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Stokes Phenomena and Non-perturbative Completion in the multi-cut matrix models Hirotaka Irie (NTU) A collaboration with Chuan-Tsung Chan (THU) and Chi-Hsien Yeh (NTU) Ref) [CIY2] C.T. Chan, HI and C.H. Yeh, Stokes Phenomena and Non-perturbative Completion in the Multi-cut Two-matrix Models, arXiv:1011.5745 [hep-th]

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From String Theory to the Standard Model String theory is a promising candidate to unify the four fundamental forces in our universe. In particular, we wish to identify the SM in the string-theory landscape and understand the reason why the SM is realized in our universe. We are here? and Why? The string-theory landscape:

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There are several approaches to extract information of the SM from String Theory (e.g. F-theory GUT). One approach is to derive the SM from the first principle. That is, By studying non-perturbative structure of the string-theory landscape. We hope that study of non-critical strings and matrix models help us obtain further understanding of the string landscape From String Theory to the Standard Model

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Plan of the talk 1.Which information is necessary for the string- theory landscape? 2.Stokes phenomena and the Riemann-Hilbert approach in non-critical string theory 3.The non-perturbative completion program and its solutions 4.Summary and prospects

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1. Which information is necessary for the string- theory landscape?

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What is the string-theory moduli space? There are two kinds of moduli spaces: Non-normalizable moduli (external parameters in string theory) Normalizable moduli (sets of on-shell vacua in string theory) Scale of observation, probe fields and their coordinates, initial and/or boundary conditions, non-normalizable modes… String Thy 1String Thy 2 String Thy 4 String Thy 3 String Thy 4String Thy 3 String Thy 2 String Thy 1Potential

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In the on-shell formulation In the on-shell formulation, this can be viewed as However this picture implicitly assumes an off-shell formulation String Thy 4String Thy 3 String Thy 2 String Thy 1Potential String Thy 4 String Thy 3String Thy 2String Thy 1 Therefore, the information from the on-shell formulation are Free-energy:Instanton actions: (and their higher order corrections)

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From these information, D-instanton chemical potentials With proper D-instanton chemical potentials we can recover the partition function: String Thy 4 String Thy 3String Thy 2String Thy 1 Free-energy:Instanton actions:

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The reconstruction from perturbation theory: String Theory There are several choices of D-instantons to construct the partition function with some D-instanton chemical potentials θ are usually integration constants of the differential equations. The D-inst. Chem. Pot. Is relevant to non-perturbative behaviors Requirements of consistency constraints for Chem.Pot. = Non-perturbative completion program Requirements of consistency constraints for Chem.Pot. = Non-perturbative completion program What are the physical chemical potentials, and how we obtain?

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2. Stokes phenomena and the Riemann-Hilbert approach in non-critical string theory - D-instanton chemical potentials Stokes data -

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Multi-Cut Matrix Models Matrix model: normal matrices The matrices X, Y are normal matrices The contour γ is chosen as 3-cut matrix models

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Spectral curve and Cuts The information of eigenvalues resolvent operator V( ) Eigenvalue density This generally defines algebraic curve:

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Spectral curve and Cuts The information of eigenvalues resolvent operator cuts

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Orthonormal polynomials Orthonormal polynomial: In the continuum limit (at critical points of matrix models), The orthonormal polynomials satisfy the following ODE system: Q(t;z) and P(t;z) are polynomial in z

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ODE system in the Multi-cut case Q(t;z) is a polynomial in z The leading of Q(t;z) (Z_k symmetric critical points) k-cut case = kxk matrix-valued system There are k solutions to this ODE system k-th root of unity

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Stokes phenomena in ODE system The kxk Matrix-valued solution Asymptotic expansion around 1.Coefficients are written with coefficients of Q(t;z) 2.Matrix C labels k solutions 3.This expansion is only valid in some angular domain

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Stokes phenomena in ODE system The plane is expanded into several pieces: Even though Ψ satisfy the asym exp: After an analytic continuation, the asym exp is generally different:

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Stokes phenomena in ODE system Introduce Canonical solutions: Stokes matrices: Stokes Data D-instanton chemical potentials These matrices Sn are called Stokes Data D-instanton chemical potentials

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The Riemann-Hilbert problem For a given contour Γ and a kxk matrix valued holomorphic function G(z) on z in Γ, Find a kxk holomorphic function Z(z) on z in C - Γ which satisfies G(z) Z(z) The Abelian case is the Hilbert transformation: The solution in the general cases is also known

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The general solution to is uniquely given as G(z) Z(z)

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The RH problem in the ODE system We make a patch of canonical solutions: Then Stokes phenomena is Dicontinuity:

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The RH problem in the ODE system Therefore, the solution to the ODE system is given as With In this expression, the Stokes matrices Sn are understood as D-instanton chemical potentials (g(t;z) is an off-shell string-background)

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3. The non-perturbative completion program and its solutions

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Cuts from the ODE system The Orthonormal polynomial is Is a k-rank vector Recall The discontinuity of the function The discontinuity of the resolvent

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Non-perturbative definition of cuts The discontinuity appears when the exponents change dominance: Is a k-rank vector Therefore, the cuts should appear when

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The two-cut constraint in the two-cut case: General situation of ODE:The cuts in the resolvent: This (+ α) gives constraints on the Stokes matrices Sn the Hastings-McLeod solution (no free parameter)

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Solutions for multi-cut cases: Discrete solutions Characterized by avalanches Which is also written with Young diagrams (avalanches): Symmetric polynomials

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Solutions for multi-cut cases: Continuum solutions The polynomials Sn are related to Schur polynomials Pn:

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4. Summary 1.Here we saw how the Stokes data of orthonormal polynomials are related to the D-instanton chemical potentials 2.Non-perturbative definition of cuts on the spectral curve does not necessarily create the desired number of cuts. This gives non-perturbative consistency condition on the D-instanton chemical potentials 3.Our procedure in the two-cut case correctly fix all the chemical potentials and results in the Hastings-McLeod solution. 4.We have obtained several solutions in the multi-cut cases. The discrete solutions are labelled by Young diagrams. The continuum solutions are written with Schur polynomials. 5.It is interesting if these solutions imply some dynamical remnants of strong-coupling theory, like M/F-theory.

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A.Sako S.Kuroki T.Ishikawa Graduate school of Mathematics, Hiroshima University Graduate school of Science, Hiroshima University Higashi-Hiroshima 739-8526,Japan.

A.Sako S.Kuroki T.Ishikawa Graduate school of Mathematics, Hiroshima University Graduate school of Science, Hiroshima University Higashi-Hiroshima 739-8526,Japan.

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