perturbative Completion in the multicut matrix models Hirotaka Irie NTU A collaboration with Chuan Tsung Chan THU and Chi Hsien Yeh NTU Ref CIY2 CT Chan HI and CH ID: 557538
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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.