Calabi-Yau attractor varieties and degeneration of Hodge structure
aa r X i v : . [ m a t h . AG ] F e b SUPERSYMMETRIC FLUX COMPACTIFICATION ONHYPERGEOMETRIC FAMILY OF CY 3-FOLDS
MOHAMMAD REZA RAHMATI
Abstract.
In this paper we study the structure of string theory flux compactificationfor a general family of elliptic CY 3-folds. Such a family can be constructed by successivequadratic twists from similar families of lower dimensions, starting from family of points.In this case the periods are given by specific hypergeometric functions. We investigatethe specific form of the attractor equation in the flux compactification procedure, knownin string theory. We also discuss the concept of the flux compactification defined via theblack holes, and study the attractor varieties in the hypergeometrics case. Introduction
A Calabi-Yau manifold is a compact Kahler manifold with trivial canonical class. Anexample is a smooth projective variety defined by a homogeneous polynomial of degree( n + 2) in complex projective space P n +1 . For n = 1 this gives an elliptic curve, andfor n = 2 it yields a K3 surface. For n = 3 the classification is open. The originalquintic mirror family constructed in [5] is a family of CY 3-folds on P \ , , ∞ withHodge structure of type (1 , , ,
1) of the general fiber. The family plays a crucial rule inmirror symmetry. Later an enormous research activities were done by mathematicians toproduce new examples of this phenomena. In [1] a way to construct the mirror of a toricCY hypersurface by dual reflexive polytopes was found. CY 3-folds with h , = 1 play animportant role both in mathematics and physics. A famous example of the mirror familiesis the one parameter family(1) f = z n +10 + z n +11 + ... + z n +1 n + ( n + 1) ψz z ...z n = 0due to Dwork. For n = 4 is the aforementioned quintic family of Candelas et al. Themirror family is defined by(2) y y ...y n ( y + y + ... + y n + 1) + ( − n +1 φ ( n + 1) n +1 = 0which defines a family over P . The family maybe studied iteratively starting from n = 1,where both of the equations define a family of quadrics laid in P , and hence are isomorphicfiberwise. Key words and phrases.
String theory flux compactification, Black hole, Attractor CY variety, Degen-eration of Hodge structure, Hypergeometric functions, Periods of integrals on CY varieties.
In this article we concern the projective family of elliptically fibered CY 3-folds(3) f : X −→ P \ , , ∞ Each fiber X s = f − ( s ) is a compact complex 3-fold with trivial canonical bundle ω X s = V T ∗ X s . The family of middle cohomologies(4) H = R f ∗ C −→ P \ , , ∞ carries a flat connection called the Gauss-Manin connection. Such a family maybe con-structed iteratively from a family of 1-lower dimension by quadratic twist, see Section 1below. If ξ ∈ Γ( P \ , , ∞ , H ) then the local analytic function Π ξ : s R γ s ξ is calleda period integral. The Gauss-Manin connection ∇ reduces to a linear differential equa-tion L n ( s )Π ξ ( s ) = 0 of rank n namely Picard-Fuchs equation. There is a meromorphic( n + 1)-form(5) Θ = X j ( − j ψf ( z i dz ∧ ... ∧ c dz i ∧ ... ∧ dz n +1 )where restricts to a holomorphic n -form Ω ψ = Res X s (Θ) on the smooth fibers. Thenthe periods of the Hodge structure of the fibration are integrals of the form R Σ Ω ψ for σ ∈ H n ( X s , Z ). The Hodge filtration on H ( X ψ , C ) is given by(6) F = h Ω ψ i F = h Ω ψ , Ω ′ ψ i F = h Ω ψ , Ω ′ ψ , Ω ′′ ψ i F = h Ω ψ , Ω ′ ψ , Ω ′′ ψ , Ω ′′′ ψ i There is another method to compute the periods by solving the Picard-Fuchs equation(7) Θ n +1 − ψ n +1 Y k +1 (Θ + kn + 2 ) ! Ω ψ = 0 , Θ = ψ ddψ
By a choice of basis ( α i ) for H N ( X ψ , Z ), we can form the period vectors Π Ω = [Π i = R α i Ω ψ ].It is known that the forms Ω ψ , ΘΩ ψ , ..., Θ n Ω ψ are linearly independent in H n ( X ψ , C ). Itfollows that we may form the vectors Π Ω , ΘΠ Ω , ..., Θ n Π Ω in a matrix to obtain a partialperiod matrix, [13]. In fact this construction applies to other smooth families.We consider family of elliptic curves(8) y = 4 x − g ( t ) x − g ( t ) , t ∈ P where g and g are polynomials of degrees at most 4 and 6 respectively. Set ∆ = g − g , j = g / ∆ where g , g , ∆ , j are Weierestrass coefficients, discriminant and J -function. The Picard-Fuchs equation for the family is given by(9) ddt (cid:20) ωη (cid:21) = (cid:18) − d log δdt δ − g δ
8∆ 112 d log δdt (cid:19) (cid:20) ωη (cid:21) , δ = 3 g g ′ − g g ′ UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS3 where ω = R γ dxy , η = R γ xdxy and γ being a 1-cycle, [2]. The local system H = R π ∗ C hasa two step Hodge filtration(10) F = H ( X t , C ) ⊃ F The Gauss-Manin action on a multivalued section of the canonical extension of H is givenas(11) ∇ : α ∗ + (cid:18) n i log s π √− ψ ( s ) (cid:19) β ∗ (cid:18) n i ds π √− s + ψ ′ ( s ) ds (cid:19) β ∗ where α and β are a symplectic basis of H ( X s , C ), [7]. Now we go to a family of K y = 4 x − G ( t, s ) x − G ( t, s )where G , G are polynomials of degree at most 8 ,
12 in the affine coordinate s and suchthat they are also polynomials in t . The periods are calculated via the integral R γ ds ∧ dxy and the local systems H = R π ∗ C has a weight 2 Hodge filtration(13) F = H ( X ( t,s ) , C ) ⊃ F ⊃ F An illustration for the second family is(14) E t −−−→ E −−−→ H y y y j ( τ ) t −−−→ P −−−→ j ( t ) P j − line where j is the j -function of the fibers, cf. [7]. One may proceed inductively to 3-dimensional fibrations over surfaces,(15) y = 4 x − G ( s, t, u ) x − G ( s, t, u )having singular fibers t = 0 , , ∞ , where s and u are affine variables on a surface. Wehave X t → P . The cohomologies of the smooth fibers build up a VHS of type (1 , , , Explanation on the text and references:
Section 1 is the introduction containing alsosome basics of Hodge theory. Section two is a brief from [2] and explains hypergeometricfibration of CY varieties. Section 3 is a brief on attractors and flux compactificationmainly from [10]. Section 4 explains a partial toroidal compactification of period domainsof Hodge structure, with main references [3, 4, 7]. In Section 5 we give a method to studyattractors in the asymptotic MHS and on the boundaries of the period domains. We alsoincluded how the attractor equation maybe applied in the hypergeometric case.
MOHAMMAD REZA RAHMATI Elliptically fibered family of CY 3-folds
This section is a brief from [2] to introduce hypergeometric families of CY manifoldselliptically fibered over varieties of smaller dimensions. The reader can consult with thereference for more details. The elliptically fibered family of CY 3-folds can be analysediteratively by starting from a family of two points(16) y = 1 − s It is isomorphic to a quadric pencil. By choosing a suitable branch cut we obtain aholomorphic section y − = F (cid:0) | s (cid:1) = (1 − s ) − / , where F is the hypergeometricfunction. It is annihilated by the operator(17) L ( s ) = Θ − s (Θ + 1 / , Θ = s dds with monodromy equal to 1 , − , − , , ∞ .Applying the twist replacement(18) y y /x ( x − s )in the equation of the former family we obtain the family(19) y = x ( x − x − s ) , s ∈ P \ , , ∞ where the middle cohomologies of the fibers H s = H ( X s , C ) provides a local system ofpolarized Hodge structure of weight 1 and of type (1 , F ,s is the whole H ( X s , C )and F = H , ( X s ). the space of harmonic 1-forms is generated by the 1-form dx/y . Theperiods are defined via the integral(20) Z dx/y = Z s dx/ p x ( x − x − s ) = ( iπ ) F (cid:16) , (cid:12)(cid:12) s (cid:17) = F (cid:18) | s (cid:19) ∗ F (cid:18) | s (cid:19) where the ∗ is the Hadamard product of power series(21) ( X a n s n ) ∗ ( X b n s n ) = X a n b n s n The Picard-Fuchs operator becomes(22) L ( s ) = Θ − s (Θ + 1 / where corresponds to the corresponding Hadamard product of the differential operators.We may repeat this process by applying the same twist to get a family of K3 surfaces(23) y = x ( x − x − t ) t ( t − s )over P where the singular fibers lie over 0 , , t, ∞ . Again the middle cohomologies of thesmooth fibers defines a Hodge structure of weight 2. we can stress the periods in two ways.The first is through the twist we applied above. For suitable varying 1-cycles γ , γ thatform a basis for the cohomology of the elliptic fiber the periods of the middle cohomology UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS5 are calculated via the integral R ts dt (cid:18)R γ dx/y R γ dx/y (cid:19) . A particular choice of branches the periodsare related to(24) Z s (cid:16) dt/ p t ( t − s ) (cid:17) F (cid:16) , (cid:12)(cid:12) s (cid:17) = ( iπ ) F (cid:16) , , , (cid:12)(cid:12) s (cid:17) = F (cid:18) | s (cid:19) ∗ ( iπ ) F (cid:16) , (cid:12)(cid:12) s (cid:17) The second, the original period is given via integrating the 2-form dt ∧ dx/y over a twocycles Γ where it results(25) Z Γ dt ∧ dx/y = ( iπ ) F (cid:16) , , , (cid:12)(cid:12) s (cid:17) The Picard-Fuchs operator is given(26) L ( s ) = Θ − s (Θ + 1 / . Applying the next twist we get the family of CY 3-folds(27) y = x ( x − x − t ) t ( t − w ) w ( w − s )that are elliptically fibered over P and also fibered over K3 surfaces of Picard rank 19.We denote f : X → P to be the family parametrized with s , where the family of middlecohomologies(28) R f ∗ C → P \ , , ∞ has a Hodge structure of weight 3 and of type (1 , , , , Z s (cid:16) dt/ p w ( w − s ) (cid:17) F (cid:16) , , , (cid:12)(cid:12) s (cid:17) = ( iπ ) F (cid:16) , , , , , (cid:12)(cid:12) s (cid:17) = F (cid:18) | s (cid:19) ∗ ( iπ ) F (cid:16) , , (cid:12)(cid:12) s (cid:17) The second, the original period is given via integrating the 2-form dw ∧ dt ∧ dx/y over atwo cycles Γ where it results(30) Z Γ dw ∧ dt ∧ dx/y = ( iπ ) F (cid:16) , , , (cid:12)(cid:12) s (cid:17) The Picard-Fuchs operator is given(31) L ( s ) = Θ − s (Θ + 1 / This process continues similarly to higher dimensional fibrations.The aforementioned construction maybe done using the Weierstrass normal forms in theequation of fibration, and in that case one obtains hypergeometric functions with slightlydifferent weights. However the same relation exists between the Picard-Fuchs equationsand the periods. In general the hyper geometric function(32) F ( s ) = n F n − (cid:0) ρ ,ρ ,...,ρ n , ,..., (cid:12)(cid:12) s (cid:1) MOHAMMAD REZA RAHMATI satisfies the differential(33) [Θ n − s (Θ + ρ ) ... (Θ + ρ n )] F ( s ) = 0The construction also applies to the mirror pair. That is on the mirror family we also havean iterative formulas involving the periods and the differential operators. An example isthe mirror for the Dwork family stated in equation (2) in the introduction. In this caseaccording to [[2] theorem 10.2] we have(34) Π n − ( s ) = (2 iπ ) n F n − (cid:18) n +1 , n +1 ,..., nn +11 n , n ,..., n − n (cid:12)(cid:12) s (cid:19) ∗ Π n − ( s ) . One notes that in all the examples constructed iteratively we have singular fibers over0 , , ∞ . We can normalize the local solutions by setting(35) f α ( t ) = t αn F αn − (cid:0) ρ ,ρ ,...,ρ n , ,..., (cid:12)(cid:12) t (cid:1) = X j ( ρ + α ) j ... ( ρ n + α ) j (1 + α ) nj t j + α = n − X l =1 (2 iπα ) l f l ( t )where f l = iπ ) l l ! ∂ l ∂α l | α =0 f α ( t ) and ( ρ ) j = ( − j Γ(1 − ρ )Γ(1 − k − ρ ) . The functions ( f n − , ..., f ) forma basis of the local solutions near t = 0 and the monodromy at t = 0 in this basis is givenby(36) M = / ... / ( n − ... / ( n − ... ... ... ... ... Similarly by introducing the functions(37) F k ( β, t ) = B k t − βn F αn − (cid:18) ρ k , ρ k , ..., ρ k β k − β , ... ˆ1 ..., β k − β n (cid:12)(cid:12) t (cid:19) for specific numbers B k . According to the analysis in [2] the monodromy at ∞ with respectto the basis ( F n , ..., F ) is given by(38) M ∞ = [ e − iπβ n , e − iπβ n − , ..., e − iπβ ]The above two monodromy are related by the transition matrix between the analytic con-tinuations of the two basis, denoted by P . We have M ∞ = P.M .P − , and the monodromyat the other singularity is given by(39) M = M ∞ M − . See the ref. loc. cit. for details.
UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS7 Attractor points in moduli of complex structures
We explain two kinds of attractors in dimension 3 with origins from string theory. Letsbegin from the definition of the attractor point in the moduli of X . Definition 3.1.
A Calabi-Yau n -fold X is called an attractor if there exists an integralcohomology class γ ∈ H n ( X, Z ) such that (40) γ ⊥ H n − , under the symplectic inner product on H n ( X, C ) where we have considered the Hodge decomposition decomposition (41) H n ( X, C ) = H n, ( X ) ⊕ H n − , ( X ) ⊕ ... ⊕ H ,n ( X )Let h i,j = dim H i,j ( X ) be the Hodge numbers. The definition puts h n − , conditionson the coordinates in the moduli of Hodge structure (period domain) of X . The pointsin the moduli space corresponding to attractor varieties are called attractor points. Itis a conjecture that an attractor variety is defined over Q . The concept is reasonably ofinterest in Physics when n = 3. In this case the above conjecture is due to Moore and wassupposed that the attractors furnish special points analogous to complex multiplicationpoints on Shimura varieties.Let X be a Calabi-Yau 3-fold and M be the moduli of complex structures on X . It isknown that dim( M ) = h , . Literally the above attractor definition comes from type IIBstring theory on the variety R × X where R is the Lorenzian 4-manifold, i.e. R , and X is a CY variety of complex dimension 3 or real dimension 6. On this space we shall havea theory of supergravity which arises from a self dual 5-form in 10-real dimension calledtype IIB supergravity. This form is presented by the form(42) F ∈ Ω ( R ) ⊗ H ( X, R )called the electro-magnetic field. The self duality means(43) F = ∗ F, in 10-dimension , where ∗ is the Hodge star operator. We should decompose H ( X, R ) = g ⊕ g ∗ as a realsymplectic vector space with the symplectic product h ., . i . In a Physics theory one dealswith a complex symplectic structure on H ( X, R ) defined by a linear transformation(44) J : H ( X, C ) → H ( X, C ) , J = − , h J x, J y i = h x, y i We can define an operator ∗ T = ∗ ⊗ J . It satisfies ∗ T = +1, and F = − ∗ T F . Using thesymplectic basis mentioned above we can write(45) F = X X j α j + Y j β j On the other hand using self duality in the complex basis ( η j , η j ) we can write(46) F = X z + j η j + z − j η j The attractor equation arises from the comparison of the two above equations, in the waythat the expression in the symplectic basis should lie in H , ⊕ H , . Let w ∈ H ( X, Z ). MOHAMMAD REZA RAHMATI
Lets choose a generator Ω ∈ H , ( X ). Also let A j , B j be a symplectic basis for H ( X ),and α j , β j the dual basis. Consider the flat coordinates x j = R α j Ω , y j = R β j Ω. Then theattractor condition may be written as(47) ¯ cx j − c ¯ x j = ia j ¯ cy j − c ¯ y j = ib j , c = const , w = [ b j , a j ] T where by [ b j , a j ] we mean the symplectic coordinates of w . If two charges w , w ∈ H ( X, Z ) correspond to the same attractor then by what was said there are two com-plex numbers λ , λ such that(48) Im(2¯ λ Ω) = w , Im(2¯ λ Ω) = w It follows that(49) h w , w i = 2Im( λ λ ) √− h Ω , Ω i If w = λw , λ ∈ Q we say the attractor has rank 1. If h w , w i 6 = 0 we say it has rank 2,[10].The attractor points can be equally defined by the fixed points of the gradient flow ofthe function(50) Z w = |h w, Ω i|h Ω , Ω i on the moduli space M . Then the attractor corresponding to the cohomology class γ arethe fixed points of the flow. This process is called flux compactification in string theory. Aslightly different mechanism of Attractors also arise from 4-dimensional N = 2 black holesin the compactification of type IIB supergravity on CY 3-fold X (with Hodge structure oftype (1 , , , ds = − e U dt + e − U dx The charge of the black hole is given by γ ∈ H ( X, Z ) which Pioncare dual to the form w we used above. The complex structure of X is getting varied by the radius r described bythe differential equation(52) dU/dρ = − e U | Z γ | , dφ/dρ = h , X j =1 − e U g j ¯ j ∂ ¯ j | Z γ | where ρ = 1 /r and the index j runs through a specific basis ( η j ) of H , ( X ). The basiscan be written specifically by considering local holomorphic coordinates on the moduli M of complex structures on X . One has T , x M = H , ( X ) and the holomorphic coordinatesmaybe denoted by z i , ≤ i ≤ h , . then the basis can be chosen to be(53) η j = e K/ π , ( ∂ z i Ω) = e K/ (cid:18) ∂ z i Ω − h ∂ z i Ω , Ω ih Ω , Ω i Ω (cid:19) UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS9 where π , is the projection on the H , factor in the Hodge decomposition and the differ-entiation is considered as part of the period coordinates involving M . The function(54) K = log (cid:0) √− (cid:10) Ω , Ω (cid:11)(cid:1) is the Kahler potential of WPZ metric on M . Then g i ¯ j = −√− h η i , η j i .(55) Z γ = e √− h Ω , Ω i / Z γ Ωis a function. These equations can be understood as a gradient flow of the function | Z γ | with respect to the metric. By choosing a symplectic basis ( A i , B i ) for H ( X, Z ) we maywrite γ = P i a i A i − b i B i . Let ( α j , β j ) be the dual basis and(56) w = X b i α i − a i β i , A = ( b j , a j ) T Then we have(57) Z γ = A t ΣΠ( − i Π † ΣΠ) / where Σ is the intersection form in the symplectic basis and A is the coordinate matrix insymplectic basis. As was said the attractor points minimize | Z γ | , [10, 8].Flux vacua in type IIB theory has contribution from two fields defined by the 3-forms F (3) and H (3) and is formulated by the superpotential W which is written in terms of thecomplex axion-dilaton τ = C + √− e − φ and the field G (3) = F (3) − τ H (3) as the integral(58) W = Z X G (3) ∧ Ωwhere Ω is a holomorphic 3-form. The parameter τ lies in an upper half plane and thevacuum constraints can be written as D τ W = D i W = 0 where D τ = ∂ τ + ∂ τ K, D j = ∂ j + ∂ j K , [11].4. Degeneration of Hodge structure of CY 3-folds
In this section we review basic concepts and definitions of the boundary componentsof the period domains of Hodge structures taken from [7, 4, 3] as well as the very refer-ences there in. A polarized Hodge structure of weight n on a Q vector space V togetherwith a ( − n -symmetric non-degenerate Q -bilinear form Q : V × V → Q , is given by arepresentation(59) φ : S → Aut ( V R , Q )defined over R such that if we denote the t p ¯ t q -eigenspace of φ ( t ) by V p,q , then p + q = n forall non-zero V p,q . We denote C := φ ( i ). The adjoint action of the group G R = Aut ( V R , Q )on φ defines the period domain of polarized Hodge structures of weight n on V , denotedby D . Denote the centralizer of φ by M = Z φ ( G R ), then D = G R /M . It embeds in ˇ D = G C /P where is the centralizer of φ (parabolic in G ), and inherits a complex structurefrom this embedding.A basic example of this is the VHS V = V , ⊕ V , of weight 1, obtained from the middlecohomology of a fibration of curves of genus g . In this case(60) D = H g = { Z ∈ M g × g : Z = t Z, Im ( Z ) > } is the Siegel generalized upper half space. A variation of Hodge structure V on a quasi-projective variety S can be given by its period map(61) Π : S → Γ Z \ D where Γ Z is a discrete subgroup of G called monodromy group. Let h = ( h p,q ) p + q = n be the h -vector of Hodge numbers. • When weight = n = 2 k + 1 is odd we have(62) D h = SP n ( R ) / Y l ≤ k U ( h l,n − l ) • When n = 2 k set h odd = P l odd h l,n − l , h even = P l even h l,n − l . Then(63) D h = SO ( h odd , h even ) / SO ( h k,k × Y l 0) + 2(0 , 1) where N = 0 • (1 , , 1) and (1 , → (0 , 0) for N . • The same as previous item for N . • , → , 0) corresponding to N + N .where the multiplicities denote the dimension of the I p,q and N and N are(72) N = , N = The underlying vector space has dimension 4 and the group G is Sp . We have the above4 boundary components. This example is not CY type.Lets consider the degeneration of a Hodge structure of weight 2 and of type (1 , n, I p,q components of the degenerating MHS are • (2 , 0) + n (1 , 1) + (0 , • (1 , 0) + (0 , 1) + (2 , 1) + (1 , 2) + ( n − , • (0 , 0) + n (1 , 1) + (2 , n ≥ 2. The nilpotent transformations N are of type ( − , − 1) on eachcomponents depending to n . Each of the above 3 cases appear as different boundarycomponents of the period domain. We consider the example of a Hodge structure of weight 3 and dim V = 4 , h = (1 , , , D = Sp ( R ) /U (1) × . We shall also illustrate the ( p, q )-domain of the Delignedecomposition V = ⊕ p,q I p,q . In this case the only possibilities for N are(73) N = a c b d c − a , N = a N = (cid:18) A (cid:19) , A t = A > B ( N ) ∼ = C ∗ all 4 nonzero I p,q on the diagonal,(0 , ← (1 , ← (2 , ← (3 , 3) where N acts backward. From the result of [ ? ] exlainedabove we have B ( N ) ∼ = non compact elliptic modular surface and the points have nonzero I p,q on (3 , , (0 , 3) isolated, and (1 , ← (2 , N correspondto two pure Hodge structure (2 , ← (3 , 1) and (0 , ← (1 , B ( N ) ∼ =CM elliptic curve, [3].Consider the HS worked out by Carayol, the Q -vector space of dimension 6 and theHodge structure of weight 3 of type (1 , , , D = U (2 , /U (1) . The examplegiven by Carayol considers HS with decomposition V Q ( √− d ) = V + ⊕ V − , where V + = V − .There are 3 possible boundary components • B ( N ) ∼ = C × of points with I p,q to be (3 , → (2 , → (1 , 0) and (2 , → (1 , → (0 , • B ( N ) ∼ = CM elliptic curve of points with I p,q to be (3 , → (2 , 0) + (1 , → (0 , , → (1 , • B ( N ) ∼ = complex conjugate of B ( N ) of points with I p,q to be (3 , 0) + (0 , 3) and2(2 , → , Attractor locus in a hypergeometric case We apply the analysis in Sections 3 and 4 to the hypergeometric families explained inSection 2. Consider the fibration over P defined by(74) y = 4 x − G ( s, t, u ) x − G ( s, t, u )where exists singular fibers at t = 0 , , ∞ , such that s and u are affine variables on asurface. The cohomologies of the smooth fibers build up a VHS of type (1 , , , F ( s ) = n F n − (cid:0) ρ ,ρ ,...,ρ n , ,..., (cid:12)(cid:12) s (cid:1) In order to write a basis for the vanishing cohomology we may use the form of solutionsnear 0. That is the basis(76) f j = 1(2 iπ ) j j ! ∂ j ∂α j (cid:12)(cid:12)(cid:12) α =0 f α ( t ) , f α = t α F ( t ) , j = 3 , , , UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS13 We form the vector ̟ = [ f , f , f , f ] T namely period vector. The Hodge filtration on H ( X ) is given by(77) F = h f i F = h f , f i F = h f , f , f i F = h f , f , f , f i over the C . Then we need to write down ̟ in a symplectic basis ( α , α , β , β ). Thereis a transition function S such that Π = S̟ is the new period vector in the symplecticbasis. then the volume form on the fibers Ω t can be written as(78) Ω t = [ α , α , β , β ] . ΠIf w ∈ H ( X, Z ) then(79) w , = w , = C Ω t = C ΩTherefore w defines an attractor if(80) w = [ α , α , β , β ]( C Π + C Π)( C Π + C Π) ∈ Z \ S and also the attractor condition involved in the above computation is prac-tically studied in Physics of flux compactification and black hole attractors in specificexamples. The equation in terms of the basis around infinity can be calculated withsimilar procedure. In that case the period vector is(81) ̟ ∞ = F F F F , F k ( β, t ) = B k t − βn F αn − (cid:18) ρ k , ρ k , ..., ρ k β k − β , ... ˆ1 ..., β k − β n (cid:12)(cid:12) t (cid:19) as in Section 2. We have to write ̟ ∞ in a symplectic basis and Π ∞ = S ∞ ̟ . Thus wehave analogous equation for the attractor.Using a Frobenius method argument we may write the periods of the VHS associatedto (14) as ̟ j ( t ) = f j ( t ), that is the same as the periods are given by the hypergeometricfunctions and their derivatives. It is convenient to locally write these periods as the sumof multivalued sections as(82) ̟ j = X l =0 (cid:18) j (cid:19) f j log( t ) − j which have logarithmic singularities and extend the former solutions over the degeneracies.We wish to write the periods in a symplectic basis, denoted by Π. According to [13, 9] or by simple linear algebra we can write(83) Π Π Π Π = a , a , a , a , a , a , ̟ ̟ ̟ ̟ In theoretical Physics they study the matrix A × in the above identity in examples as A = A ζ A log , where A ζ is a matrix with etries in terms of zeta values, and A log has formalentries in terms of logarithms a log ( n ) = n log n π √− , [9].Another method to understand the exchange matrix A above is through mirror sym-metry. The variation of HS of the family of CY 3-folds corresponds to the deformation ofa germ of Frobenius algebra constructed from the cohomology ring of the fiber varieties.The product structure on these Frobenius algebra maybe explained by a quantum Kahlerpotential on the moduli of Kahler forms on the CY 3-fold. In Physics they refer to it asthe Kahler prepotential on the Kahler side of the mirror. The potential has the form(84) φ = − φ q − φ q − φ q − φ + φ np where the coefficients φ ijk are given by certain triple products of cohomology classes indegree 2, and can be calculated in terms of the derivatives of the potential itself. Thevariable q correspond to the quantum deformation on the potential. The last term φ np is called the perturbative correction and when q → √− ∞ it tends to 0 called the largecomplex structure limit. Under the mirror map taking t = 0 corresponds to take q = √− ∞ . This leads to the following formula for the matrix S , see [13, 9],(85) S = Const(2 π √− − φ − φ φ − φ − φ − φ 01 0 0 00 1 0 0 , Const ∈ Q × The concept of the large complex structure limit comes from the mirror symmetry andthe nature of the mirror map between two mirror CY varieties. The point is, under themirror map the neighborhood of zero on the complex side correspond to a neighborhoodof √− ∞ in the mirror family (Kahler side).6. Attractor points and degeneration of Hodge structure The attractor condition can be explained as a degenerating situation for the cohomologyclass w to collapse through a family to a class in H , ⊕ H , . That is we consider acohomology class w on a thickening of X over a suitable quasiprojective base as X → S ∗ which restricts to a w s on the generic fiber X s ∼ = X , and decompose it regarding the Hodge UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS15 decomposition(86) H ( X s , C ) = H , ⊕ H , ⊕ H , ⊕ H , w s = w , s ⊕ w , s ⊕ w , s ⊕ w , s Then there should exist a compactification of S ∗ and an extension of the the variation ofHodge structure of the fibers mentioned above such that(87) lim s → s w s ∈ H , ⊕ H , The aforementioned phenomenon maybe studied in a universal way, i.e over M . The limitdefinition in a one parameter family of Hodge flags defined by a nilpotent transformation N ∈ End( V ) can be expressed as(88) lim Im( z ) →∞ exp( −√− z.N ) M p + q =3 H p,q ( X z ) ! ⊂ H , ⊕ H , Because N r = 0 for some r ≤ N actually has at most 4terms. having N to be specific while the vectors in the Hodge decomposition are writtenin terms of periods, the above limit gives the equations defining attractor points.In the hypergeometric case we go through the limit process in the moduli of Hodgestructures of type (1 , , , 1) that is the last example in the previous section. In thiscase one simply notes that the boundary B ( N ) lies in the attractor locus. The analysismentioned above shows that the HS must degenerate to pure HS of weight 1 or some Tatetwist of it. Furthermore the locus is a locus of CM points and itself is special, i.e hasCM structure. The other two locus of boundaries associated to N , N never satisfy theattractor condition as it maybe read from the ( p, q ) diagram of the corresponding LMHS.Thus B ( N ) is contained in the attractor locus.We can calculate the equation of the boundary component B ( N ). In fact exp( −√− z.N ) = I + √− z.N , because N = 0. It follows that to obtain an attractor point we must have(89) lim Im( z ) →∞ (cid:18) I + (cid:18) √− z.A (cid:19)(cid:19) Π Π Π Π is of type (3,0)+(0,3)where A × is symmetric. Letting A = (cid:20) a bb d (cid:21) we see that the condition reads as(90) Π Π √− az Π + √− bz Π + Π √− bz Π + √− dz Π + Π is of type (3,0)+(0,3) It follows that the locus of attractors obtained in this way has the equation(91) Π = 0lim z →∞ az Π + bz Π + Π = 0Π = lim z →∞ √− bz Π + √− dz Π + Π where Π is symplectic period vector. It is probable that some attractor points appear in B ( N ) and B ( N ) where by the above method we are blind to them. However we maywrite specific equations for them as the one above. For example for B ( N ) with the samemethod we obtain the following equations(92) Π = Π = 0Π = lim z →∞ az Π + Π there also may exist special values giving attractor on the component B ( N ). In this case N = 0 and exp( −√− z.N ) = I − √− z.N + z N − √− z N and the condition is(93) lim z →∞ (cid:0) I − √− z.N + z N − √− z N (cid:1) Π Π Π Π is of type (3,0)+(0,3)Replacing the matrix of N from the previous section one can simply form the possibleequations of attractors that may exist on this component.In Carayol example we expect possible attractors in the locus of B ( N ) and B ( N ). Inall the 3 cases the equation of the attractors can be exactly written using the specific formof the corresponding nilpotent transformation.7. Relation with GIT and moment map Assume we have a symplectic manifold ( X, ω, J ) that is X is closed compact Kahlermanifold with symplectic form ω and J an integrable complex structure such that ω ( ., . ) isa Riemannian metric. Also assume we have an action of a Lie group G R ⊂ U ( n ) on X whichpreserves all the 3 structures ω, J and the metric. The Lie algebra g = Lie ( G ) ⊂ u ( n )has an infinitesimal action on the vector fields over X written as ξ [ X X ξ ]. Equip g with an invariant bilinear form such that the G -action gets Hamiltonian. If(94) µ : X → g is a moment map for the action, the X ξ is a Hamiltonian vector field such that L X ξ ω = dH ξ where H ξ ( . ) = h µ ( . ) , ξ i . We can write this as(95) h dµ ( x ) v, ξ i = ω ( X ξ ( x ) , v )for v ∈ T x X .Let G C be the complexification of G and g C = g + √− g its Lie algebra. Takingthe group of holomorphic automorphisms of X we obtain an action of G C on X and the UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS17 infinitesimal action gets the form ξ = α + √− β v ξ = v α + J v β , where v α is thehamiltonian vector field of the function H xi and J v β = ∇ H β is the gradient vector field ofthe function H β .An interesting function is the square of the norm of the moment map, i.e the function f : X → R defined by f ( x ) = | µ ( x ) | . The gradient of f is given by ∇ f ( x ) = J v µ ( x ) ( x ).It is convenient to denote L x ξ = v ξ ( x ). The integral curves of the differential equation(96) ( ∗ ) : ˙ x = −∇ f ( x )play a crucial role in GIT. The celebrated Kirwan-Ness inequality states that at a criticalpoint x ∈ X of the function f we have(97) | µ ( x ) | ≤ | µ ( g.x ) | for any g ∈ G C . Thus the limits of integral curves of the (*) tend to the minimum of | µ ( x ) | , [12].The case we are interested in is when X = P ( V ). In this case a moment map for theaction of G is given by(98) h µ ( x ) , ξ i = a × h v, √− ξv i| v | , x = [ v ] ∈ P ( V ) , a ∈ R + When X is a G -invariant projective subvariety of P ( V ) we still can use the above momentmap. This specially applies for the period domain of Hodge structure which are complexKahler Homogeneous varieties G C /P for a parabolic subgroup.We now come back to the functions Z γ defined by (57). We can write the absolute valueof the numerical function Z γ as(99) | Z γ | = | A t ΣΠ | (Π † ΣΠ) / = |h µ ( A t ) , Σ i| The function Z γ is defined on the whole moduli of complex structure of the CY variety. The G -action on this space is transitive. On the other hand by what we said above the criticalpoints A t of the gradient flow of −∇| Z γ | are characterized by the inequality | µ ( A t ) | ≤| µ ( g.A t ) | . It follows that the attractor points are among the point which minimize theabsolute value |h µ ( A t ) , Σ i| .An interesting fact is how the equality in (99) depends to Σ. In fact changing Σ alonga nilpotent orbit in G C as t exp( −√− t )), then the limit Σ ∞ = lim t →∞ Σ( t ) /t existsand | Σ ∞ | equals the min | µ ( g.x ) | , by the so called Kempf existence theorem. Mumforddefines the function(100) w ( x, Σ) = lim h µ (exp( √− t Σ) x, ξ i called the moment-weight function which plays a crucial role in GIT stability. This functionis closely related to the critical values of f . In fact by the same theorem one has(101) w ( x, Σ) = − min | µ ( g.x ) | In our case if λ ≤ λ ≤ λ ≤ λ are the eigenvalues of √− w ( x, Σ) = max λ i .Therefore the attractor points are among the critical points of the moment map squeredon the period domain of X .8. Relation with CY modularity conjecture The section is expository to express the relation of attractor CY varieties to the famousmodularity conjecture. If X is defined over F p , then the Artin Zeta function of X is definedby(102) Z ( X/ F p , t ) = exp X n ♯X ( F p n ) n t n ! Dwork conjectured that the generating function Z is rational i.e.(103) Z ( X/ F p , t ) = A p ( X, t ) /B p ( X, t )where(104) A p ( X, t ) = Y i P i +1 ( X, t ) , B p ( X, t ) = Y i P i ( X, t )It is well known that deg P j ( X, t ) = dim H j ( X ). In fact P i ( t ) = det(1 − tF rob ∗ p | H iet ( X, Q l ).Assume X is a CY 3-fold and consider the Galois module H ( X, Q l ). One can definean L -function at p on H iet ( X, Q l ) by(105) L p ( X, s ) = ( ∗ ) Y p good P ( X, p − s ) − where good prime means the primes that the reduction of X at p is nonsingular and( ∗ ) is a finite product of Euler factors at the bad primes. The modularity means that L p ( X, s ) = L ( f, s ) for a modular form f .The attractor formalism suggests to look for a decomposition of the motive of H ( X, Q l )in the form M flux ⊕ M rest where M flux is of Hodge type (2 , 1) + (1 , 2) and M rest is of Hodgetype (3 , 0) + (0 , H ofthe form (1 , , , Q vector space of dimension 4. It follows that one may considerthe above two pieces as a motive of an elliptic curve, where the modularity conjecturehas a solution. In this case the characteristic polynomial of the F rob p on H which isa polynomial of degree 4, is decomposed as the product of two polynomials of degree2. Therefore a premier step in checking the modularity conjecture is to look for primes p where the characteristic polynomial P ( X, t ) can be written as the product P (1)3 P (2)3 of two polynomials of degree 2. On the other hand one can read the coefficient of thepotential modular form satisfying the modularity question from the polynomials P (1)3 and P (2)3 . It follows that with a bit of chance one may be able to find out a positive answerto the modular CY 3-fold from the table of known modular forms in a specific example. UPERSYMMETRIC FLUX COMPACTIFICATION ON HYPERGEOMETRIC FAMILY OF CY 3-FOLDS19 This method has been encountered and applied to several special examples in [5, 6, 8, 10]and results provide attractor CY 3-folds in one or several parameter families which aremodular, [5, 6, 8]. References [1] V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J.Algebraic Geom. 3 (1994), no. 3, 493–535. MR1269718 (95c:14046)[2] CF Doran, A Malmendier, Calabi–Yau manifolds realizing symplectically rigid monodromy tuples,Advances in Theoretical and Mathematical Physics 23 (5), 1271-1359, 2019[3] M. Kerr, Algebraic and arithmetic properties of period maps, in ”Calabi-Yau varieties: arithmetic,geometry, and physics”, 173-208, Fields Inst. Monogr. 34, Toronto, ON, 2015.[4] M. Kerr, G. Pearlstein, Boundary components of Mumford-Tate domains, preprint, Duke Math. J.165, 661-721. 2016[5] P. Candelas, X. C. de la Ossa, P. Green and L. Parkes, A Pair of Calabi-Yau Manifolds as an ExactlySoluble Superconformal Theory, Nuclear Physics B359 (1991) 21-74.[6] P. Candelas, X. de la Ossa, M. Elmi and D. van Straten, A One Parameter Family of Calabi-YauManifolds with Attractor Points of Rank Two. arXiv:1912.06146.[7] M. Kerr, P. Griffiths, M. Green, Some enumerative global properties of variations of Hodge structure,Moscow Math. J. 9 (2009), 469-530.[8] S. Kachru, R. Nally and W. Yang, Supersymmetric Flux Compactifications and Calabi-Yau Modu-larity. arXiv:2001.06022.[9] M. Kim and W. Yang, Mirror symmetry, mixed motives and ζ (3). arXiv:1710.02344.[10] G. Moore, Arithmetic and Attractors, arXiv:hep-th/9807087.[11] R. Schimmrigk, On flux vacua and modularity, Journal of High Energy Physics, 61 (2020)[12] V. Georgoulas, J. W. Robbin, D. A. Salomon, The moment-weight inequality and the Hilbert-Mumford criterion, Preprint, ETH-Zurich, November 2013, revised 5 April 2016.[13] W. Yang, Periods of CY n-folds and mixed Tate motives, a numerical study. arXiv:1908.09965. Universidad delaSalle Bajio, Leon GTO, Mexico, Email address ::