Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk

There are two alternative approaches to the minimal gravity - direct Liouville approach and matrix models. Recently there has been a certain progress in the matrix model approach, growing out of presence of a Frobenius manifold (FM) structure embedded in the theory. The previous studies were mainly focused on the spherical topology. Essentially, it was shown that the action principle of Douglas equation allows to define the free energy and to compute the correlation numbers if the resonance transformations are properly incorporated. The FM structure allows to find the explicit form of the resonance transformation as well as the closed expression for the partition function. In this paper we elaborate on the case of gravitating disk. We focus on the bulk correlators and show that in the similar way as in the closed topology the generating function can be formulated using the set of flat coordinates on the corresponding FM. Moreover, the resonance transformations, which follow from the spherical topology consideration, are exactly those needed to reproduce FZZ result of the Liouville gravity approach.


Introduction
The conformal bootstrap solution of Liouville field theory (LFT) [1][2][3][4] leads to the possibility 1 to formulate the "direct" path integral approach to c < 1 minimal string theory, known also as two-dimensional minimal Liouville gravity (MLG) [1]. Computations in this framework are rather complicated, since in all but simplest cases they require separate analysis of the correlation functions in both gravitational (LFT) and matter (Minimal Model) sectors [7], a careful treatment of the discrete terms [8] arising in the Liouville operator product expansion, integrating over moduli spaces of curves, etc. On the other hand, the "dual" matrix models (MM) approach [9][10][11], based on the discretization of the path integral and consequent double scaling limit, provides an efficient alternative description of 2D gravity and also reveals an integrable structure of the theory through the connection with a certain class of integrable hierarchies [12][13][14][15]. Different checks in 90s (see, e.g. [16][17][18][19][20][21]) pointed out that matrix models are connected with the minimal Liouville gravity if the connection is properly understood. More elaborate checks performed on the level of correlation functions confirmed this assumption, after the discovery of Liouville higher equations of motion [22] opened the way of analytic computations of the moduli integrals [23]. In particular, for the minimal gravity models of Lee-Yang series M(2/p), corresponding to one-matrix models, the explicit form of the resonance transformations form KdV time-parameters of MM to the Liouville coupling constants of MLG has been established in [24]. Further progress [25][26][27][28][29] has been made after establishing the connection between the Douglas string equation approach [12] to the matrix models and the Frobenius manifolds [30]. This connection allowed to find explicitly the generating function of the correlation numbers for the general case of gravitational M(q/p) minimal models. However, the consideration has been so far restricted to the case of gravitational models defined on closed Riemann-surfaces such as the sphere or the torus [31][32][33][34], relevant for the sector of closed strings.
In this paper, we are interested in the boundary minimal Liouville gravity (BMLG), relevant for open minimal strings. The worldsheet approach to BMLG requires analyzing boundary minimal models, LFT and ghosts and seems even more complicated compared to the closed case. In addition to the bulk MLG content, the classification of physical fields, or BRST cohomologies, in BMLG is specified by a set of boundary changing fields leading to admissible boundary conditions [35]. Constructing correlation numbers, even before taking moduli integrals, requires knowledge of the structure constants appearing in the operator product expansion of two bulk fields, the structure constants of boundary fields, the couplings between bulk and boundary fields and the one point functions of the identity operator for different boundary conditions [36][37][38][39][40].
In the dual approach the boundary effect was first considered in [21]. Since then, it has been investigated in many different contexts. One may refer to some of the previous studies in RSOS models and O(N) fluctuating models [41][42][43], in loop gas models [44,45] and in (one-) [46] and (two-) [47,48] matrix models. The boundary effect for the bulk correlation in the Lee-Yang series of the minimal gravity models was augmented by the resonance transformation in [49]. However, as mentioned above, the Lee-Yang series is represented by one matrix models and it turns out that the related FM is trivial (i.e. one-dimensional). Essentially, it means that the MM partition function found in [21] is easily translated to the MLG partition function, the only care is needed to properly take into account the resonance transformations. Other minimal models are given in terms of multi-matrix models, the corresponding Frobenius manifolds are multi-dimensional and higher Gelfand-Dickey hierarchies appear in general case [25,27]. The analysis of the boundary effect on the MLG models related to multi-dimensional Frobenius manifolds is still missing. In particular, there is no closed expression for the BMLG generating function of bulk correlation numbers available yet. In this work, we extend the analysis of Frobenius manifolds for the bulk correlation on the disk.
The paper is organized as follows. In section 2, we recall the dual approach to MLG and provide the framework of finding bulk correlation numbers on the sphere and on the disk. It turns out that the role of the Douglas equation on the disk is slightly different from that on the sphere, where explicit dependence of the Frobenius (flat) coordinates on the Liouville couplings is required to be determined. On the other hand, the resonance transformation on the disk is the same as in the spherical topology. Even though this statement is not completely obvious, 2 it is anticipated because the boundary operators that could additionally contribute to the coupling mixing are not considered here [51]. In section 3, we consider the effect of presence of nontrivial Frobenius manifolds for the unitary series. We present an Ising model case M(3/4) as an example. In section 4 we discuss non-unitary case. We start with the Lee-Yang series. Even though the Frobenius manifold is one-dimensional, the uniform FM description allows to represent the disk partition function as an integral over the (flat) coordinate and hence this case fits well into the general pattern. Then we consider M(3/5) non-unitary model. Similarly to unitary M(3/4) model, this model is based on two-dimensional Frobenius manifold. However, it turns out that the details of the construction are very different: compared to M(3/4) case, where the cosmological constant and the parameter x in the Douglas equation have the same gravitational scaling dimensions, the gravitational scaling dimension in M(3/5) of theses two parameters are different. This fact leads to an uncertainty in the choice of the integration contour in the flat coordinate space. Nevertheless, carefully employing the Douglas string equation together with the resonance transformations and with a conjectural choice of the contour allows to match this case with the FZZ Liouville results [39] as well. In section 5, we discuss some future perspectives. Some relevant formulae are listed in the appendix A.

Dual approach to MLG
In this section we summarize the basic elements of the dual approach to MLG and define a framework of computing 2D minimal gravity generating functions.

Douglas equation and Frobenius manifolds
The basic element of the matrix models approach (in the continuum limit) is Q differential operator (for reviews, see [52,53]). In A q−1 model it contains q − 1 variables u i and is represented as The set u = {u i } is assumed to be a function of variable x, representing the continuum limit of the discrete state enumeration in the basis of orthogonal polynomials. 3 The functional dependence of u on x at p-critical point is established by introducing P = Q p/q and requiring [P, Q] = 1 . (

2.2)
This constraint, so-called Douglas equation [12], in general reduces to a higher order non-linear differential equations.
On the other hand, the parameters u i can be considered as coordinates on a Frobenius manifold M with dim M = q − 1. The main property of M is the structure of a commutative and associative algebra with unity (to be specified below) defined in the tangent space at each point u and compatible in a certain way [30] with the Riemannian structure of M. By definition, Frobenius manifold is flat, so that there exist flat coordinates v i , in which the metric η lk on M is constant. For our purposes it will be convenient to use the flat basis and its tangent e i . The multiplicative rule for the elements of the tangent basis is where in our convention e 1 = 1, so that c k 1j = δ k j . Using η lk one may perform raising and lowering indices, e.g. c ijk = c l ij η lk . Defining property of M is that the structure constant c ijk is fully symmetric in the index permutation and obeys the following constraint The flat coordinates v can be constructed in terms of the coordinates u using the following explicit form of the flat metric (see, e.g. [25,27]) Here polynomial Q(y, u) = y q + q−1 i=1 u i y q−1−i (∂ is replaced by a commuting number y), e i = ∂Q(y, u)/∂v i and Q ′ (y, u) = ∂Q(y, u)/∂y. It gives the following expression for the flat coordinates with non-negative integers k and 1 ≤ i ≤ q − 1. The structure constants are also given explicitly We note that c 1ij = η ij since e 1 = 1.

Scaling dimensions and 2D gravity
Conformal field theoretical approach to 2D gravity emerged from the path integral formulation [1]. Liouville gravity is constructed as a direct product of three CFT's: In fact, this condition guarantees that the theory admits a nilpotent BRST operator.
In the minimal Liouville gravity the matter sector is represented by M(q/p) minimal model (with q and p coprime, q < p). The Liouville central charge c L = 1 + 6Q 2 L , where the background charge Q L = b + 1/b and b is the Liouville coupling constant, the minimal model cenrtral charge c M is conveniently parametrized as c M = 1 − 6q 2 M with q M = 1/β − β and β = q/p or explicitly c M = 1 − 6(p − q) 2 /qp. The total central charge balance condition (2.9) then requires b = β.
In the minimal Liouville gravity the field content of the matter minimal model is coupled to the Liouville vertex operators in the gravitational sector, such that the resulting "dressed" operator is BRST invarant. We consider the following BRST-invariant operators W m,n = c ·c · U m,n , W m,n = d 2 z U m,n , (2.10) where U m,n = Φ m,n · V a L is constructed from the primary fields Φ m,n and V a L in the matter and Liouville sectors respectively and c (c) is the (−1)-weight ghost field. The conformal dimension of Φ m,n = Φ q−m,p−n (1 ≤ m < q, 1 ≤ n < p) is ∆ m,n = α m,n (q M + α m,n ), where (2.11) and the Liouville primary field V a L has ∆ L a L = a L (Q L −a L ). The total conformal dimension vanishes ∆(W m,n ) = ∆( W m,n ) = 0, as a consequence of the BRST symmetry, leading to the constraint a L = α m,n + β −1 . (2.12) The conformal property of the Liouville gravity induces the gravitation scaling property, defining the reaction of the theory on the re-scaling of the (bulk) cosmological constant µ. The partition function Z L on the sphere has the following scaling behavior [11] Z L ∼ µ Q L /b , (2.13) so that its scaling dimension (g-dim) is Q L /b = (p + q)/q, while the scaling dimensions of the physical fields (2.10) are equal to a L /b. One can assign g-dim to the coupling constant λ mn accompanying fields in the MLG correlation numbers generating function Z L (λ) = exp m,n λ mn W m,n , where λ = {λ mn }: The scaling properties play a key role in relating the observables of the matrix models with those of the Liouville gravity. For example, in the spherical topology the Douglas equation (2.2) is conveniently formulated using the action principle [54] ∂S(u)/∂u i = 0 for the so-called Douglas action Here the summation over m and n is restricted to the region 1 ≤ m < q, 1 ≤ n < p modulo equivalence t (mn) = t (q−m,p−n) . If one assigns g-dim [Q] = 1/2, then taking into account that each term in (2.15) has the same g-dim one gets [t (mn) ] = [λ mn ] defined in (2.14). This allows to relate the coupling constants λ mn of the Liouville gravity with the parameters t (mn) of the matrix models, up to some resonance terms to be discussed shortly.
The susceptibility condition imposes where F is the free energy of the matrix model and v * 1 is a specific solution to the Douglas equation. Since F has the same g-dim as the Liouville partition function Z L , x becomes the parameter of the highest g-dim which is equal to (p + q − 1)/q; [x] = [t (m 1 ,n 1 ) ] with m 1 , n 1 subject to the constraint |pm 1 − qn 1 | = 1.
The free energy F (t) has contributions from all genus partition functions. To get a particular genus part one may introduce a formal genus parameter both in the free energy and in the Douglas string equation (leading to the genus expansion of its solution) and to combine then relevant terms. For the genus 0 case, according to this procedure, we should replace ∂ in Q in (2.15) by a commuting coordinate y. Using the notation θ i,k introduced in (2.6) one can write the action S(u) in the form where p = sq + p 0 with a non-negative integer s. The terms θ i,k =0 are collected in H(v). We note that the indices i, k and m, n are related as i = |pm − qn| (mod q) and In what follows, in spite of this one-to-one correspondence, we keep sometimes (for convenience) double labeling t Even though g-dim of t (mn) and λ mn are the same, one cannot identify these two quantities. There are two reasons which prohibit such simple identification: the different normalization of the two approaches and the appearance of the resonances between operators, emerging from contact term interactions and not violating the scaling property Here A (m,n) (m 1 ,n 1 ),(m 2 ,n 2 ) are dimensionless constants and the sum goes over all possible combinations respecting the scaling. Since powers of λ 11 ∝ µ provide scaling dimensions, it is convenient to reformulate the resonance transformation as follows where the non-vanishing coefficient appear when the power of µ is a non-negative integer.
There are different solutions to the Douglas equation ∂S(v)/∂v i | 0 * = 0, where the condition 0 * stands for the case, where all the couplings λ mn vanish except λ 11 . It is known [27] that the minimal gravity is described by one particular solution such that v i>1 | 0 * = 0. In what follows, we call this on-shell solution and denote it v * . Thus v * i>1 = 0 and the value v * 1 is determined from the Douglas equation. On the other hand, ∂S/∂v 1 = 0 is the integrated form of (2.2), whose integration variable is x. Since x and t 1 have the same g-dim, two variables are closely related. For later convenience, we define x which gives implicit relation between v i 's and x. If one takes the derivative with respect to x, eq.(2.20) reduces to eq.(2.2).
The distinction of x from t 1 is significant because the functional relation of v with x holds even off-shell, where t 1 (considered as coupling constant) is to be identified with λ m 1 n 1 .
The free energy F has to satisfy the susceptibility condition (2.16). From this property, the free energy is [26] Because the integrand is the closed one-form [25], taking into account the property of the relevant solution v * , it is convenient to chooses the integration path dv i>0 = 0, then the free energy has simpler form Bulk correlation numbers are obtained by differentiating with respect to the corresponding coupling constants. Thus, (2.22) is the generating function of the bulk correlation on the sphere.

Bulk generating function on the disk
The bulk free energy on the disk is obtained using the free energy idea of one matrix model. It is shown in [46] that the boundary free energy with h ≥ 1 holes (boundaries) is given by where the expectation value is evaluated with respect to the bulk interaction and the subscript c refers to the connected part. Tr acts on the matrix components. With the one boundary loop (h=1) one has the free energy on the disk.
Here we use the prescription C(M) = µ B − M which describes the case with no boundary operators with µ B boundary cosmological constant. The derivative of F 1 with respect to µ B is the one-point resolvent expectation value. It is noted that the integration over l is the Laplace transform, and the integration path is chosen form 0 to ∞ assuming the contribution due to Tr e lM is convergent. If not, the integration range is properly re-defined so that the integration converged. The expectation value Tr e lM is the main element for finding the free energy on the disk. At the continuum limit, M is replaced by Q. In [21] the element is treated as the expectation value of the loop operator w(l) . We will denote the expectation value as W(l).
where Q is given in x-representation. Tr becomes the integration over x and the integration interval x 1 is to be identified with t (m 1 ,n 1 ) . The generating function on the disk with genus 0 is obtained if ∂ is replaced by y in Q. Since y represents the Fourier space, we have put Q = Q(y, v(x)) and integration of y is performed over iR. In addition, the flat coordinates v have x-dependence through (2.20), which is different from on-shell value v * . On the sphere the bulk generating function does not contain explicit x-dependence of v. However, on the disk explicit x-dependence is necessary. On the disk we need Q(y, v(x, t)) and more information about x and coupling constant dependence is required in order to get the generating function W(l, t).
The bulk correlation on the disk is given as a certain derivatives of the bulk generating function (2.25). For example, one-point correlation is given as where κ 2 = µ/ sin(πb 2 ) and the order of the Bessel function is given as ν m,n = Q L −2αm,n b = |mp−nq| q . More specifically for the minimal model, FZZ result has the form where e α L is a certain numerical constant independent of κ and l 0 . 5 Explicit checks aimed at testing (2.26) against (2.28) will be the subject of the next two sections.

Unitary models
In this section we compute one-point functions for the unitary series of Minimal Gravity in the "dual" aproach using the existing results on the resonance transformation. We compare our results with the ones from the Liouville gravity.
where (m, n) index has the Z 2 symmetry: t has g-dim 1 the following resonance terms are allowed (21), (21), (21) The derivatives of S are From (3.7), (3.8) it follows that the on-shell solution (all t = 0 except t (11) 11 . In addition, x-dependence of v i 's is fixed by the second equation in (3.7).
Bulk generating function on the sphere is [26] (3.10) One can check that the bulk one-point correlation numbers on the sphere vanish. Explicit evaluation shows that which is identically zero due to the on-shell value v * 2 = 0 deternined in (3.8 The non-vanishing contribution is analytic in µ and is not universal, so that it can be discarded. This shows that O 13 = 0 (mod µ 2 ). One may also check that two-point correlator O 22 O 21 vanishes automatically since v * 2 = 0. Bulk one-point correlation on the disk is defined according to (2.26). Its generating function is W(l, t) = ∞ µ dx x|e lQ |x . The polynomial Q(y, v 1 ) = y 3 + v 1 y can be written Therefore, we have where on-shell value v 2 * = 0, v 1 * = (9µ) 1/3 is used.
To evaluate other one-point functions we need to compute ∂Q(y,v(x)) ∂λm,n on-shell. For this purpose we use the Douglas equation (3.7) and (3.8) This shows that In particular we find the relation between l and l 0 from (2.28): l/l 0 = 2 3/4 / √ 3.

M(q/q+1): towards the general case
Now we describe the general unitary minimal model (q, p) = (q, q + 1). Corresponding Frobenius manifold is A q−1 and the Lax operator is Q = y q + q−1 i=1 u i y q−1−i . The flat coordinates of the q − 1-dimensional Frobenius manifold are given as v i = θ i,0 = u i + · · · and the metric is η ij = δ i+j,q .
In general the structure constants are complicated. However, the bulk correlation generating function needs the structure constants c 1 ij on-shell only, which follows from the condition v * i>1 = 0. It was shown in [26] that c 1 ij on-shell has non-vanishing components when i = j only and is given explicitly as We have the action S(v) of the form where (k = m − n, j = m) or (m = j, n = j − k). The parameter t has the symmetry t The Douglas equation ∂S/∂v i = 0 has the on-shell solution v * i>1 = 0 and on-shell value of Q is given in terms Chebyshev polynomial T q [55] The derivative of the action for (v i>0 = 0) can be found explicitly using where and (a) n = Γ(a + n)/Γ(a) is a Pochhammer symbol. The bulk generating function on the sphere is where the integration path is along v i>0 = 0. In addition, the structure constant c 1 jm in (3.18) is used. Differentiating this function and requiring one-point function vanishing, diagonality of two-point function and other fusion rules, the resonance transformation were found in [26]. To this end, we introduce new integration variable w The integration now looks like In the new variable the fusion rules become the orthogonality conditions for Jacobi poly- In particular, if β = 0 Jacobi polynomial reduces to Legendre polynomial.
The generating function on the disk is given in (2.25) and its one-point correlation is given in (2.26). Explicitly one gets (if (m, n) = (1, 1)) 6 Pre-exponent is evaluated using We use the expressions for the second and third terms from the paper [26] ∂v α /∂λ m,n = −∂ v β S (m,n) (3.31) From these expressions it is clear that differentiating wrt λ mn , m > n changes only v m if m − n is even and v q−m otherwise. Using the formulae for ∂u a /∂v b from [26] we evaluate Thus, using expression (3.28) and formulae above we obtain To get one-point functions we need to integrate µ dx the results of (3.34) and (3.36). At the moment we are not aware how to perform it in the general case, however in all concrete examples we get (3.37) At first sight it appears that the second case is in contradiction with the FZZ result.
One of the reasons may be the symmetric minimal model boundary conditions chosen by matrix model integrals, however to check this we need to fix exact normalizations between MLG and Douglas equation approach. This phenomenon resembles also certain nonvanishing correlators appearing in the spherical case, which should vanish in the CFT approach (for the discussion see e.g. [25,26]). For m = n and m = n + 1, Jacobi polynomial reduces to constant and the integrals can be taken for all q and m Therefore we get if q is even, then m is odd and . Similarly for odd q we obtain We also write down the partition function where c = 1 for even q and c = cos(π/2q) for odd q. We revisit first the Lee-Yang series on the disk in [49] and discuss then the normalization effect. The Lee-Yang series M(2/p) with p = 2s + 1 is described by one-matrix model and is based on A 1 Frobenius manifold with Q = y 2 + v. The metric is trivially given by η 1,1 = 1. Nevertheless, the Douglas action is not trivial where n k = s − k ≥ 1. We use the fact that θ 1,k (v) ∼ v k+1 and put some normalization constant for later convenience. The parameter t has Z 2 symmetry t The derivative of the action is ∂S/∂v ≡ t (1,s) − x, where which presents the implicit dependence of v on x. Using the resonance transformations x was computed in [24] (including the resonance terms of t (1s) ) where P n (z) are Legendre polynomials with z = v/v 0 and v 0 = λ (11) 2(2s−1) s(s+1) . Here N k = 2 k Γ(k + 1/2)/(Γ(k + 1)Γ(1/2)) is a nomalization factor so that v k 0 N k P k (z) = v k + · · · is a monic polynomial.
The bulk generating function on the sphere is defined as (4.6) Note that v * = ±v 0 since x(±v 0 )| 0 = 0. Here we choose v * = v 0 . The one-point function O 1,n (for 2 ≤ n ≤ s − 1) is given as which vanishes due to the orthogonality of the Legendre polynomials. In the same way, two-point correlation satisfies the orthogonality.
On the disk, the generating function W(l) in (2.25) is used. However, we need a systematic way to find the perturbative contribution using the expression (4.5). For this purpose, we interpret W(l) as where we use the notation x 0 =x(v)| λ 1n =0 in (4.5) Then, the perturbative contribution is due to δλ 1n in (4.5) and its effect on δv is calculated by constraining δx = 0.
The correlation O 1,s D is simply given as where κ = v 0 . We use v * = −v 0 to make the integral convergent.
Other correlations O 1,n =1 D are given as The variation of v is obtained from (4.5): Therefore, the correlation is given as where the integration formula is used: (4.14) The one-point correlations on the disk agree with FZZ result [39].

M(3/5): new features
The first non-trivial non-unitary example is M(3/5) gravity, which is based on A 2 Frobenius manifold. It is noted that M(3/5) gravity has very different features from the unitary case. For example, the variable x vanishes at the lowest order of the perturbation. The integration path of x cannot be chosen by putting v 2 = 0 since x ∝ v 2 vanishes identically in this case. As a result, the general framework breaks in M(3/5) gravity. The purpose of this section is to check carefully how one can cure the breakdown case by case.
(11), (14) µ) . The orthogonality condition fixes the resonance coefficient:  For two-point correlations this requirement leads to the same results as in the unitary case: the resonance transformations are given in terms of Jacobi polynomials.
As in the Lee-Yang series, we use the generating function on the disk with x 0 = −v 2 v 2 1 /6. One-point correlator O 1,2 D is easily computed and has the form similar to (3.14) where v * 2 = 0 is used. This demonstrates that O 1,2 D ∝ κ 1/3 K 1/3 (lκ). Other correlators require more detailed information about x 0 and perturbation effects due to λ 1k 's. O 11 D is obtained by the perturbation of λ 11 : The variation of v 1 and v 2 due to λ 11 perturbation can be found from the string equation (4.21): δv 1 /δλ 1,1 = −(6/v 2 1 ) and δv 2 /δλ 1,1 = 0. Therefore, we have Next step is to find the path of x 0 -integration. Suppose we try to keep v 2 = 0 as in the unitary case. Then, one has dx 0 = −dv 1 (v 1 v 2 /3) and the integration has null value. This suggests that we cannot follow the general framework used in the unitary case.
Instead, we may prescribe the integration path alternatively: keep the contour path v 1 = v * 1 and v 2 ∈ [0, ∞]. In this case, using dx = −dv 2 (v 2 1 /6) we have The result is proportional to κ 2/3 K 2/3 (lκ) as expected. Noting that the integrand is not closed, the contour integration depends on the path deformation. One needs proper integration path to achieve the right expectation value. Therefore, the choice of the integration contour looks an ad hoc prescription to get an expected answer.
It is not clear at this moment whether there exists any canonical choice of the integration path. We present further examples illustrating this problem. The similar phenomenon occurs when we consider O 13 D . Varying the string equation we get Taking the same integration contour as in (4.28) we get Then the one-point function vanishes since the measure is proportional to v 2 . One possible way out is to interpret the generating function in a different way. Instead of using the variable x 0 in the generating function (4.32), we may use variable x in defining the generating function and allow the variation of the measure dx while assuming Q remains fixed so that which would lead to in agreement with the FZZ result.

Concluding remarks
We use the matrix formalism to compute bulk correlations of minimal gravity on the sphere and on the disk and compare the result with the results of the Liouville gravity approach. We use the sphere correlation generating function proposed in [25] and describe the effect of the resonance transformations for computing bulk correlators in the presence of boundary. We clarify some subtleties in the construction of the generating function if one generalize the matrix framework to M(q/p) gravity, based on A q−1 Frobenius manifold with q > 2. We demonstrate that the matrix approach and the Liouville field approach agree with each other for the one and two correlations on the sphere and for the one-point correlation on the disk.
Even though the framework of the matrix approach looks similar for the unitary and non-unitary series, the details of the calculation show some subtleties. In the unitary series M(q/q + 1), continuous variable x of the matrix model approach has the same gravitational scaling dimension as the cosmological constant µ. For the non-unitary series, the gravitational dimension of x is greater than that of cosmological constant. On the sphere, however, this subtlety is hidden because x does not come explicitly in the expression for the generating function. On the other hand, the computation of the bulk correlation on the disk requires knowledge of the specific dependence of x on the flat coordinates according to the Douglas string equation. We provide some details of unitary series in section 3 and of non-unitary series in section 4 and find the consistency of the formalism once the subtleties are well taken care of.
It is noted that on-shell condition defined in sec. 2.2, which is equivalent to a specific solution to the Douglas equation, is enough to find the correlation on the disk for unitary series, whose calculation is simplified in the flat coordinates of the Frobenius manifold. For the non-unitary Lee-Yang series M(2/2s + 1), the Frobenius manifold is one dimensional A 1 and hence is trivial. In this case, one can simply use the off-shell condition for the flat coordinate. However, for higher q non-unitary models, when the Frobenius manifold becomes multi-dimensional, the usage of the matrix model results requires more profound analysis of the off-shell condition. The considered examples show that there is no canonical way of evaluating the one-point correlation on the disk for the non-unitary models without using some additional assumptions. Even though there are certain conjectures about the choice of the contours and the off-shell application of the string equation leading to the desired FZZ result, the analytic structure of the generating function on the disk is not fully understood and needs further investigation.
Among other possible further developments of the present analysis we mention the following natural questions. The case of non-trivial boundary conditions as well as of the other types of correlations involving boundary insertions remains to be studied. In particular, the resonance transformations in this case can be affected by the presence of boundary fields and further analysis of the Frobenius manifold structure hidden beyond the matrix model formulation is required. In this context the analysis of the connection between the intersection theory on the moduli space of Riemann surfaces with boundary and the open KdV equations might be useful (see, e.g. [56][57][58][59]). Finally, we note that the new methods of the dual approach to the minimal (open and closed) strings based on the connection with the theory of Frobenius manifolds suggests that maybe some of these results can be applied for analytic computations of string amplitudes in more realistic string models. In this respect the consideration in [60], where c = 1 string theory has been considered from the worldsheet perspective and compared with the matrix models, may be relevant. To perform the x integration we use: In particular, for M(3, p) models we have R dy e il(y 3 +v 1 y) = 2v