Matrix Thermalization
Ben Craps a, Oleg Evnin b,a, K´evin Nguyen a
a Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB), and
International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium
b Department of Physics, Faculty of Science, Chulalongkorn University,
Thanon Phayathai, Pathumwan, Bangkok 10330, Thailand
Ben.Craps@vub.ac.be, oleg.evnin@gmail.com, Kevin.Huy.D.Nguyen@vub.ac.be
ABSTRACT
Matrix quantum mechanics oﬀers an attractive environment for discussing gravitational
holography, in which both sides of the holographic duality are well-deﬁned. Similarly to
higher-dimensional implementations of holography, collapsing shell solutions in the gravi-
tational bulk correspond in this setting to thermalization processes in the dual quantum
mechanical theory. We construct an explicit, fully nonlinear supergravity solution describ-
ing a generic collapsing dilaton shell, specify the holographic renormalization prescriptions
necessary for computing the relevant boundary observables, and apply them to evaluating
thermalizing two-point correlation functions in the dual matrix theory.
arXiv:1610.05333v3 [hep-th] 13 Feb 2017

Contents
1 Introduction 1
2 Review of IIA Supergravity - Matrix Theory Duality 5
3 Dimensional Reduction of IIA Supergravity 8
4 Holographic Renormalization 10
5 Retarded Boundary-to-Bulk Propagators 13
6 Linear Response in Matrix Theory 17
A Details of the Dimensional Reduction 19
B Derivation of General Background Solution 21
C Bulk Field Asymptotics 23
D On-Shell Evaluation of the Action 26
E One-point Functions 27
F Boundary-to-Bulk Propagators in Vacuum AdS2 28
G Lowest Quasinormal Modes 30
G.1 Universality of Frequency-to-Temperature Ratio . . . . . . . . . . . . . . . . 30
G.2 Frobenius Expansion Near the Horizon . . . . . . . . . . . . . . . . . . . . . 31
G.3 Numerical Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 Introduction
In recent years, gauge/gravity duality, also known as “holography,” has emerged as a rare
tool for the study of strongly coupled systems far from equilibrium. Originally motivated by
the creation of a quark gluon plasma in ultrarelativistic heavy ion collisions, many authors
have used the AdS/CFT correspondence to study what happens when energy is suddenly
injected in a strongly coupled quantum ﬁeld theory. Interesting results include thermaliza-
tion times short enough to be compatible with experiments [1–10], a thermalization pattern
in which short-wavelength modes thermalize ﬁrst [7, 11], and new insights in the spreading
of entanglement entropy after a 1+1d quantum quench [5, 7, 12–14].
It is interesting to ask whether holography can be used to make predictions for the ther-
malization of systems that can also be studied using conventional techniques. If so, this
would provide a framework in which holography can be quantitatively tested in a far-from-
equilibrium regime. With this question in mind, we will study holographic thermalization
1

in BFSS matrix theory [15, 16], a quantum-mechanical model of N × N matrices which,
in the N → ∞ limit, has been proposed as a nonperturbative deﬁnition of M-theory in
asymptotically ﬂat backgrounds [16]. Our considerations will revolve around the relation
of this model [17] to a non-conformal version of the AdS/CFT correspondence [18–21]. It
has also appeared in recent discussions of “fast scrambling” [22], the fast spreading of in-
formation that is added “locally” (e.g., in one matrix component). While our focus will be
on far-from-equilibrium processes driven by energy injection, a simpler holographic setup
involving the same matrix theory has been previously studied, in a way involving extensive
numerical simulations, in a sequence of papers including [23–28]. In those considerations, a
stationary black hole was introduced in the gravitational bulk, corresponding to thermody-
namic equilibrium, rather than thermalization, in the dual matrix theory. Further analytic
considerations of matrix theory thermodynamics can be found in [29, 30]. In [31, 32], dy-
namics of moduli ﬁelds has been explored as a tool to probe thermal properties of higher-
dimensional super-Yang-Mills theories, though applications to the case of matrix theory are
less straightforward.
Despite the apparent simplicity of the BFSS matrix theory, which involves quantum
mechanics rather than quantum ﬁeld theory, real-time evolution of this model in the ap-
propriate strong coupling regime presently appears to be out of reach of conventional tech-
niques, even for small values of N . The nine scalar matrices and their fermionic partners
contain too many degrees of freedom to allow direct diagonalization, and the interactions
between the various matrix elements appear to be too nonlocal for variational techniques
such as tensor network methods to be directly applicable. It would be really nice if these or
other techniques could be developed up to the point where they can capture matrix theory,
ﬁrst for small N and later for larger values of N , in order to allow detailed comparison
with holography. (We would like to mention an intriguing attempt to tackle the quantum
dynamics of a simpler bosonic matrix theory undertaken in [33].) In the meantime, nu-
merical simulations have been carried out in another regime, where the matrix theory can
be treated classically [34–36]. (A similar study of the related BMN matrix model can be
found in [37].) This is a simpliﬁcation which would not arise in higher dimensions; see [36]
and references therein. One motivation is that, according to numerical simulations, there
is no phase transition between the diﬀerent regimes [23–28], so some qualitative features
can be expected to be similar [36]. Further studies of semiclassical processes in the matrix
theory revolving around the idea of continuity from weak to strong coupling can be found
in [38, 39].
In this paper, we use D0-brane holography [18–21] to study far-from-equilibrium evolu-
tion of matrix theory after a sudden injection of energy. In higher-dimensional AdS/CFT,
a simple way to inject energy in a holographic ﬁeld theory is by brieﬂy turning on and oﬀ a
homogeneous source, for instance for an anisotropic component of the stress tensor [3], for
an electric current [40] or for a scalar operator [41]. In the bulk, this corresponds to turning
on nontrivial boundary conditions for the corresponding bulk ﬁeld. For a small-amplitude
scalar perturbation, an approximate AdS-Vaidya spacetime was found [41], and this has
become an often-used toy model for homogeneous, isotropic energy injection. Interestingly,
the electric ﬁeld perturbation of [40] yields an exact AdS-Vaidya spacetime.
2

For D0-brane holography, if one restricts to an ansatz that is spherically symmetric in
the “internal” directions (transverse to the D0-brane worldvolume), the supergravity ﬁeld
equations simplify to those of a dilaton-gravity model [20, 21] coupled to an additional
scalar (the “breathing mode” of the internal sphere) [20]. We will explicitly solve the
dilaton-gravity equations (with the breathing mode set to zero) with an arbitrary boundary
proﬁle for the dilaton (corresponding to an arbitrary source for the dual operator). As a
consequence of the lack of dynamical degrees of freedom in 2d dilaton-gravity, if one turns a
source on and oﬀ, the late-time bulk metric agrees with the early-time bulk metric, and no
net energy was injected. We will ﬁnd, however, that one can inject energy in the system by
considering a boundary condition for the dilaton that is constant at early times and evolves
to a diﬀerent constant value at late times. In ﬁeld theory language, this corresponds to
starting with a thermal state and ending with a thermal state at a diﬀerent temperature
(and with a diﬀerent value of the coupling constant).
Concretely, in Appendix B we derive the following exact analytic solution of IIA su-
pergravity, expressed in a dual frame in which the 2d metric is asymptotically AdS (see
Section 2, where more details will be provided):
ds2
dual = − 1
x2
[
2 dvdx +
(
1 + 20
21 ˙φ(0)(v) x − M0 e4
3 φ(0)(v)x14/5
)
dv2
]
+ 25
4 dΩ2
8, (1)
φ(v, x) = φ(0)(v) + 21
10 log x, (2)
where M0 is a mass parameter and φ(0)(v) is a function that one is free to choose as Dirichlet
boundary condition for the dilaton ﬁeld φ and which we also call dilaton source. The metric
(1) describes a black hole with mass M = M0 e4
3 φ(0)
and is asymptotically AdS2 × S8 for
x → 0, where the timelike boundary of AdS2 is located. Provided that we have M0 6 = 0, a
non-constant dilaton boundary value φ(0)(v) will eﬀectively result in a non-constant mass
term in the metric.
Even though the above collapsing solution allows arbitrary energy injection patterns,
in this paper, we will mostly consider the thin-shell limit in which a black hole spacetime
of some initial temperature is glued to a black hole spacetime of higher temperature at a
null surface v = v0. This can be achieved by assuming the following proﬁle for the dilaton
source:
v < v0 : φ(0)(v) = φ0, (3)
v > v0 : φ(0)(v) = 0, (4)
with φ0 a negative constant. For computational simplicity, we will often consider the
φ0 → −∞ limit in which the initial temperature vanishes and the early-time geometry is
vacuum AdS2:
v < v0 :
{
φ(0)(v) → −∞,
˙φ(0)(v) = 0, (5)
3

v > v0 : φ(0)(v) = 0. (6)
At least within an energy range to be discussed in the next section, this solution is
holographically dual to matrix theory excited (“quenched”) away from equilibrium through
energy injection. However, the solution (1)-(2) does not describe propagating degrees of
freedom and will not be suﬃcient for computing non-trivial correlation functions. As a
dynamical probe of this background, we then consider ﬂuctuations in the size of the compact
S8, i.e., the breathing mode. This mode has already been considered in previous holographic
works [19,20] and has been identiﬁed in [19] to be dual to a matrix theory operator T −−, to
be deﬁned in (35), by matching of generalized conformal scaling dimensions [42–44]. Our
setup will allow us to holographically compute its retarded two-point correlation function
in the quenched dual state, thereby providing a ﬁrst non-trivial observable which, in the
future, one may hope to compare with direct matrix theory computations. Predictably,
the late-time behavior of this correlation function is dominated by the lowest quasinormal
mode of the ﬁnal state black hole.
AdS2 holography has a reputation for being very subtle and relatively poorly understood
(see [45–52] for a sampling of the literature, with an emphasis on recent discussions), so one
might wonder why we did not run into problems when considering AdS2 backgrounds and
excitations thereof. To see the diﬀerence between our D0-brane holography and what is
usually referred to as AdS2 holography, note that our AdS2 solution arises in the dual frame,
in which the eﬀective dilaton-gravity action takes the form (43) with constant b = 25/4.
This action has a nontrivial dilaton kinetic term, the removal of which would require a
dilaton-dependent rescaling of the metric. After such a rescaling, the metric would no longer
be asymptotically AdS2. This should be contrasted with conventional AdS2 holography,
which considers asymptotically AdS2 solutions in the frame without a dilaton kinetic term,
which turns out to be more subtle. More speciﬁcally, subtleties such as the absence of ﬁnite-
energy excitations for ﬁxed asymptotics arise for AdS2 solutions with constant dilaton.1 In
the theory we are considering here, the dilaton ﬁeld depends at least on the radial coordinate
for all solutions of the equations of motion, and holography works in a way similar to the
running dilaton solutions considered in [52].
The paper is organized as follows. In Section 2 we review the duality between matrix
theory and IIA supergravity originally conjectured in [18], setting thereby the conventions
that are going to be used throughout this work. In Section 3, we study the bosonic part
of type IIA supergravity with asymptotically AdS2 × S8 geometry in the dual frame. In
order to simplify the problem as much as possible, we only consider spherically symmetric
solutions, leading to a two-dimensional eﬀective theory describing the metric, the dilaton
and the breathing mode accounting for the S8 size dynamics. This mode is the only prop-
agating physical degree of freedom in that system, and will be our probe for computing
non-trivial correlations functions in the quenched dual state. It is important to note that
the breathing mode cannot be considered nonperturbatively as noted in [20], because it
deforms the boundary away from AdS2 (see Appendix C). In terms of matrix theory, its
1We thank Ioannis Papadimitriou for a useful discussion on this point.
4

dual operator T −− is irrelevant and cannot be sourced nonperturbatively. Nonetheless, a
proper perturbative treatment of the breathing mode is expected to correctly reproduce the
correlation functions of the dual matrix theory operator [20,53]. In Section 4 we perform the
holographic renormalization procedure [54]. Knowing the ﬁelds’ asymptotics near the AdS2
boundary as well as the on-shell eﬀective action, local boundary counterterms are added
in order to cancel boundary divergences. This is part of the precise holographic descrip-
tion of matrix theory. In earlier works holographic renormalization has been performed
for various cases, including non-linear gravity-dilaton solutions [21] and breathing mode
perturbations around pure AdS2 × S8 [20]. Related work also includes [55]. A general dis-
cussion of holographic renormalization in the presence of irrelevant operators deforming the
AdS boundary can be found in [53]. Here we consider breathing mode perturbations around
non-linear gravity-dilaton background solutions, allowing in particular for time-dependent
backgrounds of the form (1)-(2).
In Section 5 we compute the retarded boundary-to-bulk propagator of the breathing
mode in the case of pure AdS2 and in the more interesting case of the thin-shell solution
(1)-(6), which is dual to a quenched state in matrix theory. For the retarded propagator
in the latter case, we use numerical evolution and show that its asymptotic value near the
boundary is rapidly dominated by a single decaying and oscillating mode after crossing of
the shell located at v = v0. We also show that the associated single complex frequency
dominating the retarded two-point function in this quenched state with ﬁnal temperature
T coincides with the lowest quasinormal mode frequency of breathing mode ﬂuctuations
around a static black hole at the same temperature T . The ﬁrst and second quasinormal
mode frequencies are therefore computed in Appendix G using a numerical shooting method.
Using these results, we holographically derive in Section 6 the retarded non-equal-time
two-point function of T −−. Earlier computations of holographic two-point functions in
equilibrium states (as opposed to our far-from-equilibrium setting) can be found in [19, 56,
57].
2 Review of IIA Supergravity - Matrix Theory Duality
We start by reviewing the duality originally presented in [18], looking only at terms relevant
for the present work and setting our conventions. Useful references include [19,58,59]. The
bosonic part of the 10d type IIA supergravity action in string frame is
Sstring = 1
(2π)7g2
s α′4
∫
d10x √−g
[
e−2φ(R + 4(∂φ)2) − 1
4F 2
]
, (7)
with gs and √α′ = ls being the string coupling and the string length, respectively. This
action involves the metric, a scalar dilaton φ and a gauge potential CM with ﬁeld strength
FM N = ∂M CN − ∂N CM and density F 2 ≡ FM N F M N . This system admits a solution
representing N coincident electric D-particles at the origin [60]:
ds2
string = −H−1/2dt2 + H1/2dxidxi, (8)
5

eφ = H3/4, (9)
C0 = H−1 − 1, (10)
where H is a single-centered harmonic function on the Euclidean space labeled by Cartesian
coordinates xi, given by
H = 1 + Q
r7 , r2 ≡
9∑
i=1
x2
i , Q = 60π3gsN (α′)7/2. (11)
It has been conjectured that the near-horizon limit or decoupling limit of the above
D-particle background is dual to matrix theory [18]. Explicitly, this decoupling limit is
gs → 0, α′ → 0, U ≡ r
α′ = ﬁxed, g2
Y M N = ﬁxed, (12)
where the energy is kept ﬁxed while taking the limit, and the Yang-Mills coupling of matrix
theory is identiﬁed with
g2
Y M = 4π2gs(α′)−3/2. (13)
Performing a Weyl transformation on the string frame metric (8) while deﬁning β0 ≡
4
25 (15π)2/7, one can go to the so-called dual frame [58]
ds2
dual ≡ β−1
0 α′−10/7 (g2
Y M N eφ)−2/7 ds2
string (14)
= 25
4
[
− U 5
15πg2
Y M N dt2 + U −2dU 2 + dΩ2
8
]
, (15)
in which the action reads
Sdual = β4
0 (g2
Y M N )8/7
(2π)3(α′)9/7g4
Y M
∫
d10x √−g
[
e− 6
7 φ(R + 16
49(∂φ)2) − e6
7 φ
4β0(α′)10/7(g2
Y M N )2/7 F 2
]
.
(16)
For further simpliﬁcation, we apply the following ﬁelds redeﬁnitions,
e˜φ ≡ β−1
1 (α′)3/2(g2
Y M N )3/10 eφ, (17)
˜C0 ≡ β−1/2
0 β 6
7
1 (α′)−2(g2
Y M N )−2/5 C0, (18)
where we deﬁne β1 ≡ 5×54/5
4×21/10(3π)3/10 , bringing the action to the form
Sdual = β4
0 β− 6
7
1 (g2
Y M N )7/5
(2π)3g4
Y M
∫
d10x √−g
[
e− 6
7 ˜φ(R + 16
49(∂ ˜φ)2) − 1
4e6
7 ˜φ ˜F 2
]
. (19)
Finally, by performing the coordinate redeﬁnition
z2 = 12π
5 g2
Y M N U −5, (20)
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Matrix Thermalization

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