Stokes Phenomena and Quantum Integrability in Non-critical String/M Theory
aa r X i v : . [ h e p - t h ] D ec September 2011
Stokes Phenomena and Quantum Integrabilityin Non-critical String/M Theory
Chuan-Tsung Chan ∗ ,p , Hirotaka Irie † ,q and Chi-Hsien Yeh ‡ ,rp Department of Physics, Tunghai University, Taiwan, 40704 q National Center for Theoretical Sciences,National Tsing-Hua University, Hsinchu 30013, Taiwan, R.O.C. r Department of Physics and Center for Theoretical Sciences,National Taiwan University, Taipei 10617, Taiwan, R.O.C.
Abstract
We study Stokes phenomena of the k × k isomonodromy systems with an arbi-trary Poincar´e index r , especially which correspond to the fractional-superstring (orparafermionic-string) multi-critical points (ˆ p, ˆ q ) = (1 , r −
1) in the k -cut two-matrixmodels. Investigation of this system is important for the purpose of figuring outthe non-critical version of M theory which was proposed to be the strong-couplingdual of fractional superstring theory as a two-matrix model with an infinite numberof cuts. Surprisingly the multi-cut boundary-condition recursion equations have auniversal form among the various multi-cut critical points, and this enables us toshow explicit solutions of Stokes multipliers in quite wide classes of ( k, r ). Althoughthese critical points almost break the intrinsic Z k symmetry of the multi-cut two-matrix models, this feature makes manifest a connection between the multi-cutboundary-condition recursion equations and the structures of quantum integrablesystems. In particular, it is uncovered that the Stokes multipliers satisfy multiple Hirota equations (i.e. multiple
T-systems). Therefore our result provides a largeextension of the ODE/IM correspondence to the general isomonodromy ODE sys-tems endowed with the multi-cut boundary conditions. We also comment abouta possibility that N = 2 QFT of Cecotti-Vafa would be “topological series” innon-critical M theory equipped with a single quantum integrability. ∗ [email protected] † [email protected] ‡ [email protected] ontents k = rm + 1 . . . . . . . . . . . . . . . . . . . . . . . . 253.3.2 The case of k = rm + l , (1 < l < r ) . . . . . . . . . . . . . . . . . 283.3.3 The case of k = r ( m + 1) . . . . . . . . . . . . . . . . . . . . . . . 29 Non-perturbative aspects beyond perturbative string theory still remain some of themissing pieces in our current understanding of string theory. In early study in the ’90s,the following question about non-perturbative string theory was investigated: what isthe form of non-perturbative corrections to the perturbative expansions of string theory?
Among various investigations, non-critical string theory [1] played an important role sinceit is non-perturbatively formulated by solvable matrix models [2–44] and this fact makesit possible to see higher-order behavior of the string-theory free energy in strong stringcoupling constant g str → ∞ [15, 16, 21, 22, 24, 25]. In particular, it was found that non-perturbative corrections to the perturbative amplitudes are given by order of O ( e −∗ /g str ),1.e. open-string degree of freedom [16]: F ( g str ) = ln Z ( g str ) ≃ asym ∞ X n =0 g n − F n | {z } perturb. part + X I θ I exp hX n g n − F ( I ) n i + O ( θ I ) | {z } non-perturb. corrections , (1.1)and this fact lead to a key idea of D-branes [45]. The identification of D-branes was thenconfirmed in comparison between the Liouville-theory calculation [1,46–55], i.e. ZZ-branes[51], and succeeding calculations of matrix models [31–35,37,39]. On the other hand, sincethey describe non-perturbative corrections in the perturbative asymptotic expansion, therelative weight factors (cid:8) θ I (cid:9) I , called D-instanton chemical potentials (fugacities) , remainto be ambiguous parameters in perturbative string theory [21].A basic origin of the ambiguity is the fact that these parameters are subjected todiscrete corrections in various analytic continuations of string-theory backgrounds andthat the formal expansions of analytic equations can be performed without specifyingthe direction from which we expand the series (See Figure 1). From the viewpoint ofphysics, this kind of phenomena make it possible to describe various different perturba-tive backgrounds in a single analytic theory. Mathematically these behaviors are knownas
Stokes phenomena , a basic analytic property in the asymptotic analysis. Determiningthe Stokes phenomena for the asymptotic expansions in the string coupling constant,Eq. (1.1), (i.e. how the fugacities (cid:8) θ I (cid:9) I jump in analytic continuations) can be per-formed by resurgent analysis of the perturbative asymptotic expansion Eq. (1.1), whichwas analyzed recently [56, 57]. On the other hand, the non-perturbative ambiguities areinformation completely out of the asymptotic expansions, and therefore we have to comeback to the original definition of matrix models and need to see additional principle inorder to determine the D-instanton fugacities in string theory. This is the question weinvestigate in this paper: what is the principle to choose physical D-instanton fugacities? A reason why we consider these non-perturbative problems so significant is because webelieve that it is this principle which binds various string-theory vacua together to for-mulate non-perturbatively completed string theory and consequently which guaranteesthe string dualities in a consistent way. Therefore, this would be the direction in whichwe can quantitatively tackle the string-theory landscape from the first principle.In study of this issue about the non-perturbative ambiguity, non-critical string theoryis again expected to play an important rule. In non-critical string theory, the non-perturbative ambiguities are identified as integration constants of string equations [21].At the first sight, therefore it may seem plausible to say that the D-instanton fugacitiesare arbitrary free parameters in the non-perturbative theory. However, it is also true thatthere are several systems which prefer “physical values” rather than arbitrary values [58],and these physical values are implied in the matrix models. This kind of study has beeninvestigated from various viewpoints [58–72]. In this paper, succeeding the previous studyof the authors [72], we investigate this issue in the framework of the multi-cut two-matrixmodels [39, 73] [43, 44, 72].The multi-cut two-matrix models are the two-matrix version of the multi-cut one-matrix models [74] and realize the multi-cut critical points which naturally extend thetwo-cut critical points of the matrix models [75–80]. In particular, these critical pointsare controlled by multi-component KP hierarchy [39] and this fact makes possible var-ious quantitative analysis of the matrix models [43, 44, 72]. What is more, the two-cut2 θ I } I (1) { θ I } I { θ I } I { θ I } I { θ I } I { θ I } I { θ I } I { θ I } I (2) (3) (4) (5) (6) (7) (8) g str ↔ Z ( g str ) ≃ asym X I θ I exp h F ( I ) ( g str ) i Figure 1:
The asymptotic expansions can be different up to the D-instanton fugacities (cid:8) θ I (cid:9) I dependingon from which direction of the string coupling g str ( → × e iχ ) we expand the free energy (or partitionfunction). That is, in this figure, there are eight different sets of the D-instanton fugacities (cid:8) θ ( a ) I (cid:9) I ( a =1 , , · · · ,
8) depending on the angular domains of the string coupling g str . This mathematical propertyis called Stokes phenomenon. Therefore, determining Stokes phenomena means determining how thefugacities jump in analytic continuations crossing the domain wall (e.g. θ (8) I → θ (1) I in crossing thepositive real axes). Depending on the relative magnitude of perturbative free energy F ( I ) ( g str ) and theD-instanton fugacities, various perturbative string theories emerge as a dominant saddle point. Thiskind of phenomena can occur in any analytic continuations of the string-theory moduli spaces. critical points were found to describe minimal type 0 superstring theory [81–83], and, asan extension of this, a special kind of the multi-cut critical points (which is also shownin Section 2.2.1) were proposed to describe minimal fractional superstring theory (orminimal parafermionic string theory) [73]. An interesting new feature found by quanti-tative analyses of these critical points is the fact that these multi-cut systems generallyinclude a number of perturbative vacua in its string-theory landscape [43] and also in-clude various perturbative string-theory sectors in its wavefunctions [44]. In particular,the fractional-superstring critical points are found to describe a superposition of variousminimal fractional superstring theories, and the chain of string theories is then inter-preted as the extra dimension of the non-critical M theory [44]. This means that, in thecase of an infinite number of cuts, there naturally appears a three-dimensional universalstrong-coupling dual theory which realizes the philosophy proposed by Hoˇrava-Keeler forthe non-critical M theory [84]. In this way, various new non-perturbative phenomena ofstring theory were revealed in the multi-cut critical points, and importantly these phe-nomena are fully described only after we determine the non-perturbative ambiguities,i.e. they are governed by the information out of the perturbative description of stringtheory. Therefore, it is clear that the multi-cut matrix models provide a fruitful fieldin studying this issue, and the investigation of this system is expected to answer to thefollowing question: what is the non-perturbative completion of string theory? As a natural framework to study the ambiguity, we consider the isomonodromy de-scription [85, 86] [14, 59] [87] for the multi-cut critical points. Mathematically, the systemis a k × k isomonodromy system with an irregular singularity of Poincar´e index r (= ˆ q + 1)which corresponds to the k -cut multi-critical points (ˆ p, ˆ q ) = (1 , ˆ q ) in the multi-cut ma-trix models. In the isomonodromy study of matrix models, the D-instanton chemicalpotentials are mapped to Stokes multipliers of the linear systems in the spectral parame-ter ζ (i.e. the Baker-Akhiezer systems Eq. (2.14)) through the Riemann-Hilbert procedure F ( g str ) which is given by the Stokes multipliers of theODE systems. For instance in the case of the 2-cut (ˆ p, ˆ q ) = (1 ,
2) critical point, the RHintegral is give by the Stokes multipliers { s n,i,j } n,i,j of the corresponding ODE system as ∂ F ( t ; g str ) ∂t = (cid:2) f ( t ) (cid:3) , f ( t ) = X n s n, , Z K n dλ πi e g str (cid:0) ϕ (2) ( t ; λ ) − ϕ (1) ( t ; λ ) (cid:1) + · · · , (1.2)where t is a parameter of the string-theory moduli space (i.e. the worldsheet cosmologicalconstant). In this way, the information of the non-perturbative ambiguities is translatedinto the Stokes data of ordinary differential equation (ODE) systems as the informationwhich does not depend on the direction of asymptotic expansion (shown in Figure 1). Although the Stokes phenomena in these ODE systems are relatively simpler than thoseof non-linear differential equation systems like Painlev´e equations, less is known in thecase of large values of k and r compared with the isomonodromy systems of the secondorder Painlev´e series [87].In the previous study of the authors [72], the physical constraints (required from thematrix models) on the Stokes multipliers are formulated in this isomonodromy framework.As is reviewed in Section 2.2.3, the requirement is given as the boundary conditions forphysical cuts in the spectral curves . Interestingly these physical constraints enable us toshow explicit solutions of Stokes multipliers which satisfy high-degree algebraic equationscalled monodromy free (or cyclic) conditions (Eq. (2.65)), which are generally quite hardto solve without any help of the physical boundary conditions.In this paper, on the other hand, we extend this boundary condition to wider classesof the multi-cut critical points (general number of cuts, k , and general Poincar´e index, r ), especially the critical points which correspond to minimal fractional superstring (orparafermionic string) theory [73]. A major difference from the critical points of the pre-vious study is lack of the Z k -symmetry which is intrinsic in the multi-cut two-matrixmodels. Therefore, the number of independent Stokes multipliers are k times more thanthat of Z k symmetric critical points, and this becomes a huge predicament to direct anal-ysis of the Stokes phenomena. However, we will show that by this feature it becomesmanifest that the multi-cut boundary conditions are equivalent to discrete Hirota equa-tions (i.e. T-systems) of quantum integrable systems. That is, the physical section ofD-instanton chemical potentials are expressed by quantum integrability which would bean integrable structure of strong-coupling dual description, i.e. non-critical M theory.Organization of this paper is as follows: In Section 2, basic facts about the fractional-superstring critical points are summarized. In Section 3, the multi-cut boundary condi-tions are studied. In particular, the T-systems of the Stokes multipliers are pointed outin Section 3.2. Some explicit solutions are also shown in Section 3.3. Section 4 is devotedto conclusion and discussion. In Appendix A, a brief proof of Theorem 1 is presented. One can find an explanation of this integral in [72] and [87]. For example in [87], by evaluating this integral, they show explicit expressions of how the figacities (cid:8) θ I (cid:9) I (i.e. (cid:8) θ ( a ) I (cid:9) I for each angular domain ( a = 1 , , · · · ) in Figure 1) can be expressed by the Stokesdata { s n,i,j } n,i,j of the ODE system of the Painlev´e equations. Summary of fractional-superstring critical points
In this section, we summarize basic facts about the fractional-superstring critical points[73], especially focusing on the results from the matrix-model analysis [43, 44, 72], includ-ing the Stokes phenomena of the multi-cut matrix models which are generally developedin [72].
The multi-cut two-matrix models are given by two-matrix models, Z = Z M N ( C ( k ) ) × M N ( C ( k ) ) dXdY e − N tr (cid:2) V ( X )+ V ( Y ) − XY (cid:3) , (2.1)with integration over the following special N × N normal matrices, M N ( C ( k ) ):M N ( C ( k ) ) ≡ (cid:8) U diag ki =1 ( λ i ) U † (cid:12)(cid:12) U ∈ U ( N ) , λ i ∈ C ( k ) ⊂ C (cid:9) , (2.2)where the contour of the eigenvalues, C ( k ) , are defined as C ( k ) = k − [ n =0 ω n R , ω = e πi k , (2.3)in the case of the k -cut two-matrix models.An important ingredient of this system is the bi-orthonormal polynomial systems [88]defined by the following inner product, Z C ( k ) ×C ( k ) dxdy e − N (cid:2) V ( x )+ V ( y ) − xy (cid:3) α n ( x ) β m ( y ) = δ n,m , (2.4)between the polynomials, α n ( x ) = 1 √ h n (cid:16) x n + · · · (cid:17) , β n ( y ) = 1 √ h n (cid:16) y n + · · · (cid:17) . (2.5)Here { h n } n ∈ Z + are normalization coefficients. An interesting fact about the orthonormal-polynomial system is the following expression by the matrix integral [11]: α n ( x ) = 1 √ h n (cid:10) det (cid:0) x − X (cid:1)(cid:11) n × n , β n ( x ) = 1 √ h n (cid:10) det (cid:0) y − Y (cid:1)(cid:11) n × n , (2.6)where the expectation value (cid:10) · · · (cid:11) n × n is given by n × n truncated matrices M n ( C ( k ) ): (cid:10) O ( X, Y ) (cid:11) n × n ≡ Z M n ( C ( k ) ) × M n ( C ( k ) ) dXdY e − N tr (cid:2) V ( X )+ V ( Y ) − XY (cid:3) O ( X, Y ) . (2.7)Another important ingredient of this system is the resolvent operator R ( x ) whichcontains the spectral information of the matrix integral: R ( x ) = 1 N (cid:28) tr 1 x − X (cid:29) → Z C ( k ) dλ ρ ( λ ) x − λ ( N → ∞ ) . (2.8)5n particular, the support of the eigenvalue density function ρ ( x ) corresponds to discon-tinuity of the resolvent function: πiρ ( λ ) = R ( λ + iǫ ) − R ( λ − iǫ ) , λ ∈ supp ( ρ ) ⊂ C ( k ) . (2.9)Since the orthonormal polynomials are also written with the resolvent operator: α n ( x ) ∼ √ h N (cid:10) det (cid:0) x − X (cid:1)(cid:11) = 1 √ h N exp h N Z x dx ′ R ( x ′ ) i ( n ∼ N ) , (2.10)the above discontinuity should be observed also in the orthonormal polynomial of thelarge N limit with n/N → We move on to the continuum (ˆ p, ˆ q ) minimal fractional superstring theory [73] by takingthe scaling limit of the bi-orthonormal polynomials. The scaling limit is given by a latticespacing a as follows:Ψ orth ( t ; ζ ) ≡ ψ ( t ; ζ )... ψ k ( t ; ζ ) , ψ i ( t ; ζ ) = a ˆ p ω ǫ/ h ( − n α kn + i − ( x ) e − NV ( x ) i , (2.11)where the scaling valuables are introduced as x = a ˆ p ω ǫ/ ζ , N − = g str a ˆ p +ˆ q → , knN = exp (cid:0) − ta ˆ p +ˆ q − (cid:1) → ,k − ∂ n = − a / g str ∂ t ≡ − a / ∂ → , ( a → . (2.12)The detail of these scaling limits is discussed in [43]. As will be mentioned later, there isa parameter ǫ (= 0 , p, ˆ q )critical points. Consequently, the recursion equation of the orthonormal polynomial: xα n ( x ) = X s ∈ Z A s ( n ) α n − s ( x ) , N − ∂∂x α n ( x ) = X s ∈ Z B s ( n ) α n − s ( x ) (2.13)becomes the following k × k differential equation system [39]: ζ Ψ( t ; ζ ) = P ( t ; ∂ ) Ψ( t ; ζ ) , g str ∂∂ζ Ψ( t ; ζ ) = Q ( t ; ∂ ) Ψ( t ; ζ ) . (2.14)which is governed by k -component KP hierarchy [89]. Here we note that the scaling or-thonormal polynomial Ψ orth ( t ; ζ ) of Eq. (2.11) is a special solution to the Baker-Akhiezerfunction system Eqs. (2.14). The critical points of the multi-cut two-matrix models are classified by these lax operators( P , Q ). Fractional-superstring (ˆ p, ˆ q ) critical points are originally discussed in [73] as thecritical points which realize the operator contents of (ˆ p, ˆ q ) minimal fractional superstring6heory. From the quantitative analysis of these critical points [43] (which is the multi-cutextension of [23]), it was found that there are two kinds of fractional-superstring criticalpoints which are called ω / -rotated models and real-potential models . The parameter ǫ in Eq. (2.12) then is related with these two choices of the fractional-superstring criticalpoints: ǫ = (cid:26) ω / -rotated models1 : real-potential models . (2.15)These two kinds of solutions are perturbatively distinct when k ∈ Z ; non-perturbativelyalways distinct. Here we summarize the results of critical-potential analysis [43] and theweak-coupling spectral curves [44]:1. The first kind is derived from ω / -rotated-potential models [43], and given as P ( t ; ∂ ) = Γ ∂ ˆ p + ˆ p X n =1 H (F k P) n ( t ) ∂ ˆ p − n , Q ( t ; ∂ ) = Γ ∂ ˆ q + ˆ q X n =1 H (F k Q) n ( t ) ∂ ˆ q − n , (2.16)with the shift matrix Γ, Γ = , (2.17)and the real functions H (F k P) n ( t ) and H (F k Q) n ( t ). The weak coupling analysis g str → P ( t ; ∂ ) ≃ diag kj =1 (cid:0) P ( j ) ( t ; z ) (cid:1) , Q ( t ; ∂ ) ≃ diag kj =1 (cid:0) Q ( j ) ( t ; z ) (cid:1) [44]:Cosh solution: P ( j ) = λ ˆ p cosh (cid:0) ˆ pτ + 2 πi j − k (cid:1) , Q ( j ) = λ ˆ q cosh (cid:0) ˆ qτ + 2 πi j − k (cid:1) ; (2.18)Sinh solution: P ( j ) = λ ˆ p sinh (cid:0) ˆ pτ + 2 πi j − k (cid:1) , Q ( j ) = λ ˆ q sinh (cid:0) ˆ qτ + 2 πi j − k (cid:1) , (2.19)with λ ≡ t p +ˆ q − , z ≡ λ − (ˆ p +ˆ q ) g str t ∂ t = cosh( τ ) . (2.20)Their algebraic equations are expressed as Cosh solution: F ( P, Q ) = T p ( Q/λ ˆ q ) − T q ( P/λ ˆ p ) = 0; (2.21)Sinh solution: F ( P, Q ) = T p ( − iQ/λ ˆ q ) − ( − q − p T q ( − iP/λ ˆ p ) = 0 . (2.22) T p (cosh( τ )) ≡ cosh( pτ ) stands for the Chebyshev polynomial of the first kind p, q ) are identified with the labeling of minimal fractional super-conformal models [73] and defined by( p, q ) = (ˆ k ˆ p, ˆ k ˆ q ) , (2.23)with introducing the following integers d ˆ q − ˆ p , ˆ k and η , d ˆ q − ˆ p ≡ g.c.d. (cid:0) ˆ q − ˆ p, k (cid:1) , k = ˆ k × d ˆ q − ˆ p , ˆ q − ˆ p = η × d ˆ q − ˆ p . (2.24)2. The other kind is derived from real-potential models [43] and given by replacing theshift matrix Γ by the twisted shift matrix Γ (real) as follows: P ( t ; ∂ ) = Γ (real) ∂ ˆ p + ˆ p X n =1 H (R k P) n ( t ) ∂ ˆ p − n , Q ( t ; ∂ ) = Γ (real) ∂ ˆ q + ˆ q X n =1 H (R k Q) n ( t ) ∂ ˆ q − n , (2.25)with the twisted shift matrix Γ (real) ,Γ (real) = − , (2.26)and the real functions H (R k P) n ( t ) and H (R k Q) n ( t ). The weak coupling analysis g str → P ( t ; ∂ ) ≃ diag kj =1 (cid:0) P ( j ) ( t ; z ) (cid:1) , Q ( t ; ∂ ) ≃ diag kj =1 (cid:0) Q ( j ) ( t ; z ) (cid:1) [44]:Cosh solution: P ( j ) = λ ˆ p cosh (cid:0) ˆ pτ + 2 πi j − k (cid:1) , Q ( j ) = λ ˆ q cosh (cid:0) ˆ qτ + 2 πi j − k (cid:1) ; (2.27)Sinh solution: P ( j ) = λ ˆ p sinh (cid:0) ˆ pτ + 2 πi j − k (cid:1) , Q ( j ) = λ ˆ q sinh (cid:0) ˆ qτ + 2 πi j − k (cid:1) , (2.28)with the parameters in Eqs. (2.20). Their algebraic equations are expressed asCosh solution: F ( P, Q ) = T p ( Q/λ ˆ q ) − ( − { η } T q ( P/λ ˆ p ) = 0; (2.29)Sinh solution: F ( P, Q ) = T p ( − iQ/λ ˆ q ) − ( − { η } + q − p T q ( − iP/λ ˆ p ) = 0 . (2.30)Here η is defined in Eq. (2.24). A nontrivial feature of these explicit expressions [44] is that these algebraic equations F ( P, Q ) = 0 are generally factorized into several irreducible pieces of the algebraic curves8 ( j ) ( P, Q ) = 0 , ( j = 1 , , · · · , k ): F ( P, Q ) = ⌊ k ⌋ +1 Y j =1 F ( j ) ( P, Q ) = 0 , (2.31)and these irreducible pieces are equivalent to the algebraic curves of the fractional-superstring critical points (of the matrix models) with the smaller number of cuts. Forinstance, some are equivalent to the curves of minimal bosonic string theory and someare those of minimal type 0 superstring theory.The authors of [44] derived the algebraic equation as the following limit of the Baker-Akhiezer system: F ( P, Q ) = 0 s.t. F (cid:16) ζ , g str ∂∂ζ (cid:17) Ψ( t ; ζ ) = 0 g str → . (2.32)It was shown by the general arguments of topological recursions [40] that the algebraicequation has enough information to recover the perturbative corrections of the stringtheories. This means that we can reconstruct “perturbative Baker-Akhiezer functions”from the perturbative pieces of the algebraic equation:Ψ ( j )pert ( t ; ζ ) : F ( j ) (cid:16) ζ , g str ∂∂ζ (cid:17) Ψ ( j )pert ( t ; ζ ) = 0 g str → , (2.33)the leading behavior of which is given by one of the branches of the algebraic equation R ( j ) ( t ; ζ ), F (cid:0) ζ , R ( j ) ( t ; ζ ) (cid:1) = 0:Ψ ( j )pert ( t ; ζ ) ≃ asym v ( j ) ( t ; ζ ) exp h g − Z ζ dζ ′ R ( j ) ( t ; ζ ′ ) i + · · · , g str → , (2.34)where v ( j ) ( t ; ζ ) is a proper k -order vector-valued function. Interestingly, these wavefunctions Ψ ( j )orth ( j = 1 , , · · · , k ) are completely decouple to each other in all-order per-turbation theory, since the algebraic equations are factorized into these irreducible pieces.Clearly, the exact Baker-Akhiezer function Ψ( t ; ζ ) and the perturbative Baker-Akhiezerfunctions Ψ ( j )pert ( t ; ζ ) are different, but an asymptotic expansion of the exact Baker-Akhiezer function Ψ( t ; ζ ) in g str is given by a superposition of the perturbative Baker-Akhiezer functions Ψ ( j )pert ( t ; ζ ):Ψ( t ; ζ ) ≃ asym X j c j Ψ ( j )pert ( t ; ζ ) , g str → , with ( t, ζ ) ∈ ∃ D ⊂ C . (2.35)This result suggests that the total wave function of the fractional superstring theory is asuperposition of various perturbative minimal string theories [44]! One of our motivations Each equation F ( j ) ( P, Q ) = 0 corresponds to an eigenvalue of the Lax pair ( P , Q ). Then the upperindex of the equations satisfies F ( k − j ) ( P, Q ) = F ( j ) ( P, Q ) = F ( j + k ) ( P, Q ) = 0. These perturbative Baker-Akhiezer functions also receive non-perturbative corrections by ZZ-branesof the corresponding perturbative string theory. The general form of these functions is universal and isexpressed by theta functions on the spectral curve [42] (in other words, they are the τ -functions of thecorresponding integrable hierarchy).
9n this paper is to further investigate this intriguing point from the non-perturbative pointof views.In particular, from the perturbation theory, there is no way to fix the relative co-efficients { c j } kj =1 . This is the orthonormal-polynomial version of the non-perturbativeambiguity, and it is clear that these coefficients { c j } kj =1 cannot take arbitrary values,even though these parameters { c j } kj =1 are an analogy of the theta parameters in QCD.The correct values are obtained in the Z k -symmetric critical points [72] and we will showthe fractional-superstring cases in Section 3.3. Interestingly, as is shown in [72] and aswe will see in Section 3.3, determination of these coefficients { c j } kj =1 are equivalent tofixing the D-instanton fugacities. By an analytic continuation of ζ , the most dominant perturbative Baker-Akhiezer func-tions in the asymptotic expansion Eq. (2.35) will change in a discontinuous way. Suchdiscontinuities generally appear when one crosses the Stokes lines:SL j,l ≡ n ζ ∈ C ; Re Z ζ dζ ′ h R ( j ) ( t ; ζ ′ ) − R ( l ) ( t ; ζ ′ ) i = 0 o , ( j, l = 1 , , · · · , k ) . (2.36)In particular, the Stokes lines can be interpreted as physical cuts of the eigenvalue distri-bution function ρ ( λ ) of Eq. (2.8), if the exact Baker-Akhiezer function is given by thescaling orthonormal polynomials Ψ orth ( t ; ζ ) of Eq. (2.11) (and Eq. (2.10)),Ψ( t ; ζ ) = Ψ orth ( t ; ζ ) ≃ asym v ( t ; ζ ) exp h g − Z ζ dζ ′ R ( ζ ′ ) i + · · · ,g str → t, ζ ) ∈ ∃ D ⊂ C . (2.37)Here R ( ζ ) is the scaling resolvent operator, N dxR ( x ) = g − dζ R ( ζ ). From the viewpointof the large N resolvent, Eq. (2.8), it is clear that the physical cuts (around ζ → ∞ )appear along the particular angles, ζ → ∞ × e iχ n : χ n = χ + 2 πnk , χ = πk (1 − ǫ ) , (cid:0) n = 0 , , , · · · , k − (cid:1) . (2.38)However this consideration is not straightforward from the Baker-Akhiezer function sys-tem Eq. (2.14) and implies non-trivial physical constraints on the coefficients { c j } kj =1 in Eq. (2.35) and also on the Stokes phenomena which relate the coefficients in differ-ent regions D of the asymptotic expansions Eq. (2.35). This is the multi-cut boundarycondition proposed in [72].This consideration in the fractional-superstring critical points is quite interesting be-cause the perturbative minimal string theories described by the irreducible algebraiceqations, F ( j ) ( P, Q ) = 0 , ( j = 1 , , · · · , k ) are completely decouple in the all-order per-turbation theory. Therefore, the discontinuous jumps of the wave function Ψ orth ( t ; ζ )are the jumps between distinct perturbative string theories (i.e. not the jumps within The physical cuts can be curved lines if the matrix-model potential includes complex coefficients.However the eigenvalue distribution function along the lines should be a real function. This requirementis guaranteed by the definition of the Stokes lines, Eq. (2.36) (See [72]). Note that there is a nontrivial rotation of angle by ǫ , x ∼ ω ǫ/ ζ , in Eq. (2.12). ζ and they are connected by Stokesphenomena! This idea was first introduced in [44] in order to add physical cuts whichdisappear in the spectral curves F ( P, Q ) = 0. A quantitative demonstration of this ideawas then carried out in the Z k symmetric critical points [72]. Here we now analyze whatactually happens in the fractional-superstring critical points. The general frameworks of the Stokes phenomena in the multi-cut critical points aredeveloped in Section 3 of [72] (a review of basics of Stokes phenomena with relevantreferences is in [87] or in Section 2 of [72]). Here we will not repeat the discussions but therelevant results are briefly summarized in order to fix the notations and conventions. Somenew features appearing in the fractional-superstring critical points are also mentioned.For sake of simplicity, the discussion is restricted to ˆ p = 1.In the case of ˆ p = 1, the Lax operator P ( t ; ζ ) is generally given as P ( t ; ∂ ) = A ∂ + H ( t ) , det A = 0 , (2.39)and therefore the differential equation system Eqs. (2.14) is rewritten as the followingODE system of the Zakharov-Shabat eigenvalue problem [90, 91]: g str ∂∂t Ψ( t ; ζ ) = P ( t ; ζ ) Ψ( t ; ζ ) ≡ A − (cid:16) ζ − H ( t ) (cid:17) Ψ( t ; ζ ); (2.40) g str ∂∂ζ Ψ( t ; ζ ) = Q ( t ; ζ ) Ψ( t ; ζ ) ≡ Q ( t ; ∂ ) Ψ( t ; ζ ) . (2.41)The Douglas (i.e. string) equation [12] is now given by the following Zakharov-Shabatzero-curvature form of ( P , Q ): (cid:2) P ( t ; ∂ ) , Q ( t ; ∂ ) (cid:3) = g str I k ⇔ (cid:2) g str ∂∂t − P ( t ; ζ ) , g str ∂∂ζ − Q ( t ; ζ ) (cid:3) = 0 . (2.42)This system is referred to as k × k isomonodromy system, since this differential-equationsystem preserves the monodromy data and therefore Stokes data of Eq. (2.41) with respectto the flows of ( t ; ζ ).The leading behavior of the ordinary differential equation, Eq. (2.41), is given by g str ∂ Ψ( t ; ζ ) ∂ζ = (cid:0) A − γ ζ r − + · · · (cid:1) Ψ( t ; ζ ) , γ = r − . (2.43)Here r = ˆ q + 1 is the Poincar´e index of the essential singularity at ζ → ∞ . We considertwo choices of the matrix A : A = (cid:26) Γ ≃ Ω : ω / -rotated modelsΓ (real) ≃ ω / Ω : real-potential models , Ω = diag kj =1 (cid:0) ω j − (cid:1) . (2.44) In this paper, “ ≃ ” indicates equality up to some similarity transformation.
11s in [72], we also restrict ourselves to the cases of g . c . d . ( k, γ ) = 1 in order to avoidcomplexity due to degeneracy of the exponents and appearance of the subdominant ex-ponents. Interestingly, this condition coincides with the following condition on d ˆ q − ˆ p in(2.24): d ˆ q − ˆ p = g . c . d . ( k, ˆ q −
1) = 1 . (2.45)From the fractional-superstring viewpoint, this condition means the restriction to thecritical points which are free from the Z k × Z k -orbifolding [73]. This kind of orbifold-ings was first introduced in the two-cut critical points of the two-matrix models [39](i.e. Z -orbifolding in the cases) in order to keep the correspondence with perturbativetype 0 minimal superstring theories of odd sequence (ˆ q, ˆ p ∈ Z + 1). Therefore, theabove coincidence suggests that this Z k × Z k -orbifolding procedure is naturally encodedin the non-perturbative Stokes-phenomenon structure and can be imposed by just com-pletely identifying the degenerate exponents. This is very interesting observation but theinvestigation along this direction is out of scope in this paper.The basic ingredients of Stokes phenomena are following:
This ODE system Eq. (2.41) has the k independent order- k column vector solutionsΨ ( j ) ( t ; ζ ), ( j = 1 , , · · · , k ), and they are often expressed as the following matrix:Ψ( t ; ζ ) ≡ (cid:16) Ψ (1) ( t ; ζ ) , · · · , Ψ ( k ) ( t ; ζ ) (cid:17) . (2.46)The Stokes phenomena discussed in this paper originates from the asymptotic expansionaround ζ → ∞ . In term of the matrix notation Eq. (2.46), with a proper change ofnormalization of the solutions, the expansion can always be expressed asΨ asym ( t ; ζ ) ≡ Y ( t ; ζ ) e g str ϕ ( t ; ζ ) ≡ h I k + ∞ X n =1 Y n ( t ) ζ n i × exp h g str (cid:16) ϕ ln ζ − ∞ X m = − r, =0 ϕ m ( t ) m ζ m (cid:17)i . (2.47)The leading exponents are written as ϕ ( t ; ζ ) = A − γ r ζ r + O ( ζ r − ) , ζ → ∞ . (2.48)From the hermiticity of the matrix models, these matrix solutions are always real func-tions [43]. This basis of the solutions is called the matrix-model basis (or the Γ -basis ).On the other hand, the Stokes phenomena are always discussed in the diagonal basis (or the Ω -basis ): e Ψ asym ( t ; ζ ) ≡ e Y ( t ; ζ ) e g str e ϕ ( t ; ζ ) ≡ h I k + ∞ X n =1 e Y n ( t ) ζ n i × exp h g str (cid:16) e ϕ ln ζ − ∞ X m = − r, =0 e ϕ m ( t ) m ζ m (cid:17)i ≡ U † Ψ asym ( t ; ζ ) U, (2.49)12here the unitary matrix U is given as U jl = 1 √ k ω ( j − ǫ/ l − ǫ/ , A U = U (cid:0) ω ǫ/ Ω (cid:1) . (2.50)Therefore, the exponents are diagonal matrices: e ϕ ( t ; ζ ) = diag kj =1 (cid:0) ϕ ( j ) ( t ; ζ ) (cid:1) , and theleading behavior is always the same as the exponents of the perturbative Baker-Akhiezerfunctions Eq. (2.34), ϕ ( j ) ( t ; ζ ) − Z ζ dζ ′ R ( j ) ( t ; ζ ′ ) = O ( ζ ) , ζ → ∞ . (2.51) The reconstruction of the exact solutions from these asymptotic expansions is achievedin specific angular domains called
Stokes sectors D n . This can be defined by Stokes lines
Eq. (2.36), especially their leading behavior of ζ → ∞ :sl j,l ≡ n ζ ∈ C ; Re (cid:2)(cid:0) ϕ ( j ) − r − ϕ ( l ) − r (cid:1) ζ r (cid:3) = 0 o = n ζ ∈ e iθ ( n ) j,l R ; θ ( n ) j,l = kn + γ ( j + l − ǫ ) rk π, n ∈ Z o , (2.52)where ϕ ( j ) − r is the leading exponent of ϕ ( j ) ( t ; ζ ) = (1 /r ) ϕ ( j ) − r ζ r + · · · , and γ = r −
2. Becauseof g . c . d . ( k, γ ) = 1, the minimal angular domains in between the Stokes lines, δD n , aregiven by δD n = D (cid:0) π ( n − kr , πnkr (cid:1) , (2.53)which are referred to as segments . Therefore, the Stokes sectors are defined as the angulardomains which consist of ( k + 1) different segments: D n = D (cid:0) π ( n − kr , π ( n + k ) kr (cid:1) , (2.54)which are also called fine Stokes sectors . In each Stokes sector, one can uniquely recon-struct the exact (canonical) solutions e Ψ n ( t ; ζ ) from the asymptotic expansion Eq. (2.49): e Ψ n ( t ; ζ ) ≃ asym e Ψ asym ( t ; ζ ) , ζ → ∞ ∈ D n . (2.55)However, in general, these exact solutions have different normalizations: e Ψ n +1 ( t ; ζ ) = e Ψ n ( t ; ζ ) S n . (2.56)This behavior of asymptotic expansion is called Stokes phenomena in this ODE systemand the matrix S n is called (fine) Stokes matrix . The total number of segments is 2 rk and is equal to that of fine Stokes sectors. Reading the positions of non-trivial components in the Stokes matrices, S n = ( s n,i,j ), isalways a tedious problem, although there is a basic theorem which can be applied toevery ODE system in principle. Here, however, we use more efficient way which was In this paper, we use the following notation of angular domain: D ( a, b ) ≡ { a < arg( ζ ) < b } . This choice of Stokes sectors is most fundamental. For other descendant Stokes sectors, see [72]. See Theorem 2 in [72], for example. profile of dominant exponents . The profile of dominant exponents J for ( k, r ; γ ) is a sequence of integer numbers { j l,n } of the following type: J ≡ J rk − ... J J ≡ j rk − , j rk − , · · · j rk − ,k ... ... ... j , j , · · · j ,k j , j , · · · j ,k . (2.57)Each row vector J l corresponds to a segment δD l of Eq. (2.53), and the numbers in J l = (cid:2) j l, j l, · · · j l,k (cid:3) describes the profile of dominant exponents in the segment δD l : Re (cid:2) ϕ ( j l, ) − r ζ r (cid:3) < Re (cid:2) ϕ ( j l, ) − r ζ r (cid:3) < · · · < Re (cid:2) ϕ ( j l,k ) − r ζ r (cid:3) , ζ ∈ δD l . (2.58)The profile is expressed as J ( ǫ )( k,r ) in order to explicitly show the indices ( k, r ) and ǫ = 0 , γ is always γ = r −
2. As an example, a part of the profile J (0)(7 , (7-cut(ˆ p, ˆ q ) = (1 , ω / -rotated fractional-superstring critical point) is shown: J (0)(7 , = ... ... ... ... ... ... ...7 (6 1) (5 2) (4 3)(6 7) (5 1) (4 2) 36 (5 7) (4 1) (3 2)(5 6) (4 7) (3 1) 25 (4 6) (3 7) (2 1)(4 5) (3 6) (2 7) 1 . (2.59)Here we put the parentheses as ( i | j ) in the profile to indicate the pairs ( i, j ) whichchange the dominance in the next segment (i.e. there is a Stokes line of sl i,j just abovethe segment). We also use the notation ( i | j ) l to show the segment J l which the pairbelongs to: ( i | j ) l ∈ J l . Then the non-trivial claim (Theorem 4 in [72]) is s l,i,j is non-trivial component of S l ⇔ ( j | i ) l ∈ J l . (2.60)The others off-diagonal components are zero, s l,i,j = 0. From the standard theorem, thediagonal components are always s l,i,i = 1 ( i = 1 , , · · · , k ). A simple formula for the integer numbers in the profile j l,n is also obtained in [72] (The-orem 3) for the ω / -rotated models and the formula for the real-potential models is alsoobtained by a minor change: j ( ǫ ) l,n ≡ (cid:18)j ( l − ǫγ )2 k + ( − k +( l − ǫγ )+ n j k − n + 12 k(cid:19) m , mod k, (2.61) Note that the first exponent ϕ (1) − r ζ r = ω − γ/ ζ r becomes the largest when arg( ζ ) = πkr γ . Also notethat the effect of ǫ is just a total shift of θ ( n ) j,l in Eq. (2.52). m is obtained by the Euclidean algorithm of kn + γm = 1. Interestingly, thedifference of the formula for the Stokes phenomena between the ω / -rotated models andthe real-potential models is just a shift of the angular coordinate and essentially they bothhave the same structure, even though these models are distinct from the viewpoint ofperturbation theory. This implies that different perturbative non-critical string theoriesare simply unified in the strong-coupling non-critical M-theory description, as is in thecritical 11-dimensional M theory. There are basic three (or two) constraints on the Stokes multipliers. The original threeconstraints in the two-cut cases can be found in [87], and also the extension to the multi-cut cases is in [72] with referring to the matrix-model results [43]. A major difference inthe fractional-superstring critical points is that there is no Z k symmetry (or there is only Z symmetry when k ∈ Z ). These constraints are also obtained in the similar way asin [72]: Z -symmetry condition (only when k ∈ Z ): S n + rk = Γ − k S n Γ k , (cid:0) n = 0 , , · · · , rk − (cid:1) . (2.62) The hermiticity conditions: S ∗ n = ∆Γ (1 − ǫ ) S − r − k − n Γ − (1 − ǫ ) ∆ , (cid:0) n = 0 , , · · · , rk − (cid:1) , (2.63)with the matrix ∆, ∆ i,j = δ i,k − j +1 . Note that the unitary matrix U of Eq. (2.50) satisfies U = ( − ǫ ∆Γ (1 − ǫ ) . In terms of the elements, they are expressed as s ∗ n,i,j = − s − k − n, − ǫ − i, − ǫ − j , ( j | i ) n ∈ J n . (2.64) The monodromy free condition: S · S · · · S rk − = I k . (2.65)In the case of k ∈ Z , this is also expressed as h ( S · · · S rk − )Γ − k i = I k . (2.66)In the case of k ∈ Z + 1, this is also expressed as S ∗ = ∆Γ (1 − ǫ ) S ∆Γ (1 − ǫ ) , (2.67)with S ≡ S − k +12 +1 S − k +12 +2 · · · S S · · · S rk − k +12 . See also Section 2 in [72] since the convention used in the matrix models is different. The multi-cut boundary conditions (BC)
In this section, we evaluate the multi-cut boundary conditions in the fractional-superstringcritical points. The multi-cut boundary condition is first proposed in [72] (Definition 10)and also mentioned around Eq. (2.38) in this paper.One of the main differences from the Z k symmetric cases [72] is that there are a hugenumber of Stokes multipliers. It is because the smallest Poincar´e index r is r = 3(there is no (ˆ p, ˆ q ) = (1 ,
1) critical point) and there is no Z k symmetry. This causes apredicament in direct analysis of the algebraic equations of Stokes multipliers. Therefore,we should invent some systematic way to analyze these large systems.An important property of the formula for the profile components Eq. (2.61) is thefollowing vertical shift in the profile: j ( ǫ ) l +2 a,n = j ( ǫ ) l,n + a m . (3.1)This implies the following fact: s l,i,j is non-trivial (i.e. ∃ ( j | i ) l ∈ J l ) ⇔ s l +2 a,i + am ,j + am is non-trivial (i.e. ∃ ( j + am | i + am ) l +2 a ∈ J l +2 a ) . (3.2)Therefore, once we know the existence of non-trivial Stokes multipliers of s l,i,j ( l =0 , , , · · · , r − a shift rule to indicate the range2 a of making the shift.In the following discussion, it will become clear that the fundamental set of Stokesmatrices in the above sense is (cid:8) S n (cid:9) r − n =0 , therefore essentially a single symmetric Stokesmatrices S (sym)0 . Here the symmetric Stokes matrices are defined as S (sym)2 rn ≡ S rn S rn +1 · · · S r ( n +1) − , (cid:0) n = 0 , , , · · · , k − (cid:1) . (3.3)Therefore, we will start with the symmetric Stokes sectors (cid:8) D rn (cid:9) k − n =0 in the followingdiscussion of the multi-cut boundary conditions. The scaling orthonormal polynomial Ψ orth ( t ; ζ ) of Eq. (2.11) is generally expressed by thecanonical solution e Ψ rn ( t ; ζ ) of the symmetric Stokes sectors D rn asΨ orth ( t ; ζ ) = e Ψ rn ( t ; ζ ) X (2 rn ) , X (2 rn ) = x (2 rn )1 x (2 rn )2 ... x (2 rn ) k , (3.4) For example, there are seven times more than the multipliers shown in the profile, Eq. (2.59), in the7-cut r = 3 case. The name comes from the symmetric Stokes matrices in the Z k -symmetric critical points [72], al-though there is no Z k -symmetry in the fractional-superstring critical points. n = 0 , , , · · · , k −
1. Therefore, the vectors X (2 rn ) are related by sym-metric Stokes matrices of Eq. (3.3) as X (2 rn ) = S (sym)2 rn X (2 r ( n +1)) , (cid:0) n = 0 , , , · · · , k − (cid:1) . (3.5)Then the multi-cut boundary condition is imposed on the vectors (cid:8) X (2 rn ) (cid:9) k − n =0 . By fol-lowing the procedure discussed in [72], we read the boundary conditions as follows: Herewe first consider an example of 7-cut r = 3 ω / -rotated case. The profile in D and thecorresponding vector X (0) are given as D ⊃ △ △ △ × △ × × △ × × △ × △ △ , X (0) = x (0)1 = 000 x (0)4 = 0 x (0)5 x (0)6 x (0)7 . (3.6)Note that ”2 × ” means x (0)2 = 0 and “4 △ ” means x (0)4 = 0. In the middle of the aboveprofile, there is a horizontal line where “1 △ ” and “4 △ ” change their dominance, which isinterpreted as a physical cut of the resolvent operator. In principle we should performthe same procedure for all the symmetric Stokes sectors (cid:8) D rn (cid:9) k − n =0 to obtain their vec-tors (cid:8) X (2 rn ) (cid:9) k − n =0 , however the boundary conditions for all the vectors (cid:8) X (2 rn ) (cid:9) k − n =0 areautomatically obtained from the first vector X (0) in the first symmetric Stokes sector D because the multi-cut boundary conditions for the vectors (cid:8) X (2 rn ) (cid:9) k − n =0 are related to eachother by the r shift rule of Eq. (3.1): x (2 rn ) j = 0 (cid:16) or x (2 rn ) j = 0 (cid:17) ⇒ x (2 rn +2 r ) j + rm = 0 (cid:16) or x (2 rn +2 r ) j + rm = 0 (cid:17) . (3.7)For example, one can directly check the other vectors (cid:8) X (6 n ) (cid:9) n =1 of this case ( k = 7 , r =3 , m = 1) are given as X (6) = x (6)1 x (6)2 x (6)3 x (6)4 = 000 x (6)7 = 0 , X (12) = x (12)3 = 0 x (12)4 x (12)5 x (12)6 x (12)7 = 0 , X (18) = x (18)1 x (18)2 x (18)3 = 000 x (18)6 = 0 x (18)7 · · · , (3.8)and they are all consistent with Eq. (3.7). This is because we read the multi-cut boundaryconditions from the profiles which are again obtained by using the 2 r shift rule. Therefore,only the multi-cut boundary condition equation, Eq. (3.5), of n = 0 is essential and thenthe other equations are generated by using the 2 r shift rule, Eq. (3.1). This procedurereduces a great deal of the complexity. Note that this consideration does not mean there17s the Z k symmetry in the critical points. An analogy of this consideration is “a symmetryof equation of motion ,” which does not necessarily mean a symmetry of the solutions.It is not difficult to read the boundary condition for general k with r = 3, which willbe used in the practical analysis. We put k = 2 r e m + e l (1 ≤ e l ≤ r ) and there are 2 r (= 6)different cases ( e l = 1 , , · · · ,
6, respectively):1. X (0) ↔ (1 △ , × , × , △ , × , × , △ , · · · , × , × , (cid:0) e m + 1 (cid:1) △ , − − · · · − );2. X (0) ↔ (1 △ , × , × , △ , × , × , △ , · · · , × , × , (cid:0) e m + 1 (cid:1) △ , × , − − · · · − );3. X (0) ↔ (1 △ , × , × , △ , × , × , △ , · · · , × , × , (cid:0) e m + 1 (cid:1) △ , × , × , − − · · · − );4. X (0) ↔ (1 △ , × , × , △ , × , × , △ , · · · , × , × , (cid:0) e m + 1 (cid:1) △ , × , × , (cid:0) e m + 4 (cid:1) △ , − − · · · , − );5. X (0) ↔ (1 △ , × , × , △ , × , × , △ , · · · , × , × , (cid:0) e m + 1 (cid:1) △ , × , × , (cid:0) e m + 4 (cid:1) △ , − − · · · , −− );6. X (0) ↔ (1 △ , × , × , △ , × , × , △ , · · · , × , × , (cid:0) e m + 1 (cid:1) △ , × , × , (cid:0) e m + 4 (cid:1) △ , − − · · · , − − − ) . (3.9)To save the space, we used a simpler notation: “1 △ ” means x (0)1 = 0, “ × ” on the i -th component means x (0) i = 0, and “ − ” means that there is no constraint on thesecomponents. For example, the vector X (0) of Eq. (3.6) is expressed as (1 △ , × , × , △ , − −− ). Here we introduced a notation of the indices with tildes, ( e m, e l ). It is because this 2 r classification is reduced to be a simpler r classification, k = rm + l (1 ≤ l ≤ r ), in thefinal unified form of the multi-cut BC recursion equations. Finally we mention the meaning of the general 2 a shift rule for the boundary con-dition. We used the 2 r shift rule because the multi-cut boundary condition Eq. (3.6)(generally mentioned around Eq. (2.38)) only respects the 2 r shift rule. Therefore, onemay wonder if we use the other 2 a shift rules on these multi-cut boundary conditions.The answer to this question is following: The a shift rules ( a = 1 , , · · · , r − ) for themulti-cut boundary conditions generate the complementary boundary conditions. Thecomplementary boundary condition on the Stokes phenomena was first discussed in the Z k symmetric cases [72]. The physical meaning of the complementary boundary condi-tions is still unclear but these constraints provide strong equations which help us obtainexplicit expressions of the Stokes multipliers even in the general ( k, r ) isomonodromysystems. In fact, one can check that the 2 a shift rule ( a = 1 ,
2) for the the multi-cut boundary condition Eq. (3.6) results in the following two kinds of complementary Maybe of non-critical M theory. Therefore, 1 . ↔ . , 2 . ↔ . , 3 . ↔ . in Eq. (3.9) have the same form of the multi-cut BC recursions. One may expect that it is a constraint on operators in non-critical M theory which is analogous to“degenerate operators in conformal field theory”. Also note that, if the indices ( k, r ) are small integers,then the complementary boundary conditions are exactly satisfied. See [72] for explicit analysis of someconcrete examples. D ⊃ △ △ △ × △ × × △ × × △ × △ △ , X (2) = x (2)1 x (2)2 = 000 x (2)5 = 0 x (2)6 x (2)7 ; (3.10)4 shift: D ⊃ △ △ △ × △ × × △ × × △ × △ △ , X (4) = x (4)1 x (4)2 x (4)3 = 000 x (4)6 = 0 x (4)7 . (3.11)Therefore, in general, each of these 2 a complementary boundary conditions ( a = 1 , , · · · , r −
1) requires that there is a solution of the Baker-Akhiezer function system Eq. (2.14) whichhas the physical cuts along the k directions, ζ → ∞ × e iχ n , of χ n = χ + 2 πnk , χ = πk (1 − ǫ ) + 2 πark , (cid:0) n = 0 , , , · · · , k − (cid:1) . (3.12)In the following, it is also convenient not to distinguish the multi-cut boundary conditionand the 2 a complementary boundary conditions (defined by Eq. (3.12)), therefore, theyare equally called the multi-cut a boundary condition . In the a = 0 case, we skip 2 a inthe name. In the analysis of the Z k -symmetric critical points [72], it was observed that the multi-cutboundary condition equation Eq. (3.5) with the multi-cut boundary condition Eq. (2.38)only includes a particular sequence of Stokes multipliers. Motivated by this observationgiven in [72], here we also introduce the following parametrization of the fine Stokesmultipliers: θ (2 a + ǫr ) n ≡ s r +2 a − n +1+ ǫr, j ( ǫ )2 r +2 a − n +1+ ǫr,k − n +1 , j ( ǫ )2 r +2 a − n +1+ ǫr,k − n , (cid:0) n = 1 , , · · · , k − a = 0 , , · · · , rk − (cid:1) . (3.13)It is much easier to catch the meaning of the formula by drawing it on the profile. Figure2 is an example of the profile for the 11-cut ( r = 3) ω / -rotated models.In fact, the multi-cut BC recursion equations Eq. (3.5) with the multi-cut boundarycondition only include a subset of the Stokes multipliers, (cid:8) θ (2 rb ) n (cid:9) ≤ b ≤ k − ≤ n ≤ k − , as we expected.Importantly, the 2 a shift rule acts on the parameter θ (2 rb ) n as2 a shift: θ (2 rb ) n → θ (2 rb +2 a ) n , (3.14)19 a) (b)111111 222 222 333333 444444 555555 666666 77777 7888 8 8 89 9 9999 10 10 10 101010 11 11 11 11 1111 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ Figure 2:
The theta parametrization of fine Stokes multipliers in the 11-cut ( r = 3) ω / -rotatedmodels. Here only the fundamental domain S r − l =0 J l of the profile is shown. a) The profile of dominantexponents with classification by different 3 (= r ) colors. Here we skipped the parenthesis notation ( i | j )but each pair of two same-colored boxes next to each other is the parenthesis pair ( i | j ). b) The thetaparametrization θ (2 a ) n is shown by using the correspondence s l,i,j ↔ ( j | i ) l ∈ J l . The colors are drawn inaccordance with a modulo r . One can see the tilting pattern of the parametrization. therefore the multi-cut BC recursion equations Eq. (3.5) with the multi-cut 2 a boundarycondition ( a = 1 , , · · · , r −
1) include a complementary set of the Stokes multipliers, (cid:8) θ (2 rb +2 a ) n (cid:9) ≤ b ≤ k − ≤ n ≤ k − .Before we discuss the details of the multi-cut BC recursion equations, basic propertiesof the theta parameters are summarized: • The Z -symmetry condition ( k ∈ Z ) Eq. (2.62) is expressed as θ (2 a ) n = θ (2 a + rk ) n (cid:0) n = 1 , , · · · , k − a = 0 , , , · · · , rk − (cid:1) . (3.15) • The hermiticity condition Eq. (2.63) is expressed as h θ ( a ) n i ∗ = − θ ( − r − a ) k − n (cid:0) n = 1 , , · · · , k − a = 0 , , , · · · , rk − (cid:1) . (3.16) • For later convenience, we extend the parameter as θ (2 a )0 = − , θ (2 a ) k = +1 , (cid:0) a = 0 , , , · · · , rk − (cid:1) , (3.17)which is consistent with the hermiticity constraint. In this section, we discuss the multi-cut boundary condition equation Eq. (3.5) with themulti-cut 2 a boundary condition Eq. (3.12). The derivation is given in Appendix A, andtherefore we here only show the result: Theorem 1 (The multi-cut BC recursion equations)
In the fractional-superstringcritical points of an arbitrary ( k, r ) with g . c . d . ( k, γ ) = 1 ( γ = r − ), the multi-cut bound-ary condition equations Eq. (3.5) with the multi-cut a boundary condition Eq. (3.12) of = 0 , , · · · , r − result in the following recursion equations: F (2 s ) b h(cid:8) y (2 s ) n (cid:9) n ∈ Z i ≡ X ≤ r ( n − b ≤ k, n ∈ Z θ (cid:0) r ( n − s (cid:1) r ( n − b y (2 s ) n = 0 of s = rl + a + ǫ r (cid:0) ≤ l ≤ k − , ≤ b ≤ r (cid:1) , (3.18) where the sequence of numbers (cid:8) y (2 s ) n (cid:9) n ∈ Z are all non-zero and satisfy y (2 s ) n = y (2 s +2 rb ) n − b , y (2 s ) n = y (2 s ) n + k , (cid:0) n, a, b ∈ Z (cid:1) . (3.19) Therefore, there are r distinct sequences (cid:8) y (2 a ) n (cid:9) kn =1 corresponding to each boundary con-dition of a . For each boundary condition (of 2 a ) there are r recursive equations F (2 s ) b = 0 of b =1 , , · · · , r . It is suggestive to see the positions of the Stokes multipliers (included in eachequation F (2 s ) b = 0) in the profile. Here we do not show explicitly, but one can see thatthe trajectory is along the line vertical to the trajectories of colors in Figure 2.Although the formula Eq. (3.18) is simply written in a unified way, the concrete formof each equation system depends on k of modulo r : k = rm + l . Here we show each ofthem. First of all, we introduce a new parameter σ (2 a ) m − n +1 ,b ≡ θ (2 r ( n − a ) r ( n − b , m ≡ j k − r k , (3.20)and then the multi-cut BC recursion equations are F (2 s ) b h(cid:8) y (2 s ) n (cid:9) n ∈ Z i ≡ ⌊ k + r − br ⌋ X n = ⌈ r − br ⌉ σ (2 s ) m +1 − n,b y (2 s ) n = 0 . (3.21)The concrete form of each equation is shown in Table 1. Here we also show the hermiticitycondition Eq. (3.16) in terms of the sigma parameter: σ (2 s ) m − n +1 ,b = − h σ ( − r ( m +2) − s ) n − , ( l − b ) i ∗ : 1 ≤ b < l, (cid:0) n = 1 , , · · · , m + 1 (cid:1) − h σ ( − r ( m +1) − s ) n, ( r + l − b ) i ∗ : l ≤ b ≤ r, (cid:0) n = 1 , , · · · , m (cid:1) . (3.22)Therefore, the hermiticity condition relates equations of the five cases (I, II, III, IV,V)as: I ↔ I , II ↔ IV , III ↔ III , V ↔ V . (3.23)These are consistent with Eq. (3.21) (also see Table 1). Here we point out an intriguing connection to the structure of quantum integrable models.A non-trivial connection between Stokes phenomena in ordinary differential equationsand T-systems of quantum integrable systems is found in a special kind of Schr¨odinger21ases 1 ≤ b ≤ r F (2 s ) b (cid:2)(cid:8) y (2 s ) n (cid:9) n ∈ Z (cid:3) = 0 lengthI. 1 ≤ b < l m +1 X n =1 σ (2 s ) m − n +1 ,b y (2 s ) n = 0 m + 1II. b = l ( = r ) y (2 s ) m +1 + m X n =1 σ (2 s ) m − n +1 ,b y (2 s ) n = 0 m + 1III. l < b < r m X n =1 σ (2 s ) m − n +1 ,b y (2 s ) n = 0 m IV. b = r ( = l ) − y (2 s )0 + m X n =1 σ (2 s ) m − n +1 ,b y (2 s ) n = 0 m + 1V. b = r (= l ) − y (2 s )0 + y (2 s ) m +1 + m X n =1 σ (2 s ) m − n +1 ,b y (2 s ) n = 0 m + 2Table 1: The concrete forms of Eq. (3.18) (and then Eq. (3.21)) is shown with k = rm + l, (1 ≤ l ≤ r ).The case of V is the degenerate case of II and IV (i.e. r = l ). Possible cases are (II, III, IV) for l = 1, (I,II, III, IV) for 1 < l < r , and (I, V) for l = r . The “length” means the number of terms in the equation F (2 s ) b (cid:2)(cid:8) y (2 s ) n (cid:9) n ∈ Z (cid:3) = 0. equations [92, 93], which is called the ODE/IM correspondence . In this section, weprovide a new generalization of this correspondence to general ODE systems appearingin the isomonodromy theory.In general, the Stokes multipliers are distributed in the Stokes matrices (cid:8) S n (cid:9) rk − n =0 in a very complicated way. However, if one looks at the general multi-cut BC equa-tions Eq. (3.18), one notices that the Stokes multipliers are re-organized in the followingsuggestive form: For example, the equation of IV in Table 1 is expressed as Y (2 s − r ) = Φ (2 s ) Y (2 s ) , (cid:0) s = 0 , , , · · · , rk − (cid:1) , (3.24)with Y (2 s ) ≡ y (2 s )1 y (2 s )2 ... y (2 s ) m , Φ (2 s ) = τ (2 s )1 τ (2 s )2 · · · τ (2 s ) m − τ (2 s ) m · · · · · · . (3.25)Here (cid:8) τ (2 s ) n (cid:9) n ∈ Z are related to the fine Stokes multipliers (cid:8) σ (2 s ) n,b (cid:9) n Z as τ (2 s ) n = σ (2 s ) m − n +1 ,r , (cid:0) n = 1 , , · · · , m (cid:1) . (3.26)This is the same form of the recursive equations found in the study of Stokes phenom-ena for the Schr¨odinger-equation systems of the ODE/IM correspondence (See [93] forexample). Therefore, according to the discussion in [93], each sequence (cid:8) τ (2 rl +2 b ) n (cid:9) ≤ l ≤ k ≤ n ≤ m The authors would like to thank Toshio Nakatsu and Ryo Suzuki for drawing our attention to theseinteresting papers and the correspondence. b = 1 , , · · · , r ) satisfies discrete Hirota equations [94] (or a T-system) of quantum inte-grable systems. An important difference from the original correspondence is that in the k × k isomonodromy ODE system of the Poincar´e index r , there are r different recursivesystems of Stokes multipliers which result in r different T-systems . That is, the systemis described by multiple T-systems of quantum integrable systems .We note that, in our calculation, there is no explicit appearance of spectral parameters of quantum integrable systems. However the spectral parameters appear by assuming analyticity of the index s . That is, we rewrite the index s as s = ru + a (0 ≤ a ≤ r − τ (2 s ) n = τ n ( u ; a ) , y (2 s ) n = y ( u + n ; a ) , Y (2 s ) = Y ( u ; a ) . (3.27)Then the claim is that analytic continuation of the index u realizes the spectral parametersof the T-systems. This is understood as continuous deformation of the solutions for Stokesmultipliers. Because of the isomonodromy property, therefore, this spectral parameter u commutes with the original spectral parameter ζ of the resolvent operator: h ∂∂u , ∂∂ζ i Ψ( t ; ζ ; u ) = 0 . (3.28)In this sense, this is a new spectral parameter which describes the strong-coupling dy-namics of non-critical string theory, possibly, non-critical M theory. Here we skip “ a ” which does not change in each integrable model. The direct consequenceof [93] is following: T-system appears as the algebraic relations encoded in the recursionequation Eq. (3.24) and Eq. (3.25), i.e. in the algebraic relations among components (cid:8) φ n,i,j (cid:9) n ∈ Z ≤ i,j ≤ m of the multiplication matrices Φ n ( u ) = (cid:0) φ n,i,j ( u ) (cid:1) ≤ i,j ≤ m :Φ n ( u ) ≡ Φ (cid:0) u (cid:1) Φ (cid:0) u + 1 (cid:1) · · · Φ (cid:0) u + n − (cid:1) Φ (cid:0) u + n − (cid:1) . (3.29)The discrete Hirota equation (i.e. T-system) which is relevant to the algebraic structureis that of the integrable models with the quantum symmetry U q ( A (1) m − ) of q k = 1 [93]: T a,s ( u + 1) T a,s ( u −
1) = T a,s +1 ( u ) T a,s − ( u ) + T a +1 ,s ( u ) T a − ,s ( u ) , (3.30)with the following boundary condition: T − n,s ( u ) = T m + n,s ( u ) = 0 (cid:0) n ∈ N , s ∈ Z (cid:1) , (3.31) T a,s + k − m ( u ) = 0 (cid:0) < a < m, < s < m (cid:1) , (3.32) T a, ( u ) = T ,s ( u ) = 1 (0 ≤ a ≤ m, s ∈ Z ) , T a,s ( u ) = T a,s + k ( u ) , (3.33)which is also shown in Figure 3. Note that T-functions also satisfy periodicity in thespectral parameter u : T a,b ( u + 2 k ) = T a,b ( u ) . (3.34) For references of Hirota equations, see [95] for example. n ( u ) is given in thefollowing way [93]: τ n ( u ) = ( − n +1 T n, (2 u + n ) , φ n, , ( u ) = T ,n (2 u + n ) . (3.35)Notes on the boundary conditions are in order:1. Generally it is known that general T-functions of the above T-system can be ex-pressed in terms of the vertical/horizontal inputs by the Bazhanov-Reshetikhinformula [96] T a,s ( u ) = det ≤ i,j ≤ s T a + i − j, ( u + s + 1 − i − j ) , (3.36)= det ≤ i,j ≤ a T ,s + i − j ( u + a + 1 − i − j ) , (3.37)with the boundary condition Eq. (3.33).2. One can directly solve Eq. (3.29) and check that the coefficients (cid:8) φ n, , ( u ) (cid:9) ∞ n =1 are expressed by (cid:8) τ n ( u ) (cid:9) mn =1 and the expression is equivalent to the Bazhanov-Reshetikhin formula of Eq. (3.36) (and the identification Eq. (3.35)) with the fol-lowing extension of an index: τ ( u ) = − , τ − n ( u ) = τ m + n ( u ) = 0 (cid:0) n ∈ N (cid:1) . (3.38)In particular, Eq. (3.36) results in Eq. (3.31) and Eq. (3.33).3. The cyclic condition Φ k ( u ) = I m (which results from the monondromy free con-dition Eq. (2.65) in our case) gives the following constraints on the components (cid:8) φ n, , ( u ) (cid:9) n ∈ Z : φ n, , = φ n + k, , (cid:0) n ∈ Z (cid:1) , φ k − m + n, , = δ n,m (cid:0) ≤ n ≤ m (cid:1) . (3.39)Therefore, by using Eq. (3.37), one can show Eq. (3.32) and Eq. (3.33). Here we show several explicit solutions for the Stokes multipliers which generalize thediscrete solutions found in the Z k -symmetric critical points [72]. Solutions shown hereare therefore special solutions and further analysis for the complete solutions (with usingthe above T-systems and their Bathe ansatz equations) should be studied in anotherpublication. In particular, we impose the following “ Z k -symmetry condition”: S n +2 r = Γ − S n Γ (cid:0) n = 0 , , · · · , rk − (cid:1) ⇔ θ (2 s ) n = θ (2 s +2 r ) n (cid:0) ≤ n ≤ k − s = 0 , , · · · , rk − (cid:1) . (3.40)Note that this constraint on the Stokes multipliers does not necessarily mean existenceof the Z k symmetry of the system, although the monodromy free condition Eq. (2.65)becomes (cid:2) S (sym)0 Γ − (cid:3) k = I k . (3.41) The expression for more general components is also shown in [93] (i.e. Theorem 3). k-mm sa Figure 3:
The boundary condition for the T-system with U q ( A (1) m − ) of q k = 1. The lattice pointsexplicitly shown in the figure are the support of the T-function, T a,s ( u ). In particular, T ,s ( u ) = T a, ( u ) =1 ( s ∈ Z , ≤ a ≤ m ), and also there is a periodicity T a,s ( u ) = T a,s + k ( u ). The example shown here is k = 14 and m = 4 (i.e. r = 3). From the viewpoint of the discrete Hirota equation, the Z k -symmetric constraint is equiv-alent to dropping the spectral-parameter dependence and the Hirota equation then be-comes the system for characters of a corresponding Lie group. Consequently, the Stokesmultipliers (cid:8) τ n ( u ) (cid:9) mn =1 of Eq. (3.25) are given by the characters of the anti-symmetryrepresentations, χ n, ( g ): τ n ( u ) = ( − n +1 T n, ( u + 2 n ) = ( − n +1 χ n, ( g ) , (3.42)with a proper group element g ∈ GL( m ). The constraints on the group element g isfrom the algebraic relation of the T-system boundary conditions, however, we know thesolutions since the system is the same as this Z k -symmetric situation [72]. That is, thegroup elements are given by distinct roots of unity, (cid:8) ω n j (cid:9) mj =1 and the characters are givenby symmetric polynomials: χ n, ( g ) = Sym h n, (cid:8) ω n i (cid:9) mi =1 i , g ≃ diag (cid:0) ω n , ω n , · · · , ω n m (cid:1) . (3.43)This is the connection to the discrete solutions found in [72].Clearly this is not the end of story because there are now r different T-systems, τ n ( u ) → τ n ( u ; a ) (cid:0) a = 0 , , , · · · , r − (cid:1) , (3.44)and they are related by the original monodromy free condition Eq. (2.65). This conditioncan be solved by showing explicit k distinct eigenvectors of the symmetric Stokes matrix S (sym)0 [72]. This condition results in the constraints on the charges (or group elements) g ( a ) which are classified by configurations of the avalanches [72]. k = rm + 1In this case, we have three kinds of equations (II, III and IV in Table 1). Therefore, wehave the following recursion-equation systemsII : Y ( u + 1; a ) = e Φ( u ; a ) Y ( u ; a ) , (cid:0) a = 0 , , · · · , r − (cid:1) , III : 0 = v b ( u ; a ) Y ( u ; a ) , (cid:0) < b < r ; a = 0 , , · · · , r − (cid:1) , IV : Y ( u − a ) = Φ( u ; a ) Y ( u ; a ) , (cid:0) a = 0 , , · · · , r − (cid:1) . (3.45)25he matrix Φ( u ; a ) is defined by Eq. (3.25) and Eq. (3.26), and the others are defined as e Φ( u ; a ) = · · · · · · · · · − σ (2 s ) m, − σ (2 s ) m − , − σ (2 s ) m − , · · · − σ (2 s )2 , − σ (2 s )1 , ,v b ( u ; a ) = (cid:0) σ (2 s ) m,b , σ (2 s ) m − ,b , σ (2 s ) m − ,b , · · · , σ (2 s )2 ,b , σ (2 s )1 ,b (cid:1) (cid:0) < b < r (cid:1) . (3.46)We here require e Φ( u ; a ) = Φ − ( u + 1; a ) , v b ( u ; a ) = 0 , (3.47)then the recursion equations is given by a single equation Eq. (3.24) discussed in Section3.2, which means that there are r Hirota equations. In terms of the components, theseStokes multipliers are given as II : σ (2 s ) m, = − τ (2 s ) m , σ (2 s ) m − n, = τ (2 s ) n τ (2 s ) m (cid:0) n = 1 , , · · · , m − (cid:1) , III : σ (2 s ) n,b = 0 (cid:0) < b < r ; n = 1 , , · · · , m (cid:1) , IV : σ (2 s ) m − n +1 ,r = τ (2 s ) n , (cid:0) n = 1 , , · · · , m (cid:1) . (3.49)As special solutions to the Hirota equation for (cid:8) τ n ( u ; a ) (cid:9) ≤ a ≤ r − ≤ n ≤ m (discussed in the begin-ning of this section), we assign the following discrete solutions τ n ( u ; a ) = ( − n +1 Sym h n, (cid:8) ω n (2 a ) i (cid:9) mi =1 i (cid:0) a = 0 , , · · · , r − (cid:1) . (3.50)Once we derive this formula, the procedure to obtain solutions is the same as the Z k -symmetric cases [72]. Here we explicitly checked the constraints by Mathematica, andtherefore, we make the following conjecture for the conditions on the exponents of ω in Note that there are other solutions which does not satisfy this condition. For example, we can alsoconsider recursion equation of type III and define T-system of U q ( A (1) m − ). Here however, we only focuson special solutions which are relatively simpler. Note that the inverse of the matrix Φ (2 s ) is given as (cid:2) Φ (2 s ) (cid:3) − = · · · · · · · · · τ (2 s ) m − τ (2 s )1 τ (2 s ) m − τ (2 s )2 τ (2 s ) m · · · − τ (2 s ) m − τ (2 s ) m − τ (2 s ) m − τ (2 s ) m . (3.48) (cid:8) n (2 a ) j (cid:9) ≤ a ≤ r − ≤ j ≤ m . n (2 a ) i n (2 a ) j mod k (cid:0) i = j ; 0 ≤ a ≤ r − (cid:1) (3.51)2 . n (2 a ) i n ≡ − X ≤ j ≤ m ≤ b ≤ r − n (2 b ) j + (cid:16) k (cid:17) − k k (cid:0) ≤ i ≤ m ; 0 ≤ a ≤ r − (cid:1) . (3.52)3 . (cid:8) n (2 a ) j (cid:9) ≤ j ≤ m = (cid:8) n (2( r − a )) j (cid:9) ≤ j ≤ m (cid:0) ≤ a ≤ r − (cid:1) . (3.53)Some notes are in order: • In order to satisfy the monodromy free condition Eq. (3.41), one should show thatthe symmetric Stokes matrix S (sym)0 is diagonalizable. With the explicit expres-sion of the discrete solution Eq. (3.50), one can explicitly construct rm (= k − X ( a ; n (2 a ) j ) [72], S (sym)0 Γ − X ( a ; n (2 a ) j ) = ω n (2 a ) j X ( a ; n (2 a ) j ) (cid:0) ≤ j ≤ m ; 0 ≤ a ≤ r − (cid:1) . (3.54)A brief summary of the procedure is following: Each T-system of (cid:8) T n ( u ; a ) (cid:9) mn =1 (for a fixed a = 0 , , · · · , r −
1) admits m different vectors Y ( u ; a ): Y ( u ; a ) = Y ( u ; a ; n (2 a ) j ) ≡ ω − ( u +1) n (2 a ) j ω − ( u +2) n (2 a ) j ... ω − ( u + m ) n (2 a ) j (cid:0) ≤ j ≤ m (cid:1) , (3.55)which satisfy Φ( u ; a ) Y ( u ; a ; n (2 a ) j ) = ω n (2 a ) j Y ( u ; a ; n (2 a ) j ) . (3.56)By using the recursion equation Eq. (3.5), one can find the explicit expression ofeach eigenvector X ( a ; n (2 a ) j ) in terms of Y ( u ; a ; n (2 a ) j ): X ( a ; n (2 a ) j ) = M ( u ; a ) Y ( u ; a ; n (2 a ) j ) , (3.57)with a k × m matrix M ( u ; a ) which satisfies S (sym)0 Γ − M ( u ; a ) = M ( u − a )Φ( u ; a ) , (3.58)then one obtains the explicit expression for rm vectors (cid:8) X ( a ; n (2 a ) j ) (cid:9) ≤ a ≤ r − ≤ j ≤ m . Sinceeach multi-cut 2 a boundary condition is distinct to each other, the vectors are alsodistinct if and only if the condition Eq. (3.51) is satisfied. • Although the rm (= k −
1) distinct eigenvectors are explicitly constructed in theabove procedure, there remains one missing eigenvector. Noting that det (cid:2) S (sym)0 Γ − (cid:3) =27 − k − , one can conclude that the eigenvalue associated with this vector is givenby ω n of n ≡ − X ≤ j ≤ m ≤ b ≤ r − n (2 b ) j + (cid:16) k (cid:17) − k k, (3.59)and the eigenvector is given as X ( n ) = r − X a =0 c a M ( u ; a ) Y ( u ; a ; n ) . (3.60)It is clear and the constraint Eq. (3.52) is sufficient. The necessity is non-trivialbut we explicitly check it by Mathematica for several cases. • These Stokes multipliers (cid:8) τ n ( u ; a ) (cid:9) ≤ a ≤ r − ≤ n ≤ k − should be subjected to the hermiticitycondition Eq. (3.22). This constraint results in the constraint Eq. (3.53). k = rm + l , (1 < l < r )In this case, we have four kinds of recursive equations (I, II, III and IV in Table 1). wealso perform the same procedure shown above. Here we only show the identification:I : σ (2 s ) m,b σ (2 s )0 ,b = − τ (2 s ) m , σ (2 s ) m − n,b σ (2 s )0 ,b = τ (2 s ) n τ (2 s ) m (cid:0) n = 1 , , · · · , m −
1; 1 ≤ b < l (cid:1) , II : σ (2 s ) m,l = − τ (2 s ) m , σ (2 s ) m − n,l = τ (2 s ) n τ (2 s ) m (cid:0) n = 1 , , · · · , m − (cid:1) , III : σ (2 s ) n,b = 0 (cid:0) < b < r ; n = 1 , , · · · , m (cid:1) , IV : σ (2 s ) m − n +1 ,r = τ (2 s ) n , (cid:0) n = 1 , , · · · , m (cid:1) , (3.61)then we have a single recursion equation Eq. (3.24) which again results in r different T-systems. Therefore they are given by the discrete solution expression Eq. (3.50). However,the conditions for the indices are not straightforward in this case, and therefore, we showspecial solutions of simple cases. The case of ( k, r ) = (5 , (i.e. ( m, l ) = (1 , ) We found two types of solutions. First we put σ (0)0 , : free , σ (2 a )0 , = ω η a (cid:0) a = 1 , (cid:1) , n ≡ n (0)1 , e n ≡ n (2)1 = n (4)1 . (3.62)Then the constraints are given as1. n e n mod k, η = n + e n, η = − n − e n mod k, n ≡ e n mod k, η + η ≡ − n. (3.63) The case of ( k, r ) = (14 , (i.e. ( m, l ) = (4 , )
28e found the following solutions: First we put σ (2 a ) n, = 0 (cid:0) n = 0 , , , · · · , m ; 0 ≤ a ≤ r − (cid:1) ,n j ≡ n (2 a ) j (cid:0) ≤ j ≤ m ; 0 ≤ a ≤ r − (cid:1) . (3.64)Then the constraints are given as m X j =1 n j ≡ k , n j = k. (3.65) k = r ( m + 1)In this case, we have two kinds of recursive equations (I and V in Table 1). we alsoperform the same procedure shown above. Here we only show the identification:I : σ (2 s ) n,b = 0 (cid:0) ≤ b < r ; n = 1 , , · · · , m + 1 (cid:1) , V : σ (2 s ) m − n +1 ,r = τ (2 s ) n , (cid:0) n = 1 , , · · · , m (cid:1) , τ (2 s ) m +1 = 1 , (3.66)then we have a single recursion equation Eq. (3.24) which again results in r differentT-systems (i.e. each T-system is U q ( A (1) m ) of q k = 1). To avoid confusion, we here showthe boundary conditions which are slightly different from the other cases: T − n,s ( u ) = T m +1+ n,s ( u ) = 0 (cid:0) n ∈ N , s ∈ Z (cid:1) , (3.67) T a,s + k − m − ( u ) = 0 (cid:0) < a < m + 1 , < s < m + 1 (cid:1) , (3.68) T a, ( u ) = T ,s ( u ) = T m +1 ,s ( u ) = 1 (0 ≤ a ≤ m + 1 , s ∈ Z ) , (3.69)Therefore they are given by the discrete solution expression Eq. (3.50). τ n ( u ; a ) = ( − n +1 Sym h n, (cid:8) ω n (2 a ) i (cid:9) m +1 i =1 i (cid:0) a = 0 , , · · · , r − (cid:1) , (3.70)with the following conditions on the indices:1 . n (2 a ) i n (2 a ) j mod k (cid:0) i = j ; 0 ≤ a ≤ r − (cid:1) . (3.71)2 . ≡ m +1 X j =1 n (2 a ) j + r m ( m + 1)2 mod k (cid:0) ≤ a ≤ r − (cid:1) . (3.72)3 . (cid:8) n (2 a ) j (cid:9) ≤ j ≤ m = (cid:8) n (2( r − a )) j (cid:9) ≤ j ≤ m (cid:0) ≤ a ≤ r − (cid:1) . (3.73)Note that the fractional number in Eq. (3.72) is always an integer. In this paper, we analyzed Stokes phenomena in the k × k isomonodromy systems withan arbitrary Poincar´e index r , especially which correspond to the fractional-superstring(or parafermionic-string) multi-critical points (ˆ p, ˆ q ) = (1 , r −
1) in the multi-cut two-matrix models. Throughout the analysis, the multi-cut boundary conditions proposed29n [72] turn out to be a key concept in order to uncover the underling Hirota equa-tions (T-systems) and also a powerful tool in order to show explicit expressions of theStokes multipliers in quite general class of ( k, r ). Although we here focused on fractional-superstring critical points of the multi-cut two-matrix models (i.e. γ = r − γ ) of the multi-cut two-matrix models.It should be noted that the basic observables in this paper, Stokes multipliers, areintegration constants of string equations (i.e. non-perturbative ambiguities) and are in-formation completely out of the perturbative framework of string theory. Therefore, itis quite striking to show existence of a rigid mathematical structure like quantum inte-grability which governs these non-perturbative ambiguities of perturbative string theory.This fact also makes it evidential that there is a universal strong-string-coupling dualtheory which tightly connects various non-critical perturbative string theories in non-perturbative regime.At the first sight, the multi-cut boundary condition still seems like artificial constraintswhich specialize the solutions (in string equation) originated from the matrix models. Inparticular, the physical origin of the complementary boundary conditions is still unclear.In the future study, we should clarify this point and uncover the physical origin of theseconstraints. There is, however, an interesting possibility to solve this problem. It is abouta continuum description of ( l, k )-FZZT branes. From the CFT approach, the boundarystates are described by a superposition of (1 , (cid:12)(cid:12)(cid:12) τ ; ( l, k ) E FZZT = X m,n (cid:12)(cid:12)(cid:12) τ + (cid:0) mp + nq (cid:1) iπ ; (1 , E FZZT , ζ = √ µ cosh( pτ ) . (4.1)The description in the matrix models was also found and studied in [97]. From this pointof views, the general interpretation is that the (1 , it is true that all-order perturbative amplitudes of ( l, k ) -FZZT branes areequivalent to those of the (1 , -FZZT branes with the relation (4.1), however they aredescribed as non-perturbatively different objects in string theory . That is, the aboverelation is accidental in perturbation theory and these ( l, k )-FZZT branes have differentrealizations in the matrix models. If this consideration is true, then this would mean that the non-perturbative completion of string theory requires non-perturbative realization ofall the D-brane spectrum in the theory .Another approach to the physical meaning would be theoretical consistency of non-critical M theory, i.e. the requirement that non-critical M theory can be consistently re-constructed. It is because we expect that the non-perturbative ambiguities should have After publication of this paper, we were informed that it was already observed that the Seiberg-Shihrelation (4.1) does not generally hold beyond the disk-amplitude level (i.e. even at the perturbativelevel) [100, 101]. We would like to thank Max R. Atkin for drawing our attention to the papers. on-critical M theory ( r > N = 2 QFTs ( r = 1)perturbative string-theory sectors local vacua ∂ x W ( x ) = 0 ZZ-branes connecting string theories solitons connecting local vacuaStokes multipliers (D-instanton fugacity) Stokes multipliers (the number of solitons)
Table 2:
The dictionary of isomonodromy terminology between non-critical M theory and N = 2 QFTs.Local vacua in N = 2 QFTs are interpreted as perturbative string-theory sectors in non-critical M theory(of weak coupling limit). Solitons among the vacua are understood as ZZ branes among the perturbativestring-theory sectors. Interestingly, the Stokes multipliers in N = 2 QFTs are interpreted as physicalobservables. Therefore, this suggests that the D-instanton chemical potentials in string theory shouldbe naturally calculable as physical observables in non-critical M theory. definite physical meaning in the strong-coupling dual theory, although non-perturbativeambiguities cannot be fixed in the perturbative framework. An interesting clue to thisapproach is N = 2 QFT of Cecotti-Vafa [98]. Interestingly, there appear the k × k isomonodromy systems of Poincar´e index r = 1 and therefore one can find a translationinto non-critical string/M theory (shown in Table 2). As one can see, non-critical Mtheory is a simple generalization of N = 2 systems to the general Poincar´e index r . Thecritical points of r = 1 (i.e. ˆ q = 0) in the multi-cut two-matrix models are lower criticalpoints than the first critical points ( r = 2) which is the “pure-gravity” counterpart inthe multi-cut two-matrix models. This situation is an analogy of the relation betweentopological series ( p, q ) = (1 , q ) in minimal string theories and the other critical pointswhich start with the pure-gravity ( p, q ) = (2 ,
3) points (i,e. p ≥ r ≥
2) non-critical M theory is to find the multi-criticalgeneralization of N = 2 QFTs. Also in this reconstruction, it is interesting to see how thequantum integrability of the Stokes multipliers play its rule. It is also interesting if thereis a connection between quantum integrability of the multi-cut boundary conditions andthe one of Nekrasov-Shatashvili [99].The appearance of the quantum integrability in the Stokes multipliers of the generalisomonodromy systems also open interesting directions. In particular, since the spectralparameter in non-critical string theory is understood as the spacetime coordinate [25,36],it is interesting if the new spectral parameters from the T-systems are understood as thethird dimension of non-critical M theory. Acknowledgment
The authors would like to thank Jean-Emile Bourgine, Kazuyuki Furuuchi, Martin Guest,Chang-Shou Lin, Toshio Nakatsu, Ricardo Schiappa, Ryo Suzuki, Dan Tomino and Ken-taroh Yoshida for useful discussions and comments. H. Irie would like to thank organizersof Strings 2011 for giving him the opportunity of presenting basic results of this paperin the Gong Show session in the conference. H. Irie would like to thank people in ISTfor their hospitality during his visit while completing this work. C.-T. Chan and H. Irieare supported in part by National Science Council of Taiwan under the contract No. 99-2112-M-029-001-MY3 (C.-T. Chan) and No. 100-2119-M-007-001 (H. Irie). The authors Generally speaking, we require that string theory should be completed by its dual theories in everytheoretical limit of the backgrounds.
A Sketch of a proof for Theorem 1
Here we show a brief sketch of a proof for Theorem 1. The procedure is the following:1. Introduce a division the vectors (Eq. (3.9) for r = 3) and the equation Eq. (3.5) of n = 0 according to modulo r .2. Write down all the equation Eq. (3.5) of n = 0 according to the modulo- r division.3. Obtain a closed set of equations, by using proper shift rules.4. Solve them and obtain equations with non-zero x (2 s ) n . With identifying the sequence (cid:8) y (2 s ) n (cid:9) with non-zero x (2 s ) n , these equations result in Eq. (3.18). The shift rules ofthe equation then generate all the equations of Eq. (3.5).Here we demonstrate this procedure in the case of ( k, r ) = (6 e m + 1 , Step 1
Here we introduce a modulo- r division of the vectors Eq. (3.9) and Eq. (3.5)for n = 0. The division is given in the following way: X (6) = (cid:16) △ , × , · · · , × , (cid:0) e m + 1 (cid:1) △ S (sym)0 : ↓ ↓ · · · ↓ X (0) = (cid:16) △ , × , · · · , × , (cid:0) e m + 1 (cid:1) △ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : There are only non-zero components, i △ . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × , × , (cid:0) e m + 4 (cid:1) △ − , · · · , − k − , k − , k , , (cid:17) ↓ ↓ ↓ ↓ · · · ↓ ↓ ↓ ↓ ↓ ↓ ↓− , − , − − , · · · , − k − , k − , k △ , × , × (cid:17)| {z } The number of the components is a multiple of r . (A.1)Here the components (1 , ,
3) are moved to the next to the component “ k ”. The meaningof the arrow is iS (sym)0 : ↓ i ⇔ x (0) i = x (6) i + X j s (sym)0 ,i,j △ x (6) j △ . (A.2)Here “ △ ” of x (6) j △ means that the x (6) j △ = 0. One can always prove that the contributionswith Stokes multipliers are always accompanied with the non-zero component x (6 n ) j △ . Thenwe first divide them into a part which consists only of non-zero components i △ and theother part, in which the number of components is a multiple of r . We introduce modulo- r division in the latter part so that the right-hand side of k should be boundary of thedivision, “ · · · , k | , , tep 2 We write all the equation Eq. (3.5) for n = 0:I . " △ , × , · · · , × , (cid:0) e m + 1 (cid:1) △ △ , × , · · · , × , (cid:0) e m + 1 (cid:1) △ ↔ n x (0)(3 n +1) △ = x (6)(3 n +1) △ , (cid:0) n = 1 , , · · · , e m (cid:1) , II . (cid:20) , , △ , × , × (cid:21) ↔ x (6)3 + θ (0)1 x (0)4 △ x (6)2 + θ (0)2 x (0)4 △ x (0)1 △ = x (6)1 + θ (0)3 x (0)4 △ , III . (cid:20) k − n + 1 , k − n + 2 , k − n + 3 k − n + 1 , k − n + 2 , k − n + 3 (cid:21) (cid:0) n = 1 , , · · · , e m − (cid:1) ↔ (cid:26) x (0) k − n + a = x (6) k − n + a + θ (cid:0) n − (cid:1) n +1 − a x (0)(3 n +1) △ + θ (6 n )6 n +4 − a x (0)(3 n +4) △ , (cid:0) a = 1 , , (cid:1) , IV . (cid:20) × , × , (cid:0) e m + 4 (cid:1) △ e m + 2 , e m + 3 , e m + 4 (cid:21) = (cid:20) × , × , (cid:0) k − e m + 3 (cid:1) △ k − e m + 1 k − e m + 2 , k − e m + 3 (cid:21) ↔ x (0)3 e m +4 = x (6)(3 e m +4) △ + θ (cid:0) e m − (cid:1) e m − x (0)(3 e m +1) △ x (0)3 e m +3 = θ (cid:0) e m − (cid:1) e m − x (0)(3 e m +1) △ x (0)3 e m +2 = θ (cid:0) e m − (cid:1) e m x (0)(3 e m +1) △ . (A.3) Step 3
We make a shift rule to obtain equations among (cid:8) x (6 n )1 , x (6 n )2 , x (6 n )3 (cid:9) n ∈ Z .III . (cid:26) x (6 n ) k + a = x (6( n +1)) k + a + θ (cid:0) n − (cid:1) n +1 − a x (6 n )(1+6 n ) △ + θ (cid:0) n ) (cid:1) n +4 − a x (6 n )(4+6 n ) △ , (cid:0) a = 1 , , (cid:1) IV . x (6 e m )3 = x (6( e m +1))3 △ + θ (cid:0) e m − (cid:1) e m − x (6 e m )(6 e m +1) △ x (6 e m )2 = θ (cid:0) e m − (cid:1) e m − x (6 e m )(6 e m +1) △ x (6 e m )1 = θ (cid:0) e m − (cid:1) e m x (6 e m )(6 e m +1) △ . (A.4) Step 4
By solving these equations, we obtain equations for non-zero components of (cid:8) x (6 n ) i (cid:9) i,n ∈ Z : x (6( e m +1))3 △ + e m X n =1 θ (cid:0) n − (cid:1) n − x (6 n )(1+6 n ) △ + e m − X n =0 θ (cid:0) n ) (cid:1) n +1 x (6 n )(4+6 n ) △ , e m X n =1 θ (cid:0) n − (cid:1) n − x (6 n )(6 n +1) △ + e m − X n =0 θ (cid:0) n ) (cid:1) n +2 x (6 n )(6 n +4) △ ,x (0)1 △ = e m X n =1 θ (cid:0) n − (cid:1) n x (6 n )(1+6 n ) △ + e m − X n =0 θ (cid:0) n ) (cid:1) n +3 x (6 n )(4+6 n ) △ . (A.5)By introducing the following sequence (cid:8) y (0) n (cid:9) n ∈ Z , y (0) n ≡ x (6 n )(3 n +1) △ (cid:16) = x (6( n − n +1) △ = · · · = x (6( n − e m ))(3 n +1) △ (cid:17) , (A.6)33e obtain the equations of the form: F h(cid:8) y (0) n (cid:9) n ∈ Z i = y (0)2 e m +1 + e m X n =1 θ (6( n − n − y (0) n = 0 , F h(cid:8) y (0) n (cid:9) n ∈ Z i = e m X n =1 θ (6( n − n − y (0) n = 0 , F h(cid:8) y (0) n (cid:9) n ∈ Z i = − y (0)0 + e m X n =1 θ (6( n − n − y (0) n = 0 . (A.7)By taking into account Eq. (3.17), we obtain Eq. (3.18) with s = 0. Therefore, all theother equations in Eq. (3.18) are obtained by the 2 s shift rule. Note that y (2 s ) n is definedas y (2 s ) n = x (6 n +2 s )(3 n + s +1) △ . (A.8)In this sense, we use the notation of k = rm + l (1 ≤ l ≤ r ) in the following discussions.Finally we mention the extension of this procedure to all the ( k, r ) cases. We note is thatwe have used a specialty of r = 3, i.e. m = 1 (See Eq. (3.1)). Because of this specialty,the division procedure Eq. (A.1) is simpler and the proof is not so complicated. In thegeneral cases, however, we can re-organize the components of the vector X (0) as X (0) → ( x (0)1 , x (0)1+ rm , x (0)1+2 rm , · · · ) . (A.9)In this way, we can perform the same procedure in general r and can show Eq. (3.18) forgeneral ( k, r ). References [1] A. M. Polyakov, “Quantum geometry of bosonic strings,” Phys. Lett. B (1981)207; “Quantum geometry of fermionic strings,” Phys. Lett. B (1981) 211.[2] E. Brezin and V. A. Kazakov, “Exactly solvable field theories of closed strings,”Phys. Lett. B (1990) 144;M. R. Douglas and S. H. Shenker, Nucl. Phys. B (1990) 635;D. J. Gross and A. A. Migdal, “Nonperturbative Two-Dimensional Quantum Grav-ity,” Phys. Rev. Lett. (1990) 127.[3] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun.Math. Phys. (1978) 35.[4] V. A. Kazakov, “The Appearance of Matter Fields from Quantum Fluctuations of2D Gravity,” Mod. Phys. Lett. A (1989) 2125.[5] I. K. Kostov, “Strings embedded in Dynkin diagrams,” Cargese 1990, Proceedings,Random surfaces and quantum gravity, pp.135-149. See [72] for r = 2, which still looks complicated. (1991) 317.[7] E. Brezin, E. Marinari and G. Parisi, “A Nonperturbative Ambiguity Free SolutionOf A String Model,” Phys. Lett. B (1990) 35.[8] I. K. Kostov, “Strings with discrete target space,” Nucl. Phys. B (1992) 539[arXiv:hep-th/9112059].[9] T. Banks, M. R. Douglas, N. Seiberg and S. H. Shenker, “Microscopic and macro-scopic loops in nonperturbative two-dimensional gravity,” Phys. Lett. B (1990)279.[10] E. Brezin, M. R. Douglas, V. Kazakov and S. H. Shenker, “The Ising model coupledto 2-d Gravity: A nonperturbative analysis,” Phys. Lett. B (1990) 43;D. J. Gross and A. A. Migdal, “Nonperturbative Solution of the Ising Model on aRandom Surface,” Phys. Rev. Lett. (1990) 717.[11] D. J. Gross and A. A. Migdal, “A nonperturbative treatment of two-dimensionalquantum gravity,” Nucl. Phys. B (1990) 333.[12] M. R. Douglas, “Strings in less than one-dimension and the generalized KdV hier-archies,” Phys. Lett. B (1990) 176.[13] T. Tada and M. Yamaguchi, “ P and Q operator analysis for two matrix model,”Phys. Lett. B (1990) 38;M. R. Douglas, “The Two matrix model,” In *Cargese 1990, Proceedings, Randomsurfaces and quantum gravity* 77-83. (see HIGH ENERGY PHYSICS INDEX 30(1992) No. 17911) ;T. Tada, “(
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