#### Toward holographic reconstruction of bulk geometry from lattice simulations

HJE
Toward holographic reconstruction of bulk geometry from lattice simulations
Enrico Rinaldi 1 2 4 6 8 9 10 11
h Evan Berkowitz 1 2 4 7 8 9 10 11
Masanori Hanada 0 1 2 3 4 5 8 9 10 11
Jonathan Maltz 1 2 3 4 9 10 11
Pavlos Vranas 1 2 4 8 9 10 11
Livermore CA 1 2 4 9 10 11
U.S.A. 1 2 4 9 10 11
0 The Hakubi Center for Advanced Research, Kyoto University
1 Kitashirakawa Oiwakecho , Sakyo-ku, Kyoto 606-8502 , Japan
2 Stanford , CA 94305 , U.S.A
3 Stanford Institute for Theoretical Physics, Stanford University
4 Forschungszentrum Julich , 52425 Julich , Germany
5 Yukawa Institute for Theoretical Physics, Kyoto University
6 RIKEN-BNL Research Center, Brookhaven National Laboratory
7 Institut fur Kernphysik and Institute for Advanced Simulation
8 Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory
9 Berkeley , CA 94720 , U.S.A
10 Yoshida Ushinomiyacho , Sakyo-ku, Kyoto 606-8501 , Japan
11 derivative given by (D
A black hole described in SU(N ) gauge theory consists of N D-branes. By separating one of the D-branes from others and studying the interaction between them, the black hole geometry can be probed. In order to obtain quantitative results, we employ the lattice Monte Carlo simulation. As a proof of the concept, we perform an explicit calculation in the matrix model dual to the black zero-brane in type IIA string theory. We demonstrate this method actually works in the high temperature region, where the stringy correction is large. We argue possible dual gravity interpretations.
Black Holes in String Theory; Lattice Quantum Field Theory; Gauge-gravity
1 Introduction 2 3 4
realized that the eigenvalue distribution of the matrices is closely related to the geometry [2{
5]. From the point of view of superstring/M-theory, eigenvalues correspond to the positions
of D-branes and various objects can be constructed simply as bound states of D-branes
and open strings [6, 7]. The large-N volume reduction is then analogous to the emergence
of higher dimensional branes from lower dimensional branes, e.g. D(p + 2)-branes from
D(p)-branes via the Myers e ect [8].
Although such approaches have been successful for various purposes, they have not
su ciently demonstrated how to understand the emergent geometry in holography [9{11]
because it is necessary to understand how dimensions transverse to the branes emerge.
Most of the recent studies concentrate on conformal eld theories dual to AdS spaces,
and consider the construction of bulk local operators from non-local operators on the
boundary [12{15]. In this paper, we propose | or rather, revisit | a simple method,
{ 1 {
which is (at least seemingly) di erent and applicable to more generic theories. In fact, our
strategy is very straightforward: we follow the old interpretation of refs. [6, 7, 16], and we
solve the dynamics of gauge theory from
rst principles.
In the Matrix Theory proposal [7], gravitational interactions are obtained from the
interactions between D0-branes. Therefore, by looking at the interactions in a speci c
system of D0-branes | forming an extended object such as a black hole | together with a
\probe" D0-brane whose position is moved by hand, it is possible to obtain the information
about the geometry as the force acting on this probe. The same idea applies to any gauge
theory which has D-brane origins, and has also played an important role for the discovery
of gauge/gravity duality (see e.g. [
17, 18
]). In particular, the eigenvalues are expected to be
reconstructed. Further studies along this line and related directions include refs. [19{27].1
Moreover, a similar idea has been studied in the context of entanglement entropy [28, 29],
in order to see how the S5 of AdS5
S5 geometry emerges.
In the past, there have also been attempts [30] to study the \internal" structure in
a system of D0-branes, focusing on a region of parameter space (at high temperatures)
where classical or semi -classical approaches are a good approximation for the dynamics.
Our focus is a regime of temperatures where the gauge theory is strongly coupled, and, even
though the probe brane approach described in this work is intuitively simple, it remains
challenging because of the obvious di culties in calculating observables non-perturbatively.
In this paper, we employ numerical Monte Carlo methods to overcome this di culty in the
strongly coupled regime.
This paper is organized as follows. In section 2 we consider the dynamics on the gauge
theory side. Although the quantitative calculation is hard, without relying on numerics, a
qualitative picture of the gauge theory calculation is provided. Physics captured by classical
studies [30] and features added by the new full quantum treatment will be explained. In
section 3 we use numerical Monte Carlo method to con rm this picture. In section 4, we
list possible dual gravity interpretations of the calculation. Note that the parameter region
we numerically studied corresponds to a rather stringy regime on the gravity side, and
hence the dual gravity interpretation can be speculative.
2
The gauge theory picture
In this section, we describe the proposed method to investigate how the black hole geometry
can be detected directly in the gauge theory. Before discussing various interpretations of
the dual gravity theory, we de ne the problem at hand in the gauge theory picture.
As a concrete example, let us consider the matrix model of D0-branes,2 which is
numerically tractable with reasonable computational resources. The Lagrangian of the
theory is
L =
1
2gY2 M
(
Tr (DtXM )2 + [XM ; XM0 ]2 + i
Dt
+
M h
XM ;
i
)
;
(2.1)
1The Dp-brane probe has been considered in ref. [24].
2Generalizations to higher dimensions are straightforward.
{ 2 {
where XM (M = 1; 2;
; 9) are N
N Hermitian matrices and (DtXM ) is the covariant
i[At; XM ] and At is the U(N ) gauge
eld. The
gamma matrices
M (M
= 1; 2;
; 9) are the 16
16 left-handed part of the gamma
matrices in (9 + 1)-dimensions.
(
= 1; 2;
; 16) are N
N real fermionic matrices.
This Lagrangian is the dimensional reduction of 4D N = 4 super Yang-Mills theory to
(0 + 1)-dimensions.
We set the 't Hooft coupling
2
= gY M N to one unless
is explicitly shown.
Equivalently, all dimensionful quantities are measured in units of the 't Hooft coupling; for
example the temperature T actually refers to the dimensionless combination
1=3T .
In this section we will consider the micro-canonical ensemble in the theory with
HJEP02(18)4
Minkowski signature, since we will eventually be interested in the black hole geometry
in Minkowski space.
When interpreted as the low-energy e ective description of open
strings and D0-branes, the diagonal and o -diagonal elements of XM can be regarded as
the D0-branes and open strings, respectively [6]. This theory can describe multiple objects
(such as multi-graviton or black hole states) through block-diagonal matrices, where each
block corresponds to a di erent object [7]. Interactions are then mediated by the quantum
uctuations of o -diagonal elements.
Let us consider a typical matrix con guration about the trivial vacuum, which is a
bunch of N D0-branes. Separating one of the D0-branes, which is represented by the
(N; N )-element of XM , from the bunch allows us to regard this D0-brane as a probe.3 The
matrices are then of the form,
XM =
XBMH wM !
wyM xDM0
;
(2.2)
where wM describes a small uctuation of the N -th row and column, which are interpreted
as open string excitations between the probe xDM0 and the rest of the original bunch XBMH.4
When we interpret the diagonal element xDM0 to be the position of a D0-brane, we
implicitly assume that the o -diagonal elements wM are small. One possible criterion
for the smallness of wM , which we will adopt in this paper, is that O(w3) terms of the
action are negligible and wM behaves as a harmonic oscillator. When w is so large that
O(w3) terms are no longer negligible, corrections to this simple geometric picture [6] will
be needed. Note that, even when no open string is excited, jwj cannot be exactly zero,
due to the zero-point oscillations of the harmonic oscillator. The zero-point
uctuations
become large when the probe gets close to the bunch. Hence, even at zero temperature, the
o -diagonal elements become large at short distances and it is probably not appropriate
to interpret the diagonal elements as the positions of D0-branes. The crossover between
these two regimes takes place when T
1.
3More precisely, we take the At = 0 gauge, in which the structure of the physical Hilbert space has
a natural connection to open strings and D0-branes, and then separate the (N; N )-component from the
others.
4Here we have used the subscript \BH" because, later in this paper, we will interpret the bunch as a
black hole (black zero-brane) via gauge/gravity duality.
{ 3 {
One subtle point associated with such zero-point uctuations is the interpretation of
the bunch, XBH. It is highly non-commutative at any temperature. At high temperatures,
the non-commutativity is dominated by thermal excitations of open strings, which
invalidate a classical geometric picture on the gravity side. On the other hand, at su ciently low
temperatures, the main source of the non-commutativity are the \zero-point oscillations";
then, while the diagonal elements may not be the positions of D0-branes,5 the classical
geometry on the gravity side may still make sense because there are no open string
excitations. In this paper we study only T & 1, because the crossover between these two regimes
takes place at T
1.
and the center of the bunch TNrXBM1H as
Our approach in this paper is to de ne the distance between the probe D0-brane xDM0
and then numerically calculate the force applied by the bunch on the probe as a function
of this particular distance. This allows us to obtain insights of the geometry from the dual
gravity picture. Note that we take the large-N limit for a
xed value of r and therefore
wM and xDM0 can be treated as a \subsystem" interacting with a thermal bath described
As the distance between the bunch and the probe varies, the force should behave as
can be estimated as rbunch
using rbunch
rD 1
N 1
P9
rD 1 P9
N
M=1 Tr(XM )2
E
Short distance: the probe merges into the bunch of other D-branes. The o -diagonal
elements wM and wMy condense and the \position" of the probe can not be de ned
in a meaningful sense.7 We call this region the \bunch". The radius of the bunch
E when the probe is absent, or
M=1 Tr(XBMH)2 . The latter is a good estimate when N
is large and it can be used in the presence of the probe, with the caveat that the
distribution in the 9-dimensional space will be skewed in the direction of the probe
when the acting force is large (we will comment on this later). In this paper we will
use these two de nitions interchangeably.
As we have mentioned above, rbunch is non-zero even at T = 0, due to quantum
uctuations. At su ciently low temperatures the classical geometry on the gravity
side may make sense even inside the bunch.
5At short distances, higher order terms can contribute and the o -diagonal elements do not behave
as decoupled harmonic oscillators. Due to this, the simple \zero-point
uctuations" picture may not be
appropriate.
However, in [39], in a similar theory (possessing 4 supercharges rather than 16), it was
numerically observed that the higher order terms give only small contributions and \zero-point uctuations"
picture is rather good.
6This is a re nement of the idea suggested in [
32, 33
].
7This is similar to the phase transition in a related model studied in [34].
{ 4 {
Long distance: the force goes as
f (T ) N r 8 [
7, 36, 37
], where the
temperaturedependent prefactor f (T ) disappears at T = 0.
Intermediate distance: here is where non-trivial dynamics can emerge. Firstly, o
diagonal elements are not very large and the position of the probe makes
approximate sense. As the probe approaches the bunch, open string excitations become
increasingly important and numerical calculations of the force are required in order
to understand this region. This is also where perturbative analysis is expected not
to work. We expect the shape of the bunch to deform in response to the probe. In
analogy with the Moon's tidal e ect on the Earth's oceans, we expect the bunch to
1 regime where, on the gravity side, 0 corrections will become important.
When r . T , o -diagonal elements are highly excited and non-perturbative e ects
become important. A strong attractive force is expected.8
When r & T , the o -diagonal elements are exponentially suppressed as they are too
heavy and decoupled from the dynamics, making the one-loop approximation valid.
The size of the bunch scales as rbunch
T 1=4, see e.g. refs. [
32, 33
]. Therefore, the
intermediate distance region is separated into two parts: T 1=4 . r . T and r & T .
The emission of eigenvalues from r . T is entropically suppressed with a suppression
factor
e N , because an O(N ) number of o -diagonal elements must be suppressed
simultaneously [
32, 33
].9 In the large-N limit, the eigenvalues cannot escape once
they reach the region r . T . Following this reasoning, we call r
T the trapping
radius and denote this distance by rtrap. A schematic representation of the various
distances at play is shown in gure 1.
Classical simulations (e.g. see ref. [30]) should be a valid approximation to the full
quantum theory at r
T . Therefore, physics near the bunch, for example the
thermalization of a black hole [31], can be understood based on results from classical simulations.
However, the classical approximation breaks down at r & T , because the mass of strings
| the energy quanta | becomes non-negligible compared to the energy scale T . In this
region, for example, quantum e ects assist the evaporation of a black hole [
32, 33
].
The high-temperature picture should fail for T . 1, where rbunch and rtrap become of
the same order. Below that point, we can immediately imagine two natural possibilities:
either rtrap approaches rbunch and they coincide at T = 0 or rtrap coincides with rbunch at
1 and 1=p
including the open strings.
8The same dynamics has been discussed in [35] as `moduli trapping'.
9The emission's suppression is also understood as follows. As we will demonstrate numerically in
section 3, the attractive force is of order N . The mass of the brane is of order N , and hence the D0-brane
must have an order one velocity in order to escape. However, the typical energy and velocity are of order
N , respectively, because the energy is of order N 2 and there are order N 2 degrees of freedom
{ 5 {
rbunch
XBH
w
xD0
rtrap
r
HJEP02(18)4
open strings w that connect it to the black hole XBH , and the length scales r, rtrap, and rbunch.
nite temperature. Regardless of the relationship between rtrap and rbunch, the force acting
on the probe should cancel at T = 0 due to supersymmetry.
In the rest of the paper we will show that our numerical results are consistent with
these expectations.
3
The numerical demonstration
In this section, we demonstrate the scenario described above by performing explicit
calculations in the gauge theory. Although the force depends on the relative velocity between
the black hole and the probe, we will concentrate on the case with zero relative velocity
for a practical reason explained below.
To begin, we modify the potential by adding terms which will x the distance between
XBMH and xDM0, up to quantum
direction. We add10 to the action
rotational symmetry we can take the displacement of the probe to be along the M = 1
uctuations. If the black hole is not spinning then by
L =
c
(
to the Lagrangian, in order to hold the probe D0-brane near the position R = (r0; ~0),
where r0 is the coordinate in the M = 1 direction and ~0 is an eight-dimensional vector.
Hence, we are introducing three new parameters, fc; c0; r0g. The last one, r0, is xed in
each simulation to constrain the distance of the probe, while we vary the rst two in order
to check that we are in a regime where the nal results are una ected by our choice (within
our total statistical uncertainty). In particular, the last term is needed in order to remove
the unphysical longitudinal oscillation modes of the open strings, and the value of c0 is
taken to be rather large
100, and xed throughout our simulations.
10This deformation manifestly breaks U(N ) to U(N
1) U(1). In principle, we can make a
gaugeinvariant analogue of this deformed potential, for example by
xing the position of the largest eigenvalue
of X1. We chose this speci c deformation because it is technically easy.
{ 6 {
;
(3.2)
An important remark is that, because of the interaction mediated by the o -diagonal
elements and the quantum mechanical nature of the system, the measured distance
according to our de nition in (2.3) will deviate from r0 in the M = 1 direction (and also
slightly in the other directions). In the numerical simulation we measure the following
expectation values
rM=1
rM=2
TrXB1H
TrXB2H
N
N
1
1
1
xD0
2
xD0
L
where the second distance, which should be distributed around zero, is only used as a
cross-check to monitor that the deformation in (3.1) is working as expected. In all our
simulations, with varying values of c, we
nd rM=2
0 and therefore we identify the
distance in (2.3) with rM=1. At distance r, the force F between the probe and the bunch
is canceled by the additional force coming from
. Therefore we can de ne a force for
each value of the input parameters, N , r0 and c, as
F (N; r0; c) = 2c(r0
r) ;
(3.3)
up to higher order terms in r0
r, where, again, r is our primary observable that we
identify with rM=1 in (3.2). Although this should be interpreted as the force at distance
r, we took c su ciently large so that r and r0 are always very close. Hence we will regard
it as the force at distance r0 when we show it later in the paper. In appendix A we show
a typical example of our numerical simulations and we show the measured observables to
demonstrate in details all the points above.
Note that the force calculated in this manner does not contain the e ect of the velocity
of the probe. Note also that the deformation on the dual gravity theory caused by this
additional deformation term is not clear. We have introduced
L only as a trick to
determine the force on the gauge theory side. When we discuss the dual gravity interpretation,
we will only consider the standard duality, in the absence of this modi cation term.
With this deformation
L, the con guration is made static. Therefore we can Wick
rotate the system to Euclidean signature in order to measure the force.11 We perform the
path integral in imaginary time by using Monte Carlo methods,12 so the result obtained
corresponds to the canonical ensemble. At large-N , this should give the same result as the
micro-canonical ensemble.
We added the deformation term (3.1) to a lattice simulation code for the Monte Carlo
String/M-theory Collaboration [38]. We studied T = 1:0; 1:5; 2:0 and 3:0 for matrices in
SU(N ) with N = 6 to N = 16, and with a variety of lattice spacings determined by L,
going from L = 8 to L = 24. The 't Hooft coupling
= gY2 M N is set to 1.
11In the Euclidean theory at nite temperature, the gauge eld At cannot be set to zero. Instead we
have used the static diagonal gauge, At = diag( 1; 2;
; N ), where i's are t-independent and satisfy
0
i < 2 T .
12Ofer Aharony suggested this numerical experiment to M. H. in 2009. At that time M. H. did not try
it because the physical picture was not clear to him. M. H. thanks Ofer Aharony for the valuable advice.
{ 7 {
1) at T = 1:0 and T = 2:0. For the largest value of N we only have
measurements at large r0, beyond the region of the peak.
eak 1) 6
Fp N( 4
21
01
8
2
0
2
.0
iLnear ift
uQadrtic ift
=N12 L=10
21
01
eak 1) 6
Fp N( 4
8
2
0
2
.0
iLnear ift
uQadrtic ift
=N8 L=10
T
.20
.05
.10
.15
.25
.30
.35
.05
.10
.15
.25
.30
.35
T
.20
two panels correspond to two di erent temperatures, T = 1:0 and T = 2:0, and numerical
results with N = 6; 8; 12 and 16 are included. Note that for the largest value of N , we do
not have results around the peak of the force. From this numerical data of the force, we
can identify interesting features pertaining to di erent distance regimes.
At short distance, F=(N
1) takes positive values, which con rms the O(N ) attraction
region described in the previous section. There is a peak at some distance rpeak, which
we numerically determined as the interval encompassing the three largest values of the
force. We also estimate the value of the maximal force Fpeak=(N
1) and its systematic
uncertainty, due to nite r0 spacing, using the distance between the maximum and the third
largest force (note that the statistical error is always much smaller than this systematic
error). The maximal force is shown in gure 3 as a function of the temperature for N = 8
and N
= 10 at
xed lattice spacing L = 10. Simple extrapolations using linear and
quadratic ansatze indicate that the data is consistent with a null maximal force at T = 0.
In
gure 4 we summarize the various distances, or \radii", at play in the system, for
N = 8; 12 and L = 10. We can see, for example, that the peak of the force rpeak coincides,
within uncertainties, with rbunch. This suggests that the force decreases once the probe
{ 8 {
di erent N values are horizontally displaced for clarity. The values of rpeak and rtrap are only
determined as intervals between di erent simulated values of r0: the former is determined by the
interval containing the three largest values of the force, while the latter is determined by subsequent
values of r0 where the force changes sign from positive to negative.
merges into the bunch. It is easy to understand this feature of the data: when the probe
approaches the origin from the right on the positive x1 side, outside the bunch, the probe
is pulled only to the left, while in the bunch some D0-branes pull the probe in the opposite
direction. At the center of the bunch r0
0, the force should cancel due to rotational
symmetry.
At an intermediate distance, after the peak, the force crosses zero. We identify this
distance with rtrap and we conservatively de ne an uncertainty related to the interval
containing the rst point where the force changes sign from positive to negative. As shown
in
gure 4, rtrap de ned this way behaves linearly with the temperature rtrap
T at high
temperature. By de nition, rtrap cannot be smaller than rpeak. In
gure 4, rtrap goes
closer to rpeak as the temperature decreases and it is consistent with rtrap = rpeak ' rbunch
at T = 0.
At r0 > rtrap, the force is repulsive and we will comment on the implications of this
below. After the repulsive region, at very large r0, we expect F=(N
1)
our data is not precise enough to distinguish this from zero.
h (N
1
In gure 5 we plot the square radius of the bunch in the direction of the probe rM2=1 =
1) Tr(XB1H)2i and the one averaged over the orthogonal directions (M = 2 : : : 9). We
note that, when the probe is far away, the two radii are consistent, while rM2=1 quickly grows
to a maximum when the probe moves between rtrap and rpeak. This can be interpreted
as a deformation of the bunch due to the interactions with the probe similar to a tidal
e ect; in fact, when the force has a peak at rpeak, the bunch is quite prolate.
When
r0 . rpeak, the bunch relaxes back to a spherical shape and ultimately becomes oblate as
r0 vanishes, although one should take care in this regime, as the geometrical interpretation
1=r08. However,
becomes obscure.
{ 9 {
h (N1 1) Tr(XB1H)2i (red) and the squared radius averaged over the eight orthogonal directions,
2 1
raverage = h 8(N 1)
P9
M=2 Tr(XBMH)2i (green) as a function of the probe position r0. The radius
of the bunch in M=1 is larger than the one in the orthogonal directions once the probe enters rtrap
and grows to a maximum near rpeak. rtrap and rpeak are indicated by vertical colored bands, while
2 2
rM=1 and raverage are shown with error bands representing statistical uncertainties of the Monte
Carlo simulations. The data is for N = 12, L = 10 and T = 2:0, but similar features are present
for all parameters N ,L and T that we studied. The larger error bands on the M = 1 direction
compared to the orthogonal direction re ects the fact that there are 8 orthogonal directions so we
e ectively get a larger statistical sample for the orthogonal directions.
Next, let us consider the size of the uctuation of the o -diagonal elements,
X
jwj2
1
8
X
When jwM j is small enough that O(jwj3) and O(jwj4) terms in the Lagrangian are
negligible, the o -diagonal elements behave as harmonic oscillators. In this case, P jwj2 becomes
1 11+ee rr==TT (for the derivation see appendix B). Note that we have treated the length of
all open strings, connecting the probe brane and the bunch of eigenvalues, to be r; this is
valid only when r is su ciently larger than rbunch. When taking into account the
nite
extent of rbunch, the harmonic oscillator formula should be replaced by PN
i=1
1
2riN 11+ee rrii==TT ,
1
where ri is the distance between i-th D0-brane in the bunch and the probe. Of course,
due to the non-commutativity of the matrices, the \positions" of D0-branes, and hence
the distances, are ambiguous; see [39] for detailed argument with numerical inputs. Here,
for simplicity, we consider a 9-dimensional spherical surface (shell) and a 9-dimensional
spherical volume (ball) of radius rbunch.13
13Adding the probe brane breaks the SO(9) symmetry. At each N; L and T , we could use samples with
the largest values of r0, where the SO(9) symmetry is almost restored, to determine rbunch. In the following,
for our plots at T = 2:0, we used the extrapolated continuum limit value rbunch = 1:96(6) for N = 12 and
rbunch = 1:91(1) for N = 6.
M=2 R dtjwM j2 measured on the lattice as a function of r0 for N = 12,
T = 2 and various L. The continuum limit is obtained by extrapolating the
nite-L points to
L = 1 at each r0 where enough values for a robust estimate are present. Even when we can not
take a reliable continuum limit, we show the
xed-L results. The perturbative curve is obtained
following the procedure in appendix B with rbunch = 1:96, N = 12 and T = 2. The continuum
curve at r0 > 10 is agreeing nicely with the perturbative expectation, while an enhancement can
be seen at smaller r0 . 5.
In
gure 6, the values of P jwj2 as a function of r0 are plotted together with the
harmonic oscillator value estimated by including the e ects of the bunch and of thermal
uctuations. A continuum limit is also performed by using simulations at di erent lattice
spacings, from L = 8 to L = 24. For some values of r0 we are unable to reliably determine
the continuum limit, but we still plot the individual results at xed lattice spacings. First of
all, we can see that P jwj2 in the continuum limit is perfectly consistent with the harmonic
oscillators behavior at r0
10. At smaller r0 distances the o -diagonal uctuations become
larger than the perturbative estimate, which means many open strings are excited and
nonperturbative e ects are becoming important.
We emphasize that the notion of \the position of the probe" becomes obscure when
open strings are non-perturbatively excited. Figure 6 and
gure 7 suggests that the
\geometry" becomes obscure approximately at r < rtrap. At r < rbunch, the \position" does
not even make sense approximately.
3.1
Comments on the D0/D4 system
The setup discussed above resembles the Berkooz-Douglas matrix model [40], which consists
of the D0-brane matrix model plus a avor sector which describes the open strings stretched
between D0-branes and D4-branes. This
avor sector is analogous to the o -diagonal
elements in our D0-brane probe setup. The mass of the strings is the distance between
D0-branes and D4-branes, which is analogous to the distance between the bunch and the
probe in our setup. The dual gravity picture is similar to the D3/D7 system [41] which is
often used to study avor dynamics in AdS/CFT.
measured on the lattice as a function of r0 for N = 6,
T = 2 and various L. The continuum limit is obtained by extrapolating the
nite-L points to
L = 1 at each r0 where enough values for a robust estimate are present. Even when we can not
take a reliable continuum limit, we show the xed-L results. The perturbative curve is obtained
following the procedure in appendix B with rbunch = 1:91, N = 6 and T = 2. The continuum curve
at r0 > 12 is agreeing nicely with the perturbative expectation. An enhancement can be seen at
r0 . 5, though it is less clear compared with N = 12.
This D0/D4 case has been studied in a series of papers [42{44]. The gravity analysis
suggests that, like in the D3/D7 case, a phase transition takes place when the D4 comes
close to the BH and touches the horizon; see e.g. [45{47]. The large-mass (long-distance)
and small-mass (short-distance) regions are \decon ned" and \con ned" phases,
respectively. (In the holographic QCD setup by the D3/D7, the gluons are always decon ned,
but quarks can still have a con ned phase.) The order parameter is the condensation of
the strings and, in the con ned phase, strings are highly excited.
In [42{44], some gauge theory results based on Monte Carlo simulations are also shown.
They did not nd a nice agreement with the dual gravity calculation at intermediate distance, but this could be attributed to 0 corrections, given their temperature range (T = 1:0 and T = 0:8).
4
Possible dual gravity interpretations
In this section, we discuss what kind of possible dual gravity interpretations can be given
for the results or our numerical simulations. Since we have studied only T
1, which is
rather high temperature, the dual gravity theory is expected to su er from large stringy
corrections. Hence the geometric interpretations simply inspired by supergravity may not
be appropriate. In spite of this possible shortcoming, let us review the standard duality
picture and discuss alternative interpretations of the emerging geometry.
4.1
The standard duality dictionary
with
2
= gY M N
In this paper, we consider the nite temperature dynamics near the 't Hooft limit (N ! 1
xed), to which the interpretation in the context of the gauge/gravity
duality [48] can be applied. When all N eigenvalues are clumped up to form a bunch, the
dual geometry is the near-extremal, near-horizon limit of the type IIA black zero-brane,
whose metric in string frame is given by
ds2 = 0
8
<
:
U 7=2 1
where U is the radial coordinate times ( 0) 1, which has the dimension of [mass], and U0
is the horizon. The 't Hooft coupling
has the dimension of [mass]3. Note also that the
curvature radius of S8 depends on the radial coordinate. The dilaton depends on the radial
and is identi ed with the temperature of the matrix model.
The energy of the black hole at
nite temperature has been studied numerically on
the matrix model side starting in [49]; see also [50{55]. Recent Monte Carlo results in the
continuum and in nite-N limit [55] strongly support the validity of the duality, including
the string corrections.
From (4.1) and (4.3), we can see that the horizon shrinks in the string frame when the
e ective dimensionless temperature
1=3T is large, while the horizon expands in Einstein
frame due to the non-trivial behavior of the dilaton (4.2). Therefore, the 0-corrections
become larger at higher temperature. At T = 0, the black zero-brane is extremal, i.e. the
horizon and the singularity coincide. However, note that in the 't Hooft limit, N ! 1 is
taken before T ! 0, and then in Einstein frame there is a parametrically large separation
between the singularity and horizon. Note that the horizon scales as U0
T 2=5.
The probe brane is believed to be described by the Dirac-Born-Infeld (DBI) action in
this spacetime.
4.2
Where is the horizon?
Now we discuss a few possible geometric interpretations and their advantages,
disadvantages, and falsi ability. Before going into the details, let us clarify the assumptions
regarding the holographic dictionary. Firstly, the duality between the real-time theories is
only employed without the deformation term,
L
. The deformation
L was employed
on the gauge theory side just as a numerical trick to determine the force of the original
theory in Minkowski signature without the deformation. In order to relate the original
theory with Minkowski signature to string theory, we need neither the deformation nor the
Euclidean theory. On the gravity side, we consider the motion of a probe D0-brane in the black zero-brane geometry. As can be seen from the arguments and calculations in the previous sections, { 13 {
we have assumed that the (N
1) (N
1) block XBH corresponds to the black hole, and
the probe D0-brane corresponds to the (N; N )-components of the matrices. We assumed
TrXBH=(N
1) is the `center' of the black hole, in the sense that the distance between the
probe and the black hole is de ned by jTrXBH=(N
1)
xD0j. This interpretation can be
made precise as long as the stringy e ects are not too large; when many of the open strings
are excited (correspondingly, when the N -th row and column take large values), the notion
of the localized probe becomes obscure.
We have calculated the force when the relative velocity between the black hole and the
probe is zero. Without knowing the velocity dependence, we cannot follow the motion of
the probe precisely. Below we will assume that the velocity dependence does not change
the behavior of the system drastically.
For sake of clarity, let us repeat here an important di erence between the two
temperature regions, T & 1 and T . 1, which we have brie y mentioned in section 2. On the
gauge theory side, there are two di erent sources of the non-commutativity: the thermal
excitations and the zero-point oscillations. The former corresponds to the actual stringy
excitations on the gravity side, while the latter may not invalidate the classical gravity
picture based on the smooth geometry. These two contributions should become of the
same order at T
1. Our simulations have been performed for T & 1, where the bunch is
dominated by the thermal excitations. Below, we will consider T & 1 in detail, and then
brie y comment on T . 1.
4.2.1
Probably the most conservative interpretation in this high-temperature regime is that the
entire bunch, 0
r
rbunch, describes the horizon of the type IIA black zero-brane. If one
believes that all the information about the black hole is encoded in the horizon, why don't
we regard the entire bunch, which is the carrier of the information on the gauge theory
side, with the horizon? This interpretation has some other advantages as well:
If the gauge theory describes the system from the exterior observer's viewpoint, the
light modes should appear near the horizon due to the redshift. The light strings
between the bunch and the probe, which become massless when the probe reaches
r = rbunch, are natural counterparts. See [
34
] for a related consideration for a
solvable model.
On the gravity side, the dynamics at the horizon naturally explains fast
scrambling [56]. On the gauge theory side, the non-local interaction mediated by open
strings is crucial for fast scrambling. Then 0
r
rbunch, where the open strings
condense, is a natural place where fast scrambling can take place. Note however that
this argument may not exclude the possibility that r = rtrap is the horizon, because
the open string excitations are enhanced for r
rtrap.
In this interpretation, the interior of the horizon cannot be seen from the eigenvalue
distribution.
This is an advantage when we consider the theory with Euclidean
signature, whose gravity dual does not have an `interior'.
A possible di culty of this interpretation is that the physical meaning of the distance
scale rtrap is not clear. It may not be an immediate problem, especially in the D0-brane
case, in which the 0-corrections are inevitable at nite temperature. We will come back
to this point later. Note also that this di culty might be seen as good news because, in
case one determines the existence/absence of such distance scale by studying the dynamics
of the probe D-brane from string theory, it is possible to test this interpretation.
In ref. [30], the spectrum of the Dirac operator acting on the fermion
has been
studied by means of semi-classical simulations. In such simulations, a (N
1)
(N
1)
matrix XBMH was generated and a `probe D0-brane' xDM0 was introduced by hand. The
spectrum of the Dirac operator obtained from the matrix
XM =
XBMH
0
0
xDM0
!
(4.4)
was then studied. Ref. [30] identi ed the horizon with the distance scale where the Dirac
operator becomes gapless. This length scale is likely to be our rbunch.
4.2.2
T & 1: is rtrap the horizon?
Another possibility is that rtrap is the horizon. This interpretation has some favorable
features:
When r is slightly above rtrap, the force is repulsive. This is not something expected
in the interior of the black hole. It is natural to regard r > rtrap to be (at least a
part of) the exterior.
As mentioned above, the o -diagonal elements are highly excited when r < rtrap,
although they do not condense until the probe goes to r
rbunch. Such excitations
can explain why the D-branes are trapped there; see section 2. Furthermore, when a
D-brane is emitted to r > rtrap, the temperature of the black hole goes up [
32, 33
].
Note also that, if we identify rhorizon with rtrap, it is consistent with a conservative
stance on possible stringy e ects | if stringy e ects should become relevant, it should
be at r
rhorizon.
In [57], the force acting on a D0-brane probe outside the horizon was studied on the
gravity side. At the level of supergravity, the force is attractive at any distance. When
O(gs) corrections are taken into account, a repulsive force correction is added near the
horizon. The e ects from the 0 corrections and the higher order terms in gs are not
known. If rtrap is the horizon, then our result on the gauge theory side (O(N ) repulsion)
suggests that the 0 corrections lead to a repulsion near the horizon. It provides us with
the falsi ability of this interpretation.14
The disadvantages of this interpretation include the following:
This distance scale makes sense in the Euclidean theory as well. Then this
interpretation would mean that the dual Euclidean black hole geometry may somehow knows
the black hole interior, which is against usual lore.
14We would like to thank Y. Hyakutake for the discussion concerning this point.
If rtrap is the horizon, then the probe can pass through the horizon within a nite
time in gauge theory. If we then identify the time in gauge theory with the exterior
observer's time as usual, it suggests that the in-falling observer can go into the black
hole within a nite time as seen from the exterior observer's clock.15;16
An important and subtle point related to the disadvantages mentioned in the end of
section 4.2.2 is that, when r < rtrap, many strings are excited, hence it is not clear whether
a smooth geometry can make sense there. (Clearly, for r
rbunch, the geometry does not
HJEP02(18)4
make sense.) If a smooth geometry does not make sense, the \interior" of the black hole
may not make sense; it would be better to regard the entire region r < rtrap to be some
`stringy stu ' which represents the horizon. Then the disadvantages mentioned above can
be resolved.
4.2.4
Yet another possibility is the fuzzball (see e.g. [58] for a review). In this interpretation,
space itself ends at r = rbunch (or r = rtrap) due to some stringy stu . At this moment we
do not know how to distinguish this possibility from the scenarios suggested in section 4.2.1
and section 4.2.2.
4.2.5
T . 1: low-temperature region
As we have commented before, at low temperature, a large non-commutativity does not
necessarily mean the breakdown of smooth spacetime, as long as the thermal excitation
on top of the quantum
uctuation is not large; hence the geometry would make sense
even at r < rbunch. The results of ref. [21] and ref. [39] seem to be consistent with this
expectation. In this case it would be natural to expect that the horizon is hidden below
rbunch, as discussed in ref. [21]. It ts well with the standard duality dictionary, in which
the radial coordinate U in (4.1) is identi ed with r up to a constant multiplicative factor.
Note that the horizon is at U0
T 2=5, as one can see from (4.3).
As T becomes large, U0
T 2=5 increases. At T
1, it can become as large as rbunch
and rtrap. Hence it would be natural to think that rbunch and rtrap at high temperature is
related to the horizon. At this moment this is just a speculation as we have yet to study
this scenario in the low temperature region.
15We would like to thank T. Banks for pointing out this problem.
16A possible resolution in the philosophy of the Matrix Theory Conjecture | everything is made of
eigenvalues | is as follows. Suppose everything, including the in-falling and exterior observers, is made of
eigenvalues. They communicate with each other by exchanging eigenvalues. As the in-falling observer goes
parametrically close to rtrap, say the distance of order 1=N , stringy e ects turn on and make it hard to
send eigenvalues to the exterior observer. Then the exterior observer would have to wait longer to receive
the message.
In this paper, we have studied the dynamics of eigenvalues in gauge theories, particularly
in the D0-brane matrix model. We have performed explicit numerical calculations in high
temperature region. There are two length scales, which we denoted by rtrap and rbunch,
which may be related to the horizon on the gravity side.
Our study of the high temperature region has pros and cons. The biggest pro is that
the stringy e ect is large, and the largest con is that the stringy e ect is large. Stringy
e ects are something we want to learn from gauge theory, but at the same time, when
the stringy e ects are too large the gravity interpretation is not easy. As a next step,
it is necessary to study a parameter region responsible for small stringy e ects. There
are two natural approaches: (1) long distance, corresponding to far outside the bunch,
regardless of the temperature, and (2) low temperature inside the bunch. The former is
more straightforward as o -diagonal elements are suppressed. (Note that stringy e ects
are large at long distance, but as long as we only look at the dynamics of the eigenvalues
we expect the DBI action to provide an accurate description.) For the latter, we need to
resolve the problem of the non-commutativity. One possible approach is to study D0/D4
system (section 3.1).17 Here, the masses of the avor sector speci es the position of the D4
at spatial in nity, and if the critical mass agrees with the dual gravity prediction it means
that the D4 probe is actually described by the DBI action. We can also use the D1-probe
in (1 + 1)-d SYM. Fixing the two end points far outside bunch and allowing the middle
of the D1 to fall down into the bunch, the shape of the probe can be determined, (at least
outside the bunch), and it is possible to test if the DBI action is valid there.
In the correspondence between (p + 1)-dimensional SYM and the black p-brane [48],
one can probe the geometry in the same way, by using Dp-branes; see e.g. refs. [23, 24]. An
important di erence from the case of D0-branes is that various shapes can appear. Using
lattice simulations, it should be possible to see a minimal surface directly. Other probes
such as the D-instanton can also be useful. An important point, which is not apparent
from the current analysis, is whether or not the horizon depends on the kind of probe.
In the D0-brane quantum mechanics, the Schwarzschild black hole in eleven
dimensions is expected to emerge in the M-theory parameter region, which is at much lower
temperatures than the 't Hooft large-N limit. Also, 4D N = 4 SYM on S3 is expected
to contain low-energy states describing the ten-dimensional Schwarzschild black hole [59].
On the gauge theory side, they should be described by bunches of eigenvalues of scalar
elds (see e.g. refs. [60, 61]) and hence the method proposed in this paper can be applied;
it is very important to study the emergent geometries in these cases. Another important
direction is the time-dependence; see e.g. refs. [
32, 33, 35
] for previous attempts.
Acknowledgments
The authors would like to thank O. Aharony, T. Banks, D. Kabat, F. Ferrari, G.
Lifschytz, J. Maldacena, E. Martinec, Y. Nomura, D. O'Connor, K. Papadodimas, J.
Pene17We would like to thank J. Maldacena for suggesting this approach.
distance where the force becomes negative is r0 2 [9; 10].
dones, S. Shenker, H. Shimada, E. Silverstein, K. Skenderis, B. Sundborg, L. Susskind,
and V. Filev for discussions and comments. The work of M. H. is supported in part by
the Grant-in-Aid of the Japanese Ministry of Education, Sciences and Technology, Sports
and Culture (MEXT) for Scienti c Research (No. 25287046 and 17K14285). The work of
J. M. is supported by the California Alliance fellowship (NSF grant 32540). This work is
supported in part by the DFG and the NSFC through funds provided to the Sino-German
CRC 110 \Symmetries and the Emergence of Structure in QCD" (EB). E. R. is supported
by a RIKEN Special Postdoctoral fellowship. This work was performed under the auspices
of the U.S. Department of Energy by Lawrence Livermore National Laboratory under
contract DE-AC52-07NA27344. Computing was provided through the Lawrence Livermore
National Laboratory (LLNL) Institutional Computing Grand Challenge program.
A
Numerical calculation of the force
In this appendix we present a typical example of the numerical Monte Carlo simulations
described in section 3. We take a representative set of parameters fN; L; T g = f8; 8; 3:0g
where we have explicitly checked how the force F (N; r0; c) = 2c(r0
r) depends on c,
according to the potential in (3.1).
For N = 8, L = 8 and T = 3:0 we have used several values of c, going from 30 to 120,
across the whole range of \constraining" distance r0. Remember from (3.1) that r0 is the
distance along direction M = 1 where the constraining potential is centered, while c is the
strength of the quadratic potential. A summary of the values of c used for this point in
parameter space as a function of r0 is shown in gure 8.
For larger values of c, the probe will be more tightly constrained around r0 in the M = 1
direction, and around zero in the perpendicular directions. We show the observables rM=1
and rM=2 in the left and right plots of gure 9, respectively. For each panel we report the
Monte Carlo history and the histogram of the observables, after the initial 1000 samples are
two observables rM=1 and rM=2 de ned in (3.2) at r0 = 7:0, for N = 8, L = 8 and T = 3:0.
On each plot we show two values of c, c = 50 and c = 100. The blue dotted line in the left plot
corresponds to r0 = 7, while on the right plot it corresponds to zero.
r) measured for N = 8, L = 8 and T = 3:0 at two distances
r0 and
ve values of c. When distance is small (left panel) the value of c for which the force is
independent of c must be larger than c = 30. When the distance is large (right panel) all values of
c that we tried are equivalent within the statistical precision of our measurements.
discarded for thermalization. The plot of rM=1 show that, for a potential centered around
r0 = 7:0 in the M = 1 direction (and for N = 8, L = 8 and T = 3:0) the actual coordinate
of the probe in such direction is very close to r0, and more so for larger c, as expected.
Similarly for rM=2, the distribution of the samples is narrower around zero when c = 100
rather than c = 50, again con rming that our constraining potential is behaving correctly.
For both values of c 2 f50; 100g the force near distance r0 is the same, because the
expected shift of the probe r0
r is smaller for larger c, and the two e ects cancel in the
force F = 2c(r0
r). This cancellation will break down if the potential is too shallow or
the force is too strong | if the probe wanders far from the center of its potential. We
observe deviations of this kind, for example, if c = 30 is used at small r0 = 4:0 or r0 = 5:0,
where the force is large and positive. This is clearly exempli ed in gure 10. Typically, in
the region where the force is attractive, between r0 = 0 and the transition to a repulsive
0.8
0.6
)
r
0
(r0.4
0
r0=4.0
r0=5.0
r0=11.0
r0=12.0
1/1 1/
2
a function of 1=c. The slope of the linear relation de nes the force (r0
r) = F=2 1=c. A deviation
from the linear behavior is present when the force is large and c is too small. The dotted lines
represent the slopes for our typical choices of c = 100 in the small r0 regions and c = 50 in the large
r0 region (circled points).
force, we choose c = 100, while we settle on c = 50 in the region where the force is small
or almost zero.
Another equivalent way to look at this is to investigate the relation between (r0
r)
and c at xed r0: our de nition of force will be correct in the region of c where the data
is described by a linear function.
We show the probe deviation from the center of the
potential (r0
r) as a function of 1=c in gure 11, for the same four values of r0 reported
in
gure 10. We plot the force obtained from the relation F = 2c(r0
r) with c xed to
the typical values reported above at di erent r0. Note again the deviation of the c = 30
measurements from the linear behavior when the force is large.
B
Behavior of o -diagonal elements
p2N
At long distances, the o -diagonal elements are approximated by harmonic oscillators.
There are 8(N
1) complex d.o.f., and hence 16(N
1) harmonic oscillators. By writing
wM;j = (xM;j+iyM;j) , the action for the o -diagonal part can be written as
9 N 1
X
X
M=2 i=1
2
2
2
x_ M;i +
2
y_M;i +
r2x2M;i +
2
r2yM2;i
2
!
hy2i ' R dxe rx2
R dxx2e rx2
N 1
N
N
N 1 1
2r 11+ee rr==TT .
up to the higher order terms. Note that we did not rescale r. Hence x and y are harmonic
oscillators with m = 1 and ! = r. The ground state wave function is
At su ciently long distances, jwj2 is approximated by zero-point uctuations, and hx2i =
e x2=2; e y2=2.
= 21r , hjwM;ij2i ' hx2i+hy2i = 2N1 r . Therefore, 81 P9
2N
M=2
R dtjwM j2 '
21r . When the excited modes are taken into account, this expression is modi ed to
Because the bunch of D0-branes has
nite size, there is a correction to the above
expression. If one imagines the D0-branes to be distributed in a spherically symmetric
manner, the average distance between them and the probe brane is larger than r. Hence
the
uctuation of the o -diagonal elements should be slightly smaller. We have
numerically implemented two possible distributions for the D0-branes in the bunch to assess the
corrections due to non-zero bunch size:
an 8-dimensional sphere S8 of radius rbunch,
a 9-dimensional ball of radius rbunch.
The bunch e ects have been taken into account by replacing r with r
rs, where rs is the
position of a point randomly sampled according to the aforementioned two distributions,
and by taking the average of the function over the whole sample. We have checked that
the nal result does not change within the needed precision when the number of samples
is large enough.
Open Access.
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any medium, provided the original author(s) and source are credited.
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