Numerical simulation of superstring theory
at finite temperatures
Yuhma Asano (Particle Theory)
25 June 2021

Mainly based on collaborations withV. Filev, S. Kovacik, and D. O’Connor

1. Introduction
Elementary particles and String Theory• The Standard Model of particle physics cannot consistently

include gravitons.

Superstring Theory can!
• “Unification” by Superstring Theory□✔ Graviton□✔ Gauge bosons□✔ Leptons and Quarks□✔ Higgs boson…

}

closed strings
open strings + D-branes

String theory describes
Interactions, Matters and Space-timein a unified way

1. Introduction
Problem in String Theory• Spacetime dim. is restricted to 10D or 11D.

… But we want 4D spacetime

Solved if extra dimensions arecompactified in the ground state

Problem:String theory, based on perturbation theory, has almost infinitely
many candidates of the ground state (i.e. the true vacuum).
“effective potential”

false vacuum

How is our spacetime realised?

We need non-perturbative formulation

1. Introduction
Non-perturbative formulations of String Theory
String Field Theory, Matrix Model, AdS/CFT duality, …
We discuss formulations given by matrices in this talk

One way to understand such matrix formulations is a brane picture:

ji

X=

ÃX11…

Xij!

XNNbranes (soliton-like objects)
connected by strings

The idea is to describe quantum gravity by a gauge theory.

SU(N) Gauge Theory

String Theory
with gravity

The formulations have been intensively and extensively tested.

1. Introduction
Formulations by matrices• Described by a lower dimensional gauge theory• IKKT model (0 dim.)[Ishibashi, Kawai, Kitazawa, Tsuchiya ’96]

• BFSS model (1 dim.)[de Wit, Hoppe, Nicolai ’88; Banks, Fischler, Shenker, Susskind ’96]

• N=4 Super Yang-Mills (3+1 dim.)[Maldacena ’97]

• The entire spacetime is not a priori.

It is expected to emerge in the large-N limit.
• (1st principle) Numerical computation• Emergent spacetime and expanding universe in Matrix Model[Kim, Nishimura, Tsuchiya ’11;
Anagnostopoulos, Azuma, Ito, Nishimura, Okubo, Papadoudis ’20; …]• Numerical tests of AdS/CFT duality[Anagnostopoulos, Hanada, Nishimura, Takeuchi ’07;
Catterall, Wiseman ’07; Hanada, Hyakutake, Ishiki, Nishimura ’13; …]

1. Introduction
Emergence of spacetime and branes• How does spacetime emerge in matrix models?
• In string theory, there are soliton-like objects—branes.

How do they realise in matrix models?

The BMN model partially answered these questions.

BMN model[Berenstein, Maldacena, Nastase ’02]

• Considered to be non-perturbative formulation of M-theory• 1D super quantum mechanics with SU(N) gauge sym.

S=N
<latexit sha1_base64="zTa/LIoCm+E2phijB3jcag5wZuA=">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</latexit>

Zh1⇣µ⌘2 111
dt Tr (Dt X a )2 + (Dt X m )2
✏abc X c i[X a , X b ] + [X a , X n ]22
2
4 32i
1 m n 2 µ2 m m+ [X , X ]X X + fermions472<latexit sha1_base64="YKWuBH/AUh3ZTX64HH7VUBKYKKc=">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</latexit>

a,b=1,2,3, m,n=4,...,9Symmetry: R × SO(3) × SO(6)Xa, Xm and 16 fermions
are N×N matrices

mass parameter1. Introduction

Vacua and Branes• Branes are realised typically as classical configurations, inmatrix models.

• The BMN model has many discrete vacua degenerated.
[Kim, Plefka ’02; Dasgupta, Sheikh-Jabbari, Raamsdonk ’02]…

…

Each vacuum corresponds to a brane configuration.
[Maldacena, Sheikh-Jabbari, Raamsdonk ’02; Lin, Lunin, Maldacena ’04]

• The brane geometry can be reproduced around a vacuum

by matrix eigenvalues.

[Y.A., Ishiki, Shimasaki, Okada ’14;
Y.A., Ishiki, Shimasaki, Terashima ’17]

How do they dynamically realise?

2. What to Expect
How does the geometry emerge
when we scale the Temperature?

ÃXi =!

Ãphase transition?

!

just matrices
high energy scale

low energy scale※ The vacua are nearly diagonal(so are geometrical matrices in general)2. What to Expect
How does the geometry emerge
when we scale the Temperature?

ÃXi =!

phase transition?

just matrices

geometryhigh energy scale

low energy scale※ The vacua are nearly diagonal(so are geometrical matrices in general)2. What to Expect
Confinement/deconfinement transition• The gauge field A has holonomy at finite temperatures.
H1(¿: imaginary time)
Polyakov loop: P =Tr ei d⌧ AN
This is an order parameter that detects whether dominant states
behave such that the colour d.o.f. are confined or “visible.”hP i = 0hP i 6= 0… confined phase (at low temp.)… deconfined phase (at high temp.)• It corresponds to black-hole transition on the gravity side.
[Costa, Greenspan, Penedones, Santos ’14]

phase transition ofa gravity solution

BH at high temp.

no BH at low temp.

3. Lattice Simulation
Polyakov loop:h|P |i(µ=5, ¤=24, N=11)

0.8

0.6

0.4

0.2

0.2

Myers term:⇠ hTr (iX1 [X2 , X3 ])i

0.4

0.6

0.8

1.0

T

1.2

Another phase transition!

1.0
0.8
0.6
0.4
0.2

0.2

0.4

0.6

0.8

1.0T

[Y.A., Filev, Kovacik, O’Connor ’18]

3. Lattice Simulation(µ=6, ¯=1.45, ¤=24, N=8)

~ Tr[XiXi]/N

SO(3) matrices

SO(6) matrices[Y.A., Filev, Kovacik, O’Connor ’18]

3. Lattice Simulation
Myers transition• Phase transitions between different vacua of the matrix model

were observed for the first time.

… transitions b/w different brane configurations
i.e. transitions b/w spacetimes generated by different config.

transition of brane config.

3. Lattice Simulation(¤=24, N=8)

T0.8

prediction from the gravity side0.6

0.4

confinement phase transition
that exists at large µ0.2

blue points: confinement transition
purple points: Myers transition
246810µThe simulation results AGREE with theoretical predictions.
[Y.A., Filev, Kovacik, O’Connor ’18]

4. Summary•

Observed two phase transitions:confinement transition and Myers transition
They are consistent with the gravity prediction so far, anddon’t merge at least on the lattice with finite N at 3 ≲ µ ≲ 6.

•The Myers transition looks like “no geometry → geometry”.

•Since there are many discrete vacua in the matrix model, weexpect a richer structure at lower temperatures.

Bosonic BMN model
We found there are 2 would-be transitions at finite N, whichmerge into 1 confinement transition in the large-N limit.
• Finite-N effect in the transition is reasonably understood.
• No Myers transition[Y.A., Kovacik, O’Connor ’20, and to appear]•Longitudinal 5-branes•

Adding another type of brane is also interesting. A µ=0
version was numerically studied.
[Filev, O’Connor ’15; Y.A., Filev, Kovacik, O’Connor ’16]