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 progressCIY ’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 PhysicsThis is also the Fundamental Goal of High Energy PhysicsThis is also the Fundamental Goal of High Energy PhysicsThis is also the Fundamental Goal of High Energy Physics

How do we carry it out?By studying VacuaVacua of string theoryBy studying VacuaVacua of string theory

Perturbative VacuaPerturbative Vacua (Origin of “strings”)Feynman Graphs drawn by 2D surfaces:

+ + +…

X

String Consistency requires :

e.ge.g

Important point is that this results in EOM of the string modes (G,B)

(anomaly cancelled) Conformal Field Theory(anomaly cancelled) Conformal Field Theory(e.g. this results in 26/10D and so on..)

the worldsheet field theory should be

String LandscapeAllowed CFTs Allowed CFTs are are Perturbative VacuaPerturbative Vacua: : they are related to each other by deformation of BG field config., string dualities and so on..Allowed CFTs Allowed CFTs are are Perturbative VacuaPerturbative Vacua: : they are related to each other by deformation of BG field config., string dualities and so on..

11D SUGRA

IIA

IIB

I

Het O

Het EM?

An An IMAGE IMAGE of Landscape of string-theory perturbative vacuaof Landscape of string-theory perturbative vacua

Are we here?

Cf) QFT:

Which vacuum is most stable?Which vacuum is most stable?

Within perturbation theory, we have no idea!Within perturbation theory, we have no idea!

EOMEOM

What we can get from perturbation theory26D Bosonic string theory

There is a Tachyon modeThere is a Tachyon mode something like:

10D Superstring theory

Spacetime SUSY cancellationSpacetime SUSY cancellationsomething like:

People believe that People believe that non-perturbative effects non-perturbative effects can give a can give a non-trivial non-trivial potential upliftpotential uplift

We wish to know non-perturbative relationship among perturbative vacua, and possibility of non-perturbative vacua

We succeeded realizing various situations in 2D string theory!!We succeeded realizing various situations in 2D string theory!!

This has been a long standing problemThis has been a long standing problemHow does the How does the non-perturbativenon-perturbative string landscape string landscape look like?look like?

Non-perturbative string landscape and two-dimensional string theory

2D String Theory (’81-)2D String Theory (’81-) Non-perturbatively exactly solvableNon-perturbatively exactly solvable with with Matrix ModelsMatrix Models

Many people have tried to attack this issue in 2D string theory:

Moduli space of 2D string theory? Moduli space of 2D string theory? We can read from the worldsheet action:

Moduli Moduli On-shell op.(BRST coh.)On-shell op.(BRST coh.)

However, 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?

( : potential of matrix models)Configuration of eigenvaluesConfiguration of eigenvalues

Stable background 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].

For example:

Landscape of 2D (bosonic) string is like:

Can we consider more non-trivial situations?

National extension is the Multi-Cut Case

Perturbative string theory There are more D.O.F to interplay among various perturb. stringsThere are more D.O.F to interplay among various perturb. strings

Plan of the talk1. 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

String TheoryString Theory

Brief look at matrix modelsMatrix Models (N: size of Matrix)Matrix Models (N: size of Matrix)

SummationsSummations

Triangulation(Random surfaces)

Matrix ModelsMatrix Models “Multi-Cut” Matrix Models“Multi-Cut” Matrix Models

Feynman Diagram

Continuum limit

Worldsheet(2D surfaces)

Ising model+critical Ising model

TWO Hermitian matricesM: NxN Hermitian Matrix

String TheoryString Theory Fractional Superstring Theory [H.I’09]Fractional Superstring Theory [H.I’09]

This can be seen by introducing the Resolvent (Macroscopic Loop Amplitude)

V()

which gives spectral curve (generally algebraic curve):

In Large N limit (= semi-classical)

Let’s see more detailsLet’s see more details Diagonalization:

1

4

2

Continuum limit = Blow up some points of x on the spectral curveContinuum limit = Blow up some points of x on the spectral curve

The nontrivial things occur only around the turning points

N-body problem in the potential V

3

Eigenvalue densityEigenvalue density

Correspondence with string theoryCorrespondence with string theory

V()

13

4

bosonic

super

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]

2

After continuum limit,

TOO simple to claim string Landscape??

Let’s consider the Multi-Cut Critical Points:Let’s consider the Multi-Cut Critical Points:

2-cut critical points 3-cut critical points

Continuum limit = blow up [Crinkovik-Moore ‘91]

cutscuts

We can expect variety of vacua!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]the Z_k symmetric case [CISY’09]::

|t| |t|

t > 0 t < 0

(p,q) critical points with k cuts

e.g) the 3-cut cases are

the Z_k symmetric case [CISY’09]the Z_k symmetric case [CISY’09]:: (p,q) critical points with k cuts

Too many solutions!?

is natural because we have two choices

Each has different perturbative amplitudesEach has different perturbative amplitudes

Variety of solutionsVariety of solutions

This implies thatthe string Landscape of multi-cut matrix models is non-trivial

The multi-cut matrix models provide non-trivial models for non-perturbative string landscape!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]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 InfinityCuts run from Infinity to Infinity

What does it mean? What does it mean?

Factorization and Perturbative Isolation Perturbative Isolation [CIY’10]

The algebraic equation of the solution is factorized into irreducible curves:

But each curve only has cuts on real axes:

cutscutscutscuts

Didn’t we have multi-cut geometry?

????cutscuts

and

Factorization and Perturbative Isolation Perturbative Isolation [CIY’10]Our answer is:

cutscuts cutscuts

and

Recall : “Cut = discontinuity of algebraic function W(x)“

cutscuts

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]

another cutsanother cuts

Conjecture [CIY’10]

[Eynard-Orantin ‘07] All order Perturbative correlators only depend on F(x,W)=0

FactFact

IF (All-order PerturbativelyAll-order Perturbatively)

Factorization and Perturbative Isolation Perturbative Isolation [CIY’10]What is the physical meaning of these factrization?

Only non-perturbative interactions

Perturbative interactions

This system has many perturbatively isolated sectorsThis system has many perturbatively isolated sectors

Perturbative Vacua in non-perturbative string landscape

Within perturbation theory, we cannot distinguish perturbative stringsperturbative strings from perturbative isolated sectors!!perturbative isolated sectors!!

Perturbatively (all order) Bosonic string

Perturbatively (all order) type 0 Superstring

Perturbative strings Perturbative strings can appear as can appear as sectorssectors

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 systemPerturbative string theories are just segments of the whole system

What is the whole system? It’s non-critical M theory!It’s non-critical M theory!

As a MMother 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

Infinite-cut matrix model (Infinite-Fractional SST) is the Mother Theory of Fractional Superstring TheoryInfinite-cut matrix model (Infinite-Fractional SST) is the Mother Theory of Fractional Superstring Theory

In the same way, 12-FSST C 24-FSST C …… C ∞-FSST

As a non-critical MM 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

11D SUGRA

IIA

IIB

I

Het O

Het EM?

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.

As a non-critical MM 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)

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 systemDispersion relation of free fermions:

x

p

0B:

0A:

D0 brane charge ~ Centrifugal forceIn 1+2 fermion system, they can be

0A/0B superstrings from the 1+2 fermion system

0A: The polar coordinate

0B: The Cartesian coordinate

Angular momentum = D0 charge (KK momentum = D0 charge)

Hilbert space of gives 0A

Note that this is one of the models which realize their philosophy

Hilbert space of gives 0B

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

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

Multi-cut matrix models as a non-critical MM Theory [CIY’10]

As its own light, the multi-cut matrix models look like non-critical M theoryAs its own light, the multi-cut matrix models look like 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

As its own light, the multi-cut matrix models look like non-critical M theoryAs 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 theoryWe refer to the strong coupling dual theory as non-critical M theory

Summary of other prospects1. 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 non-trivial 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:

• 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?

our models should enable us to answerour models should enable us to answer

Interesting direction related to our modelsInteresting direction related to our models

• Investigation toward the non-perturbative region.• What does the corresponding Kontsevich type matrix

model look like?