a two-centered Taub-NUT space with centers
x1 and x2. There is an S
2 between two centers, and one can introduce a self-dual mag-
netic fluxes qI through it. Although the Chern-Simons interactions put some constraint
on the allowed qI , there is a supersymmetric solution of this form [44]. In this configu-
ration, the Wilson lines aI at x1 and x2 necessarily differ by the amount proportional
to the flux, and one cannot simultaneously make them zero. An important consequence
is that the magnetic charges F I4d of the nuts at x2 and x2 necessarily differ, in spite
of the fact that the geometry and the gauge fields in five dimensions are completely
symmetric under the exchange of x1 and x2.
3We noticed that a small discrepancy proportional to cIJKq
IqJqK remains, which is related to
the zero-point energy of the conformal field theory of the black string. Its effect on the entropy is
subleading in the large Q0 limit.
4The authors of [31] used the approach to the first law developed in [41]. There is another un-
derstanding of appearance of the dipole charges in the first law [42] if one follows the approach in
[43].
– 23 –
4. Summary
In this paper, we have first clarified how the near-horizon data of black objects encode
the conserved charges measured at asymptotic infinity. Namely, the existence of the
Chern-Simons coupling means that F ∧ F is a source of electric charges, thus it was
necessary to perform the partial integration once to rewrite the asymptotic electric
charge by the integral of A ∧ F over the horizon. Since F has magnetic flux through
the horizon, A∧F cannot be naively defined, and we showed how to treat it consistently.
Likewise, we obtained the formula for the angular momenta using the near-horizon data.
Then, we saw how our formula for the charges in five dimensions is related to the
four-dimensional formula under Kaluza-Klein reduction. We studied how the ambiguity
coming from large gauge transformations in five dimensions corresponds to the Witten
effect and the associated monodromy transformation in four dimensions.
It is now straightforward to obtain the correction to the entropy of the black rings,
since we now have the supersymmetric higher-derivative action [13], the near-horizon
geometry [45, 46, 47], and also the formulation developed in this paper to obtain con-
served charges from the near-horizon data alone. It would be interesting to see if it
matches with the microscopic calculation.
Acknowledgments
YT would like to thank Juan Maldacena, Masaki Shigemori and Johannes Walcher for
discussions. KH is supported by the Center-of-Excellence (COE) Program “Nanometer-
Scale Quantum Physics” conducted by Graduate Course of Solid State Physics and
Graduate Course of Fundamental Physics at Tokyo Institute of Technology. The work
of KO is supported by Japan Society for the Promotion of Science (JSPS) under the
Post-doctoral Research Program. YT is supported by the United States DOE Grant
DE-FG02-90ER40542.
A. Geometry of Concentric Black Rings
Any supersymmetric solution in the asymptotically flat R1,4 is known to be of the form
ds2 = −f 2(dt+ ω)2 + f−1ds2(R4) (A.1)
where f and ω is a function and a one-form on R4, respectively. We parametrize the
base R4 in the Gibbons-Hawking coordinate system
ds2(R4) = H [dr2 + r2(dθ2 + sin2 θdχ2)] +H−1(2dψ + cos θdχ)2 (A.2)
– 24 –
where (r, θ, φ) parametrize a flat R3, the periodicity of ψ is 2π and H = 1/r. Our
notation mostly follows the one in Ref. [34], with the change ψthere = 2ψhere. The
quantities f , ω and the gauge field F = dA are determined by three functions K, L
and M on the flat R3. The relations we need are
f−1 = H−1K2 + L, ι∂ψω = 2H
−2K3 + 3H−1KL+ 2M, (A.3)
d[f(dt+ ω)]− 1√
G+, ι∂ψG
+ = −3d(H−1K) (A.4)
where G+ = f(dω + ⋆dω)/2 is a self-dual two-form on R4.
To construct the concentric black ring solutions, we take N points xi, (i = 1, . . . , N)
at r = R2i /4, θ = π on R
3. The orbit of xi along the coordinate ψ is a ring of radius Ri
embedded in R4. We choose functions K, L and M by
K = −1
qihi, L = 1 +
(Qi − q2i )hi, M =
qi(1− |xi|hi) (A.5)
where hi(x) = 1/|x−xi| are harmonic functions on R3. For the case with a single ring,
conversion to the ring coordinate used in (2.29) can be achieved via
φ1 = ψ + χ/2, φ2 = ψ − χ/2 (A.6)
y2 − 1
x− y = 2
r sin
1− x2
x− y = 2
r cos
. (A.7)
The behavior of ω and F at the asymptotic infinity, and the near-horizon metric
(2.36) are well-known and are not repeated here. The reader is referred to the orig-
inal article Ref. [34]. The gauge potential near the horizon can be obtained by the
combination of (A.3) and (A.4). First we have
ι∂ψF =
(−dι∂ψ)[f(dt+ ω)] +
3d(KH−1). (A.8)
which can be integrated by inspection. Hence the ψ component of the gauge field is
given by
ι∂ψA =
H−1KL/2 +M
H−1K2 + L
(A.9)
for some constant c. By demanding ιψA→ 0 as r → ∞, we obtain
c = −1
qi. (A.10)
– 25 –
Thus, we have
ι∂ψA = −
Qi − q2i
. (A.11)
near the i-th horizon. The χ component of the gauge field is fixed by the magnetic
dipole through the horizon.
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