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Fractional-Superstring A mplitudes from the Multi-Cut M atri x Models

Fractional-Superstring A mplitudes from the Multi-Cut M atri x 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”,

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Fractional-Superstring A mplitudes from the Multi-Cut M atri x Models

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  1. Fractional-Superstring Amplitudesfrom the Multi-Cut Matrix Models HirotakaIrie (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. 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 (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),” arXiv:0912.****, in preparation.

  2. Collaborators of [CISY]: [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 Chuan-Tsung Chan (Tunghai University, Taiwan) Sheng-Yu Darren Shih (NTU, Taiwan  UC, Berkeley, US [Sept. 1 ~]) Chi-HsienYeh (NTU, Taiwan)

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

  4. What is the multi-cut matrix models? • What should correspond to the multi-cut matrix models? • Macroscopic Loop Amplitudes • Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models

  5. What is the multi-cut matrix model ? TWO-matrix model Size of Matrices ONE-matrix 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 • (3,4) minimal CFT 2D Gravity(Liouville) conformal ghost • (p,q) minimal CFT 2D Gravity(Liouville) conformal ghost Continuum theories at multi-critical points CUTs (p,q) minimal string theory The two-matrix model includes more !

  6. Diagonalization: Let’s see more details 2 In Large N limit (= semi-classical) N-body problem in the potential V V(l) 1 3 l This can be seen by introducing the Resolvent (Macroscopic Loop Amplitude) 4 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

  7. Correspondence with string theory 2 V(l) l bosonic super 3 The number of Cuts is important: 1 1-cut critical points (2, 3 and 4) give (p,q) minimal (bosonic) string theory 4 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 ? ? Maximally TWO cuts around the critical points Continuum limit = blow up

  8. How to define the Multi-Cut Critical Points: [Crinkovik-Moore ‘91] Continuum limit = blow up 2-cut critical points 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]

  9. A brief summary for multi-comp. KP hierarchy [Fukuma-HI ‘06] the orthonormal polynomial systemof the matrix model: Claim 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. Continuum function of “n” At critical points (continuum limit):

  10. 2. The Z_k symmetry: k distinct orthonormal polynomials at the origin: k distinct smooth continuum functions at the origin: somehow k-component KP hierarchy Smooth function in x and l with Z_k symmetric cases: Z_k symmetry breaking cases: Summary: What happens in multi-cut matrix models? Multi-cut matrix models  k-component KP hierarchy We are going to give several notes The relation between α and f: [Crinkovik-Moore ‘91](ONE), [CISY1 ‘09](TWO) Labeling of distinct scaling functions

  11. Various notes [CISY1 ‘09] e.g.) (2,2) 3-cut critical point [CISY1 ‘09] Critical potential Blue = lower / yellow = higher

  12. Various notes [CISY1 ‘09] The operators A and B are given by The coefficient matrices are all REAL The Z_k symmetric critical points are given by ( : the shift matrix)

  13. The Z_k symmetric cases can be restricted to

  14. Various notes [CISY1 ‘09] The operators A and B are given by The coefficient matrices are all REAL functions The Z_k symmetric critical points are given by We can also break Z_k symmetry (difference from Z_k symmetric cases) ( : the shift matrix)

  15. What is the multi-cut matrix models? • What should correspond to the multi-cut matrix models? • Macroscopic Loop Amplitudes • Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models

  16. 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 2-cut critical points [Takayanagi-Toumbas ‘03], [Douglas et.al. ‘03], [Klebanov et.al ‘03] Because of WS Fermion Neveu-Shwartz sector: Ramond sector: Z_2 spin structure => Selection rule => Charge ONE-cut: TWO-cut: Superstrings ! Gauge it away by superconformal symmetry

  17. 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” R0 (= NS) sector: Rn sector: Z_k spin structure => Selection rule =>Charge ONE-cut: multi-cut: Gauge it away by Generalizedsuperconformal symmetry

  18. Which ψ ? [HI ‘09] Zamolodchikov-Fateev parafermion [Zamolodchikov-Fateev ‘85] 2. Central charge: 1. Basic parafermion fields: 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-fractionalsuper-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 are the same. [HI ‘09]

  19. 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: : R2 sector Z_ksym is broken (at least to Z_2) ! : R(k-2) sector

  20. Summary of Evidences and Issues [HI ‘09] • 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) Q1 • 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 !) Q2 Q3 Let’s consider macroscopic loop amplitudes !!

  21. What is the multi-cut matrix models? • What should correspond to the multi-cut matrix models? • Macroscopic Loop Amplitudes • Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models

  22. Macroscopic Loop Amplitudes Role of Macroscopic loop amplitudes 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 1st kind 2-cut general (p,q) critical point: [Seiberg-Shih ‘03] (from Liouville theory) :the Chebyshev Polynomial of the 2nd kind

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

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

  25. The k-Cut extension [CISY1 ‘09] The coefficients and are k x k matrices 1. The 0-th order: : Simultaneous diagonalization 2. The next order: (eigenvalues: j=1, 2, … , k ) Solve this system in the week coupling limit EZJ-DKK eq. they are no longer polynomials in general the Z_k symmetric case:

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

  27. Our Ansatz: [CISY1 ‘09] The EZJ-DKK eq. gives several polynomials…

  28. (k,l)=(3,1) cases [CISY1 ‘09] are the Jacobi Polynomial We identify

  29. Our Solution: [CISY1 ‘09] Jacobi polynomials Our ansatz is applicable to every k (=3,4,…) cut cases !! : 1stChebyshev : 2ndChebyshev

  30. Geometry [CISY1 ‘09] (1,1) critical point (l=1): (1,1) critical point (l=0):

  31. What is the multi-cut matrix models? • What should correspond to the multi-cut matrix models? • Macroscopic Loop Amplitudes • Fractional-Superstring Amplitudes from the Multi-Cut Matrix Models

  32. Z_k breaking cases: [CISY2 ‘09] The coefficients and are k x k matrices 1. The 0-th order:  Simultaneous diagonalization 2. The next order: (eigenvalues: j=1, 2, … , k ) Solve this system in the week coupling limit EZJ-DKK eq. they are no longer polynomials in general Z_k symmetry breaking cases:

  33. The Z_k symmetry breaking cases have general distributions We made a jump! How do we dodiagonalization ?

  34. The general k-cut (p,q) solution: [CISY2 ‘09] 1. The COSH solution (k is odd and even): 2. The SINH solution (k iseven): The deformed Chebyshev functions The deformed Chebyshev functions are solution to EZJ-DKK equation:

  35. The general k-cut (p,q) solution: [CISY2 ‘09] 1. The COSH solution (k is odd and even): 2. The SINH solution (k iseven): The characteristic equation: Algebraic equation of (ζ,z) Coeff. are all real functions

  36. The matrix realization for the every (p,q;k) solution [CISY2 ‘09] With a replacement: Repeatedly using the addition formulae, we can show for every pair of (p,q) !

  37. To construct the general form, we introduce “Reflected Shift Matrices” Shift matrix: Algebra: Ref. matrix: “Reflected Shift Matrices”

  38. The matrix realization for the every (p,q;k) solution [CISY2 ‘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.

  39. [CISY2 ‘09] Algebraic Equations The cosh solution: The sinh solution: Here, (p,q) is CFT labeling!!! Multi-Cut Matrix Model knows (p,q) Minimal Fractional String!

  40. [CISY2 ‘09] An Example of Algebraic Curves (k=4) COSH (µ<0) • SINH • (µ>0) : Z_2 charge conjugation (k=2) is equivalent to η=-1brane of type 0 minimal superstrings k=4 MFSST  (k=2) MSST with η=±1 ??

  41. Conclusion of Z_k breaking cases:[CISY2 ‘09] • We gave the solution to the general k-cut(p,q) Z_k breaking critical points. • They are cosh/sinhsolutions. We showed all the cases have the matrix realizations. • Only the cosh/sinhsolutions have the characteristic equations which are algebraic equations with real coefficients. (The other phase shifts are not allowed.) • 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.

  42. Some of Future Issues • Now it’s time to calculate the boundary states of fractional super-Liouville field theory ! • What is “fractional super-Riemann surfaces”? • Which string theory corresponds to the Z_k symmetric critical points(the Jacobi-poly. sol.) ? [CISY1’ 09] • Annulus amplitudes ? (Sym,Asym) • k=4 MFSST  (k=2) MSST with η=±1 (cf. [HI’07]) • The Multi-Cut Matrix Models May Include Many Many New Interesting Phenomena !!!

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