FractionalSuperstring Amplitudes from the MultiCut Matrix Models Hirotaka

Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models Hirotaka Irie (National Taiwan University, Taiwan) 20 Dec. 2009 @ Taiwan String Workshop 2010 Ref) HI, “Fractional supersymmetric Liouville theory and the multi-cut matrix models”, Nucl. Phys. B 819 [PM] (2009) 351 CT-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 press CT-Chan, HI, SY-Shih and CH-Yeh, “Fractional-superstring amplitudes from the multi-cut two-matrix models (tentative), ” ar. Xiv: 0912. ****, in preparation.

Collaborators of [CISY]: [CISY 1] “Macroscopic loop amplitudes in the multi-cut two-matrix models”, Nucl. Phys. B (10. 1016/j. nuclphysb. 2009. 10. 017) [CISY 2] “Fractional-superstring amplitudes from the multi-cut twomatrix models”, ar. Xiv: 0912. ****, in preparation Chuan-Tsung Chan (Tunghai University, Taiwan) Sheng-Yu Darren Shih (NTU, Taiwan UC, Berkeley, US [Sept. 1 ~]) Chi-Hsien Yeh (NTU, Taiwan)

Over look of the story Matrix Models (N: size of Matrix) CFT on Riemann surfaces String Theory M: TWO Nx. N Hermitian matrices Matrix Integrable hierarchy (Toda / KP) Gauge fixing Summation over 2 D Riemann Manifolds Feynman Diagram Worldsheet (2 D Riemann mfd) +critical Ising model Triangulation (Random surfaces) Q. Matrix Models Continuum limit “Multi-Cut” Matrix Models

1. What is the multi-cut matrix models? 2. What should correspond to the multicut 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 Size of Matrices Matrix String ONE-matrix 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 2 D Worldsheet Gravity (3, 4) minimal CFT 2 D Gravity(Liouville) conformal ghost Continuum theories at multi-critical points (p, q) minimal CFT 2 D Gravity(Liouville) (p, q) minimal string theory conformal ghost CUTs The two-matrix model includes more !

Let’s see more details In Large N limit (= semi-classical) V(l) Diagonalization: 2 N-body problem in the potential V 1 3 l 4 This can be seen by introducing the Resolvent (Macroscopic Loop Amplitude) which gives spectral curve (generally algebraic curve): Continuum limit = Blow up some points of x on the spectral curve The nontrivial things occur only around the turning points

Correspondence with string theory 2 V(l) 1 3 4 l super The number of Cuts is important: bosonic 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? Where is multi-cut ? ? Continuum limit = blow up Maximally TWO cuts around the critical points

How to define the Multi-Cut Critical Points: Continuum limit = blow up 2 -cut critical points [Crinkovik-Moore ‘ 91] 3 -cut critical points The matrix Integration is defined by the normal matrix: We consider that this system naturally has Z_k transformation: Why is it good? This system is controlled by k-comp. KP hierarchy [Fukuma-HI ‘ 06]

A brief summary for multi-comp. KP hierarchy Claim [Fukuma-HI ‘ 06] the orthonormal polynomial system of the matrix model: Derives the Baker-Akiehzer function system of k-comp. KP hierarchy Recursive relations: Infinitely many Integrable deformation Toda hierarchy Lax operators k-component KP hierarchy k x k Matrix-Valued Differential Operators 1. Index Shift Op. -> Index derivative Op. At critical points (continuum limit): Continuum function of “n”

2. The Z_k symmetry: k distinct orthonormal polynomials at the origin: somehow k distinct smooth continuum functions at the origin: Smooth function in x and l The relation between α and f: [Crinkovik-Moore ‘ 91](ONE), [CISY 1 ‘ 09](TWO) Labeling of distinct scaling functions k-component KP hierarchy with Z_k symmetry symmetric 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

Various notes [CISY 1 ‘ 09] e. g. ) (2, 2) 3 -cut critical point [CISY 1 ‘ 09] Critical potential Blue = lower / yellow = higher

Various notes [CISY 1 ‘ 09] The operators A and B are given by 1. The coefficient matrices are all REAL 2. The Z_k symmetric critical points are given by ( : the shift matrix)

The Z_k symmetric cases can be restricted to

Various notes [CISY 1 ‘ 09] The operators A and B are given by 1. The coefficient matrices are all REAL functions 2. The Z_k symmetric critical points are given by ( : the shift matrix) 3. We can also break Z_k symmetry (difference from Z_k symmetric cases)

1. What is the multi-cut matrix models? 2. What should correspond to the multicut 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? [HI ‘ 09] Reconsider the TWO-cut cases: Z_2 transformation = (Z_2) R-R charge conjugation [Takayanagi-Toumbas ‘ 03], [Douglas et. al. ‘ 03], [Klebanov et. al ‘ 03] Neveu-Shwartz sector: 2 -cut critical points Because of WS Fermion Ramond sector: Z_2 spin structure => Selection rule => Charge ONE-cut: TWO-cut: Gauge it away by superconformal symmetry Superstrings !

What should correspond to the multi-cut matrix models? [HI ‘ 09] How about MULTI (=k) -cut cases: [HI ‘ 09] Z_k transformation =? = Z_k “R-R charge” conjugation 3 -cut critical points Because of “Generalized Fermion” R 0 (= NS) sector: Rn sector: Z_k spin structure => Selection rule => Charge ONE-cut: multi-cut: Gauge it away by Generalized superconformal symmetry

Which ψ ? [HI ‘ 09] Zamolodchikov-Fateev parafermion [Zamolodchikov-Fateev ‘ 85] 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. GKO construction Minimal k-fractional super-CFT: (k=1: bosonic (l+2, l+3), k=2: super (l+2, l+4)) Why is it good? It is because the spectrums of the (p, q) minimal fractional super-CFT and the multi-cut matrix models [HI ‘ 09] are the same.

Correspondence [HI ‘ 09] (p, q) Minimal k-fractional superstrings Z_k symmetric case: This means that Z_k symmetry is broken (at least to Z_2) ! Minimal k-fractional super-CFT requires Screening charge: : R 2 sector Z_k sym is broken : R(k-2) sector (at least to Z_2) !

Summary of Evidences and Issues [HI ‘ 09] ü Labeling of critical points (p, q) and Operator contents (ASSUME: OP of minimal CFT and minimal strings are 1 to 1) ü Residual symmetry On both sectors, the Z_k symmetry broken or to Z_2. ü String susceptibility (of cosmological constant) Q 1 p Fractional supersymmetric Ghost system bc ghost + “ghost chiral parafermion” ? Q 2 p Covariant quantization / BRST Cohomology Complete proof of operator correspondence Q 3 p Correlators (D-brane amplitudes / macroscopic loop amplitudes) Ghost might be factorized (we can compare Matrix/CFT !) Let’s consider macroscopic loop amplitudes !!

1. What is the multi-cut matrix models? 2. What should correspond to the multicut 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. 2. 3. 4. Eigenvalue density Generating function of On-shell Vertex Op: FZZT-brane disk amplitudes (Comparison to String Theory) D-Instanton amplitudes also come from this amplitude 1 -cut general (p, q) critical point: [Kostov ‘ 90] Off critical perturbation : the Chebyshev Polynomial of the 1 st kind 2 -cut general (p, q) critical point: [Seiberg-Shih ‘ 03] (from Liouville theory) : the Chebyshev Polynomial of the 2 nd kind

The Daul-Kazakov-Kostov prescription (1 -cut) [DKK ‘ 93] : Orthogonal polynomial [Gross-Migdal ‘ 90] Solve this system in the week coupling limit Solve thisk-comp. system. KP at hierarchy) the large N limit (Dispersionless More precisely [DKK ‘ 93]

The Daul-Kazakov-Kostov prescription (1 -cut) [DKK ‘ 93] Solve this system in the week coupling limit (Dispersionless k-componet KP hierarchy) Dimensionless valuable: Scaling Eynard-Zinn-Justin-Daul-Kazakov-Kostov eq. Addition formula of trigonometric functions (q=p+1: Unitary series)

The k-Cut extension [CISY 1 ‘ 09] Solve this system in the week coupling limit The coefficients and are k x k matrices 1. The 0 -th order: : Simultaneous diagonalization 2. The next order: (eigenvalues: j=1, 2, … , k ) EZJ-DKK eq. they are no longer polynomials in general the Z_k symmetric case:

The Z_k symmetric cases can be restricted to We can easily solve the diagonalization problem !

Our Ansatz: [CISY 1 ‘ 09] The EZJ-DKK eq. gives several polynomials…

(k, l)=(3, 1) cases [CISY 1 ‘ 09] We identify are the Jacobi Polynomial

Our Solution: [CISY 1 ‘ 09] Jacobi polynomials Our ansatz is applicable to every k (=3, 4, …) cut cases !! : 1 st Chebyshev : 2 nd Chebyshev

Geometry [CISY 1 ‘ 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 multicut matrix models? 3. Macroscopic Loop Amplitudes 4. Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models

Z_k breaking cases: [CISY 2 ‘ 09] Solve this system in the week coupling limit The coefficients and are k x k matrices 1. The 0 -th order: Simultaneous diagonalization 2. The next order: (eigenvalues: j=1, 2, … , k ) EZJ-DKK eq. they are no longer polynomials in general Z_k symmetry breaking cases:

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: [CISY 2 ‘ 09] 1. The COSH solution (k is odd and even): 2. The SINH solution (k is even): The deformed Chebyshev functions are solution to EZJ-DKK equation:

The general k-cut (p, q) solution: [CISY 2 ‘ 09] 1. The COSH solution (k is odd and even): 2. The SINH solution (k is even): The characteristic equation: Algebraic equation of (ζ, z) Coeff. are all real functions

The matrix realization for the every (p, q; k) solution [CISY 2 ‘ 09] With a replacement: Repeatedly using the addition formulae, we can show 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 [CISY 2 ‘ 09] In this formula we define: 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 [CISY 2 ‘ 09] The cosh solution: The sinh solution: Here, (p, q) is CFT labeling !!! Multi-Cut Matrix Model knows (p, q) Minimal Fractional String!

An Example of Algebraic Curves (k=4) [CISY 2 ‘ 09] COSH (µ<0) SINH (µ>0) : Z_2 charge conjugation (k=2) is equivalent to η=-1 brane of type 0 minimal superstrings k=4 MFSST (k=2) MSST with η=± 1 ? ?

Conclusion of Z_k breaking cases: [CISY 2 ‘ 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. ) ? [CISY 1’ 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 New Interesting Phenomena !!!