Fractional-Superstring Amplitudesfrom the Multi-Cut Matrix Models
Hirotaka Irie (National Taiwan University, Taiwan)
20 Dec. 2009 @ Taiwan String Workshop 2010Ref) HI, “Fractional supersymmetric Liouville theory and the multi-cut matrix models”, Nucl. Phys. B819 [PM] (2009) 351CT-Chan, HI,SY-Shih and CH-Yeh, “Macroscopic loop amplitudes in the multi-cut two-matrix models,” Nucl. Phys. B (10.1016/j.nuclphysb.2009.10.017), in pressCT-Chan, HI,SY-Shih and CH-Yeh, “Fractional-superstring amplitudes from the multi-cut two-matrix models (tentative),” arXiv:0912.****, in preparation.
Collaborators of [CISY]:
Chuan-Tsung Chan (Tunghai University, Taiwan)
Sheng-Yu Darren Shih (NTU, Taiwan UC, Berkeley, US [Sept. 1 ~])
Chi-Hsien Yeh (NTU, Taiwan)
[CISY1] “Macroscopic loop amplitudes in the multi-cut two-matrix models”, Nucl. Phys. B (10.1016/j.nuclphysb.2009.10.017)
[CISY2] “Fractional-superstring amplitudes from the multi-cut two-matrix models”, arXiv:0912.****, in preparation
String Theory
Over look of the storyMatrix Models (N: size of Matrix)
CFT on Riemann surfaces
Integrable hierarchy (Toda / KP)
Summation over 2D Riemann Manifolds
Triangulation(Random surfaces)
Matrix Models “Multi-Cut” Matrix ModelsQ.
Feynman Diagram
Gauge fixing
Continuum limit
Worldsheet(2D Riemann mfd)
Ising model+critical Ising model
TWO Hermitian matricesM: NxN Hermitian Matrix
1. What is the multi-cut matrix models?
2. What should correspond to the multi-cut matrix models?
3. Macroscopic Loop Amplitudes
4. Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models
What is the multi-cut matrix model ? TWO-matrix model
Matrix String Feynman Graph = Spin models on Dynamical Lattice
TWO-matrix model Ising model on Dynamical Lattice
Continuum theory at the simplest critical point
Critical Ising models coupled to 2D Worldsheet Gravity
Continuum theories at multi-critical points
(p,q) minimal string theory The two-matrix model includes more !
CUTs
ONE-matrix
(3,4) minimal CFT 2D Gravity(Liouville) conformal ghost
(p,q) minimal CFT 2D Gravity(Liouville) conformal ghost
Size of Matrices
This can be seen by introducing the Resolvent (Macroscopic Loop Amplitude)
V(l)
l
which gives spectral curve (generally algebraic curve):
In Large N limit (= semi-classical)
Let’s see more details Diagonalization:
1
4
2
Continuum limit = Blow up some points of x on the spectral curve
The nontrivial things occur only around the turning points
N-body problem in the potential V
3
Correspondence with string theory
V(l)
l
13
4
bosonic
super
The number of Cuts is important:
1-cut critical points (2, 3 and 4) give (p,q) minimal (bosonic) string theory
2-cut critical point (1) gives (p,q) minimal superstring theory (SUSY on WS) [Takayanagi-Toumbas ‘03], [Douglas et.al. ‘03], [Klebanov et.al ‘03]
What happens in the multi-cut critical points?
Continuum limit = blow up Maximally TWO cuts around the critical points
2
?
Where is multi-cut ?
How to define the Multi-Cut Critical Points:
Why is it good? This system is controlled by k-comp. KP hierarchy [Fukuma-HI ‘06]
The matrix Integration is defined by the normal matrix:
2-cut critical points 3-cut critical points
Continuum limit = blow up [Crinkovik-Moore ‘91]
We consider that this system naturally has Z_k transformation:
A brief summary for multi-comp. KP hierarchy [Fukuma-HI ‘06]
the orthonormal polynomial system of the matrix model:Claim
Derives the Baker-Akiehzer function system of k-comp. KP hierarchy
Recursive relations:
Infinitely many Integrable deformation
Toda hierarchy
k-component KP hierarchyLax operators
k x k Matrix-Valued Differential Operators
1. Index Shift Op. -> Index derivative Op.
Continuum function of “n”At critical points (continuum limit):
2. The Z_k symmetry:
k distinct orthonormal polynomials at the origin:
k distinct smooth continuum functions at the origin:somehow
Smooth function in x and lThe relation between α and f: [Crinkovik-Moore ‘91](ONE), [CISY1 ‘09](TWO)
Labeling of distinct scaling functions
with
k-component KP hierarchy
Z_k symmetric cases:Z_k symmetry breaking cases:Summary: 1. What happens in multi-cut matrix models?2. Multi-cut matrix models k-component KP hierarchy3. We are going to give several notes
Blue = lower / yellow = higher
e.g.) (2,2) 3-cut critical point [CISY1 ‘09]
Critical potential
Various notes [CISY1 ‘09]
1. The coefficient matrices are all REAL
2. The Z_k symmetric critical points are given by
The operators A and B are given by
( : the shift matrix)
Various notes [CISY1 ‘09]
The Z_k symmetric cases can be restricted to
1. The coefficient matrices are all REAL functions
2. The Z_k symmetric critical points are given by
3. We can also break Z_k symmetry (difference from Z_k symmetric cases)
The operators A and B are given by
( : the shift matrix)
Various notes [CISY1 ‘09]
1. What is the multi-cut matrix models?
2. What should correspond to the multi-cut matrix models?
3. Macroscopic Loop Amplitudes
4. Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models
What should correspond to the multi-cut matrix models?
Reconsider the TWO-cut cases:
2-cut critical points[Takayanagi-Toumbas ‘03], [Douglas et.al. ‘03], [Klebanov et.al ‘03]
Z_2 transformation = (Z_2) R-R charge conjugation
Because of WS Fermion
ONE-cut: TWO-cut:
Gauge it away by superconformal symmetry Superstrings !
[HI ‘09]
Neveu-Shwartz sector: Ramond sector:
Z_2 spin structure => Selection rule => Charge
What should correspond to the multi-cut matrix models?
How about MULTI (=k) -cut cases:
Z_k transformation =?= Z_k “R-R charge” conjugation
Because of “Generalized Fermion”
ONE-cut: multi-cut:
[HI ‘09]
R0 (= NS) sector: Rn sector:
Z_k spin structure => Selection rule => Charge
[HI ‘09]
3-cut critical points
Gauge it away by Generalized superconformal symmetry
Minimal k-fractional super-CFT:
Which ψ ? Zamolodchikov-Fateev parafermion[Zamolodchikov-Fateev ‘85]
Why is it good? It is because the spectrums of the (p,q) minimal fractional super-CFT and the multi-cut matrix models are the same. [HI ‘09]
[HI ‘09]
1. Basic parafermion fields: 2. Central charge:
3. The OPE algebra:
k parafermion fields: Z_k spin-fields:
4. The symmetry and its minimal models of :k-Fractional superconformal sym.
(k=1: bosonic (l+2,l+3), k=2: super (l+2,l+4))
GKO construction
(p,q) Minimal k-fractional superstrings
This means that Z_k symmetry is broken (at least to Z_2) !
Minimal k-fractional super-CFT requires Screening charge:
Correspondence [HI ‘09]
: R2 sector
: R(k-2) sector
Z_k sym isbroken
(at least to Z_2) !
Z_k symmetric case:
Labeling of critical points (p,q) and Operator contents (ASSUME: OP of minimal CFT and minimal strings are 1 to 1)
Residual symmetry On both sectors, the Z_k symmetry broken or to Z_2.
String susceptibility (of cosmological constant)
Summary of Evidences and Issues
Fractional supersymmetric Ghost system bc ghost + “ghost chiral parafermion” ?
Covariant quantization / BRST Cohomology Complete proof of operator correspondence
Correlators (D-brane amplitudes / macroscopic loop amplitudes) Ghost might be factorized (we can compare Matrix/CFT !)
[HI ‘09]
Q1
Q2
Q3
Let’s consider macroscopic loop amplitudes !!
1. What is the multi-cut matrix models?
2. What should correspond to the multi-cut matrix models?
3. Macroscopic Loop Amplitudes
4. Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models
Macroscopic Loop AmplitudesRole of Macroscopic loop amplitudes
1. Eigenvalue density2. Generating function of On-shell Vertex Op:3. FZZT-brane disk amplitudes (Comparison to String Theory)4. D-Instanton amplitudes also come from this amplitude
:the Chebyshev Polynomial of the 1st kind
1-cut general (p,q) critical point: [Kostov ‘90]
:the Chebyshev Polynomial of the 2nd kind
2-cut general (p,q) critical point: [Seiberg-Shih ‘03] (from Liouville theory)
Off critical perturbation
Solve this system at the large N limit
The Daul-Kazakov-Kostov prescription (1-cut) [DKK ‘93]
: Orthogonal polynomial [Gross-Migdal ‘90]
More precisely [DKK ‘93]
Solve this system in the week coupling limit(Dispersionless k-comp. KP hierarchy)
The Daul-Kazakov-Kostov prescription (1-cut) [DKK ‘93]
Eynard-Zinn-Justin-Daul-Kazakov-Kostov eq.
Solve this system in the week coupling limit(Dispersionless k-componet KP hierarchy)
Dimensionless valuable:
Addition formula of trigonometric functions (q=p+1: Unitary series)
Scaling
The k-Cut extension [CISY1 ‘09]
the Z_k symmetric case:
1. The 0-th order:
2. The next order:
: Simultaneous diagonalization
they are no longer polynomials in general
EZJ-DKK eq.
Solve this system in the week coupling limit
The coefficients and are k x k matrices
(eigenvalues: j=1, 2, … , k )
The Z_k symmetric cases can be restricted to
We can easily solve the diagonalization problem !
Our Ansatz: [CISY1 ‘09]
The EZJ-DKK eq. gives several polynomials…
are the Jacobi PolynomialWe identify
(k,l)=(3,1) cases [CISY1 ‘09]
Our Solution: [CISY1 ‘09]
Our ansatz is applicable to every k (=3,4,…) cut cases !!
: 1st Chebyshev
: 2nd Chebyshev
Jacobi polynomials
Geometry [CISY1 ‘09]
(1,1) critical point (l=1):
(1,1) critical point (l=0):
1. What is the multi-cut matrix models?
2. What should correspond to the multi-cut matrix models?
3. Macroscopic Loop Amplitudes
4. Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models
Z_k breaking cases: [CISY2 ‘09]
Z_k symmetry breaking cases:
1. The 0-th order:
2. The next order:
Simultaneous diagonalization
they are no longer polynomials in general
EZJ-DKK eq.
Solve this system in the week coupling limit
The coefficients and are k x k matrices
(eigenvalues: j=1, 2, … , k )
The Z_k symmetry breaking cases have general distributions
How do we do diagonalization ? We made a jump!
The general k-cut (p,q) solution:
1. The COSH solution (k is odd and even):
2. The SINH solution (k is even):
[CISY2 ‘09]
The deformed Chebyshev functions
The deformed Chebyshev functions are solution to EZJ-DKK equation:
The general k-cut (p,q) solution:
1. The COSH solution (k is odd and even):
2. The SINH solution (k is even):
[CISY2 ‘09]
The characteristic equation:
Algebraic equation of (ζ,z) Coeff. are all real functions
The matrix realization for the every (p,q;k) solution
With a replacement:
Repeatedly using the addition formulae, we can show
[CISY2 ‘09]
for every pair of (p,q) !
To construct the general form, we introduce “Reflected Shift Matrices”
Shift matrix: Ref. matrix:
Algebra:
“Reflected Shift Matrices”
The matrix realization for the every (p,q;k) solution
In this formula we define:
[CISY2 ‘09]
A matrix realization for the COSH solution (p,q;k):
A matrix realization for the SINH solution (p,q;k), (k is even):
The other realizations are related by some proper siminality tr.
Algebraic EquationsThe cosh solution:
The sinh solution:
Here, (p,q) is CFT labeling !!!
Multi-Cut Matrix Model knows (p,q) Minimal Fractional String!
[CISY2 ‘09]
An Example of Algebraic Curves (k=4) [CISY2 ‘09]
COSH(µ<0)
SINH(µ>0)
: Z_2 charge conjugation
is equivalent to η=-1 brane of type 0 minimal superstrings(k=2)
k=4 MFSST (k=2) MSST with η=±1 ??
Conclusion of Z_k breaking cases:[CISY2 ‘09]
1. We gave the solution to the general k-cut (p,q) Z_k breaking critical points.
2. They are cosh/sinh solutions. We showed all the cases have the matrix realizations.
3. Only the cosh/sinh solutions have the characteristic equations which are algebraic equations with real coefficients. (The other phase shifts are not allowed.)
4. This includes the ONE-cut and TWO-cut cases. The (p,q) CFT labeling naturally appears! The cosh/sinh indicates the Modular S-matrix of the CFT.
Some of Future Issues1. Now it’s time to calculate the boundary states
of fractional super-Liouville field theory !
2. What is “fractional super-Riemann surfaces”?
3. Which string theory corresponds to the Z_k symmetric critical points (the Jacobi-poly. sol.) ? [CISY1’ 09]
4. Annulus amplitudes ? (Sym,Asym)
5. k=4 MFSST (k=2) MSST with η=±1 (cf. [HI’07])
The Multi-Cut Matrix Models May Include Many Many New Interesting Phenomena !!!
Top Related