Correlation Functions in Unitary Minimal Liouville Gravity and Frobenius Manifolds

We continue to study minimal Liouville gravity (MLG) using a dual approach based on the idea that the MLG partition function is related to the tau function of the A_q integrable hierarchy via the resonance transformations, which are in turn fixed by conformal selection rules. One of the main problems in this approach is to choose the solution of the Douglas string equation that is relevant for MLG. The appropriate solution was recently found using connection with the Frobenius manifolds. We use this solution to investigate three- and four-point correlators in the unitary MLG models. We find an agreement with the results of the original approach in the region of the parameters where both methods are applicable. In addition, we find that only part of the selection rules can be satisfied using the resonance transformations. The physical meaning of the nonzero correlators, which before coupling to Liouville gravity are forbidden by the selection rules, and also the modification of the dual formulation that takes this effect into account remains to be found.


Introduction
The two-dimensional (2D) theory of Liouville gravity arises in the context of noncritical string theory [1]. This theory is called minimal Liouville gravity (MLG) in the case where the target space in the string sigma action or, equivalently, the matter sector in the worldsheet theory is represented by some minimal CFT model. Being a conformal theory, MLG can be studied by the standard methods of 2D CFT [2] (see, e.g., [3][4][5][6]). There is another, dual approach based on the natural geometric interpretation of MLG models as theories describing renormgroup fixed points of 2D quantum systems on fluctuating surfaces. respectively concerned with computations the one-, two-, three-, and four-point correlation functions. Section 8 contains concluding remarks. Some computation details are presented in Appendices A, B, and C.

Dual approach to MLG and Frobenius manifold structure
In this paper, we restrict our attention to the series of unitary models M q,q+1 coupled to Liouville gravity in the spherical topology. In this case, the approach is formulated as follows. We introduce the so-called action S, which depends on q − 1 parameters u 1 , u 2 , . . . , u q−1 , where the polynomial Q(y) = y q + u q−1 y q−2 + u q−2 y q−3 + ... + u 1 y 0 . (2. 2) The set of λ kl denotes the Liouville couplings, and the functions t mn (µ, λ) are defined by the resonance transformations discussed in more detail below. The Duglas string equation [14] at genus zero [25] has the form ∂S ∂u α = 0 , α = 1, ..., q − 1 . (2. 3) The main claim of the approach is that among the solutions of this system, there exists s special solution u * = (u 1 * , u 2 * , . . . , u q−1 * ) that can be used to construct the generating function of the correlators in MLG.
It was shown in [18] that the parameters u α can be interpreted as the coordinates on the (q − 1)-dimensional Frobenius manifold such that the metric in these coordinates is given by ∂ ∂u α , ∂ ∂u β = − res y=∞ ∂Q(y) ∂u α ∂Q(y) ∂u β dQ dy . (2.4) To define the structure of the Frobenius manifold, we associated its points u with the (q − 1)-dimensional Frobenius algebras A q (u) (depending on the parameters u α ) such that (2.4) represents a pairing of the algebra elements with the invariance property that (ab, c) = (a, bc) for arbitrary elements a, b, c of the algebra. We recall that a finite-dimensional commutative and associative algebra with unity is called a Frobenius algebra if such an additional invariant pairing is defined for its elements. In the context of MLG, we deal with the A q Frobenius algebra that is the algebra of polynomials modulo the ideal generated by Q ′ (y). Using definition (2.4), all necessary properties of the Frobenius manifold (such as flatness of the metric, existence of the Frobenius potential, etc.) can be checked in the initial coordinates u α [23]. This interpretation of the parameters u α turns out to be very efficient. It was found in [20] that the problem of choosing the relevant solution can be solved by changing from the initial u to the flat coordinates v(u). More precisely, properties of the Frobenius manifolds were used to show that the relevant solution of the string equation v * in the flat coordinates becomes v (0) * = (v * 1 , 0, 0, . . . ) in the limit where the Liouville coupling constants are equal to zero. This solution is unique, i.e., only this solution gives the generating function (defined below) for which the necessary selection rules are satisfied on the level of one-and two-point correlators: it gives zero vacuum expectation values of the physical fields (except unity), and the two-point correlators are diagonal.
2.1. Flat coordinates. It follows from the properties of the Frobenius manifold that there exists a one-parametric deformation of the flat connection defined as follows [23]. The deformed coordinates 3 are given by θ α (z) = ∞ k=0 z k θ α,k such that θ α (0) = v α . The transformation from the initial coordinates to the flat coordinates is defined by and (a) n = Γ(a + n)/Γ(a) is the Pochhammer symbol. The flatness condition is equivalent to the recurrence relation The corresponding deformation of the Levi-Civita connection is given by the modification of the Christoffel symbols The metric in the flat coordinates has the simple form It is highly nontrivial that deformed coordinates (2.5) can be regarded as local densities of the commuting Hamiltonians of the integrable Gelfand-Dikij hierarchies. An important consequence of this fact [18] is that the MLG generating function is just the logarithm of the tau function with the simple representation (2.10) Here, the upper limit v * is the appropriate solution of the Douglas string equation The integral in (2.10) is independent of the integration contour, which means that the integrand is a closed one-form. The structure constants of the Frobenius algebra in the flat coordinates C βγ α = C α,q−β,q−γ here are In particular, because C 1αβ = η αβ , we obtain It follows from the definition of the Frobenius manifold that there exists a function (2.14) Before discussing the correlation functions, the following remark is in order. The structure constants in the flat coordinates are currently unknown, but we find that with the form of generating function (2.10) and the properties of the relevant solution v * taken into account, the general expression for the structure constant is not needed for calculating the correlation function. Instead, we need the coefficients of the expansion We are here interested in the three-and four-point correlation numbers. The necessary results for the first two terms of (2.15) are presented below. For a short representation, we introduce the function χ A,B (x) = 1 if x ∈ [A, B] and zero otherwise. In the zeroth order, the structure constant in the v coordinates on the solution of the string equation itself is The first derivative ∂ ρ C αβγ is given by In (2.16) and (2.17), N is the set of nonnegative integers, and we assume the ordering ρ ≥ α ≥ β ≥ γ. Because both tensors are symmetric, this information provides a complete answer. Some details of the derivation are given in Appendix A.

Resonance transformations
In the continuous approach the integrated correlators of MLG are defined up to so-called contact terms, which are not determined by CFT methods [17]. On the other hand, any change of contact terms is equivalent to resonance transformations of the coupling parameters. Explicitly, the resonance transformation has the form and hence λ mn ∼ µ δmn . By regrouping the terms, we can write action (2.1) in the form In what follows, we call the coefficients in expansion (3.3) counterterms. 5 All required information concerning correlation functions is encoded in generating function (2.10). Namely, the correlation numbers are related to the coefficients in the coupling constant decomposition of the generating function Z = Z 0 + m 1 ,n 1 λ m 1 n 1 Z m 1 n 1 + m 1 ,n 1 ,m 2 ,n 2 λ m 1 n 1 λ m 2 n 2 Z m 1 n 1 ,m 2 ,n 2 + . . . . (3.4) In what follows, we often use the short notation λ m i ,n i = λ i and Z m i n i ,m j n j ,... = Z ij... .
Lemma 3.2. The first-order counterterms are given by

5)
where m = m 1 , n = n 1 , and A Proof. We seek solutions of the dimensional balance requirement δ m 1 n 1 + δ m 2 n 2 + N = δ mn , (3.9) where N is a nonnegative integer and n ≤ m ≤ q −1. It is convenient to set d = m−n and d i = m i − n i . Explicitly, this requirement gives (3.10) (1) Let m 1 +m 2 < q. If m+1 = q, then there is no solution because the noninteger factor m 1 +m 2 q cannot be compensated. If m + 1 < q, then This gives m = m 1 + m 2 − 1 and n = n 1 + n 2 + 1 + 2N. (2) Let m 1 +m 2 = q. If m+1 < q, then there is no solution because the noninteger factor m+1 q cannot be compensated. If m + 1 = q, then This gives again m = m 1 +m 2 −1 and n = n 1 +n 2 +1+2N, where m 1 +m 2 = q, and this case can hence be joined with the preceding case.

Partition function and one-point correlators
There is much evidence that the approach based on the Douglas string equation formulated in the preceding section provides an alternative description of MLG. A general proof of this statement is not yet available, but this conjecture can be checked by comparison with the results of computing directly in the framework of the continuous approach. Below, we provide further support of the hypothesis by performing some checks on the level of three-and four-point correlators. The last case is most important because the continuous approach here first requires nontrivial integration over the moduli space [4].
For convenience, we start our analysis of the correlation functions by presenting the results for the zero-, one-, and two-point correlators found in [20]. The consideration in this and in the following sections is based on the following statement.

Partition function.
To define normalization independent quantities (or invariant cross ratios), we need the explicit form of the partition function where v (0) * denotes the zeroth-order term of the expansion in coupling constants of the solution of the string equation.
To find it, we write the zeroth-order term of the expansion of the action explicitly in terms of deformed flat coordinates (2.5), It can be seen from (4.1) that equations (2.11) for α < q − 1 are solved automatically by the ansatz v * = (v * 1 , 0, 0, . . . ), and we are hence left with the equation This equation defines the zeroth-order term in the expansion of the appropriate solution of the string equation. The integration contour in (4.3) can be taken along the axis v 1 because on the line ∂S (0) ∂v k = 0 for k > 1, we are left with the term containing C q−1,q−1 q−1 = 1. An explicit calculation gives 4.3. One-point functions. For the general one-point correlators, we obtain Taking into account that C q−1,β α = δ αβ on the line v k>1 = 0 and using the results in the preceding subsection, we more explicitly obtain Based on Lemma 4.1, we can conclude [20] that the one-point correlation numbers are equal to zero for all fields. 6

Two-point correlators
An essential consistency requirement is that the two-point correlators be diagonal. This allows defining the first-order counterterms in the resonance transformations.
Differentiating (2.10) twice, we obtain (5.1) The second term can be nonzero only for the correlators that have integer gravitational dimensions, which we do not consider. The first term gives It was found in [20] that for v i>1 = 0, where the new variable (t) are the Jacobi polynomials (see Appendix B), and N mn denotes t-independent factors. Its explicit form is not relevant for our further consideration. With this result, the diagonality condition for the two-point correlators, becomes equivalent to the orthogonality condition for the Jacobi polynomials. Calculating the diagonal two-point functions is straightforward,

Three-point correlators
One important change when we proceed to the level of three-point functions is that the derivative of the upper integration limit should be taken into account. Also for the first time, the contribution of the second-order counterterms arise on the level of three-point correlation functions. Using the same arguments based on the string equation and nonanalyticity requirements, we obtain the expression where we replace the indices (m i , n i ) with index i and the sum ranges permutations of the set {1, 2, 3}. In (6.1) and below, we always assume that the nonintegral part is evaluated on the solution of the string equation for all couplings equal to zero. In the rest of this paper, we use Latin indices exclusively in this sense, and the index zero means the zeroth-order term in the coupling constants expansion. The other terms in (6.16) disappear because they contain ∂S (0) ∂v α (v * ) = 0. It follows from the string equation that can be calculated using (4.4). The second term does not contribute to M αβ , because For the first term, we use (4.1), With this result, we can easily find the inverse matrix T γβ : The three-point function becomes where the second term is evaluated on the solution of the string equation.
6.1. Fusion rules and three-point functions. We first formulate a useful consequence of the fusion rules for the three-point function in unitary minimal models M q,q+1 . We recall that the primary fields Φ mn are labeled by m = 1, . . . , q − 1 and n ≤ m. For the three point function where : 2 : denotes step two. We can classify different cases with respect to the parities of 3 i=1 m i and 3 i=1 n i . We can easily see the following consequence of the fusion rules for the unitary minimal models. Proposition 6.2. In the case where both i m i and i n i are even, the three-point function G = 0.
Indeed, for any choice of the reflection images of the fields, one of the parities i m i or i n i is even, which is forbidden by (6.10).
6.3. Comparing with MLG results. For the physically relevant nonanalytic correlators, there are two possibilities: (1) The fusion rules are not satisfied, the case nonphysical below. In this case, (6.7) must give zero whenever the resonance transformations permit it. This requirement allows defining the second-order counterterms S (12) . (2) The parameters {m i , n i } of the three-point function Z 123 satisfy the fusion rules described above. We call this the physical region. In this case, (6.7) must give a result the same as the result derived in the continuous approach [4]. This can be achieved if the two conditions are satisfied: (a) The integral part of (6.7) in this region is zero.
(b) The nonintegral part of (6.7) is nonzero and leads to the correct answer for the universal ratios. Without loss of generality (interchanging the pairs if necessary), we can fix 2 ≤ m 1 ≤ m 2 ≤ m 3 . The general analysis of the three-point sector requires considering four different domains: all three fields are even, one field is odd, two fields are odd, all three fields are odd. Below, we analyze (1) and (2) in detail in the domain where all fields are even. Moreover, we impose the additional constraint m 12 ≤ m 13 ≤ m 23 ≤ q. (6.11) We note that the three-point functions are always nonanalytic in this domain. Indeed, the dimension where m 1 + m 2 + m 3 ≤ 2q − 1 and hence (6.12) is not integer.
6.3.1. Nonphysical region. Using the explicit form of the structure constants and the properties of first-and second-order counterterms (5.3) and (3.8), we write the integral part IP of (6.7): To find the second-order counterterm, we take 14) It can be seen that the terms with permutations disappear. Before we use change (5.4), it is convenient to express the second-order counterterms also in terms of the dimensionless functions X (m 1 n 1 ,m 2 n 2 ) (t): The explicit form of the nonintegral part NIP in (6.7) is Combining the integral and nonintegral parts, we obtain Because m 3 − n 3 is even, n 3 − n 1 − n 2 should be odd: We consider the region n 3 ≥ n 1 + n 2 , where the three-point function should be zero according to the fusion rules. Because of the parity requirement, n 3 takes values from n 1 + n 2 + 1 to m 3 with step 2 in this region. The degree of the Jacobi polynomial in (6.17) then changes from 0 to deg X (12) with step 1. Taking the completeness property of the Jacobi polynomials into account, we obtain where the coefficients are fixed (using orthogonality (B.1)) from the requirement Z 123 = 0. Equation (B.3) can be used to obtain the representation 7 (6.21) 7 We note that according to the results in Lemma 3.4, m 1 + m 2 − n 1 − n 2 ≥ 2. Hence, all we need to verify is the case m 3 = m 1 + m 2 − 1. From the fusion rules, we have |n 1 − n 2 | + 1 ≤ n 3 ≤ n 1 + n 2 − 1 , (6.24) where the choice of the right-hand side takes n i ≤ m i and (6.11) into account. From (6.18), we derive and using the completeness and orthogonality properties of Jacobi polynomials, we conclude that the integral part is equal to zero. The final step is to check the result for three-point universal ratio. For general (q, p) [4], it has the form where . . . = ... 1 . For p = q + 1, this expression coincides with the three-point universal ratio obtained in the dual approach Here, we use (6.16), (5.6), and (4.7).

Four-point correlators
In this section, we perform some checks for the four-point correlators. We are mainly focused on the following puzzle. From the preceding consideration, it might appear that the role of the parameters n i is somewhat suppressed with respect to the role of the parameters m i . Indeed, apart from the normalization, it seems that all they do is choose the region according to the parities m i − n i , and the values of n i seem irrelevant. Below, we demonstrate how the balance between m i and n i is recovered on the four-point level. 14 The general expression for the four-point correlator is and We consider the case where there are no higher-order counterterms starting from the second order and the first derivatives of the structure constant are zero, 9 and the term with ∂C αβ is hence absent. Under these assumptions, only the first two terms in (7.2) survive: α .

(7.5)
Similar to (6.3), we can find second derivatives of the solution of the string equation Combining all together, we obtain the structure of the four-point correlator: δ . 8 To make our formulas less cumbersome, we just remember that the resulting expression is to be calculated on the solution of the string equation. Therefore, in particular, there is no term containing S Even without the contribution of the higher counterterms, we see a few new objects in this expression that require additional calculations. The details of the calculations are in Appendix C. For the third derivative S (0) ρση , we obtain (7.8) and the second derivative S (i) αβ is given by where there is no summation over m and In the calculation, we find the three basic structures θ(γ, χ, ξ)θ(q−γ, µ, ν) , (7.12) γ θ(γ, χ, ξ)θ(γ, µ, ν) . (7.13) Here, each θ(α, β, γ) is related to one of the structure constants in (7.7), (7.8), or (7.9). We recall that it is explicitly defined as a symmetric tensor such that θ(α, β, γ) = After some computations (the details can be found in Appendix C), we obtain In terms of these functions with the convention R i = R m i n i , the four-point correlator is Here, the first four terms come from the terms in (7.7) containing second derivatives of the first-order counterterms, and the last two terms come from the term with the third derivative of the action S (0) . The overall normalization factor 10 is For the four-point correlators, the universal ratio, which is independent of the normalizations, can be constructed from (7.17), (5.6), and (4 .7): This result is to be compared with the four-point correlator in MLG calculated in [4] using the standard continuous approach, (7.20) where p = q + 1 for the unitary series. One further point should be noted. Expression (7.20) also has some restrictions that should be taken into account. In particular, the active field for which the operator product expansion is used must have the smallest product m i n i among the four pairs. Moreover, the number of the conformal blocks [4] must be equal to this number. This restriction together with the requirement that higher counterterms be absent can be satisfied for general q if we consider symmetric correlation functions of the form O 4 mn norm . We note that in this case, expression (7.7) (only partially symmetric) becomes completely symmetric with respect to the permutations of the fields, as it should.
To give some reference points, we quote a few results for the gravitational Ising, tricritical Ising, and three-state Potts models corresponding to M 3,4 , M 4,5 , and M 5,6 . Two nontrivial completely symmetric four-point correlators are the correlator of four 10 Here we suppress usual dimensional factor v δ1234 * 1 . spin-density operators σ = Φ 12 and the correlator of four energy-density operators ǫ = Φ 13 dressed by the appropriate Liouville exponential fields. We find that the two expressions (7.19) and (7.20) give the same results: We note that to obtain these results, we must, as we fixed in the very beginning, take pairs (m, n) with m ≥ n in (7.19) and their reflection images (q − m, p − n) in (7.20).

Conclusions
We have partially analyzed the three-and four-point correlation functions using the dual approach to MLG. In the domain where the fusion rules are satisfied, we found agreement with the results of the continuous approach.
According to the results in Lemma 3.2 and Lemma 3.4, a rigorous analysis requires much more systematic classification. For three-point functions (6.1), for example, there exist four possible regions, depending on which interval contains the parameter q, for example, m 12 ≤ q ≤ m 23 , etc., where m ij = m i + m j and i = 1, 2, 3. Each region, in turn, contains eight subregions according to the parities of m i − n i . For all subregions, we must check whether the corresponding three-point function is analytic, check the fusion rules, calculate the second-order counterterms, and finally compare with the results of the continuous approach.
Even this partial analysis reveals the following problems. As we saw, it turns out that only a special part of the selection rules can be satisfied using the resonance transformations. Hence, the selection rules of minimal models become modified after coupling to Liouville gravity. The nature of this phenomenon is not yet clear. It is natural to assume that a possible modification of the method is to require that satisfying this special part is a necessary condition. Indeed, the selection rules uniquely fix the form of the counterterms arising on a given level in the nonphysical region. On higher levels, these counterterms already enter the expressions for the correlators in the physical region, i.e., in the region where the fusion rules are not violated, and must therefore coincide with the results in the continuous approach. We plan to check this conjecture in the near future.
Another interesting question is to explain the nature of the resonance transformations from the standpoint of the Frobenius manifold structure. We believe that answering this question may help in finding a possible modification of the A q Frobenius manifold such that this modified version would be connected to MLG without using the resonance transformations.
Acknowledgements. I am grateful to A. Belavin, B. Dubrovin, and Yu. Rud for the useful discussions. I thank Professor K. Narain for the hospitality during my visit to ICTP in 2014 and the organizers of the 4th Workshop on Geometric Correspondences of Gauge Theories at SISSA. The study of the aspects of MLG theory connected with Frobenius manifolds was supported in part by the Russian Foundation for Basic Research (Grant No. 13-01-90614). The study of the form of the resonance transformations and the computation of the correlation functions was performed with the support of Russian Science Foundation (Grant No. 14-12-01383).
Appendix A. Details of the computations of the structure constants We start with a few comments on the multiplication law on the cotangent bundle in the initial coordinates u i , On the Frobenius manifolds A q , we can construct so-called canonical 11 coordinates w i such that the metric is diagonal (but not constant) in w i . Multiplication of the tangent vectors in the canonical coordinates has the simple form On the cotangent space, if we define dQ(z) = du 1 z q−2 + du 2 z q−3 + · · · + du q−1 (A.3) using canonical coordinates, then we can easily verify the useful multiplication law property In the left-hand side of (A.4), we have where we introduce the function and u −1 = 1, u 0 = 0. On the other hand, the metric in flat coordinates is simple, and lowering an index α is just replacing it with q − α. Hence, We can write the expansion in the vicinity v * (δv = v − v * ): where the coefficients can be found in terms of binomial coefficients, In (A.14)-(A.17), if the conditions are not satisfied, then the corresponding values are equal to zero. In particular, from (A.14), we obtain u(v * ), In this notation, we have Some manipulations with the binomial coefficients give (2.16) and (2.17).
Appendix B. Some properties of the Jacobi polynomials The polynomials P (0,b) n (t) satisfy the orthogonality condition In the standard normalization, P (1) = 1, and the highest coefficient is The Jacobi orthogonal polynomials P (1) = 1.

21
Below, we list properties of the Jacobi polynomials that reveal a more transparent structure of the second-order counterterms: where the second equation can be derived from the first using integration by parts. Another useful property of the Jacobi polynomials is d dt Appendix C. Details of the calculation of the four-point correlator Third derivatives of the action. We first discuss the third derivative S ρση . Using the definition and taking our basic recursion into account, we obtain The WDVV requirement for the structure constants of the Frobenius algebra (which can be easily verified in our case) gives With this result, it is easy to find Because we are left with only the derivative with respect to v 1 , we can now set v k>1 = 0 and use (2.16) for the structure constant. After lowering the indices, we obtain expression (7.8).
Second derivatives of the first-order counterterms. We now discuss the second derivative S (i) αβ . We can split the calculation schematically into two parts. First, we find We then use the same trick as for S (1) = (b + k + 1)k 2 .
(C.5) 22 We now prove (C.4). Using the explicit form of the first-order counterterms for even m − n, we write the expansion in terms of Jacobi polynomials, The results of both calculations, for both even and odd m − n, gives (C.4).