Integrability in Noncritical StringM Theory and in the Multicut Matrix Models Hirotaka Irie Yukawa Institute for Theoretical Physics May 17 th 2012 Nagoya Univ ID: 561556
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Slide1
Stokes Phenomena and Quantum Integrability in Non-critical String/M Theory(and in the Multi-cut Matrix Models)
Hirotaka
Irie
(Yukawa Institute for Theoretical Physics)
May
17
th
2012
@
Nagoya Univ.
Based on collaborations with
Chuan-
Tsung
Chan
(THU)
and
Chi-
Hsien
Yeh
(NTU)Slide2
String theory is defined by perturbation theoryDespite of several candidates for
non-
perturbative formulations (SFT, Matrix theory…), we are still in the middle of the way: Stokes phenomenon is a bottom-up approach: Here we study non-critical string theory. In particular, we will see that the multi-cut matrix models provide a nice toy model for this fundamental investigation.
General Motivation
How 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
?Slide3
Plan of the talkMotivation for Stokes phenomenon (from physics)
a) Perturbative knowledge from matrix models b) Spectral curves in the multi-cut matrix models (new feature related to Stokes phenomena)Stokes phenomena and isomonodromy systems a) Introduction to Stokes phenomenon (of Airy function) b) General k x k ODE systemsStokes phenomena in non-critical string theory a) Multi-cut boundary condition b) Quantum IntegrabilitySummary and discussionSlide4
Main referencesIsomonodromy theory and Stokes phenomenon to matrix models (especially of Airy and Painlevé cases)
Isomonodromy theory, Stokes phenomenon and the Riemann-Hilbert (inverse monodromy) method (Painlevé cases: 2x2, Poincaré index r=2,3):
[David ‘91] [Moore '91]; [Maldacena-Moore-Seiberg-Shih '05][Its-Novokshenov '91]; [Fokas-Its-Kapaev
-Novokshenov'06]
[FIKN]Slide5
Main referencesStokes phenomena in
general kxk isomonodromy systems
corresponding to matrix models (general Poincaré index)Spectral curves in the multi-cut matrix models[Chan-HI-Yeh 2 '10] ;[Chan-HI-Yeh 3 '11]; [Chan-HI-Yeh 4 '12, in preparation] [Chan-HI-Shih-Yeh '09]
;[Chan-HI-Yeh 1 '10]
Chan
HIYeh(S.-Y. Darren) Shih[CIY][CISY]Slide6
1. Motivation for Stokes phenomenon(from physics)Ref) Spectral curves in the multi-cut matrix models: [CISY ‘09] [CIY1 ‘10]Slide7
Perturbative knowledge from matrix modelsLarge N expansion of matrix models
(Non-critical) String theory
Continuum limit
Triangulation (Lattice Gravity)
(Large N expansion
Perturbation
t
heory of string coupling g)
There are many investigation on
non-
perturbative
string theory
CFT
N x N matricesSlide8
Perturbative amplitudes of
WS
n:Non-perturbative amplitudes are D-instantons! [Shenker ’90, Polchinski ‘94]The overall weight θ
’s (=Chemical Potentials) are out of the perturbation theory
Non-perturbative corrections
perturbative correctionsnon-perturbative (instanton) corrections
D-
instanton
Chemical Potential
WS with Boundaries
=
open string theory
Let’s see more from the matrix-model viewpoints
CFT
CFTSlide9
The
Resolvent
op. allows us to read this informationV(l)
l
In Large N limit
(= semi-classical)
Spectral curve
Diagonalization
:
N-body problem in the potential V
Eigenvalue
density
spectral curve
Position of Cuts = Position of
Eigenvalues
Resolvent
:Slide10
Why is it important?
Spectral curve
Perturbative string theoryPerturbative correlatorsare 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 observables!Slide11
Why is it important?
Spectral curve
Perturbative string theoryNon-perturbative correctionsNon-perturbative partition functions: [Eynard
’08, Eynard-Marino ‘08]
V
(l)l
In Large N limit
(= semi-classical)
spectral curve
+1
-
1
w
ith some free parameters
Summation over all the possible configurations
D-
instanton
Chemical Potential
[David’91,93];[
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 observable
;
but rather
analytic one
! Slide12
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
Stokes phenomenon!
Require!Slide13
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 topic)
t
he Riemann-Hilbert problem
([FIKN] for PII, 2-cut)
ST 1
ST 2Slide14
2. Stokes phenomenon and isomonodromy systemsRef) Stokes phenomena and isomonodromy
systems
[Moore ‘91] [FIKN‘06] [CIY 2 ‘10]Slide15
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 rSlide16
Stokes phenomenon of Airy function
Airy function:
Asymptotic expansion!
This expansion is valid in
(from Wikipedia)
≈Slide17
+
≈
(from Wikipedia)
Stokes phenomenon of Airy function
Airy function:
(valid in )
(valid in )
(relatively) Exponentially small
!
Asymptotic expansions are only applied in specific angular domains (
Stokes sectors
)
Differences of the expansions in the intersections are only by
relatively
and exponentially
small terms
Stokes multiplier
Stokes sectors
Stokes sectors
Stokes Data! Slide18
Stokes phenomenon of Airy function
Airy function:
(valid in )
(valid in )
Stokes sectors
Stokes sectors
Keep using
differentSlide19
1) Complete basis of the asymptotic solutions:
Stokes phenomenon of
the ODE of the matrix models
…
1
2
0
19
3
4
5
6
…
18
17
…
D
0
D
3
12
…
D
12
2)
Stokes sectors
In the following, we skip this
3) Stokes phenomena
(relatively and exponentially small terms)Slide20
1) Complete basis of the asymptotic solutions:
Stokes phenomenon of
the ODE of the matrix modelsHere it is convenient to introduce General solutions:
…
Superposition of
wavefunction with different perturbative string theories
Spectral curve
Perturb. String TheorySlide21
Stokes sectors
…
1
2
0
19
3
4
5
6
…
18
17
…
D
0
D
3
12
…
D
12
Stokes phenomenon of
the
ODE of the matrix models
2)
Stokes sectors, and Stokes matrices
E.g.) r=2, 5 x 5,
γ
=2 (Z_5 symmetric)
Stokes matrices
0
1
3
…
…
19
18
17
12
…
4
5
6
7
8
…
2
D
0
D
3
D
12
larger
Canonical solutions (exact solutions)
How change the dominance
Keep usingSlide22
Stokes matrices
: non-trivial
Thm [CIY2 ‘10]
0
1
23
D
0
D
1
4
5
6
7
Set of Stokes multipliers !
Stokes phenomenon of
the
ODE of the matrix models
3)
How to read the Stokes matrices?
:
Prifile
of exponents
[CIY 2 ‘10]
E.g.) r=2, 5 x 5,
γ
=2 (Z_5 symmetric)Slide23
Inverse monodromy (Riemann-Hilbert) problem [FIKN]
Direct
monodromy problemGiven: Stokes matrices
Inverse
monodromy problem
GivenSolve
Obtain
WKB
RH
Solve
Obtain
Analytic problem
Consistency (Algebraic problem)
Special Stokes multipliers
which satisfy physical constraintsSlide24
Algebraic relations of the Stokes matrices
Z_k
–symmetry conditionHermiticity conditionMonodromy Free conditionPhysical constraint: The multi-cut boundary condition
This h
elps us to obtain explicit solutions for general (
k,r)
m
ost difficult part!Slide25
3. Stokes phenomenon in non-critical string theoryRef) Stokes phenomena and quantum integrability
[CIY2 ‘10][CIY3 ‘11]Slide26
Multi-cut boundary condition
3-cut case (q=1)
2-cut case (q=2:
pureSUGRA
)Slide27
≈
+
(from Wikipedia)Stokes phenomenon of Airy functionAiry function:
(valid in )
(valid in )
Change of dominance
(Stokes line)
Dominant!
Dominant! Slide28
≈
+
(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]
discontinuitySlide29
Multi-cut boundary condition [CIY 2 ‘10]
…
1
2
0
19
3
4
5
6
…
18
17
…
D
0
D
3
12
…
D
12
0
1
2
3
…
…
19
18
17
D
0
12
…
…
5
6
7
8
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!Slide30
Multi-cut boundary condition [CIY 2 ‘10]
…
1
2
0
19
3
4
5
6
…
18
17
…
D
0
D
3
12
…
D
12
0
1
2
…
…
19
18
17
3
D
0
12
…
…
5
6
7
8
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
SnSlide31
Multi-cut boundary condition
3-cut case (q=1)
2-cut case (q=2:
pureSUGRA
)Slide32
0
1
23D0
D
145
67E.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
Prifile
of dominant exponents
[CIY 2 ‘10]Slide33
Quantum integrability [CIY 3 ‘11]
0
123……
19
181712…
…5678
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 ! Slide34
Complementary Boundary cond. [CIY 3 ‘11]
0
123……
19
181712…
…5678
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)Slide35
Solutions for multi-cut cases (Ex: r=2, k=2m+1):
m
1
m-1
2
m-2
3
m-3
4
m-4
5
m-5
6
m-6
7
m-7
8
m
1
m-1
2
m-2
3
m-3
4
m
-4
5
m-5
6
m-6
7
m-7
8
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” Slide36
SummaryThe D-instanton chemical potentials are the missing information in
the perturbative string theory. 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. In non-critical string theory, this information is described by the positions of the physical cuts. 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 systemAlthough 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. Slide37
discussionsPhysical 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]; [CIY4 ’12 in progress])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?)Slide38
Thank you for your attention!