#### Tunneling into microstate geometries: quantum effects stop gravitational collapse

Accepted: June
Tunneling into microstate geometries: quantum e ects stop gravitational collapse
Iosif Bena 0 1 3 6
Daniel R. Mayerson 0 1 3 4 5
Andrea Puhm 0 1 2 3
Bert Vercnocke 0 1 3 4
Gif-sur-Yvette 0 1 3
France 0 1 3
0 Santa Barbara , CA 93106 , U.S.A
1 University of Michigan , 450 Church Street, Ann Arbor, MI 48109-1020 , U.S.A
2 Department of Physics, University of California , USA
3 Science Park 904 , Postbus 94485, 1090 GL Amsterdam , The Netherlands
4 Institute for Theoretical Physics, University of Amsterdam
5 Department of Physics and Michigan Center for Theoretical Physics , USA
6 Institut de Physique Theorique, Universite Paris Saclay
Collapsing shells form horizons, and when the curvature is small classical general relativity is believed to describe this process arbitrarily well. On the other hand, quantum information theory based (fuzzball/ rewall) arguments suggest the existence of some structure at the black hole horizon. This structure can only form if classical general relativity stops being the correct description of the collapsing shell before it reaches the horizon size. We present strong evidence that classical general relativity can indeed break down prematurely, by explicitly computing the quantum tunneling amplitude of a collapsing shell of branes into smooth horizonless microstate geometries. We show that the amplitude for tunneling into microstate geometries with a large number of topologically non-trivial cycles is parametrically larger than e SBH , which indicates that the shell can tunnel into a horizonless con guration long before the horizon has any chance to form. We also use this technology to investigate the tunneling of M2 branes into LLM bubbling geometries.
Black Holes in String Theory; Black Holes
1 Introduction and summary
Motivation
Summary of the results
1.2.1
1.2.2
Black hole microstate geometries
LLM geometries
1.3
Organization of this paper
2
Quantum tunneling of particles and branes
Tunneling of particles
Bubble nucleation for extended objects with O(d) symmetry
Particle versus O(d) tunneling
3
Tunneling of branes into black hole microstates
1.1
1.2
2.1
2.2
2.3
3.1
3.2
3.3
3.3.1
3.3.2
3.3.3
3.4.1
3.4.2
4.1.1
4.1.2
4.3.1
4.3.2
Background solution
Probe branes
Tunneling process
4.1
Background solution
4.2
4.3
Probe branes
Bubble nucleation
M-theory
IIA theory
The O(2)-invariant action describing the tunneling of F1 strings in IIA 26
The O(3)-invariant action describing the tunneling of M2 branes
1
Supertube probes and smooth microstates
Non-scaling solutions
Scaling solutions
Concluding remarks
Comparison to Kraus-Mathur
4
Tunneling of branes in LLM geometries
4.4
Metastable states & mass-deformed M2 brane theories
A
N = 2 supergravity in four and
ve dimensions
A.1 Four dimensions
A.2 Five dimensions
A.1.1
N = 2 supergravity in 4 dimensions
A.1.2 Probe particles
A.1.3
BPS multicenter supergravity solutions
A.2.1 STU model in ve dimensions
{ i {
A.2.3 Probe supertubes
A.2.4
The energy of the metastable state
33
34
Introduction and summary
The argument that black holes must have a certain nontrivial structure at the horizon to
avoid violation of quantum mechanics [1, 2] (see also [3]) is surprising on several fronts.
The rst is that, wherever this structure comes from, it must have very peculiar properties:
since the horizon is a null surface, this structure cannot come from ordinary particles, which
would travel at the speed of light and have in nite mass; nor can it come from massless
particles, which dilute in a few horizon crossing times and must be constantly replenished.
The second is that this structure must be able to account for the Bekenstein-Hawking
entropy of the black hole, and thus must have nontrivial degrees of freedom to give rise to
this entropy. The third, and perhaps most surprising one, is that an infalling shell of dust
which initially undergoes gravitational collapse a la Oppenheimer-Snyder [4] must somehow
turn into this structure at the moment it crosses the Schwarzschild radius, regardless of
how low its curvature is and how adept one might hope classical general relativity to be
for describing its physics.
The rst two points are addressed by the black hole microstate or fuzzball solutions
that one constructs in string theory (see [5{11] for reviews). The purpose of this paper is to
address the last question and show how infalling matter can tunnel into smooth geometric
microstates that give rise to structure at the scale of the would-be horizon.
{ 1 {
In string theory, and in string theory only, one can build structure at the scale of the
horizon of black holes with a large horizon area [12{16] by solving the equations of motion
of the low-energy limit of a UV-complete theory. This structure has the desired features: it
is kept from collapsing by uxes wrapping topologically-nontrivial cycles, and for extremal
supersymmetric black holes there is evidence that it has enough degrees of freedom to
reproduce the growth with charges of the Bekenstein-Hawking entropy [17].
Note that if one tries to construct such structure-at-the-horizon in other gravity
theories, one is almost automatically guaranteed to fail. For example, in four-dimensional
HJEP07(216)3
(1.1)
(1.2)
gravitational theories there are \no solitons without horizon" theorems that guarantee
that no smooth horizonless solutions with black hole charges can be built [18{22]. Making
the solutions singular and adding extra matter to source the singularities does not work
either, as the matter is generically not sti enough to prevent its collapse and the
subsequent formation of a black hole. It is only by going to higher dimensions and exploiting
the extra
uxes that are required by string theory in order to have a consistent quantum
theory of gravity, that one can create such solitons [23{26].
However, despite the existence of such structures and their ability to carry the black
hole entropy, the question still remains of how to convince a very large collapsing shell of
dust, whose Schwarzschild radius can be of the size of the Galaxy and whose curvature is
much smaller than that on the surface of the Earth, to transform itself into this structure
around the moment it reaches its Schwarzschild radius.
Of course, as the size of the
shell becomes smaller and smaller the curvature grows and, near the Planck scale, one
expects quantum gravity e ects to take over and possibly transform the shell into a string
theory-structure of the kind discussed in [5{11]. But this will be too late: the horizon of
non-extremal black holes is in the causal past of the moment when the curvature of the shell
is large enough for general relativity to break down. Hence, unless one goes backwards in
time and destroys several million of years of past history, one cannot create any structure
at the horizon.
Only one proposal has so far been put forth to explain how a collapsing shell of
dust might transform itself when reaching the size of the would-be event horizon into
a horizonless con guration or a fuzzball. In [27] (see also [28]) Mathur considered a
shell of matter of mass M , collapsing into its (classical) Schwarzschild black hole horizon.
This shell can quantum tunnel into a given horizonless con guration with an estimated
tunneling amplitude
e Stunnel
with
Stunnel =
where dimensional analysis was used to obtain Stunnel
assumption that the curvature scale of the fuzzball is given by the black hole length scale
sets the proportionality constant
to be of order one. Although the tunneling rate (1.1) is
a very small number, if one assumes that the so-called fuzzball proposal is correct and the
entropy of the black hole comes entirely from horizonless con gurations, the total number
gR
M 2
SBH. The
of states the shell can tunnel into is very large:
N = eSBH :
{ 2 {
SBH ;
R p
The two exponentials in (1.1) and (1.2) play o against each other, and if
1 tunneling
into fuzzball states is fast and takes place before a horizon can form. In a subsequent
paper, [29] Kraus and Mathur argued that
should be equal to 1, by estimating that
the probability for a shell to tunnel into a fuzzball is the same same as that for a shell
of dust to be emitted by a Schwarzschild black hole, which is exactly the exponential of
the negative of the entropy di erence
exp(
SBH). By extrapolating this result one
may then argue that the probability of the the collapsing shell to tunnel into a particular
entropy-less microstate is
exp( SBH) and therefore a collapsing shell of dust would
tunnel into some fuzzball with probability one, rather than forming a horizon.
In this paper we are able to put calculational esh on the proposal of [27], by directly
computing the amplitude (1.1) for a collapsing shell of branes to tunnel into some of the
black hole microstate solutions that have been explicitly constructed in the past [12{14].
Our results con rm some of the expectations of the proposal of [27] and also show that
some of the assumptions in the analysis of [29] are not necessary for the validity of the
overall argument. We will discuss in detail in section 3.4.2 the di erences between our
method to compute tunneling of collapsing shells and the method used in [29].
1.2
Summary of the results
The microstate geometries whose tunneling we analyze are solutions with nontrival
topology, and charges dissolved in ux. However, the technology we develop to study brane- ux
tunneling is much more general, and can be easily adapted to other solutions where the
branes can tunnel into
uxes. This technology has been used in the past to study the
tunneling of metastable probes with anti-D3 charge [30] in the KS solution [31] and for
the tunneling of probe anti-M2 branes [32] in the CGLP solution [33]. Here we will
apply this technology for general solutions of string theory and M-theory where branes are
transitioned into
ux and topology.
The rate for the tunneling decay per unit volume is given by [34]:1
=V = Ae B=~ (1 + O(~)) :
(1.3)
The tunneling parameter B, which can be determined from the Euclidean action, is the
focus of our paper. We compute the exponent B for tunneling processes that create
nontrivial topology, by placing probe strings or branes in multi-center backgrounds. We
consider probes [35, 36] in microstate geometries of three-charge black holes [12{14] and also
to study the tunneling of the recently found probe metastable states [37] in the
Lin-LuninMaldacena (LLM) geometries [38] dual to the mass-deformed M2 brane theory [39]. In
their M-theory descriptions, these two classes of geometries share several similarities. For
instance, they both account for the entropy of the dual theory: either the
BekensteinHawking entropy of the black hole or the number of vacua of the M2 brane theory [40, 41].
However, they di er in one key feature which leads to a very di erent tunneling behavior.
For black hole microstates, the tunneling calculation reduces to a quantum-particle
tunneling problem which can be tackled analytically. For the LLM geometries, one has to consider
1We will be interested in the leading-order term in (1.3) and henceforth set ~ = 1.
{ 3 {
an O(d) invariant Euclidean tunneling action which can only be treated analytically after
an approximation.
1.2.1
Black hole microstate geometries
Starting from a collapsing shell of branes, we imagine forming smooth multi-center
geometries in a stepwise quantum tunneling process, where we form a new center, and hence
a new topologically-nontrivial cycle, at every step by tunneling an amount of the initial
branes into uxes. At each step, we treat the tunneling branes as probes, which allows us
to calculate their tunneling potential from the on-shell Euclidean (probe) action explicitly.
Using this stepwise process, we determine the number of topologically non-trivial cycles at
the end of the tunneling process. Note that since we are treating the branes as probes we
can only tunnel a small amount of them at every step. However, we can still end up with
a large number of bubbles by successively tunneling small amounts of branes.
We nd that the tunneling amplitude to a nal state with N centers scale as:
exp(
0N
SBH) ;
(1.4)
where
0 is a microstate-dependent number and the exponent
is positive for all the
solutions we have considered2 and the black hole entropy is SBH = 2 pQ3, with Q the
electric charge of the black hole.
The key information lies in the prefactor
0N
, from which we extract a
universal feature of the tunneling amplitude into multi-center solutions: its dependence on
the number of centers, N . Since the power of N appearing in the exponential is negative
( > 0), the result (1.4) implies that the tunneling amplitude is enhanced when the
number of centers is large. It is also important to emphasize that, even if we do our tunneling
calculation for bubbling solutions that have U(1)
U(1) isometry in
ve dimensions, the
tunneling rate will be the same for the black hole microstate solutions one constructs by
wiggling these solutions to form superstrata and other wiggly objects; this happens because
there is no potential barrier between various wiggly solutions.
This being said, one should also emphasize that we do not know yet whether the
microstate geometries that give rise to the full black hole entropy have very few bubbles
of very many. From the study of superstrata [11, 42] it may appear that the solutions
that can reproduce the growth with charges of the black hole entropy should have only
a few centers, although in [16] it was argued that the typical superstrata may look more
like multi-center solutions than double-centered ones. Similarly, from the study of quiver
quantum mechanics one can reproduce [43] the charge growth of the supersymmetric
fourdimensional four-charge black hole entropy from the pure-Higgs states [44] of three-center
con gurations; however, this entropy is a very small fraction of the black hole entropy3
and it is not clear whether solutions with more centers could carry larger fractions of the
black hole entropy.
Hence, the detailed aspects of the physics of a collapsing shell would be determined by
the interplay between the increased tunneling rates into multi-bubble solutions, and the
2For the non-scaling and scaling solutions we considered we found, respectively,
= 3=2 and
= 0:93.
3For the three-center scaling solution of [45], it represents 4% of the black hole entropy [46].
{ 4 {
possible larger number of tunneling end-points available in few-bubble solutions.
Nevertheless, from our calculations it is clear that the tunneling rate both into solutions with small
and large numbers of bubbles will be more than enough to ensure that the shell tunnels
before a horizon can form, irrespective of the details of which solutions carry more entropy.
Finally, we wish to address two points regarding the validity of the semi-classical
approach to calculating the tunneling amplitude into microstates, and in particular if
interference of semi-classical tunneling paths is an important phenomenon.
We will rst
address the possibility of tunneling into a given microstate over a more circuitous path
than we have considered (i.e. tunneling \through" other microstate geometries), and
secondly whether we expect interference from nearby paths in the phase space to interfere
signi cantly with each other. In both cases we will conclude that these e ects are expected
to be subleading with respect to the tunneling amplitude we have computed.
The amplitude (1.4) describes the direct tunneling of a shell of branes into one
individual microstate. However, to compute the total tunneling amplitude of the shell into
this state one also has to sum over two-step tunneling processes (in which the shell tunnels
to a di erent microstate and then tunnels from that microstate to the one we consider) as
well as three-step tunneling processes, etc. Furthermore, since there are N
intermediate steps, this two-step tunneling probability could be larger than the one-step
one, thus invalidating our semiclassical approximation.4 To argue that this does not
hapeSBH possible
pen we can estimate the multi-step contributions to the tunneling process. The one-step
2p
probability is given by (1.3). The tunneling probability for a particular two-step process
is 2 and because of the existence of N possible intermediate steps his should be naively
have di erent phases, and the resulting amplitude will be reduced by a factor of p
multiplied by a factor of N . However, the amplitudes coming from the intermediate steps
cause of destructive interference.5
Hence, the two-step tunneling amplitude is of order
N
be
N . Similarly, the three-step amplitude is of order 3 for a particular tunneling path,
but because there are Np
multiplied by a factor of
2 possible possible paths with quantum interference this will be
N 2. It is not hard to see therefore that the full tunneling
amplip
N + 2
N + : : :) and thus the semiclassical
tude will be given by a convergent sum
(1 +
approximation is valid.
One may also wonder whether quantum interference between nearby tunneling paths
may a ect the amplitude. Such interference plays an important role in certain quantum
mechanical systems where tunneling paths can be made to be arbitrarily close to one
another due to a continuous in nity in the (possible) phase space of paths.6 To study this
e ect in full detail would require knowledge of the entire phase space of black hole
microstates (or at least a local section thereof) which is still largely unknown. Moreover,
the phase space needs to be quantized [57, 75, 76] and
nite-dimensional (to give a
nite
number of microstates for the black hole), which implies that the phase space should have
neither continuous nor in nite dimension. Nearby tunneling trajectories should therefore
4We would like to thank Juan Maldacena for bringing this point to our attention.
5Remember that after a random walk of N steps the average displacement is of order pN .
6We thank the anonymous referee as well as P. Kraus, A. Kusenko, and J. Maldacena for raising this
point and discussions on this issue.
{ 5 {
have a minimum separation and we should be able to treat them independently in a rst
approximation. We thus expect (destructive) interference between nearby paths to be a
subleading e ect in the individual tunneling process. Note that the multi-step tunneling
process discussed in the previous paragraph does not rely on the phase space being
continuous or in nite and is thus di erent from the potential interference e ect between nearby
trajectories. Of course, it would be interesting to check the wavefunction overlap explicitly
once a complete knowledge of the quantized phase space is available.
1.2.2
LLM geometries
The action that describes the tunneling of branes into ux and topology has been computed
in the literature before [30, 32] to estimate by now well-known decay rates of metastable
states from a Brown-Teitelboim process [47]. Along similar lines, we estimate the lifetime
of metastable brane con gurations [37] in the LLM geometries in string and M-theory.
The key di erence to the black hole microstate geometries is that the probes that tunnel
in the LLM geometries extend along one or two non-compact spatial directions (as opposed
to the branes in the black hole microstates, which only wrap compact directions). This
implies that we will need to compute the O(3) or O(2) Euclidean actions [34] corresponding
to, respectively, tunneling of probe M2 branes in the M-theory geometry or of probe F1
strings in the reduced IIA geometry. We
nd that the amplitude for metastable LLM
con gurations to tunnel into their ground state scales as:
M2
exp(
M
6=q2) (11D) ;
F 1
exp(
IIA
4=jqj) (10D) ;
(1.5)
where
is the mass deformation parameter that sets the scale of the four-cycles in the
bubbling LLM solutions and q is the charge of the tunneling probe M2 branes/F1 strings.
The parameters
M and
IIA will be computed in section 4. The main di erence with
the black hole microstate tunneling events is that the exponent B does not scale linearly
with the charges, but is inversely proportional to qd, with d the number of non-compact
directions [34, 47]. The dependence on q and
di ers between the 10D and 11D description
since the O(2) bounce computes tunneling per unit length while the O(3) bounce computes
tunneling per unit area. For both descriptions, quantum tunneling is suppressed for small
charge jqj and large mass deformation .
1.3
Organization of this paper
In section 2 we review the basics of tunneling rates for particles and the O(d) generalization
to strings and branes; readers familiar with these (standard) methods can skip this section.
In section 3 we consider the tunneling of branes into multi-center microstate geometries of
the three-charge maximally-spinning black hole, and investigate the scaling of the tunneling
amplitude with the number of centers N . In section 4, we consider the O(2) and O(3)
tunneling rates of metastable states in LLM-type backgrounds in string theory and
Mtheory. Details of the computations of section 3 and 4 can be found in appendices A, B,
and C.
{ 6 {
HJEP07(216)3
In this section, we review the tunneling decay rate for a quantum particle tunneling through
a potential barrier as detailed in Coleman's seminal paper [34], and the generalization to
strings and branes. First, we discuss a charged particle in a curved background with
a position-dependent mass, which will be relevant for section 3. Then we review the
generalization to an O(d) symmetric Euclidean action that describes the nucleation of a
bubble of true vacuum mediating the decay of a metastable vacuum, relevant for section 4.
Finally, we make a comparison and emphasize the fundamental di erence between the
particle and O(d)-symmetric strings and branes.
2.1
Tunneling of particles
Calculating the tunneling rate of a quantum-mechanical particle in a D-dimensional target
space is a standard problem discussed in [48]. In the language of [34], the tunneling
parameter B can be determined from the on-shell Euclidean action:
1. The integration is over the trajectory of the path
that is a solution to the Euclidean equations of motion starting from an initial con guration
at ti and ending in a nal con guration at tf (a path of `least resistance'). Along this path,
the Euclidean Hamiltonian is a constant of motion that can be chosen to be zero. From
the general expressions for the Hamiltonian and the momentum conjugate to xk:
HE = pkx_ k
LE = 0 ;
pk =
we get LE = pkx_ k.
becomes
For a non-relativistic particle in a potential, that is described by the Euclidean
Lagrangian LE = (m=2)gk`x_ kx_ ` + V (x), one nds that pkx_ k = jpjjx_ j. The above integral (2.1)
that HE = 0 implies further that jpj = p
usual WKB approximation [34].
where the integral is taken over a path of least resistance and jpj
pgk`dxkdx`. In other words, instead of nding the entire trajectory by solving the
equapgk`pkp`; jdxj
tions of motion, we can simply integrate the norm of the Euclidean momentum jpj in
position space from the starting point ~xi to the endpoint ~xf of the tunneling process. Note
2mV and therefore we recover the result of the
Relativistic particle.
Consider now a (relativistic) particle with position-dependent
mass m(x) that couples to a gauge
eld A with charge q, in a curved D-dimensional
background. We assume that the metric and all elds have a timelike symmetry @t and
that the metric is static. The Euclidean action is:
Z
Z
{ 7 {
(2.1)
(2.2)
(2.3)
(2.4)
After tunneling from xi to xf the particle experiences classical decay.
where integration is over the trajectory of the particle (pulling back A appropriately).
Taking the time-like coordinate t as the a ne parameter along the path we get:
Z tf
t
i
We will exclusively consider paths along which Aix_ i = 0. Then pi is proportional to gij x_ j
and we again
nd that the tunneling amplitude is given by integrating the norm of the
momentum in position space:
Z ~xf
~xi
and the norm of the momentum can be found from solving HE = 0:
jp(x)j = jgtt(x)j 1=2pjgtt(x)jm(x)2
(qAt(x))2 :
As we explain below, we can use this expression to calculate the tunneling amplitudes in
black hole microstate geometries in section 3.
2.2
Bubble nucleation for extended objects with O(d) symmetry
To describe the tunneling of extended objects like strings and branes wrapping non-compact
cycles we have to generalize the Euclidean action for a particle of section 2.1. We have
to promote the trajectory x(t) to a function of the worldvolume coordinates x( i) with
i = 0; : : : ; d
1.
We consider the O(d) invariant tunneling process7 where the trajectory only depends
on the Euclidean radius R = pPi( i)2. Then, the tunneling parameter B can be obtained
from the Euclidean action:
The (radial) Hamiltonian and momentum conjugate to x are:
Z
Sd 1
Z Rf
Ri
LE ;
p =
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
7The O(d)-symmetric tunneling process is favored over a non-symmetric one, as mentioned in [34] and
proven in [49].
{ 8 {
As for the particle, the Hamiltonian is a constant of motion that we can choose HE = 0,
so that LE = p@Rx. The expression for the tunneling parameter becomes
Z
Sd 1
B =
d d 1
Z Rf
Ri
dR R
(2.10)
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Z
Z Rf
Ri
An important di erence between this O(d) invariant action and the action (2.3) is that
now we can no longer replace the integration over the Euclidean radius R (which plays the
role of t in (2.3)) by an integration over the path jdxj as we did in (2.3), because of the
explicit factor of Rd 1 in (2.10). We will come back to the fundamental di erence between
tunneling of particles and extended objects in section 2.3.
Relativistic e ective string.
We now generalize the results of relativistic particles to
a relativistic e ective string, so that we are considering O(2)-symmetric tunneling. The
e ective string can for instance descend from a D-brane (or M-brane) wrapping compact
cycles and extending in one non-compact spatial direction.
The Euclidean action is:
q
grangian. In principle, g can be any symmetric two-tensor and A(i) can be any
antisymmetric two tensors; g; A(i) are all pulled back onto the worldvolume. The integration is
over the trajectory x(R) that connects the initial vacuum xi at large R to the nal vacuum
xf at small R via an instantonic domain wall at R = R .
We have two worldvolume directions
0
; 1 (so R = p( 0)2 + ( 1)2), and we take
an embedding in spacetime as t =
0; w =
1; x = x(R). We further assume that the
spacetime metric is diagonal in t; x; w and moreover that jgttj = gxx. There is one relevant
component of each A(i), namely At(wi). All of these assumptions will be satis ed in the O(2)
tunneling event discussed later in section 4. The O(2) symmetric action is given by:
SE = 2
dR R LE ;
LE =
1 +
gxx
gtt + (At(w1))2=gtt
x_
2
!1=2
qgt2t + (At(w1))2 + At(w2) ;
where the dots now stand for radial derivatives x_
LE = 0 we obtain the momentum conjugate to x:
p(x) = tu
v
u
gxx(x)
gtt(x) + At(w1)(x)2=gtt(x)
qgt2t + (At(w1))2
(At(w2))2 :
However, the tunneling action (2.12) cannot be written as an integral in x-space of p,
as we did for the tunneling of a particle, because of the presence of an extra factor of R.
We will use the expression (2.12) in section 4 for calculating the tunneling amplitude in
LLM geometries.
(2.12)
(2.13)
{ 9 {
Decay of metastable vacua and domain walls.
Typically, one needs to resort to
numerics in order to calculate the trajectory x(R) and then integrate p(x(R)) given in (2.13)
to obtain B. Sometimes one can use an analytical approximation instead. When we
consider the tunneling of a metastable vacuum into a stable vacuum, where the metastable
vacuum only has a small excess energy compared to the stable one, we can use the thin
domain wall approximation to evaluate the action (2.12). In this approximation, for large
R the trajectory is approximated by the metastable vacuum xi; for small R, it is
approximately the true vacuum xf ; at the domain wall at R , the trajectory is approximately
constant (since the domain wall is thin). We can consider the contribution to the Euclidean
action of each region separately:
R
R : we can estimate the energy at the metastable minimum from the e ective
Ve (x)
HE (x; @Rx = 0) = VDBI + VWZ :
Evaluating the e ective potential at the minimum xi gives the following contribution
SE R R
= VSd 1
Z
R R
dRRd 1LE
d
=
VSd 1 RdVe (xi) :
R : here, we have (approximately) the true vacuum, and since Ve (xf ) = 0, the
potential:
to the action:
R
R
where in the last step we have de ned the tension Twall of the domain wall. In
general, this tension is given by the action of a brane wrapping the contractible Sd 1
at R = R (and possibly wrapping other compact directions).
contribution to the Euclidean action is zero.
the integral and convert the latter into an integral over x:
R : we take R to be approximately constant so that we can take it outside of
SE R R
= VSd 1
Z
R R
Z xi
xf
VSd 1 Rd 1
dx p(x)
VSd 1 Rd 1Twall ;
(2.16)
(2.14)
(2.15)
(2.17)
(2.18)
We see that we are left with the action:
SE (R ) = VSd 1 Rd 1Twall
d
VSd 1 RdVe (xi);
which we must still extremize with respect to R . The nal result is:
R = (d
1)
Twall ;
Ve (xi)
Again, we note that this approximation is only valid when Ve (xi) is small and therefore
the energy di erence between xi and xf is small. More precisely, Ve (xi) must be small
compared to the potential bump in between xi and xf , so that we have B = SE
Expression (2.18) was originally found in [34, 47] and used in many other papers studying
O(d) tunneling events such as [30, 32].
2.3
We now discuss some notable di erences between the tunneling processes and the amplitude
= Ae B for particles and for extended objects.
Computing B analytically.
As discussed above, for the particle (d = 1), the Euclidean
action simpli es dramatically: instead of needing to nding the entire trajectory by solving
the equations of motion, we can simply integrate the Euclidean momentum p in position
space from the starting point xi to the endpoint xf of the tunneling process. This is the
standard textbook method of calculating tunneling amplitudes in quantum mechanics.
To compute the tunneling parameter B for an O(d) invariant Euclidean bounce, in
principle we have to solve the Euler-Lagrange equations subject to the boundary conditions
x(R
!
1) = xi; x(R
!
1) = xf .
These equations can generally only be solved
numerically as one needs to know the full trajectory x(R). An analytic estimate of tunneling
rates is possible if the energy of the metastable con guration is small enough. Then the
on-shell action is approximated by a contribution of the tension of a thin domain wall and
one from the non-zero vacuum energy of the metastable state.
Computing A and B for metastable and supersymmetric tunneling.
The
behavior of the tunneling coe cient B as a function of the energy di erence between the vacua
(or the energy of the metastable vacuum Ems
Ve (xi)) can be obtained from eq. (2.18)
for the O(d)-symmetric tunneling and from eq. (2.7) for the tunneling of particles. In
particular, from (2.7) it is clear that the particle tunneling parameter B can remain nite
when p(xi) = p(xf )(= 0). In contrast, for O(d) tunneling, B blows up8 when Ems ! 0.
We can summarize these observations in the following expansions:
Bparticle = B0 + O(Em1s);
BO(d) = B1Em(sd 1) + O(Em(sd 2)):
(2.19)
In particular this implies that for the particle, we can simply ignore and set to zero any
(small) metastability energy as the nite Ems e ects will be subleading. We do exactly
that in section 3.
Note that, while B may remain nite when Ems ! 0, we do expect the coe cient A
in (1.3) to tend to zero in this limit, even for particle tunneling. This is hardly surprising,
especially when one uses the standard quantum mechanical picture of waves tunneling
through barriers: when the energy of the incoming wave tends to zero, the incoming wave's
momentum tends to zero and there is, strictly speaking, no wave left. This means that
there is also nothing hitting the barrier and no possibility of a wave exiting the barrier. For
example, the transmission coe cient for a rectangular barrier of height V0 at
a < x < a
has the small-E expansion (when a2mV0
1):
T =
16
V0
E exp( 4ap2mV0) + O(E2; exp( 8ap2mV0)):
(2.20)
Thus, even though B tends to a
nite value as E ! 0, we still have that A ! 0. In
section 3, we will be only interested in the behavior of the particle tunneling exponent B,
so we can set Ems ! 0 to calculate B to leading order.
8This is indeed what was found in the various O(d) tunneling models for example in [30, 32].
is a probe brane which in 5D wraps a contractible S1 on the topologically non-trivially two-cycle
between two centers and which in 4D reduces to a point particle on a line between the same
two centers.
HJEP07(216)3
3
Tunneling of branes into black hole microstates
We now discuss the tunneling of branes into smooth, multi-center supergravity backgrounds
that can be interpreted as black hole microstate geometries for the three-charge black hole.
In M-theory language, the three charges correspond to M2 branes wrapping orthogonal
T 2 cycles in a compact T 6, and in
ve dimensions these solutions can be described by a
U(1)3 ungauged supergravity. Supersymmetry dictates that the ve-dimensional metric be
a bration over a four-dimensional hyper-Kahler base space [50]. When this base space is of
the Gibbons-Hawking form, there is an extra isometry and we can reduce the system along
this isometry direction to give a U(1)4 four-dimensional ungauged supergravity theory
known as the STU model. Solutions in this system are determined by eight harmonic
functions on R
3 [51{53]. In order for the solution to be smooth and devoid of closed
time-like curves the locations and residues of the poles must satisfy certain particular
relations [6, 12, 13, 51, 54].
Any probes that we introduce in this system, as long as they have no internal degrees
of freedom along the isometry direction of the ve-dimensional space, can be described as
point particle probes in four dimensions. We use this \4D/5D connection" to describe the
system in the notation/dimension most convenient to a particular aspect. See gure 2 for
an illustration.
In section 3.1 and 3.2 we use the four-dimensional language of Denef [54] to calculate
the tunneling parameter B. We then continue with the ve-dimensional conventions of [6].
We apply the expression for the tunneling amplitude in section 3.3 to calculate the scaling of
the tunneling amplitude with the number of centers in two particular choices of background
solution. We discuss these results in section 3.4 and in particular their implications for the
formation of black hole microstates. To mediate possible confusion we give an extensive
review of multi-center solutions in appendix A along with an explicit overview of how both
sets of conventions are related. More details on calculations of section 3.3 are given in
appendix B.
3.1
Background solution
We summarize the relevant information on the four-dimensional supersymmetric
multicenter solutions of [51, 54]. These can for instance be obtained after compacti cation of
Charge IIA
0
A
A
0
D6
D4
D2
D0
KK monopole
M5
M2
P
IIA string theory on a Calabi-Yau threefold. The electric and magnetic charges of the
four-dimensional solutions then correspond to D0, D2, D4 and D6 branes on the internal
space, see table 1. We refer to appendix A for more information on the background and
the uplift to M-theory with
ve non-compact directions.
The bosonic elds of the solution are the metric, gauge elds A
(A0; AI ; AI ; A0) and
complex scalars zA. Since we do not need the explicit form of the scalars for the tunneling
amplitude, we only present the metric and the gauge elds:
ds2 =
e2U (dt + !)2 + e 2U d~x d~x ;
A = At(dt + !) + Aidxi ;
where I = 1; : : : ; nV and nV the total number of vector multiplets.9
The solution is
fully determined by a set of functions, conveniently organized into a symplectic vector as
H
(H0; HI ; HI ; H0), which are harmonic on
at R
3 up to local point sources given by
electric and magnetic charges i
( 0; I
; I ; 0)i at positions ~ri:
H = h + X
N
with h = (h0; hI ; hI ; h0) a vector of integration constants. The explicit solution for the
bosonic elds (3.1) in terms of these harmonic functions is found by inverting the relation
that de nes the harmonic functions [51] in terms of the metric function U and the scalar
elds zI :
We obtain [51]:
H
2 ?3 d [Im (eU i
(z))] ;
e2U = jZj ;
At = 2Re (eU i
) ;
?3d! = hH; dHi ; dAi ^ dxi = ?3dH :
(3.1)
(3.2)
(3.3)
(3.4)
In these expressions
= arg Z( tot), with tot
and the central charge Z( ) is de ned as
PiN=1 i the total charge of the solution
Z( )
h ; (H)i ;
with
h ; ~i
0 ~0 + I ~I
I ~I
0 ~0 :
(3.5)
9In four dimensions there are nV + 1 vectors (counting the graviphoton). In the speci c solutions of
section 3.3 we have nV = 3, but we can keep nV generic for the derivation of the tunneling amplitude.
for the tunneling calculation.
The action for BPS probe particles with charge
in a four-dimensional supersymmetric
S =
1
G4
Z
jZ( )jds +
h ; Ai ;
where G4 is the 4D Newton constant. Note the position-dependent mass m(x)
We consider tunneling processes between di erent centers on a line (see gure 1), and
therefore the path of least resistance is along this line.
The gauge eld A only has a time component and an angular component, but no
component along the symmetry axis. Hence the assumption of section 2.1 that Aix_ i = 0
along the path is justi ed. Along the path also ! = 0 so there are no mixed time-spatial
components. Then we can use the results from section 2.1 and the de nitions of H in (3.3)
and Z in (3.5) to see that the Euclidean momentum takes the simple expression:
jpj = G4 1e 2U qe2U jZj2
[Re(eU i Z( )]2 =
1
2G4 jh ; Hij :
The tunneling amplitude is then computed from
(H)
(z(H)) is the symplectic vector that expresses the dependence of the scalars zI
in terms of the harmonic functions, given in (A.9). Again, we do not need its explicit form
B =
1 Z ~xf
2G4 ~xi
jdxj jh ; Hij ;
(3.6)
jZ( )j.
HJEP07(216)3
(3.7)
(3.8)
(3.9)
with jdxj
one has that
pd~x d~x.
The integrand, the conjugate momentum jpj = jh ; Hij=2G4, has a very natural
interpretation. For supersymmetric supergravity con gurations with charges i at positions ~ri,
h i; Hij~r=~ri =
at every center ~ri. These equations are obtained as integrability conditions on the rotation
one-form ! and are known as the `Denef equations' or the `bubble equations'. The Denef
equations constrain the possible supersymmetric con gurations and reduce the (3N
3)dimensional con guration space f~xiji = 1 : : : N g for N centers to a (2N
solution space.10 One can also derive these equations by treating each of the centers as a
2)-dimensional
probe in the background sourced by the others. This has been done for supertubes ( uxed
D4 branes) in [56].
3.3
Tunneling process
We want to model the tunneling of matter undergoing gravitational collapse into black hole
microstates. The matter we start from will be a collection of branes of string/M-theory
10The three centre of mass coordinates are not part of the phase space.
it into a new supertube center.
that can tunnel to form a black hole microstate. Since we restrict ourselves to microstates
of the three-charge black hole, we start from a collection of branes carrying those three
charges. In the remainder of this section we describe the physics in the M-theory frame (for
conversion see table 1). This means we start from a collection of M2 branes. The end state
will be a smooth multi-center supergravity solution or `bubbling solution'. In the M-theory
picture, these are made out of several centers carrying Kaluza-Klein monopole charge.
To enter into the realm of calculations, we imagine a thought experiment in which a
very small fraction of the M2 branes have already formed a three-center smooth bubbling
solution. This three-center solution can then serve as a catalyst to form more smooth
entropy-less centers. To tunnel all the M2 brane charge of the original stack into an N
centered microstate geometry, we go through a multi-step process (see gure 3):
(1) Bring in some of the M2 branes on top of one of the KK monopole centers ~ri of the
background.
(2) The (now) non-smooth background center ~ri can be written as the sum of
entropyless constituents. Now `pull away' the extra entropy-less constituents to create a new
smooth center (corresponding to a supertube or KK monopole).
(3) Continue repeating step (1) and (2) until there are no M2 branes left, but only smooth
geometry with a total number of N centers.
The actual number of centers N and the charge at each center depends on the details of
the process. In the examples below we will always distribute the M2 brane charge evenly
among the newly created centers.
We make some important remarks that can greatly reduce calculational e orts. First,
step (1) can be done classically (without the need to tunnel) by allowing the original M2
brane stack to have also momentum charge in the M-theory frame; this step will therefore
not reduce the tunneling rate. This is very natural in an astrophysical setting, as the
environment can serve as an angular momentum bath. By adiabatically changing the
momentum of the M2 brane stack (exchanging momentum with the surroundings), one
can vary its relative position and freely move the M2 branes over to one of the catalyst
centers.
Note also that the bubble equations of the complete system do not allow to
bring all of the M2 brane charge on top of the catalyst in one go. One really needs to
repeat the rst two steps multiple times to turn all M2-charge into geometry. This is very
satisfactory, as at every given moment the small amount of M2-charge brought over to the
catalyst immediately tunnels into smooth geometry.
Second, pulling apart the entropy-less constituents in step (2) requires a tunneling
process. It is this tunneling rate that we calculate below. For the sake of computation we
will not consider the nucleation of new Kaluza-Klein monopole centers. Rather, we will
consider entropy-less constituents made out of supertubes. These are the simplest examples
of entropy-less constituents used to build microstate geometries. Moreover, they can easily
be considered as probes in multi-center backgrounds [35, 36]. Furthermore, they themselves
backreact to smooth geometry in the D1-D5 frame and, by a spectral ow transformation,
one can turn these into Kaluza-Klein monopole centers in the M-theory frame.
In the following we will consider step (2) at an intermediate point in the process. We
take an N -center bubbling solution, and we imagine tunneling a supertube at the outermost
(leftmost) center into a new center further in the interior of the geometry (to the right).
We will use speci c examples of multi-center solutions and supertube probes to study the
process of branes tunneling into topology and
ux.
We refer to appendix A for more
details on both the probe and the background in 4D and 5D and only highlight the main
notational conventions here. The relevant parameters of the particular solutions we study
are in appendix B.
We continue with a review of the charge vectors of smooth N -center solutions and
supertube probes, in both IIA and M-theory frames. Then we calculate the tunneling
amplitude for the process explained above for two particular types of solutions. The rst
type will be a regular (non-scaling) solution and the second will be a particular scaling
solution. A multi-center solution is called `scaling' when there exists a limit in which the
details of the microstate (charges and
uxes between centers) stay
xed as the centers
move arbitrarily close together. The scaling solutions play an important role in the black
hole microstate geometry programme, since near the so-called scaling point11 the solutions
are expected to be dual to the typical states of the CFT and thus have a very large
entropy [14, 44, 57].
3.3.1
Supertube probes and smooth microstates
We rst quickly review the notation and interpretation of the charges of the background
and probe in four dimensions and their interpretation in the ve-dimensional uplift that we
will use in the remainder of this paper. The four- and ve-dimensional solutions correspond
to IIA and M-theory compacti cations on a T 6, and generalizations to other CY manifolds
are straightforward [58].
Four-dimensional solutions and their Type IIA brane interpretation. In the
four-dimensional description we will focus on the STU model (compacti cation of IIA string
theory on T 6) which has nV + 1 = 4 vectors. The background has a total of 2(nV + 1) = 8
charges in the language of section 3.2 sourced by D6, D4, D2 and D0 branes. The probes
we consider are D4 branes with two lower-dimensional induced D2 charges and D0 charge:
(0; 0; 0; 3; 1; 2; 0; 1 2= 3) :
(3.10)
11The scaling point is the limit at which the centers all sit on top of each other and the solution has an
in nite throat.
(H0; HI ; HI ; H0)
p (V; KI ; LI ; 2M ) :
1
2
The uplift of the four-dimensional probe with charge vector (3.10) becomes a supertube
with magnetic dipole charge d3, two electric charges q1; q2 and a momentum charge 2q1q2=d3
along the wrapped M-theory circle ( fth direction). The charge vector becomes
2
1
K2
1
2
p (0; 0; 0; d3; q1; q2; 0; 2q1q2=d3) :
The uplift of the particular choice of IIA background charges turns the D6 branes at ~r = ~ri
into smooth Kaluza-Klein monopoles with GH charge vi connected by two-cycles threaded
by magnetic ux set by ux numbers ki. The charge vector at each center ~ri is written as
i = p
1
2
vi; kiI ; DIJK kiJ kiK ;
vi
DIJK kiI kiJ kiK
6
(vi)2
:
Since there are no explicit brane sources the ve-dimensional solution is smooth. The
magnetic uxes source total electric charges QI and angular momentum J at in nity.
We focus on a background with the N centers on a line. In this background the
Euclidean momentum (3.7) becomes
We are interested in tunneling processes of these uxed D4 branes in N -centered
backgrounds that are only sourced by D6 and D4 branes. At each center ~ri the charge vector
takes the form
i =
J K
i0; iI ; DIJK i 0i ;
2
i
DIJK
6
I J K
i i
where DIJK = j IJK j and I = 1; 2; 3.
Five-dimensional solutions and their M-theory interpretation. In ve dimensions
(obtained for example by compactifying M-theory on a T 6) the charges corresponding to D6,
D4, D2 and D0 branes become, respectively, Kaluza-Klein monopole or Gibbons-Hawking
(GH) charge, magnetic charge (M5), electric charge (M2) and momentum charge (P) (see
with the physical (Maxwell) electric charges of the supertube given by
jpj =
4G5 jd3j jq1e q2e V
d32Z3j ;
q
1e = q1 + d3 V
; q
2e = q2 + d3 V
K1
;
and where G5 is the 5D Newton constant. Note that it is the relative sign of the Maxwell
charges with respect to the background quantity Z3=V that determines whether a supertube
is BPS or not [35, 59], and not the sign of the Page charges, q1 and q2. The latter are
oftentimes negative even when the solutions are supersymmetric.
Next we turn to two particular examples. We consider the tunneling of a supertube
probe in a non-scaling solution in section 3.3.2 and in a scaling solution in section 3.3.3. In
the rest of this section we will stick to the conventions of [6], which are more widely used
for ve-dimensional microstate geometries.
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
1's
corresponding to the GH charges; because N is odd the sum of all these GH charges is +1.
on top of the rst center it can tunnel either into a minimum on top of the next center or into a
minimum between the two centers. The former process gives a bound on the tunneling amplitude
for the latter process.
3.3.2
Non-scaling solutions
As a rst example, we take a similar background as studied numerically in [14]. The
background has N centers on a line with N odd; the centers have alternating GH charge
vj =
1 such that the total GH charge is unity: Pj vj = 1; see also gure 4. The magnetic
uxes between two centers i and j are
where k^ is related to the background dipole charge as kiI = (1
viN )k^. The asymptotic
electric charges are
We imagine lowering a supertube probe from in nity to a point on the line near one
center, say the left outermost one (see gure 5). In a gauge where there are no Dirac strings
on that center we can move the supertube on top of this center; this process does not cost
any energy. From this initial position we then compute the amplitude for tunneling into
another minimum of the supertube potential.
As discussed in section 2.3, we can compute the tunneling parameter B between initial
and nal positions where the supertube has supersymmetric minima. Note, however, that
in a physical process, to get a non-vanishing coe cient A in the tunneling rate (1.3), the
supertube's initial energy must be above that of its supersymmetric ground state. This
can be achieved by starting with a supertube in a metastable minimum close to the center,
or, by using a supertube which has some initial kinetic energy.
When the supertube has tunneled to its
nal position, wrapping an S1 between the
centers 1 and 2 as shown in
gure 5, the solution will have a new bubble with
ux. This
is easiest seen in the duality frame where the charges of the supertube correspond to D1
and D5 branes, in which the common D1-D5 direction shrinks smoothly at the supertube
(3.17)
(3.18)
HJEP07(216)3
What happens if we do not compactify on !2? Strictly speaking we would have to start
from the M5 brane action with dissolved M2 charge. However, in the limit of small !2 this
action should reduce to that of a D4 brane with dissolved F1 charge. The tension of the
domain wall is approximated by the BPS M5 brane wall obtained as the uplift of the IIA
construction. Similarly, the energy of anti-M2 branes at the odd boundary x(i) close to the
metastable minimum is obtained from the uplift of the anti-F1 strings above. So, we have
VM2 =
where 0R11 = gsls3 = lP . The contribution to the bounce action is then obtained from the
Euclidean action per unit !1
!2 area
with VS2 = 4 , which is extremized at
SE = VS2R2TwMal5l
3
VS2 R3VeM2 ;
R =
2TwMal5l
V M2
e
)
SE =
163(V(eTMwM2a)l5l2)3 :
B
16 (TwMal5l)3
3jqj(VM2(x(i)))2
M jqj2
6
:
The tunneling action from a metastable minimum close to the boundary x(i) to the
supersymmetric minimum at the boundary x(i+1) = x(i) + w is given by
The tunneling rate is again suppressed for small charge jqj and large mass deformation .
Note, however, that the scaling of the tunneling parameter B with q and
in M-theory is
di erent from tunneling in the IIA reduced background. The reason is that for the O(2)
case we compute tunneling per unit length while in the O(3) case we compute tunneling
per unit area.
4.4
Metastable states & mass-deformed M2 brane theories
For both the 11D LLM geometries and their IIA reduction we computed the leading-order
behavior of the tunneling exponents in the approximation where the the metastable state
has only small excess energy compared to the supersymmetric vacuum. The scaling of
the tunneling exponents with the charge q and mass deformation parameter
can be
summarized as:
B
2d=jqjd 1 ;
where d = 3 for the 11D solutions and d = 2 for the 10D solutions. The decay of metastable
anti-M2 branes or anti-F1 strings in the infrared of LLM geometries is thus highly
suppressed for small charge jqj. In [37] it was suggested that these long-lived con gurations
should, via the gauge/gravity duality, correspond to metastable states in the mass-deformed
M2 brane theory [39].
(4.21)
(4.22)
(4.23)
(4.24)
There is one other example where a gravity dual was used to conjecture the existence of
metastable states in a strongly-coupled 2+1 dimensional theory - in [32] it was shown that
probe M2 branes in the CGLP solution [33] can have metastable minima. However, at this
point there is no technology that may allow one to look for the existence of such minima in
the dual theory. Such a technology only exists for 3+1 dimensional theories [70], and one
may hope that the recent progress in
nding new dualities in three dimensions [71] may
allow for the development of such a technology for searching metastable vacua of CGLP
and mass-deformed M2 brane theories.
Acknowledgments
We would like to thank Jan de Boer, Freddy Cachazo, Per Kraus, Alex Kusenko, Juan
Maldacena, Stefano Massai, Thomas Van Riet and Nick Warner for enlightening
discussions. The work of I.B. was supported in part by the ERC Starting Grant 240210
String-QCD-BH, by the John Templeton Foundation Grant 48222 and by a grant from
the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon
Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 (this grant was
administered by Theiss Research). D.R.M is supported in part by NSF CAREER Grant
PHY-0953232. This work is part of the research programme of the Foundation for
Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for
Scienti c Research (NWO). The work of A.P. is supported by National Science Foundation
Grant No. PHY12-05500. B.V. is supported by the European Commission through the
Marie Curie Intra-European fellowship 328652-QM-sing. We are grateful to the Centro de
Ciencias de Benasque Pedro Pascual for hospitality during this work.
A
N = 2 supergravity in four and
ve dimensions
We compare the common conventions of the four-dimensional multi-center solutions and
those of ve-dimensional bubbling solutions. We discuss the action, the elds, the
multicenter solutions and the probe potential in both conventions. The reader with only an
interest in the matching of the two sets of conventions can skip to section A.2.5.
A.1
A.1.1
Four dimensions
N = 2 supergravity in 4 dimensions
We consider N = 2 supergravity coupled to nV vector multiplets. There are nV scalars
and nV + 1 vector elds, and the action is
S4 =
1
c2(Im N IJ ) ?4 F
^ F
c2(Re N
)F
^ F
;
(A.1)
scalars zI and their complex conjugates.
with the index I = 1 : : : nV , F
= (F 0; F I ), and c a convention-dependent constant. For
instance Denef [54] takes c = 1. The scalar metric gIJ and period matrix N depend on the
The couplings of the theory are determined from one single function, a
holomorphic prepotential F (X), that is holomorphic of degree two in the projective coordinates
(X0(z); XI (z)). The functions gIJ (z); N
niently written by introducing the symplectic section
(z) appearing in the action are most
convewhich is endowed with a natural symplectic product
0 = (X ; F ) ;
F
=
X0X~0 + XI F~I
FI X~ I + F0X~ 0 :
The metric is special Kahler, as it can be obtained from a Kahler potential K derived from
the prepotential:
HJEP07(216)3
K =
ln ih 0
; 0i ;
zI =
XI
X0
We can and will set X0 = 1. The vector kinetic couplings are
N
= F
+ 2i
(Im F
X )(Im F
0 X 0
F
=
:
(A.5)
Finally the gauge elds are determined by the electric and magnetic charges. We de ne
the dual eld strengths and potentials as
The charges are
G
dA
Re NIJ dAJ + Im NIJ ?3 dAJ :
1 Z
4
p =
F ;
q =
1 Z
4
?4G
We will package these in a symplectic charge vector
which also has a natural symplectic product (A.3)
For later use we also de ne the non-holomorphic symplectic section
One can obtain the action (A.1)from a Calabi-Yau compacti cation of type II
supergravity. Then the prepotential is
1
6
F =
DIJK
XI XJ XK
(X0)2
;
p
8
c
with DIJK triple intersection numbers. We are interested in a IIA compacti cation. Then
the charges of the solution correspond to wrapped D6, D4, D2 and D0 branes, as in table 1.
They are related to integer ux numbers N I ; NI as
p =
T V G4N ;
q =
T V G4N ;
with T ; T the tensions of the D-branes in table 1, and V ; V the volumes of the cycles
they are wrapping.
p
c
8
(p ; q ) :
eK=2 0 :
(A.2)
(A.3)
(A.4)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
Probe particles
The probe potential for a BPS particle with charges
in a supersymmetric background
solution to the N = 2 action is
Z
S =
jZ( ; t)jds
2
dx
h ; A i ds
ds :
The function Z is known as the central charge:
Z( ; t) =
1
q 43 (Im z)3
6
z
2
1 AzA +
with the de nitions
z
2
A
DABC zBzC ;
z
3
DABC zAzBzC ; (Im z)3
DABC (Im zA)(Im zB)(Im zC ) :
The potential has supersymmetric minima at positions in R3 for which
BPS multicenter supergravity solutions
The most general multi-center supersymmetric background of N = 2 supergravity is fully
determined by a symplectic vector of harmonic functions [51, 54, 72]
AzA
0 ;
(A.13)
(A.12)
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
(A.22)
H = (H0; HI ; HI ; H0)
(H) 1(dt + !)2 +
(H) dxidxi ;
(dt + !) + Aidxi ;
00 0 0
11
I = BBB00 01 10 0 CA
0 C
@
C :
?3d! = hdH; Hi :
through a single function
that is homogeneous of degree two:
ds2 =
t
A =
A = I
HA
where I is the symplectic matrix that de ned the symplectic product:
The spatial one-forms forms are de ned as
position ~ri. With harmonic functions H = h + P
The integrability condition d ?3 d! = 0 leads to h i; Hi = 0 for every charge vector i at
i i=j~r
~rij, this can be written as the
set of Denef equations:
2HAH0 + DABC HBHC :
Q3=2HA
LyA
H0Q3=2
yA
+ i Q3=2 ;
L
0
2 (dt + !) + Ad ;
HAL
yAQ3=2
2H0
(dt + !) + AdA ;
1
2
QIJ =
(yI ) 2
IJ
The explicit solutions for the scalars and the time components of the vectors are
A =
A
0 =
AA =
AA =
A0 =
We are only concerned with
ve-dimensional N = 2 supergravity with two vector
multiplets, that reduces to the four-dimensional STU model.
A.2.1
STU model in ve dimensions
The action of 5d minimal supergravity with 2 vector multiplets is (we use the conventions
?5R5
QIJ ?5 dyI ^ dyJ
QIJ ?5 F~I
^ F~J
1
^ F~J
^ A~K
;
where now I = 1; 2; 3. We put tildes on the ve-dimensional vector eld to avoid confusion
with the four-dimensional ones. The vector multiplet kinetic matrix is
(A.23)
(A.24)
(A.25)
(A.27)
(A.28)
The scalars obey the restriction
with
1
6 CIJK yI yJ yK = 1 ;
CIJK = j IJK j :
A.2.2
Multi-center solutions in ve dimensions
The metric, scalars and gauge elds of supersymmetric solutions with a timelike Killing
vector have the form [50, 73]:
HJEP07(216)3
ds52 =
yI =
AI =
(Z1Z2Z3) 2=3(dt + k)2 + (Z1Z2Z3)1=3 ds42 ;
(Z1Z2Z3)1=3
ZI
;
ZI 1(dt + k) + B
I ;
where ds24 is a four-dimensional hyper-Kahler metric. The rotation one-form k and the
magnetic potentials BI are supported and only depend on this four-dimensional base space
and also the warp factors ZI only depend on the coordinates of the base.
We focus on solutions where the 4d base is Gibbons-Hawking:
ds42 = V 1(d + A)2 + V ds32(R3) ;
?3dA =
dV;
with V a harmonic function on R3. Solutions with a GH base have a natural interpretation
upon KK reduction along the GH
bre
as four-dimensional multi-center solutions.
A generic multi-center supersymmetric 3-charge solution in
ve dimensions is
determined by 8 harmonic functions (V; KI ; LI ; M ) on R3 [52, 74]. The warp factors ZI , magnetic
elds BI and angular momentum one-form are
with
BI = V 1KI (d + A) + I ;
d I =
?3 dKI
ZI = LI +
1
2 DIJK V 1KJ KK
k = (d + A) + ! ;
1
6
= V 2CIJK KI KJ KK + V 1KI LI + M ;
?3d! = V dM
M dV + (KI dLI
LI dKI ) :
1
2
1
2
A.2.3
Probe supertubes
q1; q2 in a background A.33 is [35]:
The potential for a probe supertube with 5D dipole charge d3 and two electric charges
V =
pZ3=V
1
+
d3 Z3=V
1
d3 Z3=V
2
q~1q~2
s
2
q~12 + d23
q~1
Z1
Z3=V
Z2
2
q~2
Z2
2 s
q~22 + d23
Z3=V
2
Z2
1
d3
Z1Z2
;
(A.29)
(A.30)
(A.31)
(A.32)
(A.33)
(A.34)
(A.35)
(A.36)
(A.37)
Z2
V
Z2
q~1
q1 + d3
;
q~2
q2 + d3
The supertube potential has supersymmetric minima at positions in R3 for which
This equation is sometimes referred to as the supertube radius relation.
A.2.4
Reduction to four dimensions
We reduce the action (A.27) to four dimensions with the ansatz:
Z3 =
V
(q1 + d3Z2=V )(q2 + d3Z1=V )
(d3)2
ds52 = e2 '(dx5
F~I = d(aI (dx5
A0) + e
A0) + AI )
'ds24 ;
= daI ^ (dx5
A0) + F I
aI F 0 :
with
of (A.1):
S4 =
1
16 G4
Z
G4 = G5=(2 L5)
1 X3 ?dzI ^ dzI
2 I=1 (Im zI )2
(Im N
) ?4 F
^ F
(Re N
)F
^ F
:
The complex scalar elds zI are related to the real scalars yI ; '; aI as:
and the matrices in this action are
Up to total derivatives, this gives the action in four dimensions:
S4 =
1
16 G4
Z
?4 R4
2
3 2 ?4 d' ^ d'
QIJ ?4 dyI ^ dyJ
QIJ e 2 ' ?4 daI ^ daJ
e3 ' ?4 F 0 ^ F 0
QIJ e ' ?4 (F I
aI F 0) ^ (F J
aJ F 0)
2 CIJK aK F I ^ F J + 2 CIJK aJ aK F I ^ F 0 1
1 1
6 CIJK aI aJ aK F 0 ^ F 0 :
z
I
aI + ibI
aI + ie 'yI :
1 + ab212 + ab2
1 2
22 + ab2
3
2
3
0 2a1a2a3
a2a3
a1a3
a1a2 1
a1
a2
b
b
b
2
1
2
2
2
3
a3
0
a3
a2
a3
0
a1
and L5 is the length of the x5 circle. We rewrite the action to comply with the notation
?4 R4
1
2
Im N = b1b2b3 BB
Re N = BBB
a1a3
a1a2
1
2
a1
b
2
1
1
b
2
1
0
0
a2
a1
0
a2
b
2
2
0
1
b
2
2
0
C
C
C
A
a3 1
b
2
b
2
3
C ;
Z1
V
:
Z1
:
(A.38)
(A.39)
(A.40)
(A.41)
(A.42)
(A.43)
(A.44)
(A.45)
HJEP07(216)3
This is the STU model, in the symplectic frame with prepotential
F (X) =
1
XI XJ XK
X0
=
X1X2X3
X0
;
Up to the sign of the superpotential, we identify the model (A.1) with DIJK = CIJK ,
provided we take the convention-dependent number
c = p
1
2
A.2.5
Convention matching of solutions
The harmonic functions of 4D and 5D solutions are related as:
H0 = cV ;
HI = cKI ;
HI = cLI ;
H0 = 2cM :
(A.48)
With c = 1=p2 in the harmonic functions. We apply same convention change to charges
(note the conventional minus sign for the D2 charge).
After applying this change of conventions, one sees that the probe potentials (A.12)
and (A.37) exactly agree for supertubes. In 4D language, supertubes have the charge
assignment of a D4-brane with world-volume ux. The only non-zero charges are p3; q1; q2
and q0
q1q2=d3.
B
Particular multi-center solutions
In this appendix, we give the explicit details of the backgrounds used in section 3.3 as well
as the tunneling calculations in those backgrounds.
We write the solution in terms of explicit harmonic functions with sources at N
centers ~ri:
and M :
m0 = X X kiI ;
V = X
KI = X
N
N
k
I
i
~rij
~rij
; M = m0 + X
; LI = 1 + X
N
We want to describe microstate geometries of the three-charge black hole in asymptotically
at ve-dimensional spacetime. The only free parameters are the KK monopole charges
vi and dipole charges kiI , as smoothness at the di erent centers ~ri xes the sources of LI
`I;i =
12 CIJK kiJ kiK ;
vi
mi =
1 ki1ki2ki3
2
q
2
i
8i (no sum) :
the harmonic functions by V j1 = KI j1 = 0, LI j1 = 1 and M j1 = m0 with
Five-dimensional Minkowski asymptotics requires PiN=1 vi = 1 and xes the constants of
(A.46)
(A.47)
(B.1)
(B.2)
(B.3)
so that Pi kiI = 0 and the physical ux between two centers is
The asymptotic charges in this background are given by:
kiI =
qiN k^ + k^;
vi)k^:
Q
QI =
4 X vj ( vj N k^ + k^)2 = 4k^2(N 2
1):
j
The bubble equations simplify considerably for this simple system. The bubble
equaThe physical, asymptotic charges are normalized as
QI
QIJ ?5 F J ;
We can also express the various harmonic functions in terms of V = Pi vi=ri and
which gives asymptotically ZI = QI = 2 for the radius
in standard polar coordinates on
a constant time slice at in nity.
B.1
Non-scaling solutions
Here we explain the background and calculations used in section 3.3.2.
B.1.1
Background details
Each GH center has three equal kiI charges
HJEP07(216)3
tion for center ri gives us:
so that in particular, if vi = +1:
and if vi =
1:
V~ = Pi 1=ri:
2k^2 X vj
j6=i
rij
vi =
3
2
( viN + 1);
4k^2
4k^2
X
X
j:vj= 1 rij
j:vj=+1 rij
1
1
=
=
3
2
3
2
(N
1);
(N + 1):
KI = (V~
N V )k^;
LI = 1
(N 2 + 1)V
2N V~ k^2;
ZI = L + K2=V = 1 +
(V~ 2
V 2):
^2
k
V
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
log plot of r12 versus N , keeping k^ xed. The red dots are the actual data points
and the blue line is the linear t r12
N 1:04.
HJEP07(216)3
We take q2 = d3(N + 1)k so that the supersymmetric minimum between the rst and
second center is pushed all the way to the second center. We can wonder what happens
18For the uninitiated, R2 is the coe cient of determination and p gives the so-called p-value.
We note that:
ZI (ri) =
( 1 + 3viN );
where we used the bubble equations (B.8).
We can see how r12 scales with N , keeping k^ xed. We do this by performing a linear
t on a log-log plot of N vs. r12 for the 6 data points of N = 11 + 8j for j = 1; : : : ; 6. The
result is (inserting by dimensional-analysis the factor of k2):
r12
k^2N 1:04 :
The t is very good, as can be seen in gure 8 and by the statistical t parameters:18
1
R2
(5 10 6) and p
10 11. However, if we increase N much further, the exponent
1
2
k
2
v1
k
1
v2
might still shift slightly to become
B.1.2
Probe supertube
We take q1e (r1) = 0, such that
We also have q2e (r2) = 0, so that:
Note that:
q1 =
d3 1 = d3(N
1)k^:
q2 =
d3 2 = d3(N + 1)k^:
q1e (r2) =
2d3k^;
q2e (r1) = 2d3k^:
(B.14)
(B.15)
(B.16)
(B.17)
(B.18)
(B.19)
when we take a di erent value for q2, and thus move the supersymmetric minimum to the
left: will the tunneling amplitude go up or down? We can take:
An easy calculation shows that:
At r1, we have (using the bubble equations):
q2 = d3(N + 1)k^ + :
p(r) = jd3
V )j:
Now, the expression that holds at the SUSY minimum rSUSY is:
This translates into an expression for in function of the SUSY radius rSUSY :
p(r1) = 1
(N
1) :
3
4 d3k
d
= d^3 (V~
V ) 1 =
0
d3
j:vj= 1
r 1
j A
;
3
4k^2 (N
4
1
1
;
where of course V; V~ are evaluated on r = rSUSY . One can easily see that the function
V is everywhere positive, has local minima at the positions of all centers where vj = +1
(as we are then furthest away from all the vj =
1 centers), and goes to +1 at all centers
where vj =
1. At the local minimum, the bubble equations give us:
(V~
V )(rj) =
1);
so that we have:
0
d3k^ <
for the allowed values of . The left boundary for
gives us rSUSY = r2 while approaching
the right boundary in principle gives rSUSY ! r1. We see that clearly (B.22) satis es:
p(r)
jd3j:
Moreover, we note that if
> 0 equation (B.23) implies that p(rSUSY ) = 0.
We can thus conclude that p 6=0(r1) < p =0(r1) and moreover that p 6=0(r) is a strictly
decreasing function between r1 and rSUSY (reaching zero at rSUSY ) while p =0(r) is a
constant function. Thus, the tunneling amplitude function B we calculate with
= 0 will
certainly give a strict upper bound for B, namely B 6=0 < B =0.
To calculate the tunneling amplitude for tunneling from r1 to r2, we rst note that for
this particular supertube probe:
so that p(r) is actually a constant function. Then, we simply have:
since we are tunneling between center 1 and 2.
p(r) = jd3j;
B = jd3jr12;
(B.22)
HJEP07(216)3
(B.20)
(B.21)
(B.23)
(B.24)
(B.25)
(B.26)
(B.27)
(B.28)
(B.29)
We now give more details on the background of section 3.3.3.
B.2.1
Background details
The background has N = 4n + 3 centers with charges given by:
vi = (n
f20; 20g; 12; 25; 12; f 20; 20g
n);
ki1 =
ki2 =
ki3 =
1 n
1 n
1965
2
1(n
and asymptotic constants
5
5
5
f1375; 1325g; 2 12; 2 12; 2 12; f 1325; 1375g
n ;
f
980k^;
1835
2
+ 980k^; 1965
+ 980k^
1835
2
3
8260 8380
3
2
1
3
n ;
1
3
1
3
; 12; 25; 12;
8380
3
;
8260
3
n ;
1 1 833
2
(B.30)
(B.31)
(B.32)
(B.33)
(B.34)
(B.36)
(B.37)
(v0; k01; k02; k03; l01; l02; l03; m0) = 0; 0; 0; 0; 1; 1; 1;
+ 49k^ + 310n
:
(B.35)
In this expressions, is a parameter that will be determined by demanding that the charges
remain (more or less) xed when we vary n.
This solution always has a scaling solution for k^ ! k for a particular value of k .
Since the bubble equations are:
a scaling solution is a solution to:
j6=i
rij
X h i; ji = hh; ii;
j6=i
We solved the latter equations to obtain the solution at the scaling point. This will always
give us the exact value of k^ at the scaling point, but we are always free to rescale all of the
rij as rij !
rij. In practice, we found the solutions up to n = 6 (N = 23).
We note that in these solutions (for
= 1), we have that QI
N 2 for large N .
Numerically, the convergence to N 2 actually happens very slowly: for N
100 it is still
about
N 1:96 for (I = 1; 2) or
N 2:02 (for I = 3). For our practical purposes, this means
that even by varying , we are not able to keep all three charges QI exactly the same.
Between the n = 1; 2 (N = 7; 11) solutions, the discrepancy between the charges is about
4 5% for Q1, 1
3% for Q2, 6
10% for Q3; the discrepancy does get smaller (the match
is better) when n increases. Still, the large N behavior is QI
N 2ki2, so that for example:
d3
qe =ki
N
q
e
Q1=2 ;
(B.38)
where qe is the physical e ective charge of the supertube after the tunneling process.
We can use the scaling symmetry rij !
microstate size. As a measure of the microstate size, we take the distance between the two
outer centers. We normalize to the n = 1 (7-center) solution.
One might worry whether keeping the total microstate size xed is the right thing to
do if one wants to compare di erent scaling microstates. Keeping the total charge xed,
one could also compare the depth:
rij to rescale our solutions to have the same total
d =
Z rcut o
router
(V 3Z1Z2Z3)1=6dr;
of microstate solutions with di erent values of n. We nd that the depths are approximately
the same when we keep the total microstate size xed (we get that the depth is at worst 93%
of the depth of the solution with n = 1; moreover for n large it is clear that this percentage
gets much better | comparing the solutions with n = 5 and n = 6 gives us 99,94%).
As a side note, it appears that r12 (the distance between the two outer centers)
converges to a constant value for large n, when we keep the charge and total microstate size
xed as described above. This may seem strange (since there are more and more centers,
one would expect them to get more and more squashed), but what this is telling us is that
when we increase n and keep the total microstate size xed, the inner centers get more and
more squashed together while the outer centers remain approximately at the same distance.
B.2.3
Tunneling amplitude
Having xed the charges and microstate size to be constant for di erent N , we can calculate
p(r) numerically in the usual way. Then, we can compare log N vs. log B. We will ignore
the n = 1 (N = 7) solution since N is probably not quite large enough to be representative
of the large-N scaling. Fitting the last 5 datapoints (n = 2;
; 6) gives us the scaling:
B
d3QN 1:93:
B
qQ1=2N 0:93:
The t is very good, see gure 9 (in the log-log plane): 1
R2
7:6 10 6 and p
We can try tting only the last four datapoints, which also gives N 1:93 behavior but with
slightly (4 10 6) lower R2 and slightly (1:5 10 6) worse p-value.
Using d3
N q=Q1=2 then gives us:
(B.39)
(B.40)
(B.41)
C
LLM solutions
We summarize here the relevant formulas for computing tunneling rates in the LLM
geometries. See [38] and [37] for more details.
C.1
Background details
The functions h; G; A; A~ showing up in the M-theory and type IIA reduced LLM solutions
of section 4.1 are given by
h 2 = 2y cosh G ;
G = arctanh(2z) ;
A =
z + 12 ;
y2
A~ =
z
y2
1
2 ;
(C.1)
1.0
0.5
- 0.5
- 1.0
- 1.5
log plot of B versus N , keeping the charges QI xed. The red dots are the
actual data points and the blue line is the linear t B
N 1:93.
and V is determined by the equations
The full solution is determined in terms of a single master function z(x; y) that obeys a
linear equation:
= 0 :
A general smooth solution is determined by a superposition of solutions to (C.2) and (C.3)
with the boundary value of z being either 1=2 or
1=2:
x
1
2 px2 + y2
z0(x; y) =
V0(x; y) =
1
1
2 px2 + y2
:
For the metric (4.1) to asymptote to AdS4
S7, the multi-strip solution must have a
semiin nite black region19 at one side of the y = 0 line and a semi-in nite white region on the
other. A general multi-strip solution is then obtained by superposition:
z(x; y) =
X ( 1)i+1z0(x
x(i); y) ;
V (x; y) =
X ( 1)i+1V0(x
x(i); y) ;
(C.5)
where x(i) is the position of the ith boundary and s denotes the number of pairs of white
and black strips. For odd i the boundary changes from black to white while for even i the
boundary changes from white to black. In the multi-strip solution (C.5) we get
2s+1
i=1
c3 =
c5 =
i=1
2y2
1
2z
2s+1
X ( 1)i+1 2(x
2p(x
x(i))2 + y2
x(i))2 + y2
y2 + c3H 1
h 2V :
2s+1
i=1
+ x + y2e2GV + c ;
(C.2)
(C.3)
(C.4)
(C.6)
(C.7)
and similarly for c~3 and c~5.
called \black regions.".
19In LLM jargon the regions where z is 1=2 are called \white regions" and those where it is 1=2 are
with the Dirac-Born-Infeld and Wess-Zumino actions
VDBI(x; y) = 4VS3H(x; y) 1q
VWZ(x; y) =
4VS3 [q^Bt!1(x; y) + c5(x; y)] ;
H(x; y)y3e3G(x;y) + (q^ c3(x; y))2 ;
where for convenience we have de ned a rescaled F1 string charge q^ = q=2 0 4VS3 with
4 = 1=2
0(2 lp)3 denoting the tension of the D4 brane and VS3 the volume of the
wrapped three-sphere. To avoid cumbersome notation we will also absorb factors of 4VS3
and denote the rescaled quantities with a hat.
One can show that the Hamiltonian has a minimum at y = 0, where the master
function z takes the value +1=2. The potential (C.8) becomes
s H+(x)
V^L+LM(x) = H+(x) 1
+ q^ c3+(x) 2
B+(x) q^ c3+(x)
1
The warp factor and the B eld become
We give the details about the probe F1 strings and M2 branes used in section 4.2.
C.2.1
Polarization potential
The potential for an M5 brane with dissolved M2 charge can either be obtained directly in
M-theory using the M5 brane potential of [68] or by dimensionally reducing the background
and the probe to type IIA and exploring the action of a D4 brane wrapped on a
threesphere of the internal space and which carries q units of dissolved F1 charge along !1. The
potential is given by [37]
VLLM(x; y) = VDBI(x; y) + VWZ(x; y) ;
j
+(x) =
and the functions entering in the RR potentials are
H+(x) =
V+2(x)
B+(x) =
V+(x)
c3+(x) =
c5+(x) =
2s+1
i=1
1
c3+(x)B+(x) :
x(i)j + x +
V+(x)
+(x)2 + c ;
where the integration constant c corresponds to a gauge choice. In a multi strip solution
we have
V+(x) =
1 2Xs+1 ( 1)i+1
2
i=1 jx
x(i)j ;
1 vuu2s+1
2 t
i=1
X ( 1)i+1 jx
j :
The potential (C.11) has supersymmetric and metastable minima.
(C.8)
(C.9)
(C.10)
(C.11)
(C.12)
(C.13)
(C.14)
(C.15)
Supersymmetric minima.
To satisfy VL+LM = 0 we have to impose
V+(x)
q^
V+(x)
q^ V+(x) = 0 :
(C.16)
There are two di erent ways to solve (C.16) and, correspondingly, there exist two di erent
kinds of minima: those where the probe M5 brane (D4 brane) shrinks to an M2 brane (F1
string), and those where it retains a
nite-size. The rst class of minima is obtained by
noting that at the boundaries x(i) of the strips where both S3 and S~3 shrink to zero size:
lim
x!x(i) +(x)
V+(x)
= ( 1)i :
(C.17)
which implies
0 i 1
X
2s+11
X
j=i
In this gauge the location of the supersymmetric minimum takes the simple form
A ( 1)j+1x(j)
2x(i) :
This means that the probe Hamiltonian can have degenerate supersymmetric minima
located at the boundaries x(i) if
q^+e (x(i)) > 0 (i odd)
or
q^+e (x(i)) < 0 (i even) :
where we de ned the e ective M2 brane (F1 string) charge:
q^+e (x(i)) = q^
c3+(x(i)) :
The second way to solve (C.16) is to require the expression inside the absolute value
and the brackets to vanish. The potential then has polarized supersymmetric minima inside
a white strip x(i) < xsusy < x(i+1) with i odd, located at
1
1
2
xsusy =
X
2s+1 1
X
j=1 j=i+1
A ( 1)j+1x(j)
1
cA ;
c ;
where lb and
br are the total size of the black strips that are respectively to the left and
right of the white strip in which the probe M5 brane polarizes. To have no Dirac strings
at the left boundary of the white strip, x(i) with i odd (\patch i"), we need
0 = c3+(x(i)) = c + x(i) +
X ( 1)j+1 x(i)
x(j)j ;
2s+1
j=1
0 i
X
2s+1 1
X
j=1 j=i+1
xsusy = x(i) +
q^
2
{ 43 {
(C.18)
(C.19)
(C.20)
(C.21)
(C.22)
(C.23)
Metastable minima. The potential also has metastable minima for q < 0.20 The full
Hamiltonian (C.11) is well-approximated for small x and small jqj by:
V^L+LM
q^(a1 + a2x)
4 3
x ;
q^
where
a2 =
0 2s+1
X
0 2s+1
X
j=1;j6=i jx(i)
j=i+1 (x(i)
( 1)j
1 1
x(j)j A
i 1
X
j=1 (x(i)
( 1)j
1
x(j))2 A (a1)2 :
If a2 > 0 the Hamiltonian (C.24) always has a metastable minimum at
x = j j
q^ r a2
2
Tunneling amplitude
ground (4.3){(4.5) we get:
Applying the derivation of the Euclidean momentum in section 2 to the IIA LLM
backp^(x; y) =
(q^ c3(1 + H 1h 2V 2) + c5V )2
h2y3e3G(1
yeG) ;
where we used (C.8) and gtt =
H 1, gxx = h2. In the y ! 0 limit this becomes
As a side note, equation (C.16) can be written in terms of the momentum:
At supersymmetric minima we thus have
while at a boundary we get
p^+(x) = jq^ c3+ + V+= +2j :
jp^+(x)j +
p^(x)V+ = 0
p^+(xsusy) = 0 ;
(C.24)
(C.25)
(C.26)
(C.27)
(C.28)
(C.29)
(C.30)
(C.31)
(C.32)
p^+(x(i)) = jq^ c3+(x(i))
j jq^e (x(i))j :
In a gauge where there are no Dirac strings at the boundary x(i) (given by c3+(x(i)) = 0
or eq. (C.22)) we have that q^e (x(i)) = q^ and hence p^+(x(i)) = jq^j. When q^ = 0 we get
degenerate supersymmetric minima located at boundaries and we recover (C.30).
Using (C.13), we can write the momentum more explicitly:
2s+1
i=1
p^+(x) = q^
X ( 1)i+1 x
x
c :
20Note that for a di erent gauge choice c there can be metastable minima for q > 0.
To describe a BPS domain wall between the x(i) and x(i+1) or to compute the tunneling
amplitude from a degenerate minimum at boundary x(i) to a polarized supersymmetric
minimum inside the white strip x(i) < xsusy < x(i+1) we can simplify (C.32) to
p^+(x) = jq^
2(x
x(i)) ;
where we chose c to take the value (C.22) so that there are no Dirac strings at the boundary
x(i). As a check, at the supersymmetric minimum xsusy = x(i) + q^=2 we have p^ = 0.
C.2.3
The energy of the metastable state
As noted in section 4.3, when the metastable state has only a small excess energy above
the stable vacuum we can approximate the e ective potential Ve
by the energy of the
metastable F1 strings/M2 branes at the boundary x(i).
This can be obtained from
the action
SF 1 =
2
jqj Z
g +
2
dtd!1Bt!1 ;
(C.34)
where p
g = H 1 and Bt!1 =
H 1h 2V . Because the functions entering in the metric
at the boundary x(i) is given by
and the NS-NS B eld are not the same, a probe F1 string (or M2 brane in the uplifted
solution) is not BPS everywhere. The energy for one such F1 string/anti-F1 string (jqj = 1)
VF 1(x(i)) =
2
1
0 y!0
lim H 1
Bt!1
x=x(i) =
2
1
[h2
V ] 1
x=x(i)
:
Uplifting this expression to M-theory gives the energy for one M2 brane/anti-M2 brane at
the boundary x(i):
VM2(x(i)) = VF 1(x(i))=2 R11 :
The expression (C.35) vanishes for q > 0 and i odd and for q < 0 and i odd so that F1
strings are BPS at odd boundaries while anti-F1 strings are BPS at even boundaries, and
similarly for the 11D uplift. Conversely, F1 strings and M2 branes at even boundaries while
anti-F1 strings and anti-M2 branes at odd boundaries are metastable and their energy is
given by, respectively, VF 1(x(i)) and VM2(x(i)).
Open Access.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
(C.35)
(C.36)
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