Transcript スライド 1

Model for Non-Perturbative String
Landscape and Appearance of M theory
Hirotaka Irie (CTS @ NTU)
with
Chuan-Tsung Chan (Tunghai U.)
and Chi-Hsien Yeh (NTU)
Ref)
CIY ’10, in progress
CIY ’10, “Fractional-superstring amplitudes, multi-cut matrix models and
non-critical M theory,” Nucl.Phys.B838:75-118,2010 [arXiv:1003:1626]
CISY ’09, “macroscopic loop amplitudes in the multi-cut two-matrix models,”
Nucl.Phys.B828:536-580,2010 [arXiv:0909.1197]
H.I ’09, “fractional supersymmetric Liouville theory and the multi-cut matrix
models,” Nucl.Phys.B819:351-374,2009 [arXiv:0902.1676]
Why String Theory?, What’s the Goal?
• One of the promising candidates for unification of the
four fundamental interactions (electromagnetism,
weak, strong and gravity) with a consistent quantum
theory of gravity.
• We wish to derive our universe from some theoretical
calculation of the true vacuum of string theory or
some meta-stable vacuum close to it.
• If it is possible, this can be a strong evidence for string
theory.
This is also the Fundamental Goal of High Energy Physics
How do we carry it out?
By studying Vacua of string theory
Perturbative Vacua (Origin of “strings”)
Xm
Feynman Graphs drawn by 2D surfaces:
+
+
+…
e.g
String Consistency requires :
the worldsheet field theory should be
(anomaly cancelled) Conformal Field Theory
(e.g. this results in 26/10D and so on..)
Important point is that this results in EOM of the string modes (G,B)
String Landscape
Allowed CFTs are Perturbative Vacua: they are related to each other
by deformation of BG field config., string dualities and so on..
EOM
Cf) QFT:
An IMAGE of Landscape of string-theory perturbative vacua
11D SUGRA
IIA
Which vacuum is most stable?
Het E
M?
IIB
I
Het O Within perturbation theory,
we have no idea!
Are we here?
What we can get from perturbation theory
26D Bosonic string theory
There is a Tachyon mode
10D Superstring theory
something like:
something like:
Spacetime SUSY cancellation
People believe that non-perturbative effects
can give a non-trivial potential uplift
We wish to know non-perturbative relationship among
perturbative vacua, and possibility of non-perturbative vacua
How doesThis
the has
non-perturbative
string landscape
been a long standing
problem look like?
We succeeded realizing various situations in 2D string theory!!
Non-perturbative string landscape
and two-dimensional string theory
2D String Theory (’81-)
Non-perturbatively exactly solvable
with Matrix Models
Many people have tried to attack this issue in 2D string theory:
Moduli space of 2D string theory?
We can read from the worldsheet action:
Moduli
However,
On-shell op.(BRST coh.)
are not the propagating modes in Hilbert space.
Condensation cannot happen 
are just given parameters
String-Theory Partition function does depend on background
Therefore, they should not be minimized [Seiberg-Shenker ‘92]
Then, what can be the moduli space of 2D string theory?
Configuration of eigenvalues
(
Stable background
: potential of matrix models)
Unstable background
(adding Unstable D-branes or instanton)
Summing over all the configurations
The moduli space is filling fraction:
 Background independent and modular invariant in spacetime
can follow [Eynard-Marino ‘08].
Landscape of 2D (bosonic) string is like:
Can we consider more non-trivial situations?
For example:
National extension is the Multi-Cut Case
Perturbative string theory
There are more D.O.F to interplay
among various perturb. strings
Plan of the talk
1. Introduction and Motivation
2. Review of the multi-cut matrix models
and fractional superstring theory
3. Non-perturbative String Landscape
and non-critical M theory
4. Summary and future directions
2. Review of the multi-cut matrix models
and fractional superstring theory
Brief look at matrix models
Matrix Models (N: size of Matrix)
String Theory
TWO
M:
NxN
Hermitian
Hermitian
matrices
Matrix
Summations
Feynman
Diagram
Worldsheet
(2D surfaces)
+critical Ising model
Ising model
Triangulation
(Random surfaces)
Continuum limit
Matrix Models
“Multi-Cut” Matrix Models
String Theory
Fractional Superstring Theory [H.I’09]
Let’s see more details
Diagonalization:
In Large N limit (= semi-classical)
1
V(l)
2
N-body problem in the potential V
3
l
4
This can be seen by introducing the Resolvent
(Macroscopic Loop Amplitude)
which gives spectral curve (generally algebraic curve):
Eigenvalue density
Continuum limit = Blow up some points of x on the spectral curve
The nontrivial things occur only around the turning points
Correspondence with string theory
2
V(l)
1
l
After continuum limit,
3
4
super
bosonic
1-cut critical points (2, 3 and 4) give (p,q) minimal (bosonic) string theory
2-cut critical point (1) gives (p,q) minimal superstring theory (SUSY on WS)
[Takayanagi-Toumbas ‘03], [Douglas et.al. ‘03], [Klebanov et.al ‘03]
TOO simple to claim string Landscape??
Let’s consider the Multi-Cut Critical Points:
[Crinkovik-Moore ‘91]
Continuum limit = blow up
2-cut critical points
3-cut critical points
cuts
We can expect variety of vacua!
These consideration are only qualitative discussion.
Therefore, we show quantitative results of the system.
Actual solutions in the system [CISY’09, CIY’10]
the Z_k symmetric case [CISY’09]:
(p,q) critical points with k cuts
e.g) the 3-cut cases are
|t|
|t|
t>0
t<0
the Z_k symmetric case [CISY’09]:
(p,q) critical points with k cuts
Too many solutions!?
is natural because we have two choices
Variety of solutions
Each has different perturbative amplitudes
This implies that
the string Landscape of multi-cut matrix models is non-trivial
The multi-cut matrix models provide
non-trivial models for non-perturbative string landscape!
Fractional-superstrings provide more non-trivial situations!
3. Non-Perturbative String Landscape
and Non-Critical M theory
the fractional superstring cases [CIY’10]:
(p,q) critical points with k cuts
We proposed that, the following geometry:
e.g) the 12-cut cases are
Cuts run
from Infinity to Infinity
What does it mean?
Factorization and Perturbative Isolation [CIY’10]
The algebraic equation of the solution
is factorized into irreducible curves:
But each curve only has cuts on real axes:
cuts
cuts
and
?
cuts
Didn’t we have multi-cut geometry?
Factorization and Perturbative Isolation [CIY’10]
Our answer is:
another cuts
cuts
cuts
cuts
and
Recall : “Cut = discontinuity of algebraic function W(x)“
Conjecture [CIY’10]
is patched by
in the region around
At some level, we have checked that
our system admits these boundary condition in several cases
[CIY2 ‘10]
Factorization and Perturbative Isolation [CIY’10]
What is the physical meaning of these factrization?
Fact
IF
[Eynard-Orantin ‘07] All order Perturbative correlators
only depend on F(x,W)=0
(All-order Perturbatively)
Perturbative interactions
Only non-perturbative interactions
This system has
many perturbatively isolated sectors
Perturbative Vacua
in non-perturbative string landscape
Within perturbation theory, we cannot distinguish
perturbative strings from perturbative isolated sectors!!
Perturbatively (all order)
type 0 Superstring
Perturbative strings
can appear as sectors
Perturbatively (all order) Bosonic string
Analogy to Quantum Mechanics System:
If we use perturbation theory,
we encounter two perturb. sectors:
The true vacuum is superposition of these wave functions:
We know that the coefficient is very important and non-perturbative.
In our case, the perturbative sectors are like superselection sectors,
which are separated within perturbation theory.
Perturbative string theories are just segments of the whole system
What is the whole system?
It’s non-critical M theory!
As a Mother Theory [CIY’10]
This 12-cut matrix model (12-Fractional superstring theory) includes
all the perturbative strings of
1(=Bosonic)-FSST, 2(=Super)-FSST, 3-FSST,
4-FSST, 6-FSST and 12-FSST
In the same way,
12-FSST C 24-FSST C …… C ∞-FSST
Infinite-cut matrix model (Infinite-Fractional SST)
is the Mother Theory of Fractional Superstring Theory
As a non-critical M Theory [CIY’10]
M Theory is a Mysterious theory which unifies IIA BPS spectrum:
IIA(10Dim): D0 F1 D2 D4 NS5
M(11Dim): KK
M2
M5
This lives in 11 dimensional spacetime (1Dim higher).
People believe that the fundamental DOF is Membrane (M2)
The low energy effective theory is 11D SUGRA.
11D SUGRA
IIA
Het E
M?
IIB
Het O
I
As a non-critical M Theory [CIY’10]
In 2005, Horava and Keeler proposed that by adding one angular
dimension in 2D string theory:
One can define the non-critical version of M theory (3D M theory)
Motivation: type 0A/0B 2D strings = 1+1D free fermion system
Dispersion relation of free fermions:
0B:
0A:
D0 brane charge ~ Centrifugal force
In 1+2 fermion system, they can be
p
x
0A/0B superstrings from the 1+2 fermion system
0A: The polar coordinate
Angular momentum = D0 charge
(KK momentum = D0 charge)
Hilbert space of
gives 0A
0B: The Cartesian coordinate
Hilbert space of
gives 0B
Note that this is one of the models which realize their philosophy
Similarities and differences among Multi-Cut and Horava-Keeler
limit 
The Im x = 0 section is type 0B superstring
In their philosophy, our angle ν is understood as
the third angular direction of non-critical M theory
Multi-cut matrix models
as a non-critical M Theory [CIY’10]
As its own light, the multi-cut matrix models
look like non-critical M theory
1) String-dual of the multi-cut matrix models is
2D fractional superstring theory at weak coupling region.
They have Z_k (or U(1)) charged objects (D-brane)
2) The spacetime interpretation of string theory is given by the k-th
root of the matrix-model coordinate x (or ζ) [Fukuma-H.I ‘06]
3) Z_k Charge means the sectors in the multi-cut matrix models
As its own light, the multi-cut matrix models
look like non-critical M theory
4) This is reminiscence of Kaluza-Klein reduction. This means that
the multi-cut matirx models implies the third angular direction,
as their own light.
5) Since the different angle sectors do not interact with each other
in perturbation theory.  3rd direction is non-perturbative!
6) At the strong coupling regime, the theory becomes 3D!
We refer to the strong coupling dual theory as non-critical M theory
Note) What is M/String theory limit? [Horava-Keeler ‘05]
String theory description is good in the Large radius limit
Therefore, non-critical M theory appears in the Small radius limit
Summary of other prospects
1. 3-dimension is consistent with c=1 barrier (2D barrier) of 2D
string theory since the third dimension is observed only with
non-perturbative effects.
2. In the strong coupling theory, we no longer can use Large N
expansion. Therefore, we cannot have the string picture, and
strong coupling dual theory should be described by
something other than strings.
3. Energy difference is Instanton action
Therefore, they are universal.
(doesn’t depend on regularization)
4. String theory has non-trivial DOF in non-perturb. region!
Summary
• We proposed various non-perturbative string landscape
models in the exactly solvable framework of 2D string
theory.
• Our work shows that string theory still seems to hide nontrivial dynamics in non-perturbative regime which connects
various perturbative vacua.
• So far, M theory is still a mysterious theory. But we
proposed a concrete and complete definition of M theory.
Since this is a solvable model, this should enable us to
extract all the information we wish to know.
• Non-perturbative vacua also exist in string theory. Our
work suggests that it is strong coupling dual theory which
becomes important in that study.
Future directions:
our models should enable us to answer
• What is the dynamical principle of non-perturbative
vacua? What kinds of DOF are suitable to describe
them?
• Does M stand for Membrane?
• Why does M live in 11D?
Interesting direction related to our models
• Investigation toward the non-perturbative region.
• What does the corresponding Kontsevich type matrix
model look like?