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Stokes Phenomena and Non- perturbative Completion in the multi-cut matrix models

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)

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Stokes Phenomena and Non- perturbative Completion in the multi-cut matrix models

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  1. Stokes Phenomena and Non-perturbative Completion in the multi-cut matrix models HirotakaIrie (NTU) A collaboration with Chuan-Tsung Chan (THU) and Chi-HsienYeh (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]

  2. 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. The string-theory landscape: We are here? and Why?

  3. From String Theory to the Standard Model • 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

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

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

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

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

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

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

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

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

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

  13. Spectral curve and Cuts The information of eigenvalues resolvent operator cuts

  14. 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

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

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

  17. 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:

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

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

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

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

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

  23. 3. The non-perturbative completion program and its solutions

  24. 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

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

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

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

  28. Solutions for multi-cut cases: Continuum solutions The polynomials Snare related to Schur polynomials Pn:

  29. 4. Summary • Here we saw how the Stokes data of orthonormal polynomials are related to the D-instanton chemical potentials • 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 • Our procedure in the two-cut case correctly fix all the chemical potentials and results in the Hastings-McLeod solution. • 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. • It is interesting if these solutions imply some dynamical remnants of strong-coupling theory, like M/F-theory.

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