Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models Hirotaka Irie (National Taiwan University, Taiwan) 20 Dec. Taiwan String Workshop 2010 Ref) HI, “Fractional supersymmetric Liouville theory and the multi-cut matrix models”, Nucl. Phys. B819 [PM] (2009) 351 CT-Chan, HI,SY-Shih and CH-Yeh, “Macroscopic loop amplitudes in the multi-cut two-matrix models,” Nucl. Phys. B ( /j.nuclphysb ), in press CT-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 ( /j.nuclphysb ) [CISY2] “Fractional-superstring amplitudes from the multi-cut two- matrix models”, arXiv:0912.****, in preparation

String Theory Over look of the story Matrix Models (N: size of Matrix) CFT on Riemann surfaces Integrable hierarchy (Toda / KP) Integrable hierarchy (Toda / KP) Summation over 2D Riemann Manifolds Summation over 2D Riemann Manifolds Triangulation (Random surfaces) Matrix Models “Multi-Cut” Matrix Models Q. 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( ) which gives spectral curve (generally algebraic curve): In Large N limit (= semi-classical) Let’s see more details Diagonalization: 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( ) 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 points3-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 hierarchy Lax 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 l The 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 hierarchy 3.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. It is because the spectrums of the (p,q) minimal fractional super-CFT and the multi-cut matrix models are the same. [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 is broken (at least to Z_2) ! Z_k sym is broken (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 Amplitudes Role of Macroscopic loop amplitudes 1.Eigenvalue density 2.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 Polynomial We identify (k,l)=(3,1) cases [CISY1 ‘09]

Our Solution:[CISY1 ‘09] Our ansatz is applicable to every k (=3,4,…) cut cases !! : 1 st Chebyshev : 2 nd 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 Equations The 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 Issues 1.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 !!!