Gauss-Manin Connection in Disguise: Dwork Family
aa r X i v : . [ m a t h . AG ] S e p Gauss-Manin Connection in Disguise: DworkFamily H. Movasati and Y. Nikdelan Abstract
We study the enhanced moduli space T of the Calabi-Yau n -folds arising fromDwork family and describe a unique vector field R in T with certain properties withrespect to the underlying Gauss-Manin connection. For n = 1 , R and give a solution of R in terms of quasi-modular forms. The project Gauss-Manin connection in disguise started in the articles [Mov15, AMSY16]and the book [Mov16] aims to unify modular and automorphic forms with topologicalstring partition functions of string theory. The first group has a vast amount of applicationsin number theory and so it is highly desirable to seek for such applications for the secondgroup. The main ingredient of this unification is a natural generalization of Ramanujanrelations between Eisenstein series interpreted as a vector field in a certain moduli space.This has been extensively used in transcendental number theory, see [NePh, Zud01] for anoverview of some results. The starting point is either a Picard-Fuchs equation or a familyof algebraic varieties. In direction of the first case, in [Mov16] the author has describedthe construction of vector fields attached to Calabi-Yau equations of the list in [GAZ10].In direction of the second case, in this article we are going to consider the family of n -dimensional Calabi-Yau varieties X = X ψ , ψ ∈ P − { , , ∞} obtained by a quotient anddesingularization of the so-called Dwork family:(1.1) x n +20 + x n +21 + . . . + x n +2 n +1 − ( n + 2) ψx x . . . x n = 0 , and from now on we call any X ψ a mirror (Calabi-Yau) variety (see Section 2 for moredetails). This family and its periods are also the main object of study in some physicsarticles like [GMP95]. In the present article we discuss this unification in the case of Dworkfamily, namely we explain a construction of a modular vector field R n = R attached to X ψ such that for n = 1 , n = 3the topological partition functions are rational functions in the coordinates of a solutionof R , and for n ≥ q -expansions beyond the so-far well-known special functions.It is worth pointing out that we can consider the modular vector field R as an extensionof the systems of differential equations introduced by G. Darboux [Dar78], G. H. Halphen[Hal81] and S. Ramanujan [Ram16] (for more details see [Mov12], [Nik15, § X , dim H n dR ( X ) = n + 1, where H n dR ( X ) is the n -th algebraic de Rham cohomology of X , MSC2010: 14J15, 14J32, 11Y55.Keywords: Gauss-Manin connection, Dwork family, Picard-Fuchs equation, Hodge filtration, quasi-modular form, q-expansion. Instituto de Matem´atica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil. email: [email protected] Universidade do Estado do Rio de Janeiro (UERJ), Instituto de Matem´atica e Estat´ıstica (IME),Departamento de An´alise Matem´atica, Rio de Janeiro, Brazil. e-mail: [email protected] h ij , i + j = n, are all one. For n = 3 this is also called the familyof mirror quintic. Let T = T n be the moduli of pairs ( X, [ α , · · · , α n , α n +1 ]), where α i ∈ F n +1 − i \ F n +2 − i , i = 1 , · · · , n, n + 1 , [ h α i , α j i ] = Φ n . Here F i is the i -th piece of the Hodge filtration of H n dR ( X ), h· , ·i is the intersection formin H n dR ( X ) and Φ = Φ n is an explicit constant matrix given by(1.2) Φ n := n +12 J n +12 − J n +12 n +12 ! , if n is an odd positive integer, and(1.3) Φ n := J n +1 , if n is an even positive integer, where by 0 k , k ∈ N , we mean a k × k block of zeros, and J k is the following k × k block (1.4) J k := . . . . . . . . . . . . . We construct the universal family X → T together with global sections α i , i = 1 , · · · , n +1of the relative algebraic de Rham cohomology H n dR ( X / T ). Let ∇ : H n dR ( X / T ) → Ω T ⊗ O T H n dR ( X / T ) , be the algebraic Gauss-Manin connection on H n dR ( X / T ). Then we state below our maintheorem. Theorem 1.1.
There is a unique vector field R := R n in T such that the Gauss-Maninconnection of the universal family of n -fold mirror variety X over T composed with thevector field R , namely ∇ R , satisfies: (1.5) ∇ R α α α ... α n α n +1 = · · · Y · · · Y · · · ... ... ... ... . . . ... ... · · · Y n −
00 0 0 0 · · · −
10 0 0 0 · · · | {z } Y α α α ... α n α n +1 , for some regular functions Y i in T such that Y Φ + Φ Y tr = 0 . In fact, (1.6) T := Spec( Q [ t , t , . . . , t d , t n +2 ( t n +2 − t n +21 )ˇ t ]) , here (1.7) d = d n = ( n +1)( n +3)4 + 1 , if n is odd n ( n +2)4 + 1 , if n is even , and ˇ t is a product of s variables among t i ’s, i = 1 , , . . . , d , i = 1 , n + 2 and s = n − if n is an odd integer and s = n − if n is an even integer. In the proof of Theorem 1.1 we will show more than what we declared in the statementof the theorem. Indeed we will give the regular functions Y i ’s explicitly, and we will findan algorithm to express the modular vector field R . An explicit expression for R hasbeen given in [Mov15, Mov16] by the first author. In the next theorem we find R and R explicitly and express their solutions in terms of quasi-modular forms. Theorem 1.2.
For n = 1 , the vector field R as an ordinary differential equation isrespectively given by (1.8) R : ˙ t = − t t − t − t )˙ t = 81 t ( t − t ) − t ˙ t = − t t , where ˙ ∗ = 3 · q · ∂ ∗ ∂q , and (1.9) R : ˙ t = t − t t ˙ t = 2 t − t ˙ t = − t t + 8 t ˙ t = − t t , in which ˙ ∗ = − · q · ∂ ∗ ∂q , and the following polynomial equation holds among t i ’s (1.10) t = 4( t − t ) . Moreover, for any complex number τ with Im τ > , if we set q = e πiτ , then we find thefollowing solutions of R and R respectively: (1.11) t ( q ) = (2 θ ( q ) θ ( q ) − θ ( − q ) θ ( − q )) ,t ( q ) = ( E ( q ) − E ( q )) ,t ( q ) = η ( q ) η ( q ) , and (1.12) t ( q ) = ( θ ( q ) + θ ( q )) , t ( q ) = ( E ( q ) + 2 E ( q )) , t ( q ) = η ( q ) η ( q ) , here E , η and θ i ’s are the classical Eisenstein, eta and theta series, respectively, givenas follows: E ( q ) = 1 − ∞ X k =1 σ ( k ) q k with σ ( k ) = X d | k d, (1.13) η ( q ) = q ∞ Y k =1 (1 − q k ) , (1.14) θ ( q ) = ∞ X k = −∞ q ( k +12 ) , θ ( q ) = 1 + 2 ∞ X k =1 q k . (1.15) Remark 1.1.
We recall that η and θ i ’s are modular forms, and E is a quasi-modularform. By studying of the coefficients of q-expansions of the solutions given in (1.11) and(1.12), we find some interesting enumerative properties. For example, in (1.11) the co-efficients of t ( q ) = P ∞ k =0 t ,k q k have the following enumerative property: Let k be anon-negative integer. If k = 4 m, m ∈ N , then the equation x + 3 y = k has t ,k integersolutions. Otherwise the equation has t ,k integer solutions. For more properties of thistype see Section 8. The article is organized in the following way. First, in Section 2 we review and sum-marize some basic facts, without proofs, about the structure of Dwork family from whichthe mirror variety X ψ arises. In Section 3 we introduce the notion of moduli space ofholomorphic n -form S , and we see that S is two dimensional and present a coordinatechart for it. Section 4 deals with the calculation of intersection form matrix of a givenbasis of the de Rham cohomology of mirror variety. In Section 5 we present the modulispace T and construct a complete coordinate system for T . Section 6 is devoted to thecomputing of Gauss-Manin connection of the families X / S and X / T . In Section 7 Theorem1.1 is proved and the modular vector field is explicitly computed for n = 1 , ,
4. Finally,in Section 8 after finding the solutions of R and R in terms of quasi-modular forms, weproceed with the studying of enumerative properties of the q -expansions of the solutions. Acknowledgment.
The second author thanks the ”Instituto de Matem´atica da Uni-versidade Federal do Rio de Janeiro” of Brazil, where he did a part of this work duringhis Postdoctoral research with the grant of ”CAPES”, and in particular he would likes tothank Bruno C. A. Scardua for his supports.
Let f ψ be the polynomial in the left hand side of (1.1). Let W ψ be an n -dimensionalhypersurface in P n +1 given by f ψ . We know that the first Chern class of W ψ is zero, fromwhich follows that W ψ is a Calabi-Yau manifold. Thus we have a family of Calabi-Yaumanifolds given by π : W → P , where W ⊂ P n × C , W ψ = π − ( ψ ) and W ∞ = { ( x , x , . . . , x n +1 ) | x x . . . x n +1 = 0 } . This family, which is known as n -fold Dwork family , was a favorite example of Dwork,where he was developing his ”deformation theory” about zeta function of a nonsingular4ypersurface in a projective space (see [Dwo62, Dwo66]). One can easily see that thesingular points of this family are ψ n +2 = 1 , ∞ . Let G be the following group(2.1) G := { ( ζ , ζ , . . . , ζ n +1 ) | ζ n +2 i = 1 , ζ ζ . . . ζ n +1 = 1 } , which acts on W ψ as follow(2.2) ( ζ , ζ , . . . , ζ n +1 ) . ( x , x , . . . , x n +1 ) = ( ζ x , ζ x , . . . , ζ n +1 x n +1 ) . Evidently we see that this action is well defined. We denote by Y ψ := W ψ /G the quotientspace of this action, which is quite singular. Indeed Y ψ is singular in any x ∈ W ψ thatits stabilizer in G is nontrivial. For ψ n +2 = 1 , ∞ there exist a resolution X ψ → Y ψ ofsingularities of Y ψ , such that X ψ is a Calabi-Yau n -fold with h i,j ( X ψ ) = 1 , i + j = n .Therefore we have a new family where the fibers are Calabi-Yau n -folds X ψ which is themirror family of W ψ (see [GMP95]). The standard variable which is used in literatures isdefined by z := ψ − ( n +2) . By this change of variables, f ψ changes to f z given by f z ( x , x , . . . , x n +1 ) := zx n +20 + x n +21 + x n +22 + · · · + x n +2 n +1 − ( n + 2) x x x · · · x n +1 . The new set of singularities is given by z = 0 , ∞ , and we have the families W z andits mirror X z as well. From now on we call X z (or X ψ ) the mirror variety . There is aglobal holomorphic ( n, η ∈ H n dR ( X z ) which is given by η := dx ∧ dx ∧ . . . ∧ dx n +1 df z . in the affine chart { x = 1 } . The periods R δ η, δ ∈ H n ( X z , Z ) satisfy the well-knownPicard-Fuchs equation L (cid:18)Z δ η (cid:19) = 0 , where(2.3) L := ϑ n +1 − z ( ϑ + 1 n + 2 )( ϑ + 2 n + 2 ) . . . ( ϑ + n + 1 n + 2 ) , (2.4)in which ϑ = z ∂∂z . Note that if n = 1 , X z is a family of ellipticcurves, K n -forms By moduli space of holomorphic n -forms S we mean the moduli of the pair ( X, α ), where X is an n -dimensional mirror variety and α is a holomorphic n -form on X . We know thatthe family X z is a one parameter family and the n -form α is unique, up to multiplicationby a constant, therefore dim S = 2. The multiplicative group G m := ( C ∗ , · ) acts on S by:( X, α ) • k = ( X, k − α ) , k ∈ G m , ( X, α ) ∈ S . We present a chart ( t , t n +2 ) for S . To do this, for any ( t , t n +2 ) ∈ C we define thefollowing polynomial f t ,t n +2 ( x , x , . . . , x n +1 ) := t n +2 x n +20 + x n +21 + x n +22 + · · · + x n +2 n +1 − ( n + 2) t x x x · · · x n +1 . f t ,t n +2 is given by ∆ t ,t n +2 = ( t n +2 − t n +21 ) t n +2 . Let W t ,t n +2 be thefollowing two parameter family of Calabi-Yau manifolds W t ,t n +2 := { ( x , x , . . . , x n +1 ) | f t ,t n +2 ( x , x , . . . , x n +1 ) = 0 } ⊂ P n +1 . W t ,t n +2 is singular if and only if ∆ t ,t n +2 = 0. For any( t , t n +2 ) ∈ C \ { ( t , t n +2 ) | ∆ t ,t n +2 = 0 } we let X t ,t n +2 to be the resolution of the singularities of W t ,t n +2 /G where the group G and the group action are given by (2.1) and (2.2). Next we fix the n -form ω on the family X t ,t n +2 , where ω in the affine space { x = 1 } is given by ω := dx ∧ dx ∧ . . . ∧ dx n +1 df t ,t n +2 . Proposition 3.1.
We have S = Spec( Q [ t , t n +2 , t n +21 − t n +2 ) t n +2 ]) and the morphism X → S is the the universal family of ( X, α ) , where X is an n -dimensionalmirror variety and α is a holomorphic n -form on X . Moreover, the G m -action on S isgiven by (3.1) ( t , t n +2 ) • k = ( kt , k n +2 t n +2 ) , ( t , t n +2 ) ∈ S , k ∈ G m . Proof.
We have the map f which maps a point ( t , t n +2 ) ∈ S to the pair ( X t ,t n +1 , ω ) inthe moduli space S as a set. Its inverse is given by( X z , aη ) ( a − , za − ( n +2) ) . Note that ( X t ,t n +2 , ω ) and ( X z , t − η ), where z = t n +2 t n +21 , in the moduli space S represent thesame element. The affirmation concerning the G m -action follows from the isomorphism:( X kt ,k n +2 t n +2 , kω ) ∼ = ( X t ,t n +2 , ω ) , (3.2) ( x , x , · · · , x n +1 ) ( k − x , k − x , · · · , k − x n +1 ) , given in the affine coordinates x = 1. Let X be an n -dimensional mirror variety and ξ , ξ ∈ H n dR ( X ). Then in the context ofde Rham cohomology, the intersection form of ξ and ξ , denoted by h ξ , ξ i , is given by h ξ , ξ i = 1(2 πi ) n Z X ξ ∧ ξ . We recall that h ., . i is a non-degenerate ( − n -symmetric form, and(4.1) h F i , F j i = 0 , i + j ≥ n + 1 , F • : { } = F n +1 ⊂ F n ⊂ . . . ⊂ F ⊂ F = H n dR ( X ) , dim F i = n + 1 − i , is the Hodge filtration of H n dR ( X ).Let(4.2) ∇ : H n dR ( X / S ) → Ω S ⊗ O S H n dR ( X / S )be Gauss-Manin connection of the two parameter family of varieties X / S , and ∂∂t be avector field on the moduli space S . By abuse of notation, we use the same notion ∂∂t , toshow ∇ ∂∂t which is the composition of Gauss-Manin connection ∇ with the vector field ∂∂t . Now we define new n -forms ω i , i = 1 , , . . . , n + 1, as follows(4.3) ω i := ∂ i − ∂t i − ( ω ) . Later, in Lemma 4.1 we will see that ω , ω , . . . , ω n +1 form a basis of H n dR ( X ) compatiblewith its Hodge filtration, i.e.(4.4) ω i ∈ F n +1 − i \ F n +2 − i , i = 1 , , . . . , n + 1 . We write the Gauss-Manin connection of X / S in the basis ω as follow(4.5) ∇ ω = ˜ A ω, and we denote by(4.6) Ω = Ω n := ( h ω i , ω j i ) ≤ i,j ≤ n +1 , the intersection form matrix in the basis ω . We have(4.7) d Ω = ˜ A Ω + Ω ˜ A tr . The entries of ˜ A and Ω are respectively regular differential 1-forms and functions in S . Forarbitrary n , we do not have a general formula for Ω and ˜ A . We have only an algorithmwhich computes the entries of Ω and ˜ A recursively. For n = 1 , , , n -form ω is given by ∂ n +1 ∂t n +1 = − S ( n + 2 , n + 1) t n +11 t n +21 − t n +2 ∂ n ∂t n − S ( n + 2 , n ) t n t n +21 − t n +2 ∂ n − ∂t n − − . . . (4.8) − S ( n + 2 , t t n +21 − t n +2 ∂∂t − S ( n + 2 , t t n +21 − t n +2 , where S ( r, s ) , r, s ∈ N , refers to Stirling number of the second kind which is given by(4.9) S ( r, s ) = 1 s ! s X i =0 ( − i (cid:18) si (cid:19) ( s − i ) r . This must be true for arbitrary n , however, we are only interested to compute this forexplicit n ’s and so we skip the proof for arbitrary n .7 emma 4.1. We have (i) h ω i , ω j i = 0 , if i + j ≤ n + 1 . (ii) h ω , ω n +1 i = ( − ( n + 2)) n c n t n +21 − t n +2 , where c n is a constant. (iii) h ω j , ω n +2 − j i = ( − j − h ω , ω n +1 i , for j = 1 , , . . . , n + 1 . (iv) We can determine all the rest of h ω i , ω j i ’s in a unique way.Proof. Note that the intersection form is well-defined for all points in S , and so, h ω i , ω j i ’sare regular functions in S . This implies that they have poles only along t n +2 = 0 and t n +2 − t n +21 = 0. (i) The Griffiths transversality implies that ω i ∈ F n +1 − i , i = 1 , , . . . , n + 1 . This property and the property given in (4.1) complete the proof of (i) . (ii) If we present the Picard-Fuchs equation associated with holomorphic n -form η asfollow:(4.10) ϑ n +1 = a ( z ) + a ( z ) ϑ + . . . + a n ( z ) ϑ n , then in account of (2.4) we find a n ( z ) = n + 12 z − z . One can verify the differential equation given bellow ϑ h η, ϑ n η i + 2 n + 1 a n ( z ) h η, ϑ n η i = 0 , from which we get h η, ϑ n η i = c n exp (cid:16) − n +1 R z a n ( v ) dvv (cid:17) , where c n is a constant.This yields(4.11) h η, ϑ n η i = c n − z . On the other hand in Section 3 we saw z = t n +2 t n +21 , which gets ϑ = z ∂∂z = − n +2 t ∂∂t .One can easily see that η = t ω , hence ϑ n η = ( − n + 2 t ∂∂t ) n ( t ω )= b ω + . . . + b n ω n + ( − n + 2 ) n t n +11 ω n +1 , where b j ’s are rational functions in t , t n +1 . Therefore (i) implies h η, ϑ n η i = h t ω , ( − n + 2 ) n t n +11 ω n +1 i , which completes the proof of (ii) . 8 iii) By (i) we have h ω j , ω n +1 − j i = 0 , j = 1 , , . . . , n . Thus we get ∂∂t h ω j , ω n +1 − j i = h ∂∂t ω j , ω n − j i + h ω j , ∂∂t ω n +1 − j i = h ω j +1 , ω n +1 − j i + h ω j , ω n +2 − j i = 0 , hence we obtain h ω j +1 , ω n +1 − j i = −h ω j , ω n +2 − j i , j = 1 , , . . . , n , from whichfollows (iii) . (iv) We present the desired algorithm. So far, we computed the first row of the matrixΩ. Suppose that we have the i -th row of Ω, 1 ≤ i ≤ n , and then determine ( i + 1)-throw. To compute h ω i +1 , ω j i , n + 2 − i ≤ j ≤ n + 1, we apply ∂∂t h ω i , ω j i , whichimplies h ω i +1 , ω j i = ∂∂t h ω i , ω j i − h ω i , ω j +1 i . Note that if j = n + 1, then ω n +2 = ∂ n +1 ∂t n +11 ( ω ) and we compute it by using ofPicard-Fuchs equation given in (4.8).The intersection form matrix for n = 1 , , = − c t − t c t − t ! , Ω = c t − t − c t − t c t ( t − t ) c t − t c t ( t − t ) − c t (5 t − t )( t − t ) , Ω = c t − t − c t − t × c t ( t − t ) c t − t − × c t ( t − t ) c t (7 t +20 t )( t − t ) − c t − t − × c t ( t − t ) c t (14 t − t )( t − t ) − c t (56 t +35 t t − t )( t − t ) c t − t × c t ( t − t ) c t (7 t +20 t )( t − t ) − c t (56 t +35 t t − t )( t − t ) c t (273 t +238 t t +217 t t + t )( t − t ) . By enhanced moduli space T = T n we mean the moduli of the pair ( X, [ α , α , . . . , α n +1 ]),in which X is an n -fold mirror variety and { α , α , . . . , α n +1 } is a basis of H n dR ( X )compatiblewith its Hodge filtration, and such that the intersection matrix of this basis is constant,that is,(5.1) (cid:0) h α i , α j i (cid:1) ≤ i,j ≤ n +1 = Φ . If we denote by d n := dim T n , then from [Nik15, Theorem 1 ] we get (1.7). The objectiveof this section is to construct a coordinates system for T .In Section 4 we fixed the basis { ω , ω , . . . , ω n +1 } of H n dR ( X ) that is compatible withits Hodge filtration. Let S = (cid:0) s ij (cid:1) ≤ i,j ≤ n +1 be a lower triangular matrix, whose entriesare indeterminates s ij , i ≥ j and s = 1. We define α := Sω, ω := (cid:0) ω ω . . . ω n +1 (cid:1) tr . We assume that (cid:0) h α i , α j i (cid:1) ≤ i,j ≤ n +1 = Φ, and so, we get the following equation(5.2) S Ω S tr = Φ . If we set Ψ = (cid:0) Ψ ij (cid:1) ≤ i,j ≤ n +1 := S Ω S tr , then Ψ is a ( − n -symmetric matrix and Ψ ij = 0for i = 1 , , . . . , n and j ≤ n + 1 − i . Moreover, in the case that n is an odd integer we getΨ ii = 0 , i = 1 , , . . . , n + 1. Therefore the equation (5.2) gives us d := ( n +2)( n +1)2 − d − d is given by (1.7). The next argument shows that these equationsare independent from each other and so we can express d numbers of parameters s ij ’sin terms of other d − independent parameters . Forsimplicity we write the first class of parameters as ˇ t , ˇ t , · · · , ˇ t d and the second class as t , t , . . . , t n +1 , t n +3 , . . . , t d . We put all these parameters inside S according to the followingrule which we write it only for n = 1 , , , (cid:18) t ˇ t (cid:19) , t ˇ t t ˇ t ˇ t ! , t t t t ˇ t t ˇ t ˇ t ˇ t , t t t t ˇ t t ˇ t ˇ t ˇ t t ˇ t ˇ t ˇ t ˇ t . Note that we have already used t , t n +1 as coordinates system of S in Section 3. Proposition 5.1.
The equation S Ω S tr = Φ yields (5.3) s ( n +2 − i )( n +2 − i ) = ( − n + i +1 c n ( n + 2) n t n +21 − t n +2 s ii , where i = 1 , , . . . , n +12 if n is an odd integer, and i = 1 , , . . . , n +22 if n is an even integer.Moreover, one can compute ˇ t i ’s in terms of t i ’s.Proof. Let us first count the number of equalities that we get from S Ω S tr = Φ. This is ( n +1)( n +2)2 + 1 − d . Note that the left upper triangle of this equality consisits of trivialequalities 0 = 0. The equality (5.3) follows from the ( i, n + 2 − i )-th entry of S Ω S tr = Φ.We have plugged the parameters ˇ t k = s ij inside S such that the equality corresponding tothe ( n + 2 − j, i )-th entry of S Ω S tr = Φ gives us an equation which computes ˇ t k in terms ofˇ t r , r < k and t s ’s. Note that only divisions by s ii ’s, t n +2 − t n +21 and t n +2 occurs. Anotherway to see this is to redefine S := S − and so we will have the equality S Φ S tr = Ω.For n = 1 , ,
4, we express ˇ t j ’s in terms of t i ’s as follows: • n = 1: ˇ t = − c ( t − t ) . • n = 2: ˇ t = 116 c ( t − t ) , ˇ t = − c ( t − t ) , (5.4) ˇ t = 116 c ( − c t ˇ t + 2 t ) , ˇ t = 132 c ( − c t + t ) . n = 4: ˇ t = t − t c , ˇ t = − t − t c t , ˇ t = t − t c , (5.5) ˇ t = t t + 9 t t − t t c t , ˇ t = − c t ˇ t − t c t , ˇ t = 1296 c t t ˇ t − c t t ˇ t + 3 t t + 20 t t c t , ˇ t = − c t − t c t , ˇ t = 1296 c t t − c t t − c t t t + 5 t t + 20 t t c t , ˇ t = − c t t − c t + t c . We return to the Gauss-Manin connection ∇ , that was introduced in (4.2), and we proceedwith the computation of the Gauss-Manin connection matrix ˜ A , which is given in (4.5).If we denote by A ( z ) the Gauss-Manin connection matrix of the family X ,z in thebasis { η, ∂η∂z , . . . , ∂ n η∂z n } , i.e., ∇ (cid:0) η ∂η∂z . . . ∂ n η∂z n (cid:1) tr = A ( z ) dz ⊗ (cid:0) η ∂η∂z . . . ∂ n η∂z n (cid:1) tr , then we get A ( z ) = . . .
00 0 1 0 . . .
00 0 0 1 . . . . . . b ( z ) b ( z ) b ( z ) b ( z ) . . . b n +1 ( z ) in which the functions b i ( z )’s are the coefficients of the Picard-Fuchs equation associatedwith the n -form η that follows from (2.4) given below: ∂ n +1 ∂z n +1 = b ( z ) + b ( z ) ∂ n ∂z n + . . . + b n ( z ) ∂ n ∂z n . We calculate ∇ with respect to the basis (4.3) of H n dR ( X /S ). For this purpose we returnback to the one parameter case. For z := t n +2 t n +21 , consider the map g : X t ,t n +2 → X ,z , k = t − . We have g ∗ η = t ω , where by abuse of notation we just writ η = t ω , and ∂∂z = − n + 2 t n +31 t n +2 ∂∂t . From these two equalities we obtain the base change matrix ˜ S = ˜ S ( t , t n +2 ) such that (cid:0) η ∂η∂z . . . ∂ n η∂z n (cid:1) tr = ˜ S − (cid:0) ω ω . . . ω n +1 (cid:1) tr . Thus we find Gauss-Manin connection in the basis ω i , i = 1 , , . . . , n + 1 as follow:˜ A = (cid:18) d ˜ S + ˜ S · A ( t n +2 t n +21 ) · d ( t n +2 t n +21 ) (cid:19) · ˜ S − . Let ˜ A [ i, j ] be the ( i, j )-th entry of the Gauss-Manin connection matrix ˜ A . We have˜ A [ i, i ] = − i ( n + 2) t n +2 dt n +2 , ≤ i ≤ n (6.1) ˜ A [ i, i + 1] = dt − t ( n + 2) t n +2 dt n +2 , ≤ i ≤ n , (6.2) ˜ A [ n + 1 , j ] = − S ( n + 2 , j ) t j t n +21 − t n +2 dt + S ( n + 2 , j ) t j +11 ( n + 2) t n +2 ( t n +21 − t n +2 ) dt n +2 , ≤ j ≤ n , ˜ A [ n + 1 , n + 1] = − S ( n + 2 , n + 1) t n +11 t n +21 − t n +2 dt + n ( n +1)2 t n +21 + ( n + 1) t n +2 ( n + 2) t n +2 ( t n +21 − t n +2 ) dt n +2 , where S ( r, s ) is the Stirling number of the second kind defined in (4.9), and the rest ofthe entries of ˜ A are zero. The equalities (6.1) and (6.2) are easy to check and to thosewith Stirling numbers are checked for n = 1 , , ,
4. It would be interesting to prove thisstatement for arbitrary n . We will not need such explicit expressions for the proof of ourmain theorem. The Gauss-Manin connection matrix ˜ A for n = 1 , A = − t dt dt − t t dt − t t − t dt + t t ( t − t ) dt − t t − t dt + t +2 t t ( t − t ) dt , ˜ A = − t dt dt − t t dt − t dt dt − t t dt − t t − t dt + t t ( t − t ) dt − t t − t dt + t t ( t − t ) dt − t t − t dt + t +3 t t ( t − t ) dt . Let A to be the Gauss-Manin connection matrix of the family X / T written in the basis α i , i = 1 , , . . . , α n +1 , i.e., ∇ α = A α . Then we calculate A as follow:(6.3) A = (cid:16) dS + S · ˜ A (cid:17) · S − , where S is the base change matrix α = Sω .12 Proof of Theorem 1.1
As we saw in (6.3), the Gauss-Manin connection matrix of the family X / T in the basis α is given by(7.1) A = dS · S − + S · ˜ A · S − . For a moment, let us consider the entries s ij , j ≤ i, ( i, j ) = (1 ,
1) of S as independentparameters with only the following relation:(7.2) s ( n +1)( n +1) + s nn s = 0 . We denote by ˜ T and ˜ α the corresponding family of varieties and a basis of differentialforms.The existence of a vector field in ˜ T with the desired property in relation with theGauss-Manin connection is equivalent to solve the equation(7.3) ˙ S = Y S − S · ˜ A ( R ) . Note that here ˙ x := dx ( R ) is the derivation of the function x along the vector field R in ˜ T . The equalities corresponding to the entries ( i, j ) , j ≤ i, ( i, j ) = (1 ,
1) serves asthe definition of ˙ s ij . The equality corresponding to (1 , , t = t − t t , ˙ t n +2 = − ( n + 2) t t n +2 . Recall that t = s and t = s . The equalities corresponding to ( i, i + 1)-th, i =2 , · · · , n −
1, entries compute the quantities Y i ’s:(7.4) Y i − = t s ii s ( i +1)( i +1) , i = 2 , , . . . , n − . Finally the equality corresponding to the ( n, n + 1)-th entry is given by (7.2) which isalready implemented in the definition of ˜ T . All the rest are trivial equalities 0 = 0. Weconclude the statement of Theorem 1.1 for the moduli space ˜ T .Now, let us prove the main theorem for the moduli space T . First, note that we havea map(7.5) ˜ T → Mat ( n +1) × ( n +1) ( C ) , ( t , t n +2 , S ) S Ω S tr and T is the fiber of this map over the point Φ. We prove that the vector field R is tangentto the fiber of the above map over Φ. This follows from . z }| { ( S Ω S tr ) = ˙ S Ω S tr + S ˙Ω S tr + S Ω ˙ S tr = ( Y S − S ˜ A )Ω S tr + S (˜ A Ω + Ω ˜ A tr ) S tr + S Ω( S tr Y tr − ˜ A tr S tr )= Y Φ + Φ Y tr = 0 . where ˙ x := dx ( R ) is the derivation of the function x along the vector field R in T . The lastequality follows from (7.4) and Proposition 5.1. It follows that if n is an even integer then13 i − = − Y n − i , i = 2 , . . . , n and if n is an odd integer then Y i − = − Y n − i , i = 2 , . . . , n − and Y n − = ( − n +32 c n ( n + 2) n t s n +12 n +12 t n +21 − t n +2 . In other words(7.6) Y Φ + Φ Y tr = 0 . To prove the uniqueness, first notice that (7.4) guaranties the uniqueness of Y i ’s. Supposethat there are two vector fields R and ˆ R such that ∇ R α = Y α and ∇ ˆ R α = Y α . If we set H := R − ˆ R , then(7.7) ∇ H α = 0 . We need to prove that H = 0, and to do this it is enough to verify that any integral curveof H is a constant point. Assume that γ is an integral curve of H given as follow γ : ( C , → T ; x γ ( x ) . Let’s denote by C := γ ( C , ⊂ T the trajectory of γ in T . We know that the membersof T are in the form of the pairs ( X, [ α , α , . . . , α n +1 ]), in which X is an n -fold mirrorvariety and { α , α , . . . , α n +1 } is a basis of H n dR ( X ) compatible with its Hodge filtrationand has constant intersection form matrix Φ. Thus, we can parameterize γ in such away that for any x ∈ ( C ,
0) the vector field H on C reduces to ∂∂x , and so, we have γ ( x ) = ( X ( x ) , [ α ( x ) , α ( x ) , . . . , α n +1 ( x )]). We know that X ( x ) is a member of mirrorfamily that depends only on the parameter z , hence x holomorphically depends to z .From this we obtain a holomorphic function f such that x = f ( z ). We now proceed toprove that f is constant. Otherwise, by contradiction suppose that f ′ = 0. Then we get(7.8) ∇ ∂∂x α = ∂z∂x ∇ ∂∂z α . Equation (7.7) gives that ∇ ∂∂x α = 0, but since α = ω , it follows that the right hand sideof (7.8) is not zero, which is a contradiction. Thus f is constant and X ( x ) does not dependon the parameter x . Since X ( x ) = X does not depends on x , we can write the Taylorseries of α i ( x ) , i = 1 , , , . . . , n + 1 , in x at some point x as α i ( x ) = P j ( x − x ) j α i,j , where α i,j ’s are elements in H n dR ( X ) independent of x . In this way the action of ∇ ∂∂x on α i is just the usual derivation ∂∂x . Again according to (7.7) we get ∇ ∂∂x α i = 0, and weconclude that α i ’s also do not depend on x . Therefore, the image of γ is a point. (cid:3) The modular vector field R for n = 1 , ,
4, are given as follows: • n = 1:(7.9) R : ˙ t = c ( − c t t − ( t − t ))˙ t = c ( t ( t − t ) − c t )˙ t = − t t . n = 2: We know that dim T = 3, hence the modular vector field R should havethree components, but to avoid the second root of ˇ t that comes from (5.4) we addone more variable t := ˇ t . Thus we find R as follow:(7.10) R : ˙ t = − t t + t ˙ t = − c ( t + 16 c t )˙ t = − c (16 c t t + t )˙ t = − t t , such that following equation holds among t i ’s(7.11) t = − c ( t − t ) . • n = 4: Here, analogously of the case n = 2, to avoid the second root of ˇ t given in(5.5), we add the variable t := ˇ t and we find:(7.12) R : ˙ t = t − t t ˙ t = c t t t − t t + t t t − t ˙ t = c t t t − t t t +3 t t t t − t ˙ t = − c t t t − t t t + t t t t − t ˙ t = c t t t − t t t − t t t +5 t t t +4 t t t +2 t t t t − t ) ˙ t = − t t ˙ t = c t − t c ˙ t = − t t t +3 t t t +3 t t t t − t , where(7.13) t = 11296 c ( t − t ) . In this case the functions Y and Y are given by(7.14) Y = ( − Y ) = 1296 c t t − t . q -expansions Here to find the q -expansion of the solution of R we follow the process given in [Mov16, § n = 3. Consider the vector field R as follow(8.1) R : ˙ t = f ( t , t , . . . , t d )˙ t = f ( t , t , . . . , t d )...˙ t d = f d ( t , t , . . . , t d ) , where for 1 ≤ j ≤ d , f j ∈ Q [ t , t , . . . , t d , t n +2 ( t n +2 − t n +21 )ˇ t ] , t is the same as Theorem 1.1. Let us assume that t j = ∞ X k =0 t j,k q k , j = 1 , , . . . , d , form a solution of R , where t j,k ’s are subject to be constants, and ˙ ∗ = a · q · ∂ ∗ ∂q in which a is an unknown constant. By comparing the coefficients of q k , k ≥ t j,k ’s. By notation, set p k := ( t ,k , t ,k , . . . , t d ,k ) , k = 1 , , , . . . . By comparing the coefficients of q we get that p is a singularity of R . The same for q ,gives us some constrains on t j, . Therefore, some of the coefficients t j,k , k = 0 , q -expansions. Following we state the results in the cases n = 1 , , n = 1 Considering the modular vector field R given in (7.9), we find Sing ( R ) = Sing ∪ Sing ,where Sing : t = t − t = 0 ,Sing : t = t + 3 c t = 0 . Thus we get p = ( t , , − c t , , ∈ Sing . The comparison of the coefficients of q yields a = c t , and we find p as follow: p = ( 29 t , t , , − c t , t , , t , ) . If we choose c = 3 − , t , = and t , = 1, then we find the solution given in (1.11) for R (to find this solution we use a Singular code).In order to study the enumerative properties we first state the following lemma.
Lemma 8.1.
The coefficient of q k , k = 0 , , , , . . . , in θ ( q r ) θ ( q s ) , r, s ∈ N , gives thenumber of integer solutions of equation rx + sy = k , in which x and y are unknownvariables.Proof. We know that θ ( q ) = 1 + 2 P ∞ j =1 q j , hence(8.2) θ ( q r ) θ ( q s ) = 1 + 2 ∞ X i =1 q ri + 2 ∞ X j =1 q sj + 4 ∞ X i,j =1 q ri + sj . If ( i,
0) or (0 , j ) is a solution, then ( − i,
0) or (0 , − j ), respectively, is another solution, and if( i, j ), with i = 0 , j = 0, is a solution, then ( − i, j ), ( i, − j ) and ( − i, − j ) are other solutions.Therefore, on account of the above fact, the proof follows from equation (8.2).16 orollary 8.1. The coefficient of q k , k = 0 , , , , . . . , in θ ( q ) θ ( q ) gives the numberof integer solutions of equation x + 3 y = k . For more information about the number of integer solutions of equation x + 3 y = k ,one can see [Oei, A033716] and the references given there.As we saw in (1.11), t ( q ) = (2 θ ( q ) θ ( q ) − θ ( − q ) θ ( − q )). If we denote by t ( q ) := P ∞ k =0 t ,k q k , then in the following proposition we state enumerative property of t ,k . Proposition 8.1.
Let k be a non-negative integer. If k = 4 m for some m ∈ Z , then theequation x + 3 y = k has t ,k integer solutions, otherwise the equation has t ,k integersolutions.Proof. Suppose that θ ( q ) θ ( q ) = P ∞ k =0 a k q k and θ ( − q ) θ ( − q ) = P ∞ k =0 b k q k . Fix anon-negative integer k . If k = 4 m for some m ∈ Z , then a k = b k , otherwise a k = − b k .This fact together with Corollary 8.1 complete the proof.Y. Martin in [Mar96] studied a more general class of η -quotients. By definition an η -quotient is a function f ( q ) of the form f ( q ) = s Y j =1 η r j ( q t j )where t j ’s are positive integers and r j ’s are arbitrary integers. He gives an explicit finiteclassification of modular forms of this type which is listed in [Mar96, Table I]. In (1.11)we found(8.3) t ( q ) = η ( q ) η ( q ) , which is the multiplicative η -quotient ♯ η -quotient the reader is referred to the Webpage [Oei, A106402]. Remark 8.1.
If we define P ∞ k =0 a k q k := t ( q ) = ( E ( q ) − E ( q )) , then one can seethat | a k for integers k ≥ . n = 2 From (7.10) we get
Sing ( R ) = { ( t , t , t , t ) | t = t − t t = t + 16 c t = 0 } , hence we find p = ( t , , k t , , k t , , ∈ Sing ( R ) , in which k = √− c . By comparing of the coefficients of q we get a = − t , k and p = ( − t , k t , , t , t , , t , , − t , t , k ) , where the equality t , = − t , t , k follows from (7.11). By considering c = − , t , = and t , = −
1, we find the solution given in (1.12) for R (here also we use a Singular code).17 t t t R t ( q ) t ( q ) 10 t ( q ) q q q q q q q q q q q q q q q q q q q q
13 -13 -2043 q q q q
12 -12 1092 q q q q
14 -14 1382 q q q q
24 -24 -2520
Table 1: Coefficients of q k , ≤ k ≤ , in the q-expansion of the solutions of R and R . Remark 8.2.
The demonstration of that (1.11) and (1.12) are solutions of (1.8) and (1.9) , respectively, can be done in a similar way as of Ramanujan’s or Darboux’s case, andin order to keep the article short, we skip the proofs and just mention that we checked theequality of first 100 coefficients of q-expansions, that we find by using of
Singular , with thecoefficients of (quasi-)modular forms given in (1.11) and (1.12) . in Table 1 we list thefirst 16 coefficients.
The sum of positive odd divisors of a positive integer k , which is also known as odddivisor function , was introduced by Glaisher [Gla07] in 1907. Let k be a positive integer.We denote the sum of divisors, the sum of odd divisors and the sum of even divisors of k ,by σ ( k ), σ o ( k ) and σ e ( k ), respectively, i.e., σ ( k ) = X d | k d & σ o ( k ) = X d | k d is odd d & σ e ( k ) = X d | k d is even d . It is evident that σ ( k ) = σ o ( k ) + σ e ( k ). Also one can find that σ o ( k ) = σ ( k ) − σ ( k/ , in which σ ( k/
2) := 0 if k is an odd integer. The generating function of odd divisor functionis given as follow ∞ X k =0 σ o ( k ) q k = 124 ( θ ( q ) + θ ( q )) , where by definition σ o (0) = 1 /
24. On account of (1.12), one finds that ∞ X k =0 σ o ( k ) q k = 106 t ( q
10 ) . Therefore we have the following result.
Proposition 8.2. t generates the odd divisor function. t presented in Table 1 with the integers sequencegiven in the [Oei, A215947] we find that ∞ X k =0 ( σ o (2 k ) − σ e (2 k )) q k = 104 t ( q
10 ) = 124 ( E ( q ) + 2 E ( q ))where we define σ o (0) − σ e (0) := 1 / Proposition 8.3. t generates the function of difference between the sum of the odddivisors and the sum of the even divisors of k , i.e., σ o (2 k ) − σ e (2 k ) . Another nice observation is about 10 t ( q ) = η ( q ) η ( q ). The same as t in the caseof elliptic curve (see (8.3)), we see that t is the η -quotient ♯ η -quotient appears in the work of HeekyoungHahn [Hah07]. She proved that 3 | µ k , k = 0 , , , . . . , where µ k is defined as follow ∞ X k =0 µ k q k := η ( q ) η ( q ) . Also she found some partition congruences by using the notion of colored partitions (formore details see [Hah07, § n = 4 The set of the singularities of R contains the set of ( t , t , . . . , t )’s that satisfy(8.4) t = t − t t = 6 c t − t = t − c t = t − t t = − t − t t = 0 . Hence if we fix t , and t , , then from (8.4) we get p = ( t , , t , , t , t , , − k t , , − k t , , , − c t , t , , − k t , ) , in which c = k . By comparing coefficients of q we find a = − t , , R q q q q q q q t t − t −
11 115137 2265573692 54820079452449 1477052190387154386 42523861222488896739828 t −
16 193131 3904146832 95619949713765 2594164605185043648 75018247757143686903060 t −
45 469872 9215455916 222628516313454 5992746995783064438 172421735348939185816992 − t − t − t − Table 2: Coefficients of q k , ≤ k ≤ , in the q-expansion of a solution of R .19nd p = ( 60 k t , t , , − k t , t , t , , − k t , t , t , , t , t , , t , t , , k t , t , , t , k t , t , , t , ) . After fixing k = − − , t , = , t , = − t , = we find the q -expansion of asolution of R . We list the first seven coefficients of q k ’s in Table 2. As it was expected,after multiplying t j ’s by a constant, all the coefficients are integers.If we compute the q -expansion of Y given in (7.14), then we find Y = 6 + 120960 q + 4136832000 q + 148146924602880 q + 5420219848911544320 q + 200623934537137119778560 q + 7478994517395643259712737280 q + 280135301818357004749298146851840 q + 10528167289356385699173014219946393600 q + 396658819202496234945300681212382224722560 q + 14972930462574202465673643937107499992165427200 q + . . . which is 4-point function discussed in [GMP95, Table 1, d = 4]. References [AMSY16] Murad Alim, Hossein Movasati, Emanuel Scheidegger, and Shing-Tung Yau.Gauss-Manin connection in disguise: Calabi-Yau threefolds.
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