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Slide1

Analytic Study for the String Theory Landscapes via Matrix Models(and Stokes Phenomena)

Hirotaka

Irie

Yukawa Institute for Theoretical Physics

, Kyoto Univ.

February 13

th

2013,

String Advanced Lecture

@ KEK

Based on collaborations with

Chuan-Tsung

Chan

(THU)

and

Chi-

Hsien

Yeh

(NCTS)Slide2

Perturbative string theory is well-knownDespite of several candidates for non-perturbative formulations (SFT,IKKT,BFSS,AdS/CFT…), we are still in the middle of the way: Stokes phenomenon is a bottom-up approach: especially, based on instantons and

Stokes phenomena.

In particular, within solvable/integrable string theory, we demonstrate how to understand the analytic aspects of the landscapes

General Motivation

How to define non-perturbatively complete string theory?

How to deal with the huge amount of string-theory vacua?Where is the true vacuum? Which are meta-stable vacua?How they decay into other vacua? How much is the decay rate?

How to reconstruct the non-perturbatively complete string theory from its perturbation theory?Slide3

Plan of the talkMotivation for Stokes phenomenon a) Perturbative knowledge from matrix models b) Spectral curves in the multi-cut matrix models (new feature related to Stokes phenomena)Stokes phenomena and isomonodromy systems a) Introduction to Stokes phenomenon (of Airy function) b) General k x k ODE systemsStokes phenomena in non-critical string theory a) Multi-cut boundary condition b) Quantum Integrability ---------- conclusion and discussion 1 ----------Analytic aspects of the string theory landscapes ---------- conclusion and discussion 2 ----------Slide4

Main referencesIsomonodromy theory and Stokes phenomenon to matrix models (especially of Airy and Painlevé cases)　　　　　　　 [Moore ’91]; [David ‘91] 　　　　　　　　　　　　　　　　　　　　　　[Maldacena-Moore-Seiberg-Shih ‘05]Isomonodromy theory, Stokes phenomenon and the Riemann-Hilbert (inverse monodromy) method (Painlevé cases: 2x2, Poincaré index r=2,3): 　　　　　　　　　　　　　[Its-Novokshenov '91]; [Fokas-Its-Kapaev-Novokshenov'06]

[FIKN]Slide5

Main referencesProposal of a first principle analysis for the string theory landscape [Chan-HI-Yeh 4 '12];[Chan-HI-Yeh 5 ‘13 in preparation] Stokes phenomena in general kxk isomonodromy systems corresponding to matrix models (general Poincaré index r)　[Chan-HI-Yeh 2 ‘10] ;[Chan-HI-Yeh 3 ’11

]

; 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 　　[Chan-HI-Yeh 4 '12] Spectral curves in the multi-cut matrix models　　　　　　 　[HI ‘09]; [Chan-HI-Shih-Yeh '09] ;[Chan-HI-Yeh 1 '10]

Chan

HI

Yeh(S.-Y. Darren) Shih[CIY][CISY]Slide6

1. Motivation for Stokes phenomenonRef) Spectral curves in the multi-cut matrix models: [CISY ‘09] [CIY1 ‘10]Slide7

Perturbative knowledge from matrix modelsLarge N expansion of matrix models(Non-critical) String theory

Continuum limit

Triangulation (Lattice Gravity)

(Large N expansion

 Perturbation

theory of string coupling g)We have known further more on non-perturbative string theory

CFTN x N matricesSlide8

Perturbative amplitudes of WSn:Non-perturbative amplitudes are D-instantons! [Shenker ’90, Polchinski ‘94]The overall weight θ’s (=Chemical Potentials) are out of the perturbation theory

Non-perturbative corrections

perturbative corrections

non-

perturbative (

instanton) correctionsD-instanton Chemical Potential

WS with Boundaries

=

open string theory

essential information

for

the

NonPert

. completion

CFT

CFT

Let’s see it more from the matrix-model viewpointsSlide9

The Resolvent op. allows us to read this informationV(l)l

In Large N limit

(= semi-classical)

Spectral curve

Diagonalization

:

N-body problem in the potential V

Eigenvalue

density

spectral curve

Position of Cuts = Position of

Eigenvalues

Resolvent

:Slide10

Why is it important?Spectral curve  Perturbative string theoryPerturbative correlatorsare all obtained recursively from the resolvent (S-D eqn., Loop eqn

…)

Therefore, we symbolically write the free energy as

Topological Recursions

[Eynard’04,

Eynard-Orantin ‘07]

Input::Bergman KernelEverything is algebraic geometric observables!Slide11

[David ‘91]Why is it important?Spectral curve  Perturbative string theoryNon-perturbative correctionsNon-perturbative partition functions:

[

Eynard ’08, Eynard-Marino ‘08]V

(l)

l

In Large N limit

(= semi-classical)

spectral curve

+1

-

1

w

ith some free parameters

Summation over all the possible configurations

D-

instanton

Chemical Potential

[David’91,93];[

Fukuma-Yahikozawa

‘96-’99];[

Hanada

-Hayakawa-

Ishibashi

-Kawai-Kuroki-

Matuso

-Tada ‘04];[Kawai-Kuroki-Matsuo ‘04];[Sato-Tsuchiya ‘04];[

Ishibashi

-Yamaguchi ‘05];[

Ishibashi

-Kuroki-Yamaguchi ‘05];[Matsuo ‘05];[Kuroki-

Sugino

‘06]…

This weight is not

algebraic geometric observable

;

but rather

analytic one

!

Theta function

onSlide12

the Position of “Eigenvalue” CutsWhat is the geometric meaning of the D-instanton chemical potentials?[CIY 2 ‘10]

But, we can also add

infinitely long cuts

From the Inverse

monodromy

(Riemann-Hilbert) problem [FIKN]

θ_I ≈ Stokes multipliers s_{l,I,j}“Physical cuts” as “Stokes lines of ODE”

How to distinguish them?

Later

This gives constraints on

θ

 T-systems on Stokes multipliers

Related to

Stokes phenomenon!

Require!



section 4Slide13

Why this is interesting?The multi-cut extension [Crinkovic-Moore ‘91];[Fukuma-HI ‘06];[HI ‘09] !1) Different string theories (ST) in spacetime [CIY 1 ‘10];[CIY 2 ‘10];[CIY 3 ‘11]

ST 1

ST 2

2) Different

perturbative

string-theory

vacua

in the landscape:

[CISY ‘09]; [CIY 2 ‘10]

We can study

the string-theory landscape

from the first principle

!



Gluing the spectral curves (STs)

Non-

perturbatively

(

Today’s first topic

)

 t

he Riemann-Hilbert problem

(

Today’s second topic

in sec. 4

)

ST 1

ST 2Slide14

2. Stokes phenomenon and isomonodromy systemsRef) Stokes phenomena and isomonodromy systems [Moore ‘91] [FIKN‘06] [CIY 2 ‘10]Slide15

The ODE systems for determinant operators (FZZT-branes)

The

resolvent, i.e. the spectral curve:Generally, this satisfies the following kind of linear ODE systems:

k-cut

 k x k matrix Q[Fukuma-HI ‘06];[CIY 2 ‘10]

For simplicity, we here assume: Poincaré index rSlide16

Stokes phenomenon of Airy functionAiry function: Asymptotic expansion!

This expansion is valid in

(from Wikipedia)

≈Slide17

+≈

(from Wikipedia)

Stokes phenomenon of Airy functionAiry function:

(valid in )

(valid in )

(relatively) Exponentially small

!

Asymptotic expansions are only applied in specific angular domains (

Stokes sectors

)

Differences of the expansions in the intersections are only by

relatively

and exponentially

small terms

Stokes multiplier

Stokes sectors

Stokes sectors

Stokes Data! Slide18

Stokes phenomenon of Airy functionAiry function: (valid in )

(valid in )

Stokes sectors

Stokes sectors

Keep using

differentSlide19

1) Complete basis of the asymptotic solutions: Stokes phenomenon of the ODE of the matrix models

…

1

2

0

19

3

4

5

6

…

18

17

…

D

0

D

3

12

…

D

12

2)

Stokes sectors

In the following, we skip this

3) Stokes phenomena

(relatively and exponentially small terms)Slide20

1) Complete basis of the asymptotic solutions: Stokes phenomenon of the ODE of the matrix modelsHere it is convenient to introduce General solutions:

…

Superposition of

wavefunction

with different

perturbative string theories

Spectral curve  Perturb. String TheorySlide21

Stokes sectors…

1

2

0

19

3

456…

18

17

…

D

0

D

3

12

…

D

12

Stokes phenomenon of

the

ODE of the matrix models

2)

Stokes sectors, and Stokes matrices

E.g.) r=2, 5 x 5,

γ

=2 (Z_5 symmetric)

Stokes matrices

0

1

3

…

…

19

18

17

12

…

4

5

6

7

8

…

2

D

0

D

3

D

12

larger

Canonical solutions (exact solutions)

How change the dominance

Keep usingSlide22

Stokes matrices: non-trivialThm [CIY2 ‘10]

0

1

2

3

D0D1456

7

Set of Stokes multipliers !

Stokes phenomenon of

the

ODE of the matrix models

3)

How to read the Stokes matrices?

:Profile of exponents

[CIY 2 ‘10]

E.g.) r=2, 5 x 5,

γ

=2 (Z_5 symmetric)Slide23

 section 4Inverse monodromy (Riemann-Hilbert) problem [FIKN]Direct monodromy problemGiven: Stokes matrices

Inverse

monodromy problem

Given

Solve

ObtainWKB

RHSolve

Obtain

Analytic problem

Consistency (Algebraic problem)

Special Stokes multipliers

which satisfy physical constraintsSlide24

Algebraic relations of the Stokes matricesZ_k –symmetry conditionHermiticity conditionMonodromy Free conditionPhysical constraint: The multi-cut boundary condition

This h

elps us to obtain explicit solutions for general (k,r)

m

ost difficult part!Slide25

3. Stokes phenomenon in non-critical string theoryRef) Stokes phenomena and quantum integrability [CIY2 ‘10][CIY3 ‘11]Slide26

Multi-cut boundary condition

3-cut case (q=1)

2-cut case (q=2:

pureSUGRA

)Slide27

≈+(from Wikipedia)Stokes phenomenon of Airy functionAiry function:

(valid in )

(valid in )

Change of dominance

(Stokes line)

Dominant!

Dominant! Slide28

≈+(from Wikipedia)Stokes phenomenon of Airy function

(valid in )

Change of dominance

(Stokes line)

Airy system

 (2,1) topological

minimal string theory



Eigenvalue

cut

of the matrix model

Dominant!

Dominant!

Physical cuts = lines with dominance change

(Stokes lines)

[MMSS ‘05]

discontinuitySlide29

Multi-cut boundary condition [CIY 2 ‘10]…

1

2

0

19

3

456…1817

…

D

0

D

3

12

…

D

12

0

1

2

3

…

…

19

18

17

D

0

12

…

…

5

6

7

8

E.g.) r=2, 5 x 5,

γ

=2 (Z_5 symmetric)

All the horizontal lines are Stokes lines!

All lines are candidates of the cuts!Slide30

Multi-cut boundary condition [CIY 2 ‘10]…

1

2

0

19

3

456…1817

…

D

0

D

3

12

…

D

12

0

1

2

…

…

19

18

17

3

D

0

12

…

…

5

6

7

8

E.g.) r=2, 5 x 5,

γ

=2 (Z_5 symmetric)

We choose “k” of them

as

physical cuts!

k-cut

 k x k matrix Q

[

Fukuma

-HI ‘06];[CIY 2 ‘10]

≠0

≠0

=

0

Constraints on

SnSlide31

Multi-cut boundary condition

3-cut case (q=1)

2-cut case (q=2:

pureSUGRA

)Slide32

0123D0

D

14567

E.g.) r=2, 5 x 5,

γ=2 (Z_5 symmetric)

: non-trivialThm [CIY2 ‘10]

Set of Stokes multipliers ! The set of non-trivial Stokes multipliers?Use Profile of dominant exponents [CIY 2 ‘10]Slide33

Quantum integrability [CIY 3 ‘11]0123……19

18

1712……56

7

8

E.g.) r=2, 5 x 5,

γ=2 (Z_5 symmetric)

This equation only includes the Stokes multipliers of

Then, the equation becomes T-systems:

cf

)

ODE/IM correspondence

[

Dorey-Tateo

‘98];[J. Suzuki ‘99]

the Stokes phenomena of special Schrodinger equations

satisfy the T-systems of quantum

integrable

models

with the boundary condition:

How about the other Stokes multipliers?

Set of Stokes multipliers ! Slide34

Complementary Boundary cond. [CIY 3 ‘11]0123……19

18

1712……56

7

8

E.g.) r=2, 5 x 5,

γ=2 (Z_5 symmetric)

This equation only includes the Stokes multipliers of

Then, the equation becomes T-systems:

with the boundary condition:

Shift the BC !

Generally there are “

r

”

such BCs

(Coupled multiple T-systems)Slide35

Solutions for multi-cut cases (Ex: r=2, k=2m+1):

m

1

m-1

2

m-2

3

m-3

4

m-4

5

m-5

6

m-6

7

m-7

8

m

1

m-1

2

m-2

3

m-3

4

m

-4

5

m-5

6

m-6

7

m-7

8

n

n

n

n

are

written with Young diagrams (

avalanches

):

(Characters of the anti-Symmetric representation of GL)

[CIY 2 ‘10] [CIY3 ‘11]

In addition, they are “coupled multiple T-systems” Slide36

4. Summary (part 1)The D-instanton chemical potentials are the missing information in the perturbative string theory. This information is responsible for the non-perturbative relationship among perturbative string-theory vacua, and important for study of the string-theory landscape from the first principle. In non-critical string theory (or generally matrix models), this information is described by the positions of the physical cuts. The multi-cut boundary conditions, which turn out to be T-systems of quantum

integrable

systems, can give a part of the constraints on the non-perturbative systemAlthough physical meaning of the complementary BC is still unclear (in progress [CIY 4 ‘12]), it allows us to obtain explicit expressions of the Stokes multipliers. Slide37

discussionsPhysical meaning of the Compl. BCs? The system is described not only by the resolvent? We need other degree of freedom to complete the system? ( FZZT-Cardy branes? [CIY 3 ‘11])D-instanton chemical potentials are determined by “strange constraints” which are expressed as quantum integrability.Are there more natural explanations of the multi-cut BC? ( Use Duality? Strong string-coupling description?

 Non-critical M theory?, Gauge theory?)Slide38

4. Analytic aspects of the string theory landscapesRef) Analytic Study for the string theory landscapes [CIY4 ‘12]Then, can we extract the analytic aspects of the landscapes i.e. true vacuum, meta-stability and decay rate ?From Stokes Data, we reconstruct string theory nonperturbatively

YES !Slide39

Reconstruction of [(p,q) minimal] string theory [CIY4 ‘12]There are p branches k = p

Spectral Curve

1st Chebyshev polynomials:

Consider

p x p Sectional Holomorphic function　

 Generally Z(x) should be sectional holomorphic functionNon-pert. Strings

ReconstructAsymp. Exp(

x  ∞

∈

C

)

s.t.

Keep using

We don’t start with ODE! Slide40

Jump lines: 　　　　　　　　　　　

∞

Essential Singularity

6

3

7

8

9

2

1

5

4

Jump line

Asymp

.

Exp

（

x



∞

∈

C

）

s.t.

Keep using

Constant Matrix

These matrices are equivalent to Stokes matricesSlide41

∞

Essential Singularity

: Constant Matrices

(



Isomonodromy

systems

)

Junctions:

In particular, at

essential singularities

, there appears the

monodromy

equation:

6

3

7

8

9

7

1

2

Jump lines:

2

1

5

4

This is what we have solved!

Jump line

Preservation of matrices: e.g.)

given

Constant Matrix



Jump lines are topological

(except for essential singularities)Slide42

∞

Essential Singularity

: Constant Matrices

(



Isomonodromy

systems

)

Junctions:

In particular, at

essential singularities

, there appears the

monodromy

equation:

6

3

7

8

9

7

1

2

Jump lines:

2

1

5

4

This is what we have solved!

Jump line

Preservation of matrices: e.g.)

given



Jump lines are topological (except for essential singularities)

Ψ(x)

can be uniquely solved by the integral equation on

：

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 （

e.g. [FIKN]

）

In fact

Obtain Ψ

RH

(x)

(Riemann-Hilbert problem)Slide43

Reconstruction and the Landscapes

∞

Essential Singularity

6

3

7

8

9

2

1

5

4

Consider deformations:

String Theory Landscape:

Land

All the

onshell

/

offshell

configurations

of

string theory background

w

hich satisfy

B.G.

indenpendence

Then

the result of RH problem Ψ

RH

(x)

is

the same

!

singular behavior

Does not

change

the singular structureSlide44

Reconstruction and the LandscapesPert. and Nonpert. Corrections

Physical Meaning of

∞

Essential Singularity

6

3

7

8

9

2

1

5

4

φ

(x)

∈　

Land

str

From Topological Recursions

How far from each other

φ'

(x)

∈　

Land

str

The same!

Different!

as “

Steepest

Descent

curves of

φ

(x)

(Anti-Stokes lines)

”

mean field path-integrals

in

matrix models

[CIY4

‘12

]Slide45

E.g.) (2,3) minimal strings (Pure-Gravity)Multi-cut BC (=matrix models) gives

Basic Sol.

∞

Essential Singularity

6

3

7

8

9

2

1

5

4

NOTE

coincide with matrix models

(a half of

[

Hanada

et.al. ‘04])

Free energy

[CIY4 ‘12]

Small

instantons



stable vacuumSlide46

E.g.) (2,5) minimal strings (Yang-Lee edge)Multi-cut BC (= matrix models) gives

∞

Essential Singularity

6

3

7

8

9

2

1

5

4

Basic Sol

Free energy

Large

instantons



unstable

( or

meta-stable

)

(1,2)

ghost

ZZ

brane

[no (1,1) ZZ

brane

]

[CIY4 ‘12]

NOTE

coincide with matrix models

( (1,2)ZZ

brane

in

[Sato-Tsuchiya ‘04]…)

Decay Rate?

Extract

meta-stable system

by

deforming

path-integral

[Coleman

]Slide47

E.g.) (2,5) minimal strings (Yang-Lee edge)Multi-cut BC (= matrix models) gives∞

Essential Singularity

6

3

7

8

9

2

1

5

4

Free energy

Decay rate

(= deform. )

NOTE

Coincide with matrix models

(

[Sato-Tsuchiya ‘04]…)

Decay rates of this string theory

(1,1) ZZ

brane

[no (1,2) ZZ

brane

]

[CIY4 ‘12]

Large

Instanton

True vacuum?

Choose

BG

in the landscape

Land

str

so that it achieves

small

instantonsSlide48

Basic SolE.g.) (2,5) minimal strings (Yang-Lee edge)Free energy

Multi-cut BC

(= matrix models) gives

∞

Essential Singularity

6

3

7

8

9

2

1

5

4

True vacuum

[CIY4 ‘12]

φ

TV

(x)

∈　

Land

str

It is not simple string theory

Deformed by

elliptic function

Large

InstantonsSlide49

Summary and conclusion, part 2D instanton chemical potentials are equivalent to Stokes data by Riemann-Hilbert methodsWith giving Stokes data, we can fix all the non-perturbative information of string theoryIn fact, we have seen that Stokes data is directly related to meta-stability/decay rate/true vacuum of the theoryInstability of minimal strings is caused by ghost D-

instantons

, whose existence is controlled by Stokes dataDiscussion:What is non-perturbative

principle of string theory?

What is the rule of duality in string landscapes?We now have all the controll

over non-perturbative string theory with description of spectral curves and resulting matrix modelsSlide50

Thank you for your attention!