Special geometry on the 101 dimesional moduli space of the quintic threefold

A new method for explicit computation of the CY moduli space metric was proposed by the authors recently. The method makes use of the connection of the moduli space with a certain Frobenius algebra. Here we clarify this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point.

where z a = Aa Ω, ∂F ∂z a = B a Ω are the period integrals of the holomorphic volume form Ω on X. Here Aa and B a form the symplectic basis in H3(X, Z). We can rewrite the expression (1) for the Kähler potential using the periods as where matrix (Σ) −1 is an intersection matrix of cycles Aa, B a equal to the symplectic unit. In practice, computation of periods in the symplectic basis is a very complicated problem and was done explicitly only in few examples [5,6,7,8]. It is due to the fact, that it requires a case by case analysis and geometric description of the symplectic basis of cycles. Recently we proposed a method [9,10] to easily compute the Kähler metric (and the symplectic basis) for a large class of CY manifolds which can be represented by specific hypersurfaces in weighted projective spaces [11]. Our method does not require the knowledge of symplectic cycles, but instead use a structure of a Frobenius algebra associated with a CY of this class and its Hodge structure. 1 Namely, let a CY manifold X be given as a solution of an equation φses(x) = 0 in some weighted projective space, where W0(x) is a quasihomogeneous function in C 5 of weight d that defines an isolated singularity at x = 0,(see [13]) which is tightly related with the underlying N = 2 superconformal theory [14,15,16]. The monomials es(x) also have weight d and correspond to deformations of the complex structure of X.
Polynomial W0(x) defines a Milnor ring R0. Inside R0 there exists a subring R Q 0 which is invariant w.r.t the action of the so-called quantum symmetry group Q. This group acts on C 5 diagonally, and preserves W (x, φ). In the cases where all complex structure deformations can be represented by polynomial deformations, dim R Q 0 = dim H 3 (X) and the ring itself has a Hodge structure in correspondence with degrees of the elements 0, d, 2d, 3d. One can introduce an invariant pairing η on R Q 0 . The pairing turns the ring to a Frobenius algebra [12] and plays an important role in the construction of our formula for e −K .
Also there exists a group of phase symmetries, which acts diagonally on C 5 and preserves W0(x). It acts naturally on the invariant ring R Q 0 , and this action respects the Hodge decomposition of R Q 0 . This allows to pick a basis eµ(x) in each of the Hodge decomposition components of R Q 0 , which consists of eigenvectors of the phase symmetry group action.
Using the invariant ring R Q 0 and differentials D± = d ± dW0∧ we construct two groups of Q−invariant cohomology H 5 D ± (C 5 )inv. These groups inherit the Hodge structure from R Q 0 . We can choose basises eµ(x) d 5 x, which also consist of eigenvectors of the phase symmetry group. As shown by Candelas [17], elements of these cohomology groups are in correspondence with harmonic forms of H 3 (X). This isomorphism allows to define a complex conjugation (we denote this operation * to distinguish from the usual complex conjugation on R Q 0 ) on the invariant cohomology. It turns out, that in the basis eµ(x) it reads * eµ(x) where eρ(x) is the unique element of degree 3d in R Q 0 , and δe µ eν ,eρ is 1 if eµ · eν = eρ and zero otherwise.
Having H 5 D ± (C 5 )inv we define a relative invariant homology group H ±,inv 5 := H5(C 5 , W0 = L, ReL → ±∞)inv inside a relative homology group H5(C 5 , W0 = L, ReL → ±∞). For this purpose we use oscillatory integrals. Using the oscillatory integral pairing we define a cycle Γ ± µ in the basis of relative invariant homology to be dual to eµ(x) d 5 x.
At last we define periods σ ± µ (φ) to be oscillatory integrals over the basis of cycles Γ ± µ . They are equal to periods of the holomorphic volume form Ω on X in a special basis of cycles H3(X, C) with complex coefficients.
Due to the phase symmetry invariance, in the chosen basis of cycles Γ ± µ the formula for Kähler potential has the following diagonal form: On the other hand, as shown in [9], matrix A = diag{A µ } is equal to the product of the matrix of the invariant pairing η in the Frobenius algebra R Q 0 and the real structure matrix M such that Matrix M can be represented as M = T −1T , where T is a transition matrix from periods in some real basis of cycles Q ± µ to periods σ ± µ (φ). Actually the real structure matrix is nothing but matrix M from (1). Using this we are able to explicitly compute the diagonal matrix elements A µ and to obtain the explicit expression for the whole e −K . In [9,10], to find the real structure, we used the knowledge of periods in some integral basis of homology cycles (e.g. from [18]). However this basis is not always known. In this paper we propose another method to compute the real structure matrix M and apply it to the 101-dimensional moduli space of the quintic threefold complex structures around the orbifold point to get an explicit exact result for the moduli space Kähler metric. Together with the knowledge of the geometry of the 1-dimensional moduli space of the quintic Kähler structures computed via the mirror symmetry in [5] it presumably gives the geometry of the full moduli space of Calabi-Yau quintic threefold.
In what follows we apply our method for the quintic threefold, however many things are true in the greater generality.
For this CY dim H3(X) = 204 and period integrals have the form where qµ ∈ H3(X, Z) and Qµ ∈ H5(C 5 \(W (x, φ) = 0), Z) are the corresponding cycles. Cohomology groups of a Kähler manifold possess a Hodge structure . Period integrals measure variation of the Hodge structure on H 3 (X) as the complex structure on X varies with φ. This Hodge structure variation is equivalent to the one on a certain ring which we will now describe.
3 Hodge structure on the invariant Milnor ring.
We can consider W0(x) as a singularity in C 5 . Then there is an associated Milnor (also Jacobi) ring We will identify its elements with unique smallest degree polynomial representatives. For the quintic threefold X its Milnor ring R0 is generated as a vector space by monomials where each variable has degree less than four, and dim R0 = 1024. Polynomial W0(x) is homogeneous and, in particular, W0(αx1, . . . , αx5) = W0(x1, . . . , x5) for α 5 = 1. This action preserves W0(x) and is trivial in the corresponding projective space and on X. Such a group with this action is called a quantum symmetry Q, in our case Q ≃ Z5. Q obviously acts on the Milnor ring R0. Now we define a subring R Q 0 in the Milnor ring R0, It is multiplicatively generated by 101 fifth-degree monomials et(x) from (2). More precisely, R Q 0 consists of elements of degree 0, 5, 10 and 15, dimensions of the corresponding subspaces are 1, 101, 101 and 1. This degree filtration defines a Hodge structure on R Q 0 . Basically R Q 0 is isomorphic to H 3 (X) and the isomorphism sends the degree filtration to the Hodge filtration on H 3 (X) [17]. Let us denote χ ī j = g ik χkj as an extrinsic curvature tensor for the hypersurface W (x, φ) = 0 in P 4 . Then the isomorphism above can be written as a map from R Q 0 to closed differential forms in H 3 (X): The details on this map can be found in [17,19]. We also introduce the notation eµ(x) for elements of the monomial basis of R Q 0 , where µ = (µ1, · · · , µ5), µi ∈ Z 5 + , eµ(x) = i x µ i i and |µ| = µi is the degree of eµ(x). In particular, ρ = (3, 3, 3, 3, 3), that is eρ(x) is a unique degree 15 element of There is a Z 5 5 phase symmetry group acting diagonally on C 5 : α · (x1, · · · , x5) = (α1x1, · · · , α5x5), α 5 i = 1. This action preserves W0 = i x 5 i . The mentioned above quantum symmetry Q is a diagonal subgroup of the phase symmetries. Basis {eµ(x)} is an eigenbasis of the phase symmetry and each eµ(x) has a unique weight. Note that phase symmetry preserves the Hodge decomposition.
One additional important fact is that on the invariant ring R Q 0 there exists a natural invariant pairing turning it into a Frobenius algebra [12,9]: .
Up to an irrelevant constant for the monomial basis it is ηµν = δµ+ν,ρ. This pairing plays a crucial role in our construction.
Let us introduce a couple of differentials [20] on differential forms on C 5 : D± = d ± dW0(x)∧. They define the cohomology groups H * D ± (C 5 ). The cohomologies are only nontrivial in the top dimension H 5 D ± (C 5 ) J ≃ R0. The isomorphism J has an explicit description We see, that Q = Z5 naturally acts on H 5 D ± (C 5 ) and J sends the Q-invariant part R Q 0 to Q-invariant subspace H 5 D ± (C 5 )inv. Therefore, the latter space obtains the Hodge structure as well. Actually, this Hodge structure naturally corresponds to the Hodge structure on H 3 (X).
The complex conjugation acts on H 3 (X) so that H p,q (X) = H q,p (X), in particular H 2,1 (X) = H 1,2 (X). Through the isomorphism between R Q 0 and H 3 (X) the complex conjugation acts also on the elements of the ring R Q 0 as * eµ(x) = pµeρ−µ(x), where pµ is a constant to be determined. In particular, differential form built from eµ(x) + pµeρ−µ(x) ∈ H 3 (X, R) is real and pµpρ−µ = 1.
Using this we define two invariant homology groups 2 H ±,inv 5 as quotient of H5(C 5 , W0 = L, ReL → ±∞) with respect to the subgroups orthogonal to H 5 D ± (C 5 )inv. Now we introduce basises Γ ± µ in the homology groups H ±,inv 5 using the duality with the basises in H 5 D ± (C 5 )inv: and the corresponding periods which are understood as series expansions in φ around zero. Periods σ ± µ (φ) satisfy the same differential equation as periods ωµ(φ) of the holomorphic volume form on X. Moreover, these sets of periods span same subspaces as functions of φ. It follows, that we can define cycles Q ± µ ∈ H ±,inv and periods ω ± αµ (φ) are given by the integrals over these cycles analogous to (4). With these notations the idea of computation of periods [21] σ can be stated as follows.

Writing (4) explicitly we have
Let mssi = 5ni + νi, νi < 5. Therefore we want to expand Note that Therefore in D+ cohomology we have By induction we obtain where (a)n = Γ(a + n)/Γ(a). Using (4) once again, we see that if any νi = 4 then the differential form is trivial and the integral is zero. Hence, rhs of (4) is proportional to eν (x) and gives the desired expression. Plugging (4) into (4) and integrating over Γ + µ gives the answer Further we will also use the periods with slightly different normalization, which turn out to be convenientσ

Computation of the Kähler potential
Pick any basis Q ± µ of cycles with integer or real coefficients as in (4). Then for the Kähler potential we have the formula in which the matrix C µν is related with the Frobenius pairing η as The last expression is due to [22,23]. Let also T ± be a coordinate change matrix Q ± µ = (T ± ) ν µ Γ ± ν . Then M = (T − ) −1 T − is a real structure matrix, that is MM = 1 and by construction M doesn't depend on the choice of basis Q ± µ . M is only defined by our choice of Γ ± µ . In [9] we deduced from (5) and (5) the formula where η µν = ηµν = δµ,ρ−ν. In that papers our method to compute the real structure matrix M used the knowledge of the periods in some basis qµ computed using the residue formula and monodromy considerations. However, this method gives only 4 out of 204 linearly independent periods for the quintic threefold X. Therefore we propose here a different method to find M .
Proof. We may extend the action of the phase symmetry group to the action A on the parameter space {φs} such that W = W0 + s φses(x) is invariant under this new action. Each es(x) has a unique weight under this group action. Action A can be compensated using the coordinate tranformation and therefore is trivial on the moduli space of the quintic (implying that point W = W0 is an orbifold point of the moduli space). In particular, e −K = X Ω ∧Ω is A invariant. Consider e −K = σµA µν σν as a series in φs, φt Each monomial has a certain weight under A . For the series to be invariant, each monomial must have weight 0. But weight of σµσν equals to µ − ν and due to non-degeneracy of weights of σµ only the ones with µ = ν have weight zero.
Due to symmetry we have a ρ−t ′ = aρ−t in each case. From (5) it follows that the product of the coefficients at |σµ| 2 and |σρ−µ| 2 in the expression for e −K should be 1: Due to reflection formula at = ± i sin(π(ti + 1)/5) up to a common factor of π. The sign turns out to be minus for Kähler metric to be positive definite in the origin. Therefore Finally the Kähler potential becomes where γ(x) = Γ(x) Γ(1−x) .
Complex conjugation sends (2, 1)-forms to (1, 2)-forms. Similarly it extends to a mapping on the dual homology cycles γµ. In the real basis of cycles a version of the formula (5) takes an especially simple form, because the real structure matrix M becomes an identity. Proof. We perform a proof for the cohomology classes represented by differential forms. For onedimensional H 3,0 (X) and H 0,3 (X) it is obvious. Let Ω2,1 := et(x) χ l i Ω ljk ∈ H 2,1 (X).
The group of phase symmetries modulo common factor acts by isomorphisms on X. Therefore, it also acts on the differential forms. Lhs and rhs of the previous equation should have the same weigth under this action, and weight of the lhs is equal −t modulo (1, 1, 1, 1, 1). It follows that P (x) = pt eρ−t(x) with some constant pt.
Using this lemma and applying the complex conjugation of cycles to the formula (5) to obtain it follows that A µ = ±1/pµ. Now formula (5) implies

Conclusions
The method for computing the Kähler potential on the CY moduli space from [9] modified in this paper does not require knowledge of periods in some real homology basis. Instead, we use some simple monodromy considerations to fix the real structure matrix. Another possible interesting method would be to determine this matrix by direct computation of coefficients (6) of the complex conjugation in the basis eµ(x). In this paper we use our modified method to compute Weil-Peterson metric on the whole 101-dimensional complex structure moduli space of the quintic threefold around the orbifold point (5). Together with the computation of the moduli space geometry of the Kähler structures through the mirror map [5] it describes the Special geometry of all Ricci flat deformations of CY metric in the region.
Though we present our result for the quintic threefold, our method should be applicable to a bigger class of models, which are connected with Landau-Ginzburg description, in particular hypersurfaces in toric varieties. We plan to consider possible generalizations in the future publications.