Flat structures on the deformations of Gepner chiral rings

We propose a simple method for the computation of the flat coordinates and Saito primitive forms on Frobenius manifolds of the deformations of Jacobi rings associated with isolated singularities. The method is based on using a conjecture about integral representations for the flat coordinates and on the Saito cohomology theory. This reduces the computation to a simple linear problem. We consider the case of the deformed Gepner chiral rings. The knowledge of the flat structures of Frobenius manifolds can be used for exact solution of the models of the topological conformal field theories corresponding to these chiral rings.


Introduction
In this paper we study flat structures [1] on Frobenius manifolds (FMs) [2] arising on the deformations of isolated singularities. The physical interest to the singularity theory [3] is motivated by its connection [4,5] with the chiral rings of N = 2 superconformal field theories (SCFTs).
The models of N = 2 SCFT, classified and explicitly constructed in [6], are important due to their role in string theory. It was shown in [7] that N = 2 superconformal symmetry on the world sheet after compactifying six of the ten dimensions is the necessary condition for the space-time supersymmetry. The requirement of N = 2 supersymmetry in the compact sector is equivalent [8] to the geometrical condition [9] of the compactification on Calabi-Yau (CY) manifolds. The massless sector of string theory corresponds to the set of chiral fields of N = 2 SCFT and after Witten's twist [10] it is described by the topological CFT (TCFT). The properties of the Lagrangian of the sigma model describing this massless sector are completely defined in terms of the CY manifold and its mirror pair [12,13].
The FM structure in TCFT was initially discovered in [14] and later mathematically developed in [2]. In [14] it was shown that the exact solution of TCFT, that is the computation of the correlation functions, is reduced to the computation of the flat coordinates on the corresponding FM. This is a specific property ofÛ(2) k ×Û(1) 1 /Û (1) k+1 models considered in [14]. In the general case of N = 2 SCFT, associated with some isolated singularity one needs also to know the Saito primitive form [1].
It was established recently [15][16][17][18][19][20][21][22][23] that the knowledge of FM structure allows to find exact solution of a special class of models of non-critical string theory (or models of Minimal Liouville gravity) [24][25][26][27]. It is natural to assume that the same approach can also help to solve the W generalizations of Liouville gravity. The solution of the Douglas string equation [28], required for the construction of the free energy, takes simple form only in the flat coordinates on the associated FM [19]. Otherwise the integral representation of the free energy is too complicated to be of any practical use.
The explicit form of the flat coordinates is important also for the solution of the TCFT models related to the Gepner chiral rings [29]. It was shown in [29] that these models, described by the Kazama-Suzuki (KS) cosets A(N, k) =Û (N) k ×Û (N − 1) 1 /Û(N − 1) k+1 , are connected with a certain class of isolated singularities. The super-potentials in this case is (1.1) Thus, the computation of the flat coordinates and the Saito primitive form for the Frobenius manifolds represents a significant part of the solution of the above mentioned problems.
In this paper we suggest a new simple method for computing flat coordinates on the FM arising on the deformations of the isolated singularities. This problem was discussed earlier in [1,[30][31][32][33]. Our approach is based on the conjecture about integral representations for the flat coordinates and on the Saito cohomology theory [1]. This method allows to reduce the problem of the computation of the flat coordinates and the Saito primitive form [1] to a linear problem. Namely, we show that the coefficients of the decomposition of the flat coordinates in a sum of monomials, constructed from the deformation parameters, are simply connected with the coefficients of the decomposition of the dual to these monomials differential forms in a sum over basis elements in the space of the Saito cohomologies.
We apply this approach to the computation of the flat coordinates and the primitive form for Gepner chiral rings in the particular case of A(3, k) =Û (3) k ×Û (2) 1 /Û (2) k+1 models. This series is interesting because, unlike previously studied A(2, k) series, in this case the marginal and irrelevant deformations are present in the spectrum. We demonstrate the general construction in the particular case k = 3 and find explicit formulae for the flat coordinates and the primitive form in terms of hypergeometric functions. The linear system for the computation of the flat coordinates together with the conjecture about the integral representation for them is the main result of the present paper.
The rest of this paper is organized as follows. In section 2 we discuss the Jacobi rings and recall some known facts about Frobenius manifolds and their relations with the Saito theory. In section 3 we develop our method for computing the flat coordinates and discuss the problem of finding the primitive form. We formulate here the conjecture about the integral representation for the flat coordinates for isolated singularity, whose Jacoby ring includes the marginal deformations in addition to the relevant ones. Further, we describe the linear system for the computation of the flat coordinates in terms of the deformation parameters. In section 4 we implement the proposed procedure to calculate the flat structure of the FM associated with the Gepner chiral rings and present the details of the computation for the SU(3) 3 case. In section 5 we conclude by summarizing and discussing some future perspectives.

Singularities, Jacobi rings and Frobenius manifolds
Here we recall the main facts about the connection between Frobenius manifolds and topological conformal field theories relevant for our further discussion.
Dijkgraaf, Verlinde, and Verlinde (DVV) have found [14] exact solution for the minimal models of TCFT. These models are Landau-Ginzburg models whose superpotential The correlation functions in these models are defined by the free energy (also called frobenius prepotential) F (s 1 , ...s k+1 ), where s µ are the coupling constants of the theory and s 1 corresponds to the unity operator. The perturbed three-point function is given by It has been shown in [14] that because of remarkable properties of TCFT all the information about arbitrary n-point correlation functions is encoded in the two basic objects: the three-and the twopoint functions. The perturbed two-point function η αβ given explicitly by where Φ α (z) is a chiral field and Φ 1,1 σ (z) is its superpartner. Using CFT Ward identities it has been proven in [14] that η αβ and C αβγ (s) should satisfy to the following set of the constraints Here, in the second relation (associativity condition), raising indices is performed by the inverse tensor η λρ . From the eqs.(2.5-2.8) it follows that C λ µν is a structure constant of some Frobenius algebra 1 In particular, one of the main properties of the Frobenius algebra, eq.(2.3), follows from (2.5) and(2.8). These relations mean that we are dealing with a Riemannian manifold, provided with some additional algebraic structure, which is compatible with the Riemannian one. The parameters s α play role of the coordinates on this manifold and η ρν defines its metric. We note that the third relation (2.7) states that this manifold is flat. In fact, the relations (2.5-2.8) define the structure of the Frobenius underlying TCFT.

Dubrovin axioms
The parameters s µ define a special choice of flat coordinates on M. The metric η µν in these coordinates is constant. In some general coordinate frame t = {t µ } the Frobenius manifold axioms look like where ∇ µ is the covariant derivative and R µνλσ is Riemann curvature tensor and the structure constants in the new coordinates are denoted by C γ αβ (t). It has been proven in [14] for the case of the minimal models that the constants C λ µν (s) are the structure constants in the flat coordinates of the Fobenius manifold connected with the deformed isolated singularity where e µ (x) = x µ−1 and the deformation parameters t µ represent some coordinates on the FM. The parameter t 1 here and in what follows corresponds to the unity operator e 1 of the Jacobi ring.
In the t-frame the structure constants are defined by e µ e ν = C λ µν (t)e λ mod ∂W /∂x . (2.14) The metric is where Ω is the Saito primitive form. In the case of minimal models According to the statement formulated above, in order to solve the model we need to know the flat coordinates s µ (t) on the FM as functions of the deformation parameters. If the metric is flat, the flat coordinates s µ (t) are given by the solutions of the equation where Γ γ αβ is Levi-Civita connection for g αβ (t). The free energy F is defined from and C µνλ ∂s µ ∂t α ∂s ν ∂t β ∂s λ ∂t γ = Res x=∞ e α e β e γ Ω ∂W /∂x 1 · · · ∂W /∂x n .
(2. 19) In fact, the eqs. (2.18) and (2.19) give us the exact solution of the minimal models of TCFT as soon as we know the explicit expression for the flat coordinates as functions of the deformation parameters.

K. Saito primitive form
Let W 0 (x 1 , ..., x n ) be a polynomial associated with an isolated quasihomogenous singularity with integer weights By definition, the corresponding Jacobi ring is given by In the analogous way as in the case of one variable, the versal deformation of the ring R 0 is given by where e α , α = 1, ..., M, represent a basis of the corresponding deformed Jacobi ring and M is the Milnor number (the dimension of the Jacobi ring) We denote the weights of the homogeneous basis elements [e α ] = σ α . In our convention e 1 is the unity in the ring. The following dimensional classification will be relevant. The elements e α with σ α < d, σ α = d and σ α > d are called respectively relevant, marginal and irrelevant deformations. The generalization of the eq.(2.14) is straightforward, while the analogue of (2.15) is where the primitive form We will use the following important statement proved by M. Saito [34].
Theorem. If W 0 is an isolated singularity, then there exists such a n-form Ω[x] or, equivalently, a function λ(x, t), [λ] = 1, that the above defined C γ αβ and g αβ satisfy all FM axioms (2.5-2.8). We note that in the DVV case, λ(x, t) = 1. In the general case, if there are marginal and/or irrelevant deformations, λ(x, t) = 1.
From this theorem it follows that there exist flat coordinates s . Additionally, the coordinate s 1 (t) is fixed by the condition that it represents a coupling of the unity operator, that is As a consequence of this definition, one gets 3 Approach for the computation of the flat coordinates

Integral representation for flat coordinates on FM
It was argued in [33], using the connection between the Jacobi rings of the deformed singularities and the Gauss-Manin (GM) systems, that the flat coordinates s µ considered as functions of the parameters t µ accept a simple integral representation. It was verified in [33] for the case of simple singularities, that is when the Jacobi ring includes only relevant deformations.
Here we conjecture that similar integral representations exist also in general case when the Jacobi ring includes the marginal deformations and probably irrelevant ones. The integration contours should be closed and chosen to belong to a basis in the homologies H n of C n \ {W = 0} In what follows we focus on the case where all elements e α of the basis of the Jacobi ring are relevant or marginal. In this case Ω(t, x) is given by (2.25), λ(t) is a function of only those parameters t α which correspond to the marginal operators and does not depend on x i . Therefore (3.1) can be rewritten in the following form On the other hand, from the dimensional arguments it is obvious that in general form In what follows we show how to compute the coefficients C µ ( m).

Saito cohomologies and exact expressions for Flat coordinates
For explicit computations of the integral in (3.1) we will use the following two facts. Firstly, two following integrals are equal where the K. Saito differential 3 and U is some (n − 1)-form For convenience reason in (3.1) and in what follows in order to avoid additional brackets we replace sometimes superscripts by subscripts t α ≡ t α and s α ≡ s α . This operation has nothing to do with raising and lowering indices by the metric. 3 Here d is the ordinary de Rham differential. We note that D 2 W0 = 0. and contour γ is closed. Secondly, the Saito differential defines cohomologies on the space of the differential forms. The dimension H n -Saito cohomology group of the highest degree is equal M, i.e., the dimension of the Jacoby ring of the isolated singularity W 0 . As a basis H n can be choosen the n-forms e µ dx 1 ∧ ... ∧ dx n or e µ dx in short, where e µ is a basis of the Jacobi ring. Since the forms P (x)dx 1 ∧ ... ∧ dx n belong to H n they can be decomposed in this basis Such an expansion exists and is unique. We will use this fact to find the coefficients C µ ( m) from the equation This equation can be written more explicitly as Assuming that the basis contours γ µ are chosen to be dual with elements e µ dx in the basis of the cohomology space, i.e., γµ e ν dx = δ µ,ν , (3.12) and substituting the RHS in (3.10) into the integrals in (3.1) we arrive to explicit expression for the flat coordinates where the coefficients C µ ( m) are unique solution of the system of linear equations (3.10).
If the isolated singularity is homogeneous, the (n − 1)-form U can be represented as a sum of homogeneous terms L l=0 U l and the equation (3.10) takes the form dW 0 U 0 = λ e m λ λ dx , (3.14) where l = 1, ..., L and L = m α − 1. In the next section we will demonstrate how this procedure works in the case of the Gepner rings. It follows [29] from the fact that these chiral rings are connected with the isolated homogeneous singularity and are isomorphic to the Jacoby rings where x 1 , ..., x N −1 are elementary symmetric polynomials of q 1 , ..., q N −1 : Note that the dimension of the ring is Taking into account symmetric property of the ring space it is clear that Schur polynomials can be chosen as a natural homogeneous basis in R 0 (N, k). We remind that λ in (4.5) stands for the

Computation of the flat coordinates for SU (3) 3 chiral ring
For k ≥ 3 the consideration of the models associated with the Gepner chiral rings (4.2) become more involved because marginal deformations appear in the spectrum. Let us consider the particular case k = 3 where W 0 (x 1 , x 2 ) = 1 6 (q 6 1 + q 6 2 ) . For convenience we will use the following identification e 1 ≡ e ∅ = 1 , e 2 ≡ e = q 1 + q 2 , e 3 ≡ e = q 1 q 2 . e 4 ≡ e = q 2 1 + q 1 q 2 + q 2 2 , e 5 ≡ e = q 1 q 2 (q 1 + q 2 ) , e 6 ≡ e = q 3 1 + q 2 1 q 2 + q 1 q 2 2 + q 3 2 , e 7 ≡ e = q 2 1 q 2 2 , e 8 ≡ e = q 1 q 2 (q 2 1 + q 1 q 2 + q 2 2 ) , e 9 ≡ e = q 2 1 q 2 2 (q 1 + q 2 ) , e 10 ≡ e = q 3 1 q 3 2 . (4.7) After the versal deformation the potential reads W (q 1 , q 2 , t) = q 6 1 + q 6 Note that in this particular case there is only one marginal deformation which is presented by e 10 , all other perturbations e 1 , ..., e 9 in (4.8) are relevant. Therefore we have to allow the primitive form (2.25) to be a function of the dimensionless coupling t 10 . The coordinates in the constant metric coordinate system are given by the following general expression where Σ µ is defined in (3.2). In accordance with our main conjecture (3.1) the coefficients C µ (m λ |n) are fixed by the following condition λ e m λ λ e n where 11) and the sum in (4.10) is over those µ, which are compatible with the dimensional restriction (3.2) for given set of m λ . The 1-form U can be conveniently represented as such that B(q 1 , q 2 ) = A(q 2 , q 1 ). More explicitly we find Taking into account the argumentation of section 3, we can transform this relation into the set of recurrence relations for its decomposition into homogeneous pieces where L is defined from the condition The first relation of the recurrence prescription (3.14-3.16) reads The second one gives In our first example we compute the contribution to s 1 which contains t 1 . It has the following form s 1 = λ(t 10 )f 1 (t 10 )t 1 + ... , (4.39) where dots stand for possible contributions of other t ν with ν = 1. Our goal now is to find f 1 (t 10 ). Since e 1 = 1, we have to solve It is convenient to derive the following recurrence relation for the coefficients C(n + 1) = P (n)C(n) . Since e 10 = q 3 1 q 3 2 , we get q 6n+6 1 q 6n+6 2 (q 1 − q 2 ) = q 6n 1 q 6n 2 (q 1 − q 2 ) + ... , (4.44) which is to be presented in the form ] . (4.45) The coefficients are easily found A 0 = −q 6n+6 which in turn is to be equal to (4.52) Our second example demonstrates the computation of the flat coordinate  It follows that where together with the statement about the form of the flat coordinates, (3.1), (3.10), are the main results of this paper.
An important feature of these models is the presence of marginal and irrelevant perturbations. The proposed method allows to systematically analyse such situations. It gives the recurrence formula for the flat coordinates and the method to construct the primitive form. This result gives the complete solution of the model under consideration. Among the open questions we would like to mention the following. Since the mathematical proof of the main conjecture is lacking, the direct computation of the flat coordinates, in the spirit of [35], is desirable. In the case where only relevant and marginal perturbations are presented, the check was performed in [38]. It is important to investigate the case of irrelevant perturbations.
Our results can be used in order to define the generating function of the physical amplitudes in the minimal models of 2d W gravity. Note that the generalization of the discrete formulation of the matrix models possessing W symmetry is not available at the moment. It is also worth noting that the analogue of the continuous approach, based on the explicit construction of the correlators of the chiral fields, is also absent. 4 Finally, maybe the most interesting is the possibility to investigate a new class of Calabi-Yau geometries, using the approach discussed in this paper.