Special geometry on the moduli space for the two-moduli non-Fermat Calabi-Yau

We clarify the recently proposed method to compute a Special K\"ahler metric on a Calabi-Yau complex structures moduli space that uses the fact that the moduli space is a subspace of specific Frobenius manifold. We apply this method to computing the Special K\"ahler metric in a two-moduli non-Fermat model which has been unknown until now.


Introduction
To find the low-energy Lagrangian of string theory compactified on a Calabi-Yau (CY) manifold X, one must know the special Kähler geometry on the moduli space of the CY manifold [1,2,3,4]. An alternative method for computing the Kähler potential in the case where the CY manifold is given by a hypersurface W0(x) = 0 in a weighted projective space was recently presented in [5]. In this paper, we briefly review this method and use it to compute the special Kähler geometry on the moduli space of a two-moduli non-Fermat threefold considered in [6]. This approach is based on the fact that the moduli space of a CY manifold is a subspace in the particular Frobenius manifold (FM) [7] that arises from the deformation space of the singularity defined by the LG superpotential W0(x). In this paper, we discuss this FM and its connection with the geometry of the CY moduli space in more detail.
It is known, that an isolated singularity W0(x) defines an object called a Milnor ring R0, Endowed with an invariant pairing this ring becomes a Frobenius algebra. Its elements define versal deformations of the singularity. It is well known that the space of deformations admits a FM structure. In our case, W0(x) is a quasihomogeneous polynomial, which defines a CY manifold in the weighted projective space. For our purposes of computation of the CY moduli space geometry we actually need the Frobenius subalgebra R Q 0 of R0, that is multiplicatively generated only by polynomials of the same degree d as W0(x). These polynomials are related to the complex structure deformations. The subalgebra R Q 0 consists of elements of degrees 0, d, 2d and 3d, and each homogeneous component of this algebra is naturally a subgroup of the respective middle Hodge cohomology H 3,0 (X), H 2,1 (X), H 1,2 (X) and H 0,3 (X). In our examples, R Q 0 is actually isomorphic to H 3 (X). The FM in which we are interested is precisely the submanifold of the FM on the versal deformations of W0(X) restricted to the deformations by the elements of R Q 0 . The relation to the FM structure allows obtaining the following explicit expression [5] for the Kähler potential in terms of the holomorphic FM metric ηµν in the flat coordinates and periods of the holomorphic 3-form Ω: Here, φ is a parameter on the moduli space, and σ ± ν (φ) are periods of the holomorphic 3-form Ω, defined below as the integrals of Ω over some special basis of cycles γµ in the homology group H3(X), which are defined in terms of the connection with the FM.
The constant nondegenerate complex matrix T connects the basis σ ± ν (φ) with another basis ω ± µ (φ) defined in [8,6] as integrals of Ω over another special basis of the cycles qν by the linear relation The explicit expression for the Kähler potential is obtained as follows. The basis ωµ is defined as integrals over the homology cycles qµ with real coefficients. These cycles have a well-defined intersection matrix Cµν = qµ ∩ qν . The matrix M and formula (1) are independent of the specific choice of ωµ. The Kähler potential K(φ) on the moduli space with the coordinates φ and the holomorphic FM metric η αβ can be expressed in the terms of Cµν , the periods ωµ, and some additional periods ωαµ(φ) (also defined below) as and Relation (3) was proved in [1,2,3,4] (see [9,10] and also [5] for the proof of relation (4)). Using these two formulas together with relation (2) between the periods ω ± µ (φ) and σ ± ν (φ), we obtain formula (1) for the Kähler potential.
In the beginning of the paper, we briefly review our method [5] for computing the moduli space geometry and clarify the role of the invariant Milnor ring R Q 0 , which was not explicit in [5]. Then we demonstrate its efficiency by computing the complex structure manifold geometry for a oneparameter family of quintic threefolds, which was previously found in [8] by an different approach. For the reader's convenience, we present our construction for this simple example, which we studied with our method in [5]. Our intention is to illustrate the relation between the Milnor ring, the invariant Milnor ring, and the cohomology for the quintic and its quotient and to show its importance for a proper understanding of the method.
We then use our method to compute a new case of the complex structure geometry for a twoparameter family of CY threefolds of a non-Fermat type considered in [6].

Special geometry
We recall the basic facts about the special Kähler geometry and how it arises on the moduli space of complex structures on CY manifolds (see [1,2,3,4]).
Let M be a Kähler n-dimensional manifold and z 1 · · · z n+1 be a set of holomorphic projective coordinates on it. Then M is called a special Kähler manifold with special coordinates z i if there exists a holomorphic homogeneous function F (z) of degree two in z, called a prepotential, such that the Kähler potential K(z) of the moduli space metric is given by Let X be a CY threefold with complex coordinates y µ (µ = 1, 2, 3) and the holomorphic 3-form Ω. The moduli space of complex structures is the space of perturbations of the metric on X that preserve the Ricci flatness of the metric. These deformations are related to the harmonic forms χa ∈ H 2,1 (X) as δagᾱβ → χ a,µνβ ∼ Ω µνλ g λᾱ δagᾱβ.
The Weil-Peterson metric on the CY moduli space is G ab = X d 6 y g 1/2 g µσ g νρ δagµνδbgσρ .
Here, a andb are indices of complex coordinates in the moduli space of the CY complex structure. It follows from the Kodaira lemma [4], that this is a Kähler metric: It is convenient to use the basis of periods that are integrals over the Poincaré dual symplectic basis By defining periods in the symplectic basis as It also follows from the Kodaira lemma that Hence, G ab is the special Kähler metric with the prepotential F (z).
Using the notation Π = Fα, z b for the vector of periods, we write the expression for the Kähler potential as where the symplectic unit Σ is an inverse intersection matrix for the cycles (A a , Ba). We can therefore transform this expression to form (3): where ωµ(φ) are the integrals of Ω over the arbitrary homology basis q µ .

The CY manifold as a hypersurface in a weighted projective space
In what follows, we concentrate on the case where the CY manifold is realized as a hypersurface in a weighted projective space or its quotient by some discrete group. Let x1, . . . , x5 be homogeneous coordinates in a weighted projective space P 4 (k 1 ,...,k 5 ) and Also let W0(x) be a quasihomogeneous polynomial defining an isolated singularity at the origin, ki.
The moduli space of complex structures is then given by (a quotient of) the space of homogeneous polynomial deformations of this singularity modulo coordinate transformations: where N is a number of polynomial deformations and es(x) are polynomials of the same weight (degree) as W0(x). They are invariant under the Z d action α · eµ(xi) = eµ(α k i xi), α d = 1.
The holomorphic volume form Ω can be written [8,6] as a residue of a 5-form in the underlying affine space C 5 : Taking this explicit expression for Ω, we obtain a basis of periods ωµ(φ) as follows [8,6]. We take a so-called fundamental cycle q1, which can be described as a torus in the large complex structure limit (LCSL) φ0 ≫ 1 (for simplicity, the other parameters φ s = 0): In the same limit, Q1 := {|xi| = δi} ∈ H5(C 5 \W (x) = 0) is a 5-dimensional torus surrounding the hypersurface W (x) = 0 in C 5 . It is chosen to represent the integral over a three-dimensional torus q1 ∈ H3(X). The fundamental period is then and is given by a residue as a series in 1/φ0.
More periods ωµ can be obtained as analytic continuations of ω1 in φ. This can be done by continuing ω1(φ) in a small φ0 region using Barnes' trick and subsequently using the symmetry of W0(x) [6].
There is a group of phase symmetries ΠX defined as a group acting diagonally on xi and preserving W0(x). When W0(x) is deformed, this group acts on the parameter space with an action A such that The moduli space is then at most a quotient {φ s } 0≤s≤N−1 /A of the parameter space. We can thus define a set of periods by analytic continuation, In examples, this construction gives all the periods of a volume form of X. We can also consider quotients X/H where H ⊂ ΠX is an admissible subgroup, which is a subgroup that preserves the volume form Ω. In this case, we should consider only H-invariant deformations and corresponding quantities.

Frobenius manifold
It is known that compactification of the superstring theory on a CY manifold is deeply connected with the compactification on the N =2 superconformal theories [14]. When the latter are Landau-Ginzburg theories, they are connected with singularity theory [15,16,17].
Using this connection, we can extract the information about the special geometry of the CY moduli space from the Milnor ring of the defining singularity W0(X) [11,18]: .
In fact, we need to consider not the whole Milnor ring but its Q := Z d -invariant subring R Q 0 . This subring is multiplicatively generated by marginal deformations of the singularity, i.e., by those deformations that have the same weight d as W0(x) and correspond to the complex structure moduli of CY. This subring consists of the elements of the Milnor ring R0 whose weights are integer multiples of d, that is 0, d, 2d, 3d. We note that in many cases, the dimension of the subring R Q 0 is equal to the dimension of the homology group H3(X), where X is the CY defined using the polynomial W0(x) in (5). But in general, dim R Q 0 ≤ dim H 3 (X). In the case of the strict inequality, we restrict our attention to the subspace of H 3 (X) that is isomorphic to R Q 0 without noting this explicitly. We let es(x) (with Latin indices s) denote the elements that correspond to the complex structure deformations of X and eµ(x) (with Greek indices µ) denote all elements of the basis of R Q 0 . In the example where X is a quintic threefold, we have the dimension dim R0 = 1024 of the whole Milnor ring, the dimension dim R Q 0 = dim H3(X) = 204 of the Q-invariant subring, and the dimension dim Mc = 101 of the subspace of marginal deformations.
There exists a natural multiplication with structure constants C σ µν in R Q 0 and a pairing ηµν that makes R Q 0 a Frobenius algebra: We consider the space of deformations of the singularity where eµ(x) belongs to the Q-invariant subring R Q 0 of the Milnor ring. The structure of the FM MF [7] with multiplication structure constants C ρ µν (t) in the ring R Q defined by the deformed singularity W (x) arises on the space with the parameters t µ : It has a Riemanian flat metric hµν (t) and the metric hµν (t = 0) equal to ηµν . The structure constants are the third derivatives of the Frobenius potential F (t). This Frobenius potential coincides with the holomorphic prepotential of the special Kähler geometry when restricted to the marginal subspace. They are naturally the Yukawa couplings of the corresponding fields [4]. We note that the FM in question is naturally a complex manifold and the metric hµν (t) and structure constants C ρ µν (t) are holomorphic.
We note that the small FM MF is a submanifold of the total FM arising from the whole Milnor ring R0. The manifold MF is connected with the cohomology of X and is used in our approach.
Only quasihomogeneous deformations W (x) define a hypersurface in a weighted projective space. The marginal deformations W0(x) + φ s es(x) define a subspace of the FM connected with W0. This subspace Mc of the FM coincides with the moduli space of the CY manifold (at least locally and maybe after some orbifolding). This fact is very important for computing the special geometry: it allows expressing the matrix Cµν in terms of the FM metric ηµν [5].

The idea for computing the periods
We can now relate the oscillatory form of the period integrals (6) to the FM structure and to the holomorphic metric ηµν .
We consider the differentials D + and D − given by They define cohomology subgroups H 5 D ± (C 5 ) w∈d·Z on the space of differential forms of the weight d · Z. These subgroups are isomorphic as linear spaces to the ring Moreover, R Q 0 acts naturally on H 5 D ± (C 5 ) w∈d·Z by multiplication (in terms of representatives). The cohomology subgroups H 5 D ± (C 5 ) w∈d·Z are dual to the homology subgroups H5( The We have a group isomorphism H 5 D ± (C 5 , Re W0(x) = ±∞) w∈d·Z = H3(X) and therefore R Q ≃ H 3 (X). This isomorphism maps the weight filtration in the left-hand side to the Hodge filtration in the right-hand side, i.e., cycles of weight k · d correspond to differential forms in H 3−d,d (X).
A possible choice of cycles Γ ± µ is A convenient computation technique can be used to find the periods represented as the oscillatory integrals Expanding the integrand in a series in φ s , we obtain integrals of the type Γ ± µ P (x)e −W 0 (x) d 5 x. Here, P (x) is a product of es(x) and is hence a Q-invariant monomial.
The technique for computing such integrals, previously used to compute the flat coordinates in the topological CFT [19,20,13,21], is based on the fact that if the forms in the integrands are equivalent in the D ± cohomology, Using this, we can easily see that an arbitrary Q-invariant form P (x)d 5 x is reducible to a linear combination of eµ(x)d 5 x, where the eµ(x) form the basis of R Q 0 . Computing integrals of the type Γ ± µ P (x)e −W 0 (x) d 5 x thus becomes a linear problem of expanding P (x)d 5 x over the basis of H 5 D ± (C 5 ) w∈d·Z .

Finding C µν and the Kähler potential
We use the relation of the CY moduli space to the FM structure to find the intersection matrix of the cycles Cµν = qµ ∩ qν = Q + µ ∩ Q − ν . For this, we introduce a few new sets of periods ω ± α,µ (φ) as integrals of eα(x)d 5 x ∈ H 5 D ± (C 5 , Re W0(x) = ±∞) w∈d·Z over the cycles Q ± µ ∈ H5(C 5 , Re W0(x) = ±∞) w∈d·Z defined above: The periods ω ± 1µ (φ) coincide with the periods ω ± µ (φ) because we assume that e1(x) = 1 denotes the unity in the ring R.
The additional periods allow computing C µν because of its relation to the FM metric η αβ [9,10]: From this formula, we can obtain the expression for C µν if we know the values of ω + α,µ (t = 0) for all α. As follows from the definition of ω ± αµ (φ), we can express ω + α,µ (φ = 0) in terms of the derivatives of the periods ω ± µ (φ) with respect to φ up to the third order at φ = 0 because the basis elements eα(x) of the invariant subring R Q 0 of the Milnor ring can be chosen as products of the marginal deformations es(x), which are related by Setting ω ± α,µ (φ = 0) := (T ± ) α µ , we rewrite the above relation as which gives the expression for the intersection matrix C ρσ in terms of η µν and the matrix T . The result can be substituted in the Kähler potential formula to obtain the explicit expression for K(φ).
Having computed the matrix T µ µ , we use (7) to express the intersection matrix C ρσ in terms of this matrix and the known Frobenius metric η µν .
We thus obtain the main statement that where the matrix M a b = (T −1 ) a cT c b . This expresses the Kähler potential K explicitly in terms of the periods σµ(φ), the FM metric ηµν , and the matrix T µ ν . All these data can be computed exactly, as explained above.
We also note two points. First, we can also find a symplectic basis of cycles by applying the Gram-Schmidt process to the obtained intersection matrix. Second, formula (10) can be used without explicitly computing T if the real structure matrix M can be found from some other argument.

Example: Quintic threefold
We consider the one-parameter family of CY manifolds defined as which was considered in detail in [8]. The phase symmetry in this case is ΠX = Z 5 5 . The full Milnor ring R0 is 1024-dimensional and consists of all polynomials in five variables where each of them has the degree less than four.
As explained above, we need not the whole Milnor ring but only its Z5-invariant subring, which has the dimension dim R Q 0 = 204 and is isomorphic to H3(X). It comprises the polynomials in R0 of degrees 0, 5, 10, and 15. The 101 fifth-degree polynomials (marginal deformations) correspond to the complex structure moduli.
To build the one-dimensional family (11), we take a subgroup H = Z 3 5 ⊂ ΠX of phase symmetries and considerX = X/H, which turns out to be a mirror of X. Family (11) is the maximal deformation surviving this factorization. The induced action A of the phase symmetry group on the one-dimensional space {ψ} is Z5 : ψ → e 2πi/5 ψ. The invariant Milnor ring is four-dimensional: Having the invariant Milnor ring, we define the corresponding cohomology group and dual cycles Γ ± µ . Using the recursion procedure for (8), we obtain The fundamental period for the quintic is a residue of a holomorphic 3-form Ω, defined as an integral over a cycle q1. Its analytic continuations give the whole basis of periods in terms of integrals over a basis of cycles ∈ H3(X): The matrix T is given by The holomorphic metric ηµν on the corresponding FM is just the coefficient of the element of maximum degree in the decomposition of eµ(x) · eν(x) in the monomial basis ofR Q 0 . In our case, it is η = antidiag(1, 1, 1, 1).
This example, which was considered in [6], is interesting because it is not of the Fermat type. 1 The weight of the singularity is equal to d = 21. The phase symmetry is Z 2 21 × Z 7 . We again consider a quotientX = X/H by the H = Z21 action H := (Z21 : 12, 2, 0, 7, 0) The Hodge numbers are h1,1(X) = 95 and h2,1(X) = 2. The two-parameter family (12) is the maximum deformation surviving the factorization. The induced action A on the two-dimensional space {φ0, φ1} is Z7 : φ0 → αφ0, φ1 → α 3 φ1, where α 7 = 1 is a primitive root. We note that 7 is a weight of the mirror singularity, as explained in [6].
Analytic continuations of the fundamental period give the full basis of periods in a basis of cycles with integral coefficients, |φ0|, |φ1| ≪ 1.
We now perform the Milnor ring computations to obtain the metric η. If we set then the H-invariant subring of R Q 0 is generated by e2 and e3. It is easy to compute the relations e 2 3 = 0 e 3 2 = 0.
The last one is of the highest degree 63, and the metric in this basis is therefore η ≃ antidiag(1, 1, 1, 1, 1, 1). Taking the first four terms of the expansion of the above periods, we obtain