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 !!!