DOI 10.1515/CRELLE.2008.021 angewandte Mathematik (Walter de Gruyter
Berlin New York 2008
Monodromy of Picard-Fuchs di¤erential
equations for Calabi-Yau threefolds
By Yao-Han Chen and Yifan Yang at Hsinchu, and Noriko Yui at KingstonWith an appendix by Cord Erdenberger at Hannover
Abstract. In this paper we are concerned with the monodromy of Picard-Fuchs dif-ferential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy rel-ative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suit-able change of bases the monodromy groups are contained in certain congruence sub-groups of Spð4; ZÞ of finite index and whose levels are related to the geometric invariants of the Calabi-Yau threefolds.
1. Introduction
Let Mz be a family of Calabi-Yau n-folds parameterized by a complex variable
z A P1ðCÞ, and oz be the unique holomorphic di¤erential n-form on Mz (up to a scalar).
Then the standard theory of Gauss-Manin connections asserts that the periods Ð
gz
oz
satisfy certain linear di¤erential equations, called the Picard-Fuchs di¤erential equations, where gzare r-cycles on Mz.
When n¼ 1, Calabi-Yau onefolds are just elliptic curves. A classical example of Picard-Fuchs di¤erential equations is
ð1 zÞy2f zyf z
4f ¼ 0; y¼ zd=dz; ð1Þ
satisfied by the periods
Y.-H. Chen and Yifan Yang were supported by Grant 94-2115-M-009-012 of the National Science Council (NSC) of the Republic of China (Taiwan). N. Yui was supported in part by Discovery Grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada.
fðzÞ ¼ Ð y 1 dx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðx 1Þðx zÞ p
of the family of elliptic curves Ez: y2¼ xðx 1Þðx zÞ.
When n¼ 2, Calabi-Yau manifolds are either 2-dimensional complex tori or K3 sur-faces. When the Picard number of a one-parameter family of K3 surfaces is 19, the Picard-Fuchs di¤erential equation has order 3. One of the simplest examples is
x14þ x24þ x 4 3þ x
4
4 z1x1x2x3x4¼ 0 H P3;
whose Picard-Fuchs di¤erential operator is
y3 4zð4y þ 1Þð4y þ 2Þð4y þ 3Þ: ð2Þ
Another well-known example is
ð1 34z þ z2Þy3þ ð3z2 51zÞy2þ ð3z2 27zÞy þ ðz2 5zÞ; ð3Þ
which is the Picard-Fuchs di¤erential operator for the family of K3 surfaces 1 ð1 XY ÞZ zXYZð1 X Þð1 Y Þð1 ZÞ ¼ 0:
(See [7].) This di¤erential equation appeared in Ape´ry’s proof of irrationality of zð3Þ. (See [5].)
When n¼ 3 and Calabi-Yau threefolds have the Hodge number h2; 1 equal to 1, the
Picard-Fuchs di¤erential equations have order 4. One of the most well-known examples of such Calabi-Yau threefolds is the quintic threefold
x51þ x25þ x 5 3þ x 5 4þ x 5 5 z1x1x2x3x4x5¼ 0 H P4:
In [9], it is shown that the Picard-Fuchs di¤erential operator for this family of Calabi-Yau threefolds is
y4 5zð5y þ 1Þð5y þ 2Þð5y þ 3Þð5y þ 4Þ: ð4Þ
Actually, it is the mirror partner of the quintic Calabi-Yau threefolds that has Hodge number h2; 1¼ 1 and hence the Picard-Fuchs di¤erential equation is of order 4. But the
mirror pair of Calabi-Yau threefolds share the same ‘‘principle periods’’. This means that the Picard-Fuchs di¤erential equation of the original quintic Calabi-Yau threefold of order 204 contains the above order 4 equation as a factor and the factors corresponding to the remaining 200 ‘‘semiperiods’’.
In this article we are concerned with the monodromy aspect of the Picard-Fuchs dif-ferential equations. Let
be a di¤erential operator with regular singularities. Let z0be a singular point and S be the
solution space of L at z0. Then analytic continuation along a closed curve g circling z0gives
rise to an automorphism of S, called a monodromy. If a basisf f1; . . . ; fng of S is chosen,
then we have a matrix representation of the monodromy. Suppose that fi becomes
ai1f1þ þ ainfnafter completing the loop g, that is, if
f1 .. . fn 0 B B @ 1 C C A 7! a11 a1n .. . .. . an1 ann 0 B @ 1 C A f1 .. . fn 0 B B @ 1 C C A;
then the matrix representation of the monodromy along g relative to the basisf fig is the
matrixðaijÞ. The group of all such matrices is referred to as the monodromy group relative
to the basisf fig of the di¤erential equation. Clearly, two di¤erent choices of bases may
re-sult in two di¤erent matrix representations for the same monodromy. However, it is easily seen that they are connected by conjugation by the matrix of basis change. Thus, the mono-dromy group is defined up to conjugation. In the subsequent discussions, for the ease of ex-position, we may often drop the phrase ‘‘up to conjugation’’ about the monodromy groups, when there is no danger of ambiguities.
It is known that for one-parameter families of Calabi-Yau varieties of dimension one and two (i.e., elliptic curves and K3 surfaces, respectively), the monodromy groups are very often congruence subgroups of SLð2; RÞ. For instance, the monodromy group of (1) is Gð2Þ, while those of (2) and (3) are G0ð2Þ þ o2 and G0ð6Þ þ o6, respectively, where od denotes
the Atkin-Lehner involution. (Technically speaking, the monodromy groups of (2) and (3) are subgroups of SLð3; RÞ since the order of the di¤erential equations is 3. But because (2) and (3) are symmetric squares of second-order di¤erential equations, we may describe the monodromy in terms of the second-order ones.) Moreover, suppose that y0ðzÞ ¼ 1 þ is
the unique holomorphic solution at z¼ 0 and y1ðzÞ ¼ y0ðzÞ log z þ gðzÞ is the solution with
logarithmic singularity. Set t¼ cy1ðzÞ=y0ðzÞ for a suitable complex number c. Then z, as a
function of t, becomes a modular function, and y0zðtÞbecomes a modular form of weight
1 for the order 2 cases and of weight 2 for the order 3 cases. For example, a classical result going back to Jacobi states that
y23¼2F1 1 2; 1 2;1; y42 y43 ! ; where y2ðtÞ ¼ q1=8 P n A Z qnðnþ1Þ=2; y3ðtÞ ¼ P n A Z qn2=2; q¼ e2pit;
or equivalently, that the modular form yðtÞ ¼ y32, as a function of zðtÞ ¼ y24=y34, satisfies (1). Here2F1denotes the Gauss hypergeometric function.
In this paper we will address the monodromy problem for Calabi-Yau threefolds. At first, given the experience with the elliptic curve and K3 surface cases, one may be tempted to guess that the monodromy group of such a di¤erential equation will be the symmetric cube of some congruence subgroup of SLð2; RÞ. After all, there is a result by Stiller [23] (see also [27]) asserting that if tðtÞ is a non-constant modular function and F ðtÞ is a
ular form of weight k on a subgroup of SLð2; RÞ commensurable with SLð2; ZÞ, then F ; tF ; . . . ; tkF , as functions of t, are solutions of aðk þ 1Þ-st order linear di¤erential equa-tion with algebraic funcequa-tions of t as coe‰cients. However, this is not the case in general. A quick way to see this is that the coe‰cients of the symmetric cube of a second order dif-ferential equation y00þ r
1ðtÞy0þ r0ðtÞy ¼ 0 are completely determined by r1and r0, but the
coe‰cients of the Picard-Fuchs di¤erential equations, including (4), do not satisfy the re-quired relations. (The exact relations can be computed using Maple’s command symmetric _power.) Nevertheless, in the subsequent discussion we will show that, with a suitable choice of bases, the monodromy groups for Calabi-Yau threefolds are contained in certain con-gruence subgroups of Spð4; ZÞ whose levels are somehow described in terms of the geomet-ric invariants of the manifolds in question. This is proved rigorously for the hypergeometgeomet-ric cases and verified numerically for other (e.g., non-hypergeometric) cases. Furthermore, our computation in the hypergeometric cases shows that the matrix representation of the mono-dromy around the finite singular point (di¤erent from the origin) relative to the Frobenius basis at the origin can be expressed completely using the geometric invariants of the asso-ciated Calabi-Yau threefolds. This phenomenon is also verified numerically in the non-hypergeometric cases. Although it is highly expected that geometric invariants will enter into the picture, in reality, geometry will dominate the entire picture in the sense that every entry of the matrix is expressed exclusively in terms of the geometric invariants.
The monodromy problem in general has been addressed by a number of authors. Pa-pers relevant to our consideration include [6], [9], [11], [16], and [26], to name a few. In [6], Beukers and Heckman studied monodromy groups for the hypergeometric functionsnFn1.
They showed that the Zariski closure of the monodromy groups of (4) is Spð4; CÞ. The same is true for other Picard-Fuchs di¤erential equations for Calabi-Yau threefolds that are hypergeometric. In [9], Candelas et al. obtained precise matrix representations of mono-dromy for (4). Then Klemm and Theisen [16] applied the same method as that of Candelas et al. to deduce monodromy groups for three other hypergeometric cases. In [11] Doran and Morgan determined the monodromy groups for all the hypergeometric cases. Their matrix representations also involve geometric invariants of the Calabi-Yau threefolds. For Picard-Fuchs di¤erential equations of non-hypergeometric type, there is not much known in liter-ature. In [26] van Enckevort and van Straten computed the monodromy matrices numeri-cally for a large class of di¤erential equations. In many cases, they are able to find bases such that the monodromy matrices have rational entries. We will discuss the above results in more detail in Sections 3–5.
Our motivations of this paper may be formulated as follows. Modular functions and modular forms have been extensively investigated over the years, and there are great body of literatures on these subjects. As we illustrated above, the monodromy groups of Picard-Fuchs di¤erential equations for families of elliptic curves and K3 surfaces are congruence subgroups of SLð2; RÞ. This modularity property can be used to study properties of the dif-ferential equations and the associated manifolds. For instance, in [18] Lian and Yau gave a uniform proof of the integrality of Fourier coe‰cients of the mirror maps for several fam-ilies of K3 surfaces using the fact that the monodromy groups are congruence subgroups of SLð2; RÞ. For such an application, it is important to express monodromy groups in a proper way so that properties of the associated di¤erential equations can be more easily discussed and obtained. Thus, the main motivation of our investigation is to find a good representation for monodromy groups from which further properties of Picard-Fuchs di¤erential equations for Calabi-Yau threefolds can be derived.
The terminology ‘‘modularity’’ has been used for many di¤erent things. One aspect of the modularity that we would like to address is the modularity question of the Galois rep-resentations attached to Calabi-Yau threefolds, assuming that Calabi-Yau threefolds in question are defined over Q. Let X be a Calabi-Yau threefold defined over Q. We consider the L-series associated to the third e´tale cohomology group of X . It is expected that the L-series should be determined by some modular (automorphic) forms. The examples of Calabi-Yau threefolds we treat in this paper are those with the third Betti number equal to 4. It appears that Calabi-Yau threefolds with this property are rather scarce. Batyrev and Straten [4] considered 13 examples of Calabi-Yau threefolds with Picard number h1; 1¼ 1.
Then their mirror partners will fulfill this requirement. (We note that more examples of such Calabi-Yau threefolds were found by Borcea [8].) All these 13 Calabi-Yau threefolds are defined as complete intersections of hypersurfaces in weighted projective spaces, and they have defining equations defined over Q.
To address the modularity, we ought to have some ‘‘modular groups’’, and this paper o¤ers candidates for appropriate modular groups via the monodromy group of the associ-ated Picard-Fuchs di¤erential equation (of order 4). In these cases, we expect that modular forms of more variables, e.g., Siegel modular forms associated to the modular groups for our congruence subgroups would enter the scene.
In general, the third Betti numbers of Calabi-Yau threefolds are rather large, and consequently, the dimension of the associated Galois representations would be rather high. To remedy this situation, we first decompose Calabi-Yau threefolds into motives, and then consider the motivic Galois representations and their modularity. Especially, when the prin-cipal motives (e.g., the motives that are invariant under the mirror maps) are of dimension 4, the modularity question for such motives should be accessible using the method devel-oped for the examples discussed in this paper.
The modularity questions will be treated in subsequent papers.
2. Statements of results
To state our first result, let us recall that among all the Picard-Fuchs di¤erential equa-tions for Calabi-Yau threefolds, there are 14 equaequa-tions that are hypergeometric of the form
y4 Czðy þ AÞðy þ 1 AÞðy þ BÞðy þ 1 BÞ:
Their geometric descriptions and references are given in the following Table 1.
K A B C description H3 c 2 H c3 ref 1 1/5 2/5 3125 Xð5Þ H P4 5 50 200 [9] 2 1/10 3/10 8 105 Xð10Þ H P4ð1; 1; 1; 2; 5Þ 1 34 288 [20] 3 1/2 1/2 256 Xð2; 2; 2; 2Þ H P7 16 64 128 [19] 4 1/3 1/3 729 Xð3; 3Þ H P5 9 54 144 [19]
5 1/3 1/2 432 Xð2; 2; 3Þ H P6 12 60 144 [19] 6 1/4 1/2 1024 Xð2; 4Þ H P5 8 56 176 [19] 7 1/8 3/8 65536 Xð8Þ H P4ð1; 1; 1; 1; 4Þ 2 44 296 [20] 8 1/6 1/3 11664 Xð6Þ H P4ð1; 1; 1; 1; 2Þ 3 42 204 [20] 9 1/12 5/12 126 Xð2; 12Þ H P5ð1; 1; 1; 1; 4; 6Þ 1 46 484 [11] 10 1/4 1/4 4096 Xð4; 4Þ H P5ð1; 1; 1; 1; 2; 2Þ 4 40 144 [17] 11 1/4 1/3 1728 Xð4; 6Þ H P5ð1; 1; 1; 2; 2; 3Þ 6 48 156 [17] 12 1/6 1/4 27648 Xð3; 4Þ H P5ð1; 1; 1; 1; 1; 2Þ 2 32 156 [17] 13 1/6 1/6 28 36 Xð6; 6Þ H P5ð1; 1; 2; 2; 3; 3Þ 1 22 120 [17] 14 1/6 1/2 6912 Xð2; 6Þ H P5ð1; 1; 1; 1; 1; 3Þ 4 52 256 [17]
Some comments might be in order for the notations in the table. We employ the notations of van Enckevort and van Straten [26]. Xðd1; d2; . . . ; dkÞ H Pnðw0; . . . ; wnÞ stands for a
com-plete intersection of k hypersurfaces of degree d1; . . . ; dk in the weighted projective space
with weightðw0; . . . ; wnÞ. For instance, X ð3; 3Þ H P5 is a complete intersection of two
cu-bics in the ordinary projective 5-space P5defined by fY13þ Y 3 2 þ Y 3 3 3fY4Y5Y6¼ 0g X f3fY1Y2Y3þ Y43þ Y 3 5 þ Y 3 6 ¼ 0g:
Slightly more generally, Xð4; 4Þ H P5ð1; 1; 2; 1; 1; 2Þ denotes a complete intersection of two
quartics in the weighted projective 5-space P5ð1; 1; 2; 1; 1; 2Þ and may be defined by the equations fY4 1 þ Y 4 2 þ Y 2 3 4fY4Y5Y6¼ 0g X f4fY1Y2Y3þ Y44þ Y 4 5 þ Y 2 6 ¼ 0g:
We note that all these examples of Calabi-Yau threefolds M have the Picard number h1; 1ðMÞ ¼ 1. Let OðHÞ be the ample generator of the Picard group PicðMÞ F Z. The basic
invariants for such a Calabi-Yau threefold M are the degree d :¼ H3, the second Chern
number c2 H and the Euler number c3 (the Euler characteristic of M). The equations are
numbered in the same way as in [1].
In [9], using analytic properties of hypergeometric functions, Candelas et al. proved that with respect to a certain basis, the monodromy matrices around z¼ 0 and z ¼ 1=3125 for the quintic threefold case (Equation 1 from Table 1) are
51 90 25 0 0 1 0 0 100 175 49 0 75 125 35 1 0 B B B @ 1 C C C A and 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A;
respectively. (Note that these two matrices are both in Spð4; ZÞ.) Applying the same method as that of Candelas et al., Klemm and Theisen [16] also obtained the monodromy of the one-parameter families of Calabi-Yau threefolds for Equations 2, 7, and 8.
Presum-ably, their method should also work for several other hypergeometric cases. However, the method fails when the indicial equation of the singularity y has repeated roots. To be more precise, it does not work for Equations 3–6, 10, 13 and 14. Moreover, the method uses the explicit knowledge that the singular point z¼ 1=C is of conifold type. (Note that in geometric terms, a conical singularity is a regular singular point whose neighborhood looks like a cone with a certain base. For instance, a 3-dimensional conifold singularity is locally isomorphic to XY ZT ¼ 0 or equivalently, to X2þ Y2þ Z2þ T2¼ 0.
Reflect-ing to the Picard-Fuchs di¤erential equations, this means that the local monodromy is uni-potent of index 1.) Thus, it can not be applied immediately to study monodromy of general hypergeometric di¤erential equations.
In [11] Doran and Morgan proved that if the characteristic polynomial of the mono-dromy around y is
x4þ ðk 4Þx3þ ð6 2k þ dÞx2þ ðk 4Þx þ 1;
then there is a basis such that the monodromy matrices around z¼ 0 and z ¼ 1=C are
1 1 0 0 0 1 d 0 0 0 1 1 0 0 0 1 0 B B B @ 1 C C C A and 1 0 0 0 k 1 0 0 1 0 1 0 1 0 0 1 0 B B B @ 1 C C C A; ð5Þ
respectively. It turns out that these numbers d and k both have geometric interpre-tation. Namely, the number d ¼ H3 is the degree of the associated threefolds and
k¼ c2 H=12 þ H3=6 is the dimension of the linear systemjHj. Doran and Morgan’s
rep-resentation has the advantage that the geometric invariants can be read o¤ from the ma-trices directly (although there is no way to extract the Euler number c3from the matrices),
but has the disadvantage that the matrices are no longer in the symplectic group (in the strict sense).
Before we state our Theorem 1, let us recall the definition of Frobenius basis. Since the only solution of the indicial equation at z¼ 0 for each of the cases is 0 with multiplicity 4, the monodromy around z¼ 0 is maximally unipotent. (See [20] for more detail.) Then the standard method of Frobenius implies that at z¼ 0 there are four solutions yj,
j ¼ 0; 1; 2; 3; with the property that
y0¼ 1 þ ; y1¼ y0log zþ g1; y2¼ 1 2y0log 2z þ g1log zþ g2; ð6Þ y3¼ 1 6y0log 3z þ1 2g1log 2z þ g2log zþ g3;
where gi are all functions holomorphic and vanishing at z¼ 0. We remark that these
solu-tions satisfy the relation
y0 y3 y0 0 y30 ¼ y1 y2 y0 1 y20 ;
and therefore the monodromy matrices relative to the ordered basisfy0; y2; y3; y1g are in
Spð4; CÞ, as predicted by [6]. Now we can present our first theorem. Theorem 1. Let
L : y4 Czðy þ AÞðy þ 1 AÞðy þ BÞðy þ 1 BÞ
be one of the 14 hypergeometric equations, and H3, c2 H, and c3 be geometric invariants
of the associated Calabi-Yau threefolds given in the table above. Let yj, j¼ 0; . . . ; 3,
be the Frobenius basis specified by (6). Then with respect to the ordered basis fy3=ð2piÞ3; y2=ð2piÞ2; y1=ð2piÞ; y0g, the monodromy matrices around z ¼ 0 and z ¼ 1=C are
1 1 1=2 1=6 0 1 1 1=2 0 0 1 1 0 0 0 1 0 B B B @ 1 C C C A and 1þ a 0 ab=d a2=d b 1 b2=d ab=d 0 0 1 0 d 0 b 1 a 0 B B B @ 1 C C C A; ð7Þ respectively, where a¼ c3 ð2piÞ3zð3Þ; b¼ c2 H=24; d ¼ H 3 :
Remark 1. We remark that by conjugating the matrices in (7) by the matrix
d 0 b a 0 d d=2 d=6þ b 0 0 1 1 0 0 0 1 0 B B B @ 1 C C C A;
we do recover Doran and Morgan’s representation (5). Thus, our Theorem 1 strengthens the results of Doran and Morgan [11]. Although the referee suggested that Theorem 1 might be a reformulation of the results of Doran and Morgan, we do not believe that is the case. For one thing, the argument of Doran-Morgan is purely based on Linear Algebra. It might be possible to derive our Theorem 1 combining the results of Doran-Morgan and those of Kontsevich; we will not address this question here, but left to future investiga-tions.
The appearance of the geometric invariants c2, c3, H and d is not so surprising. In [9],
it was shown that the conifold period, defined up to a constant as the holomorphic solution fðzÞ ¼ a1ðz 1=CÞ þ a2ðz 1=CÞ2þ at z ¼ 1=C that appears in the unique solution
fðzÞ logðz 1=CÞ þ gðzÞ with logarithmic singularity at z ¼ 1=C, is asymptotically H3 6ð2piÞ3 log 3z þc2 H 48pi log zþ c3 ð2piÞ3zð3Þ þ ð8Þ
near z¼ 0. (See also [15].) Therefore, it is expected that the entries of the monodromy ma-trices should contain the invariants. However, it is still quite remarkable that the matrix is
determined completely by the invariants exclusively. We have numerically verified the phenomenon for other families of Calabi-Yau threefolds, and also for general di¤erential equations of Calabi-Yau type. (See [1] for the definition of a di¤erential equation of Calabi-Yau type. See also Section 5 below.) It appears that if the di¤erential equation has at least one singularity with exponents 0, 1, 1, 2, then there is always a singularity whose monodromy relative to the Frobenius basis is of the form stated in the theorem. Thus, this gives a numerical method to identify the possible geometric origin of a di¤erential equation of Calabi-Yau type.
We emphasize that our proof of Theorem 1 is merely a verification. That is, we can prove it, but unfortunately it does not give any geometric insight why the matrices are in this special form.
Acutally, the referee has pointed out that such a geometric interpretation seems to ex-ist by Kontsevich. In the framework of ‘‘homological mirror symmetry’’ of Kontsevich, the first matrix in Theorem 1 would be the matrix associated to tensoring by the hyperplane line bundle in the bounded derived category of sheaves on the Calabi-Yau variety. In gen-eral, the matrices in Theorem 1 describe the cohomology action of certain Fourier-Mukai functors. In particular, this explains why the matrices are determined by topological invari-ants of the underlying Calabi-Yau manifolds. The paper of van Enckevort and van Straten [26] addressed monodromy calculations of fourth order equations of Calabi-Yau type based on homological mirror symmetry. The reader is referred to the article [26] for full details about geometric interpretations of matrices. We wonder, though, if the Kontsevich’s results fully explain why there are no ‘‘non-geometric’’ numbers in the second matrices. To be more precise, here is our question. Since the second matrix M is unipotent of rank 1, we know that the rows of M Id are all scalar multiples of a fixed row vector. We probably can deduce from Kontsevich’s result that the fourth row is ðd; 0; b; aÞ, but why the first three rows of M Id are a=d, b=a, and 0 times this vector (but not other ‘‘non-geometric’’ scalars)?
Now conjugating the matrices in (7) by
0 0 1 0 0 0 0 1 0 d d=2 b d 0 b a 0 B B B @ 1 C C C A; ð9Þ
we can bring the matrices into the symplectic group Spð4; ZÞ. The results are
1 1 0 0 0 1 0 0 d d 1 0 0 k 1 1 0 B B B @ 1 C C C A and 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A
for z¼ 0 and z ¼ 1=C, respectively, where k ¼ 2b þ d=6. Since the monodromy group is generated by these two matrices, we see that the group is contained in the congruence sub-group Gd; gcdðd; kÞ, where the notation Gðd1; d2Þ with d2j d1represents
Gðd1; d2Þ ¼ g A Spð4; ZÞ : g 1 1 0 0 0 1 0 0 0 B B B @ 1 C C C Amod d1 8 > > > < > > > : 9 > > > = > > > ; X g A Spð4; ZÞ : g 1 1 0 1 0 0 1 0 0 0 1 0 B B B @ 1 C C C Amod d2 8 > > > < > > > : 9 > > > = > > > ; :
We remark that the entries of the matrices in Gðd1; d2Þ satisfy certain congruence relations
inferred from the symplecticity of the matrices. To be more explicit, let us recall that the symplectic group is characterized by the property that
g¼ A B
C D
ASpð2n; CÞ; where A, B, C, and D are n n blocks, if and only if
g1 ¼ D t Bt Ct At : Thus, for a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 0 B B B @ 1 C C C A
to be in Gðd1; d2Þ, the integers aij should satisfy the implicit conditions
a22a44 a24a4211; a231a14a22 a12a24; a431a14a42 a12a44 mod d1;
and
a121a43mod d2:
We now summarize our finding in the following theorem. Theorem 2. Let
y4 Czðy þ AÞðy þ 1 AÞðy þ BÞðy þ 1 BÞ
be one of the 14 hypergeometric equations. Let yj, j¼ 0; . . . ; 3, be the Frobenius basis. Then
relative to the ordered basis y1 2pi; y0; H3 2ð2piÞ2y2þ H3 4piy1 c2 H 24 y0; H3 6ð2piÞ3y3 c2 H 48pi y1 c3 ð2piÞ3zð3Þy0; the monodromy matrices around z¼ 0 and z ¼ 1=C are
1 1 0 0 0 1 0 0 d d 1 0 0 k 1 1 0 B B B @ 1 C C C A and 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A ð10Þ with d ¼ H3, k¼ H3=6þ c
2 H=12, respectively. They are contained in the congruence
sub-groups Gðd1; d2Þ for the 14 cases in the table below.
K A B d1 d2 K A B d1 d2 1 1/5 2/5 5 5 8 1/6 1/3 3 1 2 1/10 3/10 1 1 9 1/12 5/12 1 1 3 1/2 1/2 16 8 10 1/4 1/4 4 4 4 1/3 1/3 9 3 11 1/4 1/3 6 1 5 1/3 1/2 12 1 12 1/6 1/4 2 1 6 1/4 1/2 8 2 13 1/6 1/6 1 1 7 1/8 3/8 2 2 14 1/6 1/2 4 1
Remark. We remark that what we show in Theorem 2 is merely the fact that the monodromy groups are contained in the congruence subgroups Gðd1; d2Þ. Although the
congruence subgroups Gðd1; d2Þ are of finite index in Spð4; ZÞ (see the appendix by Cord
Erdenberger for the index formula), the monodromy groups themselves may not be so. At this moment, we cannot say anything definite about monodromy groups, e.g., their finite indexness. In fact, there are two opposing speculations (one by the authors, and the other by Zudilin) about monodromy groups. We believe, based on a result (Theo-rem 13.3) of Sullivan [24], that it might be justified to claim that the monodromy group is an arithmetic subgroup of the congruence subgroup Gðd1; d2Þ, and hence is of finite index.
(Andrey Todorov pointed out to us Sullivan’s theorem, though we must confess that we do not fully understand the paper of Sullivan.) The a‰rmative answer would imply the finite-ness of the total number of connected vaccua in IIB string theory. As opposed to our belief, Zudilin has indicated to us via e-mail that a heuristic argument suggests that the mono-dromy groups are too ‘‘thin’’ to be of finite index.
It would not be of much significance if the hypergeometric equations are the only cases where the monodromy groups are contained in congruence subgroups. Our numerical computation suggests that there are a number of further examples where the monodromy groups continue to be contained in congruence subgroups of Spð4; ZÞ. However, the general picture is not as simple as that for the hypergeometric cases.
As mentioned earlier, our numerical data suggest that the Picard-Fuchs di¤erential equations for Calabi-Yau threefolds known in the literature all have bases relative to which the monodromy matrices around the origin and some singular points of conifolds take the form (7) described in Theorem 1. Thus, with respect to the basis given in Theorem 2, the matrices around the origin and the conifold points again have the form (10). However,
with this basis change, the monodromy matrices around other singularities may not be in Spð4; ZÞ, but in Spð4; QÞ instead, although the entries still satisfy certain congruence rela-tions. Furthermore, in most cases, we can still realize the monodromy groups in congruence subgroups of Spð4; ZÞ, by a suitable conjugation.
Example 1. Consider the di¤erential equation 25y4 15zð51y4þ 84y3þ 72y2þ 30y þ 5Þ
þ 6z2ð531y4þ 828y3þ 541y2þ 155y þ 15Þ 54z3ð423y4
þ 2160y3þ 4399y2þ 3795y þ 1170Þ þ 243z4ð279y4
þ 1368y3þ 2270y2þ 1586y þ 402Þ 59049z5ðy þ 1Þ4
:
In [4] it is shown that this is the Picard-Fuchs di¤erential equation for the Calabi-Yau threefolds defined as the complete intersection of three hypersurfaces of degreeð1; 1; 1Þ in P2 P2 P2. The invariants are H3¼ 90, c
2 H ¼ 108, and c3¼ 90. There are 6
singu-larities 0, 1=27, Gi=pffiffiffiffiffi27, 5=9, and y for the di¤erential equation. Among them, the local exponents at z¼ 5=9 are 0, 1, 3, 4 and we find that the monodromy around z ¼ 5=9 is the identity. For others, our numerical computation shows that relative to the basis in Theorem 2 the monodromy matrices are
T0¼ 1 1 0 0 0 1 0 0 90 90 1 0 0 24 1 1 0 B B B @ 1 C C C A; T1=27¼ 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A; Ti=pffiffiffiffi27 ¼ 17 3 1=3 1 54 10 1 3 972 162 19 54 162 27 3 8 0 B B B @ 1 C C C A; Ti=pffiffiffiffi27 ¼ 11 3 1=3 1 36 8 1 3 432 108 13 36 108 27 3 10 0 B B B @ 1 C C C A: From these, we see that the monodromy group is contained in the following group:
ðaijÞ A Spð4; QÞ : aij AZ Eði; jÞ 3 ð1; 3Þ; a13A 1 3Z; a21; a31; a41; a32; a3410 mod 18; a11; a3311 mod 6; a4210; a22; a4411 mod 3 : Conjugating by 3 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A;
Example 2. Consider the di¤erential equation
9y4 3zð173y4þ 340y3þ 272y2þ 102y þ 15Þ
2z2ð1129y4þ 5032y3þ 7597y2þ 4773y þ 1083Þ þ 2z3ð843y4þ 2628y3þ 2353y2þ 675y þ 6Þ
z4ð295y4þ 608y3þ 478y2þ 174y þ 26Þ þ z5ðy þ 1Þ4:
This is the Picard-Fuchs di¤erential equation for the complete intersection of 7 hyper-planes with the Grassmannian Gð2; 7Þ with the invariants H3¼ 42, c
2 H ¼ 84, and
c3¼ 98. (See [3].) The singularities are 0, 3, y, and the three roots z1¼ 0:01621 . . . ,
z2¼ 0:2139 . . . , and z3¼ 289:197 . . . of z3 289z2 57z þ 1. The monodromy around
z¼ 3 is the identity. The others have the matrix representations
T0¼ 1 1 0 0 0 1 0 0 42 42 1 0 0 14 1 1 0 B B B @ 1 C C C A; Tz1 ¼ 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A; Tz2 ¼ 13 7 1 2 28 13 2 4 196 98 15 28 98 49 7 15 0 B B B @ 1 C C C A; Tz3 ¼ 1 0 0 0 42 1 0 9 196 0 1 42 0 0 0 1 0 B B B @ 1 C C C A: Thus, the monodromy group is contained in the subgroup Gð14; 7Þ.
3. A general approach Let
yðnÞþ rn1yðn1Þþ þ r1y0þ r0y¼ 0; ri ACðzÞ;
be a linear di¤erential equation with regular singularities. Then the monodromy around a singular point z0with respect to the local Frobenius basis at z0is actually very easy to
de-scribe, as we shall see in the following discussion.
Consider the simplest cases where the indicial equation at z0 has n distinct roots
l1; . . . ;lnsuch that li lj BZfor all i 3 j. In this case, the Frobenius basis consists of
yjðzÞ ¼ ðz z0ÞljfjðzÞ; j ¼ 1; . . . ; n;
where fjðzÞ are holomorphic near z0 and have non-vanishing constant terms. It is easy to
see that the matrix of the monodromy around z0with respect tofyjg is simply
e2pil1 0 0 0 e2pil2 0 .. . .. . .. . 0 0 e2piln 0 B B B B @ 1 C C C C A:
Now assume that the indicial equation at z0 has l1; . . . ;lk, with multiplicities
e1; . . . ; ek, as solutions, where e1þ þ ek ¼ n and li lj BZfor all i 3 j. Then for each
lj, there are ej linearly independent solutions
yj; 0¼ ðz z0Þ lj fj; 0; yj; 1¼ yj; 0logðz z0Þ þ ðz z0Þljfj; 1; yj; 2¼ 1 2yj; 0log 2 ðz z0Þ þ ðz z0Þljfj; 1logðz zjÞ þ ðz z0Þljfj; 2; .. . .. . yj; ej1 ¼ ðz z0Þ lj P ej1 h¼0 1 h!fj; ej1hlog h ðz z0Þ;
where fj; h are holomorphic near z¼ z0and satisfy fj; 0ðz0Þ ¼ 1 and fj; hðz0Þ ¼ 0 for h > 0.
Since fj; hare all holomorphic near z0, the analytic continuation along a small closed curve
circling z0does not change fj; h. For other factors, circling z0once in the counterclockwise
direction results in
ðz z0Þlj 7! e2piljðz z0Þlj
and
logðz z0Þ 7! logðz z0Þ þ 2pi:
Thus, the behaviors of yj; hunder the monodromy around z0are governed by
yj; 0 yj; 1 .. . yj; ej1 0 B B B B @ 1 C C C C A7! oj 0 0 2pioj oj 0 .. . .. . .. . ð2piÞej1 ðej 1Þ! oj ð2piÞej2 ðej 2Þ! oj oj 0 B B B B B B B @ 1 C C C C C C C A yj; 0 yj; 1 .. . yj; ej1 0 B B B B @ 1 C C C C A; where oj ¼ e2pilj.
When the indicial equation of z0has distinct roots liand ljsuch that li ljAZ, there
are many possibilities for the monodromy matrix relative to the Frobenius basis, but in any case, the matrix still consists of blocks of entries that take the same form as above.
From the above discussion we see that monodromy matrices with respect to the local Frobenius bases are very easy to describe. Therefore, to find monodromy matrices uni-formly with respect to a given fixed basis, it su‰ces to find the matrix of basis change be-tween the fixed basis and the Frobenius basis at each singularity. When the di¤erential equation is hypergeometric, this can be done using the (refined) standard analytic method, in which we first express the Frobenius basis at z¼ 0 as integrals of Barnes-Mellin type and then use contour integration to obtain the analytic continuation to a neighborhood of
z¼ y. This gives us the monodromy matrices around z ¼ 0 and z ¼ y. Since the mono-dromy group is generated by these two matrices, the group is determined.
When the di¤erential equation is not hypergeometric, we are unable to determine the matrices of basis change precisely. To obtain the matrices numerically we use the following idea. Let z1and z2be two singularities andfyig and f ~yyjg, i; j ¼ 1; . . . ; n, be their Frobenius
bases. Observe that if yi¼ ai1~yy1þ þ ainyy~n, then we have
y1 y10 y ðn1Þ 1 y2 y20 y ðn1Þ 2 .. . .. . .. . yn yn0 y ðn1Þ n 0 B B B B B @ 1 C C C C C A ¼ a11 a12 a1n a21 a22 a2n .. . .. . .. . an1 an2 ann 0 B B B B @ 1 C C C C A ~ y y1 ~yy10 ~yy ðn1Þ 1 ~ y y2 ~yy20 ~yy ðn1Þ 2 .. . .. . .. . ~ y yn ~yyn0 ~yy ðn1Þ n 0 B B B B B @ 1 C C C C C A :
Thus, to determine the matrixðaijÞ it su‰ces to evaluate yðkÞi and ~yy ðkÞ
i at a common point.
To do it numerically, we expand the Frobenius bases into power series and assume that the domains of convergence for the power series have a common point z0. We then truncate and
evaluate the series at z0. This gives us approximation of the matrices of basis changes. We
will discuss some practical issues of this method in Section 5.
4. The hypergeometric cases
Throughout this section, we fix the branch cut of log z to beðy; 0 so that the argu-ment of a complex variable z is betweenp and p.
Recall that a hypergeometric function pFp1ða1; . . . ;ap;b1; . . . ;bp1; zÞ is defined for
bi30;1; 2; . . . by pFp1ða1; . . . ;ap;b1; . . . ;bp1; zÞ ¼ Py n¼0 ða1Þn. . .ðapÞn ð1Þnðb1Þn. . .ðbp1Þn zn; where ðaÞn¼ aða þ 1Þ . . . ða þ n 1Þ; if n > 0; 1; if n¼ 0:
It satisfies the di¤erential equation
½yðy þ b1 1Þ . . . ðy þ bp1 1Þ zðy þ a1Þ . . . ðy þ apÞ f ¼ 0:
ð11Þ
Moreover, it has an integral representation 1 2pi Gðb1Þ . . . Gðbp1Þ Gða1Þ . . . GðapÞ Ð C Gðs þ a1Þ . . . Gðs þ apÞ Gðs þ b1Þ . . . Gðs þ bp1Þ GðsÞðzÞsds
forjargðzÞj < p, where C is any path from iy to iy such that the poles of GðsÞ lie on the right of C and the poles of Gðs þ akÞ lie on the left of C. (See [21], Chapter 5.) Then one
can obtain the analytic continuation ofpFp1 by moving the path of integration to the far
left of the complex plane and counting the residues arising from the process. It turns out that this method can be generalized.
Lemma 1. Let m be the number of 1’s among bk. Set
Fðh; zÞ ¼ P y n¼0 ða1þ hÞn. . .ðapþ hÞn ð1 þ hÞnðb1þ hÞn. . .ðbp1þ hÞn znþh:
Then, for j¼ 0; . . . ; m, the functions qj
qhjFðh; zÞjh¼0
are solutions of (11). Moreover, ifjargðzÞj < p and h is a small quantity such that akþ h are
not zero or negative integers, then Fðh; zÞ has the integral representation Fðh; zÞ ¼ z h 2pi Gðb1þ hÞ . . . Gðbp1þ hÞGð1 þ hÞ Gða1þ hÞ . . . Gðapþ hÞ Ð C Gðs þ a1þ hÞ . . . Gðs þ apþ hÞ Gðs þ b1þ hÞ . . . Gðs þ bp1þ hÞGðs þ 1 þ hÞ p sin psðzÞ s ds;
where C is any path from iy to iy such that the integers 0; 1; 2; . . . lie on the right of C and the poles of Gðs þ akþ hÞ lie on the left of C.
Proof. The first part of the lemma is just a specialization of the Frobenius method (see [14]) to the hypergeometric cases. We have
yðy þ b1 1Þ . . . ðy þ bp1 1ÞF ðh; zÞ ¼ hðh þ b1 1Þ . . . ðh þ bp1 1Þz h þP y n¼1 ða1þ hÞn. . .ðapþ hÞ ð1 þ hÞn1ðb1þ hÞn1. . .ðbp1þ hÞn1 znþh and zðy þ a1Þ . . . ðy þ apÞF ðh; zÞ ¼ P y n¼0 ða1þ hÞnþ1. . .ðapþ hÞnþ1 ð1 þ hÞnðb1þ hÞn. . .ðbp1þ hÞn znþ1þh: It follows that
½yðy þ b1 1Þ . . . ðy þ bp1 1Þ zðy þ a1Þ . . . ðy þ apÞF ðh; zÞ
¼ hðh þ b1 1Þ . . . ðh þ bp1 1Þzh:
If the number of 1’s among bk is m, then the first non-vanishing term of the Taylor
qj
qhjFðh; zÞjh¼0
are solutions of (11) for j¼ 0; . . . ; m.
The proof of the second part about the integral representation is standard. We refer the reader to [21], Chapter 5. r
We now prove Theorem 1. Here we will only discuss the cases ðA; BÞ ¼ ð1=2; 1=2Þ; ð1=3; 1=3Þ; ð1=4; 1=2Þ; and ð1=6; 1=3Þ;
representing the four classes whose indicial equations at z¼ y have one root with multi-plicity 4, two distinct roots, each of which has multimulti-plicity 2, one repeated root and two other distinct roots, and four distinct roots, respectively. The other cases can be proved in the same fashion.
Proof of the case ðA; BÞ ¼ ð1=6; 1=3Þ. Let h denote a small real number, and let Fðh; zÞ be defined as in Lemma 1 with p ¼ 4, a1¼ 1=6, a2¼ 1=3, a3¼ 2=3, a4¼ 5=6, and
bk ¼ 1 for all k. Then, by Lemma 1, the functions
yjðzÞ ¼ 1 j! qj qhj ChFðh; CzÞ; j¼ 0; . . . ; 3; are solutions of
y4 11664zðy þ 1=6Þðy þ 1=3Þðy þ 2=3Þðy þ 5=6Þ;
where C¼ 11664. In fact, by considering the contribution of the first term, we see that these four functions make up the Frobenius basis at z¼ 0.
We now express ChFðh; CzÞ using Lemma 1. By the Gauss multiplication theorem
we have Gðs þ 1=6ÞGðs þ 1=3ÞGðs þ 2=3ÞGðs þ 5=6Þ ¼ Q6 k¼1 Gðs þ k=6Þ Gðs þ 1=2ÞGðs þ 1Þ¼ ð2pÞ5=261=26sGð6s þ 1Þ ð2pÞ1=221=22sGð2s þ 1Þ:
Thus, restricting z to the lower half-planep < arg z < 0, by Lemma 1, we may write
ChFðh; CzÞ ¼ z h 2pi Gð1 þ hÞ4Gð1 þ 2hÞ Gð1 þ 6hÞ Ð C Gð6s þ 1 þ 6hÞ Gðs þ 1 þ hÞ4Gð2s þ 1 þ 2hÞ p sin pse piszsds;
where C is the vertical line Re s ¼ 1=12. Now move the line of integration to Re s¼ 13=12. This is justified by the fact that the integrand tends to 0 as Im s tends to infinity. The integrand has four simple poles s¼ n=6 h, n ¼ 1; 2; 4; 5, between these two lines. The residues are
ð1Þn1 6GðnÞ
pepiðn=6þhÞ
Gð1 n=6Þ4Gð1 n=3Þ sin pðn=6 þ hÞz
n=6h:
Thus, we see that the analytic continuation of ChFðh; zÞ to jzj > 1 with p < arg z < 0 is
given by
ChFðh; zÞ ¼ P
n¼1; 2; 4; 5
anBnðhÞzn=6þ ðhigher order terms in 1=zÞ;
where an¼ ð1Þnpepin=6 6GðnÞGð1 n=6Þ4Gð1 n=3Þ; BnðhÞ ¼ Gð1 þ hÞ4Gð1 þ 2hÞepih Gð1 þ 6hÞ sin pðn=6 þ hÞ:
On the other hand, since the local exponents at z¼ y are 1=6, 1=3, 2=3, and 5=6, the Fro-benius basis at z¼ y consists of
~ y
ynðzÞ ¼ zn=6gnð1=zÞ; n¼ 1; 2; 4; 5;
where gn¼ 1 þ are functions holomorphic at 0. It follows that for z with p < arg z < 0
yjðzÞ ¼ 1 j! P n¼1; 2; 4; 5 anBð jÞn ðhÞ ~yynðzÞ:
Set fjðzÞ ¼ yjðzÞ=ð2piÞj for j¼ 0; . . . ; 3 and ~ffn¼ an~yyn=sinðnp=6Þ for n ¼ 1; 2; 4; 5. Then
using the evaluation
G0ð1Þ ¼ g; G00ð1Þ ¼ g2þ zð2Þ; G000ð1Þ ¼ g3 3zð2Þg 2zð3Þ; we find f3 f2 f1 f0 0 B B B @ 1 C C C A¼ M ~ ff1 ~ ff2 ~ ff4 ~ ff5 0 B B B @ 1 C C C A; where M¼ h io=4 hþ 5pffiffiffi3io2=36 hþ 5pffiffiffi3io4=36 h io5=4 5=12 io=2 1=4 io2=2pffiffiffi3 1=4 io4=2pffiffiffi3 5=12 io5=2 io io2=pffiffiffi3 io4=pffiffiffi3 io5 1 1 1 1 0 B B B @ 1 C C C A with
o¼ epi=6; h¼68zð3Þ
ð2piÞ3:
Now let P be the path traveling along the real axis with arg z¼ pþ from z ¼ 2 to y and then coming back along the real axis with arg z¼ p to z ¼ 2. The monodromy ef-fect on ~yynðzÞ ¼ zn=6gnð1=zÞ is
~ y
ynðzÞ 7! ~yynðe2pizÞ ¼ e2pin=6yy~nðzÞ:
Therefore, the matrix representation of the monodromy along P relative to the ordered basisf f3; f2; f1; f0g is Ty¼ M o2 0 0 0 0 o4 0 0 0 0 o4 0 0 0 0 o2 0 B B B @ 1 C C C AM 1:
Now the path P is equivalent to that of circling once around z¼ 1=C and then once around z¼ 0, both in the counterclockwise direction. Therefore, if we denote by T0 and T1=C the
monodromy matrices relative the basis f f3; f2; f1; f0g around z ¼ 0 and z ¼ 1=C,
respec-tively, then we have
Ty¼ T1=CT0:
Since T0is easily seen to be
T0¼ 1 1 1=2 1=6 0 1 1 1=2 0 0 1 1 0 0 0 1 0 B B B @ 1 C C C A; we find T1=C ¼ M o2 0 0 0 0 o4 0 0 0 0 o4 0 0 0 0 o2 0 B B B @ 1 C C C AM 1T1 0 ¼ 1þ a 0 ab=d a2=d b 1 b2=d ab=d 0 0 1 0 d 0 b 1 a 0 B B B @ 1 C C C A; where a¼ 204 ð2piÞ3zð3Þ; b¼ 7 4; d ¼ 3:
Comparing these numbers with the invariants, we find the matrix T1=C indeed takes the
form (7) specified in the statement of Theorem 1. This proves the caseðA; BÞ ¼ ð1=6; 1=3Þ. r
Proof of the case ðA; BÞ ¼ ð1=4; 1=2Þ. Apply Lemma 1 with p¼ 4, a1¼ 1=4,
a2¼ 3=4, a3¼ a4¼ 1=2, bk¼ 1 for all k, and set C ¼ 1024. Then
yjðzÞ ¼ 1 j! qj qhj ChFðh; CzÞ; j¼ 0; . . . ; 3;
form the Frobenius basis for
y4 1024zðy þ 1=4Þðy þ 3=4Þðy þ 1=2Þ2: Assuming thatp < arg z < 0, we have
ChFðh; CzÞ ¼ z h 2pi Gð1 þ hÞ6 Gð1 þ 2hÞGð1 þ 4hÞ Ð C Gð4s þ 1 þ 4hÞGð2s þ 1 þ 2hÞ Gðs þ 1 þ hÞ6 p sin pse piszsds;
where C is the vertical line Re s ¼ 1=8. The integrand has simple poles at k h 1=4 and k h 3=4, and double poles at k h 1=2 for k ¼ 0; 1; 2; . . . : The residues at s¼ h n=4, n ¼ 1; 3, are anCnðhÞzhn=4, where an¼ ð1Þðnþ1Þ=2 pGð1=2Þepin=4 4Gð1 n=4Þ6 ; CnðhÞ ¼ epih sin pðh þ n=4Þ: At s¼ h 1=2 we have Gð4s þ 1 þ 4hÞGð2s þ 1 þ 2hÞ Gðs þ 1 þ hÞ6 ¼ 1 8Gð1=2Þ6ðs þ h þ 1=2Þ 2 3 log 2þ 1 2Gð1=2Þ6 ðs þ h þ 1=2Þ 1 þ ; p sin ps¼ p cos phþ p 2 sin ph cos2phðs þ h þ 1=2Þ þ ; and
episzs¼ z1=2hepiðhþ1=2Þ1þ ðpi þ log zÞðs þ h þ 1=2Þ þ : Thus, the residue at s¼ h 1=2 is
pepiðhþ1=2Þ
8Gð1=2Þ6cos ph piþ log z þ 12 log 2 þ 4 p sin ph cos ph zh1=2: Set a2¼ pepi=2 8Gð1=2Þ6; C2ðhÞ ¼ epih cos ph; C
2ðhÞ ¼ C2ðhÞðpi þ 12 log 2 þ 4 p tan phÞ:
ChFðh; CzÞ ¼ a1B1ðhÞz1=4 a2B2ðhÞz1=2log z a2B2ðhÞz 1=2
a3B3ðhÞz3=4þ ðhigher order terms in 1=zÞ;
where BnðhÞ ¼ Gð1 þ hÞ6 Gð1 þ 2hÞGð1 þ 4hÞCnðhÞ; B 2ðhÞ ¼ Gð1 þ hÞ6 Gð1 þ 2hÞGð1 þ 4hÞC 2ðhÞ:
Let yjðzÞ, j ¼ 0; . . . ; 3, be the Frobenius basis at z ¼ 0, and
~ y y1ðzÞ ¼ z1=4ð1 þ Þ; ~yy3ðzÞ ¼ z3=4ð1 þ Þ; ~ y y2ðzÞ ¼ z1=2ð1 þ Þ; ~yy2ðzÞ ¼ log zþ gð1=zÞ~yy2ðzÞ
be the Frobenius basis at y, where gðtÞ is a function holomorphic and vanishing at t ¼ 0. Set fjðzÞ ¼ yjðzÞ=ð2piÞj for j¼ 0; . . . ; 3, ~ffnðzÞ ¼ an~yynðzÞ=sin pðn=4Þ for n ¼ 1; 2; 3, and
~ ff
2ðzÞ ¼ a2~yy2ðzÞ. Using the fact that
yjðzÞ ¼ 1 j! qj qhj ChFðh; CzÞ; we find f3 f2 f1 f0 0 B B B @ 1 C C C A¼
hþ ð1 iÞ=48 h 5=48 5m=12 þ pih þ 4mh h þ ð1 þ iÞ=48
ð1 6iÞ=24 7=24 pi=24þ 7m=6 ð1 þ 6iÞ=24
ði 1Þ=2 1=2 2m ði þ 1Þ=2 1 1 piþ 4m 1 0 B B B @ 1 C C C A ~ ff1 ~ ff2 ~ ff2 ~ ff3 0 B B B @ 1 C C C A where m¼ 3 log 2 þ 1; h¼22zð3Þ ð2piÞ3:
Let P be the path from z¼ 1 with argument p to y and then back to z ¼ 1 with argument p. The monodromy matrix for P relative to the ordered basisf ~ff1; ~ff2; ~ff
2; ~ff3g is i 0 0 0 0 1 2pi 0 0 0 1 0 0 0 0 i 0 B B B @ 1 C C C A:
Thus, the matrix with respect to the ordered basisf f3; f2; f1; f0g is
Ty ¼ 1 8h 1 8h 1=2 19h=3 1=6 11h=3 þ 8h2 7=3 4=3 61=72 41=72 þ 7h=3 0 0 1 1 8 8 19=3 8=3 þ 8h 0 B B B @ 1 C C C A:
Finally, it is easy to see that the monodromy around z¼ 0 with respective to f f3; f2; f1; f0g is T0¼ 1 1 1=2 1=6 0 1 1 1=2 0 0 1 1 0 0 0 1 0 B B B @ 1 C C C A;
and after a short computation we find that the monodromy T1=C ¼ TyT01around z¼ 1=C
indeed takes the form claimed in the statement of Theorem 1. r
Proof of the case ðA; BÞ ¼ ð1=3; 1=3Þ. Let z be a complex number with p < arg z < 0. By the same argument as before, we find that the Frobenius basis fyjg at
z¼ 0 can be expressed as yjðzÞ ¼ 1 j! qj qhj ChFðh; CzÞ; where C ¼ 729 and ChFðh; CzÞ ¼ z h 2pi Gð1 þ hÞ6 Gð1 þ 3hÞ2 Ð C Gð3s þ 1 þ 3hÞ2 Gðs þ 1 þ hÞ6 p sin pse pis zsds:
Here h is assumed to be a real number and C denotes the vertical line Re s ¼ 1=6. Set an¼
pepin=3
9Gð1 n=3Þ6; n¼ 1; 2: The residues at z¼ 1=3 h and z ¼ 2=3 h are
a1
piþ log z þ 9 log 3 ppffiffiffi3þ cot pð1=3 þ hÞz1=3hepih and a2piþ log z þ 9 log 3 þ p ffiffiffi 3 p þ 6 þ cot pð2=3 þ hÞz2=3hepih; respectively. Let BnðhÞ ¼ Gð1 þ hÞ6 Gð1 þ 3hÞ2 epih sin pðn=3 þ hÞ; n¼ 1; 2; and B1ðhÞ ¼ B1ðhÞpiþ 9 log 3 p ffiffiffi 3 p þ p cot pð1=3 þ hÞ; B2ðhÞ ¼ B2ðhÞpiþ 9 log 3 þ p ffiffiffi 3 p þ 6 þ p cot pð2=3 þ hÞ: Then we have
ChFðh; CzÞ ¼ P 2 n¼1 an BnðhÞzn=3log zþ BnðhÞzn=3
þ ðhigher order termsÞ:
Now the Taylor expansions of BnðhÞ and BnðnÞ are
sinp 3B1 h 2pi ¼ 1 þ ioffiffiffi 3 p h io 2p þffiffiffi3 1 12 h2þ io 12p þ hffiffiffi3 h3þ ; sinp 3B2 h 2pi ¼ 1 þio 2 ffiffiffi 3 p h io 2 2p þffiffiffi3 1 12 h2þ io 2 12p þ hffiffiffi3 h3þ ; sinp 3B 1 h 2pi ¼ m1þ2poffiffiffi 3 p þ im1ffiffiffio 3 p þ2pio 3 h im1o 2p þffiffiffi3 m1 12þ po 2pffiffiffi3 h2 þ m1hþ im1o 12p þffiffiffi3 2pho ffiffiffi 3 p pio 6 h3þ ; sinp 3B 2 h 2pi ¼ m2þ 2po2 ffiffiffi 3 p þ 6 þ im2o 2 ffiffiffi 3 p 2pio 2 3 þ 2io 2 ffiffiffi 3 p h im2o 2 2pffiffiffi3 þ m2 12þ po2 2p offiffiffi3 23 2 h2 þ m2hþim2o 2 12p þffiffiffi3 2pho2 ffiffiffi 3 p þpio 2 6 þ 6h þ io2 2pffiffiffi3 h3þ ; where o¼ epi=3; h¼16zð3Þ ð2piÞ3; m1¼ 9 log 3 p ffiffiffi 3 p ; m2¼ 9 log 3 þ p ffiffiffi 3 p :
From these we can deduce the matrix of basis change between the Frobenius basis
fjðzÞ ¼ 1 ð2piÞjj! qj qhjC hFðh; CzÞ; j ¼ 0; . . . ; 3;
and the basis ~ ff1ðzÞ ¼ a1z1=3ð1 þ Þ; ff~2ðzÞ ¼ a2z2=3ð1 þ Þ; ~ ff1¼ log zþ g1ð1=zÞ ~ ff1ðzÞ; ff~2¼ log zþ g2ð1=zÞ ~ ff2ðzÞ;
where g1ðtÞ and g2ðtÞ are functions holomorphic and vanishing at t ¼ 0. The monodromy
matrix around y with respect to the ordered basisf ~ff1; ~ff1; ~ff2; ~ff2g is easily seen to be
e2pi=3 2pie2pi=3 0 0 0 e2pi=3 0 0 0 0 e2pi=3 2pie2pi=3
0 0 0 e2pi=3 0 B B B @ 1 C C C A:
By the same argument as before, we find that the monodromy matrix with respect to the basis f f3; f2; f1; f0g indeed takes the form claimed in the statement. This proves the case
ðA; BÞ ¼ ð1=3; 1=3Þ. r
Proof of the case ðA; BÞ ¼ ð1=2; 1=2Þ. Let z be a complex number such that p < arg z < 0. We find that the Frobenius basis fyjg at z ¼ 0 can be expressed as
yjðzÞ ¼ 1 j! qj qhj ChFðh; CzÞ; where C ¼ 256 and ChFðh; CzÞ ¼ z h 2pi Gð1 þ hÞ8 Gð1 þ 2hÞ4 Ð C Gð2s þ 1 þ 2hÞ4 Gðs þ 1 þ hÞ8 p sin pse piszsds:
Here h is assumed to be a small real number and C denotes the vertical line Re s ¼ 1=4. The integrand has quadruple poles at s¼ k 1=2 h for non-positive integers k. Moving the line of integration to Re s¼ 3=4 and computing the residue at s ¼ 1=2 h, we see that ChFðh; CzÞ ¼ a1P 3 n¼0 BnðhÞ n! z 1=2ðlog zÞn
þ ðhigher order terms in 1=zÞ; where a1¼ pepi=2 16Gð1=2Þ8; B3ðhÞ ¼ Gð1 þ hÞ8epih Gð1 þ 2hÞ4cos ph; B2ðhÞ ¼ B3ðhÞðm p tan phÞ; B1ðhÞ ¼ B3ðhÞ 7 6p 2 þm 2 2 pm tan ph þ p 2 sec2ph ; and B0ðhÞ ¼ B3ðhÞ m3 6 p 2m 2tan ph
þ ðsec2ph 7=6Þp2mþ ð5=6 sec2phÞp3tan phþ 8zð3Þ
;
where m¼ 16 log 2 þ pi. Let ~ ff0ðzÞ ¼ z1=2ð1 þ Þ; ff~1ðzÞ ¼ 1 2pi log zþ g1ð1=zÞ ~ ff0ðzÞ; ~ ff2ðzÞ ¼ 1 ð2piÞ2 log2z=2þ g1ð1=zÞ log z þ g2ð1=zÞff~0ðzÞ; ~ ff3ðzÞ ¼ 1 ð2piÞ3
log3z=6þ g1ð1=zÞ log2z=2þ g2ð1=zÞ log z þ g3ð1=zÞff~0ðzÞ
be the Frobenius basis at z¼ y with gnð0Þ ¼ 0. Using the evaluation
G0ð1Þ ¼ g; G00ð1Þ ¼p 2 12þ g2 2 ; G 000ð1Þ ¼ 1 3zð3Þ p2g 12 g3 6 ;
we can find the analytic continuation of the Frobenius at z¼ 0 in terms of ~ffnðzÞ. Now the
monodromy around y relative to the basisf ~ff3ðzÞ; ~ff2ðzÞ; ~ff1ðzÞ; ~ff0ðzÞg is
1 1 1=2 1=6 0 1 1 1=2 0 0 1 1 0 0 0 1 0 B B B @ 1 C C C A:
From this we can determine the monodromy matrix around z¼ 1=C with respect to the Frobenius basis at z¼ 0. We find that the result agrees with the general pattern depicted in Theorem 1, although the detailed computation is too complicated to be presented here. r
Of course, there is no reason why our approach should be applicable only to order 4 cases. Consider the hypergeometric di¤erential equations of order 5 of the form
L : y5 zðy þ 1=2Þðy þ AÞðy þ 1 AÞðy þ BÞðy þ 1 BÞ: ð12Þ
The cases ðA; BÞ ¼ ð1=2; 1=2Þ; ð1=4; 1=2Þ; ð1=6; 1=4Þ; ð1=4; 1=3Þ; ð1=6; 1=3Þ, and ð1=8; 3=8Þ have been used by Guillera [12], [13] to construct series representations for 1=p2. Applying the above method, we determine the monodromy of these di¤erential equations in the fol-lowing theorem whose proof will be omitted.
Theorem 3. Let L be one of the di¤erential equations in (12). Let yi, i¼ 0; . . . ; 4, be
the Frobenius basis at 0. Then the monodromy matrices around z¼ 0 and z ¼ 1=C with re-spect to the ordered basisfy4=ð2piÞ4; y3=ð2piÞ3; y2=ð2piÞ2; y1=ð2piÞ; y0g are
1 1 1=2 1=6 1=24 0 1 1 1=2 1=6 0 0 1 1 1=2 0 0 0 1 1 0 0 0 0 1 0 B B B B B @ 1 C C C C C A ; a2 0 ab ð1 a2Þx b2=2 c2x=2 1 acx c2x2=2 ð1 a2Þx
ac 0 1 2a2 acx ab
0 0 0 1 0 c2=2 0 ac c2x=2 a2 0 B B B B B @ 1 C C C C C A ;
respectively, where x is an integer multiple of zð3Þ=ð2piÞ3, a and c are positive real numbers such that a2, ac, and c2 are rational numbers, and b is a real number satisfying a2þ bc ¼ 1.
The exact values of a, c, and x0¼ ð2piÞ3x=zð3Þ are given in the following table.
A B a2 c2 x0 1/2 1/2 25/36 64 10 1/2 1/4 8/9 32 24 1/4 1/6 289/288 8 80 1/3 1/4 27/32 24 28 1/3 1/6 75/64 12 70 1/8 3/8 529/288 8 150
5. Di¤erential equations of Calabi-Yau type
The Picard-Fuchs di¤erential equations for families of Calabi-Yau threefolds known in the literature have the common features that
(a) the singular points are all regular,
(b) the indicial equation at z¼ 0 has 0 as its only solution,
(c) the indicial equation at one of the singularities has solutions 0, 1, 1, 2, corre-sponding to a conifold singularity,
(d) the unique holomorphic solution y around 0 with yð0Þ ¼ 1 has integral coe‰-cients in its power series expansion,
(e) the solutions l1el2el3el4 of the indicial equation at t¼ y are positive
ra-tional numbers and satisfy l1þ l4¼ l2þ l3¼ r for some r A Q, and the characteristic
polynomial of the monodromy around t¼ y is a product of cyclotomic polynomials. (f) the coe‰cients riðzÞ, i ¼ 1; 2; 3; of the di¤erential equation satisfy
r1¼ 1 2r2r3 1 8r 3 3þ r 0 2 3 4r 0 3r3 1 2r 00 3;
(g) the instanton numbers are integers.
In [1] a fourth order linear di¤erential equation satisfying all conditions except (c) is said to be of Calabi-Yau type. Using various techniques, Almkvist etc. found more than 300 such equations. (See [2], Section 5, for an overview of strategies of finding Calabi-Yau equations. The paper also contains a ‘‘superseeker’’ that tabulates the known Calabi-Yau equations, sorted according to the instanton numbers.) Among them, there are 178 equations that have singularities with exponents 0, 1, 1, 2. It is speculated that all such equations should have geometric origins.
In [26] van Enckevort and van Straten numerically determined the monodromy for these 178 equations. They were able to find rational bases for 145 of them, among which there are 64 cases that are integral. Their method goes as follows. Let z1; . . . ; zk be the
sin-gularities of a Calabi-Yau di¤erential equation. They first chose a reference point p and piecewise linear loops each of which starts from p and encircles exactly one of zi. Then the
problem of determining analytic continuation becomes that of solving several initial value problems in sequences. This was done numerically using the dsolve function in Maple. Then they used the crucial observation that the Jordan form for the monodromy around a coni-fold singularity is unipotent of index one to find a rational basis. Finally, assuming that (5) and (8) hold for general di¤erential equations of Calabi-Yau type, conjectural values of ge-ometric invariants can be read o¤.
Here we present a di¤erent method of computing monodromy based on the approach described in Section 3. Let 0¼ z0; z1; . . . ; zn be the singular points of a Calabi-Yau
zi. According to Section 3, to find the matrix of basis change betweenf fi; kg and f fj; kg, we
only need to evaluate fi; kðmÞand fj; kðmÞ at a common point z where the power series expansions of the functions involved all converge. In practice, the choice of z is important in order to achieve required precision in a reasonable amount of time.
Let Ridenote the radius of convergence of the power series expansions of the
Frobe-nius basis at zi. In general, Ri is equal to the distance from zi to the nearest singularity
zj3zi, meaning that if we truncate the power series expansion of fi; k at the nth term, the
resulting error is Oe ð1 þ eÞn jz zijn Rn i :
Of course, the O-constants depend on the di¤erential equation and zi. Since we do not have
any control over them, in practice we just choose z in a way such that jz zij
Ri
¼jz zjj Rj
:
If this does not yield needed precision, we simply replace n by a larger integer and do the computation again.
Example. Consider
y4 5ð5y þ 1Þð5y þ 2Þð5y þ 3Þð5y þ 4Þ:
The singularities are z0¼ 0, z1¼ 1=3125, and z2¼ y. The radii of convergence for the
Frobenius bases at 0 and 1=3125 are both 1=3125. Thus, to find the matrix of basis change, we expand the Frobenius bases, say, for 30 terms, and evaluate the Frobenius bases and their derivatives at z¼ 1=6250. Then we use the idea in Section 3 to compute the mono-dromy matrix around z1with respective to the Frobenius basis at 0. We find that the
com-putation agrees with (7) in Theorem 1 up to 7 digits.
The above method works quite well if the singularities of a di¤erential equation are reasonably well spaced. However, it occurs quite often that a Calabi-Yau di¤erential equa-tion has a cluster of singular points near 0, and a couple of singular points that are far away. For example, consider Equation K19 in [1]:
529y4 23zð921y4þ 2046y3þ 1644y2þ 621y þ 92Þ
z2ð380851y4þ 1328584y3þ 1772673y2þ 1033528y þ 221168Þ 2z3ð475861y4
þ 1310172y3þ 1028791y2þ 208932y 27232Þ 68z4ð8873y4þ 14020y3þ 5139y2 1664y 976Þ
þ 6936z5ðy þ 1Þ2ð3y þ 2Þð3y þ 4Þ:
The singularities are z0¼ 0, z1¼ 1=54, z2¼ ð11 5
ffiffiffi 5 p Þ=2 ¼ 0:090 . . . , z3¼ 23=34, and z4¼ ð11 þ 5 ffiffiffi 5 p
Þ=2 ¼ 11:09 . . . : In order to determine the monodromy matrix around z4,
we need to compute the matrix of basis change between the Frobenius basis at 1=54 and that at z4. The radius of convergence for the Frobenius basis at 1=54 is 1=54, while that at
z4is z4 1=54 ¼ 11:07 . . . : Even if we choose z optimally, we still need to expand the
Fro-benius bases for thousands of terms in order to achieve a precision of a few digits. In such situations, we can choose several points lying between the two singularities, compute bases for each of them, and then use the same idea as before to determine the matrices of basis change.
Take Equation 19 above as an example. We choose wk¼ ð1 þ 3kÞ=54 and
zk¼ ð1 þ 3k=2Þ=54 for k ¼ 0; . . . ; 5. The radius of convergence for the basis at wk is
3k=54. Thus, evaluating the first n terms of the power series expansions at z
kand zkþ1 will
result in an error of
Oe
ð1=2 þ eÞn; which is good enough in practice.
Using the above ideas we computed the monodromy groups of the di¤erential equa-tions of Calabi-Yau type that have at least one conifold singularity.1) Our result shows that if a di¤erential equation comes from geometry, then the monodromy matrix around one of the conifold singularities with respect to the Frobenius basis at the origin takes the form (7). We then conjugate the monodromy matrices by the matrix (9) and find that the other ma-trices are also in Spð4; QÞ. We now tabulate the results for the equations coming from geo-metry in the following table. Note that the notations Gðd1; d2Þ and Gðd1; d2; d3Þ, d2; d3j d1,
represent the congruence subgroups
Gðd1; d2Þ ¼ g A Spð4; ZÞ : g 1 1 0 0 0 1 0 0 0 B B B @ 1 C C C Amod d1 8 > > > < > > > : 9 > > > = > > > ; X g A Spð4; ZÞ : g 1 1 0 1 0 0 1 0 0 0 1 0 B B B @ 1 C C C Amod d2 8 > > > < > > > : 9 > > > = > > > ; and Gðd1; d2; d3Þ ¼ ðaijÞ A Spð4; QÞ : aij AZ Eði; jÞ 3 ð1; 3Þ; a13A 1 d3 Z; a21; a31; a41; a32; a3410 mod d1; a4210; a22; a4411 mod d2; a11; a3311 mod d1 d3 :
Note also that since the matrix 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A
is always in the monodromy groups, it is not listed in the table. The reader should be mind-ful of this omission. The number under the column K is referred to that in [1].
K H3 c2 H c3 generators in ref 15 18 72 162 1 1 0 0 0 1 0 0 18 18 1 0 0 9 1 1 0 B B B @ 1 C C C A 2 3 1=2 1 6 5 1 2 18 18 4 6 18 18 3 7 0 B B B @ 1 C C C A Gð6; 3; 2Þ [4] 16 48 96 128 1 1 0 0 0 1 0 0 48 48 1 0 0 16 1 1 0 B B B @ 1 C C C A 5 2 1=4 1 24 7 1 4 144 48 7 24 48 16 2 9 0 B B B @ 1 C C C A Gð24; 8; 4Þ [4] 17 90 108 90 1 1 0 0 0 1 0 0 90 90 1 0 0 24 1 1 0 B B B @ 1 C C C A 17 3 1=3 1 54 10 1 3 972 162 19 54 162 27 3 8 0 B B B @ 1 C C C A 11 3 1=3 1 36 8 1 3 432 108 13 36 108 27 3 10 0 B B B @ 1 C C C A Gð18; 6; 3Þ [4] 18 40 88 128 1 1 0 0 0 1 0 0 40 40 1 0 0 14 1 1 0 B B B @ 1 C C C A 5 4 1=2 1 12 7 1 2 72 48 7 12 48 32 4 9 0 B B B @ 1 C C C A Gð4; 2; 2Þ [4] 19 46 88 106 1 1 0 0 0 1 0 0 46 46 1 0 0 15 1 1 0 B B B @ 1 C C C A 6 4 1=2 1 14 7 1 2 98 56 8 14 56 32 4 9 0 B B B @ 1 C C C A 45 12 2 6 138 35 6 18 1058 276 47 138 276 72 12 37 0 B B B @ 1 C C C A Gð2; 2; 2Þ [4] 20 54 72 18 1 1 0 0 0 1 0 0 54 54 1 0 0 15 1 1 0 B B B @ 1 C C C A 7 1 1=6 1 6 1 0 2 126 18 2 24 36 6 1 5 0 B B B @ 1 C C C A Gð6; 3; 6Þ [4]
21 80 104 88 1 1 0 0 0 1 0 0 80 80 1 0 0 22 1 1 0 B B B @ 1 C C C A 11 5 1=2 1 24 9 1 2 288 120 13 24 120 50 5 11 0 B B B @ 1 C C C A 19 4 1=2 2 80 15 2 8 800 160 21 80 160 32 4 17 0 B B B @ 1 C C C A Gð8; 2; 2Þ [4] 22 70 100 100 1 1 0 0 0 1 0 0 70 70 1 0 0 20 1 1 0 B B B @ 1 C C C A 1 0 0 0 10 1 0 2 50 0 1 10 0 0 0 1 0 B B B @ 1 C C C A 9 5 1=2 1 20 9 1 2 200 100 11 20 100 50 5 11 0 B B B @ 1 C C C A Gð10; 10; 2Þ [4] 23 96 96 32 1 1 0 0 0 1 0 0 96 96 1 0 0 24 1 1 0 B B B @ 1 C C C A 9 1 1=8 1 8 1 0 2 288 32 3 40 64 8 1 7 0 B B B @ 1 C C C A Gð8; 8; 8Þ [4] 24 15 66 150 1 1 0 0 0 1 0 0 15 15 1 0 0 8 1 1 0 B B B @ 1 C C C A 5 5 1 2 12 9 2 4 36 30 7 12 30 25 5 11 0 B B B @ 1 C C C A Gð3; 1Þ [3] 25 20 68 120 1 1 0 0 0 1 0 0 20 20 1 0 0 9 1 1 0 B B B @ 1 C C C A 7 5 1 2 16 9 2 4 64 40 9 16 40 25 5 11 0 B B B @ 1 C C C A Gð4; 1Þ [3] 26 28 76 116 1 1 0 0 0 1 0 0 28 28 1 0 0 11 1 1 0 B B B @ 1 C C C A 9 6 1 2 20 11 2 4 100 60 11 20 60 36 6 13 0 B B B @ 1 C C C A Gð4; 1Þ [3] 27 42 84 98 1 1 0 0 0 1 0 0 42 42 1 0 0 14 1 1 0 B B B @ 1 C C C A 1 0 0 0 42 1 0 9 196 0 1 42 0 0 0 1 0 B B B @ 1 C C C A 13 7 1 2 28 13 2 4 196 98 15 28 98 49 7 15 0 B B B @ 1 C C C A Gð14; 7Þ [3, 22] 28 42 84 96 1 1 0 0 0 1 0 0 42 42 1 0 0 14 1 1 0 B B B @ 1 C C C A 41 12 2 6 126 35 6 18 882 252 43 126 252 72 12 37 0 B B B @ 1 C C C A Gð42; 2Þ [3]
186 57 90 84 1 1 0 0 0 1 0 0 57 57 1 0 0 17 1 1 0 B B B @ 1 C C C A 53 12 2 6 162 35 6 18 1458 324 55 162 324 72 12 37 0 B B B @ 1 C C C A 17 8 1 2 36 15 2 4 324 144 19 36 144 64 8 17 0 B B B @ 1 C C C A Gð3; 1Þ [25]
In the second table we list a few equations whose monodromy matrices with respect to our bases have integers as entries. Note that the numbers H3, c
2 H, and c3 are all
con-jectural, obtained from evaluation of the monodromy around a singularity of conifold type. Note, again, that the matrix
1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 B B B @ 1 C C C A is omitted from the table.
K H3 c 2 H c3 generators in 29 24 72 116 1 1 0 0 0 1 0 0 24 24 1 0 0 10 1 1 0 B B B @ 1 C C C A 47 20 4 10 120 49 10 25 576 240 49 120 240 100 20 51 0 B B B @ 1 C C C A Gð24; 2Þ 33 6 36 72 1 1 0 0 0 1 0 0 6 6 1 0 0 4 1 1 0 B B B @ 1 C C C A 1 0 0 0 2 1 0 2 2 0 1 2 0 0 0 1 0 B B B @ 1 C C C A Gð2; 2Þ 42 32 80 116 1 1 0 0 0 1 0 0 32 32 1 0 0 12 1 1 0 B B B @ 1 C C C A 15 6 1 3 48 17 3 9 256 96 17 48 96 36 6 19 0 B B B @ 1 C C C A Gð16; 4Þ 51 10 64 200 1 1 0 0 0 1 0 0 10 10 1 0 0 7 1 1 0 B B B @ 1 C C C A 3 5 1 2 8 9 2 4 16 20 5 8 20 25 5 11 0 B B B @ 1 C C C A Gð2; 1Þ 63 5 62 310 1 1 0 0 0 1 0 0 5 5 1 0 0 6 1 1 0 B B B @ 1 C C C A 1 5 1 2 4 9 2 4 4 10 3 4 10 25 5 11 0 B B B @ 1 C C C A Gð1; 1Þ
73 9 30 12 1 1 0 0 0 1 0 0 9 9 1 0 0 4 1 1 0 B B B @ 1 C C C A 2 0 0 1 3 2 1 0 0 3 2 3 3 0 0 2 0 B B B @ 1 C C C A Gð3; 1Þ 99 13 58 120 1 1 0 0 0 1 0 0 13 13 1 0 0 7 1 1 0 B B B @ 1 C C C A 5 4 1 2 12 7 2 4 36 24 7 12 24 16 4 9 0 B B B @ 1 C C C A Gð1; 1Þ 100 36 72 72 1 1 0 0 0 1 0 0 36 36 1 0 0 12 1 1 0 B B B @ 1 C C C A 1 0 0 0 12 1 0 4 36 0 1 12 0 0 0 1 0 B B B @ 1 C C C A 11 6 1 2 24 11 2 4 144 72 13 24 72 36 6 13 0 B B B @ 1 C C C A Gð12; 12Þ 101 25 70 100 1 1 0 0 0 1 0 0 25 25 1 0 0 10 1 1 0 B B B @ 1 C C C A 19 10 2 4 40 19 4 8 200 100 21 40 100 50 10 21 0 B B B @ 1 C C C A 1 0 0 0 60 1 0 16 225 0 1 60 0 0 0 1 0 B B B @ 1 C C C A Gð5; 5Þ 109 7 46 120 1 1 0 0 0 1 0 0 7 7 1 0 0 5 1 1 0 B B B @ 1 C C C A 3 3 1 2 8 5 2 4 16 12 5 8 12 9 3 7 0 B B B @ 1 C C C A Gð1; 1Þ 117 12 36 32 1 1 0 0 0 1 0 0 12 12 1 0 0 5 1 1 0 B B B @ 1 C C C A 1 0 0 0 4 1 0 4 4 0 1 4 0 0 0 1 0 B B B @ 1 C C C A 59 21 9 18 120 43 18 36 400 140 61 120 140 49 21 41 0 B B B @ 1 C C C A Gð4; 1Þ
118 10 40 50 1 1 0 0 0 1 0 0 10 10 1 0 0 5 1 1 0 B B B @ 1 C C C A 1 0 0 0 30 1 0 18 50 0 1 30 0 0 0 1 0 B B B @ 1 C C C A 19 10 4 8 40 19 8 16 100 50 21 40 50 25 10 21 0 B B B @ 1 C C C A Gð10; 5Þ 185 36 84 120 1 1 0 0 0 1 0 0 36 36 1 0 0 13 1 1 0 B B B @ 1 C C C A 11 7 1 2 24 13 2 4 144 84 13 24 84 49 7 15 0 B B B @ 1 C C C A Gð12; 1Þ
Acknowledgments. The second author (Yifan Yang) would like to thank Wadim Zudilin for drawing his attention to the monodromy problems and for many interesting and fruitful discussions. This whole research project started out as the second author and Zudilin’s attempt to give a rigorous and uniform proof of Guillera’s 1=p2 formulas [12], [13] in a way analogous to the modular-function approach in [10]. (See also [27].) For this purpose, it was natural to consider the monodromy of the fifth-order hypergeometric di¤er-ential equations, and hence it led the second author to consider monodromy of general dif-ferential equations of Calabi-Yau type. The second author would also like to thank Duco van Straten for his interest in this project and for clarifying some questions about the dif-ferential equations.
During the preparation of this paper (at the final stage), the third author (N. Yui) was a visiting researcher at Max-Planck-Institut fu¨r Mathematik Bonn in May and June 2006. Her visit was supported by Max-Planck-Institut. She thanks Don Zagier, Andrey Todorov and Wadim Zudilin for their interest in this project and discussions and suggestions on the topic discussed in this paper. She is especially indebted to Cord Erdenberger of University of Hannover for his supplying the index calculation for the congruence subgroup Gðd1; d2Þ
in Spð4; ZÞ.
The final version was prepared at IHES in January 2007 where the third author was a visiting member. For the preparation of the final version, the comments and suggestions of the referee were very helpful as well as discussions with Maxim Kontsevich. We thank them for their help.
Appendix. The index of G(d1, d2) in Sp(4, Z)
By Cord Erdenberger at Hannover
In this appendix we will show that the groups Gðd1; d2Þ are indeed congruence
sub-groups in Spð4; ZÞ and provide a formula for their index.
Recall that for n A N the principal congruence subgroup of level n is defined by
GðnÞ :¼ fM A Spð4; ZÞ j N 1 I4ðmod nÞg:
It is the kernel of the map from Spð4; ZÞ to Spð4; Z=nZÞ given by reduction modulo n and thus a normal subgroup in Spð4; ZÞ. It is a well-known fact that this map is surjective and hence the sequence
I4! GðnÞ ,! Spð4; ZÞ ! Spð4; Z=nZÞ ! I4
is exact. So the index of GðnÞ in Spð4; ZÞ is just the order of Spð4; Z=nZÞ which is known to be
½Spð4; ZÞ : GðnÞ ¼ jSpð4; Z=nZÞj ¼ n10Qð1 p2Þð1 p4Þ;
where the product runs over all primes p such that pj n. For d1; d2AN, define ~ G G1ðd1Þ :¼ M A Spð4; ZÞ : M 1 1 0 0 0 1 0 0 0 B B B @ 1 C C C Amod d1 8 > > > < > > > : 9 > > > = > > > ; ; ~ G G2ðd2Þ :¼ M A Spð4; ZÞ : M 1 1 0 1 0 0 1 0 0 0 1 0 B B B @ 1 C C C Amod d2 8 > > > < > > > : 9 > > > = > > > ; and set Gðd1; d2Þ :¼ ~GG1ðd1Þ X ~GG2ðd2Þ: Note that Gðd1Þ H ~GG1ðd1Þ and Gðd2Þ H ~GG2ðd2Þ: Hence GðdÞ ¼ Gðd1Þ X Gðd2Þ H ~GG1ðd1Þ X ~GG2ðd2Þ ¼ Gðd1; d2Þ;
where d is the least common multiple of d1and d2. This shows that Gðd1; d2Þ is a
congru-ence subgroup, i.e. it contains a principal congrucongru-ence subgroup as a normal subgroup of finite index. Moreover, this implies that Gðd1; d2Þ has finite index in Spð4; ZÞ and an upper
bound is given by the index of GðdÞ as given above.
We will from now on restrict to the case relevant to this paper, namely d2j d1. Then
Gðd1; d2Þ is in fact a subgroup of ~GG1ðd1Þ, namely
Gðd1; d2Þ ¼ a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 0 B B B @ 1 C C C AA ~GG1ðd1Þ : a22 a24 a42 a44 1 1 0 1 mod d2 8 > > > < > > > : 9 > > > = > > > ; :
To obtain a formula for the index of this group in Spð4; ZÞ, we first calculate the index of ~GG1ðd1Þ. Note that ~ G G1ðd1Þ=Gðd1Þ < Spð4; ZÞ=Gðd1Þ F Spð4; Z=d1ZÞ and hence ½Spð4; ZÞ : ~GG1ðd1Þ ¼ ½Spð4; Z=d1ZÞ : ~GG1ðd1Þ=Gðd1Þ:
The quotient ~GG1ðd1Þ=Gðd1Þ considered as a subgroup of Spð4; Z=d1ZÞ via the above
isomor-phism is given by ~ G G1ðd1Þ=Gðd1Þ F M A Spð4; Z=d1ZÞ : M ¼ 1 0 0 0 1 0 0 0 B B B @ 1 C C C A 8 > > > < > > > : 9 > > > = > > > ; :
An element of this group has the following form
M ¼ 1 a12 a13 a14 0 a a23 b 0 0 1 0 0 g a43 d 0 B B B @ 1 C C C A: Let J4:¼ 0 I2 I2 0
. The symplectic relation that tMJ
4M ¼ J4 then implies that
a b
g d
ASL2ðZ=d1ZÞ. Furthermore it gives rise to the following linear system: a12þ aa43 ga23¼ 0;
a14þ ba43 da23 ¼ 0:
Writing this in matrix form, we have
a g b d a43 a23 ¼ a12 a14 :
If we choose a12, a13, a14 freely, the above linear system has a unique solution a23, a43 as
a b
g d
is in SL2ðZ=d1ZÞ. This shows that
j~GG1ðd1Þ=Gðd1Þj ¼ d13 jSL2ðZ=d1ZÞj ¼ d16
Q
ð1 p2Þ
where the product runs over all primes p dividing d1. So we have the index formula
½Spð4; ZÞ : ~GG1ðd1Þ ¼ ½Spð4; Z=d1ZÞ : ~GG1ðd1Þ=Gðd1Þ ¼ d14
Q
pj d1
ð1 p4Þ:
Now we are ready to calculate the index of Gðd1; d2Þ in Spð4; ZÞ. Since we assume that
d2j d1, we have the following chain of subgroups:
