Frobenius manifolds, Integrable Hierarchies and Minimal Liouville Gravity
aa r X i v : . [ h e p - t h ] O c t FIAN-TD-2014-10
Frobenius manifolds, Integrable Hierarchiesand Minimal Liouville Gravity
A. A. Belavin ∗ and V. A. Belavin † L. D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia P. N. Lebedev Physical Institute, 119991 Moscow, Russia Institute for Information Transmission Problems, 127994 Moscow, Russia Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
Abstract.
We use the connection between the Frobrenius manifold and the Dou-glas string equation to further investigate Minimal Liouville gravity. We search asolution of the Douglas string equation and simultaneously a proper transformationfrom the KdV to the Liouville frame which ensure the fulfilment of the conformaland fusion selection rules. We find that the desired solution of the string equationhas explicit and simple form in the flat coordinates on the Frobenious manifold inthe general case of (p,q) Minimal Liouville gravity. Introduction
The purpose of this paper is to further study Minimal Liouville Gravity (MLG)[1] using an approach based on the Douglas string equation [2]. This study is acontinuation of earlier works [3–6].The Liouville Gravity represents a consistent example of the noncritical Stringtheory. In the initial continuous approach the Liouville Gravity is formulated asa BRST invariant theory composed of the matter sector, the Liouville theory andthe ghosts system. MLG represents the theory, where the matter sector is taken tobe a ( p, q ) Minimal Model of CFT [7]. The main problem of MLG is to calculatecorrelation functions of BRST invariant observables, which are given by integralsover moduli of Riemannian surfaces. Usually they are called the correlation numbers.Numerous examples show that the solution of the problem is quite nontrivial withinthe framework of the continuous approach.An alternative approach to MLG has grown up from the idea of triangulationsof two-dimensional surfaces realized in terms of Matrix Models [8–14]. One of themost important points of the approach is the String equation which was derived by ∗ [email protected] † [email protected] ouglas [2] in Matrix Models approach to two dimensional gravity. The subjectof the String equation is the generating function of the correlation numbers whichdepends on the parameters of the problem (the so called KdV times). In our work,following [4–6], we will conjecture that the Douglas equation is applicable to theMinimal Liouville gravity as well as to Matrix Models of 2D gravity.This conjecture requires the following two questions to be answered: how to choosethe desired solution of the Douglas string equation and an appropriate form of theso called resonance transformation [3] from the KdV times to the Liouville couplingconstants. Once these two questions are answered, the generating function of thecorrelation functions in MLG is given explicitly as an integrated one-form defineduniquely for each ( p, q ) MLG model and coincides with a special choice of the tau-function of the dispersionless limit [15, 16] of the generalized KdV hierarchy.In this paper, using the connection [5] of the approach to MLG [4] based on theString equation with the Frobenius manifold structure, we find the necessary solutionof the String equation. We also show that this solution together with the suitablechoosen resonance transformation lead to the results which are consistent with themain requirements of ( p, q ) models of MLG (the so called selection rules). It isremarkable that the needed solution of the Douglas equation has a very simple formin the flat coordinates on the Frobenious manifold in the general case of (p,q) MinimalLiouville as well as it has been found recently in the case of Unitary models of MLG [6].The paper is organized as follows. In Section we recall briefly the notion ofthe Frobenius manifold and discuss its basic properties. In Section we discuss theFrobenius manifold that appears in the context of Minimal Liouville Gravity. Section is devoted to the connection between the Frobenius manifold structures, Integrablestructures and the Douglas string equation. In Section we focus on the ( p, q ) modelsof MLG and discuss the problem of the resonance transformations. The idea of theapproach based on the String equation to ( p, q ) MLG is formulated in Section . Theappropriate solution of the Douglas string equation is discussed in Section . Therest of the paper is devoted to the analysis of the correlation functions. We show thatthe special choice of the solution of the String equation together with the resonancetransformation encoded in terms of Jacobi polynomials ensure fulfilling the necessaryselection rules for the correlation numbers in ( p, q ) MLG.2. Frobenius manifolds
In this and two next sections we give the definition and a short review of mainproperties of the Frobenius manifolds needed for our purposes. Here we follow thepaper by B.Dubrovin [16], see also [5].By definition a commutative associative algebra A with unity equipped with anondegenerate invariant bilinear form ( , ) is called Frobenius algebra. The invarianceof the bilinear form means that for any three vectors a, b, c in A :( a · b, c ) = ( a, b · c ) . (2.1) et M be n -dimensional manifold with a flat metric η αβ dv α dv β which is constantin the flat coordinates v α .We introduce in the tangent space T v M the structure of the Frobenius algebra bythe following identification of the bases ∂∂v α → e α , (2.2)Thus, we can multiply tangent vectors at any point of Me α e β = C γαβ e γ . (2.3)The structure constants C γαβ may depend on v α . Such manifold M can be calledquasi-Frobenius manifold. Definition 2.1.
The manifold M is called Frobenius manifold if these two structuresare adjusted with each other in such a way that(1) the invariant bilinear form ( ∂∂v α , ∂∂v β ) is identical to the metric η αβ ;(2) the structure of the Frobenius algebra at each point of M and the metric on M are constrainted by the following relation ∇ ρ C αβγ = ∇ α C ρβγ . (2.4)The last requirement is equivalent to the requirement that there exists a function F on M which is connected with the structure constants of the Frobenius algebra as C αβγ = ∂ F∂v α ∂v β ∂v γ , (2.5)where C αβγ = η αρ C ρβγ . (2.6)Function F is called Frobenius potential. The consistency of this property with theassociativity of the Frobenius algebra is known as WDVV condition [17] ∂ F∂v α ∂v β ∂v ρ η ρλ ∂ F∂v λ ∂v µ ∂v ν = ∂ F∂v ν ∂v β ∂v ρ η ρλ ∂ F∂v λ ∂v µ ∂v α . (2.7)The following statement [16] follows from these properties of the Frobenius manifold M . There exist an one-parametric flat deformation e ∇ α of the connection ∇ α e ∇ α x γ = ∇ α x γ − zC γαβ x β , (2.8)or, equivalently, [ e ∇ α ( z ) , e ∇ β ( z )] = 0 . (2.9)The proof is based on the associativity of the Frobenius algebra and the equation(2.4). As a consequence of (2.9), there exist n linear independent solutions θ α ( v, z ) = ∞ X k =0 θ αk ( v ) z k , (2.10) f the equation e ∇ α dθ λ ( v, z ) = 0, which is equivalent to ∂ θ λ ∂v α ∂v β ( v, z ) = zC γαβ ∂θ λ ∂v γ ( v, z ) , (2.11)or ∂ θ λk +1 ∂v α ∂v β ( v ) = C γαβ ∂θ λk ∂v γ ( v ) . (2.12)The functions θ α ( v, z ) can be considered as the flat coordinates of the deformedconnection e ∇ α ( z ). We choose θ λ ( v, z ) so that θ λ ( v,
0) = θ λ ( v ) = v λ . From (2.12) itfollows, that ∇ ( ∇ θ α ( v, z ) , ∇ θ β ( v, z )) = ( z + z ) ∇ θ α ( v, z ) · ∇ θ β ( v, z ) , (2.13)and, hence, the scalar product ( ∇ θ α ( v, z ) , ∇ θ β ( v, − z )) = Const ( z ) does not dependon v α . For z = 0 we find Const (0) = η αβ . Equation (2.12) is invariant with respectto the transformation θ µ ( v, z ) → A µν ( z ) θ ν ( v, z ) , (2.14)where A µν (0) = δ µν . Using these transformations one can fix the normalization in sucha way that ( ∇ θ α ( v, z ) , ∇ θ β ( v, − z )) = η αβ . (2.15)3. Main example: Frobenius manifold of A q − -type Our main example is A q − Frobenius manifold [17]. Let Q ( y ) be a polynomial of yQ ( y ) = y q + u y q − + ... + u q − , (3.1)and { u α } represent some coordinates on M . We call { u α } the canonical coordinates. Definition 3.1. A q − Frobenius algebra is the space of polynomials modulo polyno-mial dQdy : A q − ( u ) = C [ y ] / dQdy . (3.2) The corresponding manifold M is called the Frobenius manifold of A q − type The polynomials P α ( y ) = ∂Q∂u α , (3.3)form a basis in the tangent space T v M . An invariant bilinear form (which is equivalentto the metric) is defined by( P α , P β ) = res y = ∞ (cid:18) P α ( y ) P β ( y ) dQdy ( y ) (cid:19) . (3.4)With this definition one can verify that the corresponding metric is flat and C αβγ = ∇ α ∇ β ∇ γ F ( u ) . (3.5) o this end we perform the transformation from the canonical coordinates { u α } tothe new coordinates { v α } by means of the following relation y = z − q (cid:18) v q − z + v q − z + · · · + v z q − (cid:19) + O (cid:18) z q +1 (cid:19) , (3.6)where z q = Q ( y ).Some useful properties of the new coordinates are formulated in the following Theorem 3.2.
From the transformation (3.6) it follows that (1) v α form flat coordinates, i.e., the metric in this coordinates is constant and η αβ = − q (cid:18) ∂Q∂v α , ∂Q∂v β (cid:19) = δ α + β,q , (3.7)(2) C αβγ = − q res y = ∞ (cid:18) ∂Q∂v α ∂Q∂v β ∂Q∂v γ dQdy (cid:19) = ∂ F∂v α ∂v β ∂v γ . (3.8)(3) θ α,k = − c α,k res y = ∞ Q k + αq ( y ) , (3.9) where c α,k = Γ( αq )Γ( αq + k + 1) . (3.10)To prove these statements it is convenient to use the basis elements of A q − in flatcoordinates defined by Φ α ( y ) = ∂Q ( y ) ∂v α which possess the following propertyΦ α ( y ) = 1 α ddy (cid:18) Q αq (cid:19) + . (3.11)In what follows we use the following convention θ µ,k = θ µ − q ⌊ µ/q ⌋ ,k + ⌊ µ/q ⌋ , (3.12)where ⌊ µ/q ⌋ is the integer part of µ/q . It is clear that (3.12) agrees with the definition(3.9). 4. Frobenius manifolds and Douglas string equation
Integrable hierarchies.
Let M be a space of functions of x taking values in M . Let I and J be functionals on M . We define the Poisson bracket on M as { I, J } = Z δIδv α ( x ) η αβ ddx δIδv β ( x ) dx, (4.1)or { v α ( x ) , v β ( y ) } = η αβ δ ′ ( x − y ) , (4.2) here, as usual in the calculus of variations, the integrand is defined modulo totalderivatives. The functionals H α,k = Z θ α,k +1 ( ~v ( x )) dx, α = 1 , ..., n, k ≥ , (4.3)mutually commute among themselves { H α,k , H β,l } = 0 . (4.4)As a result, the Hamiltonian flows ∂v µ ∂t αk = { v µ , H α,k } = η µν ∂∂x ∂θ α,k +1 ∂v ν = C µρλ ∂θ α,k ∂v ρ ∂v λ ∂x . (4.5)commute, i.e., ∂∂t βl ∂~v∂t αk = ∂∂t αk ∂~v∂t βl . (4.6)It follows from (4.5) that t = x .4.2. Douglas String Equation.
Let us define a function S ( v, t ) on M which de-pends on the additional papametres { t αk } S ( v, t αk ) = n X α =1 X k ≥ t αk θ α,k ( v ) . (4.7)The equation ∂S∂v α = 0 , (4.8)is called a string equation. In the case of Frobenius manifold of A q − type it is nothingbut the Douglas string equation written in the form of the principle of least Stringaction [19]. It can be shown that solutions ~v ( t αk ) of the string equation (4.8) satisfyalso (4.5).4.3. Equation for Tau-function.
We define the function Z [ t ] = log τ ( t ), where Z [ t ] = 12 Z v = v ∗ ( t )0 Ω , (4.9)and Ω = C βγα ( v ) ∂S ( v, t ) ∂v β ∂S ( v, t ) ∂v γ dv α , (4.10)is the differential form and v ∗ ( t ) is one of the solutions of the string equation (4.8).From the associativity of the algebra A q − and the equations (2.12) it follows that Ωis closed one-form. emma 4.4. On the solution of the string equation Z(t) satisfies ∂ Z ( t ) ∂t αk ∂t = θ α,k ( v ( t )) . (4.11) In particular, v α ( t ) = η αβ ∂ Z∂t β ∂t , (4.12) and for v q − ( t ) = u ( t ) ∂ Z∂x = u ( t ) . (4.13) Proof.
Differentiating with respect to t αk and t and taking into account the stringequation, we find ∂ Z∂t αk ∂t = Z v ∗ ( t )0 C βγλ ∂θ α,k ∂v β ∂θ , ∂v γ dv λ = Z v ∗ ( t )0 ∂θ α,k ∂v λ dv λ = θ α,k . (4.14)Here we used that θ , = v = v q − , C β,q − λ = δ βλ . (cid:3) Taking into account θ α, = v α we obtain from Lemma 4.4 that ∂ Z ( t ) ∂t α ∂t = v α ( t ) . (4.15)Since Z satisfy equations (4.8) and (4.13), it is a tau-function of the integrable hier-archy connected with the corresponding Frobenius manifold.4.5. A q − FM and dispersionless limit of Gelfand-Dikij Hierarchy.
The dis-persionless limit of the Gelfand-Dikij equations is formulated as follows: ∂Q∂t αk = [ A α,k , Q ] = ∂A α,k ∂x ∂Q∂y − ∂A α,k ∂y ∂Q∂x , (4.16)where Q = y q + u ( x ) y q − + ... + u q − , (4.17)and A α,k = 1 q c α,k (cid:18) Q k + αq (cid:19) + . (4.18)One can show that these equations are equivalent to the Hamiltonian equation ∂v µ ∂t αk = η µν ∂∂x ∂θ α,k +1 ∂v ν . (4.19) To simplify our expressions we write v ( t ) instead of v ∗ ( t ) when it is clear from the context. .6. Formula for tau-function.
As it was derived above the logarithm of the tau-function Z [ { t αk } ] is given by Z [ { t αk } ] = 12 Z v ∗ ( t )0 C βγα ∂S∂v β ∂S∂v γ dv α , (4.20)where S = q − X α =1 X k t αk θ α,k . (4.21)5. Resonance problem in ( p, q ) MLG
Homogeneity property of string equation. Spectrum for ( p, q ) case. Letnow only the finite number of the parameters { t αk } be nonzero. One of them we takeequal to one and others be enumerated by two integers ( m, n ). Here 1 ≤ m ≤ q − ≤ n ≤ p −
1, where p, q are two coprime integers, p > q and q is a degree of thepolynomial Q defined in (4.17). Hence, the set of the parameters { t αk } is replaced bythe set { t mn } . Let us take the action in the form S = res y = ∞ [ Q p + qq + pm − qn> X m,n t mn Q pm − qnq ] , (5.1)It is easy to check that Q [ y, u α ] and S [ u α , t mn ] are quasi-homogeneous functions Q [ ρy, ρ r α u α ] = ρ q Q [ y, u α ] , S [ ρ r α u α , ρ σ mn t mn ] = ρ p + q S [ u α , t mn ] . (5.2)Here we denote r α = q − α − , σ mn = p + q − | pm − qn | . (5.3)We call { σ mn } the set of the scaling indices of the set { t mn } . As it was found byDouglas [2], the numbers δ mn = σ mn q coincide with the gravitational dimensions of thephysical fields in ( p, q ) Minimal Liouvillle gravity [18].The function Z [ t mn ] is a quasi-homogeneous function Z [ ρ qδ mn t mn ] = ρ p + q ) Z [ t mn ] . (5.4)5.2. The group of the resonance transformations.
Since the scaling indices areinteger, the following relation can take place σ mn = σ k l + σ k l + ... + σ k N l N . (5.5)This is known as a resonance condition. The number of possible resonances growswhen p and q increase. A transformation t mn → λ mn of the form t mn = λ mn + X k ,l ,k ,l A k l ; k ,l mn λ k ,l λ k ,l + X k ,l ,k ,l ,k ,l A k l ; k ,l ; k ,l mn λ k ,l λ k ,l λ k ,l + ..., (5.6) s called resonance transformation if (5.5) is satisfied for each term. Besides, bydefinition, we suggest that the scaling index of λ mn equals to the one of t mn .It is obvious that t mn ( { ρ σ kl λ kl } ) = ρ σ mn t mn ( { λ kl } ) , (5.7)and that the resonance transformation does not change the homogeneity property ofthe partition function Z [ t mn ( { λ kl } )] = e Z [ λ mn ] e Z [ { ρ σ mn λ mn } ] = ρ p + q e Z [ { λ mn } ] . (5.8)Hence, if we find some solution of the string equation (4.8) and construct Z [ t mn ],then we get a family of the solutions e Z [ { λ mn } ] = Z [ { t mn ( { λ kl ) }} ] having the samehomogeneity properties with respect to the resonance transformations.5.3. The choice of the resonance transformation and of the solution of thestring equation.
Now we are in the position to formulate the following problem. Weare looking for solutions of the string equation and resonance transformations whichgives function e Z [ { λ mn } ] satisfying infinite number of constraints known as fusion rulesfor observables of minimal CFT models M ( p, q ) and their for the correlators. In whatfollows we restrict ourself by considering spherical topology. Then these rules can beformulated as follows.We denote by Φ mn , where 1 ≤ m ≤ p and 1 ≤ n ≤ q , the primary fields inthe minimal model M ( p, q ) of Conformal field theory. The fields Φ m,n and Φ q − m,p − n correspond to the same primary field.The following graphical representation allows to formulate these restrictionsΦ m n Φ m N n N Φ m n Φ m n Φ m n Φ m N − n N − ........ Here the external lines represent the (arbitrary arranged) primary fields in the corre-lator h Φ m n Φ m n ... Φ m N n N i (here we assume N ≥ k i , l i ) assigned to the internal lines, for which in any vertex of the graph thefollowing condition on the three pairs ( m i , n i ) ( i = 1 , ,
3) corresponding to the linesconnected to this vertex | m − m | + 1 ≤ m ≤ min { m + m − , q + 1 − m − m } step 2 , (5.9) | n − n | + 1 ≤ n ≤ min { n + n − , p + 1 − n − n } step 2 , (5.10)can not be satisfied any permutation of the pairs. n addition, from the conformal selection rules for N = 1 it follows h Φ mn i = 0 , (5.11)for ( m, n ) = (1 ,
1) and for N = 2 h Φ m n Φ m n i = 0 , (5.12)for ( m , n ) not equal to ( m , n ) or ( q − m , p − n ).Now we are going to give a more precise formulation of our main conjecture. Conjecture 5.4.
There exist the solution of the string equation and the choice ofthe resonance transformation described above, such that the function e Z [ { λ mn } ] = h exp X m,n λ m,n O m,n i = ∞ X N =0 X m i ,n i λ m n ...λ m N n N N ! h O m n ...O m N n N i , (5.13) appeares the generating function of the correlators in the Minimal Liouville Gravity. In particular, all correlators h O m n ...O m N n N i forbidden by the conformal fusionrules vanish. 6. The plan of the solution of the problem
To solve the formulated above problem we write the action S ( v α , t mn ) and thegenerating function Z [ { t mn } ] in terms of new variables { λ mn } using the resonancechange of variables t mn = λ mn + A mn µ δ mn + δ m n ≤ δ mn X m ,n A m n mn µ δ mn − δ m n λ m n ++ δ m n + δ m n ≤ δ mn X m ,n ,m ,n A m n ,m n mn µ δ mn − δ m n − δ m n λ m n λ m n + . . . , (6.1)where µ = λ is called the cosmological constant in the continuum approach to MLG.After performing this transform the action takes the form e S [ v α , { λ mn } ] = S (0) ( v α ) + X m,n λ mn S ( mn ) ( v α )++ X m ,n ,m ,n λ m n λ m n S ( m n ,m n ) ( v α ) + . . . . (6.2)The information about the form of the resonance transformation is encoded in thecoefficients of S (0) , S ( mn ) , etc. From (3.9) and (12.1) we find S (0) = res y = ∞ (cid:20) Q p + qq + s X l =1 A l µ l +12 Q p − qlq (cid:21) , (6.3) here we introduced the new positive integer numbers s and p such that p = sq + p and 0 < p < q . S ( mn ) = res y = ∞ (cid:20) Q pm − qnq + sm + ⌊ p mq ⌋ X l = n +2 A mnml µ l − n Q pm − qlq (cid:21) , (6.4)where A mnkl are the coefficients of the resonance relations and ( l − n ) is even. Thehigher coefficients can also be easily written in terms of the coefficients A { m i n i } kl .The generating function is given by e Z [ { λ mn } ] = 12 Z v ∗ C βγα ( v ) ∂ e S∂v β ∂ e S∂v γ dv α , (6.5)where v ∗ is defined as a function of the parameters { λ mn } of the Douglas stringequation (4.8).From now on we will skip the tilde over the functions e S ( { u α } , { λ mn } ) and e Z ( { λ mn } ).7. Appropriate solution
To compute the one-point function which is given by the integral h O mn i = Z v α C αβγ ∂S (0) ∂v β ∂S ( mn ) ∂v γ dv α , (7.1)we need to know the upper limit in this integral v α which is the solution of the stringequation for all couplings (except λ = µ ) equal to zero v α = v ∗ α ( λ mn ) (cid:12)(cid:12)(cid:12)(cid:12) λ mn =0 ,λ = µ . (7.2)Explicitly, v α satisfies ∂S (0) ∂v µ (cid:12)(cid:12)(cid:12)(cid:12) v α = v α = 0 . (7.3)Using (6.3), (6.4) and (3.9), S (0) and S ( mn ) can be written as S (0) = − θ p ,s +1 c p ,s +1 − s X l =1 A l µ l +12 θ p ,s − l c p ,s − l , (7.4) S ( mn ) = − θ p m,sm − n c p m,sm − n − sm + ⌊ p mq ⌋ X l = n +2 A mnml µ l +12 θ p m,sm − l c p m,sm − l . (7.5)We will use the following proposition from [6]: roposition 7.1. On the line v i> = 0 , k even : ∂θ λ,k ∂v α = δ λ,α x λ,k (cid:0) − v q (cid:1) k q ,k odd : ∂θ λ,k ∂v α = δ λ,q − α y λ,k (cid:0) − v q (cid:1) k − q + λ , (7.6) where x α,k = Γ( αq )Γ( αq + k ) (cid:0) k (cid:1) ! and y λ,k = − Γ( αq )Γ( αq + k +12 ) (cid:0) k − (cid:1) ! . (7.7)Using this statement together with (7.4) it is not difficult to see that the stringequation (7.3) has the solutions of the form v α = 0 for α = 1 and the coordinate v is a root of the equation ∂S (0) ∂v p = 0 , if s − odd , (7.8)or ∂S (0) ∂v q − p = 0 , if s − even . (7.9)Here we assume that after taking derivative we set all v α for α = 1 to zero. Moreexplicitly these equations can be written as X k = − s B odd p ,k (cid:18) − v q (cid:19) s − k q = 0 , if s − odd , (7.10)or X k = − s B odd p ,k (cid:18) − v q (cid:19) s − k − q = 0 , if s − even , (7.11)where B odd p ,k = x p ,s − k c p ,s − k A ,k µ k +12 , (7.12)and B even p ,k = y p ,s − k c p ,s − k A ,k µ k +12 , (7.13)where A , − = 1. 8. One-point functions
As it was shown in [6], the structure constant in the flat coordinates on the line v α> = 0, for α ≥ β ≥ γ C αβγ = (cid:0) − v q (cid:1) α + β + γ − q − (8.1)if α + β + γ − q − ∈ N and α + β − γ ∈ [1 , q − , otherwise 0 , here N is the set of non-negative integers. Using (7.6) we find for s odd and ( sm − n )even h O mn i = Z v C q − ,p ,p m ∂S (0) ∂v p ∂S ( mn ) ∂v p m dv . (8.2)Taking into account (8.1) we conclude that the correlation function is zero for m = 1.Hence, in this case from the selection rules we obtain h O n i = Z v (cid:0) − v q (cid:1) p − ∂S (0) ∂v p ∂S (1 n ) ∂v p dv = 0 . (8.3)For s odd and ( sm − n ) odd, h O mn i = Z v C q − ,p ,q − p m ∂S (0) ∂v p ∂S ( mn ) ∂v q − p m dv , (8.4)and the structure constant here is not equal to zero only if q − p m = p as it followsfrom (8.1). Therefore the gravitational dimension[ h O mn i ] = p + qq − δ mn = sm − n s + 12 + p m + p q , (8.5)is integer, the correlation function is analytic and we shell not consider it [5].Similarly, for s even and ( sm − n ) even, we obtain the following consequence of theselection rules h O n i = Z v (cid:0) − v q (cid:1) q − p − ∂S (0) ∂v q − p ∂S (1 n ) ∂v q − p dv = 0 . (8.6)Finally, if s even and ( sm − n ) odd, we find again that the expressions for the onepoint correlation functions are analytic.Simple analysis shows that the number of these equations is equal to the numberof the coefficients arising in the first order in the resonance relation. Hence therequirement of absence of the one point functions fixes uniquely unknown coefficients B p ,k in the expressions (7.10) and (7.11).Thus we arrive to the conclusion that the special solution of the string equationconsidered above ensure the requirements of the selection rules in agreement with thegeneral prescription described in the previous section.We note also that the variety of ( p, q ) models of minimal Liouville Gravity is splittedin two subclasses according to the condition that ⌊ p/q ⌋ be either even or odd. In eachcase we find distinct sets of requirements formulated above leading to zero valued onepoint functions. . Two-point functions
We are now going to consider the two-point function. From (6.5) we find h O m n O m n i = q − X γ =1 Z v dv (cid:0) − v q (cid:1) γ − ∂S ( m n ) ∂v γ ∂S ( m n ) ∂v γ . (9.1)It follows from (7.6) that ∂S ( mn ) ∂v γ = 0 if one of the following two conditions is satisfied1) γ = mp mod q and ( sm − n ) − even , γ = q − mp mod q and ( sm − n ) − odd . (9.2)Similarly to the consideration in the previous section we find four cases where the twopoint function can be non-zero. In two cases: where the first pair ( m , n ) satisfiesfirst condition while the second pair ( m , n ) is subject of the second condition andvice versa, we find the regular expression for the two point function. Thus, we are leftwith the two options where both pairs satisfy either the first or the second conditionin (9.2).Explicitly, in the case when both ( sm − n ) and ( sm − n ) are even we get thefollowing requirement h O mn O mn i = Z v dv (cid:0) − v q (cid:1) mp − ∂S ( mn ) ∂v mp ∂S ( mn ) ∂v mp = 0 if n = n . (9.3)Making the substitution t = 2 (cid:18) v v (cid:19) q − , (9.4)and denoting ∂S ( mn ) ∂v mp = L sm − n ( t ) , (9.5)we find the following consequence of the diagonality condition h O mn O mn i = Z − dt (1 + t ) mp − qq L sm − n ( t ) L sm − n ( t ) = 0 if n = n . (9.6)Hence, the selection rules for the two-point correlation numbers requires that thepolynomials L sm − n form an orthogonal set of Jacobi polynomials ∂S ( mn ) ∂v mp = pm − qnq P (0 , mp − qq ) sm − n ( t ) , for ( sm − n ) − even . (9.7)In the second case, where both ( sm − n ) and ( sm − n ) are odd, we have h O mn O mn i = Z v dv (cid:0) − v q (cid:1) q − mp − ∂S ( mn ) ∂v q − mp ∂S ( mn ) ∂v q − mp = 0 if n = n . (9.8) enoting ∂S ( mn ) ∂v q − mp = (1 + t ) mp q L sm − n − ( t ) , (9.9)we find the following consequence of the diagonality condition for the two-point cor-relation function in this case h O mn O mn i = Z − dt (1 + t ) mp q L sm − n − ( t ) L sm − n − ( t ) = 0 if n = n . (9.10)It means that ∂S ( mn ) ∂v q − mp = pm − qnq (1 + t ) mp q P (0 , mp q ) sm − n − ( t ) for ( sm − n ) − odd . (9.11)At last, inserting these explicit expressions for the derivatives of S ( mn ) to the equations(8.3) and (8.6) we arrive to the condition h O n i = Z − (1 + t ) p − qq L s +12 ( t ) P (0 , p − qq ) s − n ( t ) dt = 0 , (9.12)in the case where s is odd and n is odd and greater than 1. And h O n i = Z − (1 + t ) p q L s ( t ) P (0 , p q ) s − n − ( t ) dt = 0 , (9.13)in case where s is even and n is odd and greater than 1. Here we introduced thepolynomial L n ( t ) ∂S (0) ∂v p ( t ) = L s +12 ( t ) , (9.14)for s odd, ∂S (0) ∂v q − p ( t ) = (1 + t ) p q L s ( t ) . (9.15)for s even.Taking to the account these equations, the order of the polynomials ∂S (0) ∂v p and ∂S (0) ∂v q − p and the string equations (7.8), (7.9) we obtain the following explicit expressions ∂S (0) ∂v p = p + qq (cid:18) P (0 , p − qq ) s +12 ( t ) − P (0 , p − qq ) s − ( t ) (cid:19) , (9.16)if s is odd and ∂S (0) ∂v q − p = p + qq (1 + t ) p q (cid:18) P (0 , p q ) s ( t ) − P (0 , p q ) s − ( t ) (cid:19) , (9.17)if s is even. Conclusions
In this paper we have described the relation between the approach to ( p, q ) modelsof Minimal Liouville gravity based on the Douglas string equation, on one hand, andthe Frobenius manifolds of A q − type on the other. As a result of this relation thegenerating function of correlation numbers in MLG is represented by the logarithm ofthe tau-function of the corresponding integrable hierarchy. All necessary informationis encoded in the solution of the Douglas string equation and in the resonance rela-tions between the parameters of the integrable hierarchy and the coupling constantsof MLG. Using this relation and some special properties of the flat coordinates onthe Frobenius manifold, we have found the appropriate solution of the Douglas stringequation. This result generalizes analogues result found recently for Unitary models ofMinimal Liouville gravity [6]. We have shown that the appropriate solution is consis-tent with the basic requirements of the conformal selection rules arising on the levelsof one- and two-point correlation functions. Namely, the number of the parameters ofthe resonance transformations is exactly the number of the constraints following fromthe selection rules. Resolving these constraints we have found explicit form of theresonance transformations in terms of Jacoby polynomials. It would be interestingto investigate if this matching persists for multi-point correlation functions when thefusion rules of the underlying minimal models of CFT should be taken into account.This analysis requires also knowing the explicit form of the structure constants of theFrobenius algebra in the flat coordinates. We plan to study these questions in thenear future. Another possible extension of our study is to consider different general-izations of the Minimal Liouville Gravity in the context of the Douglas string equationapproach and its relations with different types of Frobenius manifolds. In particular,it would be interesting to understand what kind of the Frobenius manifold is relevantfor W N Minimal Liouville gravity.
Acknowledgements.
We thank Boris Dubrovin, Michael Lashkevich, Yaroslav Pu-gai and Grisha Tarnopolsky for useful discussions. The research was performed undera grant funded by Russian Science Foundation (project No. 14-12-01383).
References [1] A. M. Polyakov,
Quantum Geometry of Bosonic Strings , Phys.Lett.
B103 (1981) 207–210.[2] M. R. Douglas,
Strings in less than one-dimension and the generalized KdV hierarchies , Phys.Lett.
B238 (1990) 176.[3] G. W. Moore, N. Seiberg, and M. Staudacher,
From loops to states in 2-D quantum gravity , Nucl.Phys.
B362 (1991) 665–709.[4] A. Belavin and A. Zamolodchikov,
On Correlation Numbers in 2D Minimal Gravity andMatrix Models , J.Phys.
A42 (2009) 304004, [ arXiv:0811.0450 ].[5] A. Belavin, B. Dubrovin, B. Mukhametzhanov,
Minimal Liouville Gravity correlation numbersfrom Douglas string equation , JHEP (2014) 156. [ arXiv:1310.5659 ].
6] V. Belavin,
Unitary Minimal Liouville Gravity and Frobenius Manifolds , [ arXiv:1405.4468 ].[7] A. Belavin, A. M. Polyakov, and A. Zamolodchikov,
Infinite Conformal Symmetry inTwo-Dimensional Quantum Field Theory , Nucl.Phys.
B241 (1984) 333–380.[8] V. Kazakov, A. A. Migdal, and I. Kostov,
Critical Properties of Randomly TriangulatedPlanar Random Surfaces , Phys.Lett.
B157 (1985) 295–300.[9] V. Kazakov,
Ising model on a dynamical planar random lattice: Exact solution , Phys.Lett.
A119 (1986) 140–144.[10] V. Kazakov,
The Appearance of Matter Fields from Quantum Fluctuations of 2D Gravity , Mod.Phys.Lett. A4 (1989) 2125.[11] M. Staudacher, The Yang-Lee edge singularity on a dynamical planar random surface , Nucl.Phys.
B336 (1990) 349.[12] E. Brezin and V. Kazakov,
Exactly solvable field theories of closed strings , Phys.Lett.
B236 (1990) 144–150.[13] M. R. Douglas and S. H. Shenker,
Strings in Less Than One-Dimension , Nucl.Phys.
B335 (1990) 635.[14] D. J. Gross and A. A. Migdal,
Nonperturbative Two-Dimensional Quantum Gravity , Phys.Rev.Lett. (1990) 127.[15] I. Krichever, The Dispersionless Lax equations and topological minimal models , Commun.Math.Phys. (1992) 415–429.[16] B. Dubrovin,
Integrable systems in topological field theory , Nucl.Phys.
B379 (1992) 627–689.[17] R. Dijkgraaf, H. L. Verlinde, and E. P. Verlinde,
Topological strings in d less than 1 , Nucl.Phys.
B352 (1991) 59–86.[18] V. Knizhnik, A. M. Polyakov, and A. Zamolodchikov,
Fractal Structure of 2D QuantumGravity , Mod.Phys.Lett. A3 (1988) 819.[19] P. H. Ginsparg, M. Goulian, M. Plesser, and J. Zinn-Justin, (p, q) String actions , Nucl.Phys.
B342 (1990) 539–563.[20] P. Di Francesco and D. Kutasov,
Unitary Minimal Models Coupled To 2-d Quantum Gravity , Nucl.Phys.
B342 (1990) 589–624(1990) 589–624