Stokes Phenomena and Non‐perturbative Completion in the multi‐cut matrix models
Hirotaka Irie (NTU) A collaboration with
Chuan‐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 talk
1. 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 1 String Thy 2
String Thy 4
String Thy 3
String Thy 4 String Thy 3
String Thy 2 String Thy 1
Potential
In the on‐shell formulation, this can be viewed as
However 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)
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:
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 Ù 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 Í resolvent operator
V(λ) λ
Eigenvalue density
This generally defines algebraic curve:
Spectral curve and Cuts
The information of eigenvalues Í 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
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 solutions
3. 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
Ù 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)
Γ
The Abelian case is the Hilbert transformation:
The solution in the general cases is also known
Γ
The general solution to is uniquely given as
G(z)
Z(z)
Γ
Γ
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 Æ 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: