Analytic Study for the String Theory Landscapes via Matrix Models
(and Stokes Phenomena)
Hirotaka Irie Yukawa Institute for Theoretical Physics, Kyoto Univ.
February 13th 2013, String Advanced Lecture @ KEK
Based on collaborations withChuan-Tsung Chan (THU) and Chi-Hsien Yeh (NCTS)
• Perturbative string theory is well-known• Despite 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 MotivationHow 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?
Plan of the talk1. Motivation for Stokes phenomenon
a) Perturbative knowledge from matrix models b) Spectral curves in the multi-cut matrix models (new feature related to Stokes phenomena)
2. Stokes phenomena and isomonodromy systems a) Introduction to Stokes phenomenon (of Airy function) b) General k x k ODE systems
3. Stokes phenomena in non-critical string theory a) Multi-cut boundary condition b) Quantum Integrability ---------- conclusion and discussion 1 ----------
4. Analytic aspects of the string theory landscapes ---------- conclusion and discussion 2 ----------
Main references
• Isomonodromy 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]
Main references• Proposal 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]
1. Motivation for Stokes phenomenonRef) Spectral curves in the multi-cut matrix models:
[CISY ‘09] [CIY1 ‘10]
Perturbative knowledge from matrix models
Large 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
CFT
N x N matrices
1. Perturbative amplitudes of WSn:
2. Non-perturbative amplitudes are D-instantons! [Shenker ’90, Polchinski ‘94]
3. The overall weight θ’s (=Chemical Potentials) are out of the perturbation theory
Non-perturbative corrections
perturbative corrections non-perturbative (instanton) corrections
D-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 viewpoints
The Resolvent op. allows us to read this information
V(l)
l
In Large N limit (= semi-classical)
Spectral curve
Diagonalization:
N-body problem in the potential V
Eigenvalue density
spectral curvePosition of Cuts = Position of Eigenvalues
Resolvent:
Why is it important? Spectral curve Perturbative string theoryPerturbative correlators
are 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 Kernel
Everything is algebraic geometric observables!
[David ‘91]
Why is it important? Spectral curve Perturbative string theory
Non-perturbative corrections
Non-perturbative partition functions: [Eynard ’08, Eynard-Marino ‘08]
V(l)
l
In Large N limit (= semi-classical)
spectral curve
+1-1
with 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 functionon
the Position of “Eigenvalue” Cuts
What 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 4
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)
the Riemann-Hilbert problem (Today’s second topic in sec. 4)
ST 1
ST 2
2. Stokes phenomenon and isomonodromy systems
Ref) Stokes phenomena and isomonodromy systems [Moore ‘91] [FIKN‘06] [CIY 2 ‘10]
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 r
Stokes phenomenon of Airy functionAiry function:
Asymptotic expansion! This expansion is valid in
(from Wikipedia)
≈
+≈
(from Wikipedia)
Stokes phenomenon of Airy functionAiry function:
(valid in )
(valid in )
(relatively) Exponentially small !
1. Asymptotic expansions are only applied in specific angular domains (Stokes sectors)
2. Differences of the expansions in the intersections are only by relatively and exponentially small terms
Stokes multiplier Stokes sectors
Stokes sectors
Stokes Data!
Stokes phenomenon of Airy functionAiry function:
(valid in )
(valid in )
Stokes sectors
Stokes sectors
Keep usingdifferent
1) Complete basis of the asymptotic solutions:
Stokes phenomenon of the ODE of the matrix models
… 12
019
3456…
1817…
D0
D3
12…
D12
2) Stokes sectors
In the following, we skip this
3) Stokes phenomena (relatively and exponentially small terms)
1) Complete basis of the asymptotic solutions:
Stokes phenomenon of the ODE of the matrix models
Here it is convenient to introduce
General solutions: …
Superposition of wavefunction with different perturbative string theories
Spectral curve Perturb. String Theory
Stokes sectors
…
12
019
3456…
1817…
D0
D3
12…
D12
Stokes phenomenon of the ODE of the matrix models2) Stokes sectors, and Stokes matrices
E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric)
Stokes matrices
01
3
……
19
1817
12
…
4
56
78
…
2D0
D3
D12
larger
Canonical solutions (exact solutions)
How change the dominance
Keep using
Stokes matrices
: non-trivial
Thm [CIY2 ‘10] 0
1
2
3
D0
D1
4
5
6
7
Set of Stokes multipliers !
Stokes phenomenon of the ODE of the matrix models3) How to read the Stokes matrices? :Profile of exponents [CIY 2 ‘10]
E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric)
section 4
Inverse monodromy (Riemann-Hilbert) problem [FIKN]Direct monodromy problem
Given: Stokes matrices
Inverse monodromy problem
Given
Solve
Obtain
WKBRH
Solve
Obtain
Analytic problem
Consistency (Algebraic problem)
Special Stokes multipliers which satisfy physical constraints
Algebraic relations of the Stokes matrices
1. Z_k –symmetry condition
2. Hermiticity condition
3. Monodromy Free condition
4. Physical constraint: The multi-cut boundary condition
This helps us to obtain explicit solutions for general (k,r)
most difficult part!
3. Stokes phenomenon in non-critical string theory
Ref) Stokes phenomena and quantum integrability [CIY2 ‘10][CIY3 ‘11]
Multi-cut boundary condition
3-cut case (q=1) 2-cut case (q=2: pureSUGRA)
≈ +
(from Wikipedia)
Stokes phenomenon of Airy functionAiry function:
(valid in )
(valid in )
Change of dominance (Stokes line)
Dominant!
Dominant!
≈ +
(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]
discontinuity
Multi-cut boundary condition [CIY 2 ‘10]
…
12
019
3456…
1817…
D0
D3
12…
D12
012
3
……
19
1817
D0
12
……
56
78
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!
Multi-cut boundary condition [CIY 2 ‘10]
…
12
019
3456…
1817…
D0
D3
12…
D12
012
……
19
1817
3
D0
12
……
56
78
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 Sn
Multi-cut boundary condition
3-cut case (q=1) 2-cut case (q=2: pureSUGRA)
0
1
2
3
D0
D1
4
5
6
7
E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric)
: non-trivial
Thm [CIY2 ‘10]
Set of Stokes multipliers !
The set of non-trivial Stokes multipliers?Use Profile of dominant exponents [CIY 2 ‘10]
Quantum integrability [CIY 3 ‘11]
012
3
……
19
1817
12
……
56
78
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 !
Complementary Boundary cond. [CIY 3 ‘11]
012
3
……
19
1817
12
……
56
78
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)
Solutions for multi-cut cases (Ex: r=2, k=2m+1):
m1
m-12
m-23
m-34
m-45
m-56
m-67
m-78
m1
m-12
m-23
m-34
m-45
m-56
m-67
m-78
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”
4. Summary (part 1)1. The D-instanton chemical potentials are the missing
information in the perturbative string theory. 2. 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.
3. In non-critical string theory (or generally matrix models), this information is described by the positions of the physical cuts.
4. 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 system
5. Although 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.
discussions1. Physical 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])
2. 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?)
4. Analytic aspects of the string theory landscapes
Ref) 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 !
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 function
Non-pert. StringsReconstruct
Asymp. Exp( x ∞ ∈ C)
s.t.
Keep using
We don’t start with ODE!
Jump lines: ∞
Essential Singularity
6
3
78
9
2
1
54
Jump line
Asymp. Exp （ x ∞ ∈ C ）
s.t.
Keep using
Constant Matrix
These matrices are equivalent to Stokes matrices
∞
Essential Singularity
1. : Constant Matrices ( Isomonodromy systems) 2. Junctions:
3. In particular, at essential singularities, there appears the monodromy equation:
6
3
78
9
7 1
2
Jump lines:
2
1
54
This is what we have solved!
Jump line
Preservation of matrices: e.g.)
givenConstant Matrix
Jump lines are topological (except for essential singularities)
∞
Essential Singularity
1. : Constant Matrices ( Isomonodromy systems) 2. Junctions:
3. In particular, at essential singularities, there appears the monodromy equation:
6
3
78
9
7 1
2
Jump lines:
2
1
54
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 ΨRH(x) (Riemann-Hilbert problem)
Reconstruction and the Landscapes
∞
Essential Singularity
6
3
78
9
2
1
54
Consider deformations:
String Theory Landscape: LandAll the onshell/offshell configurations of string theory background
which satisfy
1. B.G. indenpendence
Then the result of RH problem ΨRH(x) is the same!
singular behavior
Does not change the singular structure
Reconstruction and the Landscapes
2. Pert. and Nonpert. Corrections
3. Physical Meaning of
∞
Essential Singularity
6
3
78
9
2
1
54
φ(x) ∈ Landstr
From Topological Recursions How far from each other
φ'(x) ∈ Landstr
The same! Different!
as “Steepest Descent curves of φ(x) (Anti-Stokes lines)” mean field path-integrals in matrix models [CIY4 ‘12]
E.g.) (2,3) minimal strings (Pure-Gravity)
Multi-cut BC (=matrix models) gives
Basic Sol. ∞
Essential Singularity
6
3
78
9
2
1
54
NOTE coincide with matrix models (a half of [Hanada et.al. ‘04])Free energy
[CIY4 ‘12]
Small instantons stable vacuum
E.g.) (2,5) minimal strings (Yang-Lee edge)
Multi-cut BC (= matrix models) gives
∞
Essential Singularity
6
3
78
9
2
1
54Basic 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]
E.g.) (2,5) minimal strings (Yang-Lee edge)
Multi-cut BC (= matrix models) gives
∞
Essential Singularity
6
3
78
9
2
1
54
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 Landstrso that it achieves small instantons
Basic Sol
E.g.) (2,5) minimal strings (Yang-Lee edge)
Free energy
Multi-cut BC (= matrix models) gives
∞
Essential Singularity
6
3
78
9
2
1
54True vacuum
[CIY4 ‘12]
φTV(x) ∈ Landstr
It is not simple string theory
Deformed by elliptic function
Large Instantons
Summary and conclusion, part 21. D instanton chemical potentials are equivalent to Stokes data
by Riemann-Hilbert methods2. With giving Stokes data, we can fix all the non-perturbative
information of string theory
3. In fact, we have seen that Stokes data is directly related to meta-stability/decay rate/true vacuum of the theory
4. Instability of minimal strings is caused by ghost D-instantons, whose existence is controlled by Stokes data
Discussion:1. What is non-perturbative principle of string theory? 2. 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 models
Thank you for your attention!
