Stokes Phenomena and Non- - PowerPoint Presentation

Slide1

Stokes Phenomena and Non-perturbative Completion in the multi-cut matrix models

Hirotaka Irie (NTU)A collaboration withChuan-Tsung Chan (THU) and Chi-Hsien Yeh (NTU)

Ref)

[CIY2] C.T. Chan, HI and C.H.

Yeh

, “Stokes Phenomena and Non-

perturbative

Completion in the Multi-cut Two-matrix Models,” arXiv:1011.5745 [

hep-th

]Slide2

From String Theory to the Standard ModelString theory is a promising candidate to unify the four fundamental forces in our universe. In particular, we wish to identify

the SM in the string-theory landscape and understand the reason why the SM is realized in our universe.

We are here?

a

nd Why?

The s

tring-theory landscape:Slide3

There are several approaches to extract information of the SM from String Theory (e.g. F-theory GUT).One approach is to derive the SM from the first principle.

That is, By studying non-perturbative structure of the string-theory landscape. We hope that study of non-critical strings and matrix models help us obtain further understanding of the string landscape

F

rom String

T

heory to the Standard ModelSlide4

Plan of the talkWhich information is necessary for the string-theory landscape?Stokes phenomena and the Riemann-Hilbert approach in non-critical string theory

The non-perturbative completion program and its solutions Summary and prospectsSlide5

1. Which information is necessary for the string-theory landscape?Slide6

What is the string-theory moduli space? There are two kinds of moduli spaces:

Non-normalizable moduli (external parameters in string theory)Normalizable moduli (sets of on-shell vacua in string theory)

Scale

of observation, probe fields and their coordinates,

initial and/or boundary conditions, non-

normalizable

modes…

String Thy 1

String Thy 2

String Thy 4

String Thy 3

String Thy 4

String Thy 3

String Thy 2

String Thy 1

PotentialSlide7

In the on-shell formulation, this can be viewed asHowever this picture implicitly assumes an off-shell formulation

String Thy 4

String Thy 3

String Thy 2

String Thy 1

Potential

String Thy 4

String Thy 3

String Thy 2

String Thy 1

Therefore, the information from the on-shell formulation are

Free-energy:

Instanton

actions:

(and their higher order corrections)Slide8

From these information,

D-instanton chemical potentials

With proper D-

instanton

chemical potentials

we can recover the partition function:

String Thy 4

String Thy 3

String Thy 2

String Thy 1

Free-energy:

Instanton

actions:Slide9

The reconstruction from perturbation theory:

String Theory

There are several choices

of D-

instantons

to construct

the partition function with

some

D-

instanton

chemical potentials

θ

are

usually integration constants of the differential equations.

The

D-inst. Chem. Pot. Is relevant to non-

perturbative

behaviors

Requirements of consistency constraints for

Chem.Pot

.

=

Non-

perturbative

completion program

What are the physical chemical potentials,

and how we obtain?Slide10

2. Stokes phenomena and the Riemann-Hilbert approach in non-critical string theory- D-instanton chemical potentials  Stokes data -Slide11

Multi-Cut Matrix ModelsMatrix model:The matrices X, Y are

normal matrices The contour γ is chosen as

3

-cut matrix modelsSlide12

Spectral curve and CutsThe information of eigenvalues

 resolvent operator

V

(

l

)

l

Eigenvalue

density

This generally defines algebraic curve: Slide13

Spectral curve and CutsThe information of eigenvalues

 resolvent operator

cutsSlide14

Orthonormal polynomialsOrthonormal

polynomial:In the continuum limit (at critical points of matrix models),

The

orthonormal

polynomials satisfy the following ODE system:

Q(

t;z

) and P(

t;z

) are polynomial in zSlide15

ODE system in the Multi-cut case

Q(t;z) is a polynomial in zThe leading of Q(t;z) (“Z_k symmetric critical points”)

k-cut case =

kxk

matrix-valued system

There are k solutions to this ODE system

k-

th

root of unitySlide16

Stokes phenomena in ODE system

The kxk Matrix-valued solutionAsymptotic expansion around

Coefficients are written with coefficients of Q(

t;z

)

Matrix C

labels k solutions

This expansion is only valid in some angular domainSlide17

Stokes phenomena in ODE system

The plane is expanded int

o several pieces:

Even though

Ψ

satisfy the

asym

exp:

After an analytic continuation, the

asym

exp is generally different: Slide18

Stokes phenomena in ODE system

Introduce Canonical solutions:

Stokes matrices:

These matrices

Sn

are called

Stokes Data

 D-

instanton

chemical potentialsSlide19

The Riemann-Hilbert problemFor

a given contour Γ and a kxk matrix valued holomorphic function G(z) on z in Γ,Find a kxk holomorphic function Z(z)

on z in

C -

Γ

which satisfies

G(z)

Z(z)

G

The

A

belian

case is the Hilbert transformation:

The solution in the general cases is also known

GSlide20

The general solution to

is uniquely given as

G(z)

Z(z)

G

GSlide21

The RH problem in the ODE system

We

make a patch of

c

anonical solutions:

Then Stokes phenomena is

Dicontinuity

:Slide22

The RH problem in the ODE system

Therefore, the solution to the ODE system is given as

With

In this expression, the Stokes matrices

Sn

are understood as D-

instanton

chemical potentials

(g(

t;z

) is an off-shell string-background)Slide23

3. The non-perturbative completion program and its solutions Slide24

Cuts from the ODE system

The Orthonormal polynomial is Is a k-rank vector

Recall

The discontinuity of the function

The discontinuity of the

resolventSlide25

Non-perturbative definition of cutsThe discontinuity appears when the exponents change dominance:

Is a k-rank vector

Therefore, the cuts should appear when Slide26

The two-cut constraint in the two-cut case:General situation of ODE:The cuts in the

resolvent:

This (+

α

)

gives constraints on the Stokes matrices

Sn

 the

Hastings-McLeod solution

(no free parameter)Slide27

Solutions for multi-cut cases:Discrete solutions

Characterized by

Which is also written with Young diagrams (

avalanches

):

Symmetric polynomialsSlide28

Solutions for multi-cut cases:Continuum solutions

The polynomials

Sn

are

related to

Schur

polynomials Pn

:Slide29

4. SummaryHere we saw how the Stokes data of orthonormal polynomials are related to the D-instanton chemical potentials

Non-perturbative definition of cuts on the spectral curve does not necessarily create the desired number of cuts. This gives non-perturbative consistency condition on the D-instanton chemical potentialsOur procedure in the two-cut case correctly fix all the chemical potentials and results in the Hastings-McLeod solution.We have obtained several solutions in the multi-cut cases. The discrete solutions are labelled by Young diagrams. The continuum solutions are written with Schur polynomials.It is interesting if these solutions imply some dynamical remnants of strong-coupling theory, like M/F-theory.