SOME MONODROMY GROUPS OF FINITE INDEX IN Sp 4 (Z)

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1 J. Aust. Math. Soc. 99 (5) 8 6 doi:.7/s SOME MONODROMY GROUPS OF FINITE INDEX IN Sp (Z) JÖRG HOFMANN and DUCO VAN STRATEN (Received 6 January ; accepted June ; first published online March 5) Communicated by B. Martin Abstract We determine the index of five of the seven hypergeometric Calabi Yau operators that have finite index in Sp (Z) and in two cases give a complete description of the monodromy group. Furthermore we find six nonhypergeometric Calabi Yau operators with finite index in Sp (Z) most notably a case where the index is one. Mathematics subject classification: primary D5; secondary F6 J. Keywords and phrases: monodromy groups arithmetic groups Fuchsian differential equations Calabi Yau equations.. Introduction The hypergeometric fourth-order operators related to mirror symmetry for complete intersections in weighted projective space have always been treated as a single group with very similar properties. An explicit description of monodromy matrices has been known for a long time. It came therefore as a surprise to us that recently Singh and Venkataramana [9] showed that in at least three of the cases the monodromy is of finite index in Sp (Z). On the other hand the work of Brav and Thomas [6] showed that in at least seven of the cases the monodromy is of infinite index. In a further paper Singh [] has shown that the monodromy is finite in the four remaining cases. So an interesting dichotomy has arisen in the class of Calabi Yau operators. In this note we give a precise determination of two of the groups of finite index and determine the index in three more cases. Furthermore six nonhypergeometric Calabi Yau operators are identified which have finite index in Sp (Z).. The hypergeometric families The general quintic hypersurface in P and the remarkable enumerative properties of the Picard Fuchs operator of the mirror family θ 5 5 x(θ + 5 )(θ + 5 )(θ + 5 )(θ + 5 ) c 5 Australian Mathematical Publishing Association Inc /5 $6. 8

2 [] Some monodromy groups of finite index in Sp (Z) 9 discovered by Candelas et al. [7] stand at the beginning of much of the interest in the mirror symmetry phenomenon that continues up to the present day. The above example was readily generalised to the case of smooth Calabi Yau threefolds in weighted projective space producing three further cases [ 6]. Then Libgober and Teitelbaum [5] produced mirror families for the other four Calabi Yau complete intersections in ordinary projective spaces. A final generalisation consisted of looking at smooth complete intersection Calabi Yau threefolds in weighted projective spaces leading to a further five cases []. In all these cases the Picard Fuchs operator is hypergeometric and takes the form [] θ Nz(θ + α )(θ + α )(θ + α )(θ + α ). It was remarked by several authors that in fact there is an overlooked th case corresponding to the complete intersection of hypersurfaces of degree and in P 5 ( 6) which represents a Calabi Yau threefold with a singularity [ 9 8]. From the point of view of differential equations the hypergeometric equations are characterised as fourth-order hypergeometrics with exponents at that carry a monodromy invariant lattice. This leads to a monodromy group that is (conjugate to) a subgroup of Sp (Z) and a necessary (and after the fact sufficient) condition for this to happen is that the characteristic polynomial of the monodromy around is a product of cyclotomic polynomials which leads immediately to the cases. We summarise the situation in Table. The last column gives the number as it appears in the table in the paper by Almkvist van Enckevort van Straten and Zudilin (AESZ) []. The factor N is introduced to make the power series expansion around of the holomorphic solution have integral coefficients in a minimal way. We call N the discriminant of the operator; the critical point is then located at x = /N =: x c. In terms of the exponents α α α = α α = α it can be given as (see [5]) where so N = N(α i ) i= ( r N := m(s) m(s) := s s) p s s /p s m(s) / 5 5/ / 5 5/ /. Monodromy matrices The explicit description of the monodromy of the general hypergeometric operator (θ + β ) (θ + β n ) x(θ + α ) (θ + α n )

3 5 J. Hofmann and D. van Straten [] Table. The hypergeometric cases. Case N α α α α AESZ P [5] P ( )[6] P ( )[8] P ( 5)[] P 5 [ ] 6 P 5 [ ] 6 P 6 [ ] 5 P 7 [ ] 8 P 5 ( )[ ] P 5 ( )[ ] 6 P 5 ( )[ 6] P 5 ( )[6 6] P 5 ( )[ 6] P 5 ( 6)[ ] has a long history. In his thesis Levelt [] showed the existence of a basis where the monodromies around and are given by the companion matrices of the characteristic polynomials f (T) = n (T e πiα k ) g(t) = k= n (T e πiβ k ). However for our purpose it is natural to work with other bases. First of all for all our operators there is a unique Frobenius basis of solutions around of the form Φ (x) = f (x) Φ (x) = log(x) f (x) + f (x) Φ (x) = log(x) f (x) + log(x) f + f (x) Φ (x) = 6 log(x) f (x) + log (x) f (x) + log(x) f (x) + f (x) k=

4 [] Some monodromy groups of finite index in Sp (Z) 5 where f = + Z[[x]] and f f f xq[[x]]. The basis of solutions y k (x) := (πi) k Φ k(x) is called the normalised Frobenius basis; the monodromy around in this basis is given by M F =. In this basis the monodromy invariant symplectic form is given by S F = and the monodromy around x c is a symplectic reflection v v C v C d in a vector C that represents the vanishing cycle and which has the form C = (d b a) where d := H is the degree of the ample generator b := c (X)H/ and a := λc (X) are the characteristic numbers of the corresponding Calabi Yau threefold X and λ = ζ() (πi). A further important invariant is the number k = H 6 + c (X) H = d 6 + b which is equal to the dimension of the linear system H. The base-change by the matrix A = d d/ b d b a conjugates the matrices M F and N F to M := AM F A = N := AN d d F A = k 6

5 5 J. Hofmann and D. van Straten [5] which are now in the integral symplectic group Sp (Z) = {M M t S M = I} realised as set of integral matrices that preserve the standard symplectic form S :=. This is the form of the generators that can be found in [8]. So the monodromy group G(d k) of the differential operator is the group generated by these two matrices M and N. It was observed in [8] that the monodromy group in fact is contained in a congruence subgroup G(d k) Γ(d gcd(d k)) where Γ(d d ) d d consist of those matrices A in Sp (Z) for which A mod d A mod d. The index of this group in Sp (Z) was computed by Erdenberger [8 Appendix] as Sp (Z) : Γ(d d ) = d ( p )d ( p ) p d p d where the product runs over the primes dividing d and d respectively. The parameters (d k) suggest a natural way to order the list of hypergeometric cases (see Table ). Remarkably this ordering coincides with the one obtained by using either the first instanton number n (rational curves of degree one) or the discriminant N. We remark further that the invariants d and k can be expressed directly in terms of the defining exponents α α as d = ( cos(πα ))( cos(πα )) k = cos(πα ) cos(πα ) which can be expressed as saying that cos(πα ) and cos(πα ) are roots of the quadratic polynomial X kx + d =.

6 [6] Some monodromy groups of finite index in Sp (Z) 5 Table. Invariants for the hypergeometric cases. (d k) α α H c H c n N AESZ ( ) ( ) 88 8 ( ) ( ) ( ) ( ) ( 5) ( ) 7 96 (5 5) (6 5) (8 6) (9 6) ( 7) (6 8) Results During the last year important progress has been made in understanding the nature of the monodromy group G(d k). Theorem. [6]. The group G(k d) has infinite index for the seven pairs (d k) = ( ) ( ) ( 5) (5 5) (8 6) ( 7) (6 8). Theorem. [9 ]. The group G(k d) has finite index for the other seven pairs To these results we add (d k) = ( ) ( ) ( ) ( ) ( ) (6 5) (9 6). Theorem.. The index Sp (Z) : G(d k) is given by the following table: (d k) ( ) ( ) ( ) ( ) ( ) (6 5) (9 6) Index G(d k) (?) 8 5 (?) Index Γ(d gcd(d k))

7 5 J. Hofmann and D. van Straten [7] The index of the last two entries is at least as big as the number indicated. For easy comparison we have also included the index of the corresponding group Γ(d gcd(d k)) in Sp (Z). On the first two groups we can be very precise: Theorem.. (i) The group G( ) of index 6 in Sp (Z) is exactly the group of matrices A Sp (Z) with the property the that A mod preserves the quintuple of vectors of (Z/) (ii) The group G( ) of index in Sp (Z) is exactly the group of matrices A Sp (Z) with the property the that A mod preserves the pair of triples of vectors of (Z/). 5. Explanation of Theorems. and.. In order to determine the index of a subgroup in a given group there is the classical method of Todd and Coxeter called coset enumeration. This has been developed into an effective computational tool that is implemented in GAP [] the main tool for computational group theory. For details on this circle of ideas we refer to [7]. For this to work one needs a good presentation of Sp (Z) in terms of generators and relations. We used a presentation of Sp (Z) described by Behr in [] that uses six generators and 8 relations and that is based on the root system for the symplectic group. The six generating matrices are: x α = x β = x α+β = x α+β = w α = w β =. We used results by Hua and Curtis [] to extract an algorithm that expresses an arbitrary element A Sp (Z) as a word in certain generators which were then reexpressed into the Behr generators x α x β x α+β x α+β w α w β.

8 [8] Some monodromy groups of finite index in Sp (Z) 55 For example the group of generators of G(d k) can be written as g = x β g = (w α w β ) x d α+β xk β x α w α xα (w α w β ). Hence if the generators of a finite index subgroup M = A... A n of Sp (Z) are given we can try to use algorithms from computational group theory for finitely presented groups to compute the index [Sp (Z) : M]. In this way the results of Theorem. were found. To understand Theorem. one has to look a bit closer to the geometry associated to the finite symplectic group. It is a classical fact that Sp (Z/) the reduction of Sp (Z) mod is isomorphic to the permutation group S 6. A way to realise Sp (Z/) naturally as a permutation group of six objects is as follows. The 5 points of P := P (Z/) correspond to the 5 transpositions in S 6 ; the point pairs having symplectic scalar product equal to one correspond to transpositions with a common index. The six quintuples of transpositions all having a common index thus correspond to six quintuples of points in P that have pairwise symplectic scalar product equal to one. Let us call such quintuples a pentade of points. These six pentades are permuted by Sp (Z/) thus defining an isomorphism with the permutation group S 6. A subgroup fixing such a pentade has index six and is a copy of S 5. Furthermore there are synthemes that is ways to divide six elements into two subsets of cardinality three. These correspond however precisely to the pairs of triples of elements of P with the property that the elements have symplectic scalar product one if they belong to the same triple and zero otherwise. The stabiliser of such a syntheme is a subgroup of index. To make this explicit let us label the elements of P by the letters from a to o: a = ( ) b = ( ) c = ( ) d = ( ) e = ( ) f = ( ) g = ( ) h = ( ) i = ( ) j = ( ) k = ( ) l = ( ) m = ( ) n = ( ) o = ( ). One verifies at once that the six pentades are given by = {a d g m o} = {a e f l n} = {b h k n o} = {b i j l m} 5 = {c d e i k} 6 = {c f g h j}. These are permuted by Sp (Z/). Indeed a transvection mod of an element p P T p : v v + (v p)p acts as a transposition on the set { 5 6}. For example one verifies that T a acts as the transposition ( ). For the matrices with d = k = mod one finds M a = a M d = o M g = m M g = m M m = d M o = g

9 56 J. Hofmann and D. van Straten [9] so that M maps the pentade to itself M =. In a similar way we obtain M = M = M = 6 M = 5 M 5 = M 6 = N = 5 N = N = N = N 5 = N 6 = 6 so that only the pentade = {a e f l n} is fixed by both M and N and one readily verifies that they generate the stabiliser. The synthemes given as pairs of triples are given by I = {{a d e} {b h j}} III = {{a l m} {c h k}} II = {{a f g} {b i k}} IV = {{a n o} {c i j}} V = {{b l n} {c d g}} VI = {{b m o} {c e f }} VII = {{d i m} { f h n}} IX = {{e i l} {g h o}} VIII = {{d k o} { f j l}} X = {{e k n} {g j m}}. The group Sp (Z/) permutes these synthemes and one verifies that in the case where d = mod and k = mod the matrix M induces the permutation (I IV II III) (VII X IX IIIV) and N the permutation (II VI) (III IX) (VI X) so that precisely the syntheme V = {{b l n} {c d g}} is preserved. Remark. There is another set of six objects that Sp (Z/) permutes which reflects the famous outer automorphism of S 6. In the finite symplectic geometry these correspond to disjoint quintuples of Lagrangian lines. In the notation used above these are = {{a b c} {d h l} {e j o} { f k m} {g i n}} = {{a b c} {d j n} {e h m} { f i o} {g k l}} = {{a j k} {b e g} {c m n} {d h l} { f i o}} = {{a h i} {b d f } {c m n} {e j o} {g k l}} 5 = {{a h i} {b e g} {c l o} {d j n} { f k m}} 6 = {{a j k} {b d f } {c l o} {e h m} {g i n}}. The stabiliser of such a pentade of lines is also isomorphic to S 5 but is not conjugate to the stabiliser of a pentade of points. The fact that the monodromy group G( ) preserves a pentade of points rather than a pentade of lines is an intrinsic property and is independent of any choices. 6. An observation The dichotomy between cases of finite and infinite index is rather mysterious. The finiteness of the index does not seem to correlate to any simple geometrical invariant

10 [] Some monodromy groups of finite index in Sp (Z) 57 of the Calabi Yau. On the other hand when we plot the cases in a diagram where black boxes represent the cases of infinite index a pattern arises: k d There is a tendency for the finite index cases to be lie under the infinite cases. Also in the cases of finite index the index increases monotonously with d. Apparently one may look at the quantity 7k d Λ := so that the cases with Λ > have infinite index and those with Λ < have finite index. There are three cases where Λ = ( ) (9 6) (6 8) of which only (9 6) has finite index. 7. Nonhypergeometric operators with finite index An obvious question to ask is which cases of Calabi Yau operators from the list [] have finite and which infinite index. Many of these are conifold operators which means that the singularity nearest to the origin has exponents. In such cases one can define the invariants d and k and one is tempted to make the following wild guess. Let G Sp (Z) be the monodromy group of a conifold Calabi Yau operator. If Λ > then the index is infinite and if Λ < then the index is finite. Using this heuristic we went through the list of Calabi Yau operators and discovered the following result. Theorem 7.. The following nonhypergeometric operators have monodromy of finite index in Sp (Z). AESZ H = d k c H c Index G(d k)-index ? We included the index of the corresponding G(d k)-group as far as we could determine it. Note that these groups do not belong to the family of. We note that

11 58 J. Hofmann and D. van Straten [] Table. Monodromy matrices for nonhypergeometric cases. Case Extra matrix Reflection vector 5 89 ( / 8 / / / ) ( ) 5 ( ) 57 ( ) ( ) ( / / ) the cases appearing here are all rather similar: all operators have apart from and two conifold points (exponents ) and a further apparent singularity (exponents ). In Table we list here the monodromy matrices around the extra conifold point in the basis explained in Section. This monodromy transformation is also a symplectic reflection; we list the corresponding reflection vector. Case 7 is remarkable in apparently having the full Sp (Z) as monodromy group. The index of G(5 ) is rather large so in this case the extra monodromy matrix makes a big difference. On the other hand for case 57 the extra monodromy transformation does nothing as in this case the index is the same as for the group G( ). We believe that there are many more of cases of finite index in the list; this is currently under investigation. No geometrical incarnation of these operators on the A- side of mirror symmetry that is coming from quantum cohomology is known to us although we believe they should exist.

12 [] Some monodromy groups of finite index in Sp (Z) 59 Operator AESZ 89 and Riemann symbol θ x(θ + 7θ + 75θ + 9θ + ) + 5 x ( 7θ 688θ + 8θ + 8θ + 77) + 5x ( 688θ + 56θ + 8θ + 6θ + 7) x (θ + )(θ + ) (θ + ) Operator AESZ 9 and Riemann symbol 9θ x(66θ + 798θ + 57θ + 8θ + 6) + 9 x (59 8θ θ 6 8θ 6 87θ 5) + 6 7x ( 9θ + θ + 5θ + 5θ + 7) 7 x (θ + )(θ + ) (θ + ) Operator AESZ and Riemann symbol θ x(5θ + 6θ + θ + θ + ) + x (8θ + 76θ 99θ 7θ 5) + 7 x ( 6θ + 6θ + 8θ + 5θ + 5) x (θ + )(θ + ) (θ + ) Operator AESZ 57 and Riemann symbol θ x(θ + 6θ + 8θ + 7θ + 7) + x ( 656θ 896θ + 6θ + 6θ + ) x (96θ + θ + θ + 6θ + ) x (θ + ) Operator AESZ 7 and Riemann symbol 5θ 5x(8θ + 6θ + θ + 9θ + ) + 5 x ( 8θ θ + θ θ 7) 8 5 x (77θ + 86θ + 997θ + 8θ + ) + x 8 x (θ + ) (θ + ) Operator AESZ and Riemann symbol θ x(θ + 56θ + θ + 9θ + ) + 9 x (58θ + 58θ + θ 7θ ) 6 x (θ + 9θ + θ + 8θ + 7) + x (θ + )

13 6 J. Hofmann and D. van Straten [] Table. Indices in Sp (Z/N). N ( ) ( ) ( ) ( ) ( ) ( ) ( 5) ( ) (5 5) (6 5) (8 6) (9 6) ( 7) (6 8)

14 [] Some monodromy groups of finite index in Sp (Z) 6 8. Monodromy group mod N Using GAP we can also try to determine the structure of the monodromy group in Sp (Z/NZ) for various N. Note that Sp (Z/NZ) = N ( p )( p ). p N For the convenience of the reader in Table we list the result of a GAP-computation. Table contains some redundancies: if N and M have no common factor the index in Sp (Z/NM) is the product of the indices in Sp (Z/N) and Sp (Z/M). The table also shows some remarkable phenomena. The case ( ) is of infinite index in Sp (Z) but the reductions mod N suggest that the index is 6 when considered -adically that is in the group Sp (Z ). The columns ( ) ( ) ( ) look very similar but here the index in Sp (Z) is indeed 6 96 respectively. For (5 5) the numbers probably will grow further; note the prime number entering in the index. All other columns have only and 5 appearing in the prime factorisation. The column (9 6) shows that the index in the last case of finite index is at least 9 78 = 667 = 8 5 and might very well be equal to this number. Acknowledgements This work was begun in October during a stay of the authors at the MSRI in Berkeley. We thank that institution for its hospitality and its excellent research environment. We thank S. Singh H. Thomas and W. Zudilin for showing an interest in early versions of this work. Thanks also to C. Doran who has indicated that he has work in preparation that is related to this paper. References [] G. Almkvist Strängar i månsken I Normat 5 () ; II Normat 5() () [] G. Almkvist C. van Enckevort D. van Straten and W. Zudilin Tables of Calabi Yau equations arxiv:math/57 Online version: [] V. Batyrev and D. van Straten Generalized hypergeometric functions and rational curves on Calabi Yau complete intersections in toric varieties Comm. Math. Phys. 68 (995) 9 5. [] H. Behr Eine endliche Präsentation der symplektischen Gruppe Sp (Z) Math. Z. (975) [5] M. Bogner On differential operators of Calabi Yau type Doctoral Thesis Mainz. [6] C. Brav and H. Thomas Thin monodromy in Sp() Compositio Math. to appear arxiv:.5. [7] P. Candelas X. de la Ossa P. Green and L. Parkes A pair of Calabi Yau manifolds as an exactly soluble superconformal theory Nucl. Phys. B 59 (99) 7. [8] Y. Chen Y. Yang and N. Yui Monodromy of Calabi Yau differential equations (With an appendix by Cord Erdenberger) J. reine angew. Math. 66 (8) 67. [9] C. Doran and J. Morgan Mirror symmetry and integral variations of Hodge structure underlying one parameter families of Calabi Yau threefolds in: Mirror Symmetry V AMS/IP Studies in Advanced Mathematics 8 (American Mathematical Society Providence RI 6).

15 6 J. Hofmann and D. van Straten [5] [] The GAP Group GAP Groups Algorithms and Programming Version.6.5 ( [] L. K. Hua and I. Reiner On the generators of the symplectic modular group Trans. Amer. Math. Soc. 65 (99) 5 6. [] A. Klemm and S. Theisen Consideration of one modulus Calabi Yau compactification: Picard Fuchs equation Kähler potentials and mirror maps Nucl. Phys. B 89 (99) 75. [] A. Klemm and S. Theisen Mirror maps and instanton sums for intersections in weighed projective space Modern Phys. Lett. A 9 (99) [] A. H. M. Levelt Hypergeometric functions Doctoral Thesis University of Amsterdam 96. [5] A. Libgober and J. Teitelbaum Lines on Calabi Yau complete intersections mirror symmetry and Picard Fuchs equations Int. Math. Res. Not. 9 (99) 9 9. [6] D. Morrison Picard Fuchs equations and mirror maps for hypersurfaces in: Essays on Mirror Manifolds (ed. S.-T. Yau) (International Press Hong Kong 99) 6. [7] L. Neubüser An elementary introduction to coset table methods in computational group theory in: Groups St Andrews 98 London Math. Society Lecture Notes 7 (eds. C. M. Campbell and E. F. Robertson) (Cambridge University Press Cambridge 98) 5. [8] F. Rodrigues-Villegas Hypergeometric families of Calabi Yau manifolds in: Calabi Yau Manifolds and Mirror Symmetry Fields Institute Communications 8 (eds. N. Yui and J. D. Lewis) (American Mathematical Society Providence RI ). [9] S. Singh and T. N. Venkataramana Arithmeticity of certain symplectic hypergeometric groups Duke Math. J. 6() () [] S. Singh Arithmeticity of hypergeometric monodromy groups associated to the Calabi Yau threefolds Int. Math. Res. Not. IMRN (): doi:.9/imrn/rnu7. JÖRG HOFMANN Institut für Mathematik Johannes Gutenberg University 5599 Mainz Germany hofmann joerg@gmx.net DUCO VAN STRATEN Institut für Mathematik Johannes Gutenberg University 5599 Mainz Germany straten@mathematik.uni-mainz.de