1. Analytic Study for the String Theory Landscapes via Matrix Models (and Stokes Phenomena) String Advanced Lecture Hirotaka Irie Yukawa Institute for Theoretical Physics, Kyoto Univ. February 13 th 2013, String Advanced Lecture @ KEK Based on collaborations with Chuan-Tsung Chan (THU) and Chi-Hsien Yeh (NCTS)

2. Perturbative string theory Perturbative string theory is well-known non-perturbative formulations Despite of several candidates for non-perturbative formulations (SFT,IKKT,BFSS,AdS/CFT…), we are still in the middle of the way: Stokes phenomenon Stokes phenomenon is a bottom-up approach: especially, based on instantons and Stokes phenomena. especially, based on instantons and Stokes phenomena. In particular, within solvable/integrable string theory, we demonstrate how to understand the analytic aspects of the landscapes In particular, within solvable/integrable string theory, we demonstrate how to understand the analytic aspects of the landscapes General Motivation non-perturbatively complete string theory How to define non-perturbatively complete string theory? with the huge amount of string-theory vacua? the true vacuummeta-stable vacua How they decay the decay rate How to deal with the huge amount of string-theory vacua? Where is the true vacuum? Which are meta-stable vacua? How they decay into other vacua? How much is the decay rate? the non-perturbatively complete string theory How to reconstruct the non-perturbatively complete string theory from its perturbation theory?

3. Plan of the talk 1.Motivation for Stokes phenomenon a) Perturbative knowledge from matrix models b) Spectral curves in the multi-cut matrix models (new feature related to Stokes phenomena) 2.Stokes phenomena and isomonodromy systems a) Introduction to Stokes phenomenon (of Airy function) b) General k x k ODE systems 3.Stokes phenomena in non-critical string theory a) Multi-cut boundary condition b) Quantum Integrability ---------- conclusion and discussion 1 ---------- 4.Analytic aspects of the string theory landscapes ---------- conclusion and discussion 2 ----------

4. Main references Isomonodromy theory and Stokes phenomenon to matrix models (especially of Airy and Painlevé cases) [Moore ’91]; [David ‘91] [Maldacena-Moore- Seiberg-Shih ‘05] Isomonodromy theory, Stokes phenomenon and the Riemann-Hilbert (inverse monodromy) method (Painlevé cases: 2x2, Poincaré index r=2,3): [Its-Novokshenov '91]; [Fokas-Its-Kapaev- Novokshenov'06] [FIKN]

5. Main references a first principle analysis for the string theory landscape Proposal of a first principle analysis for the string theory landscape [Chan-HI-Yeh 4 '12];[Chan-HI-Yeh 5 ‘13 in preparation] general kxk isomonodromy systems general Poincaré index r Stokes phenomena in general kxk isomonodromy systems corresponding to matrix models (general Poincaré index r) [Chan-HI-Yeh 2 ‘10] ;[Chan-HI-Yeh 3 ’11]; [Chan-HI-Yeh 4 '12] Spectral curves in the multi-cut matrix models [HI ‘09]; [Chan-HI-Shih-Yeh '09] ;[Chan-HI-Yeh 1 '10] ChanHIYeh (S.-Y. Darren) Shih [CIY][CISY]

6. 1. Motivation for Stokes phenomenon Ref) Spectral curves in the multi-cut matrix models: [CISY ‘09] [CIY1 ‘10]

7. Perturbative knowledge from matrix models Large N expansion of matrix models (Non-critical) String theory Continuum limit Triangulation (Lattice Gravity) (Large N expansion  Perturbation theory of string coupling g) We have known further more on non-perturbative string theory CFT N x N matrices

8. 1.Perturbative amplitudes of WS n : 2.Non-perturbative amplitudes are D-instantons! [Shenker ’90, Polchinski ‘94] 3.The overall weight θ’s (=Chemical Potentials) are out of the perturbation theory Non-perturbative corrections perturbative corrections non-perturbative (instanton) corrections D-instanton Chemical Potential WS with Boundaries = open string theory essential information for the NonPert. completion CFT Let’s see it more from the matrix-model viewpoints

9. The Resolvent op. allows us to read this information V( ) In Large N limit (= semi-classical) Spectral curve Diagonalization: N-body problem in the potential V Eigenvalue density spectral curve Position of Cuts = Position of Eigenvalues Resolvent:

10. Why is it important? Spectral curve  Perturbative string theory Perturbative correlators are all obtained recursively from the resolvent (S-D eqn., Loop eqn…) Therefore, we symbolically write the free energy as Topological Recursions [Eynard’04, Eynard-Orantin ‘07] Input::Bergman Kernel Everything is algebraic geometric observables!

11. [David ‘91] Why is it important? Spectral curve  Perturbative string theory Non-perturbative corrections Non-perturbative partition functions: [Eynard ’08, Eynard-Marino ‘08] V( ) In Large N limit (= semi-classical) spectral curve +1 -1 with some free parameters Summation over all the possible configurations D-instanton Chemical Potential [David’91,93];[Fukuma-Yahikozawa ‘96-’99];[Hanada-Hayakawa-Ishibashi-Kawai-Kuroki- Matuso-Tada ‘04];[Kawai-Kuroki-Matsuo ‘04];[Sato-Tsuchiya ‘04];[Ishibashi-Yamaguchi ‘05];[Ishibashi-Kuroki-Yamaguchi ‘05];[Matsuo ‘05];[Kuroki-Sugino ‘06]… algebraic geometric observableanalytic one This weight is not algebraic geometric observable; but rather analytic one! Theta function on

12. the Position of “Eigenvalue” Cuts What is the geometric meaning of the D-instanton chemical potentials? [CIY 2 ‘10] But, we can also add infinitely long cuts θ_I ≈ Stokes multipliers s_{l,I,j} From the Inverse monodromy (Riemann-Hilbert) problem [FIKN] θ_I ≈ Stokes multipliers s_{l,I,j} “Physical cuts” as “Stokes lines of ODE” How to distinguish them? Later This gives constraints on θ  T-systems on Stokes multipliers Related to Stokes phenomenon! Require!  section 4

13. Why this is interesting? The multi-cut extension [Crinkovic-Moore ‘91];[Fukuma-HI ‘06];[HI ‘09] ! 1) Different string theories (ST) in spacetime [CIY 1 ‘10];[CIY 2 ‘10];[CIY 3 ‘11] ST 1 ST 2 2) Different perturbative string-theory vacua in the landscape: [CISY ‘09]; [CIY 2 ‘10] the string-theory landscape from the first principle We can study the string-theory landscape from the first principle! Today’s first topic  Gluing the spectral curves (STs) Non-perturbatively (Today’s first topic) Today’s second topic in sec. 4  the Riemann-Hilbert problem (Today’s second topic in sec. 4) ST 1 ST 2

14. 2. Stokes phenomenon and isomonodromy systems Ref) Stokes phenomena and isomonodromy systems [Moore ‘91] [FIKN‘06] [CIY 2 ‘10]

15. The ODE systems for determinant operators (FZZT-branes) The resolvent, i.e. the spectral curve: linear ODE systems: Generally, this satisfies the following kind of linear ODE systems: k-cut  k x k matrix Q [Fukuma-HI ‘06];[CIY 2 ‘10] For simplicity, we here assume: Poincaré index r

16. Stokes phenomenon of Airy function Airy function: Asymptotic expansion! This expansion is valid in (from Wikipedia) ≈

17. + ≈ Stokes phenomenon of Airy function Airy function: (valid in ) (relatively) Exponentially small ! Stokes sectors 1.Asymptotic expansions are only applied in specific angular domains (Stokes sectors) relatively and exponentially small terms 2.Differences of the expansions in the intersections are only by relatively and exponentially small terms Stokes sectors 1.Asymptotic expansions are only applied in specific angular domains (Stokes sectors) relatively and exponentially small terms 2.Differences of the expansions in the intersections are only by relatively and exponentially small terms Stokes multiplier Stokes sectors Stokes Data!

18. Stokes phenomenon of Airy function Airy function: (valid in ) Stokes sectors Keep using different

19. 1) Complete basis of the asymptotic solutions: Stokes phenomenon of the ODE of the matrix models … 1 2 0 19 3 4 56 … 18 17 … D0D0 D3D3 12 … D 12 2) Stokes sectors In the following, we skip this 3) Stokes phenomena (relatively and exponentially small terms)

20. 1) Complete basis of the asymptotic solutions: Stokes phenomenon of the ODE of the matrix models Here it is convenient to introduce General solutions: … Superposition of wavefunction with different perturbative string theories Spectral curve  Perturb. String Theory

21. Stokes sectors … 1 2 0 19 3 4 56 … 18 17 … D0D0 D3D3 12 … D 12 Stokes phenomenon of the ODE of the matrix models 2) Stokes sectors, and Stokes matrices E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric) Stokes matrices 0 1 3 … … 19 18 17 12 … 4 5 6 7 8 … 2 D0D0 D3D3 D 12 larger Canonical solutions (exact solutions) How change the dominance Keep using

22. Stokes matrices : non-trivial Thm [CIY2 ‘10] 0 1 2 3 D0D0 D1D1 4 5 6 7 Set of Stokes multipliers ! Stokes phenomenon of the ODE of the matrix models 3) How to read the Stokes matrices? :Profile of exponents [CIY 2 ‘10] E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric)

23.  section 4 Inverse monodromy (Riemann-Hilbert) problem [FIKN] Direct monodromy problem Stokes matrices Given: Stokes matrices Inverse monodromy problem Given Solve Obtain WKB RH SolveObtain Analytic problem Consistency (Algebraic problem) Special Stokes multipliers which satisfy physical constraints

24. Algebraic relations of the Stokes matrices 1.Z_k –symmetry condition 2.Hermiticity condition 3.Monodromy Free condition 4.Physical constraint: The multi-cut boundary condition This helps us to obtain explicit solutions for general (k,r) most difficult part!

25. 3. Stokes phenomenon in non-critical string theory Ref) Stokes phenomena and quantum integrability [CIY2 ‘10][CIY3 ‘11]

26. Multi-cut boundary condition 3-cut case (q=1)2-cut case (q=2: pureSUGRA)

27. ≈ + (from Wikipedia) Stokes phenomenon of Airy function Airy function: (valid in ) Change of dominance (Stokes line) Change of dominance (Stokes line) Dominant!

28. ≈ + (from Wikipedia) Stokes phenomenon of Airy function (valid in ) Change of dominance (Stokes line) Change of dominance (Stokes line) Airy system  (2,1) topological minimal string theory  Eigenvalue cut of the matrix model Dominant! (Stokes lines) Physical cuts = lines with dominance change (Stokes lines) [MMSS ‘05] discontinuity

29. Multi-cut boundary condition [CIY 2 ‘10] … 1 2 0 19 3 4 56 … 18 17 … D0D0 D3D3 12 … D 12 0 1 2 3 … … 19 18 17 D0D0 12 … … 5 6 7 8 E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric) All the horizontal lines are Stokes lines! All lines are candidates of the cuts!

30. Multi-cut boundary condition [CIY 2 ‘10] … 1 2 0 19 3 4 56 … 18 17 … D0D0 D3D3 12 … D 12 0 1 2 … … 19 18 17 3 D0D0 12 … … 5 6 7 8 E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric) We choose “k” of them as physical cuts! k-cut  k x k matrix Q [Fukuma-HI ‘06];[CIY 2 ‘10] ≠0 =0=0 Constraints on Sn

31. Multi-cut boundary condition 3-cut case (q=1)2-cut case (q=2: pureSUGRA)

32. 0 1 2 3 D0D0 D1D1 4 5 6 7 E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric) : non-trivial Thm [CIY2 ‘10] Set of Stokes multipliers ! The set of non-trivial Stokes multipliers? Use Profile of dominant exponents [CIY 2 ‘10]

33. Quantum integrability [CIY 3 ‘11] 0 1 2 3 … … 19 18 17 12 … … 5 6 7 8 E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric) This equation only includes the Stokes multipliers of Then, the equation becomes T-systems: cf) ODE/IM correspondence [Dorey-Tateo ‘98];[J. Suzuki ‘99] the Stokes phenomena of special Schrodinger equations satisfy the T-systems of quantum integrable models cf) ODE/IM correspondence [Dorey-Tateo ‘98];[J. Suzuki ‘99] the Stokes phenomena of special Schrodinger equations satisfy the T-systems of quantum integrable models with the boundary condition: How about the other Stokes multipliers? Set of Stokes multipliers !

34. Complementary Boundary cond. [CIY 3 ‘11] 0 1 2 3 … … 19 18 17 12 … … 5 6 7 8 E.g.) r=2, 5 x 5, γ=2 (Z_5 symmetric) This equation only includes the Stokes multipliers of Then, the equation becomes T-systems: with the boundary condition: Shift the BC ! (Coupled multiple T-systems) Generally there are “r” such BCs (Coupled multiple T-systems)

35. Solutions for multi-cut cases (Ex: r=2, k=2m+1): m 1 m-1 2 m-2 3 m-3 4 m-4 5 m-5 6 m-6 7 m-7 8 m 1 m-1 2 m-2 3 m-3 4 m-4 5 m-5 6 m-6 7 m-7 8 nnnn avalanches are written with Young diagrams (avalanches): (Characters of the anti-Symmetric representation of GL) [CIY 2 ‘10] [CIY3 ‘11] In addition, they are “coupled multiple T-systems”

36. 4. Summary (part 1) 1.The D-instanton chemical potentials are the missing information in the perturbative string theory. responsible for the non-perturbative relationship among perturbative string-theory vacua 2.This information is responsible for the non-perturbative relationship among perturbative string-theory vacua, and important for study of the string-theory landscape from the first principle. 3.In non-critical string theory (or generally matrix models), this information is described by the positions of the physical cuts. T- systems of quantum integrable systems 4.The multi-cut boundary conditions, which turn out to be T- systems of quantum integrable systems, can give a part of the constraints on the non-perturbative system explicit expressions of the Stokes multipliers 5.Although physical meaning of the complementary BC is still unclear (in progress [CIY 4 ‘12]), it allows us to obtain explicit expressions of the Stokes multipliers.

37. discussions other degree of freedom 1.Physical meaning of the Compl. BCs?  The system is described not only by the resolvent? We need other degree of freedom to complete the system? (  FZZT-Cardy branes? [CIY 3 ‘11]) Use Duality? Strong string-coupling description? 2.D-instanton chemical potentials are determined by “strange constraints” which are expressed as quantum integrability. Are there more natural explanations of the multi-cut BC? (  Use Duality? Strong string-coupling description?  Non-critical M theory?, Gauge theory?)

38. 4. Analytic aspects of the string theory landscapes Ref) Analytic Study for the string theory landscapes [CIY4 ‘12] Then, can we extract the analytic aspects of the landscapes i.e. true vacuum, meta-stability and decay rate ? Stokes Data, From Stokes Data, we reconstruct string theory nonperturbatively YES !

39. Reconstruction of [(p,q) minimal] string theory [CIY4 ‘12] There are p branches k = p Spectral Curve 1 st Chebyshev polynomials: Consider p x p Sectional Holomorphic function Z(x)  Generally Z(x) should be sectional holomorphic function Non-pert. StringsReconstruct C Asymp. Exp( x  ∞ ∈ C ) s.t. Keep using We don’t start with ODE!

40. Jump lines: ∞ Essential Singularity 6 3 7 8 9 2 1 5 4 Jump line C Asymp. Exp （ x  ∞ ∈ C ） s.t. Keep using Constant Matrix These matrices are equivalent to Stokes matrices

41. ∞ Essential Singularity 1. : Constant Matrices (  Isomonodromy systems) 2.Junctions: 3.In particular, at essential singularities, there appears the monodromy equation: 6 3 7 8 9 7 1 2 Jump lines: 2 1 5 4 This is what we have solved! Jump line Preservation of matrices: e.g.) given Constant Matrix  Jump lines are topological (except for essential singularities)

42. ∞ Essential Singularity 1. : Constant Matrices (  Isomonodromy systems) 2.Junctions: 3.In particular, at essential singularities, there appears the monodromy equation: 6 3 7 8 9 7 1 2 Jump lines: 2 1 5 4 This is what we have solved! Jump line Preservation of matrices: e.g.) given  Jump lines are topological (except for essential singularities) Ψ(x) can be uniquely solved by the integral equation on ： （ e.g. [FIKN] ） In fact Obtain Ψ RH (x) (Riemann-Hilbert problem)

43. Reconstruction and the Landscapes ∞ Essential Singularity 6 3 7 8 9 2 1 5 4 Consider deformations: String Theory Landscape: Land All the onshell/offshell configurations of string theory background which satisfy 1.B.G. indenpendence Then the result of RH problem Ψ RH (x) is the same! singular behavior Does not change the singular structure

44. Reconstruction and the Landscapes 2.Pert. and Nonpert. Corrections 3.Physical Meaning of ∞ Essential Singularity 6 3 7 8 9 2 1 5 4 φ(x) ∈ Land str From Topological Recursions How far from each other φ'(x) ∈ Land str The same! Different! Steepest Descent as “Steepest Descent curves of φ(x) (Anti-Stokes lines) ” mean field path-integrals in matrix models [CIY4 ‘12]

45. E.g.) (2,3) minimal strings (Pure-Gravity) Multi-cut BC (=matrix models) Multi-cut BC (=matrix models) gives Basic Sol. ∞ Essential Singularity 6 3 7 8 9 2 1 5 4 NOTE coincide with matrix models (a half of [Hanada et.al. ‘04]) Free energy [CIY4 ‘12] Small instantons  stable vacuum

46. E.g.) (2,5) minimal strings (Yang-Lee edge) Multi-cut BC (= matrix models) Multi-cut BC (= matrix models) gives ∞ Essential Singularity 6 3 7 8 9 2 1 5 4 Basic Sol Free energy Large instantons  unstable ( or meta-stable) (1,2) ghost ZZ brane [no (1,1) ZZ brane] [CIY4 ‘12] NOTE coincide with matrix models ( (1,2)ZZ brane in [Sato-Tsuchiya ‘04]…) Decay Rate? Extract meta-stable system by deforming path-integral Extract meta-stable system by deforming path-integral [Coleman]

47. E.g.) (2,5) minimal strings (Yang-Lee edge) Multi-cut BC (= matrix models) Multi-cut BC (= matrix models) gives ∞ Essential Singularity 6 3 7 8 9 2 1 5 4 Free energy Decay rate (= deform. ) NOTE Coincide with matrix models ([Sato-Tsuchiya ‘04]…) Decay rates of this string theory (1,1) ZZ brane [no (1,2) ZZ brane] [CIY4 ‘12] Large Instanton True vacuum? Choose BG in the landscape so that it achieves small instantons Choose BG in the landscape Land str so that it achieves small instantons

48. Basic Sol E.g.) (2,5) minimal strings (Yang-Lee edge) Free energy Multi-cut BC (= matrix models) Multi-cut BC (= matrix models) gives ∞ Essential Singularity 6 3 7 8 9 2 1 5 4 True vacuum [CIY4 ‘12] φ TV (x) ∈ Land str It is not simple string theory Deformed by elliptic function Large Instantons

49. Summary and conclusion, part 2 1.D instanton chemical potentials are equivalent to Stokes data by Riemann-Hilbert methods 2.With giving Stokes data, we can fix all the non-perturbative information of string theory meta- stabilitydecay ratetrue vacuum 3.In fact, we have seen that Stokes data is directly related to meta- stability/decay rate/true vacuum of the theory 4.Instabilityghost D-instantons Stokes data 4.Instability of minimal strings is caused by ghost D-instantons, whose existence is controlled by Stokes data Discussion: non-perturbative principle 1.What is non-perturbative principle of string theory? duality 2.What is the rule of duality in string landscapes? We now have all the controll over non-perturbative string theory with description of spectral curves and resulting matrix models

50. Thank you for your attention!