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

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]

From String Theory to the Standard Model

• String 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? and Why?

The string-theory landscape:

• 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

From String Theory to the Standard Model

Plan of the talk1. Which information is necessary for the string-

theory landscape?

2. Stokes phenomena and the Riemann-Hilbert approach in non-critical string theory

3. The non-perturbative completion program and its solutions

4. Summary and prospects

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

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 1String Thy 2String Thy 4

String Thy 3

String Thy 4String Thy 3String Thy 2String Thy 1

Potential

In the on-shell formulation, this can be viewed as

However this picture implicitly assumes an off-shell formulation

String Thy 4String Thy 3String Thy 2String Thy 1

Potential

String Thy 4String Thy 3String Thy 2String Thy 1

Therefore, the information from the on-shell formulation are

Free-energy: Instanton actions:

(and their higher order corrections)

From these information,

D-instanton chemical potentials

With proper D-instanton chemical potentials

we can recover the partition function:

String Thy 4String Thy 3String Thy 2String Thy 1

Free-energy: Instanton actions:

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?

2. Stokes phenomena and the Riemann-Hilbert approach in non-critical string theory

- D-instanton chemical potentials Stokes data -

Multi-Cut Matrix Models

Matrix model:

The matrices X, Y are normal matrices

The contour γ is chosen as

3-cut matrix models

Spectral curve and Cuts

The information of eigenvalues resolvent operator

V(l)

l

Eigenvalue density

This generally defines algebraic curve:

Spectral curve and Cuts

The information of eigenvalues resolvent operator

cuts

Orthonormal polynomials

Orthonormal 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 z

ODE system in the Multi-cut case

Q(t;z) is a polynomial in z

The 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 unity

Stokes phenomena in ODE system

The kxk Matrix-valued solution

Asymptotic expansion around

1. Coefficients are written with coefficients of Q(t;z)2. Matrix C labels k solutions3. This expansion is only valid in some angular domain

Stokes phenomena in ODE system

The plane is expanded into several pieces:

Even though Ψ satisfy the asym exp:

After an analytic continuation, the asym exp is generally different:

Stokes phenomena in ODE system

Introduce Canonical solutions:

Stokes matrices:

These matrices Sn are called Stokes Data D-instanton chemical potentials

The Riemann-Hilbert problem

For 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 Abelian case is the Hilbert transformation:

The solution in the general cases is also known

G

The general solution to

is uniquely given as

G(z)

Z(z)

G

G

The RH problem in the ODE system

We make a patch of canonical solutions:

Then Stokes phenomena is Dicontinuity:

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)

3. The non-perturbative completion program and its solutions

Cuts from the ODE system

The Orthonormal polynomial is

Is a k-rank vector

Recall

The discontinuity of the function

The discontinuity of the resolvent

Non-perturbative definition of cuts

The discontinuity appears when the exponents change dominance:

Is a k-rank vector

Therefore, the cuts should appear when

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)

Solutions for multi-cut cases:

Discrete solutions

Characterized by

Which is also written with Young diagrams (avalanches):

Symmetric polynomials

Solutions for multi-cut cases:

Continuum solutions

The polynomials Sn are related to Schur polynomials Pn:

4. Summary1. Here we saw how the Stokes data of orthonormal polynomials

are related to the D-instanton chemical potentials2. 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 potentials

3. Our procedure in the two-cut case correctly fix all the chemical potentials and results in the Hastings-McLeod solution.

4. 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.

5. It is interesting if these solutions imply some dynamical remnants of strong-coupling theory, like M/F-theory.