January 1999 hep-th/9902001 Algebraic Structure in 0 < c 1 9 ≤ 9 Open-Closed String Field Theories 9 1 n a J Daiji Ennyu1 0 3 Hiroshima National College of Maritime Technology 1 Toyota-Gun, Hiroshima 725-0200, Japan v 1 Hiroshi Kawabe2 0 0 Yonago National College of Technology 2 0 Yonago 683-8502, Japan 9 9 / and h t Naohito Nakazawa3 - p e Department of Physics, Faculty of Science and Engineering, Shimane University h : v Matsue 690-8504, Japan i X r a Abstract We apply stochastic quantization method to Kostov’s matrix-vector models for the second quantization of orientable strings with Chan-Paton like factors, including both open and closed strings. The Fokker-Planck hamiltonian deduces an orientable open- closed string field theory atthe doublescaling limit. There appears analgebraicstructure in the continuum F-P hamiltonian including a Virasoro algebra and a SU(r) current algebra. 1e-mail address: ennyu@hiroshima-cmt.ac.jp 2e-mail address: kawabe@yonago-k.ac.jp 3e-mail address: nakazawa@ifse1.riko.shimane-u.ac.jp The explicit construction of non-critical string field theories via the double scaling limit of the matrix models [1] provides not only the basis for the non-perturbative analysis of string theories but also the clear understanding of the origin of the constraints realized in the algebraic structure of the string field theoretic hamiltonian [1-10]. The algebraic structure in non-critical string field theories has been investigated in this context for orientable closed strings [2, 3, 4], non-orientable closed strings [6], orientable open- closed strings [7, 8]and non-orientable open-closed strings[10]. For orientable 2D surfaces with boundaries, an interesting algebraic structure, sl(r,C) sl(r,C) chiral current al- × gebra including SU(r) current algebra, has been found in a field theoretic hamiltonian of orientable open-closed str ings[9]. The observation indicates the relation between the algebraic structure and a Chan-Paton like realization of gauge groups in open string theories. While the SO(r) current algebra has been found in the string field theoretic hamiltonian for non-orientable open-closed strings[10]. In this note, we construct non-critical orientable open-closed string field theories for 0 < c 1 byapplying stochastic quantization methodto Kostov’s matrix-vector model[7, ≤ 11]. To introduce Chan-Paton like factors, we slightly modify Kostov’s model. The continuumlimitofthestringfieldtheoryistakenatthedoublescalinglimitofthismatrix- vector model. We introduce the scale parameters on the boundaries which characterize thescaling behaviour of thetimeevolution alongtheboundaries. Weshow thatthe string field theoretic hamiltonian, which is the continuum limit of the loop space Fokker-Planck hamiltonian in stochastic quantization[3, 6], appears in the form of a linear combination of three deformation generators of orientabl e strings which satisfy algebraic relations including a Virasoro algebra and a SU(r) current algebra. For orientable 2D surfaces with boundaries, the sum of triangulated surfaces is re- constructed by the hermitian matrix-vector models[11, 12]. To introduce the orientable open-closed string interactions for 0 < c 1, we consider Kostov’s matrix-vector model ≤ with Chan-Paton like factors. The action is given by[7] 1 1 g S1 = Tr( Axx′Ax′x Axx′Ax′x′′Ax′′x) , 2 xX,x′ − 3√N x,Xx′,x′′ 1 r ga S2 = Xx,x′aX=1Vxa∗(δxx′ − √BNAxx′)Vxa′ , (1) where theinteractions ofmatrices, Axx′, take placeonly for x x′ 1inthetarget space | − | ≤ Z, i.e. x,x Z. Namely, the relevant matrices are A , which are N N hermitian ′ xx ij ∈ × matrices, and Axx+1 ij = A†x+1x ij. Vxai and Vxa∗i are N-dimensional complex vectors with a Chan-Paton like factor a which runs from 1 to r. The partition function, Z, and the free energy, F, of this model are given by Z dAdVdV e S1 S2 , ∗ − − ≡ Z F lnZ ≡ r = (ga)LaBgSNχ , (2) B Y a=1 where S and La are the total area and the total length of the boundary with the B “colour”index a of the 2D surface, respectively. χ is the Eular number of the triangulated 2D surface with boundaries, χ 2 2#(handles) #(boundaries). By integrating out ≡ − − all the non-hermitian matrices, Axx+1 ij and A†xx+1 ji, we obtain an effective action as a function of the hermitian matrix M A √Nδ , and the vectors, Va and Va . x ij ≡ xx ij − 2g ij x i x∗i We denote the effective action as S (M,V,V ) [7]. eff ∗ We define the time evolution, M (τ + ∆τ) M (τ) +∆M (τ) and Va (τ + x ij ≡ x ij x ij x i ∆τ) Va (τ) + ∆Va (τ) , in terms of Ito’s calculus[13] with respect to the stochastic ≡ x i x i time τ which is discretized with the unit time step ∆τ. The time evolution is given by the following Langevin equations, ∂ ∆M (τ) = S (M,V,V )(τ)∆τ +∆ξ (τ) , x ij eff ∗ x ij −∂M x ji ∂ ∆Va (τ) = λa S (M,V,V )(τ)∆τ +∆ηa (τ) , x i − x∂Va eff ∗ x i x∗i ∂ ∆Va (τ) = λa S (M,V,V )(τ)∆τ +∆ηa (τ) . (3) x∗i − x∂Va eff ∗ x∗i x i We have introduced the scale parameter λa in the Langevin equation of vector variables, x Va and Va . At the double scaling limit, the parameter λa defines the time scale for x i x∗i x the stochastic time evolution of the open string end-point along the boundaries with the “colour”index a. 2 The correlations of the white noise ∆ξ and ∆ηa are defined by x ij x i < ∆ξx ij(τ)∆ξx′ kl(τ) >ξ = 2∆τδilδjkδxx′ , < ∆ηxa∗i(τ)∆ηxb′ j(τ) >η = 2λax∆τδabδijδxx′ . (4) We will see later that the λa-independence of the equilibrium is realized as the alge- x braic relation between the constraints in the string field theoretic hamiltonian which is equivalent to the integrability condition of the multi-time evoluti on of the Langevin equations. Now we define the closed strings, φ (L) = N 1tr(eLN−1/2Mx) , and the open strings, x − ψab(L) = N 1(Va eLN−1/2MxVb) . Following to Ito’s calculus, we calculate the time de- x − ∗ velopment of these string variables[6], L ∆φx(L) = ∆τL{Z0 dL′φx(L′)φx(L−L′)+Xx′ Z0∞dL′Cx(cx)′φx(L+L′)φx′(L′) 1 + gagb ∞dLLC(o)ψab(L+L)ψba(L) Ng B BZ ′ ′ xx′ x ′ x′ ′ aX,b,x′ 0 1 r 1 ∂2 + gaψaa(L) φ (L)+g φ (L) +∆ζ (L) , N X B x − 4g x ∂L2 x } x a=1 ga ∂ 1 ∆ψab(L) = λa∆τ ( 1+ B +ga )ψab(L)+ gagc ∞dLC(o)ψcb(L+L)ψac(L) x x { − 2g B∂L x g Xc,x′ B BZ0 ′ xx′ x ′ x′ ′ } gb ∂ 1 + λb∆τ ( 1+ B +gb )ψab(L)+ gb gc ∞dLC(o)ψac(L+L)ψcb(L) x { − 2g B∂L x g Xc,x′ B BZ0 ′ xx′ x ′ x′ ′ } + 2∆τλaδabφ (L) x x L ∂2 + ∆τ{− 4gψxab(L)+Lg∂L2ψxab(L)+LXx′ Z0∞dL′Cx(cx)′ψxab(L+L′)φx′(L′) L + gc dLψac(L)ψcb(L L) Xc BZ0 ′ x ′ x − ′ gc gd L + B B dL ∞dL ∞dL C(o)ψad(L +L )ψcb(L L +L )ψdc(L +L ) cX,d,x′ g Z0 ′Z0 ′′Z0 ′′′ xx′ x ′ ′′ x − ′ ′′′ x′ ′′ ′′′ L + 2 dLLψab(L)φ (L L) +∆ζab(L) . (5) Z ′ ′ x ′ x − ′ } x 0 (c) (o) Here we have introduced the adjacency matrices, Cxx′ = Cxx′ = δxx′+1 + δxx′ 1. In (5), − the new noise variables are defined by ∆ζ (L) LN 3/2tr(eLN−1/2Mx∆ξ ) , x − x ≡ 3 L ∆ζab(L) N 3/2 dLVa eL′N−1/2Mx∆ξ e(L L′)N−1/2MxVb x ≡ − Z ′ x∗ x − x 0 + N 1 Va eLN−1/2Mx∆ηb +∆ηa eLN−1/2MxVb . (6) − { x∗ x x∗ x} We notice that < ∆ζ (L) > =< ∆ζab(L) > = 0 hold in the sense of Ito’s calculus. x ξη x ξη Especially, from eqs. (5), the λ-independence of the equilibrium limit deduces a S-D equation which ensures the SU(r) current algebra. We can describe a class of matrix-vector models for the central charge 0 < c 1 with ≤ the following adjacency matrices[7]. (c) Cxx′ = cos(πp0)(δxx′+1 +δxx′ 1) , − (o) Cxx′ = cos(πp0/2)(δxx′+1 +δxx′ 1) . (7) − The case, p = 1/m, corresponds to the central charge, c = 1 6 . 0 − m(m+1) The stochastic process isinterpreted asthe timeevolution inastring field theory. The corresponding non-critical orientable open-closed string field theory is defined by the F- P hamiltonian operator. In terms of the expectation value of an observable O(φ,ψ), a function of φ (L)’s and ψab(L)’s, the F-P hamiltonian operator Hˆ is defined by[6] x x FP < φ(0),ψ(0) e−τHˆFPO(φˆ,ψˆ) 0 > < O(φξη(τ),ψξη(τ)) >ξη . (8) | | ≡ In R.H.S., φ (τ) and ψ (τ) denote the solutions of the Langevin equations (5) with the ξη ξη appropriate initial configuration φ (L;τ = 0) and ψab(L;τ = 0). In L.H.S., Hˆ is given x x FP by the differential operator in the well-known Fokker-Planck equation for the expectation value of the observable O(φ,ψ) by replacing the closed ( open ) string variable φ (L) ( x ψab(L) ) to the creation operator φˆ (L) ( ψâb(L) ) and the differential ∂ ( ∂ ) x x x ∂φx(L) ∂ψxab(L) to the annihilation operator πˆ (L) ( πâb(L) ), respectively. x x ˆ The continuum limit of H is taken by introducing a length scale “ǫ ” which defines FP the physical length of the strings as l Lǫ. At the double scaling limit ǫ 0, we keep ≡ → the string coupling constant, G N 2ǫ 2D , to be finite. The continuum stochastic time − − ≡ is given by dτ ǫ 2+D∆τ . While we define the scaling of the scale parameter λa by ≡ − x λa ǫ 1/2 D/2λa . We also redefine field variables as follows. ≡ − − x Φ (l) ǫ Dφˆ (L) , x − x ≡ 4 Π (l) ǫ 1+Dπˆ (L) , x − x ≡ Ψab(l) ǫ 1/2 D/2ψâb(L) , x ≡ − − x Πab(l) ǫ 1/2+D/2πâb(L) . (9) x ≡ − x The commutation relations are [Πx(l),Φx′(l′)] = δxx′δ(l − l′) and [Πaxb(l),Φcxd′(l′)] = δacδbdδxx′δ(l l′) . Then we obtain the continuum F-P hamiltonian, FP, from HˆFP, − H HFP = Xx Z0∞dll{Xx′ Cxx′Z0∞dl′Φx(l+l′)Φx′(l′)+√GaX,b,x′Cxx′Z0∞dl′l′Ψaxb(l+l′)Ψbxa′(l′) l + dl Φ (l)Φ (l l)+G ∞dllΦ (l+l )Π (l ) Π (l) Z ′ x ′ x − ′ Z ′ ′ x ′ x ′ } x 0 0 + aX,b,xZ0∞dl{lXx′ Cxx′Z0∞dl′Ψaxb(l+l′)Φx′(l′) l + cX,d,x′Cxx′Z0 dl1Z0∞dl2Z0∞dl3Ψaxd(l1 +l2)Ψcxb(l−l1 +l3)Ψdxc′(l2 +l3) l + 2 dll Ψab(l )Φ (l l) Z ′ ′ x ′ x − ′ 0 + λa Cxx′Z ∞dl′Ψcxb(l+l′)Ψaxc′(l′)+λb Cxx′Z ∞dl′Ψaxc(l+l′)Ψcxb′(l′) Xc,x′ 0 Xc,x′ 0 + (λa +λb)δabΦ (l) Πab(l) x } x l l′ + √G ∞dl ∞dl dl dl Ψad(l +l )Ψcb(l+l l l )Πab(l)Πcd(l) X Z0 Z0 ′Z0 1Z0 1′ x 1 1′ x ′ − 1 − 1′ x x ′ a,b,c,d,x + √G ∞dl ∞dl λaΨcb(l+l )Πab(l)Πca(l)+λbΨac(l+l )Πab(l)Πbc(l) X Z0 Z0 ′{ x ′ x x ′ x ′ x x ′ } a,b,c,x + 2G ∞dl ∞dlll Ψab(l+l)Π (l)Πab(l ) . (10) Z Z ′ ′ x ′ x x ′ X 0 0 a,b,x The continuum F-P hamiltonian (10) takes the form of a linear combination of three continuum generators of string deformation, = ∞dll (l)Π (l)+ ∞dl λa ab(l)+λb ba (l) Πab(l) HFP Xx Z0 Lx x aX,b,xZ0 { Jx Jx ∗ } x l + ∞dl l ab(l) dl l cb(l )Ψac(l l) aX,b,xZ0 n Kx −Xc Z0 ′ ′(cid:16)Jx ′ x − ′ + ca (l )Ψbc (l l ) Πab(l) . (11) Jx ∗ ′ x∗ − ′ (cid:17)o x The explicit forms of these generators are given by Lx(l) = Xx′ Cxx′Z0∞dl′Φx(l+l′)Φx′(l′) 5 l + √GaX,b,x′Cxx′Z0∞dl′l′Ψaxb(l+l′)Ψbxa′(l′)+Z0 dl′Φx(l′)Φx(l−l′) + G ∞dll Φ (l+l)Π (l )+G ∞dllΨab(l+l )Πab(l) , Z ′ ′ x ′ x ′ Z ′ ′ x ′ x ′ 0 X 0 a,b Kxab(l) = Xx′ Cxx′Z0∞dl′Ψaxb(l+l′)Φx′(l′) l+l1 + GZ0∞dl′l′Ψaxb(l+l′)Πx(l′)+cX,d,x′Cxx′Z0∞dl1Z0 dl2Ψaxc(l2)Ψdxb(l+l1 −l2)Ψcxd′(l1) l+l1 + √G ∞dl dl Ψac(l )Ψdb(l+l l )Πdc(l ) XZ0 1Z0 2 x 2 x 1 − 2 x 1 c,d l + 2 dl Ψab(l )Φ (l l) , Z ′ x ′ x − ′ 0 Jxab(l) = δabΦx(l)+Xc,x′Cxx′Z0∞dl′Ψcxb(l+l′)Ψaxc′(l′) + √G ∞dlΨcb(l+l )Πca(l) . (12) Z ′ x ′ x ′ Xc 0 These generators satisfy the following algebra including the Virasoro algebra and the SU(r) current algebra, [ x(l), x′(l′)] = (l l′)Gδxx′ x(l+l′) , L L − − L [Lx(l),Kxab′(l′)] = −(l−l′)Gδxx′Kxab(l+l′) l + Gδxx′Xc Z0 du(l−u){Jxcb(l+l′ −u)Ψaxc(u)+Jxca∗(l+l′ −u)Ψbxc∗(u)} , [Lx(l),Jxa′b(l′)] = l′Gδxx′Jxab(l+l′) , [Jxab(l),Kxcd′(l′)] = √Gδadδxx′Kxcb(l+l′) l − √Gδadδxx′Xe Z0 duJxec∗(l+l′ −u)Ψbxe∗(u) l − √Gδxx′Z duJxad(l+l′ −u)Ψcxb(u) , 0 [Jxab(l),Jxc′d(l′)] = √Gδadδxx′Jxcb(l+l′)−√Gδcbδxx′Jad(l+l′) , l′ l′ u [Kab(l),Kcd(l′)] = √Gδxx′Xe Z0 duZ0 − dv{Jxeb(l+l′ −u−v)Ψaxd(u)Ψcxe(v) + ea (l+l u v)Ψde (u)Ψcb(v) Jx ∗ ′ − − x ∗ x } l l u − √Gδxx′Xe Z0 duZ0− dv{Jxed(l+l′ −u−v)Ψcxb(u)Ψaxe(v) + ec (l+l u v)Ψad(u)Ψbe (v) . (13) Jx ∗ ′ − − x x∗ } 6 In the precise sense, these commutation relations are not an algebra because the open string creation operators Ψab(l) appear in the R.H.S. of (13). The algebraic structure (13) is understood as the consistency condition for the constraints on the equilibrium expectation value or “integrable condition”of the stochastic time evolution. The hermitian part of the current, ab(l) + ba(l), generates the S-D equation which implies ∗ J J the λ-independence of the equilibrium distribution. While the anti-hermitian part of the current, ab(l) ba(l), satisfies SU(r) current algebra. The similar algebraic structure ∗ J −J is also found in Ref.[9] in a discretized form without continuum limit. In conclusion, we have derived the continuum F-P hamiltonian which defines the non-critical orientable open-closed string field theory for 0 < c 1 by applying SQM to ≤ Kostov’s matrix-vector model. The origin of the algebraic structure (13) can be traced to the noise correlations which generate the time evolution in the Langevin equation. Namely, the noise correlations realize the deformation of open and closed strings which are equivalent to those generated by the three constraints, (l), ab(l) and ab(l). We L J K have shown that the structure to be universal representing the scale invariance in the non-critical orientable open-closed string theories. We hope that our approach is also usefull to the non-perturbative analysis of superstring theories such as type IIA, IIB and type I theories for which matrix models are proposed [14, 15] by the large N reduction of super Yang-Mills theory. N. N. would like to thank I. Kostov for valuable comments. This work was par- tially supported by the Minisitry of Education, Science and Culture, Grant-in-Aid for Exploratory Research, 09874063, 1998. References [1] E. Brézin and V. Kazakov, Phys. Lett. B236(1990) 144; M. Douglas and S. Shenker, Nucl. Phys. B335(1990) 635; D. Gross and A. Migdal, Phys. Rev. Lett. 64(1990) 127; Nucl. Phys. B340(1990) 333. 7 [2] N. Ishibashi and H. 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