Model for Non-Perturbative String Landscape and Appearance of M theory Hirotaka Irie NTU) with Chuan-Tsung Chan (Tunghai U.) and Chi-Hsien Yeh (NTU) Ref) CIY ’10, in progress CIY ’10, “Fractional-superstring amplitudes, multi-cut matrix models and non-critical M theory,” Nucl.Phys.B838:75-118,2010 [arXiv:1003:1626] CISY ’09, “macroscopic loop amplitudes in the multi-cut two-matrix models,” Nucl.Phys.B828: ,2010 [arXiv: ] H.I ’09, “fractional supersymmetric Liouville theory and the multi-cut matrix models,” Nucl.Phys.B819: ,2009 [arXiv: ]

Why String Theory?, What’s the Goal? One of the promising candidates for unification of the four fundamental interactions (electromagnetism, weak, strong and gravity) with a consistent quantum theory of gravity. We wish to derive our universe from some theoretical calculation of the true vacuum of string theory or some meta-stable vacuum close to it. If it is possible, this can be a strong evidence for string theory. This is also the Fundamental Goal of High Energy Physics

How do we carry it out? Vacua By studying Vacua of string theory Perturbative Vacua (Origin of “strings”) Feynman Graphs drawn by 2D surfaces: + ++… XX String Consistency requires : e.g Important point is that this results in EOM of the string modes (G,B) (anomaly cancelled) Conformal Field Theory (e.g. this results in 26/10D and so on..) the worldsheet field theory should be

String Landscape Allowed CFTs are Perturbative Vacua: Allowed CFTs are Perturbative Vacua: they are related to each other by deformation of BG field config., string dualities and so on.. 11D SUGRA IIA IIB I Het O Het E M? An IMAGE of Landscape of string-theory perturbative vacua Are we here? Cf) QFT: Which vacuum is most stable? Within perturbation theory, we have no idea! EOM

What we can get from perturbation theory 26D Bosonic string theory There is a Tachyon mode something like: 10D Superstring theory Spacetime SUSY cancellation something like: People believe that non-perturbative effects can give a non-trivial potential uplift We wish to know non-perturbative relationship among perturbative vacua, and possibility of non-perturbative vacua We succeeded realizing various situations in 2D string theory!! This has been a long standing problem How does the non-perturbative string landscape look like?

Non-perturbative string landscape and two-dimensional string theory 2D String Theory (’81-) Non-perturbatively exactly solvable with Matrix Models with Matrix Models Many people have tried to attack this issue in 2D string theory: Moduli space of 2D string theory? We can read from the worldsheet action: Moduli On-shell op.(BRST coh.) However, are not the propagating modes in Hilbert space. Condensation cannot happen  are just given parameters String-Theory Partition function does depend on background Therefore, they should not be minimized [Seiberg-Shenker ‘92]

Then, what can be the moduli space of 2D string theory? ( : potential of matrix models) Configuration of eigenvalues Stable background Unstable background (adding Unstable D-branes or instanton) Summing over all the configurations The moduli space is filling fraction:  Background independent and modular invariant in spacetime can follow [Eynard-Marino ‘08].

For example: Landscape of 2D (bosonic) string is like: Can we consider more non-trivial situations? National extension is the Multi-Cut Case Perturbative string theory There are more D.O.F to interplay among various perturb. strings

Plan of the talk 1.Introduction and Motivation 2.Review of the multi-cut matrix models and fractional superstring theory 3.Non-perturbative String Landscape and non-critical M theory 4.Summary and future directions

2. Review of the multi-cut matrix models and fractional superstring theory

String Theory Brief look at matrix models Matrix Models (N: size of Matrix) Summations Triangulation (Random surfaces) Matrix Models “Multi-Cut” Matrix Models Feynman Diagram Continuum limit Worldsheet (2D surfaces) Ising model +critical Ising model TWO Hermitian matricesM: NxN Hermitian Matrix String Theory Fractional Superstring Theory [H.I’09]

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

Correspondence with string theory V( ) bosonic super 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] 2 After continuum limit, TOO simple to claim string Landscape??

Let’s consider the Multi-Cut Critical Points: 2-cut critical points3-cut critical points Continuum limit = blow up [Crinkovik-Moore ‘91] cuts We can expect variety of vacua! These consideration are only qualitative discussion. Therefore, we show quantitative results of the system.

Actual solutions in the system [CISY’09, CIY’10] the Z_k symmetric case [CISY’09]: |t| t > 0 t < 0 (p,q) critical points with k cuts e.g) the 3-cut cases are

the Z_k symmetric case [CISY’09]: (p,q) critical points with k cuts Too many solutions!? is natural because we have two choices Each has different perturbative amplitudes Variety of solutions

This implies that the string Landscape of multi-cut matrix models is non-trivial The multi-cut matrix models provide non-trivial models for non-perturbative string landscape! Fractional-superstrings provide more non-trivial situations!

3. Non-Perturbative String Landscape and Non-Critical M theory

the fractional superstring cases [CIY’10]: (p,q) critical points with k cuts We proposed that, the following geometry: e.g) the 12-cut cases are Cuts run from Infinity to Infinity What does it mean?

Perturbative Isolation Factorization and Perturbative Isolation [CIY’10] The algebraic equation of the solution is factorized into irreducible curves: But each curve only has cuts on real axes: cuts cuts Didn’t we have multi-cut geometry? ?? cuts and

Perturbative Isolation Factorization and Perturbative Isolation [CIY’10] Our answer is: cutscuts and Recall : “Cut = discontinuity of algebraic function W(x) “ cuts is patched by in the region around At some level, we have checked that our system admits these boundary condition in several cases [CIY2 ‘10] another cuts Conjecture [CIY’10]

[Eynard-Orantin ‘07] All order Perturbative correlators only depend on F(x,W)=0 Fact IF All-order Perturbatively (All-order Perturbatively) Perturbative Isolation Factorization and Perturbative Isolation [CIY’10] What is the physical meaning of these factrization? Only non-perturbative interactions Perturbative interactions This system has many perturbatively isolated sectors

Perturbative Vacua in non-perturbative string landscape perturbative stringsperturbative isolated sectors!! Within perturbation theory, we cannot distinguish perturbative strings from perturbative isolated sectors!! Perturbatively (all order) Bosonic string Perturbatively (all order) type 0 Superstring Perturbative strings can appear as sectors

Analogy to Quantum Mechanics System: If we use perturbation theory, we encounter two perturb. sectors: The true vacuum is superposition of these wave functions: We know that the coefficient is very important and non-perturbative. In our case, the perturbative sectors are like superselection sectors, which are separated within perturbation theory. Perturbative string theories are just segments of the whole system What is the whole system? It’s non-critical M theory!

M As a M other Theory [CIY’10] This 12-cut matrix model (12-Fractional superstring theory) includes all the perturbative strings of 1(=Bosonic)-FSST, 2(=Super)-FSST, 3-FSST, 4-FSST, 6-FSST and 12-FSST Infinite-cut matrix model (Infinite-Fractional SST) is the Mother Theory of Fractional Superstring Theory In the same way, 12-FSST C 24-FSST C …… C ∞-FSST

M As a non-critical M Theory [CIY’10] M Theory is a Mysterious theory which unifies IIA BPS spectrum: IIA(10Dim): D0 F1 D2 D4 NS5 M(11Dim): KK M2 M5 11D SUGRA IIA IIB I Het O Het E M? This lives in 11 dimensional spacetime (1Dim higher). People believe that the fundamental DOF is Membrane (M2) The low energy effective theory is 11D SUGRA.

M As a non-critical M Theory [CIY’10] In 2005, Horava and Keeler proposed that by adding one angular dimension in 2D string theory: One can define the non-critical version of M theory (3D M theory) Motivation: type 0A/0B 2D strings = 1+1D free fermion system Dispersion relation of free fermions: x p 0B: 0A: D0 brane charge ~ Centrifugal force In 1+2 fermion system, they can be

0A/0B superstrings from the 1+2 fermion system 0A: The polar coordinate 0B: The Cartesian coordinate Angular momentum = D0 charge (KK momentum = D0 charge) Hilbert space of gives 0A Note that this is one of the models which realize their philosophy Hilbert space of gives 0B

Similarities and differences among Multi-Cut and Horava-Keeler limit  The Im x = 0 section is type 0B superstring In their philosophy, our angle ν is understood as the third angular direction of non-critical M theory

1)String-dual of the multi-cut matrix models is 2D fractional superstring theory at weak coupling region. They have Z_k (or U(1)) charged objects (D-brane) 2)The spacetime interpretation of string theory is given by the k-th root of the matrix-model coordinate x (or ζ) [Fukuma-H.I ‘06] 3)Z_k Charge means the sectors in the multi-cut matrix models M Multi-cut matrix models as a non-critical M Theory [CIY’10] As its own light, the multi-cut matrix models look like non-critical M theory

Note) What is M/String theory limit? [Horava-Keeler ‘05] String theory description is good in the Large radius limit Therefore, non-critical M theory appears in the Small radius limit As its own light, the multi-cut matrix models look like non-critical M theory 4)This is reminiscence of Kaluza-Klein reduction. This means that the multi-cut matirx models implies the third angular direction, as their own light. 5)Since the different angle sectors do not interact with each other in perturbation theory.  3 rd direction is non-perturbative! 6)At the strong coupling regime, the theory becomes 3D! We refer to the strong coupling dual theory as non-critical M theory

Summary of other prospects 1.3-dimension is consistent with c=1 barrier (2D barrier) of 2D string theory since the third dimension is observed only with non-perturbative effects. 2.In the strong coupling theory, we no longer can use Large N expansion. Therefore, we cannot have the string picture, and strong coupling dual theory should be described by something other than strings. 3.Energy difference is Instanton action Therefore, they are universal. (doesn’t depend on regularization) 4.String theory has non-trivial DOF in non-perturb. region!

Summary We proposed various non-perturbative string landscape models in the exactly solvable framework of 2D string theory. Our work shows that string theory still seems to hide non- trivial dynamics in non-perturbative regime which connects various perturbative vacua. So far, M theory is still a mysterious theory. But we proposed a concrete and complete definition of M theory. Since this is a solvable model, this should enable us to extract all the information we wish to know. Non-perturbative vacua also exist in string theory. Our work suggests that it is strong coupling dual theory which becomes important in that study.

Future directions: What is the dynamical principle of non-perturbative vacua? What kinds of DOF are suitable to describe them? Does M stand for Membrane? Why does M live in 11D? our models should enable us to answer Interesting direction related to our models Investigation toward the non-perturbative region. What does the corresponding Kontsevich type matrix model look like?