States and Correlation Function of the B-model

In the B-model one considers a family of n-folds Xzfibered over the complex structure moduli

space _Mcs, Xz → Mcs. The states16 in the B-model are elements Bk(j) of the cohomology

5.3. MIRROR SYMMETRY FOR CALABI-YAU FOURFOLDS 113 groups Hj_(X

z,VjT ). Their cubic forms are defined as

C(B_a(i), B_b(j), B_c(k)) = Z

X

Ωn(B(i)a ∧ Bb(j)∧ Bc(k))∧ Ω , (5.42)

and vanish unless i + j + k = n. Here Ωn is the unique holomorphic (n, 0)-form and

Ωn(B(i)a ∧ B(j)_b ∧ Bc(j)) is the contraction of the n upper indices of Ba(i) ∧ B_b(j) ∧ B(j)c ∈

Hn_(X

z,VnT Xz) with Ωn, which produces an anti-holomorphic (0, n)-form on Xz. Note

that this is the standard isomorphism Hi(Xz,VjT Xz) ∼= H_H(n−j,i)(Xz) obtained by contrac-

tion with Ωn. It is important to emphasize that not all elements of Hn(Xz) can be reached as

variations of Ωn with respect to the complex structure, in contrast to the Calabi-Yau three-

fold case, which makes it necessary to introduce the primary horizontal subspace H_Hn(Xz) of

Hn(Xz). Its complement is the vertical subspace17 HVn(Xz), that is generated by powers of

the K¨ahler cone generators in H(1,1)_(X

z) [93]. We denote the image of B_k(i) in H_H(n−i,i)(Xz)

by b(i)_k =Ω(B_k(i)) and the inverse18 by B_k(i)= (b(i)_k )Ω. We define the hermitian metric

G(B_c(i), ¯B_d(i)) = Z

X

b(i)_c ∧ ¯b(i)d . (5.43)

The image b(i)a of the states Ba(i) lies in the horizontal subspace H_H(n−j,i)(Xz) and the elements

b(i)a form a basis of this space. For Bc(1) ∈ H1(Xz, T Xz) the image spans all of H(n−1,1)(Xz)

and (5.43) is the Weil-Petersson metric on _Mcs.

The integrals (5.42) and (5.43) are calculable by period integrals of the holomorphic n- form Ωn. It is in general hard to integrate the periods directly. They encode however, as in the

Calabi-Yau threefold case in section 5.2.2, the variations of Hodge structures of the family Xz → Mcs, which leads to the Picard-Fuchs differential equations on Mcs. The periods

are therefore determined as solutions of the latter up to linear combination. The precise identification as addressed in section 5.3.3 is an important problem, in particular for physical applications like the determination of the flux superpotential (4.37) on the fourfold X.

For a given base point z = z0 in the complex structure moduli space Mcs with fiber

Xz0, one fixes a graded topological basis ˆγ

(p)

a of the primary horizontal subspace HHn(Xz0, Z).

Here a = 1, . . . , h(n−p,p)_H labels the basis ˆγa(p) for fixed p = 0, . . . , n of each graded piece

H(n−p,p)(Xz). In addition, these forms can be chosen to satisfy19

Z X4 ˆ γ_a(i)∧ ˆγ(4−i)b = η (i) ab , (5.44) Z X4 ˆ γ_a(i)_{∧ ˆγb}(j) = 0 for i + j > 4 .

17_{This is completely analogous to the two-dimensional case of K3, where H}(1,1)

V spans the Picard group. 18_{The inversion is just the contraction (b}(i)

k ) Ω₌ 1

||Ω||2Ω¯

a1...an−ib1...bi_(b(i)

k )a1...an−i¯b1...¯bi such that (Ω)

Ω_{= 1.}

Formally it is the multiplication with the inverse L−1_{of the K¨ahler line bundle L = hΩi, see e.g. [82].} 19_{The generalization from fourfolds to n-folds is the obvious one.}

Later on in section 5.3.3 we will identify this grading by p with the natural grading on the observables of the A-model which are given by the vertical subspaces H_V(p,p)( ˜X) of the mirror cohomology.

Note that the ˆγa(p) basis serves as a local frame of the vector bundle Hn_H(X)→ Mcs over

the moduli space whose fiber at the point z _{∈ M}cs is HHn(Xz). Thus, the bundle HHn(X)

is flat and admits a flat connection ∇ denoted the Gauss-Manin connection. However, the individual cohomology groups H(n,n−p)(Xz) form no holomorphic vector bundles over Mcs

since holomorphic and anti-holomorphic coordinates are mixed under a change of complex structure z. Analogously to the threefold case, cf. section 5.2.2, only the horizontal parts of Fk =_⊕k_p=0H(n−p,p)(Xz) form holomorphic vector bundles for which we introduce frames βa(k)

specified by the basis expansion

β_a(k)= ˆγ_a(k)+X

p>k

Π(p,k) c_a ˆγ_c(p) . (5.45)

In special coordinates ta at the point of maximal unipotent monodromy, cf. (5.49), this can be written as _{β(0) = Ωn, βa(1) = ∂aΩn, . . .}. We note that for this basis choice we fixed the

overall normalization of each β(k)a such that the coefficient of ˆγa(k) is unity. This is needed to

obtain the right inhomogeneous flat coordinates onMcsand to make contact with enumerative

predictions for the A-model, see section 5.3.3. For grade k = 0 it corresponds to the familiar gauge choice in the vacuum line bundle _{L generated by Ω}n, cf. section 5.2.3 for the threefold

case and (5.49) for fourfolds. We also introduce a basis of integral homology cycles γa(p) dual

to ˆγa(p) obeying _Z γa(i)

ˆ

γ_b(j)= δijδab . (5.46)

Then, by construction of the basis (5.45), the period matrix P defined by period integrals takes an upper triangular form in this basis,

P = Z γ(p)a β_b(k)=        Π(p,k) a_b , p > k , δab , p = k , 0 , p < k , (5.47)

where (p, k) is the bi-grade of the non-trivial periods Π(p,k) a_b . Since ˆγ(p)a are topological and

thus are locally constant on Mcs, the moduli dependence of (5.45) is captured by the moduli

dependence of the period matrix P _{≡ P (z)}20. For Calabi-Yau fourfolds we summarize the basis choices and the periods in Table 5.1. The (n, 0)-form Ωn can always be expanded over

the topological basis ˆγ(p)a of H_Hn(Xz, Z) as

Ωn= Π(p,0) a1 γˆa(p)≡ Π(p) a γˆa(p), (5.48)

where we introduced a simplified notation Π(p) a for the periods Π(p,0) a₁ of the holomorphic n-form already given in (5.45) for k = 0. For an arbitrarily normalized n-form Ωn, the periods

20_{By means of ∇}

aβbk= ∇aΠ(k+1,k) cb β (k+1)

5.3. MIRROR SYMMETRY FOR CALABI-YAU FOURFOLDS 115 Dimension 1 h(3,1)(X4) h(2,2)(X4) h(3,1)(X4) 1 Basis of γˆa(0)0 γˆ (1) a ˆγ_β(2) γˆa(3) ˆγa(4)0 H4 H(X4, Z) βa(0)0 β (1) a β_β(2) βa(3) β(4)a0

Table 5.1: Topological ˆγ and moduli-dependent β basis of H4

H(X4, Z)

(X0_{, X}a_{) = (Π}(0) 1_{, Π}(1) a_{) with a = 1, . . . , h}(n,1)_(X

z0) at the fixed reference point z0 form for

all Calabi-Yau n-folds homogeneous projective coordinates of the complex moduli spaceMcs.

The choice of inhomogeneous coordinates set by ta= X a X0 = R γ(1)a Ω R γ(0)₁ Ω , (5.49)

which agrees with the basis choice in (5.45), corresponds to a gauge in the K¨ahler line bundle hΩni. At the point of maximal unipotent monodromy X0 and Xa are distinguished by their

monodromy, as X0 _{is holomorphic and single-valued and X}a _{∼ log(z}a_{) has monodromy}

Xa7→ Xa_{+ 1. Below the t}a_{in (5.49) are identified with the complexified K¨ahler parameters}

of the mirror ˜X.

The ta defined in (5.49) are flat coordinates for the Gauss-Manin connection ∇a, i.e. the

latter becomes just the ordinary differential _∂t∂

a. This can be seen from the gauge choice

reflecting in the basis (5.45) combined with the Griffiths transversality constraint ∇a(Fk)⊂

Fk+1/_Fk which together imply [93] _∇atb = δba, i.e ∇a= ∂a. In these coordinates the three-

point coupling becomes

C_abc(1,k,n−k−1)= C((β_a(1))Ω, (β_b(k))Ω, (β_c(n−k−1))Ω) = Z

X

β_c(n−k−1)_{∧ ∂}aβ_b(k). (5.50)

Here we use (5.45), (5.47) and the fact that (β(1)a )Ω∧(β_b(k))Ωcan be replaced by (∇aβ(k)_b )Ω [93]

under the integral (5.42) to obtain (5.50). This triple coupling is a particularly important example of (5.42). Furthermore, one can show from the properties of the Frobenius algebra that all other triple couplings in (5.42) can be expressed in terms of (5.50), see section 5.3.2.

The holomorphic topological two-point couplings of (5.44) in this basis trivially read η(k)_ab =

Z

X

β_b(n−k)_{∧ βa}(k) (5.51)

since only the lowest ˆγ(p) for p = k in the upper-triangular basis transformation (5.45) con- tributes to the integral due to the second property of (5.44). In particular η_ab(k) is moduli independent. From the above it is easy to see the basis expansion at grade k + 1,

∂aβb(k)= Cabc(1,k,n−k−1)ηcd(n−k−1)β (k+1)

d , (5.52)

Let us finish this section with some comments on general properties of the periods and their implications on theN = 1 effective action. The period integrals (5.47) obey differential and algebraic relations, which are different from the special geometry relations of Calabi-Yau threefold periods in section 5.2.3. The differential relations have however exactly the same origin namely the Griffiths transversality constraints R_XΩn∧ ∂i1. . . ∂ikΩn = 0 for k < n.

However since a_{∧ b = (−1)}nb_{∧ a for a, b real n-forms one has additional algebraic relations} from R_XΩn∧ Ωn = 0 between the periods Π(p) a for n even like n = 4, which are absent for

n odd, in particular the threefold case. _{N = 2 compactifications of Type II on Calabi-Yau} threefolds to four dimensions inherit the special geometry structure induced by the above differential relations in the vector moduli space. In contrast for a genericN = 1 supergravity theory in 4d there is no special structure beyond K¨ahler geometry.