#### Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

HJE
Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds
Cesar Fierro Cota 0 1 3
Albrecht Klemm 0 1 2 3
Thorsten Schimannek 0 1 3
0 Endenicher Allee 62 , D-53115 Bonn , Germany
1 Nussallee 12 , D-53115 Bonn , Germany
2 Hausdorff Center for Mathematics, Universita ̈t Bonn
3 Bethe Center for Theoretical Physics, Universita ̈t Bonn
We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the K¨ahler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order K¨ahler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.
Flux compactifications; Topological Strings; F-Theory
1 Introduction
1.1
Mathematical and physical structures on the moduli space
2
The period geometry of Calabi-Yau fourfolds 3
Elliptically fibered Calabi-Yau fourfolds
The structure of H4(W, Z)
Fixing an integral basis
B-branes and the asymptotic behaviour of the central charge
Geometry of non-singular elliptic Calabi-Yau fourfolds
Fourier-Mukai transforms and the SL(2, Z) monodromy
Toric construction of mirror pairs
Toric geometry of elliptic fibrations
Picard-Fuchs operators
4
Amplitudes, geometric invariants and modular forms
Review of genus zero invariants
Genus one invariants
Quasi modular forms and holomorphic anomaly equations
Modularity on the fourfold X24(1, 1, 1, 1, 8, 12)
Derivation of modular anomaly equations
4.5.1
4.5.2
4.5.3
Modular anomaly equations for periods over π-vertical 4-cycles
Genus one modular anomaly equation
4-point coupling modular anomaly equation
5
6
Horizontal flux vacua for X2∗4
5.1
5.2
Conifold C1 Orbifold O1
Conclusions and outlook
A Supplementary data
A.1 Curve counting invariants for one parameter fourfolds
A.2 Toric data for MPE18×P2
A.3 Geometric invariants for MPE18×P2
A.4 Modular expressions for X24(1, 1, 1, 1, 8, 12)
A.5 Modular expressions for MPE18×P2
A.6 Analytic continuation data for X24(1, 1, 1, 1, 8, 12)
2.1
2.2
2.3
3.1
3.2
3.3
3.4
3.5
4.1
4.2
4.3
4.4
4.5
HJEP01(28)6
Introduction
At present F-theory compactifications on elliptic Calabi-Yau fourfolds provide the richest
class of explicit N = 1 effective theories starting from string theory. The reason is that the
construction of Calabi-Yau fourfolds as algebraic varieties in a projective ambient space is
very simple and toric, or more generally non-abelian gauged linear σ-model descriptions
provide immediately trillions of geometries [1].
In fact, geometric classifications of certain compactifications with restricted physical
features seem possible even though this has been achieved mostly for elliptic Calabi-Yau
threefolds, where it has been argued that there exists only a finite number of topological
types in this class [2].
Most of the generic compact toric examples allow for elliptic fibrations and in addition
for each of them there is a huge degeneracy of possible flux choices, which together with
non-perturbative effects have been argued to solve the moduli stabilization problem by
driving the theory to a particular vacuum. Ignoring the details of how this happens for the
concrete geometry under consideration it has been shown that by degenerating the fourfold
in a controlled way viable phenomenological low energy particle spectra will emerge in four
dimensions as was worked out in the F-theory revival starting with the papers of [3–6].
An additional nice feature of F-theory is a largely unified description of gauge- or
brane moduli in terms of the complex structure moduli space of the fourfold. Together
with mirror symmetry this results in a large variety of geometrical tools that can be used
to study the physically relevant structures on these moduli spaces. In this paper we want
to improve on these tools following the line of the papers [7–12].
Of particular interest when studying the F-theory effective action associated to a given
Calabi-Yau fourfold are the admissible fluxes. There are two different types, namely
horizontal and vertical fluxes, and in general both are necessary to construct
phenomenologically viable models. While determining a basis of fluxes over C is relatively straightforward,
it has been shown that the fluxes are quantized [13] and finding the proper sublattice —
in particular for the horizontal part — is more involved. However, horizontal fluxes on
a Calabi-Yau fourfold W can be identified with the charges of topological B-branes on a
mirror manifold M . In this work we use the derived category description of the latter and
the asymptotic charge formula in terms of the Gamma class [14–16] to determine properly
quantized fluxes on W . We provide formulas that allow to write down the integral fluxes
— and in many cases an integral basis — in terms of the intersection data on M .
We then restrict to the case of non-singular elliptic Calabi-Yau fourfolds and find
explicit expressions for several elements of the monodromy group ΓM . We show that a generic
subgroup of the monodromy generates the SL(2, Z) action on the K¨ahler modulus of the
fiber. This explains certain modular properties of the topological string amplitudes on
M that we also analyze in detail. We find that the genus zero amplitudes in the type II
– 2 –
language that determine the K¨ahler potential and the superpotential are SL(2, Z)
quasimodular forms, extending results of [17]. We also show that similar features hold for the
genus one amplitude, which is conjectured to be related to the gauge kinetic terms. As in
the Calabi-Yau threefold case we find that these amplitudes are related via certain
holomorphic anomaly equations, from which they can be reconstructed in simple situations [18–22].
Finally, we study the global structure of the properly quantized horizontal flux
superpotential for a particular example. To this end we analytically continue the integral
periods to the generic conifold locus, the generic orbifold and the Gepner point. We find
that aligned flux stabilizes the theory at the conifold where the scalar potential vanishes.
Somewhat surprisingly the complex 8 × 8 continuation matrix can be expressed analytically
later sections, where we add to this discussion.
Note added: after this article appeared on the arxiv, Georg Oberdieck pointed out
that our results in section 4.5 match with his and Aaron Pixton’s conjectured holomorphic
anomaly equation on Calabi-Yau n-folds appearing in [23, 24]. Moreover, he explained to us
the explicit form of the generalized holomorphic anomaly equation for the Gromov-Witten
potentials on Calabi-Yau fourfolds, which we include now in appendix B. We performed
further non-trivial checks of his conjecture with our data beyond the material that appeared
already in appendix A.5.
1.1
Mathematical and physical structures on the moduli space
Let us give a very short account of the complex structure moduli space of Calabi-Yau
fourfolds W , its algebraic and differential structures and their physical interpretation.
As far as the differential structure and some aspects of mirror symmetry are concerned
this is based on the analysis of [7–9]. The analysis can be viewed as a generalization of the
ones that lead to special geometry for Calabi-Yau threefolds [25] and was discussed with
emphasis on mirror symmetry in [26].
moduli space M is unobstructed and of complex dimension h3,1(W ). Further key
structures are the bilinear intersection form on the horizontal cohomology αpq, βrs ∈ Hh4or(W ) =
Z
W
hαpq, βrsi =
αpq ∧ βrs = 0
unless p = s and q = r ,
(1.1)
which is even as the dimension is even and transversal with respect to the Hodge type as
indicated.
Moreover there is a positive real structure
R(α) = ip−qhα, α¯i > 0 ,
– 3 –
(1.2)
where α is a primitive form in Hp,q with p + q = n. In particular
e−K(z) = R(Ω(z)) ,
(1.3)
Z
W
Z
W
defines the real K¨ahler potential K for the Weil-Petersson metric Gi¯ = ∂j ∂¯¯K, which is
closely related to kinetic terms of the moduli fields in the N = 1 4d effective action. Here
∂j = ∂zi or ∂¯¯ are the derivatives with respect to the generic coordinates zi on M and
∂
their complex conjugates.
Because the intersection (1.1) is even on fourfolds one gets a mixture of algebraic and
differential conditions on the periods and if we consider the cohomology over Z we get
lattice structures somewhat similar to that of K3 surfaces. In particular the relations
Ω ∧ Ω = 0 ,
Ω ∧ ∂i1 . . . ∂in Ω = 0 , for n ≤ 3 ,
(1.4)
lead to non-trivial constraints on the periods. In [12] these relations have been used to
fix an integral basis for particular one parameter Calabi-Yau fourfolds.
Moreover, the
authors used the Gamma class formula for the 8-brane charge as a non-trivial check of
their results. We verified that the algebraic constraints can be used to fix an integral basis
for the mirror of the two-parameter elliptic Calabi-Yau fourfold X24 but found that this
method quickly becomes unpractical if the number of moduli increases. Our approach is
somewhat complementary in that we use the Gamma class formula to fix integral periods
and the constraints (1.4) can be used to supplement our technique and as a non-trivial
check. In particular, this approach scales well with the number of moduli.
Other immediate data are the 4-point couplings
Cijkl(z) = hΩ, ∂i∂j ∂k∂lΩ(z)i .
By the usual relation of the horizontal and vertical cohomology rings of W to the
(chiral,chiral) and (chiral,anti-chiral) rings of the N = (
2, 2
) superconformal theory on the
worldsheet — with their U(1)l × U(1)r charge bigrading corresponding to the Hodge type
grading1 — and the axioms of the CFT one sees however that these 4-point couplings are
not fundamental, but factorize into three-point couplings
Cijkl(z) = Ciαj (z)ηˆα(2β)Cβkl(z) = Ciαj (z)Cαpk(z)ηˆp(1l) ,
with the independent associativity condition
Ciαj (z)ηˆα(2β)Ckβl(z) = Ciαk(z)ηˆα(2β)Cjβl(z) .
(1.5)
(1.6)
(1.7)
Here the latin indices run over the moduli fields associated to either the complex structure
moduli on W whose tangent space is associated to harmonic forms in H3,1(W ) (dual to
H1,3(W )) or K¨ahler moduli on M whose tangent space is associated to harmonic forms in
H1,1(M ) (dual to H3,3(M )). The greek indices are associated to elements in Hh2o,2r(W ) and
1The exchange of this identification is the essence of mirror symmetry between W and M .
– 4 –
Hv2e,2rt(M ), respectively. The ηˆ’s define a constant intersection form with respect to a fixed
basis of H4hor(W ) or a suitable K-theory basis extending Hv,ert(M ).
∗ ∗
More specifically we can identify ηˆ(
2
) in a reference complex structure near large radius
with the inverse of the pairing on Hh2o,2r(W ) and ηˆ(1) with the inverse pairing on H3,1(W ) ⊕
H1,3(W ), which by (1.1) is block diagonal. This property is maintained throughout the
moduli space due to the charge grading.
The basic idea of mirror symmetry is to calculate these couplings, which are nontrivial
sections of tensor bundles over M, from the periods of Ω. The latter can be obtained as the
solutions of the Picard-Fuchs differential equations. We denote an integral basis of periods
This is physically relevant as the flux superpotential
Z
W
W (z) =
G4 ∧ Ω(z) = nκΠκ(z) ,
(1.8)
is given with respect to this basis by (half)2 integer flux quanta nκ
Dirac-Zwanziger quantization condition and additional constraints discussed in [13]. The
analysis of attractor points and cosmologically suitable minima of the associated scalar
∈ Z, quantized due to a
potential relies therefore crucially on this basis.
R
Interpreted in the A-model the triple couplings Ciαj (t) in the flat coordinates given by
the mirror map tk(z) ∝ [Ck](ω + iB), where [Ck] is an integral curve class on M and B
is the Neveu-Schwarz B-field, encode the quantum cohomology of M . In particular each
coefficient of the Fourier expansion Ciαj (e2πitk ) counts the contribution of a holomorphic
worldsheet instanton in a given topological class. These contributions are directly related
to Gromov-Witten invariants at genus zero. Gromov-Witten invariants at genus one can be
calculated from the Ray-Singer Torsion, starting with the genus zero data. Both genus zero
and genus one worldsheet instanton series give rise to a remarkable integrality structure in
terms of additional geometric invariants of embedded curves [27].
An interesting aspect of these generating functions is that they are modular forms of
the monodromy group Γ preserving the intersection form in the integral basis. For generic
Calabi-Yau fourfolds this aspect is too difficult to appreciate in the sense that not much is
known about the corresponding automorphic forms, but for elliptically fibered Calabi-Yau
spaces, there is a subgroup of Γ which acts as the modular group on the K¨ahler modulus τ of
the elliptic fiber in M . The precise way this subgroup is embedded in Γ can be inferred using
specific auto-equivalences of the derived category of B-branes, as we will see in section 3.2.
It turns out that there is a clash between holomorphicity and modularity in the τ
dependence of the triple couplings and the Ray-Singer torsion, which leads for Calabi-Yau
threefolds to the holomorphic anomaly equations. We will discuss analogous holomorphic
anomaly equations for fourfolds in section 4.
2As pointed out in [13] the combination hG4 − c2(2M) i ∈ H4(M, Z) has to be integral. However, in the
concrete examples discussed below c2(M ) is even.
– 5 –
In this section we show how to determine integral horizontal fluxes on a Calabi-Yau fourfold
W . To this end we interpret the flux lattice as the charge lattice of A-branes on W . This
in turn is related via homological mirror symmetry to the charge lattice of B-branes on
a mirror manifold M . B-branes on M form the bounded derived category of coherent
sheaves Db(M ). Given a brane E • ∈ Db(M ) the asymptotic behaviour of the charge can
be calculated using the Γ-class. Moreover, a C-basis of fluxes on W can be obtained as the
solution to a set of differential equations, the Picard-Fuchs system. Integral generators are
then linear combinations of solutions with the correct asymptotic behaviour.
A similar calculation has been used in [28] to obtain the quantum corrected A-model
cohomology ring for certain non-complete intersection Calabi-Yau fourfolds. In some cases
the asymptotic behaviour was not sufficient to uniquely determine integral elements. As
was pointed out in [28], the Jurkiewicz-Danilov theorem and the Lefschetz hyperplane
theorem prevent this behaviour for the induced cohomology on complete intersections in toric
ambient spaces. In general algebraic constraints on the periods can be used to supplement
the above procedure.
2.1
The structure of H4(W, Z)
The structure of H4(W, Z) for a Calabi-Yau fourfold is surprisingly subtle and in this paper
we will only be interested in finding an integral basis for the period lattice. However, even
this notion demands justification.
We first discuss the structure of H4(W, C). By the definition of a Calabi-Yau manifold,
H4,0(W, C) is generated by a unique, holomorphic 4-form that we call Ω. Then H3,1(W, C)
is generated by first-order derivatives ∂zi Ω — modulo a part in H4,0(W, C) — where zi
are complex structure coordinates.
Due to the existence of the harmonic (4, 0) form,
H3,1(W, C) can be identified with the first order deformations of the complex structures
and by the Tian-Todorov theorem the latter are unobstructed. H1,3(W, C) and H0,4(W, C)
are obtained from these spaces by complex conjugation.
The interesting part is thus H2,2(W, C). By Lefschetz decomposition the cohomology
splits into
Here the subgroup of primitive classes is given by
H2,2(W, C) = Hp2r,2im(W, C) ⊕ HV2,2(W, C) .
Hp2r,2im(W, C) = {α ∈ H2,2(W, C) | ω ∧ α = 0} ,
where ω is the K¨ahler form. On the other hand the so-called primary vertical cohomology
is generated by the SL(2, Z) Lefschetz action from the primitive classes in H1,1(W, C), i.e.
HV2,2(W, C) = {ω ∧ β | β ∈ H1,1(W, C) , ω3 ∧ β = 0} .
We now denote the subspace of cohomology generated by derivatives ∂zi1 · · · ∂zin Ω of the
holomorphic 4-form as the primary horizontal cohomology HH4 (W, C). Since the K¨ahler
class is independent of the complex structure, it follows from
ω ∧ Ω = 0 ,
– 6 –
(2.1)
(2.2)
(2.3)
(2.4)
in [29], there can be additional primitive classes in Hp2r,2im(W, C)\HH2,2(W, C). The structure
is thus
vertical.
H2,2(W, C) = HH2,2(W, C) ⊕ HR2,M2 (W, C) ⊕ HV2,2(W, C) ,
(2.5)
where HR2,M2 (W, C) is the subgroup of primitive classes that are neither horizontal nor
The naive expectation that mirror symmetry maps vertical into horizontal classes and
vice versa while the remaining component maps into itself can not hold. It would lead to
a contradiction when applied to the geometry studied in [28], where additional “vertical”
cycles appear in the quantum deformed A-model intersections. A true statement about the
relation under mirror symmetry would therefore require a more refined notion of verticality.
This subtlety is avoided when phrasing the problem in terms of branes and homological
mirror symmetry.
2.2
Fixing an integral basis
A 4-cycle Σ dual to an element in HH4 (W, C) ∩ H4(W, Z) is calibrated symplectically, i.e.
(2.6)
(2.7)
(2.8)
and the K¨ahler class restricts to zero ω|Σ = 0. In other words, Σ is a special lagrangian
cycle that can be wrapped by a topological A-brane L. The central charge of this brane is
then given by the period
Note that this is equal to the superpotential generated by a flux quantum along Σ.
By homological mirror symmetry [30, 31], the topological A-branes on W are related
to B-branes on the mirror M . The latter correspond to elements in Db(M ), the bounded
derived category of coherent sheaves on M . Given a B-brane that corresponds to a complex
E • ∈ Db(M ), the asymptotic behaviour of the central charge is
Z
M
ZBasy(E •) =
eJ ΓC(M ) (ch E •)∨ ,
where J is the K¨ahler class on M . The details of this formula will be discussed in the next
section. The crucial fact is that the central charges of A- and B-branes are identified via
the mirror map. While a construction for all objects in Db(M ) is in general not available,
the central charge only depends on the K-theory charge of a complex of sheaves.
Our approach to fix an integral basis for the period lattice will be to construct elements
E • in Db(M ) that generate the algebraic K-theory group Ka0lg(M ) and calculate the
asymptotic behaviour of the central charges. Using the mirror map, these can be interpreted as
the leading logarithmic terms of generators of the period lattice. The subleading terms are
given by the corresponding solutions to the Picard-Fuchs equations.
Re eiθΩ
Σ
= 0 ,
ZA(L) =
Ω .
Z
Σ
– 7 –
For a Calabi-Yau manifold M , the topological B-branes and the open string states stretched
between them are encoded in the bounded derived category of coherent sheaves Db(M ).
The objects of this category are equivalence classes of bounded complexes of coherent
sheaves
E
• = . . .
E
E
E
d−2 // E −1 d−1 // E 0 d0 // E 1
E
A set of maps fi : E i → F i, such that the fi commute with the coboundary maps,
corresponds to an element f ∈ Hom(E •, F •). Objects as well as morphisms are identified
under certain equivalence relations but a more detailed discussion of topological branes
and Db(M ) is outside the scope of this paper and can be found e.g. in [32].
HJEP01(28)6
However, we note that if there is an exact sequence
Now given the K¨ahler class J , the asymptotic charge of a B-brane that corresponds to
Z
M
Zasy(E •) =
eJ ΓC(M ) (ch E •)∨ .
The characteristic class ΓC(M ) can be expressed in terms of the Chern classes of M and
for a Calabi-Yau manifold the expansion reads
ΓC(M ) = 1 +
1
iζ(3)
24 c2 − 8π3 c3 +
1
5760
(7c22 − 4c4) + . . . .
The Chern character of the complex is given by
. . .
// E −1
// E 0
E
• = . . .
// E −1
/
/ F
// E 0
where F is a coherent sheaf and E i are locally free sheaves, i.e. equivalent to vector bundles,
then the complex
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
is equivalent to F inside Db(M ).
the complex E • is given by
Ca• = ι!OCa (K1a/2) ,
C
– 8 –
ch(E •) = . . . − ch(E−1) + ch(E0) − ch(E1) + ch(E2) − . . . ,
(. . . )∨ acts on an element β ∈ H2k(M ) as β∨ = (−1)kβ.
where Ei is the vector bundle corresponding to the locally free sheaf E i and the involution
A general basis of 0-, 2-, 6- and 8-branes has been constructed in [28]. The 8-brane
corresponds to the structure sheaf OM and the 6-branes are generated by locally free
resolutions of sheaves OJi , where the divisors Ji generate the K¨ahler cone. The 0-brane is
represented by the skyscraper sheaf Opt.. A basis of 2-branes was constructed as
where ι is the inclusion of the curve Ca that is part of a basis for the Mori cone and K1a/2
C
is a spin structure on Ca. The asymptotic charges have been calculated in [28] and for the
readers convenience they are reproduced below.
i∈I
0
❛❛❛❛00 ⊕ OM (−Di)
/
/ OM
− P Di
i∈I
/
/ OM
0
/
/ OM (−D)
We now describe a construction of 4-branes which in many cases leads to an integral
M and S = T Di, the Koszul sequence
basis. Given effective divisors Di, i ∈ I that correspond to codimension one subvarieties of
is exact and provides a locally free resolution of the coherent sheaf OS. When I contains
only one element, this is just the familiar short exact sequence
of complexes in Db(M ). This is the locally free resolution employed in [28] to calculate the
central charges for a basis of 6-branes.
of elements in H2,2(PΔ, C).
More generally, we can use the Koszul sequence to describe branes wrapped on
arbitrary cycles that are intersections of subvarieties of codimension one. If a basis of
HV2,2(M, C) ∩ H4(M, Z) can be constructed this way, then, as we described above, this
leads to an integral basis of the period lattice in the mirror. In particular the
asymptotic behaviour then uniquely singles out a solution to the Picard-Fuchs system. For a
Calabi-Yau hypersurface M in a toric variety PΔ, the cohomology of the ambient spaces is
generated by elements in H1,1(PΔ). As was pointed out by the authors of [28], the quantum
Lefschetz hyperplane theorem then guarantees that HV2,2(M, C) is generated by restrictions
The formula for the asymptotic central charge gives the following results:
// . . .
/
• 6-brane wrapped on Ja:
Zasy (OJa ) =
eJ ΓC(M ) [1 − ch (OM (Ja))]
cia =
1
24 Ca0aaa + c0a
c2(M )JaJi ,
,
c0a =
1
4! Ci0jkltitj tktl +
2 cij titj + citi + c0 ,
c3(M )Ji ,
c0 =
7c2(M )2 − 4c4(M )
Z
Z
M
M
Z
M
−
Z
M
cij =
c2(M )JiJj ,
(2.19)
• 4-brane wrapped on H = Da ∩ Db:
Zasy (ODa∩Db ) =
hij titj + hiti + h ,
hij =
h =
M
• 2-brane wrapped on Ca dual to Ja:
DaDbJiJj ,
hi =
DaDb(Da + Db)Ji ,
(2.21)
DaDb(2Da2 + 3DaDb + 2Db2) +
c2(M )DaDb
the K¨ahler cone by Ji and the K¨ahler form is given by J = tiJi.
The charge of the 0-brane is universally Zasy(Opt.) = 1. We denoted the generators of
Finally we need the intersection matrix of the 4-cycles mirror dual to the B-branes.
They are not given by the classical intersection numbers in the A-model but rather by the
open string index
χ(E •, F •) =
Td(M ) (ch E •)∨ ch F • .
The Todd class Td(M ) is for a Calabi-Yau fourfold given by
Z
M
E
where V is the volume form. Note that if we construct a basis of B-branes
and introduce the intersection matrix ηij = χ(vi, vj ), the inverse matrix η−1 will act on the
period vector Π corresponding to the mirror dual cycles. For example
Td(M ) = 1 +
+ 2V ,
c2(M )
12
~v = (E1•, . . . , En•),
Z
W
Ω ∧ Ω = 0
→
ΠT η−1Π = 0 .
3
Elliptically fibered Calabi-Yau fourfolds
Although the methods to find integral generators of the period lattice are applicable to
general Calabi-Yau manifolds we now restrict to elliptic fibrations
such that for a general choice of complex structure on M the fiber exhibits at most I1
singularities over loci of codimension 1 in the base B. In particular we require the presence
of a section. Fourfolds of this type have been previously studied in [17]. It turns out that
the intersection ring and the relevant topological invariants are completely determined by
the base. Note that this geometric setup is completely analogous to the threefolds studied
in [19, 21].
3.1
As far as it carries over to the fourfold case, we follow the notation in [21] which we now
quickly review. The generators of the Mori cone of the base B are given by {[C˜′k]}, k =
1, . . . , h11(B) = h11(M ) − 1 and the dual basis of the K¨ahler cone is {[Dk′]}. In particular
we assume that the Mori cone is simplicial. Let E be the section so that its divisor class
is given by [E].
We now obtain curves
˜k = E · π−1C˜′k , k = 1, . . . , h11(B) ,
C
by the dual basis {[D˜ e], [D˜ k]}, where
on M for some representatives C˜′k of [C˜′k]. A basis for the Mori cone on M is given by
{[C˜k], [C˜e]}, where [C˜e] is the class of the generic fiber. The K¨ahler cone of M is generated
[D˜ k] = π∗[Dk′] , [D˜ e] = [E] + π∗c1(B) .
In the following we will mostly drop the square brackets and assume that the distinction
between subvarieties and corresponding classes is clear from the context. The intersection
ring of M is determined in terms of intersections on B via
Z
M
Z
M
D˜ e · P (D˜ e, D˜ 1, . . . , D˜ h11(B)) =
P (c1(B), D1′, . . . , Dh′11(B)) ,
Z
B
P (1, D˜ 1, . . . , D˜ h11(B)) = 0 ,
where P is any polynomial in h11(B) + 1 variables.
We denote the complexified areas of the curves in the base by
Z
Z
T˜k =
C
˜k B + iω ,
τ˜ =
C
˜e B + iω .
where ω is the K¨ahler class and B is the Neveu-Schwarz B-field. The complexified area of
the fiber will be called
The generators of the Mori cone and the dual generators of the K¨ahler cone provide
a natural choice of basis for divisors and curves from the geometric perspective. However,
as was already observed for elliptically fibered threefolds, the SL(2, Z) subgroup of the
monodromy acts more naturally in a different choice of basis. We introduce
with
and the dual basis
[Ce] = [C˜e] , [Ck] = [C˜k] +
a
k
2 [C˜e] ,
ak =
Z
C
˜k c1(B) ,
De = D˜ e − 2
2
1 π∗c1(B) = E +
1 π∗c1(B) ,
Dk = D˜ k .
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
HJEP01(28)6
with similar definitions for Qk and qe.
We also define the topological invariants of the base
a = c1(B)3 ,
ai = c1(B)2 · Di′ ,
aij = c1(B) · Di′ · Dj′ , cijk = Di′ · Dj′ · Dk′ ,
(3.12)
and denote the k-th degree component of ch(F •) by chk(F •).
The definitions above are straightforward extensions of the corresponding threefold
expressions introduced in [21]. For Calabi-Yau fourfolds a basis of middle-dimensional
cycles has to be specified as well. It turns out that for elliptically fibered fourfolds with at
most I1 singularities in the fibers such a basis is given by
HJEP01(28)6
with
Hk = E · π−1Dk′ = E · D˜ k ,
Hk = π−1C˜′k ,
Hi · Hj = −aij ,
Hi · Hj = δij ,
Hi · Hj = 0 .
We call the 4-cycles Hk = π−1C˜′k, k = 1, . . . , h11(B) that result from lifting a curve in the
base to a 4-cycle in M the π-vertical 4-cycles. As we will see in section 4.5.1 the genus
zero amplitudes that correspond to π–vertical 4-cycles have particularly simple modular
properties. Using the Koszul sequence (2.16) we calculate
The complexified areas corresponding to Ck and Ce are now given by
respectively. Finally we introduce the exponentiated complexified areas
τ = τ˜ and
T k = T˜k +
a
k
2
τ˜ ,
Q
˜k = exp(2πiT˜k) , q˜e = exp(2πiτ˜) ,
(3.10)
(3.11)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
ch(OHi ) = Hi − 2
ch(OHi ) = Hi − C˜e · hi ,
1 C˜k(ckii − aki) +
1
12
V (2ai − 3aii + 2ciii) ,
hi =
Z
M
E ch3(OHi ) =
2 λa,bE · (Da · Db) · (Da + Db) ,
Hi =
X λa,bD¯ a · D¯ b ,
ch(OD˜i ) = D˜ i − 2
ch(OE ) = E +
1
1
2
Hkckii +
Hi · ai +
6
1 C˜eciii ,
6
1 C˜iai +
1
24
V · a .
with the volume form V and
where we assume that
X 1
a,b
a,b
for effective divisors D¯ a. The Chern characters of the 6-branes are given by
Moreover, ch(OM ) = 1, ch(C˜e•) = C˜e and ch(C˜k•) = C˜k.
is a Fourier-Mukai transform.
Db(X × Y ) and acts as [33, 34]3
The B-model periods are multi-valued and experience monodromies along paths encircling
special divisors in the complex structure moduli space. Homological mirror symmetry [30]
implies that the corresponding monodromies in the A-model lift to auto-equivalences of
the derived category [30, 33, 34]. Furthermore, an important theorem by Orlov states that
every equivalence of derived categories of coherent sheaves of smooth projective varieties
A Fourier-Mukai transform ΦE : Db(X) → Db(Y ) is determined by an object E ∈
F • 7→ Rπ1∗(E ⊗L Lπ2∗F •) ,
where π1 and π2 are the projections from X × Y to Y and X respectively. The object E
is called the kernel and R and L indicate that one has to take the left- or right derived
functor in place of π∗, π∗ or ⊗.
For our purpose the nice property of this picture is that certain general monodromies
correspond to generic Fourier-Mukai kernels. This allows us to write down closed forms not
only for the large complex structure monodromies but also for a certain generic conifold
monodromy and a third type that is special to elliptically fibered Calabi-Yau.
Let D be one of the generators of the K¨ahler cone and C the dual curve. The limit
in which C becomes large corresponds to a divisor in the K¨ahler moduli space. It is well
known [33] that the Fourier-Mukai transform corresponding to the monodromy around this
large radius divisor acts as
We choose a basis of branes
E • 7→ O(D) ⊗ E • .
OM , OE, ODi , OHi , OHi , C˜i, C˜e, Opt. ,
and calculate the monodromy for the large radius divisor corresponding to Dj ,
T˜j = 00 00 00
1 0 −δjk
acting on the vector of charges. One can obtain a similar expression for the monodromy
T˜e, corresponding to D˜e.
3An accessible explanation for physicists of how these calculations are performed can be found in [35].
Another auto-equivalence, the Seidel-Thomas twist, corresponds to the locus where,
given a suitable loop based on the point of large radius, the D8-brane becomes massless.
Its action on the brane charges is given by
Z(E•) 7→ Z(E•) − χ(E•, OM )Z(OM ) .
(3.23)
As was explained in [12], for a Calabi-Yau fourfold χ(OM , OM ) = 2. This implies that
Z(OM ) transforms into −Z(OM ) and this monodromy is of order two.
Elliptically fibered Calabi-Yau manifolds with at most I1 singularities exhibit yet
another type of auto-equivalence. Physically it corresponds to T-duality along both circles of
the fiber torus. The corresponding action Φ on the derived category was first studied by
Bridgeland [36] in the context of elliptic surfaces. Calculations for Calabi-Yau threefolds
can be found in [37] and were elaborated on in the subsequent review [38]. In full
generality the auto-equivalences and their implications for the modularity of the amplitudes on
elliptic Calabi-Yau threefolds with I1 singularities [39] have been presented in [40].
We can decompose the Chern character of a general brane E• as
Here we introduced n, nE, ne, s ∈ Q, and Fi, Bi are pullbacks of forms in Hi,i(B, C). The
volume form on M is denoted by V .
Adapting the calculation in [38] to Calabi-Yau
fourfolds, we find that the Chern character of the transformed brane is given by
ch0(Φ(E•)) = nE ,
1
1
ch1(Φ(E•)) = B1 − 2 nE c1 − n · E ,
ch2(Φ(E•)) = B2 − 2 B1 · c1 +
ch3(Φ(E•)) = − 2 B2 · c1 +
1
ch4(Φ(E•)) = −ne V − 61 c12 · F1 · E + 2 c1 · F2 · E +
1
1
24
n c13 · E ,
1
12 nE c12 − F1 · E + 2 n c1 · E ,
1
1
12 B1 · c12 + s C˜e +
1
2 c1 · F1 · E − F2 · E − 6
1
n c12 · E ,
with c1 = π∗c1(B). Using the formulae for the Chern characters of the basis of branes
introduced above, this translates into the matrix
0 12 ckiiai − ak
112 (3aiiai − 2ciiiai − a)
for the corresponding monodromy.
We can now explicitly calculate that
and another careful calculation reveals
It follows that
(S˜ · T˜e−1)3 = −I .
S =
h11(B)
Y
i=1
i
T˜−ai/2
S˜ ,
T =
h11(B)
Y
i=1
T
˜ai/2
i
T˜e−1 ,
generate a group isomorphic to PSL(2, Z), the modular group. In particular, Qk, q are
invariant under T , while some of the Q˜k obtain a sign under T -transformations if the
canonical class of the base is not even. As was already noted by [21], this makes Qk and q
the correct expansion parameters for the topological string amplitudes to exhibit modular
properties.
3.3
Toric construction of mirror pairs
To fix conventions we will briefly review the Batyrev construction of Calabi-Yau n-fold
mirror pairs (M, W ) as hypersurfaces in toric ambient spaces [41].
The data of the mirror pair is encoded in an n+1-dimensional reflexive lattice polytope
Δ ⊂ Γ and the choice of a regular star triangulation of Δ and the polar polytope
Δ∗ = {p ∈ Γ∗R | hq, pi ≥ −1, ∀q ∈ Δ} ,
(3.30)
that is embedded in the dual lattice Γ∗. We denoted the real extensions of the lattices by
ΓR and Γ∗R respectively. The triangulation of Δ∗ leads to a fan by taking the cones over
the facets that in turn is associated to a toric variety PΔ. The family M of Calabi-Yau
n-folds is given by the vanishing loci of sections PΔ ∈ O(KΔ∗ )
PΔ =
X
Y
ν∈Δ∩Γ ν∗∈Δ∗∩Γ∗
aν xhνν∗,ν∗i+1 = 0 .
The mirror family W is obtained by exchanging Δ ↔ Δ∗.
Even for a generic choice of section the Calabi-Yau varieties thus constructed might
be singular. For n ≤ 3 the singularities can be resolved by blowing up the ambient space.
This is not always possible for fourfolds. However, all models studied in this paper can be
fully resolved by toric divisors.
Toric geometry of elliptic fibrations
For F-theory we need Calabi-Yau manifolds that are elliptically fibered. One way to
construct these is by taking a torically fibered ambient space such that the hypersurface
constraint cuts out a genus one curve from the fiber [42]. Toric fibrations can be understood
in terms of toric morphisms. A toric morphism φ : PΔ → PΔB in turn is encoded in a
lattice morphisms
φ : Γ → ΓB ,
such that the image of every cone in Σ is completely contained inside a cone of ΣB. We
obtain a fibration with the fan of the generic fiber given by ΣF ∈ ΓF if the morphism
φ : Γ → ΓB is surjective and the sequence
0 → ΓF ֒−→ Γ −φ→B ΓB → 0 ,
In the following we will consider fibers constructed as E8 hypersurfaces
EE8 :
X6(
1, 2, 3
) = {(x, y, z) ⊂ P2(
1, 2, 3
) : x6 + y3 + z2 − sxyz = 0}.
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
(3.36)
We can now obtain elliptically fibered mirror pairs (M, W ) from the following
construction [12]. First we combine a base polytope ΔB and a reflexive fiber polytope ΔF and
embed them into a n + 1-dimensional polytope Δ as follows:
ν∗ ∈ Δ∗
νiF ∗
0
ΔF ∗ 0
ΔB∗
sij ΔB .
.
νjF
.
νjF
ΔF
ν ∈ Δ
For a fixed νiF ∗ ∈ ΔF ∗ and νjF
∈ ΔF we introduced sij = hνjF , νiF ∗i + 1 ∈ Z>0. This
describes a reflexive pair of polytopes (Δ, Δ∗) given by the convex hulls of the points
appearing in (3.34). Using the Batyrev construction one gets an n-fold M from the locus
given by (3.31) on the ambient space PΔ. As mentioned above, M inherits a fibration
structure from the ambient spaces PΔ → PΔB and we can identify a map
M = {x ⊂ PΔ|PΔ (x) = 0} −→ B = PΔB
π
One can obtain a fibration using the E8 fiber and a base B from the following toric data:
div.
ν¯∗
i
l(e)
0 −6
D1
In particular, fibrations of this type have a section and at most I1 singularities in the fiber.
We will denote as MBE8 the elliptically fibered n-fold given by the fibration EE8 ֒→ MBE8 →
Picard-Fuchs operators
The periods of the holomorphic n-form on a Calabi-Yau n-fold are annihilated by a set of
differential operators, the Picard-Fuchs system. For Calabi-Yau varieties constructed as
hypersurfaces in a toric ambient space it is easy to write down differential equations for
which the solution set is in general larger than that spanned by the periods. However, in
many cases the solution sets are equal and it is sufficient to study the so-called GKZ-system.
How to derive the GKZ-system from the toric data and the relation to the Picard-Fuchs
system is explained e.g. in [43].
B.
maps
(3.37)
Amplitudes, geometric invariants and modular forms
The topological string A-model encodes Gromov-Witten invariants, counting holomorphic
f : Σg,p¯ → M ,
from pointed curves Σg,p¯ of genus g into M . The general formula for the virtual dimension
of the moduli stack of stable maps4 into a Calabi-Yau M is given by
vir dim M¯ g,n(M, β) = (dim M − 3)(1 − g) + n ,
M¯ g,n(M, β) and Σ the domain of f .
where n is the number of marked points and we require f∗[Σ] = β ∈ H2(M ) for f ∈
While for Calabi-Yau threefolds the virtual dimension is zero at all genera with n = 0,
in the case of fourfolds it is non-negative only when g = 0, 1. A positive virtual dimension
4A map is stable if it has at most a finite number of non-trivial automorphisms that preserve marked
and nodal points.
N0,β(γ) =
ev∗1(γ) ,
Z
ξ
β≥0
with ξ = [M¯ 0,1(M, β)]virt.. From the topological string theory perspective they are encoded
in the instanton part of the normalized double-logarithmic quantum periods
F (0) = classical + X N0,β(γ)qβ .
γ
In particular, the classical terms corresponding to Fγ(0) are determined by Zasy(Oγ ). While
the Gromov-Witten invariants are in general rational numbers, they are conjecturally
related to integral instanton numbers n0,β via
X N0,β(γ)qβ =
β≥0
X n0,β(γ) X∞ qdβ
β≥0
d=1
The Gromov-Witten invariants can also be related to meeting invariants mβ1,β2 [27],
which for β1, β2 ∈ H2(M, Z) virtually enumerate rational curves of class β1 meeting rational
curves of class β2. They are recursively defined via the following rules.
can be compensated by intersecting with classes on M pulled back along the evaluation
maps
evi = f (pi) : M¯ g,n → M , i ∈ 0, . . . , n .
On the other hand, intersecting with the pull-back of the fundamental class [M ] leads
to vanishing invariants. The latter property of Gromov-Witten invariants is called the
Fundamental class axiom. It follows that for fourfolds the invariants with g ≥ 2 vanish.
We will now review the Calabi-Yau fourfold invariants for g = 0, 1 and how they are
encoded in various observables of the topological A-model.
Review of genus zero invariants
H2,2(M, Z) we obtain well-defined invariants
From the general virtual dimension formula we find vir dim M¯ 0,1 = 2 , and given γ ∈
mβ1,β2 = mβ2,β1 .
Z
γi ∪ γj .
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
1. The invariants are symmetric,
2. If either deg(β1) ≤ 0 or deg(β2) ≤ 0, then mβ1,β2 = 0.
3. If β1 6= β2, then
mβ1,β2 =
X n0,β1 (γi)η(
2
),ij n0,β2 (γj ) + mβ1,β2−β1 + mβ1−β2,β2 ,
i,j
where γi ∈ HV4 (M, Z) form a basis mod torsion and
Genus one invariants for Calabi-Yau fourfolds haven been calculated for example in [27, 28].
At genus one, the virtual dimension vanishes for Calabi-Yau manifolds of any dimension.
The corresponding invariants are encoded in the holomorphic limit of the genus one free
F (1) = classical + X N1,βqβ .
β≥0
Assuming h2,1 = 0 it has the general form
In this expression χ is the Euler characteristic of M , Δ is the discriminant and z(t) is
the mirror map in terms of the algebraic coordinates z and the flat coordinates t. The
coefficients bi can be fixed by the limiting behaviour of F (1) in the moduli space.
Assuming that the coordinates z are chosen such that zi(t) = ti + O(t2), the large
t→∞
lim F (1) = − 24
1
c3(M ) ∪ Ji ti + regular ,
bi = − 24
c3(M ) ∪ Ji − 1 .
At genus one, the conjectured relation of the Gromow-Witten numbers to integral
invariants n1,β is more involved and has been worked out in [27]. It involves the meeting
invariants as well as the genus zero Gromov-Witten invariants and is given by
energy
radius limit
implies
X
i
Z
M
1 Z
M
β>0
+
1
24
1
In appendix A.1 we provide genus one invariants of the one parameter fourfold
geometries discussed in [12]. In the following, apart from studying the modular properties
of the amplitudes, we calculate the integral invariants for E8 fibrations with bases P3 and
P1 × P2. We provide some of the invariants in appendix A.3. To our knowledge the latter
case has not been studied in the literature before and provides further evidence supporting
the conjectured relations.
where Gi¯j is the inverse of the metric Gi¯j = ∂i∂¯jK. We restrict to tc2 = 0 and introduce
Re(tc2) = x, Im(tc2) = y. Then the leading terms of the scalar potential are
v = 0.020174px2 + y2 + 0.31715x2 + 0.31715y2 − 2.8019xpx2 + y2 + O(x3, y3) . (5.25)
A plot is shown in figure 2. We checked that this is the dominant contribution at least up
to order seven, where we calculated the coefficients to a precision of twenty digits. Deep
1
inside the radius of convergence |tc | ≈ |c1| < 1/256 the potential is well approximated by
the leading order v ≈ 0.020174 · |c1|. Our findings are in agreement with [12] where it was
argued that for Calabi-Yau fourfolds the Conifold is generically stabilized by aligned flux.
5.2
To expand around O1 ∩ LR2 we use the variables zb and
We find a vector of solutions to the transformed Picard-Fuchs system with leading terms
Πo = o51, o51 log (zb) , o15 log2 (zb) , o15 log3 (zb) ,
o1, o1 log (zb) , o1 log2 (zb) , o1 log3 (zb) + O(o17, z) .
It is related to the integral basis at large complex structure via
However, in contrast to the analytic continuation matrix to the conifold, To can be
determined exactly with the help of the Barnes integral method. The latter has been discussed
for one-parameter models in [50] and can be adapted to this two-parameter model. We
give the analytic expression in the Mathematica worksheet that can be found online [48].
ΠLR = To · Πo .
(5.26)
(5.27)
(5.28)
the orbifold O1 in coordinates o1 = x + Iy, o2 = 0.
The monodromy acting on ΠLR when transported along a lasso wrapping O1 is of order
six and given by
Πo = To′ · Πo′ ,
To analyze the possible fluxes we will again move away from the large complex structure
divisor and introduce the variable
Solutions in the new variables are
Πo′ = o1, o17, o112, o119, o15, o111, o117, o123 + O(o12, o2) ,
We demand that the leading monomial of each period is absent from the other solutions
to specify the vector uniquely. It is related to the previous basis via
where the numerical expression for To′ has been calculated with a precision of around fifty
digits.
(5.29)
(5.30)
(5.31)
(5.32)
(5.33)
(5.34)
From the solution vector it follows that every choice of flux leads to a vanishing
superpotential at o1 = 0. Moreover, our numerical analysis shows that Dσi W = 0, i = 1, 2 is
generically satisfied at O1. If one chooses the flux superpotential
W = To−′ 1To−1ΠLR,0 = −(0.237201 − 0.907908i)o1
+ (97.5605 − 9.49343i)o1o2 − (24181.7 + 1211.32i)o1o22 + O(o4) ,
this leads to the scalar potential
v = 0.011139161558549787439 + O(x2, y2) .
in terms of o1 = x + Iy at o2 = 0. A plot of the potential, expanded to order eleven, is
shown in figure 3. Note that the radius of convergence is o1 < 216 · (2 − 23/4) ≈ 69.
We did a Monte Carlo scan over non-vanishing flux vectors and found that the scalar
potential was always positive at x = y = 0. Moreover, the behaviour close to the origin
was qualitatively the same in that the gradient vanished at x = y = 0 but the Hessian was
undefined.
We also performed an analytic continuation to the special locus P where the Calabi-Yau
becomes a Gepner model. However, the behaviour of the scalar potential was qualitatively
the same as for a generic point on O1. For some recent discussion of moduli stabilization
at the point of large complex structure in a particular example see also [51].
6
Conclusions and outlook
We described a very efficient method to obtain the integral flux superpotential using the
central charge formula defined in terms of the Γˆ class . This method is simple enough to
be applied to multi moduli cases. In particular if the Calabi-Yau fourfold is embedded in
a toric ambient space it is in general straightforward to find a basis by toric intersection
calculus and the Frobenius method for constructing the periods at the points of maximal
unipotent monodromy. Example calculations in Sage can be found on our homepage [48].
We then restrict to non-singular elliptic Calabi-Yau fourfolds and study universal
monodromies in the integral basis of the horizontal cohomology and the dual homology. Using
this basis we provide general expressions for the monodromies corresponding to Ti-shifts,
that act as ti → ti + 1 on the K¨ahler moduli. In the derived category these correspond to
the auto-equivalences induced by tensoring with the line bundles of the dual divisor.
Physically this is the integral Neveu-Schwarz B-field shift and the action on the periods follows
directly from their leading logarithms which are determined again by A-model intersection
numbers. In particular, the Te-shift acts as the parabolic operator T in SL(2, Z) on the
fiber parameter.
More non-trivially we extend Bridgelands construction of an auto-equivalence of
elliptic surfaces to the class of elliptic Calabi-Yau fourfolds with at most I1 singularities in
the fibers. This provides an action of the order two element S in SL(2, Z) on the fiber
parameter. Apart from being a non-trivial check of the integrality of our periods, these
auto-equivalences generate the full PSL(2, Z) action on the elliptic parameter. This gives
rise to modular properties of the genus zero and holomorphic genus one amplitudes as well
as a holomorphic anomaly that we analyze in detail.
Let us summarize the types of the amplitudes and the results. The virtual dimension
formula (4.2) is positive for genus zero. Therefore we need a meeting condition for rational
curves with γ ∈ H4(M, Z) (mod torsion) and get different amplitudes Fγ(0)(q) for each γ,
whose geometry with respect to the fibration structure plays an important role. In genus
one the virtual dimension is zero and we get a universal amplitude F (1). For g > 1 the
dimension is negative and hence all higher genus amplitudes vanish. Finally one can also
consider the modular properties of the 4-point functions. The clearest situation arises
for the genus zero amplitudes associated to π-vertical 4-cycles Hk and for the genus one
amplitude as well as for the 4-point functions with all legs in the base. In each case we
get a complete and universal answer for the holomorphic anomaly equations which can be
derived using the methods in [21, 39].
For genus zero amplitudes over 4-cycles that are not π-vertical we observe a modular
anomaly equation only for the E8 fibration over P3. However, we argue that this is a
consequence of the modular anomaly equation of the 4-point function which factorizes for
two-parameter families.
We also check the integrality of the curve counting invariants
of [27] at genus one for various new cases.
In order to study the global properties of the horizontal flux superpotential relevant
for F-theory compactifications, we analytically continued the periods of the mirror X2∗4
to the following critical divisors displayed in figure 1, whose symmetry implies that we
only need to consider the left half of it. We first studied the conifold divisor C1. Here we
could determine an analytic expression for the 8 × 8 continuation matrix Tc in (5.9) up
to five numerical coefficients.7
We also generalized the result of [12] that flux along the
vanishing cycle stabilizes the theory at this divisor. We further analyzed the possible flux
superpotentials at the generic orbifold divisor O1 and its special locus P .
The most obvious generalization of this work is to include singular elliptic fibrations.
Our formalism for fixing the integral periods explained in section 2.3 will work essentially
unchanged and with the same technical tools as long as we have Calabi-Yau spaces
embedded into toric varieties and the resolutions of the singularities can be described torically.
This will be essential to probe in a quantitative way the flux stabilization mechanism of
realistic F-theory vacua. The generalization of the construction of the Bridgeland
autoequivalence should also be possible in principle. In fact at least in the Calabi-Yau threefold
case the results for the all genus amplitudes which can be expressed in terms of
Weylinvariant Jacobi-Forms [52, 53] indicate that the affine Weyl-group of the singularity will
appear as part of the auto-equivalences of the derived category of the A-model.
7Further details about this highly non-trivial analytic continuation can be found at [48].
M/n0,d(J 2)
Acknowledgments
1582400
791944986400
783617464399966400
031333248042176116592000
17510976
48263250
71161472
7677952
4347594
1919808
852480
6255156277440
117715791990353760
36449586432
181688069500
354153540352
9408504320
3794687028
988602624
259476480
100346754888576
905026660335000
2336902632563200
15215566524416
4368985908840
669909315456
103646279680
HJEP01(28)6
We would like to thank Andreas Gerhardus, Thomas Grimm, Babak Haghighat, Min-Xin
Huang, Hans Jockers, Amir-Kian Kashani-Poor, Sheldon Katz and Georg Oberdieck for
discussions and comments. In particular we would like to thank Georg Oberdieck for notes
he provided shortly after this paper appeared on the arxiv. AK would like to thank Min-Xin
Huang and Sheldon Katz for comments on the draft and the IHES for hospitality during
the time when this work was initiated. CFC would like to thank the financial support from
the fellowship “Regierungsstipendiaten CONACYT-DAAD mit Mexiko” under the grant
number 2014 (50015952) and the BCGS for their generous support. TS would like to thank
the BCGS for their generous support as well as Amir-Kian Kashani-Poor and the ENS for
hospitality during part of the work on this project.
A
Supplementary data
A.1
Curve counting invariants for one parameter fourfolds
Here we report the genus zero and genus one curve counting invariants for nine one
parameter fourfolds in toric ambient spaces with generalized hypergeometric type Picard-Fuchs
equations. The genus zero invariants agree with the ones calculated in [12]. The genus
one invariants provide a new test for the multi covering formula derived in [27]. Similar
checks for one parameter Calabi-Yau spaces in Grassmannian ambient spaces with Apery
type Picard-Fuchs operators were provided in [28].
2734099200
2813440
9058000
146150912
47104
53928
1024
3779200
26873294164654597632
1006848150400512
20201716419250520
132534541018149888
42843921424384
7776816583356
338199639552
27474707200000
HJEP01(28)6
Here we consider the hypersurface MPE18×P2 . This arises from an E8 fibration over the base
Hence the polytope Δ∗ corresponding to the fibration over Δ∗E8 is given by
div.
D0′
D2′
D2′
D1′
D1′
D2′ −1
1 −1
0 −1
ν¯∗B
i
0
1
0
0
0 −1
0
1
0
0
0
2
2
2
2
2
2
0 −1
ν¯∗
i
0
0
0
1
0
0
0
0
1
0
0
0
0
3
3
3
3
3
div. coord.
KM
2D˜ e
3D˜ e
E
x0
x
y
z
˜
˜
˜
˜
˜
D2 u1
D2 u2
D2 u3
D1 u4
D1 u5
1
1
1
1
1
1
1
1
3 −1
0 −1
(A.1)
(A.2)
387176346729900
81906297984
845495712250
5670808217856
4277292544
1203128235
65526084
15090827264
l′(1) l′(
2
)
0 −2 −3
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0 −1
The intersections among divisors lead to the constants in (3.12),
(
2
3
cijk =
c122 = c212 = c221 = 1 ,
0 otherwise ,
a = 54, ai =
, ai =
9 12 , aij =
The genus zero amplitudes in the basis (A.5) read
The Picard-Fuchs equations read
L1 = θe(θe − 2θ1 − 3θ2) − 12ze(6θe + 5)(6θe + 1) ,
L2 = θ12 − z1(θe − 2θ1 − 3θ2)(θe − 2θ1 − 3θ2 − 1) ,
L3 = θ23 − z2(θe − 2θ1 − 3θ2)(θe − 2θ1 − 3θ2 − 1)(θe − 2θ1 − 3θ2 − 2) .
Here the coordinates za are determined by (4.29). The discriminants of the Picard-Fuchs
equations are given by
HJEP01(28)6
Δ1 = (−1 + 4z1)3 − 54(1 + 12z1)z2 − 729z22 ,
Δ2 = −h − 1 + 864ze 1 + 216ze(−1 + 4z1) i3 + 4738381338321616896ze6z22
+ 4353564672ze3(−1 + 432ze)h1 + 864ze − 1 + 216ze(1 + 12z1) iz2 .
Using the choice of basis for 4-cycles in (3.13) we obtain
H1 = E · D˜1,
H2 = E · D˜2,
H1 = D˜22,
H2 = D˜1D˜2 .
Hence the (3.14) intersections follow as
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
η(
2
) = −3 −2 0 1 .
1 0 0 0
F H(01) = τ 2 + τ t1 + 1 + F H(01),inst(qe, Qe1, Qe2) ,
2
F H(02) = 3 τ 2 + τ t2 + 1 τ +
+ F H(02),inst(qe, Qe1, Qe2) .
2
5
4
3
2
In appendix A.5 we show some of the instanton expansions of the above expressions in
terms of quasi-modular forms. Note that we make use of a ‘pure modular’ basis — as in
the case of X24 — to compute the modular weight zero components of the F H(0i) periods.
We define such a basis as follows
F H(01◦) ≡ F H(01) +
32 F H(02) ,
F H(02◦) ≡ F H(02) + F H(02) +
23 F H(01) .
(A.8)
(A.10)
(A.11)
constants related to the Chern classes are
1
1
b1 = − 24 c3 TMPE18×P2 · D1 − 1 =
be = − 24 c3 TMPE18×P2 · De − 1 =
,
539
4
This leads to the genus one amplitude
1
b2 = − 24 c3 TMPE18×P2 · D2 − 1 = 29 ,
˜
, χ = 19728 .
where we give part of the expansion of F (1),inst in terms of quasi-modular forms in
appendix A.5.
The second Chern class of MPE18×P2 can be written in terms of the basis (A.5) as
We compute the genus zero Gromov-Witten invariants of (A.9) in appendix A.3. Further
c2 TMPE18×P2
= 24H1 + 36H2 + 102H1 + 138H2 .
(A.9)
Geometric invariants for MPE18×P2
-731942730 605426932980 -139049122837500 17515925402297760
3
-243
-49734
-1719756
-27555633
-277450434
3
3
138240
34833240
1396005840
25003580040
274895998560
-35363790
-12410449830
-583905569940
4
2304
904500
56117574
1515365226
24502800744
4
4
-1698840
-786936060
-55422152100
-1654348658580
-29038118214600
578799000
342447273720
27847911802680
HJEP01(28)6
n0,(0,d1,d2)(H2) d2 = 0
n0,(1,d1,d2)(H2) d2 = 0
n0,(2,d1,d2)(H2) d2 = 0
6
0
0
0
0
0
0
0
0
0
720
-720
1440
-1440
1
-24
-192
-744
-2040
-4560
1
6120
67680
314280
961920
2313000
1
-1036800
-13718160
-69796080
-230700960
2
114
4440
55050
390744
1973472
2
-43920
-2349360
-34350480
-274751280
-1517061600
2
8217540
660289320
11223041760
99434663640
3
-864
-93744
-2528040
-34977312
-318919680
3
3
495360
65718720
2043688320
31523616000
313418304000
-131045040
-23552058960
-851198459760
4
8808
1898622
92087760
2139264666
31152820512
4
4
-6528960
-1654942320
-90803818800
-2326758388560
-36732061356480
2264001480
724510733760
45595230845400
-14568373007280
1290110994869760
-588578400 593689222980 -157013407044000 22030115559925320
0
0
0
0
0
0
0
0
0
0
0
0
0
1
6
30
84
180
330
1
-1800
-12240
-39960
-93600
-181800
1
377460
2668140
9566100
2
-30
-870
-8682
-51600
-224112
2
12240
493920
5793120
38578320
182164320
2
-2483820
-147669480
-1999807560
3
234
20196
460512
5535630
44650908
3
3
d2 = 0
d1 = 0
720
0
-3
0
0
0
0
0
0
0
0
0
0
3
81
16578
573252
9185211
92483478
3
-51840
-12547080
-492862320
-8722465560
-95139102240
3
15155910
4798354950
4
-768
-301500
-18705858
-505121742
-8167600248
4
4
617760
279335520
19402918200
574019167320
10012773524400
-231519600
-129069932760
2
3
4
1
2
3
4
1
2
3
4
0
0
0
0
0
0
0
-18
-18
0
0
0
0
0
1
36
288
972
2304
4500
1
-11772
-123228
-498420
-1313172
-2743308
0
0
1161
15174
110151
2
-90
-5166
-681210
-10098990
-80360496
2
47052
2995704
208221228
3364230240
70
-990
-183402
-4442538
-56477430
3
-30744
698400
138997944
3786456528
52885199952
3
6547608
-257783256
-54043640640
-1643238648792
-1602
12496941
487139904
199225723852
4
4
706572
-43754310
-11571378390
-504941463486
-218756626565280
-221005710
18293975928
5483304374166
265936355246088
965359376676 -273025875142044
36253634952195918
(0, 1, 0)
(1, 0, 0)
(
0, 0, 2
)
(0, 1, 1)
(
0, 2, 0
)
(1, 0, 1)
(1, 1, 0)
(
2, 0, 0
)
β2
-180
(0, 0, 1) (0, 1, 0) (1, 0, 0) (
0, 0, 2
) (0, 1, 1) (
0, 2, 0
)
378
-2016
72
-342
0
-1422
6894
648
-15012
0
0
0
0
0
5400
-24840
-2160
52920
0
38800
(1, 0, 1)
36720
-18360
376920
0
(1, 1, 0)
-11880
57240
-85320
0
0
38800
(
2, 0, 0
)
10800
-49680
-4320
105840
0
38800
-1744200
516240
-7069680 3656800 -3499200
FHb
2
Q
η48 e
Q
!2
(0)
+P70 (Hb)
η48 e
Q
!3
+···
(A.12)
(0)
P22 (Hb) = − 18
(0)
P46 (Hb) = − 12
5
1
E4E6(35E43 +37E62)
(0)
E2 P22 (Hb)
2
5E4E6
− 2985984
+208392741E43E64 +27245569E66
(0)
P70 (Hb) =
1
1671768834048
29908007E49 +207234483E46E62
q
2
Q
+P48 (Hb )
q
2
Q
+P72 (Hb )
q
2
Q
!3
+··· (A.13)
FHb◦
0
(Hb )+P24 (Hb )
P
0
(Hb ) = 960
P24 (Hb ) =
P48 (Hb ) =
5
10368
5
13759414272
∞
d=1
X σ3(d) d
qe ,
10321E46 +1680E2E44E6 +59182E43E62 +1776E2E4E63 +9985E64
34974695189E412 +955855257580E49E62 +2375228903358E46E64
+576E2E4E6 19602269E49 +134498081E46E62 +137176487E43E64 +18933723E66
2
!
P72 (Hb ) =
5
12999674453557248
169868512046891311E418 +10991441298020921814E415E62
+154580302495588543E612 +16124313600E23E43E63 35E43 +37E62
3
+8398080E22E42E62 35E43 +37E62
44357407E49 +305364363E46E62 +309961101E43E64
+42023369E66
+432E2E4E6 188980289153801E415 +4041754304722571E412E62
+171728558335663E610
– 46 –
d=1
qe ,
−10321E46 +34440E2E44E6 −59182E43E62 +36408E2E4E63 −9985E64
η48 e
Q
η48 e
Q
η48 e
Q
+···
(A.14)
−8718461011E412 −238460285300E49E62 −592848334770E46E64
−239525096180E43E66 −8301513619E68 +7649280E22E42E62(35E43 +37E62)2
!
0
P
0
P24
P48
= −2
A.5
Modular expressions for MP1×P2
P72
5
2166612408926208
−54494943725199823E418 −3526301098569327294E415E62
26341691595249846705E46E68 −3475678553808910878E43E610
−50493219640852471E612 +337714790400E23E43E63 35E43 +37E62
+1399680E22E42E62(35E43 +37E62)(3711620489E49 +25730000061E46E62
3
+25856467947E43E64 +3371520463E66)+108E2E4E6 5231073695092861E415
FHi
P
(0)
6(a1d1+a2d2)−2(Hi)
P16(H1) = E4 + E4E62 , P16(H2) =
P10(H1) = 0,
P22(H1) = 0,
P28(H1) =
P28(H2) =
P34(H1) = − 9216
P34(H2) = − 110592
1
1
9
2
3
8
X
d1,d2
3
2
85
48
1
192
X
d1,d2
FHi◦
P
6(a1d1+a2d2)(Hi )
E8
η12
12 !a d1+a2d2
1
Qe1d1 Qe2d2
P10(H2) = −3E4E6 ,
P22(H2) = − 32
E42E6 − 32
E4E63 ,
16
E4E64
qe
η12
21 !a d1+a2d2
1
Qe1d1 Qe2d2
E4 48359E46E6 +161426E43E63 +39047E65 +24E2E4 E43 +3E62
31E43 +113E62
E4 208991E46E6 +755906E43E63 +196319E65 +4E2E4 31E43 +113E62
!
(A.15)
2!
(A.16)
∞
d=1
d2 e
P0(H1 )=−24(b1+1)X σ3(d) d
◦
q , P0(H2 )=−24(b2+1)X σ3(d) d
◦
q ,
∞
d=1
d2 e
49E43+ 94E62,
P12(H2 )=
23E43+ 1E2E4E6+E62,
2
−31E2E44−926E43E6−113E2E4E62−226E63 ,
−137E43E6−47E63−2E2 E44+3E4E62
,
51E46+17E2E44E6+199E43E62+7E2E4E63+14E64 ,
−648E2E47−E45 31E22+52747E4 E6−4444E2E44E62
−386E2E47−E45 72E22+32849E4 E6−5002E2E44E62
P36(H1 )=
628895E49+438639E2E47E6+9743040E46E62+1649058E2E44E63
P36(H2 )=
333303E49+54875E2E47E6+5411350E46E62+220490E2E44E63
P12(H1 )=
P18(H1 )=
P18(H2 )=
P24(H1 )=
P24(H2 )=
P30(H1 )=
1
1
16
128
1
1
384
1
2304
1
1152
1
1327104
1
221184
(1)
P0 =−2
(1) 3
(1)
P18 =
(1)
P24 =
(1)
P30 =
X
d1,d2
F (1),inst =
(1)
P6(a1d1+a2d2) η12
21 !a1d1+a2d2
Qe1d1Qe2d2
d
∞
d=1
qe ,
24 −h11
−6E43+10E2E4E6−5E62 ,
33E46−24E2E44E6+2E42 −2E22+71E4 E62−16E2E4E63+13E64 ,
– 48 –
(A.17)
108E22E42 31E43+113E62 577E43+1871E62 +36E2 3099607E47E6
+10537042E44E63+2578903E4E65 .
A.6 Analytic continuation data for X24(1, 1, 1, 1, 8, 12)
We provide the numerical and — as far as we know them — analytic expressions for the
continuation matrices Tc, Tc′, To, To′ in a Mathematica worksheet on the webpage [48]. Due
to their special importance we reproduce here the intersection matrix at c1 = c2 = 0 as
well as the entries of the continuation matrix to the point ze = c1 = 0:
(TcTc′)T η−1TcTc′ = κ
204448 −30916905725128 2836622426104377556352
135
−466556
0
33392
204448
−46656
5
−3083774976
25
0
0
0
0
0
2519424
−308372754976
32651573504
0
0
5
0
0
where
0
1
1
−604656176
24428335104
−15768933728256 0
125
0
0
0
0
32
(A.18)
1
κ= 1572864π4 .
f1,1 = −2916π12r44+10260π6r42−27π4r5r4+144r12−32r22−7129
1152
f2,1 = 3
1(6r1+4ir2+3)
f3,1 = 3184 216π6r42+48r1+32ir2+3iπr5−404
f7,1 =−12i(2r2−9π3r4)
f5,1 = 48 −54π6r42−36iπ3r4−12r1+8ir2+101
f1,2 = 214 3r1r3−√2r2r3+384π2r4−24r5
16π
f5,2 =−−i√2πr3+πr3+768π3r4+256i
8π
– 49 –
−5760π4r12r3 +1280π4r22r3 +285160π4r3 +349440π2r3 +4464π4r3r5 /(768π4r3)
32π4r3
128π4r3
f3,3 = 15i√2π4r32 +15π4r32 +17280π10r42r3 −428544iπ9r42r3 +3840π4r1r3 +2560iπ4r2r3
+142848π6r4r3 −92160iπ5r4r3 +240iπ5r5r3 −32320π4r3 +738048iπ3r3 +15360π2r3
15 3i√2π4r32 +2π4r32 +512π4r1r3 +512iπ4r2r3 +512π4r3 +786432i√2−524288
f5,3 = − −15i√2π4r32 +15π4r32 +17280π10r42r3 +3840π4r1r3 −2560iπ4r2r3 +11520iπ7r4r3
+142848π6r4r3 −32320π4r3 +47616iπ3r3 +15360π2r3 −3932160i√2
−3932160)/(256π4r3)
5i 3√2π4r32 +512π4r2r3 +786432√2
64π4r3
256π4r3
f2,3 =
f4,3 =
f6,3 =
f7,3 =
f1,4 =
1
8π
r2 = −1. 29219644630091977480074761037
r4 = −0.00948778220735050311547607017424
B
General genus zero modular anomaly equation
After releasing a pre-print of this paper it was brought to our attention by Georg Oberdieck
that there is a conjectured modular anomaly equation for elliptic Calabi-Yau n-folds in [23,
24]. For n = 4 the conjecture implies the modular anomaly equations for the
GromovWitten potentials associated to π-vertical cycles and for genus one free energies that we
derived in this paper. For the non π-vertical cycles the conjectured anomaly equation
agrees with our results for MPE38 and we also checked it for the Gromov-Witten potentials of
MPE18×P2. Our results on the modular structure can therefore be seen as a partial derivation
and non-trivial check of the holomorphic anomaly equation conjectured in [23, 24] for
Calabi-Yau fourfolds. We will now briefly describe the general form of the holomorphic
anomaly equations for genus zero Gromov-Witten potentials.
Let Fγ(1g,)...,γm be the string amplitude associated to the Gromov-Witten invariants
Ng,κ(γ1, . . . , γm) =
Z
[Mg,n(M,κ)]vir i
Y evi∗(γi) ,
as these fulfill the intersection relations
Hi · Hj◦ = δji ,
Hi · Hj = 0 ,
Hi◦ · Hj◦ = 0 .
where κ ∈ H2(M, Z) and γi ∈ H∗(M, Z). On the one hand, given β ∈ H2(B, Z) conjecture
A in [23, 24] implies that
Coeff(Fγ(1g,)...,γm , Qβ) ∈ η12c1(B)·β C[E2, E4, E6] ,
1
which matches with our Ansatz given in expression (4.24). On the other hand, conjecture
B of [23, 24] implies a general modular anomaly equation for Fγ(1g,)...,γm .
Following the discussion of sections 3.1 and 4.4, we can make a generalization of the
pure modular basis by taking the 4-cycles,
Hi = aij akD˜ j D˜ k ,
Hi◦ = DeD˜ i , i = 1, . . . , h1,1(B),
Note that Hi
∈ H4(M, Z) while in general Hi◦ ∈/ H4(M, Z). Let ℓ ∈ H2(B) such that
hβ, ℓi 6= 0. Then for a given γ ∈ H2,2(M, C) Georg Oberdieck pointed out to us that
conjecture B of [23, 24] implies a modular anomaly equation for Fγ(0), which in the modular
basis (B.3) reads
∂Fγ(0,β) = − 12
∂E2
1 "
X
βi′Fγ(0,β)′ F
Hi,β′′ −
hβ′,ℓi2hβ′′,π∗γi+hβ′′,ℓi2hβ′,π∗γi F H(0i),β′ F H(0i◦),β′′
hβ,ℓi2
hβ,ℓi Fπ(0∗)(π∗(γ)∪ℓ),β −
hhπβ∗,γℓ,iβ2i Fπ(0∗)ℓ2,β
From the properties of the Gysin morphisms it follows that π∗Hi◦ = Di′ and π∗Hi = 0.
Hence for a π-vertical 4-cycle Hi, the modular anomaly equation (B.5) of its corresponding
string amplitude F H(0i) reduces to equation (4.54).
Now we consider the 4-cycles Hi◦ where equation (B.5.4) becomes more involved. It
is easy to verify for MPE38 that (B.5) reduces to (4.38).
Moreover, when h1,1(B) ≥ 2
equation (B.5) in general implies multiple relations, since it depends on the choice of ℓ.
We checked equation (B.5) for MPE18×P2 of which we include the toric data in appendix A.2.
We also provide some modular expressions for the corresponding amplitudes F (0◦) and F (0◦)
H1
H2
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
(B.1)
(B.3)
(B.5)
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