Frobenius manifolds, Integrable Hierarchies and Minimal Liouville Gravity

We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the 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 a continuation of earlier works [3][4][5][6].
The Liouville Gravity represents a consistent example of the noncritical String theory. In the initial continuous approach the Liouville Gravity is formulated as a BRST invariant theory composed of the matter sector, the Liouville theory and the ghosts system. MLG represents the theory, where the matter sector is taken to be a (p, q) Minimal Model of CFT [7]. The main problem of MLG is to calculate correlation functions of BRST invariant observables, which are given by integrals over moduli of Riemannian surfaces. Usually they are called the correlation numbers. Numerous examples show that the solution of the problem is quite nontrivial within the framework of the continuous approach.
An alternative approach to MLG has grown up from the idea of triangulations of two-dimensional surfaces realized in terms of Matrix Models [8][9][10][11][12][13][14]. One of the most important points of the approach is the String equation which was derived by * belavin@itp.ac.ru † belavin@lpi.ru Douglas [2] in Matrix Models approach to two dimensional gravity. The subject of the String equation is the generating function of the correlation numbers which depends on the parameters of the problem (the so called KdV times). In our work, following [4][5][6], we will conjecture that the Douglas equation is applicable to the Minimal Liouville gravity as well as to Matrix Models of 2D gravity.
This conjecture requires the following two questions to be answered: how to choose the desired solution of the Douglas string equation and an appropriate form of the so called resonance transformation [3] from the KdV times to the Liouville coupling constants. Once these two questions are answered, the generating function of the correlation functions in MLG is given explicitly as an integrated one-form defined uniquely for each (p, q) MLG model and coincides with a special choice of the taufunction 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 the String equation with the Frobenius manifold structure, we find the necessary solution of the String equation. We also show that this solution together with the suitable choosen resonance transformation lead to the results which are consistent with the main requirements of (p, q) models of MLG (the so called selection rules). It is remarkable that the needed solution of the Douglas equation has a very simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville 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 2 we recall briefly the notion of the Frobenius manifold and discuss its basic properties. In Section 3 we discuss the Frobenius manifold that appears in the context of Minimal Liouville Gravity. Section 4 is devoted to the connection between the Frobenius manifold structures, Integrable structures and the Douglas string equation. In Section 5 we focus on the (p, q) models of MLG and discuss the problem of the resonance transformations. The idea of the approach based on the String equation to (p, q) MLG is formulated in Section 6. The appropriate solution of the Douglas string equation is discussed in Section 7. The rest of the paper is devoted to the analysis of the correlation functions. We show that the special choice of the solution of the String equation together with the resonance transformation encoded in terms of Jacobi polynomials ensure fulfilling the necessary selection rules for the correlation numbers in (p, q) MLG.

Frobenius manifolds
In this and two next sections we give the definition and a short review of main properties of the Frobenius manifolds needed for our purposes. Here we follow the paper by B.Dubrovin [16], see also [5].
By definition a commutative associative algebra A with unity equipped with a nondegenerate invariant bilinear form ( , ) is called Frobenius algebra. The invariance of the bilinear form means that for any three vectors a, b, c in A: (2.1) Let M be n-dimensional manifold with a flat metric η αβ dv α dv β which is constant in the flat coordinates v α .
We introduce in the tangent space T v M the structure of the Frobenius algebra by the following identification of the bases 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 where C αβγ = η αρ C ρ βγ .
(2.6) Function F is called Frobenius potential. The consistency of this property with the associativity of the Frobenius algebra is known as WDVV condition [17] (2.7) The following statement [16] follows from these properties of the Frobenius manifold M. There exist an one-parametric flat deformation ∇ α of the connection ∇ α 8) or, equivalently, [ ∇ α (z), ∇ 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 of the equation ∇ α dθ λ (v, z) = 0, which is equivalent to (2.11) or The functions θ α (v, z) can be considered as the flat coordinates of the deformed connection ∇ α (z). We choose (2.13) and, hence, the scalar product (∇θ Using these transformations one can fix the normalization in such a way that

Main example: Frobenius manifold of A q−1 -type
Our main example is A q−1 Frobenius manifold [17]. Let Q(y) be a polynomial of y and {u α } represent some coordinates on M. We call {u α } the canonical coordinates.
Definition 3.1. A q−1 Frobenius algebra is the space of polynomials modulo polynomial dQ dy : The corresponding manifold M is called the Frobenius manifold of A q−1 type The polynomials form a basis in the tangent space T v M. An invariant bilinear form (which is equivalent to the metric) is defined by .

(3.4)
With this definition one can verify that the corresponding metric is flat and To this end we perform the transformation from the canonical {u α } to the new coordinates {v α } by means of the following relation where z q = Q(y). Some useful properties of the new coordinates are formulated in the following

the metric in this coordinates is constant and
(2) . (3.10) To prove these statements it is convenient to use the basis elements of A q−1 in flat coordinates defined by Φ α (y) = ∂Q(y) ∂v α which possess the following property In what follows we use the following convention where ⌊µ/q⌋ is the integer part of µ/q. It is clear that (3.12) agrees with the definition (3.9).

Frobenius manifolds and Douglas string equation
where, as usual in the calculus of variations, the integrand is defined modulo total derivatives. The functionals mutually commute among themselves As a result, the Hamiltonian flows It follows from (4.5) that t 1 0 = x.

Douglas String Equation.
Let us define a function S(v, t) on M which depends on the additional papametres The equation is called a string equation. In the case of Frobenius manifold of A q−1 type it is nothing but the Douglas string equation written in the form of the principle of least String action [19]. It can be shown that solutions v(t α k ) of the string equation (4.8) satisfy also (4.5).

Equation for
Tau-function. We define the function and 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−1 and the equations (2.12) it follows that Ω is closed one-form.

Lemma 4.4. Z(t) satisfies
and Proof. Differentiating with respect to t α k and t 1 0 and taking into account the string equation, we find (4.14) Here we used that Since Z satisfy equations (4.8) and (4.13), it is a tau-function of the integrable hierarchy connected with the corresponding Frobenius manifold.

4.5.
A q−1 FM and dispersionless limit of Gelfand-Dikij Hierarchy. The dispersionless limit of the Gelfand-Dikij equations is formulated as follows: where v * (t) is the solution of the string equation Here we denote We call {σ mn } the set of the scaling indices of the set {τ mn }. As it was found by Douglas [2], the numbers δ mn = σmn 2q coincide with the gravitational dimensions of the physical fields in (p, q) Minimal Liouvillle gravity [18]. The This is known as a resonance condition. The number of possible resonances grows when p and q increase. A transformation τ mn → λ mn of the form In what follows we restrict ourself by considering spherical topology. Then these rules can be formulated as follows.
We denote by Φ mn , where 1 ≤ m ≤ p and 1 ≤ n ≤ q, the primary fields in the 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 restrictions Here the external lines represent the (arbitrary arranged) primary fields in the correlator Φ m 1 n 1 Φ m 2 n 2 ...Φ m N n N (here we assume N ≥ 3). The fusion rules result to the requirement that the correlation function must be equal to zero if there are no sets of pairs (k i , l i ) assigned to the internal lines, for which in any vertex of the graph the following condition on the three pairs (m i , n i ) (i = 1, 2, 3) corresponding to the lines connected to this vertex can not be satisfied any permutation of the pairs. In addition, from the conformal selection rules for N = 1 it follows Φ mn = 0, (5.11) for (m, n) = (1, 1) and for N = 2 Φ m 1 n 1 Φ m 2 n 2 = 0, (5.12) for (m 1 , n 1 ) not equal to (m 2 , n 2 ) or (q − m 2 , p − n 2 ). Now we are going to give a more precise formulation of our main conjecture. where µ = λ 11 is called the cosmological constant in the continuum approach to MLG.
After performing this transform the action takes the form The information about the form of the resonance transformation is encoded in the coefficients of S (0) , S (mn) , etc. From (3.9) and (12.1) we find where A mn kl are the coefficients of the resonance relations. The higher coefficients can also be easily written in terms of the coefficients A The generating function is given by where v * is defined as a function of the parameters {λ mn } of the Douglas string equation (4.8).
From now on we will skip the tilde over the functions S({u α }, {λ mn }) and Z({λ mn }).

Appropriate solution
To compute the one-point function which is given by the integral we need to know the upper limit in this integral v 0 α which is the solution of the string equation for all couplings (except λ 11 = µ) equal to zero We introduce the new positive integer numbers s and p 0 such that p = sq + p 0 and 0 < p 0 < q. Using (6.3), (6.4) and (3.9), S (0) and S (mn) can be written as We will use the following proposition from [6]: Using this statement together with (7.4) it is not difficult to see that the string equation (7.3) has the solutions of the form v 0 α = 0 for α = 1 and the coordinate v 0 1 is a root of the equation or Here we assume that after taking derivative we set all v α for α = 1 to zero. More explicitly these equations can be written as and B even p 0 ,k = where A 1,−1 = 1.

One-point functions
As it was shown in [6], the structure constant in the flat coordinates on the line v α>0 = 0, for α ≥ β ≥ γ where N 0 is the set of non-negative integers. Here Θ A, and is zero otherwise. Using (7.6) we find for s odd and (sm − n) even 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 For s odd and (sm − n) odd, the gravitational dimension is integer, the correlation function is analytic and, therefore, should not be considered [5]. Similarly, for s even and (sm − n) even, we obtain the following consequence of the selection rules Finally, if s even and (sm − n) odd, we find again that the expressions for the one point correlation functions are analytic. Simple analysis shows that the number of these equations is equal to the number of the coefficients arising in the first order in the resonance relation. Hence the requirement of absence of the one point functions fixes uniquely unknown coefficients B p 0 k in the expressions (7.10) and (7.11).
Thus we arrive to the conclusion that the special solution of the string equation considered above ensure the requirements of the selection rules in agreement with the general prescription described in the previous section.
We note also that the variety of (p, q) models of minimal Liouville Gravity is splitted in two subclasses according to the condition that ⌊p/q⌋ be either even or odd. In each case we find distinct sets of requirements formulated above leading to zero valued one point functions.

Two-point functions
We are now going to consider the two-point function. From (6.5) we find It follows from (7.6) that ∂S (mn) ∂v γ = 0 if one of the following two conditions is satisfied 1) γ = mp 0 mod q and (sm − n) − even, 2) γ = q − mp 0 mod q and (sm − n) − odd. (9.2) Similarly to the consideration in the previous section we find four cases where the two point function can be non-zero. In two cases: where the first pair (m 1 , n 1 ) satisfies first condition while the second pair (m 2 , n 2 ) is subject of the second condition and vice versa, we find the regular expression for the two point function. Thus, we are left with the two options where both pairs satisfy either the first or the second condition in (9.2). Explicitly, in the case when both (sm − n 1 ) and (sm − n 2 ) are even we get the following requirement Making the substitution and denoting ∂S (mn) ∂v mp 0 = Lsm−n 2 (t), (9.5) we find the following consequence of the diagonality condition Hence, the selection rules for the two-point correlation numbers requires that the polynomials Lsm−n 2 form an orthogonal set of Jacobi polynomials In the second case, where both (sm − n 1 ) and (sm − n 2 ) are odd, we have we find the following consequence of the diagonality condition for the two-point correlation function in this case It means that At last, inserting these explicit expressions for the derivatives of S (mn) to the equations (8.3) and (8.5) we arrive to the condition (t)dt = 0, (9.12) in the case where s is odd and n is odd and greater than 1. And (t)dt = 0, (9.13) in case where s is even and n is odd and greater than 1. Here we introduced the polynomial L n (t) for s odd, ∂v q−p 0 and the string equations (7.8), (7.9) we obtain the following explicit expressions if s is odd and if s is even.

Conclusions
In this paper we have described the relation between the approach to (p, q) models of Minimal Liouville gravity based on the Douglas string equation, on one hand, and the Frobenius manifolds of A q−1 type on the other. As a result of this relation the generating function of correlation numbers in MLG is represented by the logarithm of the tau-function of the corresponding integrable hierarchy. All necessary information is encoded in the solution of the Douglas string equation and in the resonance relations between the parameters of the integrable hierarchy and the coupling constants of MLG. Using this relation and some special properties of the flat coordinates on the Frobenius manifold, we have found the appropriate solution of the Douglas string equation. This result generalizes analogues result found recently for Unitary models of Minimal Liouville gravity [6]. We have shown that the appropriate solution is consistent with the basic requirements of the conformal selection rules arising on the levels of one-and two-point correlation functions. Namely, the number of the parameters of the resonance transformations is exactly the number of the constraints following from the selection rules. Resolving these constraints we have found explicit form of the resonance transformations in terms of Jacoby polynomials. It would be interesting to investigate if this matching persists for multi-point correlation functions when the fusion 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 the Frobenius algebra in the flat coordinates. We plan to study these questions in the near future. Another possible extension of our study is to consider different generalizations of the Minimal Liouville Gravity in the context of the Douglas string equation approach and its relations with different types of Frobenius manifolds. In particular, it would be interesting to understand what kind of the Frobenius manifold is relevant for W N Minimal Liouville gravity.